3.3000 \(\int \frac{e+f x}{\sqrt [3]{a+b x} (c+d x)^{2/3}} \, dx\)

Optimal. Leaf size=200 \[ -\frac{\log (c+d x) (-a d f-2 b c f+3 b d e)}{6 b^{4/3} d^{5/3}}-\frac{(-a d f-2 b c f+3 b d e) \log \left (\frac{\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{b} \sqrt [3]{c+d x}}-1\right )}{2 b^{4/3} d^{5/3}}-\frac{(-a d f-2 b c f+3 b d e) \tan ^{-1}\left (\frac{2 \sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt{3} \sqrt [3]{b} \sqrt [3]{c+d x}}+\frac{1}{\sqrt{3}}\right )}{\sqrt{3} b^{4/3} d^{5/3}}+\frac{f (a+b x)^{2/3} \sqrt [3]{c+d x}}{b d} \]

[Out]

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

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Rubi [A]  time = 0.297264, antiderivative size = 200, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ -\frac{\log (c+d x) (-a d f-2 b c f+3 b d e)}{6 b^{4/3} d^{5/3}}-\frac{(-a d f-2 b c f+3 b d e) \log \left (\frac{\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{b} \sqrt [3]{c+d x}}-1\right )}{2 b^{4/3} d^{5/3}}-\frac{(-a d f-2 b c f+3 b d e) \tan ^{-1}\left (\frac{2 \sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt{3} \sqrt [3]{b} \sqrt [3]{c+d x}}+\frac{1}{\sqrt{3}}\right )}{\sqrt{3} b^{4/3} d^{5/3}}+\frac{f (a+b x)^{2/3} \sqrt [3]{c+d x}}{b d} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 17.0718, size = 194, normalized size = 0.97 \[ \frac{f \left (a + b x\right )^{\frac{2}{3}} \sqrt [3]{c + d x}}{b d} + \frac{3 \left (- b d e + \frac{f \left (a d + 2 b c\right )}{3}\right ) \log{\left (-1 + \frac{\sqrt [3]{d} \sqrt [3]{a + b x}}{\sqrt [3]{b} \sqrt [3]{c + d x}} \right )}}{2 b^{\frac{4}{3}} d^{\frac{5}{3}}} + \frac{\left (- b d e + \frac{f \left (a d + 2 b c\right )}{3}\right ) \log{\left (c + d x \right )}}{2 b^{\frac{4}{3}} d^{\frac{5}{3}}} + \frac{\sqrt{3} \left (- b d e + \frac{f \left (a d + 2 b c\right )}{3}\right ) \operatorname{atan}{\left (\frac{\sqrt{3}}{3} + \frac{2 \sqrt{3} \sqrt [3]{d} \sqrt [3]{a + b x}}{3 \sqrt [3]{b} \sqrt [3]{c + d x}} \right )}}{b^{\frac{4}{3}} d^{\frac{5}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.222653, size = 99, normalized size = 0.5 \[ \frac{\sqrt [3]{c+d x} \left (\sqrt [3]{\frac{d (a+b x)}{a d-b c}} (-a d f-2 b c f+3 b d e) \, _2F_1\left (\frac{1}{3},\frac{1}{3};\frac{4}{3};\frac{b (c+d x)}{b c-a d}\right )+d f (a+b x)\right )}{b d^2 \sqrt [3]{a+b x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.042, size = 0, normalized size = 0. \[ \int{(fx+e){\frac{1}{\sqrt [3]{bx+a}}} \left ( dx+c \right ) ^{-{\frac{2}{3}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.231874, size = 360, normalized size = 1.8 \[ \frac{\sqrt{3}{\left (6 \, \sqrt{3} \left (b d^{2}\right )^{\frac{1}{3}}{\left (b x + a\right )}^{\frac{2}{3}}{\left (d x + c\right )}^{\frac{1}{3}} f + \sqrt{3}{\left (3 \, b d e -{\left (2 \, b c + a d\right )} f\right )} \log \left (\frac{b d^{2} x + a d^{2} + \left (b d^{2}\right )^{\frac{1}{3}}{\left (b x + a\right )}^{\frac{2}{3}}{\left (d x + c\right )}^{\frac{1}{3}} d + \left (b d^{2}\right )^{\frac{2}{3}}{\left (b x + a\right )}^{\frac{1}{3}}{\left (d x + c\right )}^{\frac{2}{3}}}{b x + a}\right ) - 2 \, \sqrt{3}{\left (3 \, b d e -{\left (2 \, b c + a d\right )} f\right )} \log \left (-\frac{b d x + a d - \left (b d^{2}\right )^{\frac{1}{3}}{\left (b x + a\right )}^{\frac{2}{3}}{\left (d x + c\right )}^{\frac{1}{3}}}{b x + a}\right ) + 6 \,{\left (3 \, b d e -{\left (2 \, b c + a d\right )} f\right )} \arctan \left (\frac{2 \, \sqrt{3} \left (b d^{2}\right )^{\frac{1}{3}}{\left (b x + a\right )}^{\frac{2}{3}}{\left (d x + c\right )}^{\frac{1}{3}} + \sqrt{3}{\left (b d x + a d\right )}}{3 \,{\left (b d x + a d\right )}}\right )\right )}}{18 \, \left (b d^{2}\right )^{\frac{1}{3}} b d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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