3.2319 \(\int \frac{x^{9/2}}{\left (a+b x+c x^2\right )^3} \, dx\)

Optimal. Leaf size=389 \[ \frac{3 \left (-\frac{44 a^2 b c^2-11 a b^3 c+b^5}{\sqrt{b^2-4 a c}}+28 a^2 c^2-9 a b^2 c+b^4\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{4 \sqrt{2} c^{5/2} \left (b^2-4 a c\right )^2 \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{3 \left (\frac{44 a^2 b c^2-11 a b^3 c+b^5}{\sqrt{b^2-4 a c}}+28 a^2 c^2-9 a b^2 c+b^4\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{4 \sqrt{2} c^{5/2} \left (b^2-4 a c\right )^2 \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{3 b \sqrt{x} \left (b^2-8 a c\right )}{4 c^2 \left (b^2-4 a c\right )^2}+\frac{x^{7/2} (2 a+b x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac{x^{3/2} \left (b x \left (b^2-16 a c\right )+a \left (b^2-28 a c\right )\right )}{4 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )} \]

[Out]

(-3*b*(b^2 - 8*a*c)*Sqrt[x])/(4*c^2*(b^2 - 4*a*c)^2) + (x^(7/2)*(2*a + b*x))/(2*
(b^2 - 4*a*c)*(a + b*x + c*x^2)^2) + (x^(3/2)*(a*(b^2 - 28*a*c) + b*(b^2 - 16*a*
c)*x))/(4*c*(b^2 - 4*a*c)^2*(a + b*x + c*x^2)) + (3*(b^4 - 9*a*b^2*c + 28*a^2*c^
2 - (b^5 - 11*a*b^3*c + 44*a^2*b*c^2)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]
*Sqrt[x])/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(4*Sqrt[2]*c^(5/2)*(b^2 - 4*a*c)^2*Sqrt[
b - Sqrt[b^2 - 4*a*c]]) + (3*(b^4 - 9*a*b^2*c + 28*a^2*c^2 + (b^5 - 11*a*b^3*c +
44*a^2*b*c^2)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b + Sqrt
[b^2 - 4*a*c]]])/(4*Sqrt[2]*c^(5/2)*(b^2 - 4*a*c)^2*Sqrt[b + Sqrt[b^2 - 4*a*c]])

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Rubi [A]  time = 3.40763, antiderivative size = 389, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{3 \left (-\frac{44 a^2 b c^2-11 a b^3 c+b^5}{\sqrt{b^2-4 a c}}+28 a^2 c^2-9 a b^2 c+b^4\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{4 \sqrt{2} c^{5/2} \left (b^2-4 a c\right )^2 \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{3 \left (\frac{44 a^2 b c^2-11 a b^3 c+b^5}{\sqrt{b^2-4 a c}}+28 a^2 c^2-9 a b^2 c+b^4\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{4 \sqrt{2} c^{5/2} \left (b^2-4 a c\right )^2 \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{3 b \sqrt{x} \left (b^2-8 a c\right )}{4 c^2 \left (b^2-4 a c\right )^2}+\frac{x^{7/2} (2 a+b x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac{x^{3/2} \left (b x \left (b^2-16 a c\right )+a \left (b^2-28 a c\right )\right )}{4 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[x^(9/2)/(a + b*x + c*x^2)^3,x]

[Out]

(-3*b*(b^2 - 8*a*c)*Sqrt[x])/(4*c^2*(b^2 - 4*a*c)^2) + (x^(7/2)*(2*a + b*x))/(2*
(b^2 - 4*a*c)*(a + b*x + c*x^2)^2) + (x^(3/2)*(a*(b^2 - 28*a*c) + b*(b^2 - 16*a*
c)*x))/(4*c*(b^2 - 4*a*c)^2*(a + b*x + c*x^2)) + (3*(b^4 - 9*a*b^2*c + 28*a^2*c^
2 - (b^5 - 11*a*b^3*c + 44*a^2*b*c^2)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]
*Sqrt[x])/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(4*Sqrt[2]*c^(5/2)*(b^2 - 4*a*c)^2*Sqrt[
b - Sqrt[b^2 - 4*a*c]]) + (3*(b^4 - 9*a*b^2*c + 28*a^2*c^2 + (b^5 - 11*a*b^3*c +
44*a^2*b*c^2)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b + Sqrt
[b^2 - 4*a*c]]])/(4*Sqrt[2]*c^(5/2)*(b^2 - 4*a*c)^2*Sqrt[b + Sqrt[b^2 - 4*a*c]])

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(9/2)/(c*x**2+b*x+a)**3,x)

[Out]

Timed out

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Mathematica [A]  time = 2.27536, size = 452, normalized size = 1.16 \[ \frac{-\frac{4 \sqrt{x} \left (a^2 c (2 c x-3 b)+a b^2 (b-4 c x)+b^4 x\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))^2}+\frac{3 \sqrt{2} \sqrt{c} \left (28 a^2 c^2 \sqrt{b^2-4 a c}-44 a^2 b c^2+11 a b^3 c-9 a b^2 c \sqrt{b^2-4 a c}+b^4 \sqrt{b^2-4 a c}-b^5\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\left (b^2-4 a c\right )^{5/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{3 \sqrt{2} \sqrt{c} \left (28 a^2 c^2 \sqrt{b^2-4 a c}+44 a^2 b c^2-11 a b^3 c-9 a b^2 c \sqrt{b^2-4 a c}+b^4 \sqrt{b^2-4 a c}+b^5\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\left (b^2-4 a c\right )^{5/2} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{2 \sqrt{x} \left (48 a^2 b c^2-44 a^2 c^3 x-17 a b^3 c+37 a b^2 c^2 x+2 b^5-5 b^4 c x\right )}{\left (b^2-4 a c\right )^2 (a+x (b+c x))}}{8 c^3} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(9/2)/(a + b*x + c*x^2)^3,x]

[Out]

((2*Sqrt[x]*(2*b^5 - 17*a*b^3*c + 48*a^2*b*c^2 - 5*b^4*c*x + 37*a*b^2*c^2*x - 44
*a^2*c^3*x))/((b^2 - 4*a*c)^2*(a + x*(b + c*x))) - (4*Sqrt[x]*(b^4*x + a*b^2*(b
- 4*c*x) + a^2*c*(-3*b + 2*c*x)))/((b^2 - 4*a*c)*(a + x*(b + c*x))^2) + (3*Sqrt[
2]*Sqrt[c]*(-b^5 + 11*a*b^3*c - 44*a^2*b*c^2 + b^4*Sqrt[b^2 - 4*a*c] - 9*a*b^2*c
*Sqrt[b^2 - 4*a*c] + 28*a^2*c^2*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[
x])/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(5/2)*Sqrt[b - Sqrt[b^2 - 4*a*c
]]) + (3*Sqrt[2]*Sqrt[c]*(b^5 - 11*a*b^3*c + 44*a^2*b*c^2 + b^4*Sqrt[b^2 - 4*a*c
] - 9*a*b^2*c*Sqrt[b^2 - 4*a*c] + 28*a^2*c^2*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*
Sqrt[c]*Sqrt[x])/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(5/2)*Sqrt[b + Sqr
t[b^2 - 4*a*c]]))/(8*c^3)

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Maple [B]  time = 0.165, size = 5474, normalized size = 14.1 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(9/2)/(c*x^2+b*x+a)^3,x)

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \frac{3 \,{\left (b^{3} c - 8 \, a b c^{2}\right )} x^{\frac{9}{2}} +{\left (b^{4} - 11 \, a b^{2} c - 44 \, a^{2} c^{2}\right )} x^{\frac{7}{2}} + 2 \,{\left (a b^{3} - 22 \, a^{2} b c\right )} x^{\frac{5}{2}} +{\left (a^{2} b^{2} - 28 \, a^{3} c\right )} x^{\frac{3}{2}}}{4 \,{\left (a^{2} b^{4} c - 8 \, a^{3} b^{2} c^{2} + 16 \, a^{4} c^{3} +{\left (b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}\right )} x^{4} + 2 \,{\left (b^{5} c^{2} - 8 \, a b^{3} c^{3} + 16 \, a^{2} b c^{4}\right )} x^{3} +{\left (b^{6} c - 6 \, a b^{4} c^{2} + 32 \, a^{3} c^{4}\right )} x^{2} + 2 \,{\left (a b^{5} c - 8 \, a^{2} b^{3} c^{2} + 16 \, a^{3} b c^{3}\right )} x\right )}} + \int -\frac{3 \,{\left ({\left (b^{3} - 8 \, a b c\right )} x^{\frac{3}{2}} +{\left (a b^{2} - 28 \, a^{2} c\right )} \sqrt{x}\right )}}{8 \,{\left (a b^{4} c - 8 \, a^{2} b^{2} c^{2} + 16 \, a^{3} c^{3} +{\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{2} +{\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(9/2)/(c*x^2 + b*x + a)^3,x, algorithm="maxima")

[Out]

1/4*(3*(b^3*c - 8*a*b*c^2)*x^(9/2) + (b^4 - 11*a*b^2*c - 44*a^2*c^2)*x^(7/2) + 2
*(a*b^3 - 22*a^2*b*c)*x^(5/2) + (a^2*b^2 - 28*a^3*c)*x^(3/2))/(a^2*b^4*c - 8*a^3
*b^2*c^2 + 16*a^4*c^3 + (b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*x^4 + 2*(b^5*c^2 -
8*a*b^3*c^3 + 16*a^2*b*c^4)*x^3 + (b^6*c - 6*a*b^4*c^2 + 32*a^3*c^4)*x^2 + 2*(a*
b^5*c - 8*a^2*b^3*c^2 + 16*a^3*b*c^3)*x) + integrate(-3/8*((b^3 - 8*a*b*c)*x^(3/
2) + (a*b^2 - 28*a^2*c)*sqrt(x))/(a*b^4*c - 8*a^2*b^2*c^2 + 16*a^3*c^3 + (b^4*c^
2 - 8*a*b^2*c^3 + 16*a^2*c^4)*x^2 + (b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*x), x)

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Fricas [A]  time = 0.673859, size = 5771, normalized size = 14.84 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(9/2)/(c*x^2 + b*x + a)^3,x, algorithm="fricas")

[Out]

-1/8*(3*sqrt(1/2)*(a^2*b^4*c^2 - 8*a^3*b^2*c^3 + 16*a^4*c^4 + (b^4*c^4 - 8*a*b^2
*c^5 + 16*a^2*c^6)*x^4 + 2*(b^5*c^3 - 8*a*b^3*c^4 + 16*a^2*b*c^5)*x^3 + (b^6*c^2
- 6*a*b^4*c^3 + 32*a^3*c^5)*x^2 + 2*(a*b^5*c^2 - 8*a^2*b^3*c^3 + 16*a^3*b*c^4)*
x)*sqrt(-(b^9 - 21*a*b^7*c + 189*a^2*b^5*c^2 - 840*a^3*b^3*c^3 + 1680*a^4*b*c^4
+ (b^10*c^5 - 20*a*b^8*c^6 + 160*a^2*b^6*c^7 - 640*a^3*b^4*c^8 + 1280*a^4*b^2*c^
9 - 1024*a^5*c^10)*sqrt((b^8 - 22*a*b^6*c + 219*a^2*b^4*c^2 - 1078*a^3*b^2*c^3 +
2401*a^4*c^4)/(b^10*c^10 - 20*a*b^8*c^11 + 160*a^2*b^6*c^12 - 640*a^3*b^4*c^13
+ 1280*a^4*b^2*c^14 - 1024*a^5*c^15)))/(b^10*c^5 - 20*a*b^8*c^6 + 160*a^2*b^6*c^
7 - 640*a^3*b^4*c^8 + 1280*a^4*b^2*c^9 - 1024*a^5*c^10))*log(27/2*sqrt(1/2)*(b^1
3 - 31*a*b^11*c + 413*a^2*b^9*c^2 - 3012*a^3*b^7*c^3 + 12496*a^4*b^5*c^4 - 27584
*a^5*b^3*c^5 + 25088*a^6*b*c^6 - (b^14*c^5 - 30*a*b^12*c^6 + 416*a^2*b^10*c^7 -
3360*a^3*b^8*c^8 + 16640*a^4*b^6*c^9 - 49664*a^5*b^4*c^10 + 81920*a^6*b^2*c^11 -
57344*a^7*c^12)*sqrt((b^8 - 22*a*b^6*c + 219*a^2*b^4*c^2 - 1078*a^3*b^2*c^3 + 2
401*a^4*c^4)/(b^10*c^10 - 20*a*b^8*c^11 + 160*a^2*b^6*c^12 - 640*a^3*b^4*c^13 +
1280*a^4*b^2*c^14 - 1024*a^5*c^15)))*sqrt(-(b^9 - 21*a*b^7*c + 189*a^2*b^5*c^2 -
840*a^3*b^3*c^3 + 1680*a^4*b*c^4 + (b^10*c^5 - 20*a*b^8*c^6 + 160*a^2*b^6*c^7 -
640*a^3*b^4*c^8 + 1280*a^4*b^2*c^9 - 1024*a^5*c^10)*sqrt((b^8 - 22*a*b^6*c + 21
9*a^2*b^4*c^2 - 1078*a^3*b^2*c^3 + 2401*a^4*c^4)/(b^10*c^10 - 20*a*b^8*c^11 + 16
0*a^2*b^6*c^12 - 640*a^3*b^4*c^13 + 1280*a^4*b^2*c^14 - 1024*a^5*c^15)))/(b^10*c
^5 - 20*a*b^8*c^6 + 160*a^2*b^6*c^7 - 640*a^3*b^4*c^8 + 1280*a^4*b^2*c^9 - 1024*
a^5*c^10)) + 27*(21*a^2*b^8 - 447*a^3*b^6*c + 4189*a^4*b^4*c^2 - 19208*a^5*b^2*c
^3 + 38416*a^6*c^4)*sqrt(x)) - 3*sqrt(1/2)*(a^2*b^4*c^2 - 8*a^3*b^2*c^3 + 16*a^4
*c^4 + (b^4*c^4 - 8*a*b^2*c^5 + 16*a^2*c^6)*x^4 + 2*(b^5*c^3 - 8*a*b^3*c^4 + 16*
a^2*b*c^5)*x^3 + (b^6*c^2 - 6*a*b^4*c^3 + 32*a^3*c^5)*x^2 + 2*(a*b^5*c^2 - 8*a^2
*b^3*c^3 + 16*a^3*b*c^4)*x)*sqrt(-(b^9 - 21*a*b^7*c + 189*a^2*b^5*c^2 - 840*a^3*
b^3*c^3 + 1680*a^4*b*c^4 + (b^10*c^5 - 20*a*b^8*c^6 + 160*a^2*b^6*c^7 - 640*a^3*
b^4*c^8 + 1280*a^4*b^2*c^9 - 1024*a^5*c^10)*sqrt((b^8 - 22*a*b^6*c + 219*a^2*b^4
*c^2 - 1078*a^3*b^2*c^3 + 2401*a^4*c^4)/(b^10*c^10 - 20*a*b^8*c^11 + 160*a^2*b^6
*c^12 - 640*a^3*b^4*c^13 + 1280*a^4*b^2*c^14 - 1024*a^5*c^15)))/(b^10*c^5 - 20*a
*b^8*c^6 + 160*a^2*b^6*c^7 - 640*a^3*b^4*c^8 + 1280*a^4*b^2*c^9 - 1024*a^5*c^10)
)*log(-27/2*sqrt(1/2)*(b^13 - 31*a*b^11*c + 413*a^2*b^9*c^2 - 3012*a^3*b^7*c^3 +
12496*a^4*b^5*c^4 - 27584*a^5*b^3*c^5 + 25088*a^6*b*c^6 - (b^14*c^5 - 30*a*b^12
*c^6 + 416*a^2*b^10*c^7 - 3360*a^3*b^8*c^8 + 16640*a^4*b^6*c^9 - 49664*a^5*b^4*c
^10 + 81920*a^6*b^2*c^11 - 57344*a^7*c^12)*sqrt((b^8 - 22*a*b^6*c + 219*a^2*b^4*
c^2 - 1078*a^3*b^2*c^3 + 2401*a^4*c^4)/(b^10*c^10 - 20*a*b^8*c^11 + 160*a^2*b^6*
c^12 - 640*a^3*b^4*c^13 + 1280*a^4*b^2*c^14 - 1024*a^5*c^15)))*sqrt(-(b^9 - 21*a
*b^7*c + 189*a^2*b^5*c^2 - 840*a^3*b^3*c^3 + 1680*a^4*b*c^4 + (b^10*c^5 - 20*a*b
^8*c^6 + 160*a^2*b^6*c^7 - 640*a^3*b^4*c^8 + 1280*a^4*b^2*c^9 - 1024*a^5*c^10)*s
qrt((b^8 - 22*a*b^6*c + 219*a^2*b^4*c^2 - 1078*a^3*b^2*c^3 + 2401*a^4*c^4)/(b^10
*c^10 - 20*a*b^8*c^11 + 160*a^2*b^6*c^12 - 640*a^3*b^4*c^13 + 1280*a^4*b^2*c^14
- 1024*a^5*c^15)))/(b^10*c^5 - 20*a*b^8*c^6 + 160*a^2*b^6*c^7 - 640*a^3*b^4*c^8
+ 1280*a^4*b^2*c^9 - 1024*a^5*c^10)) + 27*(21*a^2*b^8 - 447*a^3*b^6*c + 4189*a^4
*b^4*c^2 - 19208*a^5*b^2*c^3 + 38416*a^6*c^4)*sqrt(x)) + 3*sqrt(1/2)*(a^2*b^4*c^
2 - 8*a^3*b^2*c^3 + 16*a^4*c^4 + (b^4*c^4 - 8*a*b^2*c^5 + 16*a^2*c^6)*x^4 + 2*(b
^5*c^3 - 8*a*b^3*c^4 + 16*a^2*b*c^5)*x^3 + (b^6*c^2 - 6*a*b^4*c^3 + 32*a^3*c^5)*
x^2 + 2*(a*b^5*c^2 - 8*a^2*b^3*c^3 + 16*a^3*b*c^4)*x)*sqrt(-(b^9 - 21*a*b^7*c +
189*a^2*b^5*c^2 - 840*a^3*b^3*c^3 + 1680*a^4*b*c^4 - (b^10*c^5 - 20*a*b^8*c^6 +
160*a^2*b^6*c^7 - 640*a^3*b^4*c^8 + 1280*a^4*b^2*c^9 - 1024*a^5*c^10)*sqrt((b^8
- 22*a*b^6*c + 219*a^2*b^4*c^2 - 1078*a^3*b^2*c^3 + 2401*a^4*c^4)/(b^10*c^10 - 2
0*a*b^8*c^11 + 160*a^2*b^6*c^12 - 640*a^3*b^4*c^13 + 1280*a^4*b^2*c^14 - 1024*a^
5*c^15)))/(b^10*c^5 - 20*a*b^8*c^6 + 160*a^2*b^6*c^7 - 640*a^3*b^4*c^8 + 1280*a^
4*b^2*c^9 - 1024*a^5*c^10))*log(27/2*sqrt(1/2)*(b^13 - 31*a*b^11*c + 413*a^2*b^9
*c^2 - 3012*a^3*b^7*c^3 + 12496*a^4*b^5*c^4 - 27584*a^5*b^3*c^5 + 25088*a^6*b*c^
6 + (b^14*c^5 - 30*a*b^12*c^6 + 416*a^2*b^10*c^7 - 3360*a^3*b^8*c^8 + 16640*a^4*
b^6*c^9 - 49664*a^5*b^4*c^10 + 81920*a^6*b^2*c^11 - 57344*a^7*c^12)*sqrt((b^8 -
22*a*b^6*c + 219*a^2*b^4*c^2 - 1078*a^3*b^2*c^3 + 2401*a^4*c^4)/(b^10*c^10 - 20*
a*b^8*c^11 + 160*a^2*b^6*c^12 - 640*a^3*b^4*c^13 + 1280*a^4*b^2*c^14 - 1024*a^5*
c^15)))*sqrt(-(b^9 - 21*a*b^7*c + 189*a^2*b^5*c^2 - 840*a^3*b^3*c^3 + 1680*a^4*b
*c^4 - (b^10*c^5 - 20*a*b^8*c^6 + 160*a^2*b^6*c^7 - 640*a^3*b^4*c^8 + 1280*a^4*b
^2*c^9 - 1024*a^5*c^10)*sqrt((b^8 - 22*a*b^6*c + 219*a^2*b^4*c^2 - 1078*a^3*b^2*
c^3 + 2401*a^4*c^4)/(b^10*c^10 - 20*a*b^8*c^11 + 160*a^2*b^6*c^12 - 640*a^3*b^4*
c^13 + 1280*a^4*b^2*c^14 - 1024*a^5*c^15)))/(b^10*c^5 - 20*a*b^8*c^6 + 160*a^2*b
^6*c^7 - 640*a^3*b^4*c^8 + 1280*a^4*b^2*c^9 - 1024*a^5*c^10)) + 27*(21*a^2*b^8 -
447*a^3*b^6*c + 4189*a^4*b^4*c^2 - 19208*a^5*b^2*c^3 + 38416*a^6*c^4)*sqrt(x))
- 3*sqrt(1/2)*(a^2*b^4*c^2 - 8*a^3*b^2*c^3 + 16*a^4*c^4 + (b^4*c^4 - 8*a*b^2*c^5
+ 16*a^2*c^6)*x^4 + 2*(b^5*c^3 - 8*a*b^3*c^4 + 16*a^2*b*c^5)*x^3 + (b^6*c^2 - 6
*a*b^4*c^3 + 32*a^3*c^5)*x^2 + 2*(a*b^5*c^2 - 8*a^2*b^3*c^3 + 16*a^3*b*c^4)*x)*s
qrt(-(b^9 - 21*a*b^7*c + 189*a^2*b^5*c^2 - 840*a^3*b^3*c^3 + 1680*a^4*b*c^4 - (b
^10*c^5 - 20*a*b^8*c^6 + 160*a^2*b^6*c^7 - 640*a^3*b^4*c^8 + 1280*a^4*b^2*c^9 -
1024*a^5*c^10)*sqrt((b^8 - 22*a*b^6*c + 219*a^2*b^4*c^2 - 1078*a^3*b^2*c^3 + 240
1*a^4*c^4)/(b^10*c^10 - 20*a*b^8*c^11 + 160*a^2*b^6*c^12 - 640*a^3*b^4*c^13 + 12
80*a^4*b^2*c^14 - 1024*a^5*c^15)))/(b^10*c^5 - 20*a*b^8*c^6 + 160*a^2*b^6*c^7 -
640*a^3*b^4*c^8 + 1280*a^4*b^2*c^9 - 1024*a^5*c^10))*log(-27/2*sqrt(1/2)*(b^13 -
31*a*b^11*c + 413*a^2*b^9*c^2 - 3012*a^3*b^7*c^3 + 12496*a^4*b^5*c^4 - 27584*a^
5*b^3*c^5 + 25088*a^6*b*c^6 + (b^14*c^5 - 30*a*b^12*c^6 + 416*a^2*b^10*c^7 - 336
0*a^3*b^8*c^8 + 16640*a^4*b^6*c^9 - 49664*a^5*b^4*c^10 + 81920*a^6*b^2*c^11 - 57
344*a^7*c^12)*sqrt((b^8 - 22*a*b^6*c + 219*a^2*b^4*c^2 - 1078*a^3*b^2*c^3 + 2401
*a^4*c^4)/(b^10*c^10 - 20*a*b^8*c^11 + 160*a^2*b^6*c^12 - 640*a^3*b^4*c^13 + 128
0*a^4*b^2*c^14 - 1024*a^5*c^15)))*sqrt(-(b^9 - 21*a*b^7*c + 189*a^2*b^5*c^2 - 84
0*a^3*b^3*c^3 + 1680*a^4*b*c^4 - (b^10*c^5 - 20*a*b^8*c^6 + 160*a^2*b^6*c^7 - 64
0*a^3*b^4*c^8 + 1280*a^4*b^2*c^9 - 1024*a^5*c^10)*sqrt((b^8 - 22*a*b^6*c + 219*a
^2*b^4*c^2 - 1078*a^3*b^2*c^3 + 2401*a^4*c^4)/(b^10*c^10 - 20*a*b^8*c^11 + 160*a
^2*b^6*c^12 - 640*a^3*b^4*c^13 + 1280*a^4*b^2*c^14 - 1024*a^5*c^15)))/(b^10*c^5
- 20*a*b^8*c^6 + 160*a^2*b^6*c^7 - 640*a^3*b^4*c^8 + 1280*a^4*b^2*c^9 - 1024*a^5
*c^10)) + 27*(21*a^2*b^8 - 447*a^3*b^6*c + 4189*a^4*b^4*c^2 - 19208*a^5*b^2*c^3
+ 38416*a^6*c^4)*sqrt(x)) + 2*(3*a^2*b^3 - 24*a^3*b*c + (5*b^4*c - 37*a*b^2*c^2
+ 44*a^2*c^3)*x^3 + (3*b^5 - 20*a*b^3*c - 4*a^2*b*c^2)*x^2 + (6*a*b^4 - 49*a^2*b
^2*c + 28*a^3*c^2)*x)*sqrt(x))/(a^2*b^4*c^2 - 8*a^3*b^2*c^3 + 16*a^4*c^4 + (b^4*
c^4 - 8*a*b^2*c^5 + 16*a^2*c^6)*x^4 + 2*(b^5*c^3 - 8*a*b^3*c^4 + 16*a^2*b*c^5)*x
^3 + (b^6*c^2 - 6*a*b^4*c^3 + 32*a^3*c^5)*x^2 + 2*(a*b^5*c^2 - 8*a^2*b^3*c^3 + 1
6*a^3*b*c^4)*x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{\frac{9}{2}}}{\left (a + b x + c x^{2}\right )^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(9/2)/(c*x**2+b*x+a)**3,x)

[Out]

Integral(x**(9/2)/(a + b*x + c*x**2)**3, x)

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GIAC/XCAS [A]  time = 159.18, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(9/2)/(c*x^2 + b*x + a)^3,x, algorithm="giac")

[Out]

Done