3.2996 \(\int \frac{\sqrt [3]{a+b x}}{\sqrt [3]{c+d x} (e+f x)^3} \, dx\)

Optimal. Leaf size=386 \[ -\frac{f (a+b x)^{4/3} (c+d x)^{2/3}}{2 (e+f x)^2 (b e-a f) (d e-c f)}+\frac{\sqrt [3]{a+b x} (c+d x)^{2/3} (-2 a d f-b c f+3 b d e)}{3 (e+f x) (b e-a f) (d e-c f)^2}-\frac{(b c-a d) \log (e+f x) (-2 a d f-b c f+3 b d e)}{18 (b e-a f)^{5/3} (d e-c f)^{7/3}}+\frac{(b c-a d) (-2 a d f-b c f+3 b d e) \log \left (\frac{\sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt [3]{d e-c f}}-\sqrt [3]{a+b x}\right )}{6 (b e-a f)^{5/3} (d e-c f)^{7/3}}+\frac{(b c-a d) (-2 a d f-b c f+3 b d e) \tan ^{-1}\left (\frac{2 \sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt{3} \sqrt [3]{a+b x} \sqrt [3]{d e-c f}}+\frac{1}{\sqrt{3}}\right )}{3 \sqrt{3} (b e-a f)^{5/3} (d e-c f)^{7/3}} \]

[Out]

-(f*(a + b*x)^(4/3)*(c + d*x)^(2/3))/(2*(b*e - a*f)*(d*e - c*f)*(e + f*x)^2) + (
(3*b*d*e - b*c*f - 2*a*d*f)*(a + b*x)^(1/3)*(c + d*x)^(2/3))/(3*(b*e - a*f)*(d*e
- c*f)^2*(e + f*x)) + ((b*c - a*d)*(3*b*d*e - b*c*f - 2*a*d*f)*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))])/(3*Sqrt[3]*(b*e - a*f)^(5/3)*(d*e - c*f)^(7/3)) - ((b*c - a*d)*(3*b*d*e -
b*c*f - 2*a*d*f)*Log[e + f*x])/(18*(b*e - a*f)^(5/3)*(d*e - c*f)^(7/3)) + ((b*c
- a*d)*(3*b*d*e - b*c*f - 2*a*d*f)*Log[-(a + b*x)^(1/3) + ((b*e - a*f)^(1/3)*(c
+ d*x)^(1/3))/(d*e - c*f)^(1/3)])/(6*(b*e - a*f)^(5/3)*(d*e - c*f)^(7/3))

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Rubi [A]  time = 0.672101, antiderivative size = 386, 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 \[ -\frac{f (a+b x)^{4/3} (c+d x)^{2/3}}{2 (e+f x)^2 (b e-a f) (d e-c f)}+\frac{\sqrt [3]{a+b x} (c+d x)^{2/3} (-2 a d f-b c f+3 b d e)}{3 (e+f x) (b e-a f) (d e-c f)^2}-\frac{(b c-a d) \log (e+f x) (-2 a d f-b c f+3 b d e)}{18 (b e-a f)^{5/3} (d e-c f)^{7/3}}+\frac{(b c-a d) (-2 a d f-b c f+3 b d e) \log \left (\frac{\sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt [3]{d e-c f}}-\sqrt [3]{a+b x}\right )}{6 (b e-a f)^{5/3} (d e-c f)^{7/3}}+\frac{(b c-a d) (-2 a d f-b c f+3 b d e) \tan ^{-1}\left (\frac{2 \sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt{3} \sqrt [3]{a+b x} \sqrt [3]{d e-c f}}+\frac{1}{\sqrt{3}}\right )}{3 \sqrt{3} (b e-a f)^{5/3} (d e-c f)^{7/3}} \]

Antiderivative was successfully verified.

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

[Out]

-(f*(a + b*x)^(4/3)*(c + d*x)^(2/3))/(2*(b*e - a*f)*(d*e - c*f)*(e + f*x)^2) + (
(3*b*d*e - b*c*f - 2*a*d*f)*(a + b*x)^(1/3)*(c + d*x)^(2/3))/(3*(b*e - a*f)*(d*e
- c*f)^2*(e + f*x)) + ((b*c - a*d)*(3*b*d*e - b*c*f - 2*a*d*f)*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))])/(3*Sqrt[3]*(b*e - a*f)^(5/3)*(d*e - c*f)^(7/3)) - ((b*c - a*d)*(3*b*d*e -
b*c*f - 2*a*d*f)*Log[e + f*x])/(18*(b*e - a*f)^(5/3)*(d*e - c*f)^(7/3)) + ((b*c
- a*d)*(3*b*d*e - b*c*f - 2*a*d*f)*Log[-(a + b*x)^(1/3) + ((b*e - a*f)^(1/3)*(c
+ d*x)^(1/3))/(d*e - c*f)^(1/3)])/(6*(b*e - a*f)^(5/3)*(d*e - c*f)^(7/3))

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Rubi in Sympy [A]  time = 76.2613, size = 345, normalized size = 0.89 \[ - \frac{f \left (a + b x\right )^{\frac{4}{3}} \left (c + d x\right )^{\frac{2}{3}}}{2 \left (e + f x\right )^{2} \left (a f - b e\right ) \left (c f - d e\right )} + \frac{\sqrt [3]{a + b x} \left (c + d x\right )^{\frac{2}{3}} \left (2 a d f + b c f - 3 b d e\right )}{3 \left (e + f x\right ) \left (a f - b e\right ) \left (c f - d e\right )^{2}} - \frac{\left (a d - b c\right ) \left (2 a d f + b c f - 3 b d e\right ) \log{\left (e + f x \right )}}{18 \left (a f - b e\right )^{\frac{5}{3}} \left (c f - d e\right )^{\frac{7}{3}}} + \frac{\left (a d - b c\right ) \left (2 a d f + b c f - 3 b d e\right ) \log{\left (- \sqrt [3]{a + b x} + \frac{\sqrt [3]{c + d x} \sqrt [3]{a f - b e}}{\sqrt [3]{c f - d e}} \right )}}{6 \left (a f - b e\right )^{\frac{5}{3}} \left (c f - d e\right )^{\frac{7}{3}}} + \frac{\sqrt{3} \left (a d - b c\right ) \left (2 a d f + b c f - 3 b d e\right ) \operatorname{atan}{\left (\frac{\sqrt{3}}{3} + \frac{2 \sqrt{3} \sqrt [3]{c + d x} \sqrt [3]{a f - b e}}{3 \sqrt [3]{a + b x} \sqrt [3]{c f - d e}} \right )}}{9 \left (a f - b e\right )^{\frac{5}{3}} \left (c f - d e\right )^{\frac{7}{3}}} \]

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)**3,x)

[Out]

-f*(a + b*x)**(4/3)*(c + d*x)**(2/3)/(2*(e + f*x)**2*(a*f - b*e)*(c*f - d*e)) +
(a + b*x)**(1/3)*(c + d*x)**(2/3)*(2*a*d*f + b*c*f - 3*b*d*e)/(3*(e + f*x)*(a*f
- b*e)*(c*f - d*e)**2) - (a*d - b*c)*(2*a*d*f + b*c*f - 3*b*d*e)*log(e + f*x)/(1
8*(a*f - b*e)**(5/3)*(c*f - d*e)**(7/3)) + (a*d - b*c)*(2*a*d*f + b*c*f - 3*b*d*
e)*log(-(a + b*x)**(1/3) + (c + d*x)**(1/3)*(a*f - b*e)**(1/3)/(c*f - d*e)**(1/3
))/(6*(a*f - b*e)**(5/3)*(c*f - d*e)**(7/3)) + sqrt(3)*(a*d - b*c)*(2*a*d*f + b*
c*f - 3*b*d*e)*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)))/(9*(a*f - b*e)**(5/3)*(c*f - d*e)**(7/3))

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Mathematica [C]  time = 1.16912, size = 212, normalized size = 0.55 \[ \frac{\sqrt [3]{a+b x} \left (2 (e+f x)^2 (b c-a d) (2 a d f+b c f-3 b d e) \sqrt [3]{\frac{(c+d x) (b e-a f)}{(e+f x) (b c-a d)}} \, _2F_1\left (\frac{1}{3},\frac{1}{3};\frac{4}{3};\frac{(c f-d e) (a+b x)}{(b c-a d) (e+f x)}\right )+(c+d x) (b e-a f) ((e+f x) (-4 a d f+b c f+3 b d e)+3 (b e-a f) (d e-c f))\right )}{6 \sqrt [3]{c+d x} (e+f x)^2 (b e-a f)^2 (d e-c f)^2} \]

Antiderivative was successfully verified.

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

[Out]

((a + b*x)^(1/3)*((b*e - a*f)*(c + d*x)*(3*(b*e - a*f)*(d*e - c*f) + (3*b*d*e +
b*c*f - 4*a*d*f)*(e + f*x)) + 2*(b*c - a*d)*(-3*b*d*e + b*c*f + 2*a*d*f)*(((b*e
- a*f)*(c + d*x))/((b*c - a*d)*(e + f*x)))^(1/3)*(e + f*x)^2*Hypergeometric2F1[1
/3, 1/3, 4/3, ((-(d*e) + c*f)*(a + b*x))/((b*c - a*d)*(e + f*x))]))/(6*(b*e - a*
f)^2*(d*e - c*f)^2*(c + d*x)^(1/3)*(e + f*x)^2)

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Maple [F]  time = 0.082, size = 0, normalized size = 0. \[ \int{\frac{1}{ \left ( fx+e \right ) ^{3}}\sqrt [3]{bx+a}{\frac{1}{\sqrt [3]{dx+c}}}}\, dx \]

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)^3,x)

[Out]

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

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

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)^3),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.258156, size = 1683, normalized size = 4.36 \[ \text{result too large to display} \]

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)^3),x, algorithm="fricas")

[Out]

1/54*sqrt(3)*(3*sqrt(3)*(b^2*d*e^3 - a^2*c*f^3 - (b^2*c + 2*a*b*d)*e^2*f + (2*a*
b*c + a^2*d)*e*f^2)^(1/3)*(6*b*d*e^2 + 3*a*c*f^2 - (2*b*c + 7*a*d)*e*f + (3*b*d*
e*f + (b*c - 4*a*d)*f^2)*x)*(b*x + a)^(1/3)*(d*x + c)^(2/3) - sqrt(3)*(3*(b^2*c*
d - a*b*d^2)*e^3 - (b^2*c^2 + a*b*c*d - 2*a^2*d^2)*e^2*f + (3*(b^2*c*d - a*b*d^2
)*e*f^2 - (b^2*c^2 + a*b*c*d - 2*a^2*d^2)*f^3)*x^2 + 2*(3*(b^2*c*d - a*b*d^2)*e^
2*f - (b^2*c^2 + a*b*c*d - 2*a^2*d^2)*e*f^2)*x)*log((b^2*c*e^2 - 2*a*b*c*e*f + a
^2*c*f^2 + (b^2*d*e^3 - a^2*c*f^3 - (b^2*c + 2*a*b*d)*e^2*f + (2*a*b*c + a^2*d)*
e*f^2)^(1/3)*(b*e - a*f)*(b*x + a)^(1/3)*(d*x + c)^(2/3) + (b^2*d*e^2 - 2*a*b*d*
e*f + a^2*d*f^2)*x + (b^2*d*e^3 - a^2*c*f^3 - (b^2*c + 2*a*b*d)*e^2*f + (2*a*b*c
+ a^2*d)*e*f^2)^(2/3)*(b*x + a)^(2/3)*(d*x + c)^(1/3))/(d*x + c)) + 2*sqrt(3)*(
3*(b^2*c*d - a*b*d^2)*e^3 - (b^2*c^2 + a*b*c*d - 2*a^2*d^2)*e^2*f + (3*(b^2*c*d
- a*b*d^2)*e*f^2 - (b^2*c^2 + a*b*c*d - 2*a^2*d^2)*f^3)*x^2 + 2*(3*(b^2*c*d - a*
b*d^2)*e^2*f - (b^2*c^2 + a*b*c*d - 2*a^2*d^2)*e*f^2)*x)*log(-(b*c*e - a*c*f + (
b*d*e - a*d*f)*x - (b^2*d*e^3 - a^2*c*f^3 - (b^2*c + 2*a*b*d)*e^2*f + (2*a*b*c +
a^2*d)*e*f^2)^(1/3)*(b*x + a)^(1/3)*(d*x + c)^(2/3))/(d*x + c)) + 6*(3*(b^2*c*d
- a*b*d^2)*e^3 - (b^2*c^2 + a*b*c*d - 2*a^2*d^2)*e^2*f + (3*(b^2*c*d - a*b*d^2)
*e*f^2 - (b^2*c^2 + a*b*c*d - 2*a^2*d^2)*f^3)*x^2 + 2*(3*(b^2*c*d - a*b*d^2)*e^2
*f - (b^2*c^2 + a*b*c*d - 2*a^2*d^2)*e*f^2)*x)*arctan(-1/3*(2*sqrt(3)*(b^2*d*e^3
- a^2*c*f^3 - (b^2*c + 2*a*b*d)*e^2*f + (2*a*b*c + a^2*d)*e*f^2)^(1/3)*(b*x + a
)^(1/3)*(d*x + c)^(2/3) + sqrt(3)*(b*c*e - a*c*f + (b*d*e - a*d*f)*x))/(b*c*e -
a*c*f + (b*d*e - a*d*f)*x)))/((b*d^2*e^5 - a*c^2*e^2*f^3 - (2*b*c*d + a*d^2)*e^4
*f + (b*c^2 + 2*a*c*d)*e^3*f^2 + (b*d^2*e^3*f^2 - a*c^2*f^5 - (2*b*c*d + a*d^2)*
e^2*f^3 + (b*c^2 + 2*a*c*d)*e*f^4)*x^2 + 2*(b*d^2*e^4*f - a*c^2*e*f^4 - (2*b*c*d
+ a*d^2)*e^3*f^2 + (b*c^2 + 2*a*c*d)*e^2*f^3)*x)*(b^2*d*e^3 - a^2*c*f^3 - (b^2*
c + 2*a*b*d)*e^2*f + (2*a*b*c + a^2*d)*e*f^2)^(1/3))

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

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)**3,x)

[Out]

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

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

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)^3),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError