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

Optimal. Leaf size=209 \[ \frac{a c^{7/3} \log \left (c^{2/3}-\frac{\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{3 b^{5/3}}-\frac{a c^{7/3} \log \left (\frac{b^{2/3} (c x)^{4/3}}{\left (a+b x^2\right )^{2/3}}+\frac{\sqrt [3]{b} c^{2/3} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}+c^{4/3}\right )}{6 b^{5/3}}+\frac{a c^{7/3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}+c^{2/3}}{\sqrt{3} c^{2/3}}\right )}{\sqrt{3} b^{5/3}}+\frac{c (c x)^{4/3} \sqrt [3]{a+b x^2}}{2 b} \]

[Out]

(c*(c*x)^(4/3)*(a + b*x^2)^(1/3))/(2*b) + (a*c^(7/3)*ArcTan[(c^(2/3) + (2*b^(1/3
)*(c*x)^(2/3))/(a + b*x^2)^(1/3))/(Sqrt[3]*c^(2/3))])/(Sqrt[3]*b^(5/3)) + (a*c^(
7/3)*Log[c^(2/3) - (b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3)])/(3*b^(5/3)) - (a*c^
(7/3)*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)])/(6*b^(5/3))

_______________________________________________________________________________________

Rubi [A]  time = 0.589379, antiderivative size = 209, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 10, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.526 \[ \frac{a c^{7/3} \log \left (c^{2/3}-\frac{\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{3 b^{5/3}}-\frac{a c^{7/3} \log \left (\frac{b^{2/3} (c x)^{4/3}}{\left (a+b x^2\right )^{2/3}}+\frac{\sqrt [3]{b} c^{2/3} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}+c^{4/3}\right )}{6 b^{5/3}}+\frac{a c^{7/3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}+c^{2/3}}{\sqrt{3} c^{2/3}}\right )}{\sqrt{3} b^{5/3}}+\frac{c (c x)^{4/3} \sqrt [3]{a+b x^2}}{2 b} \]

Antiderivative was successfully verified.

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

[Out]

(c*(c*x)^(4/3)*(a + b*x^2)^(1/3))/(2*b) + (a*c^(7/3)*ArcTan[(c^(2/3) + (2*b^(1/3
)*(c*x)^(2/3))/(a + b*x^2)^(1/3))/(Sqrt[3]*c^(2/3))])/(Sqrt[3]*b^(5/3)) + (a*c^(
7/3)*Log[c^(2/3) - (b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3)])/(3*b^(5/3)) - (a*c^
(7/3)*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)])/(6*b^(5/3))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 58.1946, size = 197, normalized size = 0.94 \[ \frac{a c^{\frac{7}{3}} \log{\left (- \frac{\sqrt [3]{b} \left (c x\right )^{\frac{2}{3}}}{\sqrt [3]{a + b x^{2}}} + c^{\frac{2}{3}} \right )}}{3 b^{\frac{5}{3}}} - \frac{a c^{\frac{7}{3}} \log{\left (\frac{b^{\frac{2}{3}} \left (c x\right )^{\frac{4}{3}}}{c^{\frac{4}{3}} \left (a + b x^{2}\right )^{\frac{2}{3}}} + \frac{\sqrt [3]{b} \left (c x\right )^{\frac{2}{3}}}{c^{\frac{2}{3}} \sqrt [3]{a + b x^{2}}} + 1 \right )}}{6 b^{\frac{5}{3}}} + \frac{\sqrt{3} a c^{\frac{7}{3}} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{2 \sqrt [3]{b} \left (c x\right )^{\frac{2}{3}}}{3 \sqrt [3]{a + b x^{2}}} + \frac{c^{\frac{2}{3}}}{3}\right )}{c^{\frac{2}{3}}} \right )}}{3 b^{\frac{5}{3}}} + \frac{c \left (c x\right )^{\frac{4}{3}} \sqrt [3]{a + b x^{2}}}{2 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*c**(7/3)*log(-b**(1/3)*(c*x)**(2/3)/(a + b*x**2)**(1/3) + c**(2/3))/(3*b**(5/3
)) - a*c**(7/3)*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)/(6*b**(5/3)) + sqrt(3)*a*c*
*(7/3)*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))/(3*b**(5/3)) + c*(c*x)**(4/3)*(a + b*x**2)**(1/3)/(2*b)

_______________________________________________________________________________________

Mathematica [C]  time = 0.0565378, size = 69, normalized size = 0.33 \[ \frac{c (c x)^{4/3} \left (-a \left (\frac{b x^2}{a}+1\right )^{2/3} \, _2F_1\left (\frac{2}{3},\frac{2}{3};\frac{5}{3};-\frac{b x^2}{a}\right )+a+b x^2\right )}{2 b \left (a+b x^2\right )^{2/3}} \]

Antiderivative was successfully verified.

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

[Out]

(c*(c*x)^(4/3)*(a + b*x^2 - a*(1 + (b*x^2)/a)^(2/3)*Hypergeometric2F1[2/3, 2/3,
5/3, -((b*x^2)/a)]))/(2*b*(a + b*x^2)^(2/3))

_______________________________________________________________________________________

Maple [F]  time = 0.033, size = 0, normalized size = 0. \[ \int{1 \left ( cx \right ) ^{{\frac{7}{3}}} \left ( b{x}^{2}+a \right ) ^{-{\frac{2}{3}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int((c*x)^(7/3)/(b*x^2+a)^(2/3),x)

_______________________________________________________________________________________

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)^(7/3)/(b*x^2 + a)^(2/3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

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)^(7/3)/(b*x^2 + a)^(2/3),x, algorithm="fricas")

[Out]

Timed out

_______________________________________________________________________________________

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

[Out]

Timed out

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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