3.984 \(\int \frac{x^3}{\left (a+b x^2\right )^{5/2} \sqrt{c+d x^2}} \, dx\)

Optimal. Leaf size=89 \[ \frac{a \sqrt{c+d x^2}}{3 b \left (a+b x^2\right )^{3/2} (b c-a d)}-\frac{\sqrt{c+d x^2} (3 b c-a d)}{3 b \sqrt{a+b x^2} (b c-a d)^2} \]

[Out]

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

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Rubi [A]  time = 0.21821, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ \frac{a \sqrt{c+d x^2}}{3 b \left (a+b x^2\right )^{3/2} (b c-a d)}-\frac{\sqrt{c+d x^2} (3 b c-a d)}{3 b \sqrt{a+b x^2} (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]  Int[x^3/((a + b*x^2)^(5/2)*Sqrt[c + d*x^2]),x]

[Out]

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

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Rubi in Sympy [A]  time = 21.1663, size = 71, normalized size = 0.8 \[ - \frac{a \sqrt{c + d x^{2}}}{3 b \left (a + b x^{2}\right )^{\frac{3}{2}} \left (a d - b c\right )} + \frac{\sqrt{c + d x^{2}} \left (a d - 3 b c\right )}{3 b \sqrt{a + b x^{2}} \left (a d - b c\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3/(b*x**2+a)**(5/2)/(d*x**2+c)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.0882337, size = 54, normalized size = 0.61 \[ \frac{\sqrt{c+d x^2} \left (-2 a c+a d x^2-3 b c x^2\right )}{3 \left (a+b x^2\right )^{3/2} (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3/((a + b*x^2)^(5/2)*Sqrt[c + d*x^2]),x]

[Out]

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

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Maple [A]  time = 0.009, size = 63, normalized size = 0.7 \[ -{\frac{-ad{x}^{2}+3\,c{x}^{2}b+2\,ac}{3\,{a}^{2}{d}^{2}-6\,cabd+3\,{b}^{2}{c}^{2}}\sqrt{d{x}^{2}+c} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(b*x^2+a)^(5/2)/(d*x^2+c)^(1/2),x)

[Out]

-1/3*(d*x^2+c)^(1/2)*(-a*d*x^2+3*b*c*x^2+2*a*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^3/((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 = 0.267402, size = 173, normalized size = 1.94 \[ -\frac{{\left ({\left (3 \, b c - a d\right )} x^{2} + 2 \, a c\right )} \sqrt{b x^{2} + a} \sqrt{d x^{2} + c}}{3 \,{\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2} +{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{4} + 2 \,{\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/((b*x^2 + a)^(5/2)*sqrt(d*x^2 + c)),x, algorithm="fricas")

[Out]

-1/3*((3*b*c - a*d)*x^2 + 2*a*c)*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. \[ \int \frac{x^{3}}{\left (a + b x^{2}\right )^{\frac{5}{2}} \sqrt{c + d x^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3/(b*x**2+a)**(5/2)/(d*x**2+c)**(1/2),x)

[Out]

Integral(x**3/((a + b*x**2)**(5/2)*sqrt(c + d*x**2)), x)

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GIAC/XCAS [A]  time = 0.257286, size = 289, normalized size = 3.25 \[ -\frac{2 \,{\left (3 \, \sqrt{b d} b^{5} c^{2} - 4 \, \sqrt{b d} a b^{4} c d + \sqrt{b d} a^{2} b^{3} d^{2} - 6 \, \sqrt{b d}{\left (\sqrt{b x^{2} + a} \sqrt{b d} - \sqrt{b^{2} c +{\left (b x^{2} + a\right )} b d - a b d}\right )}^{2} b^{3} c + 3 \, \sqrt{b d}{\left (\sqrt{b x^{2} + a} \sqrt{b d} - \sqrt{b^{2} c +{\left (b x^{2} + a\right )} b d - a b d}\right )}^{4} b\right )}}{3 \,{\left (b^{2} c - a b d -{\left (\sqrt{b x^{2} + a} \sqrt{b d} - \sqrt{b^{2} c +{\left (b x^{2} + a\right )} b d - a b d}\right )}^{2}\right )}^{3} b{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/((b*x^2 + a)^(5/2)*sqrt(d*x^2 + c)),x, algorithm="giac")

[Out]

-2/3*(3*sqrt(b*d)*b^5*c^2 - 4*sqrt(b*d)*a*b^4*c*d + sqrt(b*d)*a^2*b^3*d^2 - 6*sq
rt(b*d)*(sqrt(b*x^2 + a)*sqrt(b*d) - sqrt(b^2*c + (b*x^2 + a)*b*d - a*b*d))^2*b^
3*c + 3*sqrt(b*d)*(sqrt(b*x^2 + a)*sqrt(b*d) - sqrt(b^2*c + (b*x^2 + a)*b*d - a*
b*d))^4*b)/((b^2*c - a*b*d - (sqrt(b*x^2 + a)*sqrt(b*d) - sqrt(b^2*c + (b*x^2 +
a)*b*d - a*b*d))^2)^3*b*abs(b))