3.3009 \(\int \frac{1}{(a+b x) \sqrt [3]{c+d x} \sqrt [3]{b c+a d+2 b d x}} \, dx\)

Optimal. Leaf size=178 \[ -\frac{\log (a+b x)}{2 b^{2/3} (b c-a d)^{2/3}}+\frac{3 \log \left (\frac{b^{2/3} (c+d x)^{2/3}}{\sqrt [3]{b c-a d}}-\sqrt [3]{a d+b c+2 b d x}\right )}{4 b^{2/3} (b c-a d)^{2/3}}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{2 b^{2/3} (c+d x)^{2/3}}{\sqrt{3} \sqrt [3]{b c-a d} \sqrt [3]{a d+b c+2 b d x}}+\frac{1}{\sqrt{3}}\right )}{2 b^{2/3} (b c-a d)^{2/3}} \]

[Out]

-(Sqrt[3]*ArcTan[1/Sqrt[3] + (2*b^(2/3)*(c + d*x)^(2/3))/(Sqrt[3]*(b*c - a*d)^(1
/3)*(b*c + a*d + 2*b*d*x)^(1/3))])/(2*b^(2/3)*(b*c - a*d)^(2/3)) - Log[a + b*x]/
(2*b^(2/3)*(b*c - a*d)^(2/3)) + (3*Log[(b^(2/3)*(c + d*x)^(2/3))/(b*c - a*d)^(1/
3) - (b*c + a*d + 2*b*d*x)^(1/3)])/(4*b^(2/3)*(b*c - a*d)^(2/3))

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Rubi [A]  time = 0.223969, antiderivative size = 178, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.03 \[ -\frac{\log (a+b x)}{2 b^{2/3} (b c-a d)^{2/3}}+\frac{3 \log \left (\frac{b^{2/3} (c+d x)^{2/3}}{\sqrt [3]{b c-a d}}-\sqrt [3]{a d+b c+2 b d x}\right )}{4 b^{2/3} (b c-a d)^{2/3}}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{2 b^{2/3} (c+d x)^{2/3}}{\sqrt{3} \sqrt [3]{b c-a d} \sqrt [3]{a d+b c+2 b d x}}+\frac{1}{\sqrt{3}}\right )}{2 b^{2/3} (b c-a d)^{2/3}} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[3]*ArcTan[1/Sqrt[3] + (2*b^(2/3)*(c + d*x)^(2/3))/(Sqrt[3]*(b*c - a*d)^(1
/3)*(b*c + a*d + 2*b*d*x)^(1/3))])/(2*b^(2/3)*(b*c - a*d)^(2/3)) - Log[a + b*x]/
(2*b^(2/3)*(b*c - a*d)^(2/3)) + (3*Log[(b^(2/3)*(c + d*x)^(2/3))/(b*c - a*d)^(1/
3) - (b*c + a*d + 2*b*d*x)^(1/3)])/(4*b^(2/3)*(b*c - a*d)^(2/3))

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Rubi in Sympy [A]  time = 16.4917, size = 165, normalized size = 0.93 \[ - \frac{\log{\left (a + b x \right )}}{2 b^{\frac{2}{3}} \left (a d - b c\right )^{\frac{2}{3}}} + \frac{3 \log{\left (- \frac{b^{\frac{2}{3}} \left (c + d x\right )^{\frac{2}{3}}}{\sqrt [3]{a d - b c}} - \sqrt [3]{a d + b c + 2 b d x} \right )}}{4 b^{\frac{2}{3}} \left (a d - b c\right )^{\frac{2}{3}}} + \frac{\sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt{3} b^{\frac{2}{3}} \left (c + d x\right )^{\frac{2}{3}}}{3 \sqrt [3]{a d - b c} \sqrt [3]{a d + b c + 2 b d x}} - \frac{\sqrt{3}}{3} \right )}}{2 b^{\frac{2}{3}} \left (a d - b c\right )^{\frac{2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 1.15527, size = 276, normalized size = 1.55 \[ -\frac{15 d (a+b x) F_1\left (\frac{2}{3};\frac{1}{3},\frac{1}{3};\frac{5}{3};\frac{a d-b c}{d (a+b x)},-\frac{b c-a d}{2 a d+2 b x d}\right )}{b \sqrt [3]{c+d x} \sqrt [3]{a d+b (c+2 d x)} \left (10 d (a+b x) F_1\left (\frac{2}{3};\frac{1}{3},\frac{1}{3};\frac{5}{3};\frac{a d-b c}{d (a+b x)},-\frac{b c-a d}{2 a d+2 b x d}\right )-(b c-a d) \left (F_1\left (\frac{5}{3};\frac{1}{3},\frac{4}{3};\frac{8}{3};\frac{a d-b c}{d (a+b x)},-\frac{b c-a d}{2 a d+2 b x d}\right )+2 F_1\left (\frac{5}{3};\frac{4}{3},\frac{1}{3};\frac{8}{3};\frac{a d-b c}{d (a+b x)},-\frac{b c-a d}{2 a d+2 b x d}\right )\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(-15*d*(a + b*x)*AppellF1[2/3, 1/3, 1/3, 5/3, (-(b*c) + a*d)/(d*(a + b*x)), -((b
*c - a*d)/(2*a*d + 2*b*d*x))])/(b*(c + d*x)^(1/3)*(a*d + b*(c + 2*d*x))^(1/3)*(1
0*d*(a + b*x)*AppellF1[2/3, 1/3, 1/3, 5/3, (-(b*c) + a*d)/(d*(a + b*x)), -((b*c
- a*d)/(2*a*d + 2*b*d*x))] - (b*c - a*d)*(AppellF1[5/3, 1/3, 4/3, 8/3, (-(b*c) +
a*d)/(d*(a + b*x)), -((b*c - a*d)/(2*a*d + 2*b*d*x))] + 2*AppellF1[5/3, 4/3, 1/
3, 8/3, (-(b*c) + a*d)/(d*(a + b*x)), -((b*c - a*d)/(2*a*d + 2*b*d*x))])))

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Maple [F]  time = 0.076, size = 0, normalized size = 0. \[ \int{\frac{1}{bx+a}{\frac{1}{\sqrt [3]{dx+c}}}{\frac{1}{\sqrt [3]{2\,bdx+ad+bc}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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