3.137 \(\int \frac{\left (3 a+b x^2\right )^2}{\left (a-b x^2\right )^{7/3}} \, dx\)

Optimal. Leaf size=44 \[ \frac{3 x \left (3 a+b x^2\right )}{2 \left (a-b x^2\right )^{4/3}}+\frac{9 x}{2 \sqrt [3]{a-b x^2}} \]

[Out]

(9*x)/(2*(a - b*x^2)^(1/3)) + (3*x*(3*a + b*x^2))/(2*(a - b*x^2)^(4/3))

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Rubi [A]  time = 0.0511061, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{3 x \left (3 a+b x^2\right )}{2 \left (a-b x^2\right )^{4/3}}+\frac{9 x}{2 \sqrt [3]{a-b x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(3*a + b*x^2)^2/(a - b*x^2)^(7/3),x]

[Out]

(9*x)/(2*(a - b*x^2)^(1/3)) + (3*x*(3*a + b*x^2))/(2*(a - b*x^2)^(4/3))

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Rubi in Sympy [A]  time = 21.476, size = 37, normalized size = 0.84 \[ \frac{9 x}{2 \sqrt [3]{a - b x^{2}}} + \frac{3 x \left (3 a + b x^{2}\right )}{2 \left (a - b x^{2}\right )^{\frac{4}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+3*a)**2/(-b*x**2+a)**(7/3),x)

[Out]

9*x/(2*(a - b*x**2)**(1/3)) + 3*x*(3*a + b*x**2)/(2*(a - b*x**2)**(4/3))

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Mathematica [A]  time = 0.0381237, size = 24, normalized size = 0.55 \[ \frac{9 a x-3 b x^3}{\left (a-b x^2\right )^{4/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(3*a + b*x^2)^2/(a - b*x^2)^(7/3),x]

[Out]

(9*a*x - 3*b*x^3)/(a - b*x^2)^(4/3)

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Maple [A]  time = 0.008, size = 24, normalized size = 0.6 \[ 3\,{\frac{x \left ( -b{x}^{2}+3\,a \right ) }{ \left ( -b{x}^{2}+a \right ) ^{4/3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+3*a)^2/(-b*x^2+a)^(7/3),x)

[Out]

3/(-b*x^2+a)^(4/3)*x*(-b*x^2+3*a)

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Maxima [A]  time = 1.48278, size = 45, normalized size = 1.02 \[ \frac{3 \,{\left (b x^{3} - 3 \, a x\right )}}{{\left (b x^{2} - a\right )}{\left (-b x^{2} + a\right )}^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + 3*a)^2/(-b*x^2 + a)^(7/3),x, algorithm="maxima")

[Out]

3*(b*x^3 - 3*a*x)/((b*x^2 - a)*(-b*x^2 + a)^(1/3))

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Fricas [A]  time = 0.261818, size = 57, normalized size = 1.3 \[ -\frac{3 \,{\left (b x^{3} - 3 \, a x\right )}{\left (-b x^{2} + a\right )}^{\frac{2}{3}}}{b^{2} x^{4} - 2 \, a b x^{2} + a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + 3*a)^2/(-b*x^2 + a)^(7/3),x, algorithm="fricas")

[Out]

-3*(b*x^3 - 3*a*x)*(-b*x^2 + a)^(2/3)/(b^2*x^4 - 2*a*b*x^2 + a^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (3 a + b x^{2}\right )^{2}}{\left (a - b x^{2}\right )^{\frac{7}{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+3*a)**2/(-b*x**2+a)**(7/3),x)

[Out]

Integral((3*a + b*x**2)**2/(a - b*x**2)**(7/3), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + 3 \, a\right )}^{2}}{{\left (-b x^{2} + a\right )}^{\frac{7}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + 3*a)^2/(-b*x^2 + a)^(7/3),x, algorithm="giac")

[Out]

integrate((b*x^2 + 3*a)^2/(-b*x^2 + a)^(7/3), x)