3.779 \(\int \frac{(c x)^{10/3}}{\left (a+b x^2\right )^{2/3}} \, dx\)

Optimal. Leaf size=421 \[ -\frac{7 a c^3 \sqrt [3]{c x} \sqrt [3]{a+b x^2}}{9 b^2}+\frac{7 a c^{7/3} \sqrt [3]{c x} \sqrt [3]{a+b x^2} \left (c^{2/3}-\frac{\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right ) \sqrt{\frac{\frac{b^{2/3} (c x)^{4/3}}{\left (a+b x^2\right )^{2/3}}+\frac{\sqrt [3]{b} c^{2/3} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}+c^{4/3}}{\left (c^{2/3}-\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )^2}} F\left (\cos ^{-1}\left (\frac{c^{2/3}-\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{b x^2+a}}}{c^{2/3}-\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{b x^2+a}}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{18 \sqrt [4]{3} b^2 \sqrt{-\frac{\sqrt [3]{b} (c x)^{2/3} \left (c^{2/3}-\frac{\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{\sqrt [3]{a+b x^2} \left (c^{2/3}-\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )^2}}}+\frac{c (c x)^{7/3} \sqrt [3]{a+b x^2}}{3 b} \]

[Out]

(-7*a*c^3*(c*x)^(1/3)*(a + b*x^2)^(1/3))/(9*b^2) + (c*(c*x)^(7/3)*(a + b*x^2)^(1
/3))/(3*b) + (7*a*c^(7/3)*(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 - Sq
rt[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])/(18*3^(1/4)*b^2*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 = 1.55699, antiderivative size = 421, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ -\frac{7 a c^3 \sqrt [3]{c x} \sqrt [3]{a+b x^2}}{9 b^2}+\frac{7 a c^{7/3} \sqrt [3]{c x} \sqrt [3]{a+b x^2} \left (c^{2/3}-\frac{\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right ) \sqrt{\frac{\frac{b^{2/3} (c x)^{4/3}}{\left (a+b x^2\right )^{2/3}}+\frac{\sqrt [3]{b} c^{2/3} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}+c^{4/3}}{\left (c^{2/3}-\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )^2}} F\left (\cos ^{-1}\left (\frac{c^{2/3}-\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{b x^2+a}}}{c^{2/3}-\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{b x^2+a}}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{18 \sqrt [4]{3} b^2 \sqrt{-\frac{\sqrt [3]{b} (c x)^{2/3} \left (c^{2/3}-\frac{\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{\sqrt [3]{a+b x^2} \left (c^{2/3}-\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )^2}}}+\frac{c (c x)^{7/3} \sqrt [3]{a+b x^2}}{3 b} \]

Antiderivative was successfully verified.

[In]  Int[(c*x)^(10/3)/(a + b*x^2)^(2/3),x]

[Out]

(-7*a*c^3*(c*x)^(1/3)*(a + b*x^2)^(1/3))/(9*b^2) + (c*(c*x)^(7/3)*(a + b*x^2)^(1
/3))/(3*b) + (7*a*c^(7/3)*(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 - Sq
rt[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])/(18*3^(1/4)*b^2*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 = 36.779, size = 411, normalized size = 0.98 \[ \frac{7 \cdot 3^{\frac{3}{4}} a^{2} c^{\frac{7}{3}} \sqrt [3]{c x} \sqrt{\frac{\frac{b^{\frac{2}{3}} \left (c x\right )^{\frac{4}{3}}}{\left (a + b x^{2}\right )^{\frac{2}{3}}} + \frac{\sqrt [3]{b} c^{\frac{2}{3}} \left (c x\right )^{\frac{2}{3}}}{\sqrt [3]{a + b x^{2}}} + c^{\frac{4}{3}}}{\left (\frac{\sqrt [3]{b} \left (c x\right )^{\frac{2}{3}} \left (- \sqrt{3} - 1\right )}{\sqrt [3]{a + b x^{2}}} + c^{\frac{2}{3}}\right )^{2}}} \left (- \frac{\sqrt [3]{b} \left (c x\right )^{\frac{2}{3}}}{\sqrt [3]{a + b x^{2}}} + c^{\frac{2}{3}}\right ) F\left (\operatorname{acos}{\left (\frac{\frac{\sqrt [3]{b} \left (c x\right )^{\frac{2}{3}} \left (-1 + \sqrt{3}\right )}{\sqrt [3]{a + b x^{2}}} + c^{\frac{2}{3}}}{\frac{\sqrt [3]{b} \left (c x\right )^{\frac{2}{3}} \left (- \sqrt{3} - 1\right )}{\sqrt [3]{a + b x^{2}}} + c^{\frac{2}{3}}} \right )}\middle | \frac{\sqrt{3}}{4} + \frac{1}{2}\right )}{54 b^{2} \sqrt{\frac{a}{a + b x^{2}}} \sqrt{- \frac{\sqrt [3]{b} \left (c x\right )^{\frac{2}{3}} \left (- \frac{\sqrt [3]{b} \left (c x\right )^{\frac{2}{3}}}{\sqrt [3]{a + b x^{2}}} + c^{\frac{2}{3}}\right )}{\sqrt [3]{a + b x^{2}} \left (\frac{\sqrt [3]{b} \left (c x\right )^{\frac{2}{3}} \left (- \sqrt{3} - 1\right )}{\sqrt [3]{a + b x^{2}}} + c^{\frac{2}{3}}\right )^{2}}} \left (a + b x^{2}\right )^{\frac{2}{3}} \sqrt{- \frac{b x^{2}}{a + b x^{2}} + 1}} - \frac{7 a c^{3} \sqrt [3]{c x} \sqrt [3]{a + b x^{2}}}{9 b^{2}} + \frac{c \left (c x\right )^{\frac{7}{3}} \sqrt [3]{a + b x^{2}}}{3 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x)**(10/3)/(b*x**2+a)**(2/3),x)

[Out]

7*3**(3/4)*a**2*c**(7/3)*(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)/(54*b**2*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)) -
7*a*c**3*(c*x)**(1/3)*(a + b*x**2)**(1/3)/(9*b**2) + c*(c*x)**(7/3)*(a + b*x**2)
**(1/3)/(3*b)

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Mathematica [C]  time = 0.0622693, size = 87, normalized size = 0.21 \[ \frac{c^3 \sqrt [3]{c x} \left (7 a^2 \left (\frac{b x^2}{a}+1\right )^{2/3} \, _2F_1\left (\frac{1}{6},\frac{2}{3};\frac{7}{6};-\frac{b x^2}{a}\right )-7 a^2-4 a b x^2+3 b^2 x^4\right )}{9 b^2 \left (a+b x^2\right )^{2/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(c*x)^(10/3)/(a + b*x^2)^(2/3),x]

[Out]

(c^3*(c*x)^(1/3)*(-7*a^2 - 4*a*b*x^2 + 3*b^2*x^4 + 7*a^2*(1 + (b*x^2)/a)^(2/3)*H
ypergeometric2F1[1/6, 2/3, 7/6, -((b*x^2)/a)]))/(9*b^2*(a + b*x^2)^(2/3))

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Maple [F]  time = 0.035, size = 0, normalized size = 0. \[ \int{1 \left ( cx \right ) ^{{\frac{10}{3}}} \left ( b{x}^{2}+a \right ) ^{-{\frac{2}{3}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x)^(10/3)/(b*x^2+a)^(2/3),x)

[Out]

int((c*x)^(10/3)/(b*x^2+a)^(2/3),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (c x\right )^{\frac{10}{3}}}{{\left (b x^{2} + a\right )}^{\frac{2}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x)^(10/3)/(b*x^2 + a)^(2/3),x, algorithm="maxima")

[Out]

integrate((c*x)^(10/3)/(b*x^2 + a)^(2/3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x)^(10/3)/(b*x^2 + a)^(2/3),x, algorithm="fricas")

[Out]

integral((c*x)^(1/3)*c^3*x^3/(b*x^2 + a)^(2/3), 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)**(10/3)/(b*x**2+a)**(2/3),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (c x\right )^{\frac{10}{3}}}{{\left (b x^{2} + a\right )}^{\frac{2}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x)^(10/3)/(b*x^2 + a)^(2/3),x, algorithm="giac")

[Out]

integrate((c*x)^(10/3)/(b*x^2 + a)^(2/3), x)