3.2994 \(\int \frac{\sqrt [3]{a+b x}}{\sqrt [3]{c+d x} (e+f x)} \, dx\)

Optimal. Leaf size=339 \[ -\frac{\sqrt [3]{b e-a f} \log (e+f x)}{2 f \sqrt [3]{d e-c f}}+\frac{3 \sqrt [3]{b e-a f} \log \left (\frac{\sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt [3]{d e-c f}}-\sqrt [3]{a+b x}\right )}{2 f \sqrt [3]{d e-c f}}+\frac{\sqrt{3} \sqrt [3]{b e-a f} \tan ^{-1}\left (\frac{2 \sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt{3} \sqrt [3]{a+b x} \sqrt [3]{d e-c f}}+\frac{1}{\sqrt{3}}\right )}{f \sqrt [3]{d e-c f}}-\frac{3 \sqrt [3]{b} \log \left (\frac{\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}-1\right )}{2 \sqrt [3]{d} f}-\frac{\sqrt{3} \sqrt [3]{b} \tan ^{-1}\left (\frac{2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt{3} \sqrt [3]{d} \sqrt [3]{a+b x}}+\frac{1}{\sqrt{3}}\right )}{\sqrt [3]{d} f}-\frac{\sqrt [3]{b} \log (a+b x)}{2 \sqrt [3]{d} f} \]

[Out]

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

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Rubi [A]  time = 0.418196, antiderivative size = 339, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ -\frac{\sqrt [3]{b e-a f} \log (e+f x)}{2 f \sqrt [3]{d e-c f}}+\frac{3 \sqrt [3]{b e-a f} \log \left (\frac{\sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt [3]{d e-c f}}-\sqrt [3]{a+b x}\right )}{2 f \sqrt [3]{d e-c f}}+\frac{\sqrt{3} \sqrt [3]{b e-a f} \tan ^{-1}\left (\frac{2 \sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt{3} \sqrt [3]{a+b x} \sqrt [3]{d e-c f}}+\frac{1}{\sqrt{3}}\right )}{f \sqrt [3]{d e-c f}}-\frac{3 \sqrt [3]{b} \log \left (\frac{\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}-1\right )}{2 \sqrt [3]{d} f}-\frac{\sqrt{3} \sqrt [3]{b} \tan ^{-1}\left (\frac{2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt{3} \sqrt [3]{d} \sqrt [3]{a+b x}}+\frac{1}{\sqrt{3}}\right )}{\sqrt [3]{d} f}-\frac{\sqrt [3]{b} \log (a+b x)}{2 \sqrt [3]{d} f} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 33.4676, size = 303, normalized size = 0.89 \[ - \frac{\sqrt [3]{b} \log{\left (a + b x \right )}}{2 \sqrt [3]{d} f} - \frac{3 \sqrt [3]{b} \log{\left (\frac{\sqrt [3]{b} \sqrt [3]{c + d x}}{\sqrt [3]{d} \sqrt [3]{a + b x}} - 1 \right )}}{2 \sqrt [3]{d} f} - \frac{\sqrt{3} \sqrt [3]{b} \operatorname{atan}{\left (\frac{2 \sqrt{3} \sqrt [3]{b} \sqrt [3]{c + d x}}{3 \sqrt [3]{d} \sqrt [3]{a + b x}} + \frac{\sqrt{3}}{3} \right )}}{\sqrt [3]{d} f} - \frac{\sqrt [3]{a f - b e} \log{\left (e + f x \right )}}{2 f \sqrt [3]{c f - d e}} + \frac{3 \sqrt [3]{a f - b e} \log{\left (- \sqrt [3]{a + b x} + \frac{\sqrt [3]{c + d x} \sqrt [3]{a f - b e}}{\sqrt [3]{c f - d e}} \right )}}{2 f \sqrt [3]{c f - d e}} + \frac{\sqrt{3} \sqrt [3]{a f - b e} \operatorname{atan}{\left (\frac{\sqrt{3}}{3} + \frac{2 \sqrt{3} \sqrt [3]{c + d x} \sqrt [3]{a f - b e}}{3 \sqrt [3]{a + b x} \sqrt [3]{c f - d e}} \right )}}{f \sqrt [3]{c f - d e}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 1.04861, size = 290, normalized size = 0.86 \[ -\frac{21 (a+b x)^{4/3} (b c-a d) (b e-a f)^2 F_1\left (\frac{4}{3};\frac{1}{3},1;\frac{7}{3};\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )}{4 b \sqrt [3]{c+d x} (e+f x) (a f-b e) \left (7 (b c-a d) (b e-a f) F_1\left (\frac{4}{3};\frac{1}{3},1;\frac{7}{3};\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )+(a+b x) \left ((3 a d f-3 b c f) F_1\left (\frac{7}{3};\frac{1}{3},2;\frac{10}{3};\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )+d (a f-b e) F_1\left (\frac{7}{3};\frac{4}{3},1;\frac{10}{3};\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(-21*(b*c - a*d)*(b*e - a*f)^2*(a + b*x)^(4/3)*AppellF1[4/3, 1/3, 1, 7/3, (d*(a
+ b*x))/(-(b*c) + a*d), (f*(a + b*x))/(-(b*e) + a*f)])/(4*b*(-(b*e) + a*f)*(c +
d*x)^(1/3)*(e + f*x)*(7*(b*c - a*d)*(b*e - a*f)*AppellF1[4/3, 1/3, 1, 7/3, (d*(a
+ b*x))/(-(b*c) + a*d), (f*(a + b*x))/(-(b*e) + a*f)] + (a + b*x)*((-3*b*c*f +
3*a*d*f)*AppellF1[7/3, 1/3, 2, 10/3, (d*(a + b*x))/(-(b*c) + a*d), (f*(a + b*x))
/(-(b*e) + a*f)] + d*(-(b*e) + a*f)*AppellF1[7/3, 4/3, 1, 10/3, (d*(a + b*x))/(-
(b*c) + a*d), (f*(a + b*x))/(-(b*e) + a*f)])))

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Maple [F]  time = 0.077, size = 0, normalized size = 0. \[ \int{\frac{1}{fx+e}\sqrt [3]{bx+a}{\frac{1}{\sqrt [3]{dx+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.273083, size = 652, normalized size = 1.92 \[ -\frac{2 \, \sqrt{3} \left (\frac{b e - a f}{d e - c f}\right )^{\frac{1}{3}} \arctan \left (\frac{\sqrt{3}{\left ({\left (d x + c\right )} \left (\frac{b e - a f}{d e - c f}\right )^{\frac{1}{3}} + 2 \,{\left (b x + a\right )}^{\frac{1}{3}}{\left (d x + c\right )}^{\frac{2}{3}}\right )}}{3 \,{\left (d x + c\right )} \left (\frac{b e - a f}{d e - c f}\right )^{\frac{1}{3}}}\right ) - 2 \, \sqrt{3} \left (-\frac{b}{d}\right )^{\frac{1}{3}} \arctan \left (-\frac{\sqrt{3}{\left ({\left (d x + c\right )} \left (-\frac{b}{d}\right )^{\frac{1}{3}} - 2 \,{\left (b x + a\right )}^{\frac{1}{3}}{\left (d x + c\right )}^{\frac{2}{3}}\right )}}{3 \,{\left (d x + c\right )} \left (-\frac{b}{d}\right )^{\frac{1}{3}}}\right ) + \left (\frac{b e - a f}{d e - c f}\right )^{\frac{1}{3}} \log \left (\frac{{\left (d x + c\right )} \left (\frac{b e - a f}{d e - c f}\right )^{\frac{2}{3}} +{\left (b x + a\right )}^{\frac{1}{3}}{\left (d x + c\right )}^{\frac{2}{3}} \left (\frac{b e - a f}{d e - c f}\right )^{\frac{1}{3}} +{\left (b x + a\right )}^{\frac{2}{3}}{\left (d x + c\right )}^{\frac{1}{3}}}{d x + c}\right ) + \left (-\frac{b}{d}\right )^{\frac{1}{3}} \log \left (\frac{{\left (d x + c\right )} \left (-\frac{b}{d}\right )^{\frac{2}{3}} -{\left (b x + a\right )}^{\frac{1}{3}}{\left (d x + c\right )}^{\frac{2}{3}} \left (-\frac{b}{d}\right )^{\frac{1}{3}} +{\left (b x + a\right )}^{\frac{2}{3}}{\left (d x + c\right )}^{\frac{1}{3}}}{d x + c}\right ) - 2 \, \left (\frac{b e - a f}{d e - c f}\right )^{\frac{1}{3}} \log \left (-\frac{{\left (d x + c\right )} \left (\frac{b e - a f}{d e - c f}\right )^{\frac{1}{3}} -{\left (b x + a\right )}^{\frac{1}{3}}{\left (d x + c\right )}^{\frac{2}{3}}}{d x + c}\right ) - 2 \, \left (-\frac{b}{d}\right )^{\frac{1}{3}} \log \left (\frac{{\left (d x + c\right )} \left (-\frac{b}{d}\right )^{\frac{1}{3}} +{\left (b x + a\right )}^{\frac{1}{3}}{\left (d x + c\right )}^{\frac{2}{3}}}{d x + c}\right )}{2 \, f} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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