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

Optimal. Leaf size=536 \[ \frac{a^{7/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} b^{3/4} (b c-a d)^2}-\frac{a^{7/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} b^{3/4} (b c-a d)^2}-\frac{a^{7/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} b^{3/4} (b c-a d)^2}+\frac{a^{7/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} b^{3/4} (b c-a d)^2}+\frac{c^{3/4} (3 b c-7 a d) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} d^{7/4} (b c-a d)^2}-\frac{c^{3/4} (3 b c-7 a d) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} d^{7/4} (b c-a d)^2}-\frac{c^{3/4} (3 b c-7 a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{4 \sqrt{2} d^{7/4} (b c-a d)^2}+\frac{c^{3/4} (3 b c-7 a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{4 \sqrt{2} d^{7/4} (b c-a d)^2}-\frac{c x^{3/2}}{2 d \left (c+d x^2\right ) (b c-a d)} \]

[Out]

-(c*x^(3/2))/(2*d*(b*c - a*d)*(c + d*x^2)) - (a^(7/4)*ArcTan[1 - (Sqrt[2]*b^(1/4
)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*b^(3/4)*(b*c - a*d)^2) + (a^(7/4)*ArcTan[1 + (Sqrt
[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*b^(3/4)*(b*c - a*d)^2) - (c^(3/4)*(3*b*c
- 7*a*d)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(4*Sqrt[2]*d^(7/4)*(b*c
- a*d)^2) + (c^(3/4)*(3*b*c - 7*a*d)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/
4)])/(4*Sqrt[2]*d^(7/4)*(b*c - a*d)^2) + (a^(7/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*
b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*b^(3/4)*(b*c - a*d)^2) - (a^(7/4)*Log[S
qrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*b^(3/4)*(b*c -
a*d)^2) + (c^(3/4)*(3*b*c - 7*a*d)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x
] + Sqrt[d]*x])/(8*Sqrt[2]*d^(7/4)*(b*c - a*d)^2) - (c^(3/4)*(3*b*c - 7*a*d)*Log
[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(8*Sqrt[2]*d^(7/4)*(b*c
- a*d)^2)

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Rubi [A]  time = 1.2961, antiderivative size = 536, normalized size of antiderivative = 1., number of steps used = 22, number of rules used = 9, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375 \[ \frac{a^{7/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} b^{3/4} (b c-a d)^2}-\frac{a^{7/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} b^{3/4} (b c-a d)^2}-\frac{a^{7/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} b^{3/4} (b c-a d)^2}+\frac{a^{7/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} b^{3/4} (b c-a d)^2}+\frac{c^{3/4} (3 b c-7 a d) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} d^{7/4} (b c-a d)^2}-\frac{c^{3/4} (3 b c-7 a d) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} d^{7/4} (b c-a d)^2}-\frac{c^{3/4} (3 b c-7 a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{4 \sqrt{2} d^{7/4} (b c-a d)^2}+\frac{c^{3/4} (3 b c-7 a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{4 \sqrt{2} d^{7/4} (b c-a d)^2}-\frac{c x^{3/2}}{2 d \left (c+d x^2\right ) (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

-(c*x^(3/2))/(2*d*(b*c - a*d)*(c + d*x^2)) - (a^(7/4)*ArcTan[1 - (Sqrt[2]*b^(1/4
)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*b^(3/4)*(b*c - a*d)^2) + (a^(7/4)*ArcTan[1 + (Sqrt
[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*b^(3/4)*(b*c - a*d)^2) - (c^(3/4)*(3*b*c
- 7*a*d)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(4*Sqrt[2]*d^(7/4)*(b*c
- a*d)^2) + (c^(3/4)*(3*b*c - 7*a*d)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/
4)])/(4*Sqrt[2]*d^(7/4)*(b*c - a*d)^2) + (a^(7/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*
b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*b^(3/4)*(b*c - a*d)^2) - (a^(7/4)*Log[S
qrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*b^(3/4)*(b*c -
a*d)^2) + (c^(3/4)*(3*b*c - 7*a*d)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x
] + Sqrt[d]*x])/(8*Sqrt[2]*d^(7/4)*(b*c - a*d)^2) - (c^(3/4)*(3*b*c - 7*a*d)*Log
[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(8*Sqrt[2]*d^(7/4)*(b*c
- a*d)^2)

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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)/(b*x**2+a)/(d*x**2+c)**2,x)

[Out]

Timed out

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Mathematica [A]  time = 0.588122, size = 527, normalized size = 0.98 \[ \frac{4 \sqrt{2} a^{7/4} d^{7/4} \left (c+d x^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )-4 \sqrt{2} a^{7/4} d^{7/4} \left (c+d x^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )-8 \sqrt{2} a^{7/4} d^{7/4} \left (c+d x^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )+8 \sqrt{2} a^{7/4} d^{7/4} \left (c+d x^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )+\sqrt{2} b^{3/4} c^{3/4} \left (c+d x^2\right ) (3 b c-7 a d) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )-\sqrt{2} b^{3/4} c^{3/4} \left (c+d x^2\right ) (3 b c-7 a d) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )-2 \sqrt{2} b^{3/4} c^{3/4} \left (c+d x^2\right ) (3 b c-7 a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )+2 \sqrt{2} b^{3/4} c^{3/4} \left (c+d x^2\right ) (3 b c-7 a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )-8 b^{3/4} c d^{3/4} x^{3/2} (b c-a d)}{16 b^{3/4} d^{7/4} \left (c+d x^2\right ) (b c-a d)^2} \]

Antiderivative was successfully verified.

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

[Out]

(-8*b^(3/4)*c*d^(3/4)*(b*c - a*d)*x^(3/2) - 8*Sqrt[2]*a^(7/4)*d^(7/4)*(c + d*x^2
)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] + 8*Sqrt[2]*a^(7/4)*d^(7/4)*(c +
d*x^2)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] - 2*Sqrt[2]*b^(3/4)*c^(3/4
)*(3*b*c - 7*a*d)*(c + d*x^2)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)] + 2*
Sqrt[2]*b^(3/4)*c^(3/4)*(3*b*c - 7*a*d)*(c + d*x^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*
Sqrt[x])/c^(1/4)] + 4*Sqrt[2]*a^(7/4)*d^(7/4)*(c + d*x^2)*Log[Sqrt[a] - Sqrt[2]*
a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] - 4*Sqrt[2]*a^(7/4)*d^(7/4)*(c + d*x^2)*Log
[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] + Sqrt[2]*b^(3/4)*c^(3/4
)*(3*b*c - 7*a*d)*(c + d*x^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sq
rt[d]*x] - Sqrt[2]*b^(3/4)*c^(3/4)*(3*b*c - 7*a*d)*(c + d*x^2)*Log[Sqrt[c] + Sqr
t[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(16*b^(3/4)*d^(7/4)*(b*c - a*d)^2*(c
+ d*x^2))

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Maple [A]  time = 0.025, size = 566, normalized size = 1.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*c/(a*d-b*c)^2*x^(3/2)/(d*x^2+c)*a-1/2*c^2/(a*d-b*c)^2/d*x^(3/2)/(d*x^2+c)*b-
7/8*c/(a*d-b*c)^2/d/(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)*a+
3/8*c^2/(a*d-b*c)^2/d^2/(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1
)*b-7/8*c/(a*d-b*c)^2/d/(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1
)*a+3/8*c^2/(a*d-b*c)^2/d^2/(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/
2)-1)*b-7/16*c/(a*d-b*c)^2/d/(c/d)^(1/4)*2^(1/2)*ln((x-(c/d)^(1/4)*x^(1/2)*2^(1/
2)+(c/d)^(1/2))/(x+(c/d)^(1/4)*x^(1/2)*2^(1/2)+(c/d)^(1/2)))*a+3/16*c^2/(a*d-b*c
)^2/d^2/(c/d)^(1/4)*2^(1/2)*ln((x-(c/d)^(1/4)*x^(1/2)*2^(1/2)+(c/d)^(1/2))/(x+(c
/d)^(1/4)*x^(1/2)*2^(1/2)+(c/d)^(1/2)))*b+1/4*a^2/(a*d-b*c)^2/b/(a/b)^(1/4)*2^(1
/2)*ln((x-(a/b)^(1/4)*x^(1/2)*2^(1/2)+(a/b)^(1/2))/(x+(a/b)^(1/4)*x^(1/2)*2^(1/2
)+(a/b)^(1/2)))+1/2*a^2/(a*d-b*c)^2/b/(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(
1/4)*x^(1/2)+1)+1/2*a^2/(a*d-b*c)^2/b/(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(
1/4)*x^(1/2)-1)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 12.2834, size = 4362, normalized size = 8.14 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8*(4*c*x^(3/2) - 16*(-a^7/(b^11*c^8 - 8*a*b^10*c^7*d + 28*a^2*b^9*c^6*d^2 - 5
6*a^3*b^8*c^5*d^3 + 70*a^4*b^7*c^4*d^4 - 56*a^5*b^6*c^3*d^5 + 28*a^6*b^5*c^2*d^6
- 8*a^7*b^4*c*d^7 + a^8*b^3*d^8))^(1/4)*(b*c^2*d - a*c*d^2 + (b*c*d^2 - a*d^3)*
x^2)*arctan((b^8*c^6 - 6*a*b^7*c^5*d + 15*a^2*b^6*c^4*d^2 - 20*a^3*b^5*c^3*d^3 +
15*a^4*b^4*c^2*d^4 - 6*a^5*b^3*c*d^5 + a^6*b^2*d^6)*(-a^7/(b^11*c^8 - 8*a*b^10*
c^7*d + 28*a^2*b^9*c^6*d^2 - 56*a^3*b^8*c^5*d^3 + 70*a^4*b^7*c^4*d^4 - 56*a^5*b^
6*c^3*d^5 + 28*a^6*b^5*c^2*d^6 - 8*a^7*b^4*c*d^7 + a^8*b^3*d^8))^(3/4)/(a^5*sqrt
(x) + sqrt(a^10*x - (a^7*b^5*c^4 - 4*a^8*b^4*c^3*d + 6*a^9*b^3*c^2*d^2 - 4*a^10*
b^2*c*d^3 + a^11*b*d^4)*sqrt(-a^7/(b^11*c^8 - 8*a*b^10*c^7*d + 28*a^2*b^9*c^6*d^
2 - 56*a^3*b^8*c^5*d^3 + 70*a^4*b^7*c^4*d^4 - 56*a^5*b^6*c^3*d^5 + 28*a^6*b^5*c^
2*d^6 - 8*a^7*b^4*c*d^7 + a^8*b^3*d^8))))) + 4*(b*c^2*d - a*c*d^2 + (b*c*d^2 - a
*d^3)*x^2)*(-(81*b^4*c^7 - 756*a*b^3*c^6*d + 2646*a^2*b^2*c^5*d^2 - 4116*a^3*b*c
^4*d^3 + 2401*a^4*c^3*d^4)/(b^8*c^8*d^7 - 8*a*b^7*c^7*d^8 + 28*a^2*b^6*c^6*d^9 -
56*a^3*b^5*c^5*d^10 + 70*a^4*b^4*c^4*d^11 - 56*a^5*b^3*c^3*d^12 + 28*a^6*b^2*c^
2*d^13 - 8*a^7*b*c*d^14 + a^8*d^15))^(1/4)*arctan(-(b^6*c^6*d^5 - 6*a*b^5*c^5*d^
6 + 15*a^2*b^4*c^4*d^7 - 20*a^3*b^3*c^3*d^8 + 15*a^4*b^2*c^2*d^9 - 6*a^5*b*c*d^1
0 + a^6*d^11)*(-(81*b^4*c^7 - 756*a*b^3*c^6*d + 2646*a^2*b^2*c^5*d^2 - 4116*a^3*
b*c^4*d^3 + 2401*a^4*c^3*d^4)/(b^8*c^8*d^7 - 8*a*b^7*c^7*d^8 + 28*a^2*b^6*c^6*d^
9 - 56*a^3*b^5*c^5*d^10 + 70*a^4*b^4*c^4*d^11 - 56*a^5*b^3*c^3*d^12 + 28*a^6*b^2
*c^2*d^13 - 8*a^7*b*c*d^14 + a^8*d^15))^(3/4)/((27*b^3*c^5 - 189*a*b^2*c^4*d + 4
41*a^2*b*c^3*d^2 - 343*a^3*c^2*d^3)*sqrt(x) - sqrt((729*b^6*c^10 - 10206*a*b^5*c
^9*d + 59535*a^2*b^4*c^8*d^2 - 185220*a^3*b^3*c^7*d^3 + 324135*a^4*b^2*c^6*d^4 -
302526*a^5*b*c^5*d^5 + 117649*a^6*c^4*d^6)*x - (81*b^8*c^11*d^3 - 1080*a*b^7*c^
10*d^4 + 6156*a^2*b^6*c^9*d^5 - 19560*a^3*b^5*c^8*d^6 + 37846*a^4*b^4*c^7*d^7 -
45640*a^5*b^3*c^6*d^8 + 33516*a^6*b^2*c^5*d^9 - 13720*a^7*b*c^4*d^10 + 2401*a^8*
c^3*d^11)*sqrt(-(81*b^4*c^7 - 756*a*b^3*c^6*d + 2646*a^2*b^2*c^5*d^2 - 4116*a^3*
b*c^4*d^3 + 2401*a^4*c^3*d^4)/(b^8*c^8*d^7 - 8*a*b^7*c^7*d^8 + 28*a^2*b^6*c^6*d^
9 - 56*a^3*b^5*c^5*d^10 + 70*a^4*b^4*c^4*d^11 - 56*a^5*b^3*c^3*d^12 + 28*a^6*b^2
*c^2*d^13 - 8*a^7*b*c*d^14 + a^8*d^15))))) - 4*(-a^7/(b^11*c^8 - 8*a*b^10*c^7*d
+ 28*a^2*b^9*c^6*d^2 - 56*a^3*b^8*c^5*d^3 + 70*a^4*b^7*c^4*d^4 - 56*a^5*b^6*c^3*
d^5 + 28*a^6*b^5*c^2*d^6 - 8*a^7*b^4*c*d^7 + a^8*b^3*d^8))^(1/4)*(b*c^2*d - a*c*
d^2 + (b*c*d^2 - a*d^3)*x^2)*log(a^5*sqrt(x) + (b^8*c^6 - 6*a*b^7*c^5*d + 15*a^2
*b^6*c^4*d^2 - 20*a^3*b^5*c^3*d^3 + 15*a^4*b^4*c^2*d^4 - 6*a^5*b^3*c*d^5 + a^6*b
^2*d^6)*(-a^7/(b^11*c^8 - 8*a*b^10*c^7*d + 28*a^2*b^9*c^6*d^2 - 56*a^3*b^8*c^5*d
^3 + 70*a^4*b^7*c^4*d^4 - 56*a^5*b^6*c^3*d^5 + 28*a^6*b^5*c^2*d^6 - 8*a^7*b^4*c*
d^7 + a^8*b^3*d^8))^(3/4)) + 4*(-a^7/(b^11*c^8 - 8*a*b^10*c^7*d + 28*a^2*b^9*c^6
*d^2 - 56*a^3*b^8*c^5*d^3 + 70*a^4*b^7*c^4*d^4 - 56*a^5*b^6*c^3*d^5 + 28*a^6*b^5
*c^2*d^6 - 8*a^7*b^4*c*d^7 + a^8*b^3*d^8))^(1/4)*(b*c^2*d - a*c*d^2 + (b*c*d^2 -
a*d^3)*x^2)*log(a^5*sqrt(x) - (b^8*c^6 - 6*a*b^7*c^5*d + 15*a^2*b^6*c^4*d^2 - 2
0*a^3*b^5*c^3*d^3 + 15*a^4*b^4*c^2*d^4 - 6*a^5*b^3*c*d^5 + a^6*b^2*d^6)*(-a^7/(b
^11*c^8 - 8*a*b^10*c^7*d + 28*a^2*b^9*c^6*d^2 - 56*a^3*b^8*c^5*d^3 + 70*a^4*b^7*
c^4*d^4 - 56*a^5*b^6*c^3*d^5 + 28*a^6*b^5*c^2*d^6 - 8*a^7*b^4*c*d^7 + a^8*b^3*d^
8))^(3/4)) + (b*c^2*d - a*c*d^2 + (b*c*d^2 - a*d^3)*x^2)*(-(81*b^4*c^7 - 756*a*b
^3*c^6*d + 2646*a^2*b^2*c^5*d^2 - 4116*a^3*b*c^4*d^3 + 2401*a^4*c^3*d^4)/(b^8*c^
8*d^7 - 8*a*b^7*c^7*d^8 + 28*a^2*b^6*c^6*d^9 - 56*a^3*b^5*c^5*d^10 + 70*a^4*b^4*
c^4*d^11 - 56*a^5*b^3*c^3*d^12 + 28*a^6*b^2*c^2*d^13 - 8*a^7*b*c*d^14 + a^8*d^15
))^(1/4)*log((b^6*c^6*d^5 - 6*a*b^5*c^5*d^6 + 15*a^2*b^4*c^4*d^7 - 20*a^3*b^3*c^
3*d^8 + 15*a^4*b^2*c^2*d^9 - 6*a^5*b*c*d^10 + a^6*d^11)*(-(81*b^4*c^7 - 756*a*b^
3*c^6*d + 2646*a^2*b^2*c^5*d^2 - 4116*a^3*b*c^4*d^3 + 2401*a^4*c^3*d^4)/(b^8*c^8
*d^7 - 8*a*b^7*c^7*d^8 + 28*a^2*b^6*c^6*d^9 - 56*a^3*b^5*c^5*d^10 + 70*a^4*b^4*c
^4*d^11 - 56*a^5*b^3*c^3*d^12 + 28*a^6*b^2*c^2*d^13 - 8*a^7*b*c*d^14 + a^8*d^15)
)^(3/4) - (27*b^3*c^5 - 189*a*b^2*c^4*d + 441*a^2*b*c^3*d^2 - 343*a^3*c^2*d^3)*s
qrt(x)) - (b*c^2*d - a*c*d^2 + (b*c*d^2 - a*d^3)*x^2)*(-(81*b^4*c^7 - 756*a*b^3*
c^6*d + 2646*a^2*b^2*c^5*d^2 - 4116*a^3*b*c^4*d^3 + 2401*a^4*c^3*d^4)/(b^8*c^8*d
^7 - 8*a*b^7*c^7*d^8 + 28*a^2*b^6*c^6*d^9 - 56*a^3*b^5*c^5*d^10 + 70*a^4*b^4*c^4
*d^11 - 56*a^5*b^3*c^3*d^12 + 28*a^6*b^2*c^2*d^13 - 8*a^7*b*c*d^14 + a^8*d^15))^
(1/4)*log(-(b^6*c^6*d^5 - 6*a*b^5*c^5*d^6 + 15*a^2*b^4*c^4*d^7 - 20*a^3*b^3*c^3*
d^8 + 15*a^4*b^2*c^2*d^9 - 6*a^5*b*c*d^10 + a^6*d^11)*(-(81*b^4*c^7 - 756*a*b^3*
c^6*d + 2646*a^2*b^2*c^5*d^2 - 4116*a^3*b*c^4*d^3 + 2401*a^4*c^3*d^4)/(b^8*c^8*d
^7 - 8*a*b^7*c^7*d^8 + 28*a^2*b^6*c^6*d^9 - 56*a^3*b^5*c^5*d^10 + 70*a^4*b^4*c^4
*d^11 - 56*a^5*b^3*c^3*d^12 + 28*a^6*b^2*c^2*d^13 - 8*a^7*b*c*d^14 + a^8*d^15))^
(3/4) - (27*b^3*c^5 - 189*a*b^2*c^4*d + 441*a^2*b*c^3*d^2 - 343*a^3*c^2*d^3)*sqr
t(x)))/(b*c^2*d - a*c*d^2 + (b*c*d^2 - a*d^3)*x^2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.336445, size = 919, normalized size = 1.71 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(a*b^3)^(3/4)*a*arctan(1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) + 2*sqrt(x))/(a/b)^(1/4)
)/(sqrt(2)*b^5*c^2 - 2*sqrt(2)*a*b^4*c*d + sqrt(2)*a^2*b^3*d^2) + (a*b^3)^(3/4)*
a*arctan(-1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) - 2*sqrt(x))/(a/b)^(1/4))/(sqrt(2)*b^
5*c^2 - 2*sqrt(2)*a*b^4*c*d + sqrt(2)*a^2*b^3*d^2) - 1/2*(a*b^3)^(3/4)*a*ln(sqrt
(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(sqrt(2)*b^5*c^2 - 2*sqrt(2)*a*b^4*c*d
+ sqrt(2)*a^2*b^3*d^2) + 1/2*(a*b^3)^(3/4)*a*ln(-sqrt(2)*sqrt(x)*(a/b)^(1/4) + x
+ sqrt(a/b))/(sqrt(2)*b^5*c^2 - 2*sqrt(2)*a*b^4*c*d + sqrt(2)*a^2*b^3*d^2) + 1/
4*(3*(c*d^3)^(3/4)*b*c - 7*(c*d^3)^(3/4)*a*d)*arctan(1/2*sqrt(2)*(sqrt(2)*(c/d)^
(1/4) + 2*sqrt(x))/(c/d)^(1/4))/(sqrt(2)*b^2*c^2*d^4 - 2*sqrt(2)*a*b*c*d^5 + sqr
t(2)*a^2*d^6) + 1/4*(3*(c*d^3)^(3/4)*b*c - 7*(c*d^3)^(3/4)*a*d)*arctan(-1/2*sqrt
(2)*(sqrt(2)*(c/d)^(1/4) - 2*sqrt(x))/(c/d)^(1/4))/(sqrt(2)*b^2*c^2*d^4 - 2*sqrt
(2)*a*b*c*d^5 + sqrt(2)*a^2*d^6) - 1/8*(3*(c*d^3)^(3/4)*b*c - 7*(c*d^3)^(3/4)*a*
d)*ln(sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/(sqrt(2)*b^2*c^2*d^4 - 2*sqrt
(2)*a*b*c*d^5 + sqrt(2)*a^2*d^6) + 1/8*(3*(c*d^3)^(3/4)*b*c - 7*(c*d^3)^(3/4)*a*
d)*ln(-sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/(sqrt(2)*b^2*c^2*d^4 - 2*sqr
t(2)*a*b*c*d^5 + sqrt(2)*a^2*d^6) - 1/2*c*x^(3/2)/((b*c*d - a*d^2)*(d*x^2 + c))