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

Optimal. Leaf size=74 \[ \frac{2 d \sqrt{c+d x^2}}{3 \sqrt{a+b x^2} (b c-a d)^2}-\frac{\sqrt{c+d x^2}}{3 \left (a+b x^2\right )^{3/2} (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.147639, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{2 d \sqrt{c+d x^2}}{3 \sqrt{a+b x^2} (b c-a d)^2}-\frac{\sqrt{c+d x^2}}{3 \left (a+b x^2\right )^{3/2} (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 15.3092, size = 61, normalized size = 0.82 \[ \frac{2 d \sqrt{c + d x^{2}}}{3 \sqrt{a + b x^{2}} \left (a d - b c\right )^{2}} + \frac{\sqrt{c + d x^{2}}}{3 \left (a + b x^{2}\right )^{\frac{3}{2}} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0628408, size = 52, normalized size = 0.7 \[ \frac{\sqrt{c+d x^2} \left (3 a d-b c+2 b d x^2\right )}{3 \left (a+b x^2\right )^{3/2} (b c-a d)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.007, size = 60, normalized size = 0.8 \[{\frac{2\,bd{x}^{2}+3\,ad-bc}{3\,{a}^{2}{d}^{2}-6\,cabd+3\,{b}^{2}{c}^{2}}\sqrt{d{x}^{2}+c} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(d*x^2+c)^(1/2)*(2*b*d*x^2+3*a*d-b*c)/(b*x^2+a)^(3/2)/(a^2*d^2-2*a*b*c*d+b^2
*c^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(x/((b*x^2 + a)^(5/2)*sqrt(d*x^2 + c)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.270516, size = 170, normalized size = 2.3 \[ \frac{{\left (2 \, b d x^{2} - b c + 3 \, a d\right )} \sqrt{b x^{2} + a} \sqrt{d x^{2} + c}}{3 \,{\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2} +{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{4} + 2 \,{\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(2*b*d*x^2 - b*c + 3*a*d)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c)/(a^2*b^2*c^2 - 2*a
^3*b*c*d + a^4*d^2 + (b^4*c^2 - 2*a*b^3*c*d + a^2*b^2*d^2)*x^4 + 2*(a*b^3*c^2 -
2*a^2*b^2*c*d + a^3*b*d^2)*x^2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.239711, size = 174, normalized size = 2.35 \[ \frac{4 \,{\left (b^{2} c - a b d - 3 \,{\left (\sqrt{b x^{2} + a} \sqrt{b d} - \sqrt{b^{2} c +{\left (b x^{2} + a\right )} b d - a b d}\right )}^{2}\right )} \sqrt{b d} b^{2} d}{3 \,{\left (b^{2} c - a b d -{\left (\sqrt{b x^{2} + a} \sqrt{b d} - \sqrt{b^{2} c +{\left (b x^{2} + a\right )} b d - a b d}\right )}^{2}\right )}^{3}{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4/3*(b^2*c - a*b*d - 3*(sqrt(b*x^2 + a)*sqrt(b*d) - sqrt(b^2*c + (b*x^2 + a)*b*d
- a*b*d))^2)*sqrt(b*d)*b^2*d/((b^2*c - a*b*d - (sqrt(b*x^2 + a)*sqrt(b*d) - sqr
t(b^2*c + (b*x^2 + a)*b*d - a*b*d))^2)^3*abs(b))