3.2990 \(\int \frac{\sqrt [3]{a+b x} (c+d x)^{2/3}}{(e+f x)^4} \, dx\)

Optimal. Leaf size=465 \[ -\frac{\sqrt [3]{a+b x} (c+d x)^{2/3} (b c-a d) (-4 a d f-5 b c f+9 b d e)}{54 (e+f x) (b e-a f)^2 (d e-c f)^2}+\frac{\sqrt [3]{a+b x} (c+d x)^{5/3} (-4 a d f-5 b c f+9 b d e)}{18 (e+f x)^2 (b e-a f) (d e-c f)^2}-\frac{f (a+b x)^{4/3} (c+d x)^{5/3}}{3 (e+f x)^3 (b e-a f) (d e-c f)}-\frac{(b c-a d)^2 \log (e+f x) (-4 a d f-5 b c f+9 b d e)}{162 (b e-a f)^{8/3} (d e-c f)^{7/3}}+\frac{(b c-a d)^2 (-4 a d f-5 b c f+9 b d e) \log \left (\frac{\sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt [3]{d e-c f}}-\sqrt [3]{a+b x}\right )}{54 (b e-a f)^{8/3} (d e-c f)^{7/3}}+\frac{(b c-a d)^2 (-4 a d f-5 b c f+9 b d e) \tan ^{-1}\left (\frac{2 \sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt{3} \sqrt [3]{a+b x} \sqrt [3]{d e-c f}}+\frac{1}{\sqrt{3}}\right )}{27 \sqrt{3} (b e-a f)^{8/3} (d e-c f)^{7/3}} \]

[Out]

-(f*(a + b*x)^(4/3)*(c + d*x)^(5/3))/(3*(b*e - a*f)*(d*e - c*f)*(e + f*x)^3) + (
(9*b*d*e - 5*b*c*f - 4*a*d*f)*(a + b*x)^(1/3)*(c + d*x)^(5/3))/(18*(b*e - a*f)*(
d*e - c*f)^2*(e + f*x)^2) - ((b*c - a*d)*(9*b*d*e - 5*b*c*f - 4*a*d*f)*(a + b*x)
^(1/3)*(c + d*x)^(2/3))/(54*(b*e - a*f)^2*(d*e - c*f)^2*(e + f*x)) + ((b*c - a*d
)^2*(9*b*d*e - 5*b*c*f - 4*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))])/(27*Sqrt[3]*(b*e - a*f)
^(8/3)*(d*e - c*f)^(7/3)) - ((b*c - a*d)^2*(9*b*d*e - 5*b*c*f - 4*a*d*f)*Log[e +
f*x])/(162*(b*e - a*f)^(8/3)*(d*e - c*f)^(7/3)) + ((b*c - a*d)^2*(9*b*d*e - 5*b
*c*f - 4*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)])/(54*(b*e - a*f)^(8/3)*(d*e - c*f)^(7/3))

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Rubi [A]  time = 1.46312, antiderivative size = 465, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ -\frac{\sqrt [3]{a+b x} (c+d x)^{2/3} (b c-a d) (-4 a d f-5 b c f+9 b d e)}{54 (e+f x) (b e-a f)^2 (d e-c f)^2}+\frac{\sqrt [3]{a+b x} (c+d x)^{5/3} (-4 a d f-5 b c f+9 b d e)}{18 (e+f x)^2 (b e-a f) (d e-c f)^2}-\frac{f (a+b x)^{4/3} (c+d x)^{5/3}}{3 (e+f x)^3 (b e-a f) (d e-c f)}-\frac{(b c-a d)^2 \log (e+f x) (-4 a d f-5 b c f+9 b d e)}{162 (b e-a f)^{8/3} (d e-c f)^{7/3}}+\frac{(b c-a d)^2 (-4 a d f-5 b c f+9 b d e) \log \left (\frac{\sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt [3]{d e-c f}}-\sqrt [3]{a+b x}\right )}{54 (b e-a f)^{8/3} (d e-c f)^{7/3}}+\frac{(b c-a d)^2 (-4 a d f-5 b c f+9 b d e) \tan ^{-1}\left (\frac{2 \sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt{3} \sqrt [3]{a+b x} \sqrt [3]{d e-c f}}+\frac{1}{\sqrt{3}}\right )}{27 \sqrt{3} (b e-a f)^{8/3} (d e-c f)^{7/3}} \]

Antiderivative was successfully verified.

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

[Out]

-(f*(a + b*x)^(4/3)*(c + d*x)^(5/3))/(3*(b*e - a*f)*(d*e - c*f)*(e + f*x)^3) + (
(9*b*d*e - 5*b*c*f - 4*a*d*f)*(a + b*x)^(1/3)*(c + d*x)^(5/3))/(18*(b*e - a*f)*(
d*e - c*f)^2*(e + f*x)^2) - ((b*c - a*d)*(9*b*d*e - 5*b*c*f - 4*a*d*f)*(a + b*x)
^(1/3)*(c + d*x)^(2/3))/(54*(b*e - a*f)^2*(d*e - c*f)^2*(e + f*x)) + ((b*c - a*d
)^2*(9*b*d*e - 5*b*c*f - 4*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))])/(27*Sqrt[3]*(b*e - a*f)
^(8/3)*(d*e - c*f)^(7/3)) - ((b*c - a*d)^2*(9*b*d*e - 5*b*c*f - 4*a*d*f)*Log[e +
f*x])/(162*(b*e - a*f)^(8/3)*(d*e - c*f)^(7/3)) + ((b*c - a*d)^2*(9*b*d*e - 5*b
*c*f - 4*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)])/(54*(b*e - a*f)^(8/3)*(d*e - c*f)^(7/3))

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Rubi in Sympy [A]  time = 123.763, size = 427, normalized size = 0.92 \[ - \frac{f \left (a + b x\right )^{\frac{4}{3}} \left (c + d x\right )^{\frac{5}{3}}}{3 \left (e + f x\right )^{3} \left (a f - b e\right ) \left (c f - d e\right )} + \frac{\left (a + b x\right )^{\frac{4}{3}} \left (c + d x\right )^{\frac{2}{3}} \left (4 a d f + 5 b c f - 9 b d e\right )}{18 \left (e + f x\right )^{2} \left (a f - b e\right )^{2} \left (c f - d e\right )} + \frac{\sqrt [3]{a + b x} \left (c + d x\right )^{\frac{2}{3}} \left (a d - b c\right ) \left (4 a d f + 5 b c f - 9 b d e\right )}{27 \left (e + f x\right ) \left (a f - b e\right )^{2} \left (c f - d e\right )^{2}} - \frac{\left (a d - b c\right )^{2} \left (4 a d f + 5 b c f - 9 b d e\right ) \log{\left (e + f x \right )}}{162 \left (a f - b e\right )^{\frac{8}{3}} \left (c f - d e\right )^{\frac{7}{3}}} + \frac{\left (a d - b c\right )^{2} \left (4 a d f + 5 b c f - 9 b d e\right ) \log{\left (- \sqrt [3]{a + b x} + \frac{\sqrt [3]{c + d x} \sqrt [3]{a f - b e}}{\sqrt [3]{c f - d e}} \right )}}{54 \left (a f - b e\right )^{\frac{8}{3}} \left (c f - d e\right )^{\frac{7}{3}}} + \frac{\sqrt{3} \left (a d - b c\right )^{2} \left (4 a d f + 5 b c f - 9 b d e\right ) \operatorname{atan}{\left (\frac{\sqrt{3}}{3} + \frac{2 \sqrt{3} \sqrt [3]{c + d x} \sqrt [3]{a f - b e}}{3 \sqrt [3]{a + b x} \sqrt [3]{c f - d e}} \right )}}{81 \left (a f - b e\right )^{\frac{8}{3}} \left (c f - d e\right )^{\frac{7}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(1/3)*(d*x+c)**(2/3)/(f*x+e)**4,x)

[Out]

-f*(a + b*x)**(4/3)*(c + d*x)**(5/3)/(3*(e + f*x)**3*(a*f - b*e)*(c*f - d*e)) +
(a + b*x)**(4/3)*(c + d*x)**(2/3)*(4*a*d*f + 5*b*c*f - 9*b*d*e)/(18*(e + f*x)**2
*(a*f - b*e)**2*(c*f - d*e)) + (a + b*x)**(1/3)*(c + d*x)**(2/3)*(a*d - b*c)*(4*
a*d*f + 5*b*c*f - 9*b*d*e)/(27*(e + f*x)*(a*f - b*e)**2*(c*f - d*e)**2) - (a*d -
b*c)**2*(4*a*d*f + 5*b*c*f - 9*b*d*e)*log(e + f*x)/(162*(a*f - b*e)**(8/3)*(c*f
- d*e)**(7/3)) + (a*d - b*c)**2*(4*a*d*f + 5*b*c*f - 9*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))/(54*(a*f - b*e)**
(8/3)*(c*f - d*e)**(7/3)) + sqrt(3)*(a*d - b*c)**2*(4*a*d*f + 5*b*c*f - 9*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)))/(81*(a*f - b*e)**(8/3)*(c*f - d*e)**(7/3))

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Mathematica [C]  time = 1.34406, size = 304, normalized size = 0.65 \[ \frac{\sqrt [3]{a+b x} \left (2 f (e+f x)^3 (b c-a d)^2 (4 a d f+5 b c f-9 b d e) \sqrt [3]{\frac{(c+d x) (b e-a f)}{(e+f x) (b c-a d)}} \, _2F_1\left (\frac{1}{3},\frac{1}{3};\frac{4}{3};\frac{(c f-d e) (a+b x)}{(b c-a d) (e+f x)}\right )-(c+d x) (b e-a f) \left (-(e+f x)^2 \left (8 a^2 d^2 f^2-4 a b d f (c f+3 d e)+b^2 \left (5 c^2 f^2-6 c d e f+9 d^2 e^2\right )\right )-3 (e+f x) (b e-a f) (d e-c f) (-2 a d f-b c f+3 b d e)+18 (b e-a f)^2 (d e-c f)^2\right )\right )}{54 f \sqrt [3]{c+d x} (e+f x)^3 (b e-a f)^3 (d e-c f)^2} \]

Antiderivative was successfully verified.

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

[Out]

((a + b*x)^(1/3)*(-((b*e - a*f)*(c + d*x)*(18*(b*e - a*f)^2*(d*e - c*f)^2 - 3*(b
*e - a*f)*(d*e - c*f)*(3*b*d*e - b*c*f - 2*a*d*f)*(e + f*x) - (8*a^2*d^2*f^2 - 4
*a*b*d*f*(3*d*e + c*f) + b^2*(9*d^2*e^2 - 6*c*d*e*f + 5*c^2*f^2))*(e + f*x)^2))
+ 2*(b*c - a*d)^2*f*(-9*b*d*e + 5*b*c*f + 4*a*d*f)*(((b*e - a*f)*(c + d*x))/((b*
c - a*d)*(e + f*x)))^(1/3)*(e + f*x)^3*Hypergeometric2F1[1/3, 1/3, 4/3, ((-(d*e)
+ c*f)*(a + b*x))/((b*c - a*d)*(e + f*x))]))/(54*f*(b*e - a*f)^3*(d*e - c*f)^2*
(c + d*x)^(1/3)*(e + f*x)^3)

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Maple [F]  time = 0.077, size = 0, normalized size = 0. \[ \int{\frac{1}{ \left ( fx+e \right ) ^{4}}\sqrt [3]{bx+a} \left ( dx+c \right ) ^{{\frac{2}{3}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int((b*x+a)^(1/3)*(d*x+c)^(2/3)/(f*x+e)^4,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{\frac{1}{3}}{\left (d x + c\right )}^{\frac{2}{3}}}{{\left (f x + e\right )}^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(1/3)*(d*x + c)^(2/3)/(f*x + e)^4,x, algorithm="maxima")

[Out]

integrate((b*x + a)^(1/3)*(d*x + c)^(2/3)/(f*x + e)^4, x)

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Fricas [A]  time = 0.301861, size = 2911, normalized size = 6.26 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(1/3)*(d*x + c)^(2/3)/(f*x + e)^4,x, algorithm="fricas")

[Out]

-1/486*sqrt(3)*(3*sqrt(3)*(18*a^2*c^2*f^3 - 9*(2*b^2*c*d + a*b*d^2)*e^3 + 2*(5*b
^2*c^2 + 29*a*b*c*d + 2*a^2*d^2)*e^2*f - 3*(11*a*b*c^2 + 10*a^2*c*d)*e*f^2 - (9*
b^2*d^2*e^2*f - 6*(b^2*c*d + 2*a*b*d^2)*e*f^2 + (5*b^2*c^2 - 4*a*b*c*d + 8*a^2*d
^2)*f^3)*x^2 - (27*b^2*d^2*e^3 - 3*(8*b^2*c*d + 13*a*b*d^2)*e^2*f + (13*b^2*c^2
+ 10*a*b*c*d + 22*a^2*d^2)*e*f^2 - 3*(a*b*c^2 + 2*a^2*c*d)*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)*(b*x + a)^
(1/3)*(d*x + c)^(2/3) - sqrt(3)*(9*(b^3*c^2*d - 2*a*b^2*c*d^2 + a^2*b*d^3)*e^4 -
(5*b^3*c^3 - 6*a*b^2*c^2*d - 3*a^2*b*c*d^2 + 4*a^3*d^3)*e^3*f + (9*(b^3*c^2*d -
2*a*b^2*c*d^2 + a^2*b*d^3)*e*f^3 - (5*b^3*c^3 - 6*a*b^2*c^2*d - 3*a^2*b*c*d^2 +
4*a^3*d^3)*f^4)*x^3 + 3*(9*(b^3*c^2*d - 2*a*b^2*c*d^2 + a^2*b*d^3)*e^2*f^2 - (5
*b^3*c^3 - 6*a*b^2*c^2*d - 3*a^2*b*c*d^2 + 4*a^3*d^3)*e*f^3)*x^2 + 3*(9*(b^3*c^2
*d - 2*a*b^2*c*d^2 + a^2*b*d^3)*e^3*f - (5*b^3*c^3 - 6*a*b^2*c^2*d - 3*a^2*b*c*d
^2 + 4*a^3*d^3)*e^2*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)*(9*(b^3*c^2*d - 2*a
*b^2*c*d^2 + a^2*b*d^3)*e^4 - (5*b^3*c^3 - 6*a*b^2*c^2*d - 3*a^2*b*c*d^2 + 4*a^3
*d^3)*e^3*f + (9*(b^3*c^2*d - 2*a*b^2*c*d^2 + a^2*b*d^3)*e*f^3 - (5*b^3*c^3 - 6*
a*b^2*c^2*d - 3*a^2*b*c*d^2 + 4*a^3*d^3)*f^4)*x^3 + 3*(9*(b^3*c^2*d - 2*a*b^2*c*
d^2 + a^2*b*d^3)*e^2*f^2 - (5*b^3*c^3 - 6*a*b^2*c^2*d - 3*a^2*b*c*d^2 + 4*a^3*d^
3)*e*f^3)*x^2 + 3*(9*(b^3*c^2*d - 2*a*b^2*c*d^2 + a^2*b*d^3)*e^3*f - (5*b^3*c^3
- 6*a*b^2*c^2*d - 3*a^2*b*c*d^2 + 4*a^3*d^3)*e^2*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*(9*(b^3*c^2
*d - 2*a*b^2*c*d^2 + a^2*b*d^3)*e^4 - (5*b^3*c^3 - 6*a*b^2*c^2*d - 3*a^2*b*c*d^2
+ 4*a^3*d^3)*e^3*f + (9*(b^3*c^2*d - 2*a*b^2*c*d^2 + a^2*b*d^3)*e*f^3 - (5*b^3*
c^3 - 6*a*b^2*c^2*d - 3*a^2*b*c*d^2 + 4*a^3*d^3)*f^4)*x^3 + 3*(9*(b^3*c^2*d - 2*
a*b^2*c*d^2 + a^2*b*d^3)*e^2*f^2 - (5*b^3*c^3 - 6*a*b^2*c^2*d - 3*a^2*b*c*d^2 +
4*a^3*d^3)*e*f^3)*x^2 + 3*(9*(b^3*c^2*d - 2*a*b^2*c*d^2 + a^2*b*d^3)*e^3*f - (5*
b^3*c^3 - 6*a*b^2*c^2*d - 3*a^2*b*c*d^2 + 4*a^3*d^3)*e^2*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^2*d^2*e^7 + a^2*c^2*e^3*f^4
- 2*(b^2*c*d + a*b*d^2)*e^6*f + (b^2*c^2 + 4*a*b*c*d + a^2*d^2)*e^5*f^2 - 2*(a*b
*c^2 + a^2*c*d)*e^4*f^3 + (b^2*d^2*e^4*f^3 + a^2*c^2*f^7 - 2*(b^2*c*d + a*b*d^2)
*e^3*f^4 + (b^2*c^2 + 4*a*b*c*d + a^2*d^2)*e^2*f^5 - 2*(a*b*c^2 + a^2*c*d)*e*f^6
)*x^3 + 3*(b^2*d^2*e^5*f^2 + a^2*c^2*e*f^6 - 2*(b^2*c*d + a*b*d^2)*e^4*f^3 + (b^
2*c^2 + 4*a*b*c*d + a^2*d^2)*e^3*f^4 - 2*(a*b*c^2 + a^2*c*d)*e^2*f^5)*x^2 + 3*(b
^2*d^2*e^6*f + a^2*c^2*e^2*f^5 - 2*(b^2*c*d + a*b*d^2)*e^5*f^2 + (b^2*c^2 + 4*a*
b*c*d + a^2*d^2)*e^4*f^3 - 2*(a*b*c^2 + a^2*c*d)*e^3*f^4)*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))

_______________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**(1/3)*(d*x+c)**(2/3)/(f*x+e)**4,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 1.05762, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(1/3)*(d*x + c)^(2/3)/(f*x + e)^4,x, algorithm="giac")

[Out]

Done