\(\int \frac {(a+b x+c x^2)^{3/2}}{(d+e x)^6} \, dx\) [2352]
Optimal result
Integrand size = 22, antiderivative size = 296 \[
\int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^6} \, dx=-\frac {3 \left (b^2-4 a c\right ) (2 c d-b e) (b d-2 a e+(2 c d-b e) x) \sqrt {a+b x+c x^2}}{128 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^2}+\frac {(2 c d-b e) (b d-2 a e+(2 c d-b e) x) \left (a+b x+c x^2\right )^{3/2}}{16 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^4}-\frac {e \left (a+b x+c x^2\right )^{5/2}}{5 \left (c d^2-b d e+a e^2\right ) (d+e x)^5}+\frac {3 \left (b^2-4 a c\right )^2 (2 c d-b e) \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{256 \left (c d^2-b d e+a e^2\right )^{7/2}}
\]
[Out]
1/16*(-b*e+2*c*d)*(b*d-2*a*e+(-b*e+2*c*d)*x)*(c*x^2+b*x+a)^(3/2)/(a*e^2-b*d*e+c*d^2)^2/(e*x+d)^4-1/5*e*(c*x^2+
b*x+a)^(5/2)/(a*e^2-b*d*e+c*d^2)/(e*x+d)^5+3/256*(-4*a*c+b^2)^2*(-b*e+2*c*d)*arctanh(1/2*(b*d-2*a*e+(-b*e+2*c*
d)*x)/(a*e^2-b*d*e+c*d^2)^(1/2)/(c*x^2+b*x+a)^(1/2))/(a*e^2-b*d*e+c*d^2)^(7/2)-3/128*(-4*a*c+b^2)*(-b*e+2*c*d)
*(b*d-2*a*e+(-b*e+2*c*d)*x)*(c*x^2+b*x+a)^(1/2)/(a*e^2-b*d*e+c*d^2)^3/(e*x+d)^2
Rubi [A] (verified)
Time = 0.14 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.00, number of
steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {744, 734, 738, 212}
\[
\int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^6} \, dx=\frac {3 \left (b^2-4 a c\right )^2 (2 c d-b e) \text {arctanh}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{256 \left (a e^2-b d e+c d^2\right )^{7/2}}-\frac {3 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} (2 c d-b e) (-2 a e+x (2 c d-b e)+b d)}{128 (d+e x)^2 \left (a e^2-b d e+c d^2\right )^3}-\frac {e \left (a+b x+c x^2\right )^{5/2}}{5 (d+e x)^5 \left (a e^2-b d e+c d^2\right )}+\frac {\left (a+b x+c x^2\right )^{3/2} (2 c d-b e) (-2 a e+x (2 c d-b e)+b d)}{16 (d+e x)^4 \left (a e^2-b d e+c d^2\right )^2}
\]
[In]
Int[(a + b*x + c*x^2)^(3/2)/(d + e*x)^6,x]
[Out]
(-3*(b^2 - 4*a*c)*(2*c*d - b*e)*(b*d - 2*a*e + (2*c*d - b*e)*x)*Sqrt[a + b*x + c*x^2])/(128*(c*d^2 - b*d*e + a
*e^2)^3*(d + e*x)^2) + ((2*c*d - b*e)*(b*d - 2*a*e + (2*c*d - b*e)*x)*(a + b*x + c*x^2)^(3/2))/(16*(c*d^2 - b*
d*e + a*e^2)^2*(d + e*x)^4) - (e*(a + b*x + c*x^2)^(5/2))/(5*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^5) + (3*(b^2 -
4*a*c)^2*(2*c*d - b*e)*ArcTanh[(b*d - 2*a*e + (2*c*d - b*e)*x)/(2*Sqrt[c*d^2 - b*d*e + a*e^2]*Sqrt[a + b*x + c
*x^2])])/(256*(c*d^2 - b*d*e + a*e^2)^(7/2))
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 734
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))
*(d*b - 2*a*e + (2*c*d - b*e)*x)*((a + b*x + c*x^2)^p/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Dist[p*((b^2
- 4*a*c)/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2))), Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2)^(p - 1), x], x] /; Free
Q[{a, b, c, d, e}, 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] && EqQ[m
+ 2*p + 2, 0] && GtQ[p, 0]
Rule 738
Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]
Rule 744
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)/((m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Dist[(2*c*d - b*e)/(2*(c*d^2 - b*d*e + a*e
^2)), Int[(d + e*x)^(m + 1)*(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] && EqQ[m + 2*p + 3, 0]
Rubi steps \begin{align*}
\text {integral}& = -\frac {e \left (a+b x+c x^2\right )^{5/2}}{5 \left (c d^2-b d e+a e^2\right ) (d+e x)^5}+\frac {(2 c d-b e) \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^5} \, dx}{2 \left (c d^2-b d e+a e^2\right )} \\ & = \frac {(2 c d-b e) (b d-2 a e+(2 c d-b e) x) \left (a+b x+c x^2\right )^{3/2}}{16 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^4}-\frac {e \left (a+b x+c x^2\right )^{5/2}}{5 \left (c d^2-b d e+a e^2\right ) (d+e x)^5}-\frac {\left (3 \left (b^2-4 a c\right ) (2 c d-b e)\right ) \int \frac {\sqrt {a+b x+c x^2}}{(d+e x)^3} \, dx}{32 \left (c d^2-b d e+a e^2\right )^2} \\ & = -\frac {3 \left (b^2-4 a c\right ) (2 c d-b e) (b d-2 a e+(2 c d-b e) x) \sqrt {a+b x+c x^2}}{128 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^2}+\frac {(2 c d-b e) (b d-2 a e+(2 c d-b e) x) \left (a+b x+c x^2\right )^{3/2}}{16 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^4}-\frac {e \left (a+b x+c x^2\right )^{5/2}}{5 \left (c d^2-b d e+a e^2\right ) (d+e x)^5}+\frac {\left (3 \left (b^2-4 a c\right )^2 (2 c d-b e)\right ) \int \frac {1}{(d+e x) \sqrt {a+b x+c x^2}} \, dx}{256 \left (c d^2-b d e+a e^2\right )^3} \\ & = -\frac {3 \left (b^2-4 a c\right ) (2 c d-b e) (b d-2 a e+(2 c d-b e) x) \sqrt {a+b x+c x^2}}{128 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^2}+\frac {(2 c d-b e) (b d-2 a e+(2 c d-b e) x) \left (a+b x+c x^2\right )^{3/2}}{16 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^4}-\frac {e \left (a+b x+c x^2\right )^{5/2}}{5 \left (c d^2-b d e+a e^2\right ) (d+e x)^5}-\frac {\left (3 \left (b^2-4 a c\right )^2 (2 c d-b e)\right ) \text {Subst}\left (\int \frac {1}{4 c d^2-4 b d e+4 a e^2-x^2} \, dx,x,\frac {-b d+2 a e-(2 c d-b e) x}{\sqrt {a+b x+c x^2}}\right )}{128 \left (c d^2-b d e+a e^2\right )^3} \\ & = -\frac {3 \left (b^2-4 a c\right ) (2 c d-b e) (b d-2 a e+(2 c d-b e) x) \sqrt {a+b x+c x^2}}{128 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^2}+\frac {(2 c d-b e) (b d-2 a e+(2 c d-b e) x) \left (a+b x+c x^2\right )^{3/2}}{16 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^4}-\frac {e \left (a+b x+c x^2\right )^{5/2}}{5 \left (c d^2-b d e+a e^2\right ) (d+e x)^5}+\frac {3 \left (b^2-4 a c\right )^2 (2 c d-b e) \tanh ^{-1}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{256 \left (c d^2-b d e+a e^2\right )^{7/2}} \\
\end{align*}
Mathematica [A] (verified)
Time = 10.94 (sec) , antiderivative size = 275, normalized size of antiderivative = 0.93
\[
\int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^6} \, dx=-\frac {e (a+x (b+c x))^{5/2}}{5 \left (c d^2+e (-b d+a e)\right ) (d+e x)^5}-\frac {(2 c d-b e) \left (\frac {2 (-b d+2 a e-2 c d x+b e x) (a+x (b+c x))^{3/2}}{(d+e x)^4}+3 \left (b^2-4 a c\right ) \left (\frac {\sqrt {a+x (b+c x)} (-2 a e+2 c d x+b (d-e x))}{4 \left (c d^2+e (-b d+a e)\right ) (d+e x)^2}+\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {-b d+2 a e-2 c d x+b e x}{2 \sqrt {c d^2+e (-b d+a e)} \sqrt {a+x (b+c x)}}\right )}{8 \left (c d^2+e (-b d+a e)\right )^{3/2}}\right )\right )}{32 \left (c d^2+e (-b d+a e)\right )^2}
\]
[In]
Integrate[(a + b*x + c*x^2)^(3/2)/(d + e*x)^6,x]
[Out]
-1/5*(e*(a + x*(b + c*x))^(5/2))/((c*d^2 + e*(-(b*d) + a*e))*(d + e*x)^5) - ((2*c*d - b*e)*((2*(-(b*d) + 2*a*e
- 2*c*d*x + b*e*x)*(a + x*(b + c*x))^(3/2))/(d + e*x)^4 + 3*(b^2 - 4*a*c)*((Sqrt[a + x*(b + c*x)]*(-2*a*e + 2
*c*d*x + b*(d - e*x)))/(4*(c*d^2 + e*(-(b*d) + a*e))*(d + e*x)^2) + ((b^2 - 4*a*c)*ArcTanh[(-(b*d) + 2*a*e - 2
*c*d*x + b*e*x)/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)])])/(8*(c*d^2 + e*(-(b*d) + a*e))^(3/2)
))))/(32*(c*d^2 + e*(-(b*d) + a*e))^2)
Maple [B] (verified)
Leaf count of result is larger than twice the leaf count of optimal. \(5188\) vs. \(2(274)=548\).
Time = 0.70 (sec) , antiderivative size = 5189, normalized size of antiderivative =
17.53
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method | result | size |
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default |
\(\text {Expression too large to display}\) |
\(5189\) |
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[In]
int((c*x^2+b*x+a)^(3/2)/(e*x+d)^6,x,method=_RETURNVERBOSE)
[Out]
result too large to display
Fricas [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 2394 vs. \(2 (274) = 548\).
Time = 44.81 (sec) , antiderivative size = 4830, normalized size of antiderivative = 16.32
\[
\int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^6} \, dx=\text {Too large to display}
\]
[In]
integrate((c*x^2+b*x+a)^(3/2)/(e*x+d)^6,x, algorithm="fricas")
[Out]
[-1/2560*(15*(2*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d^6 - (b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*d^5*e + (2*(b^4*c -
8*a*b^2*c^2 + 16*a^2*c^3)*d*e^5 - (b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*e^6)*x^5 + 5*(2*(b^4*c - 8*a*b^2*c^2 + 16*a
^2*c^3)*d^2*e^4 - (b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*d*e^5)*x^4 + 10*(2*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d^3*e
^3 - (b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*d^2*e^4)*x^3 + 10*(2*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d^4*e^2 - (b^5 -
8*a*b^3*c + 16*a^2*b*c^2)*d^3*e^3)*x^2 + 5*(2*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d^5*e - (b^5 - 8*a*b^3*c + 1
6*a^2*b*c^2)*d^4*e^2)*x)*sqrt(c*d^2 - b*d*e + a*e^2)*log((8*a*b*d*e - 8*a^2*e^2 - (b^2 + 4*a*c)*d^2 - (8*c^2*d
^2 - 8*b*c*d*e + (b^2 + 4*a*c)*e^2)*x^2 + 4*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(c*x^2 + b*x + a)*(b*d - 2*a*e + (
2*c*d - b*e)*x) - 2*(4*b*c*d^2 + 4*a*b*e^2 - (3*b^2 + 4*a*c)*d*e)*x)/(e^2*x^2 + 2*d*e*x + d^2)) - 4*(464*a^4*b
*d*e^6 - 128*a^5*e^7 - 10*(3*b^3*c^2 - 20*a*b*c^3)*d^7 + (45*b^4*c - 320*a*b^2*c^2 - 528*a^2*c^3)*d^6*e - (15*
b^5 - 100*a*b^3*c - 1424*a^2*b*c^2)*d^5*e^2 + (5*a*b^4 - 1064*a^2*b^2*c - 944*a^3*c^2)*d^4*e^3 + 2*(129*a^2*b^
3 + 724*a^3*b*c)*d^3*e^4 - 8*(73*a^3*b^2 + 68*a^4*c)*d^2*e^5 + (32*c^5*d^6*e - 96*b*c^4*d^5*e^2 + 4*(19*b^2*c^
3 + 44*a*c^4)*d^4*e^3 + 8*(b^3*c^2 - 44*a*b*c^3)*d^3*e^4 - (35*b^4*c - 256*a*b^2*c^2 - 16*a^2*c^3)*d^2*e^5 + (
15*b^5 - 80*a*b^3*c - 16*a^2*b*c^2)*d*e^6 - (15*a*b^4 - 100*a^2*b^2*c + 128*a^3*c^2)*e^7)*x^4 + 2*(80*c^5*d^7
- 248*b*c^4*d^6*e + 2*(107*b^2*c^3 + 220*a*c^4)*d^5*e^2 + (b^3*c^2 - 924*a*b*c^3)*d^4*e^3 - 2*(41*b^4*c - 334*
a*b^2*c^2 - 80*a^2*c^3)*d^3*e^4 + (35*b^5 - 174*a*b^3*c - 224*a^2*b*c^2)*d^2*e^5 - 2*(20*a*b^4 - 127*a^2*b^2*c
+ 100*a^3*c^2)*d*e^6 + (5*a^2*b^3 - 28*a^3*b*c)*e^7)*x^3 + 2*(120*b*c^4*d^7 - 2*(227*b^2*c^3 - 116*a*c^4)*d^6
*e + 9*(63*b^3*c^2 - 4*a*b*c^3)*d^5*e^2 - 3*(99*b^4*c + 152*a*b^2*c^2 + 128*a^2*c^3)*d^4*e^3 + 2*(32*b^5 + 151
*a*b^3*c + 504*a^2*b*c^2)*d^3*e^4 - 3*(29*a*b^4 + 122*a^2*b^2*c + 248*a^3*c^2)*d^2*e^5 + 3*(9*a^2*b^3 + 148*a^
3*b*c)*d*e^6 - 4*(a^3*b^2 + 32*a^4*c)*e^7)*x^2 - 2*(88*a^4*b*e^7 - 10*(b^2*c^3 + 20*a*c^4)*d^7 + (85*b^3*c^2 +
468*a*b*c^3)*d^6*e - 2*(55*b^4*c + 382*a*b^2*c^2 - 80*a^2*c^3)*d^5*e^2 + (35*b^5 + 794*a*b^3*c - 16*a^2*b*c^2
)*d^4*e^3 - 2*(134*a*b^4 + 409*a^2*b^2*c - 220*a^3*c^2)*d^3*e^4 + 3*(163*a^2*b^3 + 28*a^3*b*c)*d^2*e^5 - 8*(43
*a^3*b^2 - 10*a^4*c)*d*e^6)*x)*sqrt(c*x^2 + b*x + a))/(c^4*d^13 - 4*b*c^3*d^12*e - 4*a^3*b*d^6*e^7 + a^4*d^5*e
^8 + 2*(3*b^2*c^2 + 2*a*c^3)*d^11*e^2 - 4*(b^3*c + 3*a*b*c^2)*d^10*e^3 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^9*e^
4 - 4*(a*b^3 + 3*a^2*b*c)*d^8*e^5 + 2*(3*a^2*b^2 + 2*a^3*c)*d^7*e^6 + (c^4*d^8*e^5 - 4*b*c^3*d^7*e^6 - 4*a^3*b
*d*e^12 + a^4*e^13 + 2*(3*b^2*c^2 + 2*a*c^3)*d^6*e^7 - 4*(b^3*c + 3*a*b*c^2)*d^5*e^8 + (b^4 + 12*a*b^2*c + 6*a
^2*c^2)*d^4*e^9 - 4*(a*b^3 + 3*a^2*b*c)*d^3*e^10 + 2*(3*a^2*b^2 + 2*a^3*c)*d^2*e^11)*x^5 + 5*(c^4*d^9*e^4 - 4*
b*c^3*d^8*e^5 - 4*a^3*b*d^2*e^11 + a^4*d*e^12 + 2*(3*b^2*c^2 + 2*a*c^3)*d^7*e^6 - 4*(b^3*c + 3*a*b*c^2)*d^6*e^
7 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^5*e^8 - 4*(a*b^3 + 3*a^2*b*c)*d^4*e^9 + 2*(3*a^2*b^2 + 2*a^3*c)*d^3*e^10)
*x^4 + 10*(c^4*d^10*e^3 - 4*b*c^3*d^9*e^4 - 4*a^3*b*d^3*e^10 + a^4*d^2*e^11 + 2*(3*b^2*c^2 + 2*a*c^3)*d^8*e^5
- 4*(b^3*c + 3*a*b*c^2)*d^7*e^6 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^6*e^7 - 4*(a*b^3 + 3*a^2*b*c)*d^5*e^8 + 2*(
3*a^2*b^2 + 2*a^3*c)*d^4*e^9)*x^3 + 10*(c^4*d^11*e^2 - 4*b*c^3*d^10*e^3 - 4*a^3*b*d^4*e^9 + a^4*d^3*e^10 + 2*(
3*b^2*c^2 + 2*a*c^3)*d^9*e^4 - 4*(b^3*c + 3*a*b*c^2)*d^8*e^5 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^7*e^6 - 4*(a*b
^3 + 3*a^2*b*c)*d^6*e^7 + 2*(3*a^2*b^2 + 2*a^3*c)*d^5*e^8)*x^2 + 5*(c^4*d^12*e - 4*b*c^3*d^11*e^2 - 4*a^3*b*d^
5*e^8 + a^4*d^4*e^9 + 2*(3*b^2*c^2 + 2*a*c^3)*d^10*e^3 - 4*(b^3*c + 3*a*b*c^2)*d^9*e^4 + (b^4 + 12*a*b^2*c + 6
*a^2*c^2)*d^8*e^5 - 4*(a*b^3 + 3*a^2*b*c)*d^7*e^6 + 2*(3*a^2*b^2 + 2*a^3*c)*d^6*e^7)*x), 1/1280*(15*(2*(b^4*c
- 8*a*b^2*c^2 + 16*a^2*c^3)*d^6 - (b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*d^5*e + (2*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^
3)*d*e^5 - (b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*e^6)*x^5 + 5*(2*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d^2*e^4 - (b^5
- 8*a*b^3*c + 16*a^2*b*c^2)*d*e^5)*x^4 + 10*(2*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d^3*e^3 - (b^5 - 8*a*b^3*c +
16*a^2*b*c^2)*d^2*e^4)*x^3 + 10*(2*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d^4*e^2 - (b^5 - 8*a*b^3*c + 16*a^2*b*c
^2)*d^3*e^3)*x^2 + 5*(2*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d^5*e - (b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*d^4*e^2)*x
)*sqrt(-c*d^2 + b*d*e - a*e^2)*arctan(-1/2*sqrt(-c*d^2 + b*d*e - a*e^2)*sqrt(c*x^2 + b*x + a)*(b*d - 2*a*e + (
2*c*d - b*e)*x)/(a*c*d^2 - a*b*d*e + a^2*e^2 + (c^2*d^2 - b*c*d*e + a*c*e^2)*x^2 + (b*c*d^2 - b^2*d*e + a*b*e^
2)*x)) + 2*(464*a^4*b*d*e^6 - 128*a^5*e^7 - 10*(3*b^3*c^2 - 20*a*b*c^3)*d^7 + (45*b^4*c - 320*a*b^2*c^2 - 528*
a^2*c^3)*d^6*e - (15*b^5 - 100*a*b^3*c - 1424*a^2*b*c^2)*d^5*e^2 + (5*a*b^4 - 1064*a^2*b^2*c - 944*a^3*c^2)*d^
4*e^3 + 2*(129*a^2*b^3 + 724*a^3*b*c)*d^3*e^4 - 8*(73*a^3*b^2 + 68*a^4*c)*d^2*e^5 + (32*c^5*d^6*e - 96*b*c^4*d
^5*e^2 + 4*(19*b^2*c^3 + 44*a*c^4)*d^4*e^3 + 8*(b^3*c^2 - 44*a*b*c^3)*d^3*e^4 - (35*b^4*c - 256*a*b^2*c^2 - 16
*a^2*c^3)*d^2*e^5 + (15*b^5 - 80*a*b^3*c - 16*a^2*b*c^2)*d*e^6 - (15*a*b^4 - 100*a^2*b^2*c + 128*a^3*c^2)*e^7)
*x^4 + 2*(80*c^5*d^7 - 248*b*c^4*d^6*e + 2*(107*b^2*c^3 + 220*a*c^4)*d^5*e^2 + (b^3*c^2 - 924*a*b*c^3)*d^4*e^3
- 2*(41*b^4*c - 334*a*b^2*c^2 - 80*a^2*c^3)*d^3*e^4 + (35*b^5 - 174*a*b^3*c - 224*a^2*b*c^2)*d^2*e^5 - 2*(20*
a*b^4 - 127*a^2*b^2*c + 100*a^3*c^2)*d*e^6 + (5*a^2*b^3 - 28*a^3*b*c)*e^7)*x^3 + 2*(120*b*c^4*d^7 - 2*(227*b^2
*c^3 - 116*a*c^4)*d^6*e + 9*(63*b^3*c^2 - 4*a*b*c^3)*d^5*e^2 - 3*(99*b^4*c + 152*a*b^2*c^2 + 128*a^2*c^3)*d^4*
e^3 + 2*(32*b^5 + 151*a*b^3*c + 504*a^2*b*c^2)*d^3*e^4 - 3*(29*a*b^4 + 122*a^2*b^2*c + 248*a^3*c^2)*d^2*e^5 +
3*(9*a^2*b^3 + 148*a^3*b*c)*d*e^6 - 4*(a^3*b^2 + 32*a^4*c)*e^7)*x^2 - 2*(88*a^4*b*e^7 - 10*(b^2*c^3 + 20*a*c^4
)*d^7 + (85*b^3*c^2 + 468*a*b*c^3)*d^6*e - 2*(55*b^4*c + 382*a*b^2*c^2 - 80*a^2*c^3)*d^5*e^2 + (35*b^5 + 794*a
*b^3*c - 16*a^2*b*c^2)*d^4*e^3 - 2*(134*a*b^4 + 409*a^2*b^2*c - 220*a^3*c^2)*d^3*e^4 + 3*(163*a^2*b^3 + 28*a^3
*b*c)*d^2*e^5 - 8*(43*a^3*b^2 - 10*a^4*c)*d*e^6)*x)*sqrt(c*x^2 + b*x + a))/(c^4*d^13 - 4*b*c^3*d^12*e - 4*a^3*
b*d^6*e^7 + a^4*d^5*e^8 + 2*(3*b^2*c^2 + 2*a*c^3)*d^11*e^2 - 4*(b^3*c + 3*a*b*c^2)*d^10*e^3 + (b^4 + 12*a*b^2*
c + 6*a^2*c^2)*d^9*e^4 - 4*(a*b^3 + 3*a^2*b*c)*d^8*e^5 + 2*(3*a^2*b^2 + 2*a^3*c)*d^7*e^6 + (c^4*d^8*e^5 - 4*b*
c^3*d^7*e^6 - 4*a^3*b*d*e^12 + a^4*e^13 + 2*(3*b^2*c^2 + 2*a*c^3)*d^6*e^7 - 4*(b^3*c + 3*a*b*c^2)*d^5*e^8 + (b
^4 + 12*a*b^2*c + 6*a^2*c^2)*d^4*e^9 - 4*(a*b^3 + 3*a^2*b*c)*d^3*e^10 + 2*(3*a^2*b^2 + 2*a^3*c)*d^2*e^11)*x^5
+ 5*(c^4*d^9*e^4 - 4*b*c^3*d^8*e^5 - 4*a^3*b*d^2*e^11 + a^4*d*e^12 + 2*(3*b^2*c^2 + 2*a*c^3)*d^7*e^6 - 4*(b^3*
c + 3*a*b*c^2)*d^6*e^7 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^5*e^8 - 4*(a*b^3 + 3*a^2*b*c)*d^4*e^9 + 2*(3*a^2*b^2
+ 2*a^3*c)*d^3*e^10)*x^4 + 10*(c^4*d^10*e^3 - 4*b*c^3*d^9*e^4 - 4*a^3*b*d^3*e^10 + a^4*d^2*e^11 + 2*(3*b^2*c^
2 + 2*a*c^3)*d^8*e^5 - 4*(b^3*c + 3*a*b*c^2)*d^7*e^6 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^6*e^7 - 4*(a*b^3 + 3*a
^2*b*c)*d^5*e^8 + 2*(3*a^2*b^2 + 2*a^3*c)*d^4*e^9)*x^3 + 10*(c^4*d^11*e^2 - 4*b*c^3*d^10*e^3 - 4*a^3*b*d^4*e^9
+ a^4*d^3*e^10 + 2*(3*b^2*c^2 + 2*a*c^3)*d^9*e^4 - 4*(b^3*c + 3*a*b*c^2)*d^8*e^5 + (b^4 + 12*a*b^2*c + 6*a^2*
c^2)*d^7*e^6 - 4*(a*b^3 + 3*a^2*b*c)*d^6*e^7 + 2*(3*a^2*b^2 + 2*a^3*c)*d^5*e^8)*x^2 + 5*(c^4*d^12*e - 4*b*c^3*
d^11*e^2 - 4*a^3*b*d^5*e^8 + a^4*d^4*e^9 + 2*(3*b^2*c^2 + 2*a*c^3)*d^10*e^3 - 4*(b^3*c + 3*a*b*c^2)*d^9*e^4 +
(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^8*e^5 - 4*(a*b^3 + 3*a^2*b*c)*d^7*e^6 + 2*(3*a^2*b^2 + 2*a^3*c)*d^6*e^7)*x)]
Sympy [F]
\[
\int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^6} \, dx=\int \frac {\left (a + b x + c x^{2}\right )^{\frac {3}{2}}}{\left (d + e x\right )^{6}}\, dx
\]
[In]
integrate((c*x**2+b*x+a)**(3/2)/(e*x+d)**6,x)
[Out]
Integral((a + b*x + c*x**2)**(3/2)/(d + e*x)**6, x)
Maxima [F(-2)]
Exception generated. \[
\int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^6} \, dx=\text {Exception raised: ValueError}
\]
[In]
integrate((c*x^2+b*x+a)^(3/2)/(e*x+d)^6,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(a*e^2-b*d*e>0)', see `assume?`
for more de
Giac [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 8409 vs. \(2 (274) = 548\).
Time = 0.98 (sec) , antiderivative size = 8409, normalized size of antiderivative = 28.41
\[
\int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^6} \, dx=\text {Too large to display}
\]
[In]
integrate((c*x^2+b*x+a)^(3/2)/(e*x+d)^6,x, algorithm="giac")
[Out]
3/128*(2*b^4*c*d - 16*a*b^2*c^2*d + 32*a^2*c^3*d - b^5*e + 8*a*b^3*c*e - 16*a^2*b*c^2*e)*arctan(-((sqrt(c)*x -
sqrt(c*x^2 + b*x + a))*e + sqrt(c)*d)/sqrt(-c*d^2 + b*d*e - a*e^2))/((c^3*d^6 - 3*b*c^2*d^5*e + 3*b^2*c*d^4*e
^2 + 3*a*c^2*d^4*e^2 - b^3*d^3*e^3 - 6*a*b*c*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a^2*c*d^2*e^4 - 3*a^2*b*d*e^5 + a^3
*e^6)*sqrt(-c*d^2 + b*d*e - a*e^2)) - 1/640*(30*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*b^4*c*d*e^8 - 240*(sqrt(
c)*x - sqrt(c*x^2 + b*x + a))^9*a*b^2*c^2*d*e^8 + 480*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*a^2*c^3*d*e^8 - 15
*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*b^5*e^9 + 120*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*a*b^3*c*e^9 - 240*(
sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*a^2*b*c^2*e^9 - 1280*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*c^(11/2)*d^6*e
^3 + 3840*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*b*c^(9/2)*d^5*e^4 - 3840*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8
*b^2*c^(7/2)*d^4*e^5 - 3840*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*a*c^(9/2)*d^4*e^5 + 1280*(sqrt(c)*x - sqrt(c
*x^2 + b*x + a))^8*b^3*c^(5/2)*d^3*e^6 + 7680*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*a*b*c^(7/2)*d^3*e^6 + 270*
(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*b^4*c^(3/2)*d^2*e^7 - 6000*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*a*b^2*c
^(5/2)*d^2*e^7 + 480*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*a^2*c^(7/2)*d^2*e^7 - 135*(sqrt(c)*x - sqrt(c*x^2 +
b*x + a))^8*b^5*sqrt(c)*d*e^8 + 1080*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*a*b^3*c^(3/2)*d*e^8 + 1680*(sqrt(c
)*x - sqrt(c*x^2 + b*x + a))^8*a^2*b*c^(5/2)*d*e^8 - 1280*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*a^3*c^(5/2)*e^
9 - 2560*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*c^6*d^7*e^2 + 3840*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*b*c^5*
d^6*e^3 + 3840*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*b^2*c^4*d^5*e^4 - 7680*(sqrt(c)*x - sqrt(c*x^2 + b*x + a)
)^7*a*c^5*d^5*e^4 - 8960*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*b^3*c^3*d^4*e^5 + 3840*(sqrt(c)*x - sqrt(c*x^2
+ b*x + a))^7*a*b*c^4*d^4*e^5 + 4780*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*b^4*c^2*d^3*e^6 + 7840*(sqrt(c)*x -
sqrt(c*x^2 + b*x + a))^7*a*b^2*c^3*d^3*e^6 + 7360*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*a^2*c^4*d^3*e^6 - 330
*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*b^5*c*d^2*e^7 - 8880*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*a*b^3*c^2*d^
2*e^7 - 9120*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*a^2*b*c^3*d^2*e^7 - 70*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^
7*b^6*d*e^8 + 420*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*a*b^4*c*d*e^8 + 11520*(sqrt(c)*x - sqrt(c*x^2 + b*x +
a))^7*a^2*b^2*c^2*d*e^8 - 4800*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*a^3*c^3*d*e^8 + 70*(sqrt(c)*x - sqrt(c*x^
2 + b*x + a))^7*a*b^5*e^9 - 560*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*a^2*b^3*c*e^9 - 2720*(sqrt(c)*x - sqrt(c
*x^2 + b*x + a))^7*a^3*b*c^2*e^9 - 2560*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*c^(13/2)*d^8*e + 1280*(sqrt(c)*x
- sqrt(c*x^2 + b*x + a))^6*b*c^(11/2)*d^7*e^2 + 6400*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*b^2*c^(9/2)*d^6*e^
3 - 7680*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*a*c^(11/2)*d^6*e^3 - 1280*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6
*b^3*c^(7/2)*d^5*e^4 - 3840*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*a*b*c^(9/2)*d^5*e^4 - 7420*(sqrt(c)*x - sqrt
(c*x^2 + b*x + a))^6*b^4*c^(5/2)*d^4*e^5 + 3040*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*a*b^2*c^(7/2)*d^4*e^5 +
16960*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*a^2*c^(9/2)*d^4*e^5 + 5330*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*b
^5*c^(3/2)*d^3*e^6 + 9840*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*a*b^3*c^(5/2)*d^3*e^6 - 8160*(sqrt(c)*x - sqrt
(c*x^2 + b*x + a))^6*a^2*b*c^(7/2)*d^3*e^6 - 490*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*b^6*sqrt(c)*d^2*e^7 - 1
2420*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*a*b^4*c^(3/2)*d^2*e^7 + 3840*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*
a^2*b^2*c^(5/2)*d^2*e^7 - 18240*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*a^3*c^(7/2)*d^2*e^7 + 490*(sqrt(c)*x - s
qrt(c*x^2 + b*x + a))^6*a*b^5*sqrt(c)*d*e^8 + 11440*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*a^2*b^3*c^(3/2)*d*e^
8 + 1440*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*a^3*b*c^(5/2)*d*e^8 - 5120*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^
6*a^3*b^2*c^(3/2)*e^9 - 1024*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*c^7*d^9 - 3072*(sqrt(c)*x - sqrt(c*x^2 + b*
x + a))^5*b*c^6*d^8*e + 7936*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*b^2*c^5*d^7*e^2 + 512*(sqrt(c)*x - sqrt(c*x
^2 + b*x + a))^5*a*c^6*d^7*e^2 + 1280*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*b^3*c^4*d^6*e^3 - 24832*(sqrt(c)*x
- sqrt(c*x^2 + b*x + a))^5*a*b*c^5*d^6*e^3 - 4280*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*b^4*c^3*d^5*e^4 + 188
80*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a*b^2*c^4*d^5*e^4 + 25216*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a^2*c
^5*d^5*e^4 - 1448*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*b^5*c^2*d^4*e^5 - 8640*(sqrt(c)*x - sqrt(c*x^2 + b*x +
a))^5*a*b^3*c^3*d^4*e^5 - 12160*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a^2*b*c^4*d^4*e^5 + 2626*(sqrt(c)*x - s
qrt(c*x^2 + b*x + a))^5*b^6*c*d^3*e^6 + 6260*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a*b^4*c^2*d^3*e^6 + 21760*(
sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a^2*b^2*c^3*d^3*e^6 - 29120*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a^3*c^4
*d^3*e^6 - 128*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*b^7*d^2*e^7 - 9026*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*
a*b^5*c*d^2*e^7 - 1520*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a^2*b^3*c^2*d^2*e^7 - 11040*(sqrt(c)*x - sqrt(c*x
^2 + b*x + a))^5*a^3*b*c^3*d^2*e^7 + 256*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a*b^6*d*e^8 + 8960*(sqrt(c)*x -
sqrt(c*x^2 + b*x + a))^5*a^2*b^4*c*d*e^8 + 7680*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a^4*c^3*d*e^8 - 128*(sq
rt(c)*x - sqrt(c*x^2 + b*x + a))^5*a^2*b^5*e^9 - 2560*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a^3*b^3*c*e^9 - 38
40*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a^4*b*c^2*e^9 - 2560*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*b*c^(13/2)
*d^9 + 1280*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*b^2*c^(11/2)*d^8*e + 2560*(sqrt(c)*x - sqrt(c*x^2 + b*x + a)
)^4*a*c^(13/2)*d^8*e + 6400*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*b^3*c^(9/2)*d^7*e^2 - 8960*(sqrt(c)*x - sqrt
(c*x^2 + b*x + a))^4*a*b*c^(11/2)*d^7*e^2 - 2400*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*b^4*c^(7/2)*d^6*e^3 - 1
9200*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a*b^2*c^(9/2)*d^6*e^3 + 7680*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*
a^2*c^(11/2)*d^6*e^3 - 2860*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*b^5*c^(5/2)*d^5*e^4 + 21600*(sqrt(c)*x - sqr
t(c*x^2 + b*x + a))^4*a*b^3*c^(7/2)*d^5*e^4 + 40000*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^2*b*c^(9/2)*d^5*e^
4 + 1390*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*b^6*c^(3/2)*d^4*e^5 - 9620*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^
4*a*b^4*c^(5/2)*d^4*e^5 - 37120*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^2*b^2*c^(7/2)*d^4*e^5 - 37440*(sqrt(c)
*x - sqrt(c*x^2 + b*x + a))^4*a^3*c^(9/2)*d^4*e^5 + 640*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*b^7*sqrt(c)*d^3*
e^6 - 1390*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a*b^5*c^(3/2)*d^3*e^6 + 41840*(sqrt(c)*x - sqrt(c*x^2 + b*x +
a))^4*a^2*b^3*c^(5/2)*d^3*e^6 + 2080*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^3*b*c^(7/2)*d^3*e^6 - 2560*(sqrt
(c)*x - sqrt(c*x^2 + b*x + a))^4*a*b^6*sqrt(c)*d^2*e^7 - 5120*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^2*b^4*c^
(3/2)*d^2*e^7 - 30720*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^3*b^2*c^(5/2)*d^2*e^7 + 15360*(sqrt(c)*x - sqrt(
c*x^2 + b*x + a))^4*a^4*c^(7/2)*d^2*e^7 + 3200*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^2*b^5*sqrt(c)*d*e^8 + 7
680*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^3*b^3*c^(3/2)*d*e^8 + 3840*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a
^4*b*c^(5/2)*d*e^8 - 1280*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^3*b^4*sqrt(c)*e^9 - 2560*(sqrt(c)*x - sqrt(c
*x^2 + b*x + a))^4*a^4*b^2*c^(3/2)*e^9 - 2560*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^5*c^(5/2)*e^9 - 2560*(sq
rt(c)*x - sqrt(c*x^2 + b*x + a))^3*b^2*c^6*d^9 + 3840*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b^3*c^5*d^8*e + 51
20*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*b*c^6*d^8*e + 1600*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b^4*c^4*d^
7*e^2 - 17920*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*b^2*c^5*d^7*e^2 - 2560*(sqrt(c)*x - sqrt(c*x^2 + b*x + a
))^3*a^2*c^6*d^7*e^2 - 2080*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b^5*c^3*d^6*e^3 + 24320*(sqrt(c)*x - sqrt(c*
x^2 + b*x + a))^3*a^2*b*c^5*d^6*e^3 - 540*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b^6*c^2*d^5*e^4 + 7520*(sqrt(c
)*x - sqrt(c*x^2 + b*x + a))^3*a*b^4*c^3*d^5*e^4 + 13120*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^2*b^2*c^4*d^5
*e^4 - 12800*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^3*c^5*d^5*e^4 + 930*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3
*b^7*c*d^4*e^5 - 4760*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*b^5*c^2*d^4*e^5 - 14240*(sqrt(c)*x - sqrt(c*x^2
+ b*x + a))^3*a^2*b^3*c^3*d^4*e^5 - 42880*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^3*b*c^4*d^4*e^5 + 70*(sqrt(c
)*x - sqrt(c*x^2 + b*x + a))^3*b^8*d^3*e^6 - 2280*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*b^6*c*d^3*e^6 + 2042
0*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^2*b^4*c^2*d^3*e^6 + 20320*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^3*
b^2*c^3*d^3*e^6 + 24640*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^4*c^4*d^3*e^6 - 210*(sqrt(c)*x - sqrt(c*x^2 +
b*x + a))^3*a*b^7*d^2*e^7 - 230*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^2*b^5*c*d^2*e^7 - 26480*(sqrt(c)*x - s
qrt(c*x^2 + b*x + a))^3*a^3*b^3*c^2*d^2*e^7 - 6240*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^4*b*c^3*d^2*e^7 + 2
10*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^2*b^6*d*e^8 + 3580*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^3*b^4*c*
d*e^8 + 13760*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^4*b^2*c^2*d*e^8 - 5440*(sqrt(c)*x - sqrt(c*x^2 + b*x + a
))^3*a^5*c^3*d*e^8 - 70*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^3*b^5*e^9 - 2000*(sqrt(c)*x - sqrt(c*x^2 + b*x
+ a))^3*a^4*b^3*c*e^9 - 2400*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^5*b*c^2*e^9 - 1280*(sqrt(c)*x - sqrt(c*x
^2 + b*x + a))^2*b^3*c^(11/2)*d^9 + 2400*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b^4*c^(9/2)*d^8*e + 3840*(sqrt(
c)*x - sqrt(c*x^2 + b*x + a))^2*a*b^2*c^(11/2)*d^8*e - 160*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b^5*c^(7/2)*d
^7*e^2 - 12800*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*b^3*c^(9/2)*d^7*e^2 - 3840*(sqrt(c)*x - sqrt(c*x^2 + b*
x + a))^2*a^2*b*c^(11/2)*d^7*e^2 - 860*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b^6*c^(5/2)*d^6*e^3 + 5200*(sqrt(
c)*x - sqrt(c*x^2 + b*x + a))^2*a*b^4*c^(7/2)*d^6*e^3 + 24000*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^2*b^2*c^
(9/2)*d^6*e^3 + 1280*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^3*c^(11/2)*d^6*e^3 + 230*(sqrt(c)*x - sqrt(c*x^2
+ b*x + a))^2*b^7*c^(3/2)*d^5*e^4 + 320*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*b^5*c^(5/2)*d^5*e^4 - 5920*(sq
rt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^2*b^3*c^(7/2)*d^5*e^4 - 23040*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^3*b
*c^(9/2)*d^5*e^4 + 210*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b^8*sqrt(c)*d^4*e^5 - 1720*(sqrt(c)*x - sqrt(c*x^
2 + b*x + a))^2*a*b^6*c^(3/2)*d^4*e^5 + 2380*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^2*b^4*c^(5/2)*d^4*e^5 - 1
0720*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^3*b^2*c^(7/2)*d^4*e^5 + 7360*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^
2*a^4*c^(9/2)*d^4*e^5 - 630*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*b^7*sqrt(c)*d^3*e^6 + 4430*(sqrt(c)*x - sq
rt(c*x^2 + b*x + a))^2*a^2*b^5*c^(3/2)*d^3*e^6 + 2480*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^3*b^3*c^(5/2)*d^
3*e^6 + 22240*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^4*b*c^(7/2)*d^3*e^6 + 630*(sqrt(c)*x - sqrt(c*x^2 + b*x
+ a))^2*a^2*b^6*sqrt(c)*d^2*e^7 - 7180*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^3*b^4*c^(3/2)*d^2*e^7 - 7360*(s
qrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^4*b^2*c^(5/2)*d^2*e^7 - 11200*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^5*
c^(7/2)*d^2*e^7 - 210*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^3*b^5*sqrt(c)*d*e^8 + 6800*(sqrt(c)*x - sqrt(c*x
^2 + b*x + a))^2*a^4*b^3*c^(3/2)*d*e^8 + 3040*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^5*b*c^(5/2)*d*e^8 - 2560
*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^5*b^2*c^(3/2)*e^9 - 320*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^4*c^5*d
^9 + 640*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^5*c^4*d^8*e + 1280*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b^3*c^
5*d^8*e - 120*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^6*c^3*d^7*e^2 - 4000*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a
*b^4*c^4*d^7*e^2 - 1920*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^2*b^2*c^5*d^7*e^2 - 200*(sqrt(c)*x - sqrt(c*x^2
+ b*x + a))*b^7*c^2*d^6*e^3 + 1920*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b^5*c^3*d^6*e^3 + 9600*(sqrt(c)*x - s
qrt(c*x^2 + b*x + a))*a^2*b^3*c^4*d^6*e^3 + 1280*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^3*b*c^5*d^6*e^3 + 120*(
sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^8*c*d^5*e^4 - 260*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b^6*c^2*d^5*e^4 -
3880*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^2*b^4*c^3*d^5*e^4 - 12800*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^3*
b^2*c^4*d^5*e^4 - 640*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^4*c^5*d^5*e^4 + 15*(sqrt(c)*x - sqrt(c*x^2 + b*x +
a))*b^9*d^4*e^5 - 450*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b^7*c*d^4*e^5 + 2080*(sqrt(c)*x - sqrt(c*x^2 + b*
x + a))*a^2*b^5*c^2*d^4*e^5 + 1120*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^3*b^3*c^3*d^4*e^5 + 8960*(sqrt(c)*x -
sqrt(c*x^2 + b*x + a))*a^4*b*c^4*d^4*e^5 - 60*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b^8*d^3*e^6 + 750*(sqrt(c
)*x - sqrt(c*x^2 + b*x + a))*a^2*b^6*c*d^3*e^6 - 2820*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^3*b^4*c^2*d^3*e^6
+ 5760*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^4*b^2*c^3*d^3*e^6 - 2880*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^5*
c^4*d^3*e^6 + 90*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^2*b^7*d^2*e^7 - 750*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))
*a^3*b^5*c*d^2*e^7 + 400*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^4*b^3*c^2*d^2*e^7 - 6880*(sqrt(c)*x - sqrt(c*x^
2 + b*x + a))*a^5*b*c^3*d^2*e^7 - 60*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^3*b^6*d*e^8 + 450*(sqrt(c)*x - sqrt
(c*x^2 + b*x + a))*a^4*b^4*c*d*e^8 + 1840*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^5*b^2*c^2*d*e^8 + 2080*(sqrt(c
)*x - sqrt(c*x^2 + b*x + a))*a^6*c^3*d*e^8 + 15*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^4*b^5*e^9 - 120*(sqrt(c)
*x - sqrt(c*x^2 + b*x + a))*a^5*b^3*c*e^9 - 1040*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^6*b*c^2*e^9 - 32*b^5*c^
(9/2)*d^9 + 64*b^6*c^(7/2)*d^8*e + 160*a*b^4*c^(9/2)*d^8*e - 12*b^7*c^(5/2)*d^7*e^2 - 464*a*b^5*c^(7/2)*d^7*e^
2 - 320*a^2*b^3*c^(9/2)*d^7*e^2 - 20*b^8*c^(3/2)*d^6*e^3 + 204*a*b^6*c^(5/2)*d^6*e^3 + 1360*a^2*b^4*c^(7/2)*d^
6*e^3 + 320*a^3*b^2*c^(9/2)*d^6*e^3 + 15*b^9*sqrt(c)*d^5*e^4 - 30*a*b^7*c^(3/2)*d^5*e^4 - 532*a^2*b^5*c^(5/2)*
d^5*e^4 - 2240*a^3*b^3*c^(7/2)*d^5*e^4 - 320*a^4*b*c^(9/2)*d^5*e^4 - 60*a*b^8*sqrt(c)*d^4*e^5 + 330*a^2*b^6*c^
(3/2)*d^4*e^5 + 260*a^3*b^4*c^(5/2)*d^4*e^5 + 2560*a^4*b^2*c^(7/2)*d^4*e^5 + 64*a^5*c^(9/2)*d^4*e^5 + 90*a^2*b
^7*sqrt(c)*d^3*e^6 - 610*a^3*b^5*c^(3/2)*d^3*e^6 + 560*a^4*b^3*c^(5/2)*d^3*e^6 - 1568*a^5*b*c^(7/2)*d^3*e^6 -
60*a^3*b^6*sqrt(c)*d^2*e^7 + 450*a^4*b^4*c^(3/2)*d^2*e^7 - 976*a^5*b^2*c^(5/2)*d^2*e^7 + 288*a^6*c^(7/2)*d^2*e
^7 + 15*a^4*b^5*sqrt(c)*d*e^8 - 120*a^5*b^3*c^(3/2)*d*e^8 + 752*a^6*b*c^(5/2)*d*e^8 - 256*a^7*c^(5/2)*e^9)/((c
^3*d^6*e^4 - 3*b*c^2*d^5*e^5 + 3*b^2*c*d^4*e^6 + 3*a*c^2*d^4*e^6 - b^3*d^3*e^7 - 6*a*b*c*d^3*e^7 + 3*a*b^2*d^2
*e^8 + 3*a^2*c*d^2*e^8 - 3*a^2*b*d*e^9 + a^3*e^10)*((sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*e + 2*(sqrt(c)*x - s
qrt(c*x^2 + b*x + a))*sqrt(c)*d + b*d - a*e)^5)
Mupad [F(-1)]
Timed out. \[
\int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^6} \, dx=\int \frac {{\left (c\,x^2+b\,x+a\right )}^{3/2}}{{\left (d+e\,x\right )}^6} \,d x
\]
[In]
int((a + b*x + c*x^2)^(3/2)/(d + e*x)^6,x)
[Out]
int((a + b*x + c*x^2)^(3/2)/(d + e*x)^6, x)