\(\int (d+e x)^3 (a+b x+c x^2)^{5/2} \, dx\) [2355]
Optimal result
Integrand size = 22, antiderivative size = 400 \[
\int (d+e x)^3 \left (a+b x+c x^2\right )^{5/2} \, dx=\frac {5 \left (b^2-4 a c\right )^2 (2 c d-b e) \left (32 c^2 d^2+11 b^2 e^2-4 c e (8 b d+3 a e)\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{32768 c^6}-\frac {5 \left (b^2-4 a c\right ) (2 c d-b e) \left (32 c^2 d^2+11 b^2 e^2-4 c e (8 b d+3 a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{12288 c^5}+\frac {(2 c d-b e) \left (32 c^2 d^2+11 b^2 e^2-4 c e (8 b d+3 a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{768 c^4}+\frac {e (d+e x)^2 \left (a+b x+c x^2\right )^{7/2}}{9 c}+\frac {e \left (640 c^2 d^2+99 b^2 e^2-2 c e (243 b d+32 a e)+154 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{7/2}}{2016 c^3}-\frac {5 \left (b^2-4 a c\right )^3 (2 c d-b e) \left (32 c^2 d^2+11 b^2 e^2-4 c e (8 b d+3 a e)\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{65536 c^{13/2}}
\]
[Out]
-5/12288*(-4*a*c+b^2)*(-b*e+2*c*d)*(32*c^2*d^2+11*b^2*e^2-4*c*e*(3*a*e+8*b*d))*(2*c*x+b)*(c*x^2+b*x+a)^(3/2)/c
^5+1/768*(-b*e+2*c*d)*(32*c^2*d^2+11*b^2*e^2-4*c*e*(3*a*e+8*b*d))*(2*c*x+b)*(c*x^2+b*x+a)^(5/2)/c^4+1/9*e*(e*x
+d)^2*(c*x^2+b*x+a)^(7/2)/c+1/2016*e*(640*c^2*d^2+99*b^2*e^2-2*c*e*(32*a*e+243*b*d)+154*c*e*(-b*e+2*c*d)*x)*(c
*x^2+b*x+a)^(7/2)/c^3-5/65536*(-4*a*c+b^2)^3*(-b*e+2*c*d)*(32*c^2*d^2+11*b^2*e^2-4*c*e*(3*a*e+8*b*d))*arctanh(
1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))/c^(13/2)+5/32768*(-4*a*c+b^2)^2*(-b*e+2*c*d)*(32*c^2*d^2+11*b^2*e^2
-4*c*e*(3*a*e+8*b*d))*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c^6
Rubi [A] (verified)
Time = 0.29 (sec) , antiderivative size = 400, normalized size of antiderivative = 1.00, number of
steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {756, 793, 626, 635, 212}
\[
\int (d+e x)^3 \left (a+b x+c x^2\right )^{5/2} \, dx=-\frac {5 \left (b^2-4 a c\right )^3 (2 c d-b e) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (-4 c e (3 a e+8 b d)+11 b^2 e^2+32 c^2 d^2\right )}{65536 c^{13/2}}+\frac {5 \left (b^2-4 a c\right )^2 (b+2 c x) \sqrt {a+b x+c x^2} (2 c d-b e) \left (-4 c e (3 a e+8 b d)+11 b^2 e^2+32 c^2 d^2\right )}{32768 c^6}-\frac {5 \left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2} (2 c d-b e) \left (-4 c e (3 a e+8 b d)+11 b^2 e^2+32 c^2 d^2\right )}{12288 c^5}+\frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/2} (2 c d-b e) \left (-4 c e (3 a e+8 b d)+11 b^2 e^2+32 c^2 d^2\right )}{768 c^4}+\frac {e \left (a+b x+c x^2\right )^{7/2} \left (-2 c e (32 a e+243 b d)+99 b^2 e^2+154 c e x (2 c d-b e)+640 c^2 d^2\right )}{2016 c^3}+\frac {e (d+e x)^2 \left (a+b x+c x^2\right )^{7/2}}{9 c}
\]
[In]
Int[(d + e*x)^3*(a + b*x + c*x^2)^(5/2),x]
[Out]
(5*(b^2 - 4*a*c)^2*(2*c*d - b*e)*(32*c^2*d^2 + 11*b^2*e^2 - 4*c*e*(8*b*d + 3*a*e))*(b + 2*c*x)*Sqrt[a + b*x +
c*x^2])/(32768*c^6) - (5*(b^2 - 4*a*c)*(2*c*d - b*e)*(32*c^2*d^2 + 11*b^2*e^2 - 4*c*e*(8*b*d + 3*a*e))*(b + 2*
c*x)*(a + b*x + c*x^2)^(3/2))/(12288*c^5) + ((2*c*d - b*e)*(32*c^2*d^2 + 11*b^2*e^2 - 4*c*e*(8*b*d + 3*a*e))*(
b + 2*c*x)*(a + b*x + c*x^2)^(5/2))/(768*c^4) + (e*(d + e*x)^2*(a + b*x + c*x^2)^(7/2))/(9*c) + (e*(640*c^2*d^
2 + 99*b^2*e^2 - 2*c*e*(243*b*d + 32*a*e) + 154*c*e*(2*c*d - b*e)*x)*(a + b*x + c*x^2)^(7/2))/(2016*c^3) - (5*
(b^2 - 4*a*c)^3*(2*c*d - b*e)*(32*c^2*d^2 + 11*b^2*e^2 - 4*c*e*(8*b*d + 3*a*e))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]
*Sqrt[a + b*x + c*x^2])])/(65536*c^(13/2))
Rule 212
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])
Rule 626
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x)*((a + b*x + c*x^2)^p/(2*c*(2*p + 1
))), x] - Dist[p*((b^2 - 4*a*c)/(2*c*(2*p + 1))), Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x]
&& NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && IntegerQ[4*p]
Rule 635
Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]
Rule 756
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*(d + e*x)^(m - 1)*
((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 1))), x] + Dist[1/(c*(m + 2*p + 1)), Int[(d + e*x)^(m - 2)*Simp[c*d^2
*(m + 2*p + 1) - e*(a*e*(m - 1) + b*d*(p + 1)) + e*(2*c*d - b*e)*(m + p)*x, x]*(a + b*x + c*x^2)^p, x], x] /;
FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0]
&& If[RationalQ[m], GtQ[m, 1], SumSimplerQ[m, -2]] && NeQ[m + 2*p + 1, 0] && IntQuadraticQ[a, b, c, d, e, m,
p, x]
Rule 793
Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(b
*e*g*(p + 2) - c*(e*f + d*g)*(2*p + 3) - 2*c*e*g*(p + 1)*x))*((a + b*x + c*x^2)^(p + 1)/(2*c^2*(p + 1)*(2*p +
3))), x] + Dist[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p + 3))/(2*c^2*(2*p + 3)), Int[(
a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b^2 - 4*a*c, 0] && !LeQ[p, -1]
Rubi steps \begin{align*}
\text {integral}& = \frac {e (d+e x)^2 \left (a+b x+c x^2\right )^{7/2}}{9 c}+\frac {\int (d+e x) \left (\frac {1}{2} \left (18 c d^2-e (7 b d+4 a e)\right )+\frac {11}{2} e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{5/2} \, dx}{9 c} \\ & = \frac {e (d+e x)^2 \left (a+b x+c x^2\right )^{7/2}}{9 c}+\frac {e \left (640 c^2 d^2+99 b^2 e^2-2 c e (243 b d+32 a e)+154 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{7/2}}{2016 c^3}+\frac {\left ((2 c d-b e) \left (32 c^2 d^2+11 b^2 e^2-4 c e (8 b d+3 a e)\right )\right ) \int \left (a+b x+c x^2\right )^{5/2} \, dx}{64 c^3} \\ & = \frac {(2 c d-b e) \left (32 c^2 d^2+11 b^2 e^2-4 c e (8 b d+3 a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{768 c^4}+\frac {e (d+e x)^2 \left (a+b x+c x^2\right )^{7/2}}{9 c}+\frac {e \left (640 c^2 d^2+99 b^2 e^2-2 c e (243 b d+32 a e)+154 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{7/2}}{2016 c^3}-\frac {\left (5 \left (b^2-4 a c\right ) (2 c d-b e) \left (32 c^2 d^2+11 b^2 e^2-4 c e (8 b d+3 a e)\right )\right ) \int \left (a+b x+c x^2\right )^{3/2} \, dx}{1536 c^4} \\ & = -\frac {5 \left (b^2-4 a c\right ) (2 c d-b e) \left (32 c^2 d^2+11 b^2 e^2-4 c e (8 b d+3 a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{12288 c^5}+\frac {(2 c d-b e) \left (32 c^2 d^2+11 b^2 e^2-4 c e (8 b d+3 a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{768 c^4}+\frac {e (d+e x)^2 \left (a+b x+c x^2\right )^{7/2}}{9 c}+\frac {e \left (640 c^2 d^2+99 b^2 e^2-2 c e (243 b d+32 a e)+154 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{7/2}}{2016 c^3}+\frac {\left (5 \left (b^2-4 a c\right )^2 (2 c d-b e) \left (32 c^2 d^2+11 b^2 e^2-4 c e (8 b d+3 a e)\right )\right ) \int \sqrt {a+b x+c x^2} \, dx}{8192 c^5} \\ & = \frac {5 \left (b^2-4 a c\right )^2 (2 c d-b e) \left (32 c^2 d^2+11 b^2 e^2-4 c e (8 b d+3 a e)\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{32768 c^6}-\frac {5 \left (b^2-4 a c\right ) (2 c d-b e) \left (32 c^2 d^2+11 b^2 e^2-4 c e (8 b d+3 a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{12288 c^5}+\frac {(2 c d-b e) \left (32 c^2 d^2+11 b^2 e^2-4 c e (8 b d+3 a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{768 c^4}+\frac {e (d+e x)^2 \left (a+b x+c x^2\right )^{7/2}}{9 c}+\frac {e \left (640 c^2 d^2+99 b^2 e^2-2 c e (243 b d+32 a e)+154 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{7/2}}{2016 c^3}-\frac {\left (5 \left (b^2-4 a c\right )^3 (2 c d-b e) \left (32 c^2 d^2+11 b^2 e^2-4 c e (8 b d+3 a e)\right )\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{65536 c^6} \\ & = \frac {5 \left (b^2-4 a c\right )^2 (2 c d-b e) \left (32 c^2 d^2+11 b^2 e^2-4 c e (8 b d+3 a e)\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{32768 c^6}-\frac {5 \left (b^2-4 a c\right ) (2 c d-b e) \left (32 c^2 d^2+11 b^2 e^2-4 c e (8 b d+3 a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{12288 c^5}+\frac {(2 c d-b e) \left (32 c^2 d^2+11 b^2 e^2-4 c e (8 b d+3 a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{768 c^4}+\frac {e (d+e x)^2 \left (a+b x+c x^2\right )^{7/2}}{9 c}+\frac {e \left (640 c^2 d^2+99 b^2 e^2-2 c e (243 b d+32 a e)+154 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{7/2}}{2016 c^3}-\frac {\left (5 \left (b^2-4 a c\right )^3 (2 c d-b e) \left (32 c^2 d^2+11 b^2 e^2-4 c e (8 b d+3 a e)\right )\right ) \text {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x}{\sqrt {a+b x+c x^2}}\right )}{32768 c^6} \\ & = \frac {5 \left (b^2-4 a c\right )^2 (2 c d-b e) \left (32 c^2 d^2+11 b^2 e^2-4 c e (8 b d+3 a e)\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{32768 c^6}-\frac {5 \left (b^2-4 a c\right ) (2 c d-b e) \left (32 c^2 d^2+11 b^2 e^2-4 c e (8 b d+3 a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{12288 c^5}+\frac {(2 c d-b e) \left (32 c^2 d^2+11 b^2 e^2-4 c e (8 b d+3 a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{768 c^4}+\frac {e (d+e x)^2 \left (a+b x+c x^2\right )^{7/2}}{9 c}+\frac {e \left (640 c^2 d^2+99 b^2 e^2-2 c e (243 b d+32 a e)+154 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{7/2}}{2016 c^3}-\frac {5 \left (b^2-4 a c\right )^3 (2 c d-b e) \left (32 c^2 d^2+11 b^2 e^2-4 c e (8 b d+3 a e)\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{65536 c^{13/2}} \\
\end{align*}
Mathematica [A] (verified)
Time = 10.50 (sec) , antiderivative size = 280, normalized size of antiderivative = 0.70
\[
\int (d+e x)^3 \left (a+b x+c x^2\right )^{5/2} \, dx=\frac {e (d+e x)^2 (a+x (b+c x))^{7/2}+\frac {e (a+x (b+c x))^{7/2} \left (99 b^2 e^2+4 c^2 d (160 d+77 e x)-2 c e (243 b d+32 a e+77 b e x)\right )}{224 c^2}+\frac {3 (2 c d-b e) \left (32 c^2 d^2+11 b^2 e^2-4 c e (8 b d+3 a e)\right ) \left (2 \sqrt {c} (b+2 c x) \sqrt {a+x (b+c x)} \left (15 b^4-40 b^3 c x+32 b c^2 x \left (13 a+8 c x^2\right )+8 b^2 c \left (-20 a+11 c x^2\right )+16 c^2 \left (33 a^2+26 a c x^2+8 c^2 x^4\right )\right )-15 \left (b^2-4 a c\right )^3 \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )\right )}{65536 c^{11/2}}}{9 c}
\]
[In]
Integrate[(d + e*x)^3*(a + b*x + c*x^2)^(5/2),x]
[Out]
(e*(d + e*x)^2*(a + x*(b + c*x))^(7/2) + (e*(a + x*(b + c*x))^(7/2)*(99*b^2*e^2 + 4*c^2*d*(160*d + 77*e*x) - 2
*c*e*(243*b*d + 32*a*e + 77*b*e*x)))/(224*c^2) + (3*(2*c*d - b*e)*(32*c^2*d^2 + 11*b^2*e^2 - 4*c*e*(8*b*d + 3*
a*e))*(2*Sqrt[c]*(b + 2*c*x)*Sqrt[a + x*(b + c*x)]*(15*b^4 - 40*b^3*c*x + 32*b*c^2*x*(13*a + 8*c*x^2) + 8*b^2*
c*(-20*a + 11*c*x^2) + 16*c^2*(33*a^2 + 26*a*c*x^2 + 8*c^2*x^4)) - 15*(b^2 - 4*a*c)^3*ArcTanh[(b + 2*c*x)/(2*S
qrt[c]*Sqrt[a + x*(b + c*x)])]))/(65536*c^(11/2)))/(9*c)
Maple [B] (verified)
Leaf count of result is larger than twice the leaf count of optimal. \(1206\) vs. \(2(370)=740\).
Time = 0.39 (sec) , antiderivative size = 1207, normalized size of antiderivative =
3.02
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| method | result | size |
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| default |
\(\text {Expression too large to display}\) |
\(1207\) |
| risch |
\(\text {Expression too large to display}\) |
\(1257\) |
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[In]
int((e*x+d)^3*(c*x^2+b*x+a)^(5/2),x,method=_RETURNVERBOSE)
[Out]
d^3*(1/12*(2*c*x+b)*(c*x^2+b*x+a)^(5/2)/c+5/24*(4*a*c-b^2)/c*(1/8*(2*c*x+b)*(c*x^2+b*x+a)^(3/2)/c+3/16*(4*a*c-
b^2)/c*(1/4*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2)
))))+e^3*(1/9*x^2*(c*x^2+b*x+a)^(7/2)/c-11/18*b/c*(1/8*x*(c*x^2+b*x+a)^(7/2)/c-9/16*b/c*(1/7*(c*x^2+b*x+a)^(7/
2)/c-1/2*b/c*(1/12*(2*c*x+b)*(c*x^2+b*x+a)^(5/2)/c+5/24*(4*a*c-b^2)/c*(1/8*(2*c*x+b)*(c*x^2+b*x+a)^(3/2)/c+3/1
6*(4*a*c-b^2)/c*(1/4*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x
+a)^(1/2))))))-1/8*a/c*(1/12*(2*c*x+b)*(c*x^2+b*x+a)^(5/2)/c+5/24*(4*a*c-b^2)/c*(1/8*(2*c*x+b)*(c*x^2+b*x+a)^(
3/2)/c+3/16*(4*a*c-b^2)/c*(1/4*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+
(c*x^2+b*x+a)^(1/2))))))-2/9*a/c*(1/7*(c*x^2+b*x+a)^(7/2)/c-1/2*b/c*(1/12*(2*c*x+b)*(c*x^2+b*x+a)^(5/2)/c+5/24
*(4*a*c-b^2)/c*(1/8*(2*c*x+b)*(c*x^2+b*x+a)^(3/2)/c+3/16*(4*a*c-b^2)/c*(1/4*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c+1/
8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2)))))))+3*d*e^2*(1/8*x*(c*x^2+b*x+a)^(7/2)/c-9/
16*b/c*(1/7*(c*x^2+b*x+a)^(7/2)/c-1/2*b/c*(1/12*(2*c*x+b)*(c*x^2+b*x+a)^(5/2)/c+5/24*(4*a*c-b^2)/c*(1/8*(2*c*x
+b)*(c*x^2+b*x+a)^(3/2)/c+3/16*(4*a*c-b^2)/c*(1/4*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c+1/8*(4*a*c-b^2)/c^(3/2)*ln((
1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))))))-1/8*a/c*(1/12*(2*c*x+b)*(c*x^2+b*x+a)^(5/2)/c+5/24*(4*a*c-b^2)/c*(
1/8*(2*c*x+b)*(c*x^2+b*x+a)^(3/2)/c+3/16*(4*a*c-b^2)/c*(1/4*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c+1/8*(4*a*c-b^2)/c^
(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))))))+3*d^2*e*(1/7*(c*x^2+b*x+a)^(7/2)/c-1/2*b/c*(1/12*(2*c*x+
b)*(c*x^2+b*x+a)^(5/2)/c+5/24*(4*a*c-b^2)/c*(1/8*(2*c*x+b)*(c*x^2+b*x+a)^(3/2)/c+3/16*(4*a*c-b^2)/c*(1/4*(2*c*
x+b)*(c*x^2+b*x+a)^(1/2)/c+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))))))
Fricas [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 1058 vs. \(2 (370) = 740\).
Time = 0.57 (sec) , antiderivative size = 2119, normalized size of antiderivative = 5.30
\[
\int (d+e x)^3 \left (a+b x+c x^2\right )^{5/2} \, dx=\text {Too large to display}
\]
[In]
integrate((e*x+d)^3*(c*x^2+b*x+a)^(5/2),x, algorithm="fricas")
[Out]
[-1/8257536*(315*(64*(b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6)*d^3 - 96*(b^7*c^2 - 12*a*b^5*c^3 +
48*a^2*b^3*c^4 - 64*a^3*b*c^5)*d^2*e + 6*(9*b^8*c - 112*a*b^6*c^2 + 480*a^2*b^4*c^3 - 768*a^3*b^2*c^4 + 256*a
^4*c^5)*d*e^2 - (11*b^9 - 144*a*b^7*c + 672*a^2*b^5*c^2 - 1280*a^3*b^3*c^3 + 768*a^4*b*c^4)*e^3)*sqrt(c)*log(-
8*c^2*x^2 - 8*b*c*x - b^2 - 4*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(c) - 4*a*c) - 4*(229376*c^9*e^3*x^8 + 143
36*(54*c^9*d*e^2 + 37*b*c^8*e^3)*x^7 + 1024*(864*c^9*d^2*e + 1782*b*c^8*d*e^2 + (309*b^2*c^7 + 608*a*c^8)*e^3)
*x^6 + 256*(1344*c^9*d^3 + 8352*b*c^8*d^2*e + 18*(243*b^2*c^7 + 476*a*c^8)*d*e^2 + (5*b^3*c^6 + 3012*a*b*c^7)*
e^3)*x^5 + 128*(6720*b*c^8*d^3 + 288*(37*b^2*c^7 + 72*a*c^8)*d^2*e + 18*(3*b^3*c^6 + 1228*a*b*c^7)*d*e^2 - (11
*b^4*c^5 - 84*a*b^2*c^6 - 3840*a^2*c^7)*e^3)*x^4 + 1344*(15*b^5*c^4 - 160*a*b^3*c^5 + 528*a^2*b*c^6)*d^3 - 288
*(105*b^6*c^3 - 1120*a*b^4*c^4 + 3696*a^2*b^2*c^5 - 3072*a^3*c^6)*d^2*e + 18*(945*b^7*c^2 - 10500*a*b^5*c^3 +
37744*a^2*b^3*c^4 - 42432*a^3*b*c^5)*d*e^2 - (3465*b^8*c - 40740*a*b^6*c^2 + 162288*a^2*b^4*c^3 - 234432*a^3*b
^2*c^4 + 65536*a^4*c^5)*e^3 + 16*(1344*(27*b^2*c^7 + 52*a*c^8)*d^3 + 288*(3*b^3*c^6 + 788*a*b*c^7)*d^2*e - 18*
(27*b^4*c^5 - 216*a*b^2*c^6 - 6608*a^2*c^7)*d*e^2 + (99*b^5*c^4 - 856*a*b^3*c^5 + 1968*a^2*b*c^6)*e^3)*x^3 + 8
*(1344*(b^3*c^6 + 156*a*b*c^7)*d^3 - 288*(7*b^4*c^5 - 60*a*b^2*c^6 - 1152*a^2*c^7)*d^2*e + 18*(63*b^5*c^4 - 56
8*a*b^3*c^5 + 1392*a^2*b*c^6)*d*e^2 - (231*b^6*c^3 - 2232*a*b^4*c^4 + 6384*a^2*b^2*c^5 - 4096*a^3*c^6)*e^3)*x^
2 - 2*(1344*(5*b^4*c^5 - 48*a*b^2*c^6 - 528*a^2*c^7)*d^3 - 288*(35*b^5*c^4 - 336*a*b^3*c^5 + 912*a^2*b*c^6)*d^
2*e + 18*(315*b^6*c^3 - 3164*a*b^4*c^4 + 9552*a^2*b^2*c^5 - 6720*a^3*c^6)*d*e^2 - (1155*b^7*c^2 - 12348*a*b^5*
c^3 + 42192*a^2*b^3*c^4 - 44096*a^3*b*c^5)*e^3)*x)*sqrt(c*x^2 + b*x + a))/c^7, 1/4128768*(315*(64*(b^6*c^3 - 1
2*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6)*d^3 - 96*(b^7*c^2 - 12*a*b^5*c^3 + 48*a^2*b^3*c^4 - 64*a^3*b*c^5)*d
^2*e + 6*(9*b^8*c - 112*a*b^6*c^2 + 480*a^2*b^4*c^3 - 768*a^3*b^2*c^4 + 256*a^4*c^5)*d*e^2 - (11*b^9 - 144*a*b
^7*c + 672*a^2*b^5*c^2 - 1280*a^3*b^3*c^3 + 768*a^4*b*c^4)*e^3)*sqrt(-c)*arctan(1/2*sqrt(c*x^2 + b*x + a)*(2*c
*x + b)*sqrt(-c)/(c^2*x^2 + b*c*x + a*c)) + 2*(229376*c^9*e^3*x^8 + 14336*(54*c^9*d*e^2 + 37*b*c^8*e^3)*x^7 +
1024*(864*c^9*d^2*e + 1782*b*c^8*d*e^2 + (309*b^2*c^7 + 608*a*c^8)*e^3)*x^6 + 256*(1344*c^9*d^3 + 8352*b*c^8*d
^2*e + 18*(243*b^2*c^7 + 476*a*c^8)*d*e^2 + (5*b^3*c^6 + 3012*a*b*c^7)*e^3)*x^5 + 128*(6720*b*c^8*d^3 + 288*(3
7*b^2*c^7 + 72*a*c^8)*d^2*e + 18*(3*b^3*c^6 + 1228*a*b*c^7)*d*e^2 - (11*b^4*c^5 - 84*a*b^2*c^6 - 3840*a^2*c^7)
*e^3)*x^4 + 1344*(15*b^5*c^4 - 160*a*b^3*c^5 + 528*a^2*b*c^6)*d^3 - 288*(105*b^6*c^3 - 1120*a*b^4*c^4 + 3696*a
^2*b^2*c^5 - 3072*a^3*c^6)*d^2*e + 18*(945*b^7*c^2 - 10500*a*b^5*c^3 + 37744*a^2*b^3*c^4 - 42432*a^3*b*c^5)*d*
e^2 - (3465*b^8*c - 40740*a*b^6*c^2 + 162288*a^2*b^4*c^3 - 234432*a^3*b^2*c^4 + 65536*a^4*c^5)*e^3 + 16*(1344*
(27*b^2*c^7 + 52*a*c^8)*d^3 + 288*(3*b^3*c^6 + 788*a*b*c^7)*d^2*e - 18*(27*b^4*c^5 - 216*a*b^2*c^6 - 6608*a^2*
c^7)*d*e^2 + (99*b^5*c^4 - 856*a*b^3*c^5 + 1968*a^2*b*c^6)*e^3)*x^3 + 8*(1344*(b^3*c^6 + 156*a*b*c^7)*d^3 - 28
8*(7*b^4*c^5 - 60*a*b^2*c^6 - 1152*a^2*c^7)*d^2*e + 18*(63*b^5*c^4 - 568*a*b^3*c^5 + 1392*a^2*b*c^6)*d*e^2 - (
231*b^6*c^3 - 2232*a*b^4*c^4 + 6384*a^2*b^2*c^5 - 4096*a^3*c^6)*e^3)*x^2 - 2*(1344*(5*b^4*c^5 - 48*a*b^2*c^6 -
528*a^2*c^7)*d^3 - 288*(35*b^5*c^4 - 336*a*b^3*c^5 + 912*a^2*b*c^6)*d^2*e + 18*(315*b^6*c^3 - 3164*a*b^4*c^4
+ 9552*a^2*b^2*c^5 - 6720*a^3*c^6)*d*e^2 - (1155*b^7*c^2 - 12348*a*b^5*c^3 + 42192*a^2*b^3*c^4 - 44096*a^3*b*c
^5)*e^3)*x)*sqrt(c*x^2 + b*x + a))/c^7]
Sympy [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 11123 vs. \(2 (401) = 802\).
Time = 0.95 (sec) , antiderivative size = 11123, normalized size of antiderivative = 27.81
\[
\int (d+e x)^3 \left (a+b x+c x^2\right )^{5/2} \, dx=\text {Too large to display}
\]
[In]
integrate((e*x+d)**3*(c*x**2+b*x+a)**(5/2),x)
[Out]
Piecewise((sqrt(a + b*x + c*x**2)*(c**2*e**3*x**8/9 + x**7*(37*b*c**2*e**3/18 + 3*c**3*d*e**2)/(8*c) + x**6*(1
9*a*c**2*e**3/9 + 3*b**2*c*e**3 + 9*b*c**2*d*e**2 - 15*b*(37*b*c**2*e**3/18 + 3*c**3*d*e**2)/(16*c) + 3*c**3*d
**2*e)/(7*c) + x**5*(6*a*b*c*e**3 + 9*a*c**2*d*e**2 - 7*a*(37*b*c**2*e**3/18 + 3*c**3*d*e**2)/(8*c) + b**3*e**
3 + 9*b**2*c*d*e**2 + 9*b*c**2*d**2*e - 13*b*(19*a*c**2*e**3/9 + 3*b**2*c*e**3 + 9*b*c**2*d*e**2 - 15*b*(37*b*
c**2*e**3/18 + 3*c**3*d*e**2)/(16*c) + 3*c**3*d**2*e)/(14*c) + c**3*d**3)/(6*c) + x**4*(3*a**2*c*e**3 + 3*a*b*
*2*e**3 + 18*a*b*c*d*e**2 + 9*a*c**2*d**2*e - 6*a*(19*a*c**2*e**3/9 + 3*b**2*c*e**3 + 9*b*c**2*d*e**2 - 15*b*(
37*b*c**2*e**3/18 + 3*c**3*d*e**2)/(16*c) + 3*c**3*d**2*e)/(7*c) + 3*b**3*d*e**2 + 9*b**2*c*d**2*e + 3*b*c**2*
d**3 - 11*b*(6*a*b*c*e**3 + 9*a*c**2*d*e**2 - 7*a*(37*b*c**2*e**3/18 + 3*c**3*d*e**2)/(8*c) + b**3*e**3 + 9*b*
*2*c*d*e**2 + 9*b*c**2*d**2*e - 13*b*(19*a*c**2*e**3/9 + 3*b**2*c*e**3 + 9*b*c**2*d*e**2 - 15*b*(37*b*c**2*e**
3/18 + 3*c**3*d*e**2)/(16*c) + 3*c**3*d**2*e)/(14*c) + c**3*d**3)/(12*c))/(5*c) + x**3*(3*a**2*b*e**3 + 9*a**2
*c*d*e**2 + 9*a*b**2*d*e**2 + 18*a*b*c*d**2*e + 3*a*c**2*d**3 - 5*a*(6*a*b*c*e**3 + 9*a*c**2*d*e**2 - 7*a*(37*
b*c**2*e**3/18 + 3*c**3*d*e**2)/(8*c) + b**3*e**3 + 9*b**2*c*d*e**2 + 9*b*c**2*d**2*e - 13*b*(19*a*c**2*e**3/9
+ 3*b**2*c*e**3 + 9*b*c**2*d*e**2 - 15*b*(37*b*c**2*e**3/18 + 3*c**3*d*e**2)/(16*c) + 3*c**3*d**2*e)/(14*c) +
c**3*d**3)/(6*c) + 3*b**3*d**2*e + 3*b**2*c*d**3 - 9*b*(3*a**2*c*e**3 + 3*a*b**2*e**3 + 18*a*b*c*d*e**2 + 9*a
*c**2*d**2*e - 6*a*(19*a*c**2*e**3/9 + 3*b**2*c*e**3 + 9*b*c**2*d*e**2 - 15*b*(37*b*c**2*e**3/18 + 3*c**3*d*e*
*2)/(16*c) + 3*c**3*d**2*e)/(7*c) + 3*b**3*d*e**2 + 9*b**2*c*d**2*e + 3*b*c**2*d**3 - 11*b*(6*a*b*c*e**3 + 9*a
*c**2*d*e**2 - 7*a*(37*b*c**2*e**3/18 + 3*c**3*d*e**2)/(8*c) + b**3*e**3 + 9*b**2*c*d*e**2 + 9*b*c**2*d**2*e -
13*b*(19*a*c**2*e**3/9 + 3*b**2*c*e**3 + 9*b*c**2*d*e**2 - 15*b*(37*b*c**2*e**3/18 + 3*c**3*d*e**2)/(16*c) +
3*c**3*d**2*e)/(14*c) + c**3*d**3)/(12*c))/(10*c))/(4*c) + x**2*(a**3*e**3 + 9*a**2*b*d*e**2 + 9*a**2*c*d**2*e
+ 9*a*b**2*d**2*e + 6*a*b*c*d**3 - 4*a*(3*a**2*c*e**3 + 3*a*b**2*e**3 + 18*a*b*c*d*e**2 + 9*a*c**2*d**2*e - 6
*a*(19*a*c**2*e**3/9 + 3*b**2*c*e**3 + 9*b*c**2*d*e**2 - 15*b*(37*b*c**2*e**3/18 + 3*c**3*d*e**2)/(16*c) + 3*c
**3*d**2*e)/(7*c) + 3*b**3*d*e**2 + 9*b**2*c*d**2*e + 3*b*c**2*d**3 - 11*b*(6*a*b*c*e**3 + 9*a*c**2*d*e**2 - 7
*a*(37*b*c**2*e**3/18 + 3*c**3*d*e**2)/(8*c) + b**3*e**3 + 9*b**2*c*d*e**2 + 9*b*c**2*d**2*e - 13*b*(19*a*c**2
*e**3/9 + 3*b**2*c*e**3 + 9*b*c**2*d*e**2 - 15*b*(37*b*c**2*e**3/18 + 3*c**3*d*e**2)/(16*c) + 3*c**3*d**2*e)/(
14*c) + c**3*d**3)/(12*c))/(5*c) + b**3*d**3 - 7*b*(3*a**2*b*e**3 + 9*a**2*c*d*e**2 + 9*a*b**2*d*e**2 + 18*a*b
*c*d**2*e + 3*a*c**2*d**3 - 5*a*(6*a*b*c*e**3 + 9*a*c**2*d*e**2 - 7*a*(37*b*c**2*e**3/18 + 3*c**3*d*e**2)/(8*c
) + b**3*e**3 + 9*b**2*c*d*e**2 + 9*b*c**2*d**2*e - 13*b*(19*a*c**2*e**3/9 + 3*b**2*c*e**3 + 9*b*c**2*d*e**2 -
15*b*(37*b*c**2*e**3/18 + 3*c**3*d*e**2)/(16*c) + 3*c**3*d**2*e)/(14*c) + c**3*d**3)/(6*c) + 3*b**3*d**2*e +
3*b**2*c*d**3 - 9*b*(3*a**2*c*e**3 + 3*a*b**2*e**3 + 18*a*b*c*d*e**2 + 9*a*c**2*d**2*e - 6*a*(19*a*c**2*e**3/9
+ 3*b**2*c*e**3 + 9*b*c**2*d*e**2 - 15*b*(37*b*c**2*e**3/18 + 3*c**3*d*e**2)/(16*c) + 3*c**3*d**2*e)/(7*c) +
3*b**3*d*e**2 + 9*b**2*c*d**2*e + 3*b*c**2*d**3 - 11*b*(6*a*b*c*e**3 + 9*a*c**2*d*e**2 - 7*a*(37*b*c**2*e**3/1
8 + 3*c**3*d*e**2)/(8*c) + b**3*e**3 + 9*b**2*c*d*e**2 + 9*b*c**2*d**2*e - 13*b*(19*a*c**2*e**3/9 + 3*b**2*c*e
**3 + 9*b*c**2*d*e**2 - 15*b*(37*b*c**2*e**3/18 + 3*c**3*d*e**2)/(16*c) + 3*c**3*d**2*e)/(14*c) + c**3*d**3)/(
12*c))/(10*c))/(8*c))/(3*c) + x*(3*a**3*d*e**2 + 9*a**2*b*d**2*e + 3*a**2*c*d**3 + 3*a*b**2*d**3 - 3*a*(3*a**2
*b*e**3 + 9*a**2*c*d*e**2 + 9*a*b**2*d*e**2 + 18*a*b*c*d**2*e + 3*a*c**2*d**3 - 5*a*(6*a*b*c*e**3 + 9*a*c**2*d
*e**2 - 7*a*(37*b*c**2*e**3/18 + 3*c**3*d*e**2)/(8*c) + b**3*e**3 + 9*b**2*c*d*e**2 + 9*b*c**2*d**2*e - 13*b*(
19*a*c**2*e**3/9 + 3*b**2*c*e**3 + 9*b*c**2*d*e**2 - 15*b*(37*b*c**2*e**3/18 + 3*c**3*d*e**2)/(16*c) + 3*c**3*
d**2*e)/(14*c) + c**3*d**3)/(6*c) + 3*b**3*d**2*e + 3*b**2*c*d**3 - 9*b*(3*a**2*c*e**3 + 3*a*b**2*e**3 + 18*a*
b*c*d*e**2 + 9*a*c**2*d**2*e - 6*a*(19*a*c**2*e**3/9 + 3*b**2*c*e**3 + 9*b*c**2*d*e**2 - 15*b*(37*b*c**2*e**3/
18 + 3*c**3*d*e**2)/(16*c) + 3*c**3*d**2*e)/(7*c) + 3*b**3*d*e**2 + 9*b**2*c*d**2*e + 3*b*c**2*d**3 - 11*b*(6*
a*b*c*e**3 + 9*a*c**2*d*e**2 - 7*a*(37*b*c**2*e**3/18 + 3*c**3*d*e**2)/(8*c) + b**3*e**3 + 9*b**2*c*d*e**2 + 9
*b*c**2*d**2*e - 13*b*(19*a*c**2*e**3/9 + 3*b**2*c*e**3 + 9*b*c**2*d*e**2 - 15*b*(37*b*c**2*e**3/18 + 3*c**3*d
*e**2)/(16*c) + 3*c**3*d**2*e)/(14*c) + c**3*d**3)/(12*c))/(10*c))/(4*c) - 5*b*(a**3*e**3 + 9*a**2*b*d*e**2 +
9*a**2*c*d**2*e + 9*a*b**2*d**2*e + 6*a*b*c*d**3 - 4*a*(3*a**2*c*e**3 + 3*a*b**2*e**3 + 18*a*b*c*d*e**2 + 9*a*
c**2*d**2*e - 6*a*(19*a*c**2*e**3/9 + 3*b**2*c*e**3 + 9*b*c**2*d*e**2 - 15*b*(37*b*c**2*e**3/18 + 3*c**3*d*e**
2)/(16*c) + 3*c**3*d**2*e)/(7*c) + 3*b**3*d*e**2 + 9*b**2*c*d**2*e + 3*b*c**2*d**3 - 11*b*(6*a*b*c*e**3 + 9*a*
c**2*d*e**2 - 7*a*(37*b*c**2*e**3/18 + 3*c**3*d*e**2)/(8*c) + b**3*e**3 + 9*b**2*c*d*e**2 + 9*b*c**2*d**2*e -
13*b*(19*a*c**2*e**3/9 + 3*b**2*c*e**3 + 9*b*c**2*d*e**2 - 15*b*(37*b*c**2*e**3/18 + 3*c**3*d*e**2)/(16*c) + 3
*c**3*d**2*e)/(14*c) + c**3*d**3)/(12*c))/(5*c) + b**3*d**3 - 7*b*(3*a**2*b*e**3 + 9*a**2*c*d*e**2 + 9*a*b**2*
d*e**2 + 18*a*b*c*d**2*e + 3*a*c**2*d**3 - 5*a*(6*a*b*c*e**3 + 9*a*c**2*d*e**2 - 7*a*(37*b*c**2*e**3/18 + 3*c*
*3*d*e**2)/(8*c) + b**3*e**3 + 9*b**2*c*d*e**2 + 9*b*c**2*d**2*e - 13*b*(19*a*c**2*e**3/9 + 3*b**2*c*e**3 + 9*
b*c**2*d*e**2 - 15*b*(37*b*c**2*e**3/18 + 3*c**3*d*e**2)/(16*c) + 3*c**3*d**2*e)/(14*c) + c**3*d**3)/(6*c) + 3
*b**3*d**2*e + 3*b**2*c*d**3 - 9*b*(3*a**2*c*e**3 + 3*a*b**2*e**3 + 18*a*b*c*d*e**2 + 9*a*c**2*d**2*e - 6*a*(1
9*a*c**2*e**3/9 + 3*b**2*c*e**3 + 9*b*c**2*d*e**2 - 15*b*(37*b*c**2*e**3/18 + 3*c**3*d*e**2)/(16*c) + 3*c**3*d
**2*e)/(7*c) + 3*b**3*d*e**2 + 9*b**2*c*d**2*e + 3*b*c**2*d**3 - 11*b*(6*a*b*c*e**3 + 9*a*c**2*d*e**2 - 7*a*(3
7*b*c**2*e**3/18 + 3*c**3*d*e**2)/(8*c) + b**3*e**3 + 9*b**2*c*d*e**2 + 9*b*c**2*d**2*e - 13*b*(19*a*c**2*e**3
/9 + 3*b**2*c*e**3 + 9*b*c**2*d*e**2 - 15*b*(37*b*c**2*e**3/18 + 3*c**3*d*e**2)/(16*c) + 3*c**3*d**2*e)/(14*c)
+ c**3*d**3)/(12*c))/(10*c))/(8*c))/(6*c))/(2*c) + (3*a**3*d**2*e + 3*a**2*b*d**3 - 2*a*(a**3*e**3 + 9*a**2*b
*d*e**2 + 9*a**2*c*d**2*e + 9*a*b**2*d**2*e + 6*a*b*c*d**3 - 4*a*(3*a**2*c*e**3 + 3*a*b**2*e**3 + 18*a*b*c*d*e
**2 + 9*a*c**2*d**2*e - 6*a*(19*a*c**2*e**3/9 + 3*b**2*c*e**3 + 9*b*c**2*d*e**2 - 15*b*(37*b*c**2*e**3/18 + 3*
c**3*d*e**2)/(16*c) + 3*c**3*d**2*e)/(7*c) + 3*b**3*d*e**2 + 9*b**2*c*d**2*e + 3*b*c**2*d**3 - 11*b*(6*a*b*c*e
**3 + 9*a*c**2*d*e**2 - 7*a*(37*b*c**2*e**3/18 + 3*c**3*d*e**2)/(8*c) + b**3*e**3 + 9*b**2*c*d*e**2 + 9*b*c**2
*d**2*e - 13*b*(19*a*c**2*e**3/9 + 3*b**2*c*e**3 + 9*b*c**2*d*e**2 - 15*b*(37*b*c**2*e**3/18 + 3*c**3*d*e**2)/
(16*c) + 3*c**3*d**2*e)/(14*c) + c**3*d**3)/(12*c))/(5*c) + b**3*d**3 - 7*b*(3*a**2*b*e**3 + 9*a**2*c*d*e**2 +
9*a*b**2*d*e**2 + 18*a*b*c*d**2*e + 3*a*c**2*d**3 - 5*a*(6*a*b*c*e**3 + 9*a*c**2*d*e**2 - 7*a*(37*b*c**2*e**3
/18 + 3*c**3*d*e**2)/(8*c) + b**3*e**3 + 9*b**2*c*d*e**2 + 9*b*c**2*d**2*e - 13*b*(19*a*c**2*e**3/9 + 3*b**2*c
*e**3 + 9*b*c**2*d*e**2 - 15*b*(37*b*c**2*e**3/18 + 3*c**3*d*e**2)/(16*c) + 3*c**3*d**2*e)/(14*c) + c**3*d**3)
/(6*c) + 3*b**3*d**2*e + 3*b**2*c*d**3 - 9*b*(3*a**2*c*e**3 + 3*a*b**2*e**3 + 18*a*b*c*d*e**2 + 9*a*c**2*d**2*
e - 6*a*(19*a*c**2*e**3/9 + 3*b**2*c*e**3 + 9*b*c**2*d*e**2 - 15*b*(37*b*c**2*e**3/18 + 3*c**3*d*e**2)/(16*c)
+ 3*c**3*d**2*e)/(7*c) + 3*b**3*d*e**2 + 9*b**2*c*d**2*e + 3*b*c**2*d**3 - 11*b*(6*a*b*c*e**3 + 9*a*c**2*d*e**
2 - 7*a*(37*b*c**2*e**3/18 + 3*c**3*d*e**2)/(8*c) + b**3*e**3 + 9*b**2*c*d*e**2 + 9*b*c**2*d**2*e - 13*b*(19*a
*c**2*e**3/9 + 3*b**2*c*e**3 + 9*b*c**2*d*e**2 - 15*b*(37*b*c**2*e**3/18 + 3*c**3*d*e**2)/(16*c) + 3*c**3*d**2
*e)/(14*c) + c**3*d**3)/(12*c))/(10*c))/(8*c))/(3*c) - 3*b*(3*a**3*d*e**2 + 9*a**2*b*d**2*e + 3*a**2*c*d**3 +
3*a*b**2*d**3 - 3*a*(3*a**2*b*e**3 + 9*a**2*c*d*e**2 + 9*a*b**2*d*e**2 + 18*a*b*c*d**2*e + 3*a*c**2*d**3 - 5*a
*(6*a*b*c*e**3 + 9*a*c**2*d*e**2 - 7*a*(37*b*c**2*e**3/18 + 3*c**3*d*e**2)/(8*c) + b**3*e**3 + 9*b**2*c*d*e**2
+ 9*b*c**2*d**2*e - 13*b*(19*a*c**2*e**3/9 + 3*b**2*c*e**3 + 9*b*c**2*d*e**2 - 15*b*(37*b*c**2*e**3/18 + 3*c*
*3*d*e**2)/(16*c) + 3*c**3*d**2*e)/(14*c) + c**3*d**3)/(6*c) + 3*b**3*d**2*e + 3*b**2*c*d**3 - 9*b*(3*a**2*c*e
**3 + 3*a*b**2*e**3 + 18*a*b*c*d*e**2 + 9*a*c**2*d**2*e - 6*a*(19*a*c**2*e**3/9 + 3*b**2*c*e**3 + 9*b*c**2*d*e
**2 - 15*b*(37*b*c**2*e**3/18 + 3*c**3*d*e**2)/(16*c) + 3*c**3*d**2*e)/(7*c) + 3*b**3*d*e**2 + 9*b**2*c*d**2*e
+ 3*b*c**2*d**3 - 11*b*(6*a*b*c*e**3 + 9*a*c**2*d*e**2 - 7*a*(37*b*c**2*e**3/18 + 3*c**3*d*e**2)/(8*c) + b**3
*e**3 + 9*b**2*c*d*e**2 + 9*b*c**2*d**2*e - 13*b*(19*a*c**2*e**3/9 + 3*b**2*c*e**3 + 9*b*c**2*d*e**2 - 15*b*(3
7*b*c**2*e**3/18 + 3*c**3*d*e**2)/(16*c) + 3*c**3*d**2*e)/(14*c) + c**3*d**3)/(12*c))/(10*c))/(4*c) - 5*b*(a**
3*e**3 + 9*a**2*b*d*e**2 + 9*a**2*c*d**2*e + 9*a*b**2*d**2*e + 6*a*b*c*d**3 - 4*a*(3*a**2*c*e**3 + 3*a*b**2*e*
*3 + 18*a*b*c*d*e**2 + 9*a*c**2*d**2*e - 6*a*(19*a*c**2*e**3/9 + 3*b**2*c*e**3 + 9*b*c**2*d*e**2 - 15*b*(37*b*
c**2*e**3/18 + 3*c**3*d*e**2)/(16*c) + 3*c**3*d**2*e)/(7*c) + 3*b**3*d*e**2 + 9*b**2*c*d**2*e + 3*b*c**2*d**3
- 11*b*(6*a*b*c*e**3 + 9*a*c**2*d*e**2 - 7*a*(37*b*c**2*e**3/18 + 3*c**3*d*e**2)/(8*c) + b**3*e**3 + 9*b**2*c*
d*e**2 + 9*b*c**2*d**2*e - 13*b*(19*a*c**2*e**3/9 + 3*b**2*c*e**3 + 9*b*c**2*d*e**2 - 15*b*(37*b*c**2*e**3/18
+ 3*c**3*d*e**2)/(16*c) + 3*c**3*d**2*e)/(14*c) + c**3*d**3)/(12*c))/(5*c) + b**3*d**3 - 7*b*(3*a**2*b*e**3 +
9*a**2*c*d*e**2 + 9*a*b**2*d*e**2 + 18*a*b*c*d**2*e + 3*a*c**2*d**3 - 5*a*(6*a*b*c*e**3 + 9*a*c**2*d*e**2 - 7*
a*(37*b*c**2*e**3/18 + 3*c**3*d*e**2)/(8*c) + b**3*e**3 + 9*b**2*c*d*e**2 + 9*b*c**2*d**2*e - 13*b*(19*a*c**2*
e**3/9 + 3*b**2*c*e**3 + 9*b*c**2*d*e**2 - 15*b*(37*b*c**2*e**3/18 + 3*c**3*d*e**2)/(16*c) + 3*c**3*d**2*e)/(1
4*c) + c**3*d**3)/(6*c) + 3*b**3*d**2*e + 3*b**2*c*d**3 - 9*b*(3*a**2*c*e**3 + 3*a*b**2*e**3 + 18*a*b*c*d*e**2
+ 9*a*c**2*d**2*e - 6*a*(19*a*c**2*e**3/9 + 3*b**2*c*e**3 + 9*b*c**2*d*e**2 - 15*b*(37*b*c**2*e**3/18 + 3*c**
3*d*e**2)/(16*c) + 3*c**3*d**2*e)/(7*c) + 3*b**3*d*e**2 + 9*b**2*c*d**2*e + 3*b*c**2*d**3 - 11*b*(6*a*b*c*e**3
+ 9*a*c**2*d*e**2 - 7*a*(37*b*c**2*e**3/18 + 3*c**3*d*e**2)/(8*c) + b**3*e**3 + 9*b**2*c*d*e**2 + 9*b*c**2*d*
*2*e - 13*b*(19*a*c**2*e**3/9 + 3*b**2*c*e**3 + 9*b*c**2*d*e**2 - 15*b*(37*b*c**2*e**3/18 + 3*c**3*d*e**2)/(16
*c) + 3*c**3*d**2*e)/(14*c) + c**3*d**3)/(12*c))/(10*c))/(8*c))/(6*c))/(4*c))/c) + (a**3*d**3 - a*(3*a**3*d*e*
*2 + 9*a**2*b*d**2*e + 3*a**2*c*d**3 + 3*a*b**2*d**3 - 3*a*(3*a**2*b*e**3 + 9*a**2*c*d*e**2 + 9*a*b**2*d*e**2
+ 18*a*b*c*d**2*e + 3*a*c**2*d**3 - 5*a*(6*a*b*c*e**3 + 9*a*c**2*d*e**2 - 7*a*(37*b*c**2*e**3/18 + 3*c**3*d*e*
*2)/(8*c) + b**3*e**3 + 9*b**2*c*d*e**2 + 9*b*c**2*d**2*e - 13*b*(19*a*c**2*e**3/9 + 3*b**2*c*e**3 + 9*b*c**2*
d*e**2 - 15*b*(37*b*c**2*e**3/18 + 3*c**3*d*e**2)/(16*c) + 3*c**3*d**2*e)/(14*c) + c**3*d**3)/(6*c) + 3*b**3*d
**2*e + 3*b**2*c*d**3 - 9*b*(3*a**2*c*e**3 + 3*a*b**2*e**3 + 18*a*b*c*d*e**2 + 9*a*c**2*d**2*e - 6*a*(19*a*c**
2*e**3/9 + 3*b**2*c*e**3 + 9*b*c**2*d*e**2 - 15*b*(37*b*c**2*e**3/18 + 3*c**3*d*e**2)/(16*c) + 3*c**3*d**2*e)/
(7*c) + 3*b**3*d*e**2 + 9*b**2*c*d**2*e + 3*b*c**2*d**3 - 11*b*(6*a*b*c*e**3 + 9*a*c**2*d*e**2 - 7*a*(37*b*c**
2*e**3/18 + 3*c**3*d*e**2)/(8*c) + b**3*e**3 + 9*b**2*c*d*e**2 + 9*b*c**2*d**2*e - 13*b*(19*a*c**2*e**3/9 + 3*
b**2*c*e**3 + 9*b*c**2*d*e**2 - 15*b*(37*b*c**2*e**3/18 + 3*c**3*d*e**2)/(16*c) + 3*c**3*d**2*e)/(14*c) + c**3
*d**3)/(12*c))/(10*c))/(4*c) - 5*b*(a**3*e**3 + 9*a**2*b*d*e**2 + 9*a**2*c*d**2*e + 9*a*b**2*d**2*e + 6*a*b*c*
d**3 - 4*a*(3*a**2*c*e**3 + 3*a*b**2*e**3 + 18*a*b*c*d*e**2 + 9*a*c**2*d**2*e - 6*a*(19*a*c**2*e**3/9 + 3*b**2
*c*e**3 + 9*b*c**2*d*e**2 - 15*b*(37*b*c**2*e**3/18 + 3*c**3*d*e**2)/(16*c) + 3*c**3*d**2*e)/(7*c) + 3*b**3*d*
e**2 + 9*b**2*c*d**2*e + 3*b*c**2*d**3 - 11*b*(6*a*b*c*e**3 + 9*a*c**2*d*e**2 - 7*a*(37*b*c**2*e**3/18 + 3*c**
3*d*e**2)/(8*c) + b**3*e**3 + 9*b**2*c*d*e**2 + 9*b*c**2*d**2*e - 13*b*(19*a*c**2*e**3/9 + 3*b**2*c*e**3 + 9*b
*c**2*d*e**2 - 15*b*(37*b*c**2*e**3/18 + 3*c**3*d*e**2)/(16*c) + 3*c**3*d**2*e)/(14*c) + c**3*d**3)/(12*c))/(5
*c) + b**3*d**3 - 7*b*(3*a**2*b*e**3 + 9*a**2*c*d*e**2 + 9*a*b**2*d*e**2 + 18*a*b*c*d**2*e + 3*a*c**2*d**3 - 5
*a*(6*a*b*c*e**3 + 9*a*c**2*d*e**2 - 7*a*(37*b*c**2*e**3/18 + 3*c**3*d*e**2)/(8*c) + b**3*e**3 + 9*b**2*c*d*e*
*2 + 9*b*c**2*d**2*e - 13*b*(19*a*c**2*e**3/9 + 3*b**2*c*e**3 + 9*b*c**2*d*e**2 - 15*b*(37*b*c**2*e**3/18 + 3*
c**3*d*e**2)/(16*c) + 3*c**3*d**2*e)/(14*c) + c**3*d**3)/(6*c) + 3*b**3*d**2*e + 3*b**2*c*d**3 - 9*b*(3*a**2*c
*e**3 + 3*a*b**2*e**3 + 18*a*b*c*d*e**2 + 9*a*c**2*d**2*e - 6*a*(19*a*c**2*e**3/9 + 3*b**2*c*e**3 + 9*b*c**2*d
*e**2 - 15*b*(37*b*c**2*e**3/18 + 3*c**3*d*e**2)/(16*c) + 3*c**3*d**2*e)/(7*c) + 3*b**3*d*e**2 + 9*b**2*c*d**2
*e + 3*b*c**2*d**3 - 11*b*(6*a*b*c*e**3 + 9*a*c**2*d*e**2 - 7*a*(37*b*c**2*e**3/18 + 3*c**3*d*e**2)/(8*c) + b*
*3*e**3 + 9*b**2*c*d*e**2 + 9*b*c**2*d**2*e - 13*b*(19*a*c**2*e**3/9 + 3*b**2*c*e**3 + 9*b*c**2*d*e**2 - 15*b*
(37*b*c**2*e**3/18 + 3*c**3*d*e**2)/(16*c) + 3*c**3*d**2*e)/(14*c) + c**3*d**3)/(12*c))/(10*c))/(8*c))/(6*c))/
(2*c) - b*(3*a**3*d**2*e + 3*a**2*b*d**3 - 2*a*(a**3*e**3 + 9*a**2*b*d*e**2 + 9*a**2*c*d**2*e + 9*a*b**2*d**2*
e + 6*a*b*c*d**3 - 4*a*(3*a**2*c*e**3 + 3*a*b**2*e**3 + 18*a*b*c*d*e**2 + 9*a*c**2*d**2*e - 6*a*(19*a*c**2*e**
3/9 + 3*b**2*c*e**3 + 9*b*c**2*d*e**2 - 15*b*(37*b*c**2*e**3/18 + 3*c**3*d*e**2)/(16*c) + 3*c**3*d**2*e)/(7*c)
+ 3*b**3*d*e**2 + 9*b**2*c*d**2*e + 3*b*c**2*d**3 - 11*b*(6*a*b*c*e**3 + 9*a*c**2*d*e**2 - 7*a*(37*b*c**2*e**
3/18 + 3*c**3*d*e**2)/(8*c) + b**3*e**3 + 9*b**2*c*d*e**2 + 9*b*c**2*d**2*e - 13*b*(19*a*c**2*e**3/9 + 3*b**2*
c*e**3 + 9*b*c**2*d*e**2 - 15*b*(37*b*c**2*e**3/18 + 3*c**3*d*e**2)/(16*c) + 3*c**3*d**2*e)/(14*c) + c**3*d**3
)/(12*c))/(5*c) + b**3*d**3 - 7*b*(3*a**2*b*e**3 + 9*a**2*c*d*e**2 + 9*a*b**2*d*e**2 + 18*a*b*c*d**2*e + 3*a*c
**2*d**3 - 5*a*(6*a*b*c*e**3 + 9*a*c**2*d*e**2 - 7*a*(37*b*c**2*e**3/18 + 3*c**3*d*e**2)/(8*c) + b**3*e**3 + 9
*b**2*c*d*e**2 + 9*b*c**2*d**2*e - 13*b*(19*a*c**2*e**3/9 + 3*b**2*c*e**3 + 9*b*c**2*d*e**2 - 15*b*(37*b*c**2*
e**3/18 + 3*c**3*d*e**2)/(16*c) + 3*c**3*d**2*e)/(14*c) + c**3*d**3)/(6*c) + 3*b**3*d**2*e + 3*b**2*c*d**3 - 9
*b*(3*a**2*c*e**3 + 3*a*b**2*e**3 + 18*a*b*c*d*e**2 + 9*a*c**2*d**2*e - 6*a*(19*a*c**2*e**3/9 + 3*b**2*c*e**3
+ 9*b*c**2*d*e**2 - 15*b*(37*b*c**2*e**3/18 + 3*c**3*d*e**2)/(16*c) + 3*c**3*d**2*e)/(7*c) + 3*b**3*d*e**2 + 9
*b**2*c*d**2*e + 3*b*c**2*d**3 - 11*b*(6*a*b*c*e**3 + 9*a*c**2*d*e**2 - 7*a*(37*b*c**2*e**3/18 + 3*c**3*d*e**2
)/(8*c) + b**3*e**3 + 9*b**2*c*d*e**2 + 9*b*c**2*d**2*e - 13*b*(19*a*c**2*e**3/9 + 3*b**2*c*e**3 + 9*b*c**2*d*
e**2 - 15*b*(37*b*c**2*e**3/18 + 3*c**3*d*e**2)/(16*c) + 3*c**3*d**2*e)/(14*c) + c**3*d**3)/(12*c))/(10*c))/(8
*c))/(3*c) - 3*b*(3*a**3*d*e**2 + 9*a**2*b*d**2*e + 3*a**2*c*d**3 + 3*a*b**2*d**3 - 3*a*(3*a**2*b*e**3 + 9*a**
2*c*d*e**2 + 9*a*b**2*d*e**2 + 18*a*b*c*d**2*e + 3*a*c**2*d**3 - 5*a*(6*a*b*c*e**3 + 9*a*c**2*d*e**2 - 7*a*(37
*b*c**2*e**3/18 + 3*c**3*d*e**2)/(8*c) + b**3*e**3 + 9*b**2*c*d*e**2 + 9*b*c**2*d**2*e - 13*b*(19*a*c**2*e**3/
9 + 3*b**2*c*e**3 + 9*b*c**2*d*e**2 - 15*b*(37*b*c**2*e**3/18 + 3*c**3*d*e**2)/(16*c) + 3*c**3*d**2*e)/(14*c)
+ c**3*d**3)/(6*c) + 3*b**3*d**2*e + 3*b**2*c*d**3 - 9*b*(3*a**2*c*e**3 + 3*a*b**2*e**3 + 18*a*b*c*d*e**2 + 9*
a*c**2*d**2*e - 6*a*(19*a*c**2*e**3/9 + 3*b**2*c*e**3 + 9*b*c**2*d*e**2 - 15*b*(37*b*c**2*e**3/18 + 3*c**3*d*e
**2)/(16*c) + 3*c**3*d**2*e)/(7*c) + 3*b**3*d*e**2 + 9*b**2*c*d**2*e + 3*b*c**2*d**3 - 11*b*(6*a*b*c*e**3 + 9*
a*c**2*d*e**2 - 7*a*(37*b*c**2*e**3/18 + 3*c**3*d*e**2)/(8*c) + b**3*e**3 + 9*b**2*c*d*e**2 + 9*b*c**2*d**2*e
- 13*b*(19*a*c**2*e**3/9 + 3*b**2*c*e**3 + 9*b*c**2*d*e**2 - 15*b*(37*b*c**2*e**3/18 + 3*c**3*d*e**2)/(16*c) +
3*c**3*d**2*e)/(14*c) + c**3*d**3)/(12*c))/(10*c))/(4*c) - 5*b*(a**3*e**3 + 9*a**2*b*d*e**2 + 9*a**2*c*d**2*e
+ 9*a*b**2*d**2*e + 6*a*b*c*d**3 - 4*a*(3*a**2*c*e**3 + 3*a*b**2*e**3 + 18*a*b*c*d*e**2 + 9*a*c**2*d**2*e - 6
*a*(19*a*c**2*e**3/9 + 3*b**2*c*e**3 + 9*b*c**2*d*e**2 - 15*b*(37*b*c**2*e**3/18 + 3*c**3*d*e**2)/(16*c) + 3*c
**3*d**2*e)/(7*c) + 3*b**3*d*e**2 + 9*b**2*c*d**2*e + 3*b*c**2*d**3 - 11*b*(6*a*b*c*e**3 + 9*a*c**2*d*e**2 - 7
*a*(37*b*c**2*e**3/18 + 3*c**3*d*e**2)/(8*c) + b**3*e**3 + 9*b**2*c*d*e**2 + 9*b*c**2*d**2*e - 13*b*(19*a*c**2
*e**3/9 + 3*b**2*c*e**3 + 9*b*c**2*d*e**2 - 15*b*(37*b*c**2*e**3/18 + 3*c**3*d*e**2)/(16*c) + 3*c**3*d**2*e)/(
14*c) + c**3*d**3)/(12*c))/(5*c) + b**3*d**3 - 7*b*(3*a**2*b*e**3 + 9*a**2*c*d*e**2 + 9*a*b**2*d*e**2 + 18*a*b
*c*d**2*e + 3*a*c**2*d**3 - 5*a*(6*a*b*c*e**3 + 9*a*c**2*d*e**2 - 7*a*(37*b*c**2*e**3/18 + 3*c**3*d*e**2)/(8*c
) + b**3*e**3 + 9*b**2*c*d*e**2 + 9*b*c**2*d**2*e - 13*b*(19*a*c**2*e**3/9 + 3*b**2*c*e**3 + 9*b*c**2*d*e**2 -
15*b*(37*b*c**2*e**3/18 + 3*c**3*d*e**2)/(16*c) + 3*c**3*d**2*e)/(14*c) + c**3*d**3)/(6*c) + 3*b**3*d**2*e +
3*b**2*c*d**3 - 9*b*(3*a**2*c*e**3 + 3*a*b**2*e**3 + 18*a*b*c*d*e**2 + 9*a*c**2*d**2*e - 6*a*(19*a*c**2*e**3/9
+ 3*b**2*c*e**3 + 9*b*c**2*d*e**2 - 15*b*(37*b*c**2*e**3/18 + 3*c**3*d*e**2)/(16*c) + 3*c**3*d**2*e)/(7*c) +
3*b**3*d*e**2 + 9*b**2*c*d**2*e + 3*b*c**2*d**3 - 11*b*(6*a*b*c*e**3 + 9*a*c**2*d*e**2 - 7*a*(37*b*c**2*e**3/1
8 + 3*c**3*d*e**2)/(8*c) + b**3*e**3 + 9*b**2*c*d*e**2 + 9*b*c**2*d**2*e - 13*b*(19*a*c**2*e**3/9 + 3*b**2*c*e
**3 + 9*b*c**2*d*e**2 - 15*b*(37*b*c**2*e**3/18 + 3*c**3*d*e**2)/(16*c) + 3*c**3*d**2*e)/(14*c) + c**3*d**3)/(
12*c))/(10*c))/(8*c))/(6*c))/(4*c))/(2*c))*Piecewise((log(b + 2*sqrt(c)*sqrt(a + b*x + c*x**2) + 2*c*x)/sqrt(c
), Ne(a - b**2/(4*c), 0)), ((b/(2*c) + x)*log(b/(2*c) + x)/sqrt(c*(b/(2*c) + x)**2), True)), Ne(c, 0)), (2*(e*
*3*(a + b*x)**(13/2)/(13*b**3) + (a + b*x)**(11/2)*(-3*a*e**3 + 3*b*d*e**2)/(11*b**3) + (a + b*x)**(9/2)*(3*a*
*2*e**3 - 6*a*b*d*e**2 + 3*b**2*d**2*e)/(9*b**3) + (a + b*x)**(7/2)*(-a**3*e**3 + 3*a**2*b*d*e**2 - 3*a*b**2*d
**2*e + b**3*d**3)/(7*b**3))/b, Ne(b, 0)), (a**(5/2)*Piecewise((d**3*x, Eq(e, 0)), ((d + e*x)**4/(4*e), True))
, True))
Maxima [F(-2)]
Exception generated. \[
\int (d+e x)^3 \left (a+b x+c x^2\right )^{5/2} \, dx=\text {Exception raised: ValueError}
\]
[In]
integrate((e*x+d)^3*(c*x^2+b*x+a)^(5/2),x, algorithm="maxima")
[Out]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` f
or more deta
Giac [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 1193 vs. \(2 (370) = 740\).
Time = 0.30 (sec) , antiderivative size = 1193, normalized size of antiderivative = 2.98
\[
\int (d+e x)^3 \left (a+b x+c x^2\right )^{5/2} \, dx=\text {Too large to display}
\]
[In]
integrate((e*x+d)^3*(c*x^2+b*x+a)^(5/2),x, algorithm="giac")
[Out]
1/2064384*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(2*(4*(14*(16*c^2*e^3*x + (54*c^10*d*e^2 + 37*b*c^9*e^3)/c^8)*x +
(864*c^10*d^2*e + 1782*b*c^9*d*e^2 + 309*b^2*c^8*e^3 + 608*a*c^9*e^3)/c^8)*x + (1344*c^10*d^3 + 8352*b*c^9*d^2
*e + 4374*b^2*c^8*d*e^2 + 8568*a*c^9*d*e^2 + 5*b^3*c^7*e^3 + 3012*a*b*c^8*e^3)/c^8)*x + (6720*b*c^9*d^3 + 1065
6*b^2*c^8*d^2*e + 20736*a*c^9*d^2*e + 54*b^3*c^7*d*e^2 + 22104*a*b*c^8*d*e^2 - 11*b^4*c^6*e^3 + 84*a*b^2*c^7*e
^3 + 3840*a^2*c^8*e^3)/c^8)*x + (36288*b^2*c^8*d^3 + 69888*a*c^9*d^3 + 864*b^3*c^7*d^2*e + 226944*a*b*c^8*d^2*
e - 486*b^4*c^6*d*e^2 + 3888*a*b^2*c^7*d*e^2 + 118944*a^2*c^8*d*e^2 + 99*b^5*c^5*e^3 - 856*a*b^3*c^6*e^3 + 196
8*a^2*b*c^7*e^3)/c^8)*x + (1344*b^3*c^7*d^3 + 209664*a*b*c^8*d^3 - 2016*b^4*c^6*d^2*e + 17280*a*b^2*c^7*d^2*e
+ 331776*a^2*c^8*d^2*e + 1134*b^5*c^5*d*e^2 - 10224*a*b^3*c^6*d*e^2 + 25056*a^2*b*c^7*d*e^2 - 231*b^6*c^4*e^3
+ 2232*a*b^4*c^5*e^3 - 6384*a^2*b^2*c^6*e^3 + 4096*a^3*c^7*e^3)/c^8)*x - (6720*b^4*c^6*d^3 - 64512*a*b^2*c^7*d
^3 - 709632*a^2*c^8*d^3 - 10080*b^5*c^5*d^2*e + 96768*a*b^3*c^6*d^2*e - 262656*a^2*b*c^7*d^2*e + 5670*b^6*c^4*
d*e^2 - 56952*a*b^4*c^5*d*e^2 + 171936*a^2*b^2*c^6*d*e^2 - 120960*a^3*c^7*d*e^2 - 1155*b^7*c^3*e^3 + 12348*a*b
^5*c^4*e^3 - 42192*a^2*b^3*c^5*e^3 + 44096*a^3*b*c^6*e^3)/c^8)*x + (20160*b^5*c^5*d^3 - 215040*a*b^3*c^6*d^3 +
709632*a^2*b*c^7*d^3 - 30240*b^6*c^4*d^2*e + 322560*a*b^4*c^5*d^2*e - 1064448*a^2*b^2*c^6*d^2*e + 884736*a^3*
c^7*d^2*e + 17010*b^7*c^3*d*e^2 - 189000*a*b^5*c^4*d*e^2 + 679392*a^2*b^3*c^5*d*e^2 - 763776*a^3*b*c^6*d*e^2 -
3465*b^8*c^2*e^3 + 40740*a*b^6*c^3*e^3 - 162288*a^2*b^4*c^4*e^3 + 234432*a^3*b^2*c^5*e^3 - 65536*a^4*c^6*e^3)
/c^8) + 5/65536*(64*b^6*c^3*d^3 - 768*a*b^4*c^4*d^3 + 3072*a^2*b^2*c^5*d^3 - 4096*a^3*c^6*d^3 - 96*b^7*c^2*d^2
*e + 1152*a*b^5*c^3*d^2*e - 4608*a^2*b^3*c^4*d^2*e + 6144*a^3*b*c^5*d^2*e + 54*b^8*c*d*e^2 - 672*a*b^6*c^2*d*e
^2 + 2880*a^2*b^4*c^3*d*e^2 - 4608*a^3*b^2*c^4*d*e^2 + 1536*a^4*c^5*d*e^2 - 11*b^9*e^3 + 144*a*b^7*c*e^3 - 672
*a^2*b^5*c^2*e^3 + 1280*a^3*b^3*c^3*e^3 - 768*a^4*b*c^4*e^3)*log(abs(2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqr
t(c) + b))/c^(13/2)
Mupad [F(-1)]
Timed out. \[
\int (d+e x)^3 \left (a+b x+c x^2\right )^{5/2} \, dx=\int {\left (d+e\,x\right )}^3\,{\left (c\,x^2+b\,x+a\right )}^{5/2} \,d x
\]
[In]
int((d + e*x)^3*(a + b*x + c*x^2)^(5/2),x)
[Out]
int((d + e*x)^3*(a + b*x + c*x^2)^(5/2), x)