3.2992 \(\int \frac{\sqrt [3]{a+b x} (e+f x)}{\sqrt [3]{c+d x}} \, dx\)

Optimal. Leaf size=273 \[ \frac{(b c-a d) \log (a+b x) (-a d f-2 b c f+3 b d e)}{18 b^{5/3} d^{7/3}}+\frac{(b c-a d) (-a d f-2 b c f+3 b d e) \log \left (\frac{\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}-1\right )}{6 b^{5/3} d^{7/3}}+\frac{(b c-a d) (-a d f-2 b c f+3 b d e) \tan ^{-1}\left (\frac{2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt{3} \sqrt [3]{d} \sqrt [3]{a+b x}}+\frac{1}{\sqrt{3}}\right )}{3 \sqrt{3} b^{5/3} d^{7/3}}+\frac{\sqrt [3]{a+b x} (c+d x)^{2/3} (-a d f-2 b c f+3 b d e)}{3 b d^2}+\frac{f (a+b x)^{4/3} (c+d x)^{2/3}}{2 b d} \]

[Out]

((3*b*d*e - 2*b*c*f - a*d*f)*(a + b*x)^(1/3)*(c + d*x)^(2/3))/(3*b*d^2) + (f*(a
+ b*x)^(4/3)*(c + d*x)^(2/3))/(2*b*d) + ((b*c - a*d)*(3*b*d*e - 2*b*c*f - a*d*f)
*ArcTan[1/Sqrt[3] + (2*b^(1/3)*(c + d*x)^(1/3))/(Sqrt[3]*d^(1/3)*(a + b*x)^(1/3)
)])/(3*Sqrt[3]*b^(5/3)*d^(7/3)) + ((b*c - a*d)*(3*b*d*e - 2*b*c*f - a*d*f)*Log[a
+ b*x])/(18*b^(5/3)*d^(7/3)) + ((b*c - a*d)*(3*b*d*e - 2*b*c*f - a*d*f)*Log[-1
+ (b^(1/3)*(c + d*x)^(1/3))/(d^(1/3)*(a + b*x)^(1/3))])/(6*b^(5/3)*d^(7/3))

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Rubi [A]  time = 0.446681, antiderivative size = 273, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{(b c-a d) \log (a+b x) (-a d f-2 b c f+3 b d e)}{18 b^{5/3} d^{7/3}}+\frac{(b c-a d) (-a d f-2 b c f+3 b d e) \log \left (\frac{\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}-1\right )}{6 b^{5/3} d^{7/3}}+\frac{(b c-a d) (-a d f-2 b c f+3 b d e) \tan ^{-1}\left (\frac{2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt{3} \sqrt [3]{d} \sqrt [3]{a+b x}}+\frac{1}{\sqrt{3}}\right )}{3 \sqrt{3} b^{5/3} d^{7/3}}+\frac{\sqrt [3]{a+b x} (c+d x)^{2/3} (-a d f-2 b c f+3 b d e)}{3 b d^2}+\frac{f (a+b x)^{4/3} (c+d x)^{2/3}}{2 b d} \]

Antiderivative was successfully verified.

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

[Out]

((3*b*d*e - 2*b*c*f - a*d*f)*(a + b*x)^(1/3)*(c + d*x)^(2/3))/(3*b*d^2) + (f*(a
+ b*x)^(4/3)*(c + d*x)^(2/3))/(2*b*d) + ((b*c - a*d)*(3*b*d*e - 2*b*c*f - a*d*f)
*ArcTan[1/Sqrt[3] + (2*b^(1/3)*(c + d*x)^(1/3))/(Sqrt[3]*d^(1/3)*(a + b*x)^(1/3)
)])/(3*Sqrt[3]*b^(5/3)*d^(7/3)) + ((b*c - a*d)*(3*b*d*e - 2*b*c*f - a*d*f)*Log[a
+ b*x])/(18*b^(5/3)*d^(7/3)) + ((b*c - a*d)*(3*b*d*e - 2*b*c*f - a*d*f)*Log[-1
+ (b^(1/3)*(c + d*x)^(1/3))/(d^(1/3)*(a + b*x)^(1/3))])/(6*b^(5/3)*d^(7/3))

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Rubi in Sympy [A]  time = 27.6713, size = 260, normalized size = 0.95 \[ \frac{f \left (a + b x\right )^{\frac{4}{3}} \left (c + d x\right )^{\frac{2}{3}}}{2 b d} - \frac{\sqrt [3]{a + b x} \left (c + d x\right )^{\frac{2}{3}} \left (- b d e + \frac{f \left (a d + 2 b c\right )}{3}\right )}{b d^{2}} + \frac{\left (a d - b c\right ) \left (a d f + 2 b c f - 3 b d e\right ) \log{\left (a + b x \right )}}{18 b^{\frac{5}{3}} d^{\frac{7}{3}}} + \frac{\left (a d - b c\right ) \left (a d f + 2 b c f - 3 b d e\right ) \log{\left (\frac{\sqrt [3]{b} \sqrt [3]{c + d x}}{\sqrt [3]{d} \sqrt [3]{a + b x}} - 1 \right )}}{6 b^{\frac{5}{3}} d^{\frac{7}{3}}} + \frac{\sqrt{3} \left (a d - b c\right ) \left (a d f + 2 b c f - 3 b d e\right ) \operatorname{atan}{\left (\frac{2 \sqrt{3} \sqrt [3]{b} \sqrt [3]{c + d x}}{3 \sqrt [3]{d} \sqrt [3]{a + b x}} + \frac{\sqrt{3}}{3} \right )}}{9 b^{\frac{5}{3}} d^{\frac{7}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

f*(a + b*x)**(4/3)*(c + d*x)**(2/3)/(2*b*d) - (a + b*x)**(1/3)*(c + d*x)**(2/3)*
(-b*d*e + f*(a*d + 2*b*c)/3)/(b*d**2) + (a*d - b*c)*(a*d*f + 2*b*c*f - 3*b*d*e)*
log(a + b*x)/(18*b**(5/3)*d**(7/3)) + (a*d - b*c)*(a*d*f + 2*b*c*f - 3*b*d*e)*lo
g(b**(1/3)*(c + d*x)**(1/3)/(d**(1/3)*(a + b*x)**(1/3)) - 1)/(6*b**(5/3)*d**(7/3
)) + sqrt(3)*(a*d - b*c)*(a*d*f + 2*b*c*f - 3*b*d*e)*atan(2*sqrt(3)*b**(1/3)*(c
+ d*x)**(1/3)/(3*d**(1/3)*(a + b*x)**(1/3)) + sqrt(3)/3)/(9*b**(5/3)*d**(7/3))

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Mathematica [C]  time = 0.214958, size = 129, normalized size = 0.47 \[ \frac{(c+d x)^{2/3} \left ((b c-a d) \left (\frac{d (a+b x)}{a d-b c}\right )^{2/3} (a d f+2 b c f-3 b d e) \, _2F_1\left (\frac{2}{3},\frac{2}{3};\frac{5}{3};\frac{b (c+d x)}{b c-a d}\right )+d (a+b x) (a d f+b (-4 c f+6 d e+3 d f x))\right )}{6 b d^3 (a+b x)^{2/3}} \]

Antiderivative was successfully verified.

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

[Out]

((c + d*x)^(2/3)*(d*(a + b*x)*(a*d*f + b*(6*d*e - 4*c*f + 3*d*f*x)) + (b*c - a*d
)*(-3*b*d*e + 2*b*c*f + a*d*f)*((d*(a + b*x))/(-(b*c) + a*d))^(2/3)*Hypergeometr
ic2F1[2/3, 2/3, 5/3, (b*(c + d*x))/(b*c - a*d)]))/(6*b*d^3*(a + b*x)^(2/3))

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Maple [F]  time = 0.044, size = 0, normalized size = 0. \[ \int{(fx+e)\sqrt [3]{bx+a}{\frac{1}{\sqrt [3]{dx+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.230982, size = 502, normalized size = 1.84 \[ \frac{\sqrt{3}{\left (3 \, \sqrt{3}{\left (3 \, b d f x + 6 \, b d e -{\left (4 \, b c - a d\right )} f\right )} \left (b^{2} d\right )^{\frac{1}{3}}{\left (b x + a\right )}^{\frac{1}{3}}{\left (d x + c\right )}^{\frac{2}{3}} - \sqrt{3}{\left (3 \,{\left (b^{2} c d - a b d^{2}\right )} e -{\left (2 \, b^{2} c^{2} - a b c d - a^{2} d^{2}\right )} f\right )} \log \left (\frac{b^{2} d x + b^{2} c + \left (b^{2} d\right )^{\frac{1}{3}}{\left (b x + a\right )}^{\frac{1}{3}}{\left (d x + c\right )}^{\frac{2}{3}} b + \left (b^{2} d\right )^{\frac{2}{3}}{\left (b x + a\right )}^{\frac{2}{3}}{\left (d x + c\right )}^{\frac{1}{3}}}{d x + c}\right ) + 2 \, \sqrt{3}{\left (3 \,{\left (b^{2} c d - a b d^{2}\right )} e -{\left (2 \, b^{2} c^{2} - a b c d - a^{2} d^{2}\right )} f\right )} \log \left (-\frac{b d x + b c - \left (b^{2} d\right )^{\frac{1}{3}}{\left (b x + a\right )}^{\frac{1}{3}}{\left (d x + c\right )}^{\frac{2}{3}}}{d x + c}\right ) - 6 \,{\left (3 \,{\left (b^{2} c d - a b d^{2}\right )} e -{\left (2 \, b^{2} c^{2} - a b c d - a^{2} d^{2}\right )} f\right )} \arctan \left (\frac{2 \, \sqrt{3} \left (b^{2} d\right )^{\frac{1}{3}}{\left (b x + a\right )}^{\frac{1}{3}}{\left (d x + c\right )}^{\frac{2}{3}} + \sqrt{3}{\left (b d x + b c\right )}}{3 \,{\left (b d x + b c\right )}}\right )\right )}}{54 \, \left (b^{2} d\right )^{\frac{1}{3}} b d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/54*sqrt(3)*(3*sqrt(3)*(3*b*d*f*x + 6*b*d*e - (4*b*c - a*d)*f)*(b^2*d)^(1/3)*(b
*x + a)^(1/3)*(d*x + c)^(2/3) - sqrt(3)*(3*(b^2*c*d - a*b*d^2)*e - (2*b^2*c^2 -
a*b*c*d - a^2*d^2)*f)*log((b^2*d*x + b^2*c + (b^2*d)^(1/3)*(b*x + a)^(1/3)*(d*x
+ c)^(2/3)*b + (b^2*d)^(2/3)*(b*x + a)^(2/3)*(d*x + c)^(1/3))/(d*x + c)) + 2*sqr
t(3)*(3*(b^2*c*d - a*b*d^2)*e - (2*b^2*c^2 - a*b*c*d - a^2*d^2)*f)*log(-(b*d*x +
b*c - (b^2*d)^(1/3)*(b*x + a)^(1/3)*(d*x + c)^(2/3))/(d*x + c)) - 6*(3*(b^2*c*d
- a*b*d^2)*e - (2*b^2*c^2 - a*b*c*d - a^2*d^2)*f)*arctan(1/3*(2*sqrt(3)*(b^2*d)
^(1/3)*(b*x + a)^(1/3)*(d*x + c)^(2/3) + sqrt(3)*(b*d*x + b*c))/(b*d*x + b*c)))/
((b^2*d)^(1/3)*b*d^2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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