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

Optimal. Leaf size=122 \[ \frac{b^2 \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{\sqrt{a} (b c-a d)^{5/2}}-\frac{d x (5 b c-2 a d)}{3 c^2 \sqrt{c+d x^2} (b c-a d)^2}-\frac{d x}{3 c \left (c+d x^2\right )^{3/2} (b c-a d)} \]

[Out]

-(d*x)/(3*c*(b*c - a*d)*(c + d*x^2)^(3/2)) - (d*(5*b*c - 2*a*d)*x)/(3*c^2*(b*c -
a*d)^2*Sqrt[c + d*x^2]) + (b^2*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x
^2])])/(Sqrt[a]*(b*c - a*d)^(5/2))

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Rubi [A]  time = 0.270984, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238 \[ \frac{b^2 \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{\sqrt{a} (b c-a d)^{5/2}}-\frac{d x (5 b c-2 a d)}{3 c^2 \sqrt{c+d x^2} (b c-a d)^2}-\frac{d x}{3 c \left (c+d x^2\right )^{3/2} (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

-(d*x)/(3*c*(b*c - a*d)*(c + d*x^2)^(3/2)) - (d*(5*b*c - 2*a*d)*x)/(3*c^2*(b*c -
a*d)^2*Sqrt[c + d*x^2]) + (b^2*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x
^2])])/(Sqrt[a]*(b*c - a*d)^(5/2))

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Rubi in Sympy [A]  time = 55.9503, size = 107, normalized size = 0.88 \[ \frac{d x}{3 c \left (c + d x^{2}\right )^{\frac{3}{2}} \left (a d - b c\right )} + \frac{d x \left (2 a d - 5 b c\right )}{3 c^{2} \sqrt{c + d x^{2}} \left (a d - b c\right )^{2}} + \frac{b^{2} \operatorname{atanh}{\left (\frac{x \sqrt{a d - b c}}{\sqrt{a} \sqrt{c + d x^{2}}} \right )}}{\sqrt{a} \left (a d - b c\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d*x/(3*c*(c + d*x**2)**(3/2)*(a*d - b*c)) + d*x*(2*a*d - 5*b*c)/(3*c**2*sqrt(c +
d*x**2)*(a*d - b*c)**2) + b**2*atanh(x*sqrt(a*d - b*c)/(sqrt(a)*sqrt(c + d*x**2
)))/(sqrt(a)*(a*d - b*c)**(5/2))

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Mathematica [A]  time = 0.285055, size = 111, normalized size = 0.91 \[ \frac{b^2 \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{\sqrt{a} (b c-a d)^{5/2}}+\frac{d x \left (a d \left (3 c+2 d x^2\right )-b c \left (6 c+5 d x^2\right )\right )}{3 c^2 \left (c+d x^2\right )^{3/2} (b c-a d)^2} \]

Antiderivative was successfully verified.

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

[Out]

(d*x*(a*d*(3*c + 2*d*x^2) - b*c*(6*c + 5*d*x^2)))/(3*c^2*(b*c - a*d)^2*(c + d*x^
2)^(3/2)) + (b^2*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(Sqrt[a]
*(b*c - a*d)^(5/2))

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Maple [B]  time = 0.019, size = 1086, normalized size = 8.9 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6/(-a*b)^(1/2)/(a*d-b*c)*b/((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/
b*(-a*b)^(1/2))-(a*d-b*c)/b)^(3/2)+1/6*d/(a*d-b*c)/c/((x-1/b*(-a*b)^(1/2))^2*d+2
*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(3/2)*x+1/3*d/(a*d-b*c)/c^2/
((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(
1/2)*x+1/2/(-a*b)^(1/2)*b^2/(a*d-b*c)^2/((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/
2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)-1/2*b/(a*d-b*c)^2/c/((x-1/b*(-a*b)^
(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)*x*d-1/2/(-
a*b)^(1/2)*b^2/(a*d-b*c)^2/(-(a*d-b*c)/b)^(1/2)*ln((-2*(a*d-b*c)/b+2*d*(-a*b)^(1
/2)/b*(x-1/b*(-a*b)^(1/2))+2*(-(a*d-b*c)/b)^(1/2)*((x-1/b*(-a*b)^(1/2))^2*d+2*d*
(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))/(x-1/b*(-a*b)^(1/2)))+1/
6/(-a*b)^(1/2)/(a*d-b*c)*b/((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(
-a*b)^(1/2))-(a*d-b*c)/b)^(3/2)+1/6*d/(a*d-b*c)/c/((x+1/b*(-a*b)^(1/2))^2*d-2*d*
(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(3/2)*x+1/3*d/(a*d-b*c)/c^2/((x
+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2
)*x-1/2/(-a*b)^(1/2)*b^2/(a*d-b*c)^2/((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/
b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)-1/2*b/(a*d-b*c)^2/c/((x+1/b*(-a*b)^(1/
2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)*x*d+1/2/(-a*b
)^(1/2)*b^2/(a*d-b*c)^2/(-(a*d-b*c)/b)^(1/2)*ln((-2*(a*d-b*c)/b-2*d*(-a*b)^(1/2)
/b*(x+1/b*(-a*b)^(1/2))+2*(-(a*d-b*c)/b)^(1/2)*((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a
*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))/(x+1/b*(-a*b)^(1/2)))

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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)^(5/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.564618, size = 1, normalized size = 0.01 \[ \left [-\frac{4 \,{\left ({\left (5 \, b c d^{2} - 2 \, a d^{3}\right )} x^{3} + 3 \,{\left (2 \, b c^{2} d - a c d^{2}\right )} x\right )} \sqrt{-a b c + a^{2} d} \sqrt{d x^{2} + c} - 3 \,{\left (b^{2} c^{2} d^{2} x^{4} + 2 \, b^{2} c^{3} d x^{2} + b^{2} c^{4}\right )} \log \left (\frac{{\left ({\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} - 2 \,{\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{2}\right )} \sqrt{-a b c + a^{2} d} + 4 \,{\left ({\left (a b^{2} c^{2} - 3 \, a^{2} b c d + 2 \, a^{3} d^{2}\right )} x^{3} -{\left (a^{2} b c^{2} - a^{3} c d\right )} x\right )} \sqrt{d x^{2} + c}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right )}{12 \,{\left (b^{2} c^{6} - 2 \, a b c^{5} d + a^{2} c^{4} d^{2} +{\left (b^{2} c^{4} d^{2} - 2 \, a b c^{3} d^{3} + a^{2} c^{2} d^{4}\right )} x^{4} + 2 \,{\left (b^{2} c^{5} d - 2 \, a b c^{4} d^{2} + a^{2} c^{3} d^{3}\right )} x^{2}\right )} \sqrt{-a b c + a^{2} d}}, -\frac{2 \,{\left ({\left (5 \, b c d^{2} - 2 \, a d^{3}\right )} x^{3} + 3 \,{\left (2 \, b c^{2} d - a c d^{2}\right )} x\right )} \sqrt{a b c - a^{2} d} \sqrt{d x^{2} + c} - 3 \,{\left (b^{2} c^{2} d^{2} x^{4} + 2 \, b^{2} c^{3} d x^{2} + b^{2} c^{4}\right )} \arctan \left (\frac{{\left (b c - 2 \, a d\right )} x^{2} - a c}{2 \, \sqrt{a b c - a^{2} d} \sqrt{d x^{2} + c} x}\right )}{6 \,{\left (b^{2} c^{6} - 2 \, a b c^{5} d + a^{2} c^{4} d^{2} +{\left (b^{2} c^{4} d^{2} - 2 \, a b c^{3} d^{3} + a^{2} c^{2} d^{4}\right )} x^{4} + 2 \,{\left (b^{2} c^{5} d - 2 \, a b c^{4} d^{2} + a^{2} c^{3} d^{3}\right )} x^{2}\right )} \sqrt{a b c - a^{2} d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/12*(4*((5*b*c*d^2 - 2*a*d^3)*x^3 + 3*(2*b*c^2*d - a*c*d^2)*x)*sqrt(-a*b*c +
a^2*d)*sqrt(d*x^2 + c) - 3*(b^2*c^2*d^2*x^4 + 2*b^2*c^3*d*x^2 + b^2*c^4)*log((((
b^2*c^2 - 8*a*b*c*d + 8*a^2*d^2)*x^4 + a^2*c^2 - 2*(3*a*b*c^2 - 4*a^2*c*d)*x^2)*
sqrt(-a*b*c + a^2*d) + 4*((a*b^2*c^2 - 3*a^2*b*c*d + 2*a^3*d^2)*x^3 - (a^2*b*c^2
- a^3*c*d)*x)*sqrt(d*x^2 + c))/(b^2*x^4 + 2*a*b*x^2 + a^2)))/((b^2*c^6 - 2*a*b*
c^5*d + a^2*c^4*d^2 + (b^2*c^4*d^2 - 2*a*b*c^3*d^3 + a^2*c^2*d^4)*x^4 + 2*(b^2*c
^5*d - 2*a*b*c^4*d^2 + a^2*c^3*d^3)*x^2)*sqrt(-a*b*c + a^2*d)), -1/6*(2*((5*b*c*
d^2 - 2*a*d^3)*x^3 + 3*(2*b*c^2*d - a*c*d^2)*x)*sqrt(a*b*c - a^2*d)*sqrt(d*x^2 +
c) - 3*(b^2*c^2*d^2*x^4 + 2*b^2*c^3*d*x^2 + b^2*c^4)*arctan(1/2*((b*c - 2*a*d)*
x^2 - a*c)/(sqrt(a*b*c - a^2*d)*sqrt(d*x^2 + c)*x)))/((b^2*c^6 - 2*a*b*c^5*d + a
^2*c^4*d^2 + (b^2*c^4*d^2 - 2*a*b*c^3*d^3 + a^2*c^2*d^4)*x^4 + 2*(b^2*c^5*d - 2*
a*b*c^4*d^2 + a^2*c^3*d^3)*x^2)*sqrt(a*b*c - a^2*d))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (a + b x^{2}\right ) \left (c + d x^{2}\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.242996, size = 433, normalized size = 3.55 \[ -\frac{b^{2} \sqrt{d} \arctan \left (\frac{{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt{a b c d - a^{2} d^{2}}}\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt{a b c d - a^{2} d^{2}}} - \frac{{\left (\frac{{\left (5 \, b^{3} c^{3} d^{3} - 12 \, a b^{2} c^{2} d^{4} + 9 \, a^{2} b c d^{5} - 2 \, a^{3} d^{6}\right )} x^{2}}{b^{4} c^{6} d - 4 \, a b^{3} c^{5} d^{2} + 6 \, a^{2} b^{2} c^{4} d^{3} - 4 \, a^{3} b c^{3} d^{4} + a^{4} c^{2} d^{5}} + \frac{3 \,{\left (2 \, b^{3} c^{4} d^{2} - 5 \, a b^{2} c^{3} d^{3} + 4 \, a^{2} b c^{2} d^{4} - a^{3} c d^{5}\right )}}{b^{4} c^{6} d - 4 \, a b^{3} c^{5} d^{2} + 6 \, a^{2} b^{2} c^{4} d^{3} - 4 \, a^{3} b c^{3} d^{4} + a^{4} c^{2} d^{5}}\right )} x}{3 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-b^2*sqrt(d)*arctan(1/2*((sqrt(d)*x - sqrt(d*x^2 + c))^2*b - b*c + 2*a*d)/sqrt(a
*b*c*d - a^2*d^2))/((b^2*c^2 - 2*a*b*c*d + a^2*d^2)*sqrt(a*b*c*d - a^2*d^2)) - 1
/3*((5*b^3*c^3*d^3 - 12*a*b^2*c^2*d^4 + 9*a^2*b*c*d^5 - 2*a^3*d^6)*x^2/(b^4*c^6*
d - 4*a*b^3*c^5*d^2 + 6*a^2*b^2*c^4*d^3 - 4*a^3*b*c^3*d^4 + a^4*c^2*d^5) + 3*(2*
b^3*c^4*d^2 - 5*a*b^2*c^3*d^3 + 4*a^2*b*c^2*d^4 - a^3*c*d^5)/(b^4*c^6*d - 4*a*b^
3*c^5*d^2 + 6*a^2*b^2*c^4*d^3 - 4*a^3*b*c^3*d^4 + a^4*c^2*d^5))*x/(d*x^2 + c)^(3
/2)