3.3001 \(\int \frac{1}{\sqrt [3]{a+b x} (c+d x)^{2/3}} \, dx\)

Optimal. Leaf size=126 \[ -\frac{3 \log \left (\frac{\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{b} \sqrt [3]{c+d x}}-1\right )}{2 \sqrt [3]{b} d^{2/3}}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt{3} \sqrt [3]{b} \sqrt [3]{c+d x}}+\frac{1}{\sqrt{3}}\right )}{\sqrt [3]{b} d^{2/3}}-\frac{\log (c+d x)}{2 \sqrt [3]{b} d^{2/3}} \]

[Out]

-((Sqrt[3]*ArcTan[1/Sqrt[3] + (2*d^(1/3)*(a + b*x)^(1/3))/(Sqrt[3]*b^(1/3)*(c +
d*x)^(1/3))])/(b^(1/3)*d^(2/3))) - Log[c + d*x]/(2*b^(1/3)*d^(2/3)) - (3*Log[-1
+ (d^(1/3)*(a + b*x)^(1/3))/(b^(1/3)*(c + d*x)^(1/3))])/(2*b^(1/3)*d^(2/3))

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Rubi [A]  time = 0.0695051, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053 \[ -\frac{3 \log \left (\frac{\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{b} \sqrt [3]{c+d x}}-1\right )}{2 \sqrt [3]{b} d^{2/3}}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt{3} \sqrt [3]{b} \sqrt [3]{c+d x}}+\frac{1}{\sqrt{3}}\right )}{\sqrt [3]{b} d^{2/3}}-\frac{\log (c+d x)}{2 \sqrt [3]{b} d^{2/3}} \]

Antiderivative was successfully verified.

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

[Out]

-((Sqrt[3]*ArcTan[1/Sqrt[3] + (2*d^(1/3)*(a + b*x)^(1/3))/(Sqrt[3]*b^(1/3)*(c +
d*x)^(1/3))])/(b^(1/3)*d^(2/3))) - Log[c + d*x]/(2*b^(1/3)*d^(2/3)) - (3*Log[-1
+ (d^(1/3)*(a + b*x)^(1/3))/(b^(1/3)*(c + d*x)^(1/3))])/(2*b^(1/3)*d^(2/3))

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Rubi in Sympy [A]  time = 6.11854, size = 122, normalized size = 0.97 \[ - \frac{3 \log{\left (-1 + \frac{\sqrt [3]{d} \sqrt [3]{a + b x}}{\sqrt [3]{b} \sqrt [3]{c + d x}} \right )}}{2 \sqrt [3]{b} d^{\frac{2}{3}}} - \frac{\log{\left (c + d x \right )}}{2 \sqrt [3]{b} d^{\frac{2}{3}}} - \frac{\sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3}}{3} + \frac{2 \sqrt{3} \sqrt [3]{d} \sqrt [3]{a + b x}}{3 \sqrt [3]{b} \sqrt [3]{c + d x}} \right )}}{\sqrt [3]{b} d^{\frac{2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*log(-1 + d**(1/3)*(a + b*x)**(1/3)/(b**(1/3)*(c + d*x)**(1/3)))/(2*b**(1/3)*d
**(2/3)) - log(c + d*x)/(2*b**(1/3)*d**(2/3)) - sqrt(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**(1/3)*d**(2/3))

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Mathematica [C]  time = 0.0746757, size = 71, normalized size = 0.56 \[ \frac{3 \sqrt [3]{c+d x} \sqrt [3]{\frac{d (a+b x)}{a d-b c}} \, _2F_1\left (\frac{1}{3},\frac{1}{3};\frac{4}{3};\frac{b (c+d x)}{b c-a d}\right )}{d \sqrt [3]{a+b x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.220395, size = 239, normalized size = 1.9 \[ \frac{2 \, \sqrt{3} \arctan \left (-\frac{\sqrt{3}{\left (b d x + a d - 2 \, \left (-b d^{2}\right )^{\frac{1}{3}}{\left (b x + a\right )}^{\frac{2}{3}}{\left (d x + c\right )}^{\frac{1}{3}}\right )}}{3 \,{\left (b d x + a d\right )}}\right ) - \log \left (\frac{b d^{2} x + a d^{2} - \left (-b d^{2}\right )^{\frac{1}{3}}{\left (b x + a\right )}^{\frac{2}{3}}{\left (d x + c\right )}^{\frac{1}{3}} d + \left (-b d^{2}\right )^{\frac{2}{3}}{\left (b x + a\right )}^{\frac{1}{3}}{\left (d x + c\right )}^{\frac{2}{3}}}{b x + a}\right ) + 2 \, \log \left (\frac{b d x + a d + \left (-b d^{2}\right )^{\frac{1}{3}}{\left (b x + a\right )}^{\frac{2}{3}}{\left (d x + c\right )}^{\frac{1}{3}}}{b x + a}\right )}{2 \, \left (-b d^{2}\right )^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/((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. \[ \int \frac{1}{{\left (b x + a\right )}^{\frac{1}{3}}{\left (d x + c\right )}^{\frac{2}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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