3.37 \(\int \frac{1}{\sqrt [3]{b x+c x^2}} \, dx\)

Optimal. Leaf size=715 \[ -\frac{3 (b+2 c x) \sqrt [3]{-\frac{c \left (b x+c x^2\right )}{b^2}}}{\sqrt [3]{2} c \sqrt [3]{b x+c x^2} \left (-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}-\sqrt{3}+1\right )}+\frac{\sqrt [6]{2} 3^{3/4} b^2 \sqrt [3]{-\frac{c \left (b x+c x^2\right )}{b^2}} \left (1-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}\right ) \sqrt{\frac{2 \sqrt [3]{2} \left (-\frac{c x (b+c x)}{b^2}\right )^{2/3}+2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}+1}{\left (-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}+\sqrt{3}+1}{-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{c (b+2 c x) \sqrt [3]{b x+c x^2} \sqrt{-\frac{1-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}}{\left (-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}-\sqrt{3}+1\right )^2}}}-\frac{3 \sqrt [4]{3} \sqrt{2+\sqrt{3}} b^2 \sqrt [3]{-\frac{c \left (b x+c x^2\right )}{b^2}} \left (1-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}\right ) \sqrt{\frac{2 \sqrt [3]{2} \left (-\frac{c x (b+c x)}{b^2}\right )^{2/3}+2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}+1}{\left (-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}-\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}+\sqrt{3}+1}{-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{2 \sqrt [3]{2} c (b+2 c x) \sqrt [3]{b x+c x^2} \sqrt{-\frac{1-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}}{\left (-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}-\sqrt{3}+1\right )^2}}} \]

[Out]

(-3*(b + 2*c*x)*(-((c*(b*x + c*x^2))/b^2))^(1/3))/(2^(1/3)*c*(b*x + c*x^2)^(1/3)
*(1 - Sqrt[3] - 2^(2/3)*(-((c*x*(b + c*x))/b^2))^(1/3))) - (3*3^(1/4)*Sqrt[2 + S
qrt[3]]*b^2*(-((c*(b*x + c*x^2))/b^2))^(1/3)*(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]
*EllipticE[ArcSin[(1 + Sqrt[3] - 2^(2/3)*(-((c*x*(b + c*x))/b^2))^(1/3))/(1 - Sq
rt[3] - 2^(2/3)*(-((c*x*(b + c*x))/b^2))^(1/3))], -7 + 4*Sqrt[3]])/(2*2^(1/3)*c*
(b + 2*c*x)*(b*x + c*x^2)^(1/3)*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)]) + (2^(1/6)*3^(3/
4)*b^2*(-((c*(b*x + c*x^2))/b^2))^(1/3)*(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]*Elli
pticF[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)^(1/3)*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 = 1.66507, antiderivative size = 715, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.462 \[ -\frac{3 (b+2 c x) \sqrt [3]{-\frac{c \left (b x+c x^2\right )}{b^2}}}{\sqrt [3]{2} c \sqrt [3]{b x+c x^2} \left (-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}-\sqrt{3}+1\right )}+\frac{\sqrt [6]{2} 3^{3/4} b^2 \sqrt [3]{-\frac{c \left (b x+c x^2\right )}{b^2}} \left (1-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}\right ) \sqrt{\frac{2 \sqrt [3]{2} \left (-\frac{c x (b+c x)}{b^2}\right )^{2/3}+2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}+1}{\left (-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}+\sqrt{3}+1}{-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{c (b+2 c x) \sqrt [3]{b x+c x^2} \sqrt{-\frac{1-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}}{\left (-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}-\sqrt{3}+1\right )^2}}}-\frac{3 \sqrt [4]{3} \sqrt{2+\sqrt{3}} b^2 \sqrt [3]{-\frac{c \left (b x+c x^2\right )}{b^2}} \left (1-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}\right ) \sqrt{\frac{2 \sqrt [3]{2} \left (-\frac{c x (b+c x)}{b^2}\right )^{2/3}+2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}+1}{\left (-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}-\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}+\sqrt{3}+1}{-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{2 \sqrt [3]{2} c (b+2 c x) \sqrt [3]{b x+c x^2} \sqrt{-\frac{1-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}}{\left (-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}-\sqrt{3}+1\right )^2}}} \]

Warning: Unable to verify antiderivative.

[In]  Int[(b*x + c*x^2)^(-1/3),x]

[Out]

(-3*(b + 2*c*x)*(-((c*(b*x + c*x^2))/b^2))^(1/3))/(2^(1/3)*c*(b*x + c*x^2)^(1/3)
*(1 - Sqrt[3] - 2^(2/3)*(-((c*x*(b + c*x))/b^2))^(1/3))) - (3*3^(1/4)*Sqrt[2 + S
qrt[3]]*b^2*(-((c*(b*x + c*x^2))/b^2))^(1/3)*(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]
*EllipticE[ArcSin[(1 + Sqrt[3] - 2^(2/3)*(-((c*x*(b + c*x))/b^2))^(1/3))/(1 - Sq
rt[3] - 2^(2/3)*(-((c*x*(b + c*x))/b^2))^(1/3))], -7 + 4*Sqrt[3]])/(2*2^(1/3)*c*
(b + 2*c*x)*(b*x + c*x^2)^(1/3)*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)]) + (2^(1/6)*3^(3/
4)*b^2*(-((c*(b*x + c*x^2))/b^2))^(1/3)*(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]*Elli
pticF[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)^(1/3)*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 = 55.4489, size = 595, normalized size = 0.83 \[ - \frac{3 \cdot 2^{\frac{2}{3}} \sqrt [4]{3} 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}}} \sqrt [3]{\frac{c \left (- b x - c x^{2}\right )}{b^{2}}} \sqrt{\sqrt{3} + 2} \left (- \sqrt [3]{1 - \frac{\left (- b - 2 c x\right )^{2}}{b^{2}}} + 1\right ) E\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 )}{4 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 ) \sqrt [3]{b x + c x^{2}}} + \frac{\sqrt [6]{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}}} \sqrt [3]{\frac{c \left (- b x - c x^{2}\right )}{b^{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 ) \sqrt [3]{b x + c x^{2}}} - \frac{3 \cdot 2^{\frac{2}{3}} \sqrt [3]{\frac{c \left (- b x - c x^{2}\right )}{b^{2}}} \left (b + 2 c x\right )}{2 c \sqrt [3]{b x + c x^{2}} \left (- \sqrt [3]{1 - \frac{\left (- b - 2 c x\right )^{2}}{b^{2}}} - \sqrt{3} + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*2**(2/3)*3**(1/4)*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)**(1/3)*sqrt(sqrt(3) + 2)*(-(1 - (-b - 2*c*x)**2/b**2)
**(1/3) + 1)*elliptic_e(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))/(4*c*sqrt((
(1 - (-b - 2*c*x)**2/b**2)**(1/3) - 1)/(-(1 - (-b - 2*c*x)**2/b**2)**(1/3) - sqr
t(3) + 1)**2)*(b + 2*c*x)*(b*x + c*x**2)**(1/3)) + 2**(1/6)*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)**(1/3
)*(-(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)**(1/3)) - 3
*2**(2/3)*(c*(-b*x - c*x**2)/b**2)**(1/3)*(b + 2*c*x)/(2*c*(b*x + c*x**2)**(1/3)
*(-(1 - (-b - 2*c*x)**2/b**2)**(1/3) - sqrt(3) + 1))

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Mathematica [C]  time = 0.026365, size = 46, normalized size = 0.06 \[ \frac{3 x \sqrt [3]{\frac{b+c x}{b}} \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{5}{3};-\frac{c x}{b}\right )}{2 \sqrt [3]{x (b+c x)}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.047, size = 0, normalized size = 0. \[ \int{\frac{1}{\sqrt [3]{c{x}^{2}+bx}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(c*x^2+b*x)^(1/3),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((c*x^2 + b*x)^(-1/3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((c*x^2 + b*x)^(-1/3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((b*x + c*x**2)**(-1/3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((c*x^2 + b*x)^(-1/3), x)