3.2989 \(\int \frac{\sqrt [3]{a+b x} (c+d x)^{2/3}}{(e+f x)^3} \, dx\)

Optimal. Leaf size=325 \[ -\frac{\sqrt [3]{a+b x} (c+d x)^{2/3} (b c-a d)}{6 (e+f x) (b e-a f) (d e-c f)}+\frac{\sqrt [3]{a+b x} (c+d x)^{5/3}}{2 (e+f x)^2 (d e-c f)}-\frac{(b c-a d)^2 \log (e+f x)}{18 (b e-a f)^{5/3} (d e-c f)^{4/3}}+\frac{(b c-a d)^2 \log \left (\frac{\sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt [3]{d e-c f}}-\sqrt [3]{a+b x}\right )}{6 (b e-a f)^{5/3} (d e-c f)^{4/3}}+\frac{(b c-a d)^2 \tan ^{-1}\left (\frac{2 \sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt{3} \sqrt [3]{a+b x} \sqrt [3]{d e-c f}}+\frac{1}{\sqrt{3}}\right )}{3 \sqrt{3} (b e-a f)^{5/3} (d e-c f)^{4/3}} \]

[Out]

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

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Rubi [A]  time = 0.818368, antiderivative size = 325, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ -\frac{\sqrt [3]{a+b x} (c+d x)^{2/3} (b c-a d)}{6 (e+f x) (b e-a f) (d e-c f)}+\frac{\sqrt [3]{a+b x} (c+d x)^{5/3}}{2 (e+f x)^2 (d e-c f)}-\frac{(b c-a d)^2 \log (e+f x)}{18 (b e-a f)^{5/3} (d e-c f)^{4/3}}+\frac{(b c-a d)^2 \log \left (\frac{\sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt [3]{d e-c f}}-\sqrt [3]{a+b x}\right )}{6 (b e-a f)^{5/3} (d e-c f)^{4/3}}+\frac{(b c-a d)^2 \tan ^{-1}\left (\frac{2 \sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt{3} \sqrt [3]{a+b x} \sqrt [3]{d e-c f}}+\frac{1}{\sqrt{3}}\right )}{3 \sqrt{3} (b e-a f)^{5/3} (d e-c f)^{4/3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 54.7115, size = 272, normalized size = 0.84 \[ - \frac{\sqrt [3]{a + b x} \left (c + d x\right )^{\frac{5}{3}}}{2 \left (e + f x\right )^{2} \left (c f - d e\right )} + \frac{\sqrt [3]{a + b x} \left (c + d x\right )^{\frac{2}{3}} \left (a d - b c\right )}{6 \left (e + f x\right ) \left (a f - b e\right ) \left (c f - d e\right )} + \frac{\left (a d - b c\right )^{2} \log{\left (e + f x \right )}}{18 \left (a f - b e\right )^{\frac{5}{3}} \left (c f - d e\right )^{\frac{4}{3}}} - \frac{\left (a d - b c\right )^{2} \log{\left (- \sqrt [3]{a + b x} + \frac{\sqrt [3]{c + d x} \sqrt [3]{a f - b e}}{\sqrt [3]{c f - d e}} \right )}}{6 \left (a f - b e\right )^{\frac{5}{3}} \left (c f - d e\right )^{\frac{4}{3}}} - \frac{\sqrt{3} \left (a d - b c\right )^{2} \operatorname{atan}{\left (\frac{\sqrt{3}}{3} + \frac{2 \sqrt{3} \sqrt [3]{c + d x} \sqrt [3]{a f - b e}}{3 \sqrt [3]{a + b x} \sqrt [3]{c f - d e}} \right )}}{9 \left (a f - b e\right )^{\frac{5}{3}} \left (c f - d e\right )^{\frac{4}{3}}} \]

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)**3,x)

[Out]

-(a + b*x)**(1/3)*(c + d*x)**(5/3)/(2*(e + f*x)**2*(c*f - d*e)) + (a + b*x)**(1/
3)*(c + d*x)**(2/3)*(a*d - b*c)/(6*(e + f*x)*(a*f - b*e)*(c*f - d*e)) + (a*d - b
*c)**2*log(e + f*x)/(18*(a*f - b*e)**(5/3)*(c*f - d*e)**(4/3)) - (a*d - b*c)**2*
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)**(4/3)) - sqrt(3)*(a*d - b*c)**2*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)**(4/3))

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Mathematica [C]  time = 1.24338, size = 196, normalized size = 0.6 \[ \frac{\sqrt [3]{a+b x} \left (f (c+d x) (b e-a f) (-3 a c f+a d (e-2 f x)+b (2 c e-c f x+3 d e x))-2 f (e+f x)^2 (b c-a d)^2 \sqrt [3]{\frac{(c+d x) (b e-a f)}{(e+f x) (b c-a d)}} \, _2F_1\left (\frac{1}{3},\frac{1}{3};\frac{4}{3};\frac{(c f-d e) (a+b x)}{(b c-a d) (e+f x)}\right )\right )}{6 f \sqrt [3]{c+d x} (e+f x)^2 (b e-a f)^2 (d e-c f)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.076, size = 0, normalized size = 0. \[ \int{\frac{1}{ \left ( fx+e \right ) ^{3}}\sqrt [3]{bx+a} \left ( dx+c \right ) ^{{\frac{2}{3}}}}\, dx \]

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)^3,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.243014, size = 1278, normalized size = 3.93 \[ \text{result too large to display} \]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.953667, 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)^3,x, algorithm="giac")

[Out]

Done