3.479 \(\int \frac{1}{x^{7/2} \left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx\)

Optimal. Leaf size=618 \[ \frac{b^{13/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} a^{9/4} (b c-a d)^2}-\frac{b^{13/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} a^{9/4} (b c-a d)^2}-\frac{b^{13/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} a^{9/4} (b c-a d)^2}+\frac{b^{13/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} a^{9/4} (b c-a d)^2}+\frac{-9 a^2 d^2+4 a b c d+4 b^2 c^2}{2 a^2 c^3 \sqrt{x} (b c-a d)}-\frac{d^{9/4} (13 b c-9 a d) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} c^{13/4} (b c-a d)^2}+\frac{d^{9/4} (13 b c-9 a d) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} c^{13/4} (b c-a d)^2}+\frac{d^{9/4} (13 b c-9 a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{4 \sqrt{2} c^{13/4} (b c-a d)^2}-\frac{d^{9/4} (13 b c-9 a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{4 \sqrt{2} c^{13/4} (b c-a d)^2}-\frac{4 b c-9 a d}{10 a c^2 x^{5/2} (b c-a d)}-\frac{d}{2 c x^{5/2} \left (c+d x^2\right ) (b c-a d)} \]

[Out]

-(4*b*c - 9*a*d)/(10*a*c^2*(b*c - a*d)*x^(5/2)) + (4*b^2*c^2 + 4*a*b*c*d - 9*a^2
*d^2)/(2*a^2*c^3*(b*c - a*d)*Sqrt[x]) - d/(2*c*(b*c - a*d)*x^(5/2)*(c + d*x^2))
- (b^(13/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*a^(9/4)*(b*c
- a*d)^2) + (b^(13/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*a
^(9/4)*(b*c - a*d)^2) + (d^(9/4)*(13*b*c - 9*a*d)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sq
rt[x])/c^(1/4)])/(4*Sqrt[2]*c^(13/4)*(b*c - a*d)^2) - (d^(9/4)*(13*b*c - 9*a*d)*
ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(4*Sqrt[2]*c^(13/4)*(b*c - a*d)^2
) + (b^(13/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqr
t[2]*a^(9/4)*(b*c - a*d)^2) - (b^(13/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sq
rt[x] + Sqrt[b]*x])/(2*Sqrt[2]*a^(9/4)*(b*c - a*d)^2) - (d^(9/4)*(13*b*c - 9*a*d
)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(8*Sqrt[2]*c^(13/4
)*(b*c - a*d)^2) + (d^(9/4)*(13*b*c - 9*a*d)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/
4)*Sqrt[x] + Sqrt[d]*x])/(8*Sqrt[2]*c^(13/4)*(b*c - a*d)^2)

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Rubi [A]  time = 2.14629, antiderivative size = 618, normalized size of antiderivative = 1., number of steps used = 24, number of rules used = 10, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417 \[ \frac{b^{13/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} a^{9/4} (b c-a d)^2}-\frac{b^{13/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} a^{9/4} (b c-a d)^2}-\frac{b^{13/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} a^{9/4} (b c-a d)^2}+\frac{b^{13/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} a^{9/4} (b c-a d)^2}+\frac{-9 a^2 d^2+4 a b c d+4 b^2 c^2}{2 a^2 c^3 \sqrt{x} (b c-a d)}-\frac{d^{9/4} (13 b c-9 a d) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} c^{13/4} (b c-a d)^2}+\frac{d^{9/4} (13 b c-9 a d) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} c^{13/4} (b c-a d)^2}+\frac{d^{9/4} (13 b c-9 a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{4 \sqrt{2} c^{13/4} (b c-a d)^2}-\frac{d^{9/4} (13 b c-9 a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{4 \sqrt{2} c^{13/4} (b c-a d)^2}-\frac{4 b c-9 a d}{10 a c^2 x^{5/2} (b c-a d)}-\frac{d}{2 c x^{5/2} \left (c+d x^2\right ) (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^(7/2)*(a + b*x^2)*(c + d*x^2)^2),x]

[Out]

-(4*b*c - 9*a*d)/(10*a*c^2*(b*c - a*d)*x^(5/2)) + (4*b^2*c^2 + 4*a*b*c*d - 9*a^2
*d^2)/(2*a^2*c^3*(b*c - a*d)*Sqrt[x]) - d/(2*c*(b*c - a*d)*x^(5/2)*(c + d*x^2))
- (b^(13/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*a^(9/4)*(b*c
- a*d)^2) + (b^(13/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*a
^(9/4)*(b*c - a*d)^2) + (d^(9/4)*(13*b*c - 9*a*d)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sq
rt[x])/c^(1/4)])/(4*Sqrt[2]*c^(13/4)*(b*c - a*d)^2) - (d^(9/4)*(13*b*c - 9*a*d)*
ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(4*Sqrt[2]*c^(13/4)*(b*c - a*d)^2
) + (b^(13/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqr
t[2]*a^(9/4)*(b*c - a*d)^2) - (b^(13/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sq
rt[x] + Sqrt[b]*x])/(2*Sqrt[2]*a^(9/4)*(b*c - a*d)^2) - (d^(9/4)*(13*b*c - 9*a*d
)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(8*Sqrt[2]*c^(13/4
)*(b*c - a*d)^2) + (d^(9/4)*(13*b*c - 9*a*d)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/
4)*Sqrt[x] + Sqrt[d]*x])/(8*Sqrt[2]*c^(13/4)*(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**(7/2)/(b*x**2+a)/(d*x**2+c)**2,x)

[Out]

Timed out

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Mathematica [A]  time = 1.51029, size = 563, normalized size = 0.91 \[ \frac{1}{80} \left (\frac{20 \sqrt{2} b^{13/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{a^{9/4} (b c-a d)^2}-\frac{20 \sqrt{2} b^{13/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{a^{9/4} (b c-a d)^2}-\frac{40 \sqrt{2} b^{13/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{a^{9/4} (b c-a d)^2}+\frac{40 \sqrt{2} b^{13/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{a^{9/4} (b c-a d)^2}+\frac{160 (2 a d+b c)}{a^2 c^3 \sqrt{x}}+\frac{5 \sqrt{2} d^{9/4} (9 a d-13 b c) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{c^{13/4} (b c-a d)^2}+\frac{5 \sqrt{2} d^{9/4} (13 b c-9 a d) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{c^{13/4} (b c-a d)^2}+\frac{10 \sqrt{2} d^{9/4} (13 b c-9 a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{c^{13/4} (b c-a d)^2}+\frac{10 \sqrt{2} d^{9/4} (9 a d-13 b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{c^{13/4} (b c-a d)^2}-\frac{40 d^3 x^{3/2}}{c^3 \left (c+d x^2\right ) (b c-a d)}-\frac{32}{a c^2 x^{5/2}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^(7/2)*(a + b*x^2)*(c + d*x^2)^2),x]

[Out]

(-32/(a*c^2*x^(5/2)) + (160*(b*c + 2*a*d))/(a^2*c^3*Sqrt[x]) - (40*d^3*x^(3/2))/
(c^3*(b*c - a*d)*(c + d*x^2)) - (40*Sqrt[2]*b^(13/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)
*Sqrt[x])/a^(1/4)])/(a^(9/4)*(b*c - a*d)^2) + (40*Sqrt[2]*b^(13/4)*ArcTan[1 + (S
qrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(a^(9/4)*(b*c - a*d)^2) + (10*Sqrt[2]*d^(9/4)*
(13*b*c - 9*a*d)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(c^(13/4)*(b*c -
a*d)^2) + (10*Sqrt[2]*d^(9/4)*(-13*b*c + 9*a*d)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqr
t[x])/c^(1/4)])/(c^(13/4)*(b*c - a*d)^2) + (20*Sqrt[2]*b^(13/4)*Log[Sqrt[a] - Sq
rt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(a^(9/4)*(b*c - a*d)^2) - (20*Sqrt[2
]*b^(13/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(a^(9/4)*
(b*c - a*d)^2) + (5*Sqrt[2]*d^(9/4)*(-13*b*c + 9*a*d)*Log[Sqrt[c] - Sqrt[2]*c^(1
/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(c^(13/4)*(b*c - a*d)^2) + (5*Sqrt[2]*d^(9/4)*
(13*b*c - 9*a*d)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(c^
(13/4)*(b*c - a*d)^2))/80

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Maple [A]  time = 0.033, size = 612, normalized size = 1. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^(7/2)/(b*x^2+a)/(d*x^2+c)^2,x)

[Out]

1/2*d^4/c^3/(a*d-b*c)^2*x^(3/2)/(d*x^2+c)*a-1/2*d^3/c^2/(a*d-b*c)^2*x^(3/2)/(d*x
^2+c)*b+9/16*d^3/c^3/(a*d-b*c)^2/(c/d)^(1/4)*2^(1/2)*a*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)))+9/8*d^3/c^3/(
a*d-b*c)^2/(c/d)^(1/4)*2^(1/2)*a*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)+9/8*d^3/c
^3/(a*d-b*c)^2/(c/d)^(1/4)*2^(1/2)*a*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)-13/16
*d^2/c^2/(a*d-b*c)^2/(c/d)^(1/4)*2^(1/2)*b*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)))-13/8*d^2/c^2/(a*d-b*c)^2/
(c/d)^(1/4)*2^(1/2)*b*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)-13/8*d^2/c^2/(a*d-b*
c)^2/(c/d)^(1/4)*2^(1/2)*b*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)-2/5/a/c^2/x^(5/
2)+4/x^(1/2)/a/c^3*d+2/x^(1/2)/a^2/c^2*b+1/4*b^3/a^2/(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*b^3/a^2/(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*b^3/a^2/(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^(7/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 88.2567, size = 4652, normalized size = 7.53 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)*(d*x^2 + c)^2*x^(7/2)),x, algorithm="fricas")

[Out]

-1/40*(16*a*b*c^3 - 16*a^2*c^2*d - 20*(4*b^2*c^2*d + 4*a*b*c*d^2 - 9*a^2*d^3)*x^
4 - 16*(5*b^2*c^3 + 4*a*b*c^2*d - 9*a^2*c*d^2)*x^2 - 80*(-b^13/(a^9*b^8*c^8 - 8*
a^10*b^7*c^7*d + 28*a^11*b^6*c^6*d^2 - 56*a^12*b^5*c^5*d^3 + 70*a^13*b^4*c^4*d^4
- 56*a^14*b^3*c^3*d^5 + 28*a^15*b^2*c^2*d^6 - 8*a^16*b*c*d^7 + a^17*d^8))^(1/4)
*((a^2*b*c^4*d - a^3*c^3*d^2)*x^4 + (a^2*b*c^5 - a^3*c^4*d)*x^2)*sqrt(x)*arctan(
(a^7*b^6*c^6 - 6*a^8*b^5*c^5*d + 15*a^9*b^4*c^4*d^2 - 20*a^10*b^3*c^3*d^3 + 15*a
^11*b^2*c^2*d^4 - 6*a^12*b*c*d^5 + a^13*d^6)*(-b^13/(a^9*b^8*c^8 - 8*a^10*b^7*c^
7*d + 28*a^11*b^6*c^6*d^2 - 56*a^12*b^5*c^5*d^3 + 70*a^13*b^4*c^4*d^4 - 56*a^14*
b^3*c^3*d^5 + 28*a^15*b^2*c^2*d^6 - 8*a^16*b*c*d^7 + a^17*d^8))^(3/4)/(b^10*sqrt
(x) + sqrt(b^20*x - (a^5*b^17*c^4 - 4*a^6*b^16*c^3*d + 6*a^7*b^15*c^2*d^2 - 4*a^
8*b^14*c*d^3 + a^9*b^13*d^4)*sqrt(-b^13/(a^9*b^8*c^8 - 8*a^10*b^7*c^7*d + 28*a^1
1*b^6*c^6*d^2 - 56*a^12*b^5*c^5*d^3 + 70*a^13*b^4*c^4*d^4 - 56*a^14*b^3*c^3*d^5
+ 28*a^15*b^2*c^2*d^6 - 8*a^16*b*c*d^7 + a^17*d^8))))) - 20*((a^2*b*c^4*d - a^3*
c^3*d^2)*x^4 + (a^2*b*c^5 - a^3*c^4*d)*x^2)*sqrt(x)*(-(28561*b^4*c^4*d^9 - 79092
*a*b^3*c^3*d^10 + 82134*a^2*b^2*c^2*d^11 - 37908*a^3*b*c*d^12 + 6561*a^4*d^13)/(
b^8*c^21 - 8*a*b^7*c^20*d + 28*a^2*b^6*c^19*d^2 - 56*a^3*b^5*c^18*d^3 + 70*a^4*b
^4*c^17*d^4 - 56*a^5*b^3*c^16*d^5 + 28*a^6*b^2*c^15*d^6 - 8*a^7*b*c^14*d^7 + a^8
*c^13*d^8))^(1/4)*arctan(-(b^6*c^16 - 6*a*b^5*c^15*d + 15*a^2*b^4*c^14*d^2 - 20*
a^3*b^3*c^13*d^3 + 15*a^4*b^2*c^12*d^4 - 6*a^5*b*c^11*d^5 + a^6*c^10*d^6)*(-(285
61*b^4*c^4*d^9 - 79092*a*b^3*c^3*d^10 + 82134*a^2*b^2*c^2*d^11 - 37908*a^3*b*c*d
^12 + 6561*a^4*d^13)/(b^8*c^21 - 8*a*b^7*c^20*d + 28*a^2*b^6*c^19*d^2 - 56*a^3*b
^5*c^18*d^3 + 70*a^4*b^4*c^17*d^4 - 56*a^5*b^3*c^16*d^5 + 28*a^6*b^2*c^15*d^6 -
8*a^7*b*c^14*d^7 + a^8*c^13*d^8))^(3/4)/((2197*b^3*c^3*d^7 - 4563*a*b^2*c^2*d^8
+ 3159*a^2*b*c*d^9 - 729*a^3*d^10)*sqrt(x) - sqrt((4826809*b^6*c^6*d^14 - 200498
22*a*b^5*c^5*d^15 + 34701615*a^2*b^4*c^4*d^16 - 32032260*a^3*b^3*c^3*d^17 + 1663
2135*a^4*b^2*c^2*d^18 - 4605822*a^5*b*c*d^19 + 531441*a^6*d^20)*x - (28561*b^8*c
^15*d^9 - 193336*a*b^7*c^14*d^10 + 569868*a^2*b^6*c^13*d^11 - 955240*a^3*b^5*c^1
2*d^12 + 995926*a^4*b^4*c^11*d^13 - 661320*a^5*b^3*c^10*d^14 + 273132*a^6*b^2*c^
9*d^15 - 64152*a^7*b*c^8*d^16 + 6561*a^8*c^7*d^17)*sqrt(-(28561*b^4*c^4*d^9 - 79
092*a*b^3*c^3*d^10 + 82134*a^2*b^2*c^2*d^11 - 37908*a^3*b*c*d^12 + 6561*a^4*d^13
)/(b^8*c^21 - 8*a*b^7*c^20*d + 28*a^2*b^6*c^19*d^2 - 56*a^3*b^5*c^18*d^3 + 70*a^
4*b^4*c^17*d^4 - 56*a^5*b^3*c^16*d^5 + 28*a^6*b^2*c^15*d^6 - 8*a^7*b*c^14*d^7 +
a^8*c^13*d^8))))) - 20*(-b^13/(a^9*b^8*c^8 - 8*a^10*b^7*c^7*d + 28*a^11*b^6*c^6*
d^2 - 56*a^12*b^5*c^5*d^3 + 70*a^13*b^4*c^4*d^4 - 56*a^14*b^3*c^3*d^5 + 28*a^15*
b^2*c^2*d^6 - 8*a^16*b*c*d^7 + a^17*d^8))^(1/4)*((a^2*b*c^4*d - a^3*c^3*d^2)*x^4
+ (a^2*b*c^5 - a^3*c^4*d)*x^2)*sqrt(x)*log(b^10*sqrt(x) + (a^7*b^6*c^6 - 6*a^8*
b^5*c^5*d + 15*a^9*b^4*c^4*d^2 - 20*a^10*b^3*c^3*d^3 + 15*a^11*b^2*c^2*d^4 - 6*a
^12*b*c*d^5 + a^13*d^6)*(-b^13/(a^9*b^8*c^8 - 8*a^10*b^7*c^7*d + 28*a^11*b^6*c^6
*d^2 - 56*a^12*b^5*c^5*d^3 + 70*a^13*b^4*c^4*d^4 - 56*a^14*b^3*c^3*d^5 + 28*a^15
*b^2*c^2*d^6 - 8*a^16*b*c*d^7 + a^17*d^8))^(3/4)) + 20*(-b^13/(a^9*b^8*c^8 - 8*a
^10*b^7*c^7*d + 28*a^11*b^6*c^6*d^2 - 56*a^12*b^5*c^5*d^3 + 70*a^13*b^4*c^4*d^4
- 56*a^14*b^3*c^3*d^5 + 28*a^15*b^2*c^2*d^6 - 8*a^16*b*c*d^7 + a^17*d^8))^(1/4)*
((a^2*b*c^4*d - a^3*c^3*d^2)*x^4 + (a^2*b*c^5 - a^3*c^4*d)*x^2)*sqrt(x)*log(b^10
*sqrt(x) - (a^7*b^6*c^6 - 6*a^8*b^5*c^5*d + 15*a^9*b^4*c^4*d^2 - 20*a^10*b^3*c^3
*d^3 + 15*a^11*b^2*c^2*d^4 - 6*a^12*b*c*d^5 + a^13*d^6)*(-b^13/(a^9*b^8*c^8 - 8*
a^10*b^7*c^7*d + 28*a^11*b^6*c^6*d^2 - 56*a^12*b^5*c^5*d^3 + 70*a^13*b^4*c^4*d^4
- 56*a^14*b^3*c^3*d^5 + 28*a^15*b^2*c^2*d^6 - 8*a^16*b*c*d^7 + a^17*d^8))^(3/4)
) - 5*((a^2*b*c^4*d - a^3*c^3*d^2)*x^4 + (a^2*b*c^5 - a^3*c^4*d)*x^2)*sqrt(x)*(-
(28561*b^4*c^4*d^9 - 79092*a*b^3*c^3*d^10 + 82134*a^2*b^2*c^2*d^11 - 37908*a^3*b
*c*d^12 + 6561*a^4*d^13)/(b^8*c^21 - 8*a*b^7*c^20*d + 28*a^2*b^6*c^19*d^2 - 56*a
^3*b^5*c^18*d^3 + 70*a^4*b^4*c^17*d^4 - 56*a^5*b^3*c^16*d^5 + 28*a^6*b^2*c^15*d^
6 - 8*a^7*b*c^14*d^7 + a^8*c^13*d^8))^(1/4)*log((b^6*c^16 - 6*a*b^5*c^15*d + 15*
a^2*b^4*c^14*d^2 - 20*a^3*b^3*c^13*d^3 + 15*a^4*b^2*c^12*d^4 - 6*a^5*b*c^11*d^5
+ a^6*c^10*d^6)*(-(28561*b^4*c^4*d^9 - 79092*a*b^3*c^3*d^10 + 82134*a^2*b^2*c^2*
d^11 - 37908*a^3*b*c*d^12 + 6561*a^4*d^13)/(b^8*c^21 - 8*a*b^7*c^20*d + 28*a^2*b
^6*c^19*d^2 - 56*a^3*b^5*c^18*d^3 + 70*a^4*b^4*c^17*d^4 - 56*a^5*b^3*c^16*d^5 +
28*a^6*b^2*c^15*d^6 - 8*a^7*b*c^14*d^7 + a^8*c^13*d^8))^(3/4) - (2197*b^3*c^3*d^
7 - 4563*a*b^2*c^2*d^8 + 3159*a^2*b*c*d^9 - 729*a^3*d^10)*sqrt(x)) + 5*((a^2*b*c
^4*d - a^3*c^3*d^2)*x^4 + (a^2*b*c^5 - a^3*c^4*d)*x^2)*sqrt(x)*(-(28561*b^4*c^4*
d^9 - 79092*a*b^3*c^3*d^10 + 82134*a^2*b^2*c^2*d^11 - 37908*a^3*b*c*d^12 + 6561*
a^4*d^13)/(b^8*c^21 - 8*a*b^7*c^20*d + 28*a^2*b^6*c^19*d^2 - 56*a^3*b^5*c^18*d^3
+ 70*a^4*b^4*c^17*d^4 - 56*a^5*b^3*c^16*d^5 + 28*a^6*b^2*c^15*d^6 - 8*a^7*b*c^1
4*d^7 + a^8*c^13*d^8))^(1/4)*log(-(b^6*c^16 - 6*a*b^5*c^15*d + 15*a^2*b^4*c^14*d
^2 - 20*a^3*b^3*c^13*d^3 + 15*a^4*b^2*c^12*d^4 - 6*a^5*b*c^11*d^5 + a^6*c^10*d^6
)*(-(28561*b^4*c^4*d^9 - 79092*a*b^3*c^3*d^10 + 82134*a^2*b^2*c^2*d^11 - 37908*a
^3*b*c*d^12 + 6561*a^4*d^13)/(b^8*c^21 - 8*a*b^7*c^20*d + 28*a^2*b^6*c^19*d^2 -
56*a^3*b^5*c^18*d^3 + 70*a^4*b^4*c^17*d^4 - 56*a^5*b^3*c^16*d^5 + 28*a^6*b^2*c^1
5*d^6 - 8*a^7*b*c^14*d^7 + a^8*c^13*d^8))^(3/4) - (2197*b^3*c^3*d^7 - 4563*a*b^2
*c^2*d^8 + 3159*a^2*b*c*d^9 - 729*a^3*d^10)*sqrt(x)))/(((a^2*b*c^4*d - a^3*c^3*d
^2)*x^4 + (a^2*b*c^5 - a^3*c^4*d)*x^2)*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**(7/2)/(b*x**2+a)/(d*x**2+c)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.368401, size = 965, normalized size = 1.56 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)*(d*x^2 + c)^2*x^(7/2)),x, algorithm="giac")

[Out]

-1/2*d^3*x^(3/2)/((b*c^4 - a*c^3*d)*(d*x^2 + c)) + (a*b^3)^(3/4)*b*arctan(1/2*sq
rt(2)*(sqrt(2)*(a/b)^(1/4) + 2*sqrt(x))/(a/b)^(1/4))/(sqrt(2)*a^3*b^2*c^2 - 2*sq
rt(2)*a^4*b*c*d + sqrt(2)*a^5*d^2) + (a*b^3)^(3/4)*b*arctan(-1/2*sqrt(2)*(sqrt(2
)*(a/b)^(1/4) - 2*sqrt(x))/(a/b)^(1/4))/(sqrt(2)*a^3*b^2*c^2 - 2*sqrt(2)*a^4*b*c
*d + sqrt(2)*a^5*d^2) - 1/2*(a*b^3)^(3/4)*b*ln(sqrt(2)*sqrt(x)*(a/b)^(1/4) + x +
sqrt(a/b))/(sqrt(2)*a^3*b^2*c^2 - 2*sqrt(2)*a^4*b*c*d + sqrt(2)*a^5*d^2) + 1/2*
(a*b^3)^(3/4)*b*ln(-sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(sqrt(2)*a^3*b^
2*c^2 - 2*sqrt(2)*a^4*b*c*d + sqrt(2)*a^5*d^2) - 1/4*(13*(c*d^3)^(3/4)*b*c - 9*(
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^6 - 2*sqrt(2)*a*b*c^5*d + sqrt(2)*a^2*c^4*d^2) - 1/4*(13*(c*d
^3)^(3/4)*b*c - 9*(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^6 - 2*sqrt(2)*a*b*c^5*d + sqrt(2)*a^2*c^4
*d^2) + 1/8*(13*(c*d^3)^(3/4)*b*c - 9*(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^6 - 2*sqrt(2)*a*b*c^5*d + sqrt(2)*a^2*c^
4*d^2) - 1/8*(13*(c*d^3)^(3/4)*b*c - 9*(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^6 - 2*sqrt(2)*a*b*c^5*d + sqrt(2)*a^2*
c^4*d^2) + 2/5*(5*b*c*x^2 + 10*a*d*x^2 - a*c)/(a^2*c^3*x^(5/2))