3.3025 \(\int \frac{(a+b x)^{4/3}}{(c+d x)^{4/3} (e+f x)^3} \, dx\)

Optimal. Leaf size=434 \[ \frac{3 d (a+b x)^{7/3}}{\sqrt [3]{c+d x} (e+f x)^2 (b c-a d) (d e-c f)}-\frac{(a+b x)^{4/3} (c+d x)^{2/3} (-7 a d f+b c f+6 b d e)}{2 (e+f x)^2 (b c-a d) (d e-c f)^2}+\frac{2 \sqrt [3]{a+b x} (c+d x)^{2/3} (-7 a d f+b c f+6 b d e)}{3 (e+f x) (d e-c f)^3}-\frac{(b c-a d) \log (e+f x) (-7 a d f+b c f+6 b d e)}{9 (b e-a f)^{2/3} (d e-c f)^{10/3}}+\frac{(b c-a d) (-7 a d f+b c f+6 b d e) \log \left (\frac{\sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt [3]{d e-c f}}-\sqrt [3]{a+b x}\right )}{3 (b e-a f)^{2/3} (d e-c f)^{10/3}}+\frac{2 (b c-a d) (-7 a d f+b c f+6 b d e) \tan ^{-1}\left (\frac{2 \sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt{3} \sqrt [3]{a+b x} \sqrt [3]{d e-c f}}+\frac{1}{\sqrt{3}}\right )}{3 \sqrt{3} (b e-a f)^{2/3} (d e-c f)^{10/3}} \]

[Out]

(3*d*(a + b*x)^(7/3))/((b*c - a*d)*(d*e - c*f)*(c + d*x)^(1/3)*(e + f*x)^2) - ((
6*b*d*e + b*c*f - 7*a*d*f)*(a + b*x)^(4/3)*(c + d*x)^(2/3))/(2*(b*c - a*d)*(d*e
- c*f)^2*(e + f*x)^2) + (2*(6*b*d*e + b*c*f - 7*a*d*f)*(a + b*x)^(1/3)*(c + d*x)
^(2/3))/(3*(d*e - c*f)^3*(e + f*x)) + (2*(b*c - a*d)*(6*b*d*e + b*c*f - 7*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)^(2/3)*(d*e - c*f)^(10/3)) - ((b*c
- a*d)*(6*b*d*e + b*c*f - 7*a*d*f)*Log[e + f*x])/(9*(b*e - a*f)^(2/3)*(d*e - c*
f)^(10/3)) + ((b*c - a*d)*(6*b*d*e + b*c*f - 7*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)])/(3*(b*e - a*f)^(2/3)*(d*e -
c*f)^(10/3))

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Rubi [A]  time = 0.912767, antiderivative size = 434, 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 \[ \frac{3 d (a+b x)^{7/3}}{\sqrt [3]{c+d x} (e+f x)^2 (b c-a d) (d e-c f)}-\frac{(a+b x)^{4/3} (c+d x)^{2/3} (-7 a d f+b c f+6 b d e)}{2 (e+f x)^2 (b c-a d) (d e-c f)^2}+\frac{2 \sqrt [3]{a+b x} (c+d x)^{2/3} (-7 a d f+b c f+6 b d e)}{3 (e+f x) (d e-c f)^3}-\frac{(b c-a d) \log (e+f x) (-7 a d f+b c f+6 b d e)}{9 (b e-a f)^{2/3} (d e-c f)^{10/3}}+\frac{(b c-a d) (-7 a d f+b c f+6 b d e) \log \left (\frac{\sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt [3]{d e-c f}}-\sqrt [3]{a+b x}\right )}{3 (b e-a f)^{2/3} (d e-c f)^{10/3}}+\frac{2 (b c-a d) (-7 a d f+b c f+6 b d e) \tan ^{-1}\left (\frac{2 \sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt{3} \sqrt [3]{a+b x} \sqrt [3]{d e-c f}}+\frac{1}{\sqrt{3}}\right )}{3 \sqrt{3} (b e-a f)^{2/3} (d e-c f)^{10/3}} \]

Antiderivative was successfully verified.

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

[Out]

(3*d*(a + b*x)^(7/3))/((b*c - a*d)*(d*e - c*f)*(c + d*x)^(1/3)*(e + f*x)^2) - ((
6*b*d*e + b*c*f - 7*a*d*f)*(a + b*x)^(4/3)*(c + d*x)^(2/3))/(2*(b*c - a*d)*(d*e
- c*f)^2*(e + f*x)^2) + (2*(6*b*d*e + b*c*f - 7*a*d*f)*(a + b*x)^(1/3)*(c + d*x)
^(2/3))/(3*(d*e - c*f)^3*(e + f*x)) + (2*(b*c - a*d)*(6*b*d*e + b*c*f - 7*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)^(2/3)*(d*e - c*f)^(10/3)) - ((b*c
- a*d)*(6*b*d*e + b*c*f - 7*a*d*f)*Log[e + f*x])/(9*(b*e - a*f)^(2/3)*(d*e - c*
f)^(10/3)) + ((b*c - a*d)*(6*b*d*e + b*c*f - 7*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)])/(3*(b*e - a*f)^(2/3)*(d*e -
c*f)^(10/3))

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Rubi in Sympy [A]  time = 101.335, size = 408, normalized size = 0.94 \[ - \frac{f \left (a + b x\right )^{\frac{7}{3}}}{2 \sqrt [3]{c + d x} \left (e + f x\right )^{2} \left (a f - b e\right ) \left (c f - d e\right )} + \frac{\left (a + b x\right )^{\frac{4}{3}} \left (7 a d f - b c f - 6 b d e\right )}{6 \sqrt [3]{c + d x} \left (e + f x\right ) \left (a f - b e\right ) \left (c f - d e\right )^{2}} + \frac{2 \sqrt [3]{a + b x} \left (a d - b c\right ) \left (7 a d f - b c f - 6 b d e\right )}{3 \sqrt [3]{c + d x} \left (a f - b e\right ) \left (c f - d e\right )^{3}} - \frac{\left (a d - b c\right ) \left (7 a d f - b c f - 6 b d e\right ) \log{\left (e + f x \right )}}{9 \left (a f - b e\right )^{\frac{2}{3}} \left (c f - d e\right )^{\frac{10}{3}}} + \frac{\left (a d - b c\right ) \left (7 a d f - b c f - 6 b d e\right ) \log{\left (- \sqrt [3]{a + b x} + \frac{\sqrt [3]{c + d x} \sqrt [3]{a f - b e}}{\sqrt [3]{c f - d e}} \right )}}{3 \left (a f - b e\right )^{\frac{2}{3}} \left (c f - d e\right )^{\frac{10}{3}}} + \frac{2 \sqrt{3} \left (a d - b c\right ) \left (7 a d f - b c f - 6 b d e\right ) \operatorname{atan}{\left (\frac{\sqrt{3}}{3} + \frac{2 \sqrt{3} \sqrt [3]{c + d x} \sqrt [3]{a f - b e}}{3 \sqrt [3]{a + b x} \sqrt [3]{c f - d e}} \right )}}{9 \left (a f - b e\right )^{\frac{2}{3}} \left (c f - d e\right )^{\frac{10}{3}}} \]

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)**3,x)

[Out]

-f*(a + b*x)**(7/3)/(2*(c + d*x)**(1/3)*(e + f*x)**2*(a*f - b*e)*(c*f - d*e)) +
(a + b*x)**(4/3)*(7*a*d*f - b*c*f - 6*b*d*e)/(6*(c + d*x)**(1/3)*(e + f*x)*(a*f
- b*e)*(c*f - d*e)**2) + 2*(a + b*x)**(1/3)*(a*d - b*c)*(7*a*d*f - b*c*f - 6*b*d
*e)/(3*(c + d*x)**(1/3)*(a*f - b*e)*(c*f - d*e)**3) - (a*d - b*c)*(7*a*d*f - b*c
*f - 6*b*d*e)*log(e + f*x)/(9*(a*f - b*e)**(2/3)*(c*f - d*e)**(10/3)) + (a*d - b
*c)*(7*a*d*f - b*c*f - 6*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))/(3*(a*f - b*e)**(2/3)*(c*f - d*e)**(10/3)) + 2*s
qrt(3)*(a*d - b*c)*(7*a*d*f - b*c*f - 6*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)**(2/3)*(c*f - d*e)**(10/3))

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Mathematica [C]  time = 1.30708, size = 208, normalized size = 0.48 \[ \frac{\sqrt [3]{a+b x} \left (-\frac{4 (c+d x) (-7 a d f+b c f+6 b d e) \, _2F_1\left (\frac{1}{3},\frac{1}{3};\frac{4}{3};\frac{(c f-d e) (a+b x)}{(b c-a d) (e+f x)}\right )}{(e+f x) \left (\frac{(c+d x) (b e-a f)}{(e+f x) (b c-a d)}\right )^{2/3}}+\frac{(c+d x) (-10 a d f+7 b c f+3 b d e)}{e+f x}+\frac{3 (c+d x) (b e-a f) (d e-c f)}{(e+f x)^2}+18 d (b c-a d)\right )}{6 \sqrt [3]{c+d x} (d e-c f)^3} \]

Antiderivative was successfully verified.

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

[Out]

((a + b*x)^(1/3)*(18*d*(b*c - a*d) + (3*(b*e - a*f)*(d*e - c*f)*(c + d*x))/(e +
f*x)^2 + ((3*b*d*e + 7*b*c*f - 10*a*d*f)*(c + d*x))/(e + f*x) - (4*(6*b*d*e + b*
c*f - 7*a*d*f)*(c + d*x)*Hypergeometric2F1[1/3, 1/3, 4/3, ((-(d*e) + c*f)*(a + b
*x))/((b*c - a*d)*(e + f*x))])/((((b*e - a*f)*(c + d*x))/((b*c - a*d)*(e + f*x))
)^(2/3)*(e + f*x))))/(6*(d*e - c*f)^3*(c + d*x)^(1/3))

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

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)^3,x)

[Out]

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

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

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)^3),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.323267, size = 2352, normalized size = 5.42 \[ \text{result too large to display} \]

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)^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)*(3*a*c^2*f^2 + 6*(4*b*c*d - 3*a*d^2)*e^2 + (4*b*c^2 -
13*a*c*d)*e*f + (3*b*d^2*e*f + (25*b*c*d - 28*a*d^2)*f^2)*x^2 + (6*b*d^2*e^2 +
(43*b*c*d - 49*a*d^2)*e*f + 7*(b*c^2 - a*c*d)*f^2)*x)*(b*x + a)^(1/3)*(d*x + c)^
(2/3) + 2*sqrt(3)*(6*(b^2*c^2*d - a*b*c*d^2)*e^3 + (b^2*c^3 - 8*a*b*c^2*d + 7*a^
2*c*d^2)*e^2*f + (6*(b^2*c*d^2 - a*b*d^3)*e*f^2 + (b^2*c^2*d - 8*a*b*c*d^2 + 7*a
^2*d^3)*f^3)*x^3 + (12*(b^2*c*d^2 - a*b*d^3)*e^2*f + 2*(4*b^2*c^2*d - 11*a*b*c*d
^2 + 7*a^2*d^3)*e*f^2 + (b^2*c^3 - 8*a*b*c^2*d + 7*a^2*c*d^2)*f^3)*x^2 + (6*(b^2
*c*d^2 - a*b*d^3)*e^3 + (13*b^2*c^2*d - 20*a*b*c*d^2 + 7*a^2*d^3)*e^2*f + 2*(b^2
*c^3 - 8*a*b*c^2*d + 7*a^2*c*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)) - 4*sqrt(3)*(6*
(b^2*c^2*d - a*b*c*d^2)*e^3 + (b^2*c^3 - 8*a*b*c^2*d + 7*a^2*c*d^2)*e^2*f + (6*(
b^2*c*d^2 - a*b*d^3)*e*f^2 + (b^2*c^2*d - 8*a*b*c*d^2 + 7*a^2*d^3)*f^3)*x^3 + (1
2*(b^2*c*d^2 - a*b*d^3)*e^2*f + 2*(4*b^2*c^2*d - 11*a*b*c*d^2 + 7*a^2*d^3)*e*f^2
+ (b^2*c^3 - 8*a*b*c^2*d + 7*a^2*c*d^2)*f^3)*x^2 + (6*(b^2*c*d^2 - a*b*d^3)*e^3
+ (13*b^2*c^2*d - 20*a*b*c*d^2 + 7*a^2*d^3)*e^2*f + 2*(b^2*c^3 - 8*a*b*c^2*d +
7*a^2*c*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)) + 12*(6*(b^2*c^2*d - a*b*c*d^2)*e^3 + (b^2*c^3 -
8*a*b*c^2*d + 7*a^2*c*d^2)*e^2*f + (6*(b^2*c*d^2 - a*b*d^3)*e*f^2 + (b^2*c^2*d -
8*a*b*c*d^2 + 7*a^2*d^3)*f^3)*x^3 + (12*(b^2*c*d^2 - a*b*d^3)*e^2*f + 2*(4*b^2*
c^2*d - 11*a*b*c*d^2 + 7*a^2*d^3)*e*f^2 + (b^2*c^3 - 8*a*b*c^2*d + 7*a^2*c*d^2)*
f^3)*x^2 + (6*(b^2*c*d^2 - a*b*d^3)*e^3 + (13*b^2*c^2*d - 20*a*b*c*d^2 + 7*a^2*d
^3)*e^2*f + 2*(b^2*c^3 - 8*a*b*c^2*d + 7*a^2*c*d^2)*e*f^2)*x)*arctan(-1/3*(2*sqr
t(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)))/((c*d^3*e^5 - 3*c^2*d^2*e^4*f + 3*c
^3*d*e^3*f^2 - c^4*e^2*f^3 + (d^4*e^3*f^2 - 3*c*d^3*e^2*f^3 + 3*c^2*d^2*e*f^4 -
c^3*d*f^5)*x^3 + (2*d^4*e^4*f - 5*c*d^3*e^3*f^2 + 3*c^2*d^2*e^2*f^3 + c^3*d*e*f^
4 - c^4*f^5)*x^2 + (d^4*e^5 - c*d^3*e^4*f - 3*c^2*d^2*e^3*f^2 + 5*c^3*d*e^2*f^3
- 2*c^4*e*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))

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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)**3,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)^3),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError