3.1590 \(\int \frac{1}{(a+b x)^{2/3} \sqrt [3]{c+d x}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0634427, 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 \[ -\frac{3 \log \left (\frac{\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}-1\right )}{2 b^{2/3} \sqrt [3]{d}}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt{3} \sqrt [3]{d} \sqrt [3]{a+b x}}+\frac{1}{\sqrt{3}}\right )}{b^{2/3} \sqrt [3]{d}}-\frac{\log (a+b x)}{2 b^{2/3} \sqrt [3]{d}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 6.24435, size = 122, normalized size = 0.97 \[ - \frac{\log{\left (a + b x \right )}}{2 b^{\frac{2}{3}} \sqrt [3]{d}} - \frac{3 \log{\left (\frac{\sqrt [3]{b} \sqrt [3]{c + d x}}{\sqrt [3]{d} \sqrt [3]{a + b x}} - 1 \right )}}{2 b^{\frac{2}{3}} \sqrt [3]{d}} - \frac{\sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt{3} \sqrt [3]{b} \sqrt [3]{c + d x}}{3 \sqrt [3]{d} \sqrt [3]{a + b x}} + \frac{\sqrt{3}}{3} \right )}}{b^{\frac{2}{3}} \sqrt [3]{d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.0666333, size = 73, normalized size = 0.58 \[ \frac{3 (c+d x)^{2/3} \left (\frac{d (a+b x)}{a d-b c}\right )^{2/3} \, _2F_1\left (\frac{2}{3},\frac{2}{3};\frac{5}{3};\frac{b (c+d x)}{b c-a d}\right )}{2 d (a+b x)^{2/3}} \]

Antiderivative was successfully verified.

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

[Out]

(3*((d*(a + b*x))/(-(b*c) + a*d))^(2/3)*(c + d*x)^(2/3)*Hypergeometric2F1[2/3, 2
/3, 5/3, (b*(c + d*x))/(b*c - a*d)])/(2*d*(a + b*x)^(2/3))

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Maple [F]  time = 0.056, size = 0, normalized size = 0. \[ \int{1 \left ( bx+a \right ) ^{-{\frac{2}{3}}}{\frac{1}{\sqrt [3]{dx+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.224163, size = 239, normalized size = 1.9 \[ \frac{2 \, \sqrt{3} \arctan \left (-\frac{\sqrt{3}{\left (b d x + b c - 2 \, \left (-b^{2} d\right )^{\frac{1}{3}}{\left (b x + a\right )}^{\frac{1}{3}}{\left (d x + c\right )}^{\frac{2}{3}}\right )}}{3 \,{\left (b d x + b c\right )}}\right ) - \log \left (\frac{b^{2} d x + b^{2} c - \left (-b^{2} d\right )^{\frac{1}{3}}{\left (b x + a\right )}^{\frac{1}{3}}{\left (d x + c\right )}^{\frac{2}{3}} b + \left (-b^{2} d\right )^{\frac{2}{3}}{\left (b x + a\right )}^{\frac{2}{3}}{\left (d x + c\right )}^{\frac{1}{3}}}{d x + c}\right ) + 2 \, \log \left (\frac{b d x + b c + \left (-b^{2} d\right )^{\frac{1}{3}}{\left (b x + a\right )}^{\frac{1}{3}}{\left (d x + c\right )}^{\frac{2}{3}}}{d x + c}\right )}{2 \, \left (-b^{2} d\right )^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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