3.747 \(\int (c x)^{4/3} \sqrt [3]{a+b x^2} \, dx\)

Optimal. Leaf size=418 \[ -\frac{a \sqrt [3]{c} \sqrt [3]{c x} \sqrt [3]{a+b x^2} \left (c^{2/3}-\frac{\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right ) \sqrt{\frac{\frac{b^{2/3} (c x)^{4/3}}{\left (a+b x^2\right )^{2/3}}+\frac{\sqrt [3]{b} c^{2/3} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}+c^{4/3}}{\left (c^{2/3}-\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )^2}} F\left (\cos ^{-1}\left (\frac{c^{2/3}-\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{b x^2+a}}}{c^{2/3}-\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{b x^2+a}}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{9 \sqrt [4]{3} b \sqrt{-\frac{\sqrt [3]{b} (c x)^{2/3} \left (c^{2/3}-\frac{\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{\sqrt [3]{a+b x^2} \left (c^{2/3}-\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )^2}}}+\frac{(c x)^{7/3} \sqrt [3]{a+b x^2}}{3 c}+\frac{2 a c \sqrt [3]{c x} \sqrt [3]{a+b x^2}}{9 b} \]

[Out]

(2*a*c*(c*x)^(1/3)*(a + b*x^2)^(1/3))/(9*b) + ((c*x)^(7/3)*(a + b*x^2)^(1/3))/(3
*c) - (a*c^(1/3)*(c*x)^(1/3)*(a + b*x^2)^(1/3)*(c^(2/3) - (b^(1/3)*(c*x)^(2/3))/
(a + b*x^2)^(1/3))*Sqrt[(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))/(c^(2/3) - ((1 + Sqrt[3])*b^(1/3)*
(c*x)^(2/3))/(a + b*x^2)^(1/3))^2]*EllipticF[ArcCos[(c^(2/3) - ((1 - Sqrt[3])*b^
(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))/(c^(2/3) - ((1 + Sqrt[3])*b^(1/3)*(c*x)^(2
/3))/(a + b*x^2)^(1/3))], (2 + Sqrt[3])/4])/(9*3^(1/4)*b*Sqrt[-((b^(1/3)*(c*x)^(
2/3)*(c^(2/3) - (b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3)))/((a + b*x^2)^(1/3)*(c^
(2/3) - ((1 + Sqrt[3])*b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))^2))])

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Rubi [A]  time = 1.53783, antiderivative size = 418, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263 \[ -\frac{a \sqrt [3]{c} \sqrt [3]{c x} \sqrt [3]{a+b x^2} \left (c^{2/3}-\frac{\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right ) \sqrt{\frac{\frac{b^{2/3} (c x)^{4/3}}{\left (a+b x^2\right )^{2/3}}+\frac{\sqrt [3]{b} c^{2/3} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}+c^{4/3}}{\left (c^{2/3}-\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )^2}} F\left (\cos ^{-1}\left (\frac{c^{2/3}-\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{b x^2+a}}}{c^{2/3}-\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{b x^2+a}}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{9 \sqrt [4]{3} b \sqrt{-\frac{\sqrt [3]{b} (c x)^{2/3} \left (c^{2/3}-\frac{\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{\sqrt [3]{a+b x^2} \left (c^{2/3}-\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )^2}}}+\frac{(c x)^{7/3} \sqrt [3]{a+b x^2}}{3 c}+\frac{2 a c \sqrt [3]{c x} \sqrt [3]{a+b x^2}}{9 b} \]

Antiderivative was successfully verified.

[In]  Int[(c*x)^(4/3)*(a + b*x^2)^(1/3),x]

[Out]

(2*a*c*(c*x)^(1/3)*(a + b*x^2)^(1/3))/(9*b) + ((c*x)^(7/3)*(a + b*x^2)^(1/3))/(3
*c) - (a*c^(1/3)*(c*x)^(1/3)*(a + b*x^2)^(1/3)*(c^(2/3) - (b^(1/3)*(c*x)^(2/3))/
(a + b*x^2)^(1/3))*Sqrt[(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))/(c^(2/3) - ((1 + Sqrt[3])*b^(1/3)*
(c*x)^(2/3))/(a + b*x^2)^(1/3))^2]*EllipticF[ArcCos[(c^(2/3) - ((1 - Sqrt[3])*b^
(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))/(c^(2/3) - ((1 + Sqrt[3])*b^(1/3)*(c*x)^(2
/3))/(a + b*x^2)^(1/3))], (2 + Sqrt[3])/4])/(9*3^(1/4)*b*Sqrt[-((b^(1/3)*(c*x)^(
2/3)*(c^(2/3) - (b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3)))/((a + b*x^2)^(1/3)*(c^
(2/3) - ((1 + Sqrt[3])*b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))^2))])

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Rubi in Sympy [A]  time = 35.618, size = 403, normalized size = 0.96 \[ - \frac{3^{\frac{3}{4}} a^{2} \sqrt [3]{c} \sqrt [3]{c x} \sqrt{\frac{\frac{b^{\frac{2}{3}} \left (c x\right )^{\frac{4}{3}}}{\left (a + b x^{2}\right )^{\frac{2}{3}}} + \frac{\sqrt [3]{b} c^{\frac{2}{3}} \left (c x\right )^{\frac{2}{3}}}{\sqrt [3]{a + b x^{2}}} + c^{\frac{4}{3}}}{\left (\frac{\sqrt [3]{b} \left (c x\right )^{\frac{2}{3}} \left (- \sqrt{3} - 1\right )}{\sqrt [3]{a + b x^{2}}} + c^{\frac{2}{3}}\right )^{2}}} \left (- \frac{\sqrt [3]{b} \left (c x\right )^{\frac{2}{3}}}{\sqrt [3]{a + b x^{2}}} + c^{\frac{2}{3}}\right ) F\left (\operatorname{acos}{\left (\frac{\frac{\sqrt [3]{b} \left (c x\right )^{\frac{2}{3}} \left (-1 + \sqrt{3}\right )}{\sqrt [3]{a + b x^{2}}} + c^{\frac{2}{3}}}{\frac{\sqrt [3]{b} \left (c x\right )^{\frac{2}{3}} \left (- \sqrt{3} - 1\right )}{\sqrt [3]{a + b x^{2}}} + c^{\frac{2}{3}}} \right )}\middle | \frac{\sqrt{3}}{4} + \frac{1}{2}\right )}{27 b \sqrt{\frac{a}{a + b x^{2}}} \sqrt{- \frac{\sqrt [3]{b} \left (c x\right )^{\frac{2}{3}} \left (- \frac{\sqrt [3]{b} \left (c x\right )^{\frac{2}{3}}}{\sqrt [3]{a + b x^{2}}} + c^{\frac{2}{3}}\right )}{\sqrt [3]{a + b x^{2}} \left (\frac{\sqrt [3]{b} \left (c x\right )^{\frac{2}{3}} \left (- \sqrt{3} - 1\right )}{\sqrt [3]{a + b x^{2}}} + c^{\frac{2}{3}}\right )^{2}}} \left (a + b x^{2}\right )^{\frac{2}{3}} \sqrt{- \frac{b x^{2}}{a + b x^{2}} + 1}} + \frac{2 a c \sqrt [3]{c x} \sqrt [3]{a + b x^{2}}}{9 b} + \frac{\left (c x\right )^{\frac{7}{3}} \sqrt [3]{a + b x^{2}}}{3 c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x)**(4/3)*(b*x**2+a)**(1/3),x)

[Out]

-3**(3/4)*a**2*c**(1/3)*(c*x)**(1/3)*sqrt((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) + c**(4/3))/(b**(1/3)*
(c*x)**(2/3)*(-sqrt(3) - 1)/(a + b*x**2)**(1/3) + c**(2/3))**2)*(-b**(1/3)*(c*x)
**(2/3)/(a + b*x**2)**(1/3) + c**(2/3))*elliptic_f(acos((b**(1/3)*(c*x)**(2/3)*(
-1 + sqrt(3))/(a + b*x**2)**(1/3) + c**(2/3))/(b**(1/3)*(c*x)**(2/3)*(-sqrt(3) -
1)/(a + b*x**2)**(1/3) + c**(2/3))), sqrt(3)/4 + 1/2)/(27*b*sqrt(a/(a + b*x**2)
)*sqrt(-b**(1/3)*(c*x)**(2/3)*(-b**(1/3)*(c*x)**(2/3)/(a + b*x**2)**(1/3) + c**(
2/3))/((a + b*x**2)**(1/3)*(b**(1/3)*(c*x)**(2/3)*(-sqrt(3) - 1)/(a + b*x**2)**(
1/3) + c**(2/3))**2))*(a + b*x**2)**(2/3)*sqrt(-b*x**2/(a + b*x**2) + 1)) + 2*a*
c*(c*x)**(1/3)*(a + b*x**2)**(1/3)/(9*b) + (c*x)**(7/3)*(a + b*x**2)**(1/3)/(3*c
)

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Mathematica [C]  time = 0.0565944, size = 85, normalized size = 0.2 \[ \frac{c \sqrt [3]{c x} \left (-2 a^2 \left (\frac{b x^2}{a}+1\right )^{2/3} \, _2F_1\left (\frac{1}{6},\frac{2}{3};\frac{7}{6};-\frac{b x^2}{a}\right )+2 a^2+5 a b x^2+3 b^2 x^4\right )}{9 b \left (a+b x^2\right )^{2/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(c*x)^(4/3)*(a + b*x^2)^(1/3),x]

[Out]

(c*(c*x)^(1/3)*(2*a^2 + 5*a*b*x^2 + 3*b^2*x^4 - 2*a^2*(1 + (b*x^2)/a)^(2/3)*Hype
rgeometric2F1[1/6, 2/3, 7/6, -((b*x^2)/a)]))/(9*b*(a + b*x^2)^(2/3))

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Maple [F]  time = 0.034, size = 0, normalized size = 0. \[ \int \left ( cx \right ) ^{{\frac{4}{3}}}\sqrt [3]{b{x}^{2}+a}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x)^(4/3)*(b*x^2+a)^(1/3),x)

[Out]

int((c*x)^(4/3)*(b*x^2+a)^(1/3),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{2} + a\right )}^{\frac{1}{3}} \left (c x\right )^{\frac{4}{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(1/3)*(c*x)^(4/3),x, algorithm="maxima")

[Out]

integrate((b*x^2 + a)^(1/3)*(c*x)^(4/3), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (b x^{2} + a\right )}^{\frac{1}{3}} \left (c x\right )^{\frac{1}{3}} c x, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(1/3)*(c*x)^(4/3),x, algorithm="fricas")

[Out]

integral((b*x^2 + a)^(1/3)*(c*x)^(1/3)*c*x, x)

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Sympy [A]  time = 149.214, size = 46, normalized size = 0.11 \[ \frac{\sqrt [3]{a} c^{\frac{4}{3}} x^{\frac{7}{3}} \Gamma \left (\frac{7}{6}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{3}, \frac{7}{6} \\ \frac{13}{6} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac{13}{6}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x)**(4/3)*(b*x**2+a)**(1/3),x)

[Out]

a**(1/3)*c**(4/3)*x**(7/3)*gamma(7/6)*hyper((-1/3, 7/6), (13/6,), b*x**2*exp_pol
ar(I*pi)/a)/(2*gamma(13/6))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{2} + a\right )}^{\frac{1}{3}} \left (c x\right )^{\frac{4}{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(1/3)*(c*x)^(4/3),x, algorithm="giac")

[Out]

integrate((b*x^2 + a)^(1/3)*(c*x)^(4/3), x)