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

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

[Out]

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

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Rubi [A]  time = 0.345895, 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 \[ \frac{b x (2 b c-5 a d)}{3 a^2 \sqrt{a+b x^2} (b c-a d)^2}+\frac{d^2 \tanh ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{c} \sqrt{a+b x^2}}\right )}{\sqrt{c} (b c-a d)^{5/2}}+\frac{b x}{3 a \left (a+b x^2\right )^{3/2} (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 55.144, size = 107, normalized size = 0.88 \[ \frac{d^{2} \operatorname{atan}{\left (\frac{x \sqrt{a d - b c}}{\sqrt{c} \sqrt{a + b x^{2}}} \right )}}{\sqrt{c} \left (a d - b c\right )^{\frac{5}{2}}} - \frac{b x}{3 a \left (a + b x^{2}\right )^{\frac{3}{2}} \left (a d - b c\right )} - \frac{b x \left (5 a d - 2 b c\right )}{3 a^{2} \sqrt{a + b x^{2}} \left (a d - b c\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d**2*atan(x*sqrt(a*d - b*c)/(sqrt(c)*sqrt(a + b*x**2)))/(sqrt(c)*(a*d - b*c)**(5
/2)) - b*x/(3*a*(a + b*x**2)**(3/2)*(a*d - b*c)) - b*x*(5*a*d - 2*b*c)/(3*a**2*s
qrt(a + b*x**2)*(a*d - b*c)**2)

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Mathematica [A]  time = 0.360047, size = 112, normalized size = 0.92 \[ \frac{b x \left (-6 a^2 d+a b \left (3 c-5 d x^2\right )+2 b^2 c x^2\right )}{3 a^2 \left (a+b x^2\right )^{3/2} (b c-a d)^2}+\frac{d^2 \tan ^{-1}\left (\frac{x \sqrt{a d-b c}}{\sqrt{c} \sqrt{a+b x^2}}\right )}{\sqrt{c} (a d-b c)^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.027, size = 1070, normalized size = 8.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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)^(5/2)*(d*x^2 + c)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.546142, size = 1, normalized size = 0.01 \[ \left [\frac{4 \,{\left ({\left (2 \, b^{3} c - 5 \, a b^{2} d\right )} x^{3} + 3 \,{\left (a b^{2} c - 2 \, a^{2} b d\right )} x\right )} \sqrt{b c^{2} - a c d} \sqrt{b x^{2} + a} + 3 \,{\left (a^{2} b^{2} d^{2} x^{4} + 2 \, a^{3} b d^{2} x^{2} + a^{4} d^{2}\right )} \log \left (\frac{{\left ({\left (8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} + 2 \,{\left (4 \, a b c^{2} - 3 \, a^{2} c d\right )} x^{2}\right )} \sqrt{b c^{2} - a c d} + 4 \,{\left ({\left (2 \, b^{2} c^{3} - 3 \, a b c^{2} d + a^{2} c d^{2}\right )} x^{3} +{\left (a b c^{3} - a^{2} c^{2} d\right )} x\right )} \sqrt{b x^{2} + a}}{d^{2} x^{4} + 2 \, c d x^{2} + c^{2}}\right )}{12 \,{\left (a^{4} b^{2} c^{2} - 2 \, a^{5} b c d + a^{6} d^{2} +{\left (a^{2} b^{4} c^{2} - 2 \, a^{3} b^{3} c d + a^{4} b^{2} d^{2}\right )} x^{4} + 2 \,{\left (a^{3} b^{3} c^{2} - 2 \, a^{4} b^{2} c d + a^{5} b d^{2}\right )} x^{2}\right )} \sqrt{b c^{2} - a c d}}, \frac{2 \,{\left ({\left (2 \, b^{3} c - 5 \, a b^{2} d\right )} x^{3} + 3 \,{\left (a b^{2} c - 2 \, a^{2} b d\right )} x\right )} \sqrt{-b c^{2} + a c d} \sqrt{b x^{2} + a} + 3 \,{\left (a^{2} b^{2} d^{2} x^{4} + 2 \, a^{3} b d^{2} x^{2} + a^{4} d^{2}\right )} \arctan \left (\frac{\sqrt{-b c^{2} + a c d}{\left ({\left (2 \, b c - a d\right )} x^{2} + a c\right )}}{2 \,{\left (b c^{2} - a c d\right )} \sqrt{b x^{2} + a} x}\right )}{6 \,{\left (a^{4} b^{2} c^{2} - 2 \, a^{5} b c d + a^{6} d^{2} +{\left (a^{2} b^{4} c^{2} - 2 \, a^{3} b^{3} c d + a^{4} b^{2} d^{2}\right )} x^{4} + 2 \,{\left (a^{3} b^{3} c^{2} - 2 \, a^{4} b^{2} c d + a^{5} b d^{2}\right )} x^{2}\right )} \sqrt{-b c^{2} + a c d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.229209, size = 432, normalized size = 3.54 \[ -\frac{\sqrt{b} d^{2} \arctan \left (\frac{{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} d + 2 \, b c - a d}{2 \, \sqrt{-b^{2} c^{2} + a b c d}}\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt{-b^{2} c^{2} + a b c d}} + \frac{{\left (\frac{{\left (2 \, b^{6} c^{3} - 9 \, a b^{5} c^{2} d + 12 \, a^{2} b^{4} c d^{2} - 5 \, a^{3} b^{3} d^{3}\right )} x^{2}}{a^{2} b^{5} c^{4} - 4 \, a^{3} b^{4} c^{3} d + 6 \, a^{4} b^{3} c^{2} d^{2} - 4 \, a^{5} b^{2} c d^{3} + a^{6} b d^{4}} + \frac{3 \,{\left (a b^{5} c^{3} - 4 \, a^{2} b^{4} c^{2} d + 5 \, a^{3} b^{3} c d^{2} - 2 \, a^{4} b^{2} d^{3}\right )}}{a^{2} b^{5} c^{4} - 4 \, a^{3} b^{4} c^{3} d + 6 \, a^{4} b^{3} c^{2} d^{2} - 4 \, a^{5} b^{2} c d^{3} + a^{6} b d^{4}}\right )} x}{3 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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