3.772 \(\int \frac{(c x)^{13/3}}{\left (a+b x^2\right )^{2/3}} \, dx\)

Optimal. Leaf size=247 \[ -\frac{5 a^2 c^{13/3} \log \left (c^{2/3}-\frac{\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{18 b^{8/3}}+\frac{5 a^2 c^{13/3} \log \left (\frac{b^{2/3} (c x)^{4/3}}{\left (a+b x^2\right )^{2/3}}+\frac{\sqrt [3]{b} c^{2/3} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}+c^{4/3}\right )}{36 b^{8/3}}-\frac{5 a^2 c^{13/3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}+c^{2/3}}{\sqrt{3} c^{2/3}}\right )}{6 \sqrt{3} b^{8/3}}-\frac{5 a c^3 (c x)^{4/3} \sqrt [3]{a+b x^2}}{12 b^2}+\frac{c (c x)^{10/3} \sqrt [3]{a+b x^2}}{4 b} \]

[Out]

(-5*a*c^3*(c*x)^(4/3)*(a + b*x^2)^(1/3))/(12*b^2) + (c*(c*x)^(10/3)*(a + b*x^2)^
(1/3))/(4*b) - (5*a^2*c^(13/3)*ArcTan[(c^(2/3) + (2*b^(1/3)*(c*x)^(2/3))/(a + b*
x^2)^(1/3))/(Sqrt[3]*c^(2/3))])/(6*Sqrt[3]*b^(8/3)) - (5*a^2*c^(13/3)*Log[c^(2/3
) - (b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3)])/(18*b^(8/3)) + (5*a^2*c^(13/3)*Log
[c^(4/3) + (b^(2/3)*(c*x)^(4/3))/(a + b*x^2)^(2/3) + (b^(1/3)*c^(2/3)*(c*x)^(2/3
))/(a + b*x^2)^(1/3)])/(36*b^(8/3))

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Rubi [A]  time = 0.677233, antiderivative size = 247, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 10, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.526 \[ -\frac{5 a^2 c^{13/3} \log \left (c^{2/3}-\frac{\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{18 b^{8/3}}+\frac{5 a^2 c^{13/3} \log \left (\frac{b^{2/3} (c x)^{4/3}}{\left (a+b x^2\right )^{2/3}}+\frac{\sqrt [3]{b} c^{2/3} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}+c^{4/3}\right )}{36 b^{8/3}}-\frac{5 a^2 c^{13/3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}+c^{2/3}}{\sqrt{3} c^{2/3}}\right )}{6 \sqrt{3} b^{8/3}}-\frac{5 a c^3 (c x)^{4/3} \sqrt [3]{a+b x^2}}{12 b^2}+\frac{c (c x)^{10/3} \sqrt [3]{a+b x^2}}{4 b} \]

Antiderivative was successfully verified.

[In]  Int[(c*x)^(13/3)/(a + b*x^2)^(2/3),x]

[Out]

(-5*a*c^3*(c*x)^(4/3)*(a + b*x^2)^(1/3))/(12*b^2) + (c*(c*x)^(10/3)*(a + b*x^2)^
(1/3))/(4*b) - (5*a^2*c^(13/3)*ArcTan[(c^(2/3) + (2*b^(1/3)*(c*x)^(2/3))/(a + b*
x^2)^(1/3))/(Sqrt[3]*c^(2/3))])/(6*Sqrt[3]*b^(8/3)) - (5*a^2*c^(13/3)*Log[c^(2/3
) - (b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3)])/(18*b^(8/3)) + (5*a^2*c^(13/3)*Log
[c^(4/3) + (b^(2/3)*(c*x)^(4/3))/(a + b*x^2)^(2/3) + (b^(1/3)*c^(2/3)*(c*x)^(2/3
))/(a + b*x^2)^(1/3)])/(36*b^(8/3))

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Rubi in Sympy [A]  time = 67.4719, size = 236, normalized size = 0.96 \[ - \frac{5 a^{2} c^{\frac{13}{3}} \log{\left (- \frac{\sqrt [3]{b} \left (c x\right )^{\frac{2}{3}}}{\sqrt [3]{a + b x^{2}}} + c^{\frac{2}{3}} \right )}}{18 b^{\frac{8}{3}}} + \frac{5 a^{2} c^{\frac{13}{3}} \log{\left (\frac{b^{\frac{2}{3}} \left (c x\right )^{\frac{4}{3}}}{c^{\frac{4}{3}} \left (a + b x^{2}\right )^{\frac{2}{3}}} + \frac{\sqrt [3]{b} \left (c x\right )^{\frac{2}{3}}}{c^{\frac{2}{3}} \sqrt [3]{a + b x^{2}}} + 1 \right )}}{36 b^{\frac{8}{3}}} - \frac{5 \sqrt{3} a^{2} c^{\frac{13}{3}} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{2 \sqrt [3]{b} \left (c x\right )^{\frac{2}{3}}}{3 \sqrt [3]{a + b x^{2}}} + \frac{c^{\frac{2}{3}}}{3}\right )}{c^{\frac{2}{3}}} \right )}}{18 b^{\frac{8}{3}}} - \frac{5 a c^{3} \left (c x\right )^{\frac{4}{3}} \sqrt [3]{a + b x^{2}}}{12 b^{2}} + \frac{c \left (c x\right )^{\frac{10}{3}} \sqrt [3]{a + b x^{2}}}{4 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x)**(13/3)/(b*x**2+a)**(2/3),x)

[Out]

-5*a**2*c**(13/3)*log(-b**(1/3)*(c*x)**(2/3)/(a + b*x**2)**(1/3) + c**(2/3))/(18
*b**(8/3)) + 5*a**2*c**(13/3)*log(b**(2/3)*(c*x)**(4/3)/(c**(4/3)*(a + b*x**2)**
(2/3)) + b**(1/3)*(c*x)**(2/3)/(c**(2/3)*(a + b*x**2)**(1/3)) + 1)/(36*b**(8/3))
- 5*sqrt(3)*a**2*c**(13/3)*atan(sqrt(3)*(2*b**(1/3)*(c*x)**(2/3)/(3*(a + b*x**2
)**(1/3)) + c**(2/3)/3)/c**(2/3))/(18*b**(8/3)) - 5*a*c**3*(c*x)**(4/3)*(a + b*x
**2)**(1/3)/(12*b**2) + c*(c*x)**(10/3)*(a + b*x**2)**(1/3)/(4*b)

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Mathematica [C]  time = 0.067104, size = 87, normalized size = 0.35 \[ \frac{c^3 (c x)^{4/3} \left (5 a^2 \left (\frac{b x^2}{a}+1\right )^{2/3} \, _2F_1\left (\frac{2}{3},\frac{2}{3};\frac{5}{3};-\frac{b x^2}{a}\right )-5 a^2-2 a b x^2+3 b^2 x^4\right )}{12 b^2 \left (a+b x^2\right )^{2/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(c*x)^(13/3)/(a + b*x^2)^(2/3),x]

[Out]

(c^3*(c*x)^(4/3)*(-5*a^2 - 2*a*b*x^2 + 3*b^2*x^4 + 5*a^2*(1 + (b*x^2)/a)^(2/3)*H
ypergeometric2F1[2/3, 2/3, 5/3, -((b*x^2)/a)]))/(12*b^2*(a + b*x^2)^(2/3))

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Maple [F]  time = 0.036, size = 0, normalized size = 0. \[ \int{1 \left ( cx \right ) ^{{\frac{13}{3}}} \left ( b{x}^{2}+a \right ) ^{-{\frac{2}{3}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x)^(13/3)/(b*x^2+a)^(2/3),x)

[Out]

int((c*x)^(13/3)/(b*x^2+a)^(2/3),x)

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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((c*x)^(13/3)/(b*x^2 + a)^(2/3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x)^(13/3)/(b*x^2 + a)^(2/3),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x)**(13/3)/(b*x**2+a)**(2/3),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (c x\right )^{\frac{13}{3}}}{{\left (b x^{2} + a\right )}^{\frac{2}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x)^(13/3)/(b*x^2 + a)^(2/3),x, algorithm="giac")

[Out]

integrate((c*x)^(13/3)/(b*x^2 + a)^(2/3), x)