3.478 \(\int \frac{1}{x^{5/2} \left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx\)

Optimal. Leaf size=570 \[ \frac{b^{11/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} a^{7/4} (b c-a d)^2}-\frac{b^{11/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} a^{7/4} (b c-a d)^2}+\frac{b^{11/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} a^{7/4} (b c-a d)^2}-\frac{b^{11/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} a^{7/4} (b c-a d)^2}-\frac{d^{7/4} (11 b c-7 a d) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} c^{11/4} (b c-a d)^2}+\frac{d^{7/4} (11 b c-7 a d) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} c^{11/4} (b c-a d)^2}-\frac{d^{7/4} (11 b c-7 a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{4 \sqrt{2} c^{11/4} (b c-a d)^2}+\frac{d^{7/4} (11 b c-7 a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{4 \sqrt{2} c^{11/4} (b c-a d)^2}-\frac{4 b c-7 a d}{6 a c^2 x^{3/2} (b c-a d)}-\frac{d}{2 c x^{3/2} \left (c+d x^2\right ) (b c-a d)} \]

[Out]

-(4*b*c - 7*a*d)/(6*a*c^2*(b*c - a*d)*x^(3/2)) - d/(2*c*(b*c - a*d)*x^(3/2)*(c +
d*x^2)) + (b^(11/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*a^(
7/4)*(b*c - a*d)^2) - (b^(11/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(
Sqrt[2]*a^(7/4)*(b*c - a*d)^2) - (d^(7/4)*(11*b*c - 7*a*d)*ArcTan[1 - (Sqrt[2]*d
^(1/4)*Sqrt[x])/c^(1/4)])/(4*Sqrt[2]*c^(11/4)*(b*c - a*d)^2) + (d^(7/4)*(11*b*c
- 7*a*d)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(4*Sqrt[2]*c^(11/4)*(b*c
- a*d)^2) + (b^(11/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x
])/(2*Sqrt[2]*a^(7/4)*(b*c - a*d)^2) - (b^(11/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b
^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*a^(7/4)*(b*c - a*d)^2) - (d^(7/4)*(11*b*
c - 7*a*d)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(8*Sqrt[2
]*c^(11/4)*(b*c - a*d)^2) + (d^(7/4)*(11*b*c - 7*a*d)*Log[Sqrt[c] + Sqrt[2]*c^(1
/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(8*Sqrt[2]*c^(11/4)*(b*c - a*d)^2)

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Rubi [A]  time = 1.72427, antiderivative size = 570, normalized size of antiderivative = 1., number of steps used = 22, number of rules used = 10, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417 \[ \frac{b^{11/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} a^{7/4} (b c-a d)^2}-\frac{b^{11/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} a^{7/4} (b c-a d)^2}+\frac{b^{11/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} a^{7/4} (b c-a d)^2}-\frac{b^{11/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} a^{7/4} (b c-a d)^2}-\frac{d^{7/4} (11 b c-7 a d) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} c^{11/4} (b c-a d)^2}+\frac{d^{7/4} (11 b c-7 a d) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} c^{11/4} (b c-a d)^2}-\frac{d^{7/4} (11 b c-7 a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{4 \sqrt{2} c^{11/4} (b c-a d)^2}+\frac{d^{7/4} (11 b c-7 a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{4 \sqrt{2} c^{11/4} (b c-a d)^2}-\frac{4 b c-7 a d}{6 a c^2 x^{3/2} (b c-a d)}-\frac{d}{2 c x^{3/2} \left (c+d x^2\right ) (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^(5/2)*(a + b*x^2)*(c + d*x^2)^2),x]

[Out]

-(4*b*c - 7*a*d)/(6*a*c^2*(b*c - a*d)*x^(3/2)) - d/(2*c*(b*c - a*d)*x^(3/2)*(c +
d*x^2)) + (b^(11/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*a^(
7/4)*(b*c - a*d)^2) - (b^(11/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(
Sqrt[2]*a^(7/4)*(b*c - a*d)^2) - (d^(7/4)*(11*b*c - 7*a*d)*ArcTan[1 - (Sqrt[2]*d
^(1/4)*Sqrt[x])/c^(1/4)])/(4*Sqrt[2]*c^(11/4)*(b*c - a*d)^2) + (d^(7/4)*(11*b*c
- 7*a*d)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(4*Sqrt[2]*c^(11/4)*(b*c
- a*d)^2) + (b^(11/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x
])/(2*Sqrt[2]*a^(7/4)*(b*c - a*d)^2) - (b^(11/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b
^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*a^(7/4)*(b*c - a*d)^2) - (d^(7/4)*(11*b*
c - 7*a*d)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(8*Sqrt[2
]*c^(11/4)*(b*c - a*d)^2) + (d^(7/4)*(11*b*c - 7*a*d)*Log[Sqrt[c] + Sqrt[2]*c^(1
/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(8*Sqrt[2]*c^(11/4)*(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(1/x**(5/2)/(b*x**2+a)/(d*x**2+c)**2,x)

[Out]

Timed out

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Mathematica [A]  time = 1.63879, size = 542, normalized size = 0.95 \[ \frac{1}{48} \left (\frac{12 \sqrt{2} b^{11/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{a^{7/4} (b c-a d)^2}-\frac{12 \sqrt{2} b^{11/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{a^{7/4} (b c-a d)^2}+\frac{24 \sqrt{2} b^{11/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{a^{7/4} (b c-a d)^2}-\frac{24 \sqrt{2} b^{11/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{a^{7/4} (b c-a d)^2}+\frac{3 \sqrt{2} d^{7/4} (7 a d-11 b c) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{c^{11/4} (b c-a d)^2}+\frac{3 \sqrt{2} d^{7/4} (11 b c-7 a d) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{c^{11/4} (b c-a d)^2}+\frac{6 \sqrt{2} d^{7/4} (7 a d-11 b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{c^{11/4} (b c-a d)^2}+\frac{6 \sqrt{2} d^{7/4} (11 b c-7 a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{c^{11/4} (b c-a d)^2}+\frac{24 d^2 \sqrt{x}}{c^2 \left (c+d x^2\right ) (b c-a d)}-\frac{32}{a c^2 x^{3/2}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^(5/2)*(a + b*x^2)*(c + d*x^2)^2),x]

[Out]

(-32/(a*c^2*x^(3/2)) + (24*d^2*Sqrt[x])/(c^2*(b*c - a*d)*(c + d*x^2)) + (24*Sqrt
[2]*b^(11/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(a^(7/4)*(b*c - a*d)
^2) - (24*Sqrt[2]*b^(11/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(a^(7/
4)*(b*c - a*d)^2) + (6*Sqrt[2]*d^(7/4)*(-11*b*c + 7*a*d)*ArcTan[1 - (Sqrt[2]*d^(
1/4)*Sqrt[x])/c^(1/4)])/(c^(11/4)*(b*c - a*d)^2) + (6*Sqrt[2]*d^(7/4)*(11*b*c -
7*a*d)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(c^(11/4)*(b*c - a*d)^2) +
(12*Sqrt[2]*b^(11/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x]
)/(a^(7/4)*(b*c - a*d)^2) - (12*Sqrt[2]*b^(11/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b
^(1/4)*Sqrt[x] + Sqrt[b]*x])/(a^(7/4)*(b*c - a*d)^2) + (3*Sqrt[2]*d^(7/4)*(-11*b
*c + 7*a*d)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(c^(11/4
)*(b*c - a*d)^2) + (3*Sqrt[2]*d^(7/4)*(11*b*c - 7*a*d)*Log[Sqrt[c] + Sqrt[2]*c^(
1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(c^(11/4)*(b*c - a*d)^2))/48

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Maple [A]  time = 0.028, size = 588, normalized size = 1. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^(5/2)/(b*x^2+a)/(d*x^2+c)^2,x)

[Out]

-1/2*d^3/c^2/(a*d-b*c)^2*x^(1/2)/(d*x^2+c)*a+1/2*d^2/c/(a*d-b*c)^2*x^(1/2)/(d*x^
2+c)*b-7/8*d^3/c^3/(a*d-b*c)^2*(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^
(1/2)+1)*a+11/8*d^2/c^2/(a*d-b*c)^2*(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/
4)*x^(1/2)+1)*b-7/8*d^3/c^3/(a*d-b*c)^2*(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)
^(1/4)*x^(1/2)-1)*a+11/8*d^2/c^2/(a*d-b*c)^2*(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/
(c/d)^(1/4)*x^(1/2)-1)*b-7/16*d^3/c^3/(a*d-b*c)^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))
)*a+11/16*d^2/c^2/(a*d-b*c)^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-2/3/a/c^2/x^(3/2
)-1/4/a^2*b^3/(a*d-b*c)^2*(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*b^3/(a*d-b*c)^
2*(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)-1/2/a^2*b^3/(a*d-b*c
)^2*(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(1/((b*x^2 + a)*(d*x^2 + c)^2*x^(5/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 120.048, size = 3791, normalized size = 6.65 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)*(d*x^2 + c)^2*x^(5/2)),x, algorithm="fricas")

[Out]

-1/24*(16*b*c^2 - 16*a*c*d + 4*(4*b*c*d - 7*a*d^2)*x^2 - 48*(-b^11/(a^7*b^8*c^8
- 8*a^8*b^7*c^7*d + 28*a^9*b^6*c^6*d^2 - 56*a^10*b^5*c^5*d^3 + 70*a^11*b^4*c^4*d
^4 - 56*a^12*b^3*c^3*d^5 + 28*a^13*b^2*c^2*d^6 - 8*a^14*b*c*d^7 + a^15*d^8))^(1/
4)*((a*b*c^3*d - a^2*c^2*d^2)*x^3 + (a*b*c^4 - a^2*c^3*d)*x)*sqrt(x)*arctan((-b^
11/(a^7*b^8*c^8 - 8*a^8*b^7*c^7*d + 28*a^9*b^6*c^6*d^2 - 56*a^10*b^5*c^5*d^3 + 7
0*a^11*b^4*c^4*d^4 - 56*a^12*b^3*c^3*d^5 + 28*a^13*b^2*c^2*d^6 - 8*a^14*b*c*d^7
+ a^15*d^8))^(1/4)*(a^2*b^2*c^2 - 2*a^3*b*c*d + a^4*d^2)/(b^3*sqrt(x) + sqrt(b^6
*x + (a^4*b^4*c^4 - 4*a^5*b^3*c^3*d + 6*a^6*b^2*c^2*d^2 - 4*a^7*b*c*d^3 + a^8*d^
4)*sqrt(-b^11/(a^7*b^8*c^8 - 8*a^8*b^7*c^7*d + 28*a^9*b^6*c^6*d^2 - 56*a^10*b^5*
c^5*d^3 + 70*a^11*b^4*c^4*d^4 - 56*a^12*b^3*c^3*d^5 + 28*a^13*b^2*c^2*d^6 - 8*a^
14*b*c*d^7 + a^15*d^8))))) - 12*((a*b*c^3*d - a^2*c^2*d^2)*x^3 + (a*b*c^4 - a^2*
c^3*d)*x)*sqrt(x)*(-(14641*b^4*c^4*d^7 - 37268*a*b^3*c^3*d^8 + 35574*a^2*b^2*c^2
*d^9 - 15092*a^3*b*c*d^10 + 2401*a^4*d^11)/(b^8*c^19 - 8*a*b^7*c^18*d + 28*a^2*b
^6*c^17*d^2 - 56*a^3*b^5*c^16*d^3 + 70*a^4*b^4*c^15*d^4 - 56*a^5*b^3*c^14*d^5 +
28*a^6*b^2*c^13*d^6 - 8*a^7*b*c^12*d^7 + a^8*c^11*d^8))^(1/4)*arctan(-(b^2*c^5 -
2*a*b*c^4*d + a^2*c^3*d^2)*(-(14641*b^4*c^4*d^7 - 37268*a*b^3*c^3*d^8 + 35574*a
^2*b^2*c^2*d^9 - 15092*a^3*b*c*d^10 + 2401*a^4*d^11)/(b^8*c^19 - 8*a*b^7*c^18*d
+ 28*a^2*b^6*c^17*d^2 - 56*a^3*b^5*c^16*d^3 + 70*a^4*b^4*c^15*d^4 - 56*a^5*b^3*c
^14*d^5 + 28*a^6*b^2*c^13*d^6 - 8*a^7*b*c^12*d^7 + a^8*c^11*d^8))^(1/4)/((11*b*c
*d^2 - 7*a*d^3)*sqrt(x) - sqrt((121*b^2*c^2*d^4 - 154*a*b*c*d^5 + 49*a^2*d^6)*x
+ (b^4*c^10 - 4*a*b^3*c^9*d + 6*a^2*b^2*c^8*d^2 - 4*a^3*b*c^7*d^3 + a^4*c^6*d^4)
*sqrt(-(14641*b^4*c^4*d^7 - 37268*a*b^3*c^3*d^8 + 35574*a^2*b^2*c^2*d^9 - 15092*
a^3*b*c*d^10 + 2401*a^4*d^11)/(b^8*c^19 - 8*a*b^7*c^18*d + 28*a^2*b^6*c^17*d^2 -
56*a^3*b^5*c^16*d^3 + 70*a^4*b^4*c^15*d^4 - 56*a^5*b^3*c^14*d^5 + 28*a^6*b^2*c^
13*d^6 - 8*a^7*b*c^12*d^7 + a^8*c^11*d^8))))) + 12*(-b^11/(a^7*b^8*c^8 - 8*a^8*b
^7*c^7*d + 28*a^9*b^6*c^6*d^2 - 56*a^10*b^5*c^5*d^3 + 70*a^11*b^4*c^4*d^4 - 56*a
^12*b^3*c^3*d^5 + 28*a^13*b^2*c^2*d^6 - 8*a^14*b*c*d^7 + a^15*d^8))^(1/4)*((a*b*
c^3*d - a^2*c^2*d^2)*x^3 + (a*b*c^4 - a^2*c^3*d)*x)*sqrt(x)*log(b^3*sqrt(x) + (-
b^11/(a^7*b^8*c^8 - 8*a^8*b^7*c^7*d + 28*a^9*b^6*c^6*d^2 - 56*a^10*b^5*c^5*d^3 +
70*a^11*b^4*c^4*d^4 - 56*a^12*b^3*c^3*d^5 + 28*a^13*b^2*c^2*d^6 - 8*a^14*b*c*d^
7 + a^15*d^8))^(1/4)*(a^2*b^2*c^2 - 2*a^3*b*c*d + a^4*d^2)) - 12*(-b^11/(a^7*b^8
*c^8 - 8*a^8*b^7*c^7*d + 28*a^9*b^6*c^6*d^2 - 56*a^10*b^5*c^5*d^3 + 70*a^11*b^4*
c^4*d^4 - 56*a^12*b^3*c^3*d^5 + 28*a^13*b^2*c^2*d^6 - 8*a^14*b*c*d^7 + a^15*d^8)
)^(1/4)*((a*b*c^3*d - a^2*c^2*d^2)*x^3 + (a*b*c^4 - a^2*c^3*d)*x)*sqrt(x)*log(b^
3*sqrt(x) - (-b^11/(a^7*b^8*c^8 - 8*a^8*b^7*c^7*d + 28*a^9*b^6*c^6*d^2 - 56*a^10
*b^5*c^5*d^3 + 70*a^11*b^4*c^4*d^4 - 56*a^12*b^3*c^3*d^5 + 28*a^13*b^2*c^2*d^6 -
8*a^14*b*c*d^7 + a^15*d^8))^(1/4)*(a^2*b^2*c^2 - 2*a^3*b*c*d + a^4*d^2)) + 3*((
a*b*c^3*d - a^2*c^2*d^2)*x^3 + (a*b*c^4 - a^2*c^3*d)*x)*sqrt(x)*(-(14641*b^4*c^4
*d^7 - 37268*a*b^3*c^3*d^8 + 35574*a^2*b^2*c^2*d^9 - 15092*a^3*b*c*d^10 + 2401*a
^4*d^11)/(b^8*c^19 - 8*a*b^7*c^18*d + 28*a^2*b^6*c^17*d^2 - 56*a^3*b^5*c^16*d^3
+ 70*a^4*b^4*c^15*d^4 - 56*a^5*b^3*c^14*d^5 + 28*a^6*b^2*c^13*d^6 - 8*a^7*b*c^12
*d^7 + a^8*c^11*d^8))^(1/4)*log(-(11*b*c*d^2 - 7*a*d^3)*sqrt(x) + (b^2*c^5 - 2*a
*b*c^4*d + a^2*c^3*d^2)*(-(14641*b^4*c^4*d^7 - 37268*a*b^3*c^3*d^8 + 35574*a^2*b
^2*c^2*d^9 - 15092*a^3*b*c*d^10 + 2401*a^4*d^11)/(b^8*c^19 - 8*a*b^7*c^18*d + 28
*a^2*b^6*c^17*d^2 - 56*a^3*b^5*c^16*d^3 + 70*a^4*b^4*c^15*d^4 - 56*a^5*b^3*c^14*
d^5 + 28*a^6*b^2*c^13*d^6 - 8*a^7*b*c^12*d^7 + a^8*c^11*d^8))^(1/4)) - 3*((a*b*c
^3*d - a^2*c^2*d^2)*x^3 + (a*b*c^4 - a^2*c^3*d)*x)*sqrt(x)*(-(14641*b^4*c^4*d^7
- 37268*a*b^3*c^3*d^8 + 35574*a^2*b^2*c^2*d^9 - 15092*a^3*b*c*d^10 + 2401*a^4*d^
11)/(b^8*c^19 - 8*a*b^7*c^18*d + 28*a^2*b^6*c^17*d^2 - 56*a^3*b^5*c^16*d^3 + 70*
a^4*b^4*c^15*d^4 - 56*a^5*b^3*c^14*d^5 + 28*a^6*b^2*c^13*d^6 - 8*a^7*b*c^12*d^7
+ a^8*c^11*d^8))^(1/4)*log(-(11*b*c*d^2 - 7*a*d^3)*sqrt(x) - (b^2*c^5 - 2*a*b*c^
4*d + a^2*c^3*d^2)*(-(14641*b^4*c^4*d^7 - 37268*a*b^3*c^3*d^8 + 35574*a^2*b^2*c^
2*d^9 - 15092*a^3*b*c*d^10 + 2401*a^4*d^11)/(b^8*c^19 - 8*a*b^7*c^18*d + 28*a^2*
b^6*c^17*d^2 - 56*a^3*b^5*c^16*d^3 + 70*a^4*b^4*c^15*d^4 - 56*a^5*b^3*c^14*d^5 +
28*a^6*b^2*c^13*d^6 - 8*a^7*b*c^12*d^7 + a^8*c^11*d^8))^(1/4)))/(((a*b*c^3*d -
a^2*c^2*d^2)*x^3 + (a*b*c^4 - a^2*c^3*d)*x)*sqrt(x))

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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(1/x**(5/2)/(b*x**2+a)/(d*x**2+c)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.341204, size = 969, normalized size = 1.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)*(d*x^2 + c)^2*x^(5/2)),x, algorithm="giac")

[Out]

-(a*b^3)^(1/4)*b^2*arctan(1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) + 2*sqrt(x))/(a/b)^(1
/4))/(sqrt(2)*a^2*b^2*c^2 - 2*sqrt(2)*a^3*b*c*d + sqrt(2)*a^4*d^2) - (a*b^3)^(1/
4)*b^2*arctan(-1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) - 2*sqrt(x))/(a/b)^(1/4))/(sqrt(
2)*a^2*b^2*c^2 - 2*sqrt(2)*a^3*b*c*d + sqrt(2)*a^4*d^2) - 1/2*(a*b^3)^(1/4)*b^2*
ln(sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(sqrt(2)*a^2*b^2*c^2 - 2*sqrt(2)
*a^3*b*c*d + sqrt(2)*a^4*d^2) + 1/2*(a*b^3)^(1/4)*b^2*ln(-sqrt(2)*sqrt(x)*(a/b)^
(1/4) + x + sqrt(a/b))/(sqrt(2)*a^2*b^2*c^2 - 2*sqrt(2)*a^3*b*c*d + sqrt(2)*a^4*
d^2) + 1/4*(11*(c*d^3)^(1/4)*b*c*d - 7*(c*d^3)^(1/4)*a*d^2)*arctan(1/2*sqrt(2)*(
sqrt(2)*(c/d)^(1/4) + 2*sqrt(x))/(c/d)^(1/4))/(sqrt(2)*b^2*c^5 - 2*sqrt(2)*a*b*c
^4*d + sqrt(2)*a^2*c^3*d^2) + 1/4*(11*(c*d^3)^(1/4)*b*c*d - 7*(c*d^3)^(1/4)*a*d^
2)*arctan(-1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4) - 2*sqrt(x))/(c/d)^(1/4))/(sqrt(2)*b
^2*c^5 - 2*sqrt(2)*a*b*c^4*d + sqrt(2)*a^2*c^3*d^2) + 1/8*(11*(c*d^3)^(1/4)*b*c*
d - 7*(c*d^3)^(1/4)*a*d^2)*ln(sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/(sqrt
(2)*b^2*c^5 - 2*sqrt(2)*a*b*c^4*d + sqrt(2)*a^2*c^3*d^2) - 1/8*(11*(c*d^3)^(1/4)
*b*c*d - 7*(c*d^3)^(1/4)*a*d^2)*ln(-sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))
/(sqrt(2)*b^2*c^5 - 2*sqrt(2)*a*b*c^4*d + sqrt(2)*a^2*c^3*d^2) + 1/2*d^2*sqrt(x)
/((b*c^3 - a*c^2*d)*(d*x^2 + c)) - 2/3/(a*c^2*x^(3/2))