3.24.55 \(\int (d+e x)^3 (a+b x+c x^2)^{5/2} \, dx\) [2355]

Optimal. Leaf size=400 \[ \frac {5 \left (b^2-4 a c\right )^2 (2 c d-b e) \left (32 c^2 d^2+11 b^2 e^2-4 c e (8 b d+3 a e)\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{32768 c^6}-\frac {5 \left (b^2-4 a c\right ) (2 c d-b e) \left (32 c^2 d^2+11 b^2 e^2-4 c e (8 b d+3 a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{12288 c^5}+\frac {(2 c d-b e) \left (32 c^2 d^2+11 b^2 e^2-4 c e (8 b d+3 a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{768 c^4}+\frac {e (d+e x)^2 \left (a+b x+c x^2\right )^{7/2}}{9 c}+\frac {e \left (640 c^2 d^2+99 b^2 e^2-2 c e (243 b d+32 a e)+154 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{7/2}}{2016 c^3}-\frac {5 \left (b^2-4 a c\right )^3 (2 c d-b e) \left (32 c^2 d^2+11 b^2 e^2-4 c e (8 b d+3 a e)\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{65536 c^{13/2}} \]

[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

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Rubi [A]
time = 0.27, antiderivative size = 400, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {756, 793, 626, 635, 212} \begin {gather*} -\frac {5 \left (b^2-4 a c\right )^3 (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 ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^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} \end {gather*}

Antiderivative was successfully verified.

[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*} \int (d+e x)^3 \left (a+b x+c x^2\right )^{5/2} \, dx &=\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*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(807\) vs. \(2(400)=800\).
time = 4.71, size = 807, normalized size = 2.02 \begin {gather*} \frac {2 \sqrt {c} \sqrt {a+x (b+c x)} \left (-3465 b^8 e^3+210 b^7 c e^2 (81 d+11 e x)-84 b^6 c e \left (-485 a e^2+c \left (360 d^2+135 d e x+22 e^2 x^2\right )\right )+72 b^5 c^2 \left (-7 a e^2 (375 d+49 e x)+2 c \left (140 d^3+140 d^2 e x+63 d e^2 x^2+11 e^3 x^3\right )\right )-16 b^4 c^2 \left (10143 a^2 e^3-9 a c e \left (2240 d^2+791 d e x+124 e^2 x^2\right )+2 c^2 x \left (420 d^3+504 d^2 e x+243 d e^2 x^2+44 e^3 x^3\right )\right )+32 b^3 c^3 \left (9 a^2 e^2 (2359 d+293 e x)+8 c^2 x^2 \left (42 d^3+54 d^2 e x+27 d e^2 x^2+5 e^3 x^3\right )-4 a c \left (1680 d^3+1512 d^2 e x+639 d e^2 x^2+107 e^3 x^3\right )\right )+192 b^2 c^3 \left (1221 a^3 e^3-a^2 c e \left (5544 d^2+1791 d e x+266 e^2 x^2\right )+4 a c^2 x \left (168 d^3+180 d^2 e x+81 d e^2 x^2+14 e^3 x^3\right )+8 c^3 x^3 \left (378 d^3+888 d^2 e x+729 d e^2 x^2+206 e^3 x^3\right )\right )+128 b c^4 \left (-13 a^3 e^2 (459 d+53 e x)+6 a^2 c \left (924 d^3+684 d^2 e x+261 d e^2 x^2+41 e^3 x^3\right )+24 a c^2 x^2 \left (546 d^3+1182 d^2 e x+921 d e^2 x^2+251 e^3 x^3\right )+16 c^3 x^4 \left (420 d^3+1044 d^2 e x+891 d e^2 x^2+259 e^3 x^3\right )\right )+256 c^4 \left (-256 a^4 e^3+a^3 c e \left (3456 d^2+945 d e x+128 e^2 x^2\right )+16 c^4 x^5 \left (84 d^3+216 d^2 e x+189 d e^2 x^2+56 e^3 x^3\right )+8 a c^3 x^3 \left (546 d^3+1296 d^2 e x+1071 d e^2 x^2+304 e^3 x^3\right )+6 a^2 c^2 x \left (924 d^3+1728 d^2 e x+1239 d e^2 x^2+320 e^3 x^3\right )\right )\right )-315 \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 ) \log \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )}{4128768 c^{13/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^3*(a + b*x + c*x^2)^(5/2),x]

[Out]

(2*Sqrt[c]*Sqrt[a + x*(b + c*x)]*(-3465*b^8*e^3 + 210*b^7*c*e^2*(81*d + 11*e*x) - 84*b^6*c*e*(-485*a*e^2 + c*(
360*d^2 + 135*d*e*x + 22*e^2*x^2)) + 72*b^5*c^2*(-7*a*e^2*(375*d + 49*e*x) + 2*c*(140*d^3 + 140*d^2*e*x + 63*d
*e^2*x^2 + 11*e^3*x^3)) - 16*b^4*c^2*(10143*a^2*e^3 - 9*a*c*e*(2240*d^2 + 791*d*e*x + 124*e^2*x^2) + 2*c^2*x*(
420*d^3 + 504*d^2*e*x + 243*d*e^2*x^2 + 44*e^3*x^3)) + 32*b^3*c^3*(9*a^2*e^2*(2359*d + 293*e*x) + 8*c^2*x^2*(4
2*d^3 + 54*d^2*e*x + 27*d*e^2*x^2 + 5*e^3*x^3) - 4*a*c*(1680*d^3 + 1512*d^2*e*x + 639*d*e^2*x^2 + 107*e^3*x^3)
) + 192*b^2*c^3*(1221*a^3*e^3 - a^2*c*e*(5544*d^2 + 1791*d*e*x + 266*e^2*x^2) + 4*a*c^2*x*(168*d^3 + 180*d^2*e
*x + 81*d*e^2*x^2 + 14*e^3*x^3) + 8*c^3*x^3*(378*d^3 + 888*d^2*e*x + 729*d*e^2*x^2 + 206*e^3*x^3)) + 128*b*c^4
*(-13*a^3*e^2*(459*d + 53*e*x) + 6*a^2*c*(924*d^3 + 684*d^2*e*x + 261*d*e^2*x^2 + 41*e^3*x^3) + 24*a*c^2*x^2*(
546*d^3 + 1182*d^2*e*x + 921*d*e^2*x^2 + 251*e^3*x^3) + 16*c^3*x^4*(420*d^3 + 1044*d^2*e*x + 891*d*e^2*x^2 + 2
59*e^3*x^3)) + 256*c^4*(-256*a^4*e^3 + a^3*c*e*(3456*d^2 + 945*d*e*x + 128*e^2*x^2) + 16*c^4*x^5*(84*d^3 + 216
*d^2*e*x + 189*d*e^2*x^2 + 56*e^3*x^3) + 8*a*c^3*x^3*(546*d^3 + 1296*d^2*e*x + 1071*d*e^2*x^2 + 304*e^3*x^3) +
6*a^2*c^2*x*(924*d^3 + 1728*d^2*e*x + 1239*d*e^2*x^2 + 320*e^3*x^3))) - 315*(b^2 - 4*a*c)^3*(-2*c*d + b*e)*(3
2*c^2*d^2 + 11*b^2*e^2 - 4*c*e*(8*b*d + 3*a*e))*Log[b + 2*c*x - 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])/(4128768*c^(
13/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1206\) vs. \(2(370)=740\).
time = 0.91, size = 1207, normalized size = 3.02

method	result	size
default	\(\text {Expression too large to display}\)	\(1207\)
risch	\(\text {Expression too large to display}\)	\(1683\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^3*(c*x^2+b*x+a)^(5/2),x,method=_RETURNVERBOSE)

[Out]

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+3/16*(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))))))-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))))))+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)))))

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1039 vs. \(2 (384) = 768\).
time = 4.05, size = 2081, normalized size = 5.20 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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*(344064*c^9*d^3*x^5 + 860
160*b*c^8*d^3*x^4 + 21504*(27*b^2*c^7 + 52*a*c^8)*d^3*x^3 + 10752*(b^3*c^6 + 156*a*b*c^7)*d^3*x^2 - 2688*(5*b^
4*c^5 - 48*a*b^2*c^6 - 528*a^2*c^7)*d^3*x + 1344*(15*b^5*c^4 - 160*a*b^3*c^5 + 528*a^2*b*c^6)*d^3 + (229376*c^
9*x^8 + 530432*b*c^8*x^7 - 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 + 1024*(309*b^2*c^7 + 608*a*c^8)*x^6 + 256*(5*b^3*c^6 + 3012*a*b*c^7)*x^5 - 128*(11*b^4*c^5 - 84*a*b^2*c^6
- 3840*a^2*c^7)*x^4 + 16*(99*b^5*c^4 - 856*a*b^3*c^5 + 1968*a^2*b*c^6)*x^3 - 8*(231*b^6*c^3 - 2232*a*b^4*c^4
+ 6384*a^2*b^2*c^5 - 4096*a^3*c^6)*x^2 + 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)*x)*e^3 + 18*(43008*c^9*d*x^7 + 101376*b*c^8*d*x^6 + 256*(243*b^2*c^7 + 476*a*c^8)*d*x^5 + 128*(3*b^3*c^6 +
1228*a*b*c^7)*d*x^4 - 16*(27*b^4*c^5 - 216*a*b^2*c^6 - 6608*a^2*c^7)*d*x^3 + 8*(63*b^5*c^4 - 568*a*b^3*c^5 +
1392*a^2*b*c^6)*d*x^2 - 2*(315*b^6*c^3 - 3164*a*b^4*c^4 + 9552*a^2*b^2*c^5 - 6720*a^3*c^6)*d*x + (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 + 288*(3072*c^9*d^2*x^6 + 7424*b*c^8*d^2*x^5 +
128*(37*b^2*c^7 + 72*a*c^8)*d^2*x^4 + 16*(3*b^3*c^6 + 788*a*b*c^7)*d^2*x^3 - 8*(7*b^4*c^5 - 60*a*b^2*c^6 - 11
52*a^2*c^7)*d^2*x^2 + 2*(35*b^5*c^4 - 336*a*b^3*c^5 + 912*a^2*b*c^6)*d^2*x - (105*b^6*c^3 - 1120*a*b^4*c^4 + 3
696*a^2*b^2*c^5 - 3072*a^3*c^6)*d^2)*e)*sqrt(c*x^2 + b*x + a))/c^7, 1/4128768*(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)*arctan(1/2*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqr
t(-c)/(c^2*x^2 + b*c*x + a*c)) + 2*(344064*c^9*d^3*x^5 + 860160*b*c^8*d^3*x^4 + 21504*(27*b^2*c^7 + 52*a*c^8)*
d^3*x^3 + 10752*(b^3*c^6 + 156*a*b*c^7)*d^3*x^2 - 2688*(5*b^4*c^5 - 48*a*b^2*c^6 - 528*a^2*c^7)*d^3*x + 1344*(
15*b^5*c^4 - 160*a*b^3*c^5 + 528*a^2*b*c^6)*d^3 + (229376*c^9*x^8 + 530432*b*c^8*x^7 - 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 + 1024*(309*b^2*c^7 + 608*a*c^8)*x^6 + 256*(5*
b^3*c^6 + 3012*a*b*c^7)*x^5 - 128*(11*b^4*c^5 - 84*a*b^2*c^6 - 3840*a^2*c^7)*x^4 + 16*(99*b^5*c^4 - 856*a*b^3*
c^5 + 1968*a^2*b*c^6)*x^3 - 8*(231*b^6*c^3 - 2232*a*b^4*c^4 + 6384*a^2*b^2*c^5 - 4096*a^3*c^6)*x^2 + 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)*x)*e^3 + 18*(43008*c^9*d*x^7 + 101376*b*c^8*d*
x^6 + 256*(243*b^2*c^7 + 476*a*c^8)*d*x^5 + 128*(3*b^3*c^6 + 1228*a*b*c^7)*d*x^4 - 16*(27*b^4*c^5 - 216*a*b^2*
c^6 - 6608*a^2*c^7)*d*x^3 + 8*(63*b^5*c^4 - 568*a*b^3*c^5 + 1392*a^2*b*c^6)*d*x^2 - 2*(315*b^6*c^3 - 3164*a*b^
4*c^4 + 9552*a^2*b^2*c^5 - 6720*a^3*c^6)*d*x + (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 + 288*(3072*c^9*d^2*x^6 + 7424*b*c^8*d^2*x^5 + 128*(37*b^2*c^7 + 72*a*c^8)*d^2*x^4 + 16*(3*b^3*c
^6 + 788*a*b*c^7)*d^2*x^3 - 8*(7*b^4*c^5 - 60*a*b^2*c^6 - 1152*a^2*c^7)*d^2*x^2 + 2*(35*b^5*c^4 - 336*a*b^3*c^
5 + 912*a^2*b*c^6)*d^2*x - (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)*sqrt(c*x^2
+ b*x + a))/c^7]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (d + e x\right )^{3} \left (a + b x + c x^{2}\right )^{\frac {5}{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**3*(c*x**2+b*x+a)**(5/2),x)

[Out]

Integral((d + e*x)**3*(a + b*x + c*x**2)**(5/2), x)

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1160 vs. \(2 (384) = 768\).
time = 3.91, size = 1160, normalized size = 2.90 \begin {gather*} \frac {1}{2064384} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (2 \, {\left (4 \, {\left (14 \, {\left (16 \, c^{2} x e^{3} + \frac {54 \, c^{10} d e^{2} + 37 \, b c^{9} e^{3}}{c^{8}}\right )} x + \frac {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}}\right )} x + \frac {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}}\right )} x + \frac {6720 \, b c^{9} d^{3} + 10656 \, 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}}\right )} x + \frac {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} + 1968 \, a^{2} b c^{7} e^{3}}{c^{8}}\right )} x + \frac {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}}\right )} x - \frac {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}}\right )} x + \frac {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}}\right )} + \frac {5 \, {\left (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}\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} - b \right |}\right )}{65536 \, c^{\frac {13}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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*x*e^3 + (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))*sq
rt(c) - b))/c^(13/2)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (d+e\,x\right )}^3\,{\left (c\,x^2+b\,x+a\right )}^{5/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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