3.3026 \(\int \frac{(a+b x)^{4/3}}{(c+d x)^{4/3} (e+f x)^4} \, dx\)

Optimal. Leaf size=645 \[ \frac{\sqrt [3]{a+b x} (c+d x)^{2/3} \left (140 a^2 d^2 f^2-7 a b d f (19 c f+21 d e)+b^2 \left (2 c^2 f^2+129 c d e f+9 d^2 e^2\right )\right )}{27 (e+f x) (b e-a f) (d e-c f)^4}-\frac{2 (b c-a d) \log (e+f x) \left (35 a^2 d^2 f^2-7 a b d f (c f+9 d e)+b^2 \left (-c^2 f^2+9 c d e f+27 d^2 e^2\right )\right )}{81 (b e-a f)^{5/3} (d e-c f)^{13/3}}+\frac{2 (b c-a d) \left (35 a^2 d^2 f^2-7 a b d f (c f+9 d e)+b^2 \left (-c^2 f^2+9 c d e f+27 d^2 e^2\right )\right ) \log \left (\frac{\sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt [3]{d e-c f}}-\sqrt [3]{a+b x}\right )}{27 (b e-a f)^{5/3} (d e-c f)^{13/3}}+\frac{4 (b c-a d) \left (35 a^2 d^2 f^2-7 a b d f (c f+9 d e)+b^2 \left (-c^2 f^2+9 c d e f+27 d^2 e^2\right )\right ) \tan ^{-1}\left (\frac{2 \sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt{3} \sqrt [3]{a+b x} \sqrt [3]{d e-c f}}+\frac{1}{\sqrt{3}}\right )}{27 \sqrt{3} (b e-a f)^{5/3} (d e-c f)^{13/3}}+\frac{3 \sqrt [3]{a+b x} (b c-a d)}{d \sqrt [3]{c+d x} (e+f x)^3 (d e-c f)}+\frac{\sqrt [3]{a+b x} (c+d x)^{2/3} (-35 a d f+32 b c f+3 b d e)}{9 (e+f x)^2 (d e-c f)^3}+\frac{\sqrt [3]{a+b x} (c+d x)^{2/3} (-10 a d f+9 b c f+b d e)}{3 d (e+f x)^3 (d e-c f)^2} \]

[Out]

(3*(b*c - a*d)*(a + b*x)^(1/3))/(d*(d*e - c*f)*(c + d*x)^(1/3)*(e + f*x)^3) + ((
b*d*e + 9*b*c*f - 10*a*d*f)*(a + b*x)^(1/3)*(c + d*x)^(2/3))/(3*d*(d*e - c*f)^2*
(e + f*x)^3) + ((3*b*d*e + 32*b*c*f - 35*a*d*f)*(a + b*x)^(1/3)*(c + d*x)^(2/3))
/(9*(d*e - c*f)^3*(e + f*x)^2) + ((140*a^2*d^2*f^2 - 7*a*b*d*f*(21*d*e + 19*c*f)
+ b^2*(9*d^2*e^2 + 129*c*d*e*f + 2*c^2*f^2))*(a + b*x)^(1/3)*(c + d*x)^(2/3))/(
27*(b*e - a*f)*(d*e - c*f)^4*(e + f*x)) + (4*(b*c - a*d)*(35*a^2*d^2*f^2 - 7*a*b
*d*f*(9*d*e + c*f) + b^2*(27*d^2*e^2 + 9*c*d*e*f - 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)^(5/3)*(d*e - c*f)^(13/3)) - (2*(b*c - a*d)*(35*a^2*d
^2*f^2 - 7*a*b*d*f*(9*d*e + c*f) + b^2*(27*d^2*e^2 + 9*c*d*e*f - c^2*f^2))*Log[e
+ f*x])/(81*(b*e - a*f)^(5/3)*(d*e - c*f)^(13/3)) + (2*(b*c - a*d)*(35*a^2*d^2*
f^2 - 7*a*b*d*f*(9*d*e + c*f) + b^2*(27*d^2*e^2 + 9*c*d*e*f - 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)])/(27*(b*e
- a*f)^(5/3)*(d*e - c*f)^(13/3))

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Rubi [A]  time = 3.83432, antiderivative size = 645, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{\sqrt [3]{a+b x} (c+d x)^{2/3} \left (140 a^2 d^2 f^2-7 a b d f (19 c f+21 d e)+b^2 \left (2 c^2 f^2+129 c d e f+9 d^2 e^2\right )\right )}{27 (e+f x) (b e-a f) (d e-c f)^4}-\frac{2 (b c-a d) \log (e+f x) \left (35 a^2 d^2 f^2-7 a b d f (c f+9 d e)+b^2 \left (-c^2 f^2+9 c d e f+27 d^2 e^2\right )\right )}{81 (b e-a f)^{5/3} (d e-c f)^{13/3}}+\frac{2 (b c-a d) \left (35 a^2 d^2 f^2-7 a b d f (c f+9 d e)+b^2 \left (-c^2 f^2+9 c d e f+27 d^2 e^2\right )\right ) \log \left (\frac{\sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt [3]{d e-c f}}-\sqrt [3]{a+b x}\right )}{27 (b e-a f)^{5/3} (d e-c f)^{13/3}}+\frac{4 (b c-a d) \left (35 a^2 d^2 f^2-7 a b d f (c f+9 d e)+b^2 \left (-c^2 f^2+9 c d e f+27 d^2 e^2\right )\right ) \tan ^{-1}\left (\frac{2 \sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt{3} \sqrt [3]{a+b x} \sqrt [3]{d e-c f}}+\frac{1}{\sqrt{3}}\right )}{27 \sqrt{3} (b e-a f)^{5/3} (d e-c f)^{13/3}}+\frac{3 \sqrt [3]{a+b x} (b c-a d)}{d \sqrt [3]{c+d x} (e+f x)^3 (d e-c f)}+\frac{\sqrt [3]{a+b x} (c+d x)^{2/3} (-35 a d f+32 b c f+3 b d e)}{9 (e+f x)^2 (d e-c f)^3}+\frac{\sqrt [3]{a+b x} (c+d x)^{2/3} (-10 a d f+9 b c f+b d e)}{3 d (e+f x)^3 (d e-c f)^2} \]

Antiderivative was successfully verified.

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

[Out]

(3*(b*c - a*d)*(a + b*x)^(1/3))/(d*(d*e - c*f)*(c + d*x)^(1/3)*(e + f*x)^3) + ((
b*d*e + 9*b*c*f - 10*a*d*f)*(a + b*x)^(1/3)*(c + d*x)^(2/3))/(3*d*(d*e - c*f)^2*
(e + f*x)^3) + ((3*b*d*e + 32*b*c*f - 35*a*d*f)*(a + b*x)^(1/3)*(c + d*x)^(2/3))
/(9*(d*e - c*f)^3*(e + f*x)^2) + ((140*a^2*d^2*f^2 - 7*a*b*d*f*(21*d*e + 19*c*f)
+ b^2*(9*d^2*e^2 + 129*c*d*e*f + 2*c^2*f^2))*(a + b*x)^(1/3)*(c + d*x)^(2/3))/(
27*(b*e - a*f)*(d*e - c*f)^4*(e + f*x)) + (4*(b*c - a*d)*(35*a^2*d^2*f^2 - 7*a*b
*d*f*(9*d*e + c*f) + b^2*(27*d^2*e^2 + 9*c*d*e*f - 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)^(5/3)*(d*e - c*f)^(13/3)) - (2*(b*c - a*d)*(35*a^2*d
^2*f^2 - 7*a*b*d*f*(9*d*e + c*f) + b^2*(27*d^2*e^2 + 9*c*d*e*f - c^2*f^2))*Log[e
+ f*x])/(81*(b*e - a*f)^(5/3)*(d*e - c*f)^(13/3)) + (2*(b*c - a*d)*(35*a^2*d^2*
f^2 - 7*a*b*d*f*(9*d*e + c*f) + b^2*(27*d^2*e^2 + 9*c*d*e*f - 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)])/(27*(b*e
- a*f)^(5/3)*(d*e - c*f)^(13/3))

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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)**(4/3)/(d*x+c)**(4/3)/(f*x+e)**4,x)

[Out]

Timed out

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Mathematica [C]  time = 1.63467, size = 371, normalized size = 0.58 \[ \frac{\sqrt [3]{a+b x} \left (4 (e+f x)^3 (b c-a d) \left (-35 a^2 d^2 f^2+7 a b d f (c f+9 d e)+b^2 \left (c^2 f^2-9 c d e f-27 d^2 e^2\right )\right ) \sqrt [3]{\frac{(c+d x) (b e-a f)}{(e+f x) (b c-a d)}} \, _2F_1\left (\frac{1}{3},\frac{1}{3};\frac{4}{3};\frac{(c f-d e) (a+b x)}{(b c-a d) (e+f x)}\right )+(b e-a f) \left ((c+d x) (e+f x)^2 \left (59 a^2 d^2 f^2-2 a b d f (26 c f+33 d e)+b^2 \left (2 c^2 f^2+48 c d e f+9 d^2 e^2\right )\right )+81 d^2 (e+f x)^3 (b c-a d) (b e-a f)+3 (c+d x) (e+f x) (b e-a f) (d e-c f) (-8 a d f+5 b c f+3 b d e)+9 (c+d x) (b e-a f)^2 (d e-c f)^2\right )\right )}{27 \sqrt [3]{c+d x} (e+f x)^3 (b e-a f)^2 (d e-c f)^4} \]

Antiderivative was successfully verified.

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

[Out]

((a + b*x)^(1/3)*((b*e - a*f)*(9*(b*e - a*f)^2*(d*e - c*f)^2*(c + d*x) + 3*(b*e
- a*f)*(d*e - c*f)*(3*b*d*e + 5*b*c*f - 8*a*d*f)*(c + d*x)*(e + f*x) + (59*a^2*d
^2*f^2 - 2*a*b*d*f*(33*d*e + 26*c*f) + b^2*(9*d^2*e^2 + 48*c*d*e*f + 2*c^2*f^2))
*(c + d*x)*(e + f*x)^2 + 81*d^2*(b*c - a*d)*(b*e - a*f)*(e + f*x)^3) + 4*(b*c -
a*d)*(-35*a^2*d^2*f^2 + 7*a*b*d*f*(9*d*e + c*f) + b^2*(-27*d^2*e^2 - 9*c*d*e*f +
c^2*f^2))*(((b*e - a*f)*(c + d*x))/((b*c - a*d)*(e + f*x)))^(1/3)*(e + f*x)^3*H
ypergeometric2F1[1/3, 1/3, 4/3, ((-(d*e) + c*f)*(a + b*x))/((b*c - a*d)*(e + f*x
))]))/(27*(b*e - a*f)^2*(d*e - c*f)^4*(c + d*x)^(1/3)*(e + f*x)^3)

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Maple [F]  time = 0.091, size = 0, normalized size = 0. \[ \int{\frac{1}{ \left ( fx+e \right ) ^{4}} \left ( bx+a \right ) ^{{\frac{4}{3}}} \left ( dx+c \right ) ^{-{\frac{4}{3}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.483369, size = 5064, normalized size = 7.85 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/243*sqrt(3)*(3*sqrt(3)*(9*a^2*c^3*f^4 + 27*(4*b^2*c*d^2 - 3*a*b*d^3)*e^4 + 9*(
4*b^2*c^2*d - 22*a*b*c*d^2 + 9*a^2*d^3)*e^3*f - 2*(2*b^2*c^3 - a*b*c^2*d - 46*a^
2*c*d^2)*e^2*f^2 - 3*(a*b*c^3 + 14*a^2*c^2*d)*e*f^3 + (9*b^2*d^3*e^2*f^2 + 3*(43
*b^2*c*d^2 - 49*a*b*d^3)*e*f^3 + (2*b^2*c^2*d - 133*a*b*c*d^2 + 140*a^2*d^3)*f^4
)*x^3 + (27*b^2*d^3*e^3*f + 6*(59*b^2*c*d^2 - 68*a*b*d^3)*e^2*f^2 + (37*b^2*c^2*
d - 395*a*b*c*d^2 + 385*a^2*d^3)*e*f^3 + (2*b^2*c^3 - 37*a*b*c^2*d + 35*a^2*c*d^
2)*f^4)*x^2 + (27*b^2*d^3*e^4 + 18*(17*b^2*c*d^2 - 20*a*b*d^3)*e^3*f + (98*b^2*c
^2*d - 406*a*b*c*d^2 + 335*a^2*d^3)*e^2*f^2 - (11*b^2*c^3 + 89*a*b*c^2*d - 100*a
^2*c*d^2)*e*f^3 + 15*(a*b*c^3 - a^2*c^2*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) + 2*sqrt(3)*(27*(b^3*c^2*d^2 - a*b^2*c*d^3)*e^5 + 9*(b^3*c^3*d - 8*a*b^2*c
^2*d^2 + 7*a^2*b*c*d^3)*e^4*f - (b^3*c^4 + 6*a*b^2*c^3*d - 42*a^2*b*c^2*d^2 + 35
*a^3*c*d^3)*e^3*f^2 + (27*(b^3*c*d^3 - a*b^2*d^4)*e^2*f^3 + 9*(b^3*c^2*d^2 - 8*a
*b^2*c*d^3 + 7*a^2*b*d^4)*e*f^4 - (b^3*c^3*d + 6*a*b^2*c^2*d^2 - 42*a^2*b*c*d^3
+ 35*a^3*d^4)*f^5)*x^4 + (81*(b^3*c*d^3 - a*b^2*d^4)*e^3*f^2 + 27*(2*b^3*c^2*d^2
- 9*a*b^2*c*d^3 + 7*a^2*b*d^4)*e^2*f^3 + 3*(2*b^3*c^3*d - 30*a*b^2*c^2*d^2 + 63
*a^2*b*c*d^3 - 35*a^3*d^4)*e*f^4 - (b^3*c^4 + 6*a*b^2*c^3*d - 42*a^2*b*c^2*d^2 +
35*a^3*c*d^3)*f^5)*x^3 + 3*(27*(b^3*c*d^3 - a*b^2*d^4)*e^4*f + 9*(4*b^3*c^2*d^2
- 11*a*b^2*c*d^3 + 7*a^2*b*d^4)*e^3*f^2 + (8*b^3*c^3*d - 78*a*b^2*c^2*d^2 + 105
*a^2*b*c*d^3 - 35*a^3*d^4)*e^2*f^3 - (b^3*c^4 + 6*a*b^2*c^3*d - 42*a^2*b*c^2*d^2
+ 35*a^3*c*d^3)*e*f^4)*x^2 + (27*(b^3*c*d^3 - a*b^2*d^4)*e^5 + 9*(10*b^3*c^2*d^
2 - 17*a*b^2*c*d^3 + 7*a^2*b*d^4)*e^4*f + (26*b^3*c^3*d - 222*a*b^2*c^2*d^2 + 23
1*a^2*b*c*d^3 - 35*a^3*d^4)*e^3*f^2 - 3*(b^3*c^4 + 6*a*b^2*c^3*d - 42*a^2*b*c^2*
d^2 + 35*a^3*c*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)) - 4*sqrt(3)*(27*(b^3*c^2*d^
2 - a*b^2*c*d^3)*e^5 + 9*(b^3*c^3*d - 8*a*b^2*c^2*d^2 + 7*a^2*b*c*d^3)*e^4*f - (
b^3*c^4 + 6*a*b^2*c^3*d - 42*a^2*b*c^2*d^2 + 35*a^3*c*d^3)*e^3*f^2 + (27*(b^3*c*
d^3 - a*b^2*d^4)*e^2*f^3 + 9*(b^3*c^2*d^2 - 8*a*b^2*c*d^3 + 7*a^2*b*d^4)*e*f^4 -
(b^3*c^3*d + 6*a*b^2*c^2*d^2 - 42*a^2*b*c*d^3 + 35*a^3*d^4)*f^5)*x^4 + (81*(b^3
*c*d^3 - a*b^2*d^4)*e^3*f^2 + 27*(2*b^3*c^2*d^2 - 9*a*b^2*c*d^3 + 7*a^2*b*d^4)*e
^2*f^3 + 3*(2*b^3*c^3*d - 30*a*b^2*c^2*d^2 + 63*a^2*b*c*d^3 - 35*a^3*d^4)*e*f^4
- (b^3*c^4 + 6*a*b^2*c^3*d - 42*a^2*b*c^2*d^2 + 35*a^3*c*d^3)*f^5)*x^3 + 3*(27*(
b^3*c*d^3 - a*b^2*d^4)*e^4*f + 9*(4*b^3*c^2*d^2 - 11*a*b^2*c*d^3 + 7*a^2*b*d^4)*
e^3*f^2 + (8*b^3*c^3*d - 78*a*b^2*c^2*d^2 + 105*a^2*b*c*d^3 - 35*a^3*d^4)*e^2*f^
3 - (b^3*c^4 + 6*a*b^2*c^3*d - 42*a^2*b*c^2*d^2 + 35*a^3*c*d^3)*e*f^4)*x^2 + (27
*(b^3*c*d^3 - a*b^2*d^4)*e^5 + 9*(10*b^3*c^2*d^2 - 17*a*b^2*c*d^3 + 7*a^2*b*d^4)
*e^4*f + (26*b^3*c^3*d - 222*a*b^2*c^2*d^2 + 231*a^2*b*c*d^3 - 35*a^3*d^4)*e^3*f
^2 - 3*(b^3*c^4 + 6*a*b^2*c^3*d - 42*a^2*b*c^2*d^2 + 35*a^3*c*d^3)*e^2*f^3)*x)*l
og((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)) + 12*(27*(b^3*c^2*d^2 - a*b^2*c*d^3)*e^5 + 9*(b^3*c^3*d - 8*a*b^2*c^2*d^2
+ 7*a^2*b*c*d^3)*e^4*f - (b^3*c^4 + 6*a*b^2*c^3*d - 42*a^2*b*c^2*d^2 + 35*a^3*c
*d^3)*e^3*f^2 + (27*(b^3*c*d^3 - a*b^2*d^4)*e^2*f^3 + 9*(b^3*c^2*d^2 - 8*a*b^2*c
*d^3 + 7*a^2*b*d^4)*e*f^4 - (b^3*c^3*d + 6*a*b^2*c^2*d^2 - 42*a^2*b*c*d^3 + 35*a
^3*d^4)*f^5)*x^4 + (81*(b^3*c*d^3 - a*b^2*d^4)*e^3*f^2 + 27*(2*b^3*c^2*d^2 - 9*a
*b^2*c*d^3 + 7*a^2*b*d^4)*e^2*f^3 + 3*(2*b^3*c^3*d - 30*a*b^2*c^2*d^2 + 63*a^2*b
*c*d^3 - 35*a^3*d^4)*e*f^4 - (b^3*c^4 + 6*a*b^2*c^3*d - 42*a^2*b*c^2*d^2 + 35*a^
3*c*d^3)*f^5)*x^3 + 3*(27*(b^3*c*d^3 - a*b^2*d^4)*e^4*f + 9*(4*b^3*c^2*d^2 - 11*
a*b^2*c*d^3 + 7*a^2*b*d^4)*e^3*f^2 + (8*b^3*c^3*d - 78*a*b^2*c^2*d^2 + 105*a^2*b
*c*d^3 - 35*a^3*d^4)*e^2*f^3 - (b^3*c^4 + 6*a*b^2*c^3*d - 42*a^2*b*c^2*d^2 + 35*
a^3*c*d^3)*e*f^4)*x^2 + (27*(b^3*c*d^3 - a*b^2*d^4)*e^5 + 9*(10*b^3*c^2*d^2 - 17
*a*b^2*c*d^3 + 7*a^2*b*d^4)*e^4*f + (26*b^3*c^3*d - 222*a*b^2*c^2*d^2 + 231*a^2*
b*c*d^3 - 35*a^3*d^4)*e^3*f^2 - 3*(b^3*c^4 + 6*a*b^2*c^3*d - 42*a^2*b*c^2*d^2 +
35*a^3*c*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*c*d^4*e^8 - a*c^5*e^3*f^5 - (4*b*c^2*d^3 + a*c*d^4)*e^7*f + 2*(3*b*
c^3*d^2 + 2*a*c^2*d^3)*e^6*f^2 - 2*(2*b*c^4*d + 3*a*c^3*d^2)*e^5*f^3 + (b*c^5 +
4*a*c^4*d)*e^4*f^4 + (b*d^5*e^5*f^3 - a*c^4*d*f^8 - (4*b*c*d^4 + a*d^5)*e^4*f^4
+ 2*(3*b*c^2*d^3 + 2*a*c*d^4)*e^3*f^5 - 2*(2*b*c^3*d^2 + 3*a*c^2*d^3)*e^2*f^6 +
(b*c^4*d + 4*a*c^3*d^2)*e*f^7)*x^4 + (3*b*d^5*e^6*f^2 - a*c^5*f^8 - (11*b*c*d^4
+ 3*a*d^5)*e^5*f^3 + (14*b*c^2*d^3 + 11*a*c*d^4)*e^4*f^4 - 2*(3*b*c^3*d^2 + 7*a*
c^2*d^3)*e^3*f^5 - (b*c^4*d - 6*a*c^3*d^2)*e^2*f^6 + (b*c^5 + a*c^4*d)*e*f^7)*x^
3 + 3*(b*d^5*e^7*f - a*c^5*e*f^7 - (3*b*c*d^4 + a*d^5)*e^6*f^2 + (2*b*c^2*d^3 +
3*a*c*d^4)*e^5*f^3 + 2*(b*c^3*d^2 - a*c^2*d^3)*e^4*f^4 - (3*b*c^4*d + 2*a*c^3*d^
2)*e^3*f^5 + (b*c^5 + 3*a*c^4*d)*e^2*f^6)*x^2 + (b*d^5*e^8 - 3*a*c^5*e^2*f^6 - (
b*c*d^4 + a*d^5)*e^7*f - (6*b*c^2*d^3 - a*c*d^4)*e^6*f^2 + 2*(7*b*c^3*d^2 + 3*a*
c^2*d^3)*e^5*f^3 - (11*b*c^4*d + 14*a*c^3*d^2)*e^4*f^4 + (3*b*c^5 + 11*a*c^4*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)**(4/3)/(d*x+c)**(4/3)/(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)^(4/3)/((d*x + c)^(4/3)*(f*x + e)^4),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError