3.3002 \(\int \frac{1}{\sqrt [3]{a+b x} (c+d x)^{2/3} (e+f x)} \, dx\)

Optimal. Leaf size=197 \[ \frac{\log (e+f x)}{2 \sqrt [3]{b e-a f} (d e-c f)^{2/3}}-\frac{3 \log \left (\frac{\sqrt [3]{a+b x} \sqrt [3]{d e-c f}}{\sqrt [3]{b e-a f}}-\sqrt [3]{c+d x}\right )}{2 \sqrt [3]{b e-a f} (d e-c f)^{2/3}}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x} \sqrt [3]{d e-c f}}{\sqrt{3} \sqrt [3]{c+d x} \sqrt [3]{b e-a f}}+\frac{1}{\sqrt{3}}\right )}{\sqrt [3]{b e-a f} (d e-c f)^{2/3}} \]

[Out]

-((Sqrt[3]*ArcTan[1/Sqrt[3] + (2*(d*e - c*f)^(1/3)*(a + b*x)^(1/3))/(Sqrt[3]*(b*
e - a*f)^(1/3)*(c + d*x)^(1/3))])/((b*e - a*f)^(1/3)*(d*e - c*f)^(2/3))) + Log[e
+ f*x]/(2*(b*e - a*f)^(1/3)*(d*e - c*f)^(2/3)) - (3*Log[((d*e - c*f)^(1/3)*(a +
b*x)^(1/3))/(b*e - a*f)^(1/3) - (c + d*x)^(1/3)])/(2*(b*e - a*f)^(1/3)*(d*e - c
*f)^(2/3))

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Rubi [A]  time = 0.223432, antiderivative size = 197, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.038 \[ \frac{\log (e+f x)}{2 \sqrt [3]{b e-a f} (d e-c f)^{2/3}}-\frac{3 \log \left (\frac{\sqrt [3]{a+b x} \sqrt [3]{d e-c f}}{\sqrt [3]{b e-a f}}-\sqrt [3]{c+d x}\right )}{2 \sqrt [3]{b e-a f} (d e-c f)^{2/3}}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x} \sqrt [3]{d e-c f}}{\sqrt{3} \sqrt [3]{c+d x} \sqrt [3]{b e-a f}}+\frac{1}{\sqrt{3}}\right )}{\sqrt [3]{b e-a f} (d e-c f)^{2/3}} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b*x)^(1/3)*(c + d*x)^(2/3)*(e + f*x)),x]

[Out]

-((Sqrt[3]*ArcTan[1/Sqrt[3] + (2*(d*e - c*f)^(1/3)*(a + b*x)^(1/3))/(Sqrt[3]*(b*
e - a*f)^(1/3)*(c + d*x)^(1/3))])/((b*e - a*f)^(1/3)*(d*e - c*f)^(2/3))) + Log[e
+ f*x]/(2*(b*e - a*f)^(1/3)*(d*e - c*f)^(2/3)) - (3*Log[((d*e - c*f)^(1/3)*(a +
b*x)^(1/3))/(b*e - a*f)^(1/3) - (c + d*x)^(1/3)])/(2*(b*e - a*f)^(1/3)*(d*e - c
*f)^(2/3))

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Rubi in Sympy [A]  time = 12.0507, size = 170, normalized size = 0.86 \[ - \frac{\log{\left (e + f x \right )}}{2 \sqrt [3]{a f - b e} \left (c f - d e\right )^{\frac{2}{3}}} + \frac{3 \log{\left (\frac{\sqrt [3]{a + b x} \sqrt [3]{c f - d e}}{\sqrt [3]{a f - b e}} - \sqrt [3]{c + d x} \right )}}{2 \sqrt [3]{a f - b e} \left (c f - d e\right )^{\frac{2}{3}}} + \frac{\sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt{3} \sqrt [3]{a + b x} \sqrt [3]{c f - d e}}{3 \sqrt [3]{c + d x} \sqrt [3]{a f - b e}} + \frac{\sqrt{3}}{3} \right )}}{\sqrt [3]{a f - b e} \left (c f - d e\right )^{\frac{2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-log(e + f*x)/(2*(a*f - b*e)**(1/3)*(c*f - d*e)**(2/3)) + 3*log((a + b*x)**(1/3)
*(c*f - d*e)**(1/3)/(a*f - b*e)**(1/3) - (c + d*x)**(1/3))/(2*(a*f - b*e)**(1/3)
*(c*f - d*e)**(2/3)) + sqrt(3)*atan(2*sqrt(3)*(a + b*x)**(1/3)*(c*f - d*e)**(1/3
)/(3*(c + d*x)**(1/3)*(a*f - b*e)**(1/3)) + sqrt(3)/3)/((a*f - b*e)**(1/3)*(c*f
- d*e)**(2/3))

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Mathematica [C]  time = 1.50156, size = 108, normalized size = 0.55 \[ \frac{3 (a+b x)^{2/3} \left (\frac{(c+d x) (b e-a f)}{(e+f x) (b c-a d)}\right )^{2/3} \, _2F_1\left (\frac{2}{3},\frac{2}{3};\frac{5}{3};\frac{(c f-d e) (a+b x)}{(b c-a d) (e+f x)}\right )}{2 (c+d x)^{2/3} (b e-a f)} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b*x)^(1/3)*(c + d*x)^(2/3)*(e + f*x)),x]

[Out]

(3*(a + b*x)^(2/3)*(((b*e - a*f)*(c + d*x))/((b*c - a*d)*(e + f*x)))^(2/3)*Hyper
geometric2F1[2/3, 2/3, 5/3, ((-(d*e) + c*f)*(a + b*x))/((b*c - a*d)*(e + f*x))])
/(2*(b*e - a*f)*(c + d*x)^(2/3))

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Maple [F]  time = 0.081, size = 0, normalized size = 0. \[ \int{\frac{1}{fx+e}{\frac{1}{\sqrt [3]{bx+a}}} \left ( dx+c \right ) ^{-{\frac{2}{3}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x+a)^(1/3)/(d*x+c)^(2/3)/(f*x+e),x)

[Out]

int(1/(b*x+a)^(1/3)/(d*x+c)^(2/3)/(f*x+e),x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/((b*x + a)^(1/3)*(d*x + c)^(2/3)*(f*x + e)), x)

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Fricas [A]  time = 0.229965, size = 657, normalized size = 3.34 \[ -\frac{2 \, \sqrt{3} \arctan \left (\frac{\sqrt{3}{\left (a d e - a c f +{\left (b d e - b c f\right )} x - 2 \,{\left (-b d^{2} e^{3} + a c^{2} f^{3} +{\left (2 \, b c d + a d^{2}\right )} e^{2} f -{\left (b c^{2} + 2 \, a c d\right )} e f^{2}\right )}^{\frac{1}{3}}{\left (b x + a\right )}^{\frac{2}{3}}{\left (d x + c\right )}^{\frac{1}{3}}\right )}}{3 \,{\left (a d e - a c f +{\left (b d e - b c f\right )} x\right )}}\right ) + \log \left (\frac{a d^{2} e^{2} - 2 \, a c d e f + a c^{2} f^{2} -{\left (-b d^{2} e^{3} + a c^{2} f^{3} +{\left (2 \, b c d + a d^{2}\right )} e^{2} f -{\left (b c^{2} + 2 \, a c d\right )} e f^{2}\right )}^{\frac{1}{3}}{\left (d e - c f\right )}{\left (b x + a\right )}^{\frac{2}{3}}{\left (d x + c\right )}^{\frac{1}{3}} +{\left (b d^{2} e^{2} - 2 \, b c d e f + b c^{2} f^{2}\right )} x +{\left (-b d^{2} e^{3} + a c^{2} f^{3} +{\left (2 \, b c d + a d^{2}\right )} e^{2} f -{\left (b c^{2} + 2 \, a c d\right )} e f^{2}\right )}^{\frac{2}{3}}{\left (b x + a\right )}^{\frac{1}{3}}{\left (d x + c\right )}^{\frac{2}{3}}}{b x + a}\right ) - 2 \, \log \left (\frac{a d e - a c f +{\left (b d e - b c f\right )} x +{\left (-b d^{2} e^{3} + a c^{2} f^{3} +{\left (2 \, b c d + a d^{2}\right )} e^{2} f -{\left (b c^{2} + 2 \, a c d\right )} e f^{2}\right )}^{\frac{1}{3}}{\left (b x + a\right )}^{\frac{2}{3}}{\left (d x + c\right )}^{\frac{1}{3}}}{b x + a}\right )}{2 \,{\left (-b d^{2} e^{3} + a c^{2} f^{3} +{\left (2 \, b c d + a d^{2}\right )} e^{2} f -{\left (b c^{2} + 2 \, a c d\right )} e f^{2}\right )}^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(2*sqrt(3)*arctan(1/3*sqrt(3)*(a*d*e - a*c*f + (b*d*e - b*c*f)*x - 2*(-b*d^
2*e^3 + a*c^2*f^3 + (2*b*c*d + a*d^2)*e^2*f - (b*c^2 + 2*a*c*d)*e*f^2)^(1/3)*(b*
x + a)^(2/3)*(d*x + c)^(1/3))/(a*d*e - a*c*f + (b*d*e - b*c*f)*x)) + log((a*d^2*
e^2 - 2*a*c*d*e*f + a*c^2*f^2 - (-b*d^2*e^3 + a*c^2*f^3 + (2*b*c*d + a*d^2)*e^2*
f - (b*c^2 + 2*a*c*d)*e*f^2)^(1/3)*(d*e - c*f)*(b*x + a)^(2/3)*(d*x + c)^(1/3) +
(b*d^2*e^2 - 2*b*c*d*e*f + b*c^2*f^2)*x + (-b*d^2*e^3 + a*c^2*f^3 + (2*b*c*d +
a*d^2)*e^2*f - (b*c^2 + 2*a*c*d)*e*f^2)^(2/3)*(b*x + a)^(1/3)*(d*x + c)^(2/3))/(
b*x + a)) - 2*log((a*d*e - a*c*f + (b*d*e - b*c*f)*x + (-b*d^2*e^3 + a*c^2*f^3 +
(2*b*c*d + a*d^2)*e^2*f - (b*c^2 + 2*a*c*d)*e*f^2)^(1/3)*(b*x + a)^(2/3)*(d*x +
c)^(1/3))/(b*x + a)))/(-b*d^2*e^3 + a*c^2*f^3 + (2*b*c*d + a*d^2)*e^2*f - (b*c^
2 + 2*a*c*d)*e*f^2)^(1/3)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/((a + b*x)**(1/3)*(c + d*x)**(2/3)*(e + f*x)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/((b*x + a)^(1/3)*(d*x + c)^(2/3)*(f*x + e)), x)