3.32 \(\int \frac{1}{\left (b x+c x^2\right )^{2/3}} \, dx\)

Optimal. Leaf size=322 \[ \frac{\sqrt [3]{2} 3^{3/4} \sqrt{2-\sqrt{3}} b^2 \left (-\frac{c \left (b x+c x^2\right )}{b^2}\right )^{2/3} \left (1-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}\right ) \sqrt{\frac{2 \sqrt [3]{2} \left (-\frac{c x (b+c x)}{b^2}\right )^{2/3}+2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}+1}{\left (-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}+\sqrt{3}+1}{-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{c (b+2 c x) \left (b x+c x^2\right )^{2/3} \sqrt{-\frac{1-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}}{\left (-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}-\sqrt{3}+1\right )^2}}} \]

[Out]

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

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Rubi [A]  time = 0.788418, antiderivative size = 322, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ \frac{\sqrt [3]{2} 3^{3/4} \sqrt{2-\sqrt{3}} b^2 \left (-\frac{c \left (b x+c x^2\right )}{b^2}\right )^{2/3} \left (1-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}\right ) \sqrt{\frac{2 \sqrt [3]{2} \left (-\frac{c x (b+c x)}{b^2}\right )^{2/3}+2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}+1}{\left (-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}+\sqrt{3}+1}{-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{c (b+2 c x) \left (b x+c x^2\right )^{2/3} \sqrt{-\frac{1-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}}{\left (-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}-\sqrt{3}+1\right )^2}}} \]

Antiderivative was successfully verified.

[In]  Int[(b*x + c*x^2)^(-2/3),x]

[Out]

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

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Rubi in Sympy [A]  time = 23.123, size = 264, normalized size = 0.82 \[ \frac{\sqrt [3]{2} \cdot 3^{\frac{3}{4}} b^{2} \sqrt{\frac{\left (1 - \frac{\left (- b - 2 c x\right )^{2}}{b^{2}}\right )^{\frac{2}{3}} + \sqrt [3]{1 - \frac{\left (- b - 2 c x\right )^{2}}{b^{2}}} + 1}{\left (- \sqrt [3]{1 - \frac{\left (- b - 2 c x\right )^{2}}{b^{2}}} - \sqrt{3} + 1\right )^{2}}} \left (\frac{c \left (- b x - c x^{2}\right )}{b^{2}}\right )^{\frac{2}{3}} \sqrt{- \sqrt{3} + 2} \left (- \sqrt [3]{1 - \frac{\left (- b - 2 c x\right )^{2}}{b^{2}}} + 1\right ) F\left (\operatorname{asin}{\left (\frac{- \sqrt [3]{1 - \frac{\left (- b - 2 c x\right )^{2}}{b^{2}}} + 1 + \sqrt{3}}{- \sqrt [3]{1 - \frac{\left (- b - 2 c x\right )^{2}}{b^{2}}} - \sqrt{3} + 1} \right )}\middle | -7 + 4 \sqrt{3}\right )}{c \sqrt{\frac{\sqrt [3]{1 - \frac{\left (- b - 2 c x\right )^{2}}{b^{2}}} - 1}{\left (- \sqrt [3]{1 - \frac{\left (- b - 2 c x\right )^{2}}{b^{2}}} - \sqrt{3} + 1\right )^{2}}} \left (b + 2 c x\right ) \left (b x + c x^{2}\right )^{\frac{2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2**(1/3)*3**(3/4)*b**2*sqrt(((1 - (-b - 2*c*x)**2/b**2)**(2/3) + (1 - (-b - 2*c*
x)**2/b**2)**(1/3) + 1)/(-(1 - (-b - 2*c*x)**2/b**2)**(1/3) - sqrt(3) + 1)**2)*(
c*(-b*x - c*x**2)/b**2)**(2/3)*sqrt(-sqrt(3) + 2)*(-(1 - (-b - 2*c*x)**2/b**2)**
(1/3) + 1)*elliptic_f(asin((-(1 - (-b - 2*c*x)**2/b**2)**(1/3) + 1 + sqrt(3))/(-
(1 - (-b - 2*c*x)**2/b**2)**(1/3) - sqrt(3) + 1)), -7 + 4*sqrt(3))/(c*sqrt(((1 -
(-b - 2*c*x)**2/b**2)**(1/3) - 1)/(-(1 - (-b - 2*c*x)**2/b**2)**(1/3) - sqrt(3)
+ 1)**2)*(b + 2*c*x)*(b*x + c*x**2)**(2/3))

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Mathematica [C]  time = 0.0232084, size = 44, normalized size = 0.14 \[ \frac{3 x \left (\frac{b+c x}{b}\right )^{2/3} \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{4}{3};-\frac{c x}{b}\right )}{(x (b+c x))^{2/3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.055, size = 0, normalized size = 0. \[ \int \left ( c{x}^{2}+bx \right ) ^{-{\frac{2}{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(c*x^2+b*x)^(2/3),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((c*x^2 + b*x)^(-2/3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((c*x^2 + b*x)^(-2/3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((b*x + c*x**2)**(-2/3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((c*x^2 + b*x)^(-2/3), x)