3.2997 \(\int \frac{\sqrt [3]{a+b x}}{\sqrt [3]{c+d x} (e+f x)^4} \, dx\)

Optimal. Leaf size=591 \[ \frac{\sqrt [3]{a+b x} (c+d x)^{2/3} \left (28 a^2 d^2 f^2-a b d f (5 c f+51 d e)+b^2 \left (-5 c^2 f^2+15 c d e f+18 d^2 e^2\right )\right )}{54 (e+f x) (b e-a f)^2 (d e-c f)^3}-\frac{(b c-a d) \log (e+f x) \left (14 a^2 d^2 f^2-4 a b d f (9 d e-2 c f)+b^2 \left (5 c^2 f^2-18 c d e f+27 d^2 e^2\right )\right )}{162 (b e-a f)^{8/3} (d e-c f)^{10/3}}+\frac{(b c-a d) \left (14 a^2 d^2 f^2-4 a b d f (9 d e-2 c f)+b^2 \left (5 c^2 f^2-18 c d e f+27 d^2 e^2\right )\right ) \log \left (\frac{\sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt [3]{d e-c f}}-\sqrt [3]{a+b x}\right )}{54 (b e-a f)^{8/3} (d e-c f)^{10/3}}+\frac{(b c-a d) \left (14 a^2 d^2 f^2-4 a b d f (9 d e-2 c f)+b^2 \left (5 c^2 f^2-18 c d e f+27 d^2 e^2\right )\right ) \tan ^{-1}\left (\frac{2 \sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt{3} \sqrt [3]{a+b x} \sqrt [3]{d e-c f}}+\frac{1}{\sqrt{3}}\right )}{27 \sqrt{3} (b e-a f)^{8/3} (d e-c f)^{10/3}}+\frac{\sqrt [3]{a+b x} (c+d x)^{2/3} (-7 a d f+b c f+6 b d e)}{18 (e+f x)^2 (b e-a f) (d e-c f)^2}+\frac{\sqrt [3]{a+b x} (c+d x)^{2/3}}{3 (e+f x)^3 (d e-c f)} \]

[Out]

((a + b*x)^(1/3)*(c + d*x)^(2/3))/(3*(d*e - c*f)*(e + f*x)^3) + ((6*b*d*e + b*c*
f - 7*a*d*f)*(a + b*x)^(1/3)*(c + d*x)^(2/3))/(18*(b*e - a*f)*(d*e - c*f)^2*(e +
f*x)^2) + ((28*a^2*d^2*f^2 - a*b*d*f*(51*d*e + 5*c*f) + b^2*(18*d^2*e^2 + 15*c*
d*e*f - 5*c^2*f^2))*(a + b*x)^(1/3)*(c + d*x)^(2/3))/(54*(b*e - a*f)^2*(d*e - c*
f)^3*(e + f*x)) + ((b*c - a*d)*(14*a^2*d^2*f^2 - 4*a*b*d*f*(9*d*e - 2*c*f) + b^2
*(27*d^2*e^2 - 18*c*d*e*f + 5*c^2*f^2))*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)^(10/3)) - ((b*c - a*d)*(14*a^2*d^2*f^2 - 4*a*b*d*f*(9*d
*e - 2*c*f) + b^2*(27*d^2*e^2 - 18*c*d*e*f + 5*c^2*f^2))*Log[e + f*x])/(162*(b*e
- a*f)^(8/3)*(d*e - c*f)^(10/3)) + ((b*c - a*d)*(14*a^2*d^2*f^2 - 4*a*b*d*f*(9*
d*e - 2*c*f) + b^2*(27*d^2*e^2 - 18*c*d*e*f + 5*c^2*f^2))*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)^(10/3))

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Rubi [A]  time = 2.68231, antiderivative size = 591, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{\sqrt [3]{a+b x} (c+d x)^{2/3} \left (28 a^2 d^2 f^2-a b d f (5 c f+51 d e)+b^2 \left (-5 c^2 f^2+15 c d e f+18 d^2 e^2\right )\right )}{54 (e+f x) (b e-a f)^2 (d e-c f)^3}-\frac{(b c-a d) \log (e+f x) \left (14 a^2 d^2 f^2-4 a b d f (9 d e-2 c f)+b^2 \left (5 c^2 f^2-18 c d e f+27 d^2 e^2\right )\right )}{162 (b e-a f)^{8/3} (d e-c f)^{10/3}}+\frac{(b c-a d) \left (14 a^2 d^2 f^2-4 a b d f (9 d e-2 c f)+b^2 \left (5 c^2 f^2-18 c d e f+27 d^2 e^2\right )\right ) \log \left (\frac{\sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt [3]{d e-c f}}-\sqrt [3]{a+b x}\right )}{54 (b e-a f)^{8/3} (d e-c f)^{10/3}}+\frac{(b c-a d) \left (14 a^2 d^2 f^2-4 a b d f (9 d e-2 c f)+b^2 \left (5 c^2 f^2-18 c d e f+27 d^2 e^2\right )\right ) \tan ^{-1}\left (\frac{2 \sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt{3} \sqrt [3]{a+b x} \sqrt [3]{d e-c f}}+\frac{1}{\sqrt{3}}\right )}{27 \sqrt{3} (b e-a f)^{8/3} (d e-c f)^{10/3}}+\frac{\sqrt [3]{a+b x} (c+d x)^{2/3} (-7 a d f+b c f+6 b d e)}{18 (e+f x)^2 (b e-a f) (d e-c f)^2}+\frac{\sqrt [3]{a+b x} (c+d x)^{2/3}}{3 (e+f x)^3 (d e-c f)} \]

Antiderivative was successfully verified.

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

[Out]

((a + b*x)^(1/3)*(c + d*x)^(2/3))/(3*(d*e - c*f)*(e + f*x)^3) + ((6*b*d*e + b*c*
f - 7*a*d*f)*(a + b*x)^(1/3)*(c + d*x)^(2/3))/(18*(b*e - a*f)*(d*e - c*f)^2*(e +
f*x)^2) + ((28*a^2*d^2*f^2 - a*b*d*f*(51*d*e + 5*c*f) + b^2*(18*d^2*e^2 + 15*c*
d*e*f - 5*c^2*f^2))*(a + b*x)^(1/3)*(c + d*x)^(2/3))/(54*(b*e - a*f)^2*(d*e - c*
f)^3*(e + f*x)) + ((b*c - a*d)*(14*a^2*d^2*f^2 - 4*a*b*d*f*(9*d*e - 2*c*f) + b^2
*(27*d^2*e^2 - 18*c*d*e*f + 5*c^2*f^2))*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)^(10/3)) - ((b*c - a*d)*(14*a^2*d^2*f^2 - 4*a*b*d*f*(9*d
*e - 2*c*f) + b^2*(27*d^2*e^2 - 18*c*d*e*f + 5*c^2*f^2))*Log[e + f*x])/(162*(b*e
- a*f)^(8/3)*(d*e - c*f)^(10/3)) + ((b*c - a*d)*(14*a^2*d^2*f^2 - 4*a*b*d*f*(9*
d*e - 2*c*f) + b^2*(27*d^2*e^2 - 18*c*d*e*f + 5*c^2*f^2))*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)^(10/3))

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

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)**4,x)

[Out]

Timed out

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Mathematica [C]  time = 1.4343, size = 334, normalized size = 0.57 \[ \frac{\sqrt [3]{a+b x} \left ((c+d x) (b e-a f) \left ((e+f x)^2 \left (28 a^2 d^2 f^2-a b d f (5 c f+51 d e)+b^2 \left (-5 c^2 f^2+15 c d e f+18 d^2 e^2\right )\right )+3 (e+f x) (b e-a f) (d e-c f) (-7 a d f+b c f+6 b d e)+18 (b e-a f)^2 (d e-c f)^2\right )-2 (e+f x)^3 (b c-a d) \left (14 a^2 d^2 f^2+4 a b d f (2 c f-9 d e)+b^2 \left (5 c^2 f^2-18 c d e f+27 d^2 e^2\right )\right ) \sqrt [3]{\frac{(c+d x) (b e-a f)}{(e+f x) (b c-a d)}} \, _2F_1\left (\frac{1}{3},\frac{1}{3};\frac{4}{3};\frac{(c f-d e) (a+b x)}{(b c-a d) (e+f x)}\right )\right )}{54 \sqrt [3]{c+d x} (e+f x)^3 (b e-a f)^3 (d e-c f)^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^(1/3)/((c + d*x)^(1/3)*(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)*(6*b*d*e + b*c*f - 7*a*d*f)*(e + f*x) + (28*a^2*d^2*f^2 - a*
b*d*f*(51*d*e + 5*c*f) + b^2*(18*d^2*e^2 + 15*c*d*e*f - 5*c^2*f^2))*(e + f*x)^2)
- 2*(b*c - a*d)*(14*a^2*d^2*f^2 + 4*a*b*d*f*(-9*d*e + 2*c*f) + b^2*(27*d^2*e^2
- 18*c*d*e*f + 5*c^2*f^2))*(((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*(b*e - a*f)^3*(d*e - c*f)^3*(c + d*x)^(1/3)*(e + f*x)^
3)

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Maple [F]  time = 0.083, size = 0, normalized size = 0. \[ \int{\frac{1}{ \left ( fx+e \right ) ^{4}}\sqrt [3]{bx+a}{\frac{1}{\sqrt [3]{dx+c}}}}\, dx \]

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)^4,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.363699, size = 3586, normalized size = 6.07 \[ \text{result too large to display} \]

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)^4),x, algorithm="fricas")

[Out]

1/486*sqrt(3)*(3*sqrt(3)*(54*b^2*d^2*e^4 + 18*a^2*c^2*f^4 - 18*(2*b^2*c*d + 7*a*
b*d^2)*e^3*f + (10*b^2*c^2 + 103*a*b*c*d + 67*a^2*d^2)*e^2*f^2 - 3*(11*a*b*c^2 +
19*a^2*c*d)*e*f^3 + (18*b^2*d^2*e^2*f^2 + 3*(5*b^2*c*d - 17*a*b*d^2)*e*f^3 - (5
*b^2*c^2 + 5*a*b*c*d - 28*a^2*d^2)*f^4)*x^2 + (54*b^2*d^2*e^3*f + 3*(5*b^2*c*d -
47*a*b*d^2)*e^2*f^2 - (13*b^2*c^2 - 26*a*b*c*d - 77*a^2*d^2)*e*f^3 + 3*(a*b*c^2
- 7*a^2*c*d)*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)*(b*x + a)^(1/3)*(d*x + c)^(2/3) - sqrt(3)*(27*(b^3*c*d^2
- a*b^2*d^3)*e^5 - 18*(b^3*c^2*d + a*b^2*c*d^2 - 2*a^2*b*d^3)*e^4*f + (5*b^3*c^
3 + 3*a*b^2*c^2*d + 6*a^2*b*c*d^2 - 14*a^3*d^3)*e^3*f^2 + (27*(b^3*c*d^2 - a*b^2
*d^3)*e^2*f^3 - 18*(b^3*c^2*d + a*b^2*c*d^2 - 2*a^2*b*d^3)*e*f^4 + (5*b^3*c^3 +
3*a*b^2*c^2*d + 6*a^2*b*c*d^2 - 14*a^3*d^3)*f^5)*x^3 + 3*(27*(b^3*c*d^2 - a*b^2*
d^3)*e^3*f^2 - 18*(b^3*c^2*d + a*b^2*c*d^2 - 2*a^2*b*d^3)*e^2*f^3 + (5*b^3*c^3 +
3*a*b^2*c^2*d + 6*a^2*b*c*d^2 - 14*a^3*d^3)*e*f^4)*x^2 + 3*(27*(b^3*c*d^2 - a*b
^2*d^3)*e^4*f - 18*(b^3*c^2*d + a*b^2*c*d^2 - 2*a^2*b*d^3)*e^3*f^2 + (5*b^3*c^3
+ 3*a*b^2*c^2*d + 6*a^2*b*c*d^2 - 14*a^3*d^3)*e^2*f^3)*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)*(27*(b^3*c*d^2 - a*b^2*d^3)*e^5 - 18*(b^3*c^2*d + a*b^2*c*d^2 - 2*a^2*b
*d^3)*e^4*f + (5*b^3*c^3 + 3*a*b^2*c^2*d + 6*a^2*b*c*d^2 - 14*a^3*d^3)*e^3*f^2 +
(27*(b^3*c*d^2 - a*b^2*d^3)*e^2*f^3 - 18*(b^3*c^2*d + a*b^2*c*d^2 - 2*a^2*b*d^3
)*e*f^4 + (5*b^3*c^3 + 3*a*b^2*c^2*d + 6*a^2*b*c*d^2 - 14*a^3*d^3)*f^5)*x^3 + 3*
(27*(b^3*c*d^2 - a*b^2*d^3)*e^3*f^2 - 18*(b^3*c^2*d + a*b^2*c*d^2 - 2*a^2*b*d^3)
*e^2*f^3 + (5*b^3*c^3 + 3*a*b^2*c^2*d + 6*a^2*b*c*d^2 - 14*a^3*d^3)*e*f^4)*x^2 +
3*(27*(b^3*c*d^2 - a*b^2*d^3)*e^4*f - 18*(b^3*c^2*d + a*b^2*c*d^2 - 2*a^2*b*d^3
)*e^3*f^2 + (5*b^3*c^3 + 3*a*b^2*c^2*d + 6*a^2*b*c*d^2 - 14*a^3*d^3)*e^2*f^3)*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*(27*(b^3*c*d^2 - a*b^2*d^3)*e^5 - 18*(b^3*c^2*d + a*b^2*c*d^2 - 2*a
^2*b*d^3)*e^4*f + (5*b^3*c^3 + 3*a*b^2*c^2*d + 6*a^2*b*c*d^2 - 14*a^3*d^3)*e^3*f
^2 + (27*(b^3*c*d^2 - a*b^2*d^3)*e^2*f^3 - 18*(b^3*c^2*d + a*b^2*c*d^2 - 2*a^2*b
*d^3)*e*f^4 + (5*b^3*c^3 + 3*a*b^2*c^2*d + 6*a^2*b*c*d^2 - 14*a^3*d^3)*f^5)*x^3
+ 3*(27*(b^3*c*d^2 - a*b^2*d^3)*e^3*f^2 - 18*(b^3*c^2*d + a*b^2*c*d^2 - 2*a^2*b*
d^3)*e^2*f^3 + (5*b^3*c^3 + 3*a*b^2*c^2*d + 6*a^2*b*c*d^2 - 14*a^3*d^3)*e*f^4)*x
^2 + 3*(27*(b^3*c*d^2 - a*b^2*d^3)*e^4*f - 18*(b^3*c^2*d + a*b^2*c*d^2 - 2*a^2*b
*d^3)*e^3*f^2 + (5*b^3*c^3 + 3*a*b^2*c^2*d + 6*a^2*b*c*d^2 - 14*a^3*d^3)*e^2*f^3
)*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^3*e^8
- a^2*c^3*e^3*f^5 - (3*b^2*c*d^2 + 2*a*b*d^3)*e^7*f + (3*b^2*c^2*d + 6*a*b*c*d^
2 + a^2*d^3)*e^6*f^2 - (b^2*c^3 + 6*a*b*c^2*d + 3*a^2*c*d^2)*e^5*f^3 + (2*a*b*c^
3 + 3*a^2*c^2*d)*e^4*f^4 + (b^2*d^3*e^5*f^3 - a^2*c^3*f^8 - (3*b^2*c*d^2 + 2*a*b
*d^3)*e^4*f^4 + (3*b^2*c^2*d + 6*a*b*c*d^2 + a^2*d^3)*e^3*f^5 - (b^2*c^3 + 6*a*b
*c^2*d + 3*a^2*c*d^2)*e^2*f^6 + (2*a*b*c^3 + 3*a^2*c^2*d)*e*f^7)*x^3 + 3*(b^2*d^
3*e^6*f^2 - a^2*c^3*e*f^7 - (3*b^2*c*d^2 + 2*a*b*d^3)*e^5*f^3 + (3*b^2*c^2*d + 6
*a*b*c*d^2 + a^2*d^3)*e^4*f^4 - (b^2*c^3 + 6*a*b*c^2*d + 3*a^2*c*d^2)*e^3*f^5 +
(2*a*b*c^3 + 3*a^2*c^2*d)*e^2*f^6)*x^2 + 3*(b^2*d^3*e^7*f - a^2*c^3*e^2*f^6 - (3
*b^2*c*d^2 + 2*a*b*d^3)*e^6*f^2 + (3*b^2*c^2*d + 6*a*b*c*d^2 + a^2*d^3)*e^5*f^3
- (b^2*c^3 + 6*a*b*c^2*d + 3*a^2*c*d^2)*e^4*f^4 + (2*a*b*c^3 + 3*a^2*c^2*d)*e^3*
f^5)*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(-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)**(1/3)/(f*x+e)**4,x)

[Out]

Timed out

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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)^4),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError