3.763 \(\int (c x)^{4/3} \left (a+b x^2\right )^{4/3} \, dx\)

Optimal. Leaf size=448 \[ -\frac{8 a^2 \sqrt [3]{c} \sqrt [3]{c x} \sqrt [3]{a+b x^2} \left (c^{2/3}-\frac{\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right ) \sqrt{\frac{\frac{b^{2/3} (c x)^{4/3}}{\left (a+b x^2\right )^{2/3}}+\frac{\sqrt [3]{b} c^{2/3} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}+c^{4/3}}{\left (c^{2/3}-\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )^2}} F\left (\cos ^{-1}\left (\frac{c^{2/3}-\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{b x^2+a}}}{c^{2/3}-\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{b x^2+a}}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{135 \sqrt [4]{3} b \sqrt{-\frac{\sqrt [3]{b} (c x)^{2/3} \left (c^{2/3}-\frac{\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{\sqrt [3]{a+b x^2} \left (c^{2/3}-\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )^2}}}+\frac{16 a^2 c \sqrt [3]{c x} \sqrt [3]{a+b x^2}}{135 b}+\frac{(c x)^{7/3} \left (a+b x^2\right )^{4/3}}{5 c}+\frac{8 a (c x)^{7/3} \sqrt [3]{a+b x^2}}{45 c} \]

[Out]

(16*a^2*c*(c*x)^(1/3)*(a + b*x^2)^(1/3))/(135*b) + (8*a*(c*x)^(7/3)*(a + b*x^2)^
(1/3))/(45*c) + ((c*x)^(7/3)*(a + b*x^2)^(4/3))/(5*c) - (8*a^2*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])/(135*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 = 1.6542, antiderivative size = 448, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263 \[ -\frac{8 a^2 \sqrt [3]{c} \sqrt [3]{c x} \sqrt [3]{a+b x^2} \left (c^{2/3}-\frac{\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right ) \sqrt{\frac{\frac{b^{2/3} (c x)^{4/3}}{\left (a+b x^2\right )^{2/3}}+\frac{\sqrt [3]{b} c^{2/3} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}+c^{4/3}}{\left (c^{2/3}-\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )^2}} F\left (\cos ^{-1}\left (\frac{c^{2/3}-\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{b x^2+a}}}{c^{2/3}-\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{b x^2+a}}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{135 \sqrt [4]{3} b \sqrt{-\frac{\sqrt [3]{b} (c x)^{2/3} \left (c^{2/3}-\frac{\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{\sqrt [3]{a+b x^2} \left (c^{2/3}-\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )^2}}}+\frac{16 a^2 c \sqrt [3]{c x} \sqrt [3]{a+b x^2}}{135 b}+\frac{(c x)^{7/3} \left (a+b x^2\right )^{4/3}}{5 c}+\frac{8 a (c x)^{7/3} \sqrt [3]{a+b x^2}}{45 c} \]

Antiderivative was successfully verified.

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

[Out]

(16*a^2*c*(c*x)^(1/3)*(a + b*x^2)^(1/3))/(135*b) + (8*a*(c*x)^(7/3)*(a + b*x^2)^
(1/3))/(45*c) + ((c*x)^(7/3)*(a + b*x^2)^(4/3))/(5*c) - (8*a^2*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])/(135*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 = 43.9759, size = 430, normalized size = 0.96 \[ - \frac{8 \cdot 3^{\frac{3}{4}} a^{3} \sqrt [3]{c} \sqrt [3]{c x} \sqrt{\frac{\frac{b^{\frac{2}{3}} \left (c x\right )^{\frac{4}{3}}}{\left (a + b x^{2}\right )^{\frac{2}{3}}} + \frac{\sqrt [3]{b} c^{\frac{2}{3}} \left (c x\right )^{\frac{2}{3}}}{\sqrt [3]{a + b x^{2}}} + c^{\frac{4}{3}}}{\left (\frac{\sqrt [3]{b} \left (c x\right )^{\frac{2}{3}} \left (- \sqrt{3} - 1\right )}{\sqrt [3]{a + b x^{2}}} + c^{\frac{2}{3}}\right )^{2}}} \left (- \frac{\sqrt [3]{b} \left (c x\right )^{\frac{2}{3}}}{\sqrt [3]{a + b x^{2}}} + c^{\frac{2}{3}}\right ) F\left (\operatorname{acos}{\left (\frac{\frac{\sqrt [3]{b} \left (c x\right )^{\frac{2}{3}} \left (-1 + \sqrt{3}\right )}{\sqrt [3]{a + b x^{2}}} + c^{\frac{2}{3}}}{\frac{\sqrt [3]{b} \left (c x\right )^{\frac{2}{3}} \left (- \sqrt{3} - 1\right )}{\sqrt [3]{a + b x^{2}}} + c^{\frac{2}{3}}} \right )}\middle | \frac{\sqrt{3}}{4} + \frac{1}{2}\right )}{405 b \sqrt{\frac{a}{a + b x^{2}}} \sqrt{- \frac{\sqrt [3]{b} \left (c x\right )^{\frac{2}{3}} \left (- \frac{\sqrt [3]{b} \left (c x\right )^{\frac{2}{3}}}{\sqrt [3]{a + b x^{2}}} + c^{\frac{2}{3}}\right )}{\sqrt [3]{a + b x^{2}} \left (\frac{\sqrt [3]{b} \left (c x\right )^{\frac{2}{3}} \left (- \sqrt{3} - 1\right )}{\sqrt [3]{a + b x^{2}}} + c^{\frac{2}{3}}\right )^{2}}} \left (a + b x^{2}\right )^{\frac{2}{3}} \sqrt{- \frac{b x^{2}}{a + b x^{2}} + 1}} + \frac{16 a^{2} c \sqrt [3]{c x} \sqrt [3]{a + b x^{2}}}{135 b} + \frac{8 a \left (c x\right )^{\frac{7}{3}} \sqrt [3]{a + b x^{2}}}{45 c} + \frac{\left (c x\right )^{\frac{7}{3}} \left (a + b x^{2}\right )^{\frac{4}{3}}}{5 c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-8*3**(3/4)*a**3*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)/(405*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)) + 1
6*a**2*c*(c*x)**(1/3)*(a + b*x**2)**(1/3)/(135*b) + 8*a*(c*x)**(7/3)*(a + b*x**2
)**(1/3)/(45*c) + (c*x)**(7/3)*(a + b*x**2)**(4/3)/(5*c)

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Mathematica [C]  time = 0.0631214, size = 96, normalized size = 0.21 \[ \frac{c \sqrt [3]{c x} \left (-16 a^3 \left (\frac{b x^2}{a}+1\right )^{2/3} \, _2F_1\left (\frac{1}{6},\frac{2}{3};\frac{7}{6};-\frac{b x^2}{a}\right )+16 a^3+67 a^2 b x^2+78 a b^2 x^4+27 b^3 x^6\right )}{135 b \left (a+b x^2\right )^{2/3}} \]

Antiderivative was successfully verified.

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

[Out]

(c*(c*x)^(1/3)*(16*a^3 + 67*a^2*b*x^2 + 78*a*b^2*x^4 + 27*b^3*x^6 - 16*a^3*(1 +
(b*x^2)/a)^(2/3)*Hypergeometric2F1[1/6, 2/3, 7/6, -((b*x^2)/a)]))/(135*b*(a + b*
x^2)^(2/3))

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Maple [F]  time = 0.031, size = 0, normalized size = 0. \[ \int \left ( cx \right ) ^{{\frac{4}{3}}} \left ( b{x}^{2}+a \right ) ^{{\frac{4}{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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)**(4/3)*(b*x**2+a)**(4/3),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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