3.3014 \(\int \frac{a+b x}{\sqrt [3]{c+d x} (b c+a d+2 b d x)^{4/3}} \, dx\)

Optimal. Leaf size=32 \[ \frac{3 (c+d x)^{2/3}}{2 d^2 \sqrt [3]{a d+b c+2 b d x}} \]

[Out]

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

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Rubi [A]  time = 0.0431683, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.032 \[ \frac{3 (c+d x)^{2/3}}{2 d^2 \sqrt [3]{a d+b c+2 b d x}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)/((c + d*x)^(1/3)*(b*c + a*d + 2*b*d*x)^(4/3)),x]

[Out]

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

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Rubi in Sympy [A]  time = 7.89092, size = 31, normalized size = 0.97 \[ \frac{3 \left (c + d x\right )^{\frac{2}{3}}}{2 d^{2} \sqrt [3]{a d + b c + 2 b d x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)/(d*x+c)**(1/3)/(2*b*d*x+a*d+b*c)**(4/3),x)

[Out]

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

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Mathematica [A]  time = 0.096631, size = 32, normalized size = 1. \[ \frac{3 (c+d x)^{2/3}}{2 d^2 \sqrt [3]{a d+b (c+2 d x)}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)/((c + d*x)^(1/3)*(b*c + a*d + 2*b*d*x)^(4/3)),x]

[Out]

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

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Maple [A]  time = 0.009, size = 27, normalized size = 0.8 \[{\frac{3}{2\,{d}^{2}} \left ( dx+c \right ) ^{{\frac{2}{3}}}{\frac{1}{\sqrt [3]{2\,bdx+ad+bc}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)/(d*x+c)^(1/3)/(2*b*d*x+a*d+b*c)^(4/3),x)

[Out]

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

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Maxima [A]  time = 1.52187, size = 35, normalized size = 1.09 \[ \frac{3 \,{\left (d x + c\right )}^{\frac{2}{3}}}{2 \,{\left (2 \, b d x + b c + a d\right )}^{\frac{1}{3}} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)/((2*b*d*x + b*c + a*d)^(4/3)*(d*x + c)^(1/3)),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.321613, size = 59, normalized size = 1.84 \[ \frac{3 \,{\left (2 \, b d x + b c + a d\right )}^{\frac{2}{3}}{\left (d x + c\right )}^{\frac{2}{3}}}{2 \,{\left (2 \, b d^{3} x + b c d^{2} + a d^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)/((2*b*d*x + b*c + a*d)^(4/3)*(d*x + c)^(1/3)),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{a + b x}{\sqrt [3]{c + d x} \left (a d + b c + 2 b d x\right )^{\frac{4}{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)/(d*x+c)**(1/3)/(2*b*d*x+a*d+b*c)**(4/3),x)

[Out]

Integral((a + b*x)/((c + d*x)**(1/3)*(a*d + b*c + 2*b*d*x)**(4/3)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{b x + a}{{\left (2 \, b d x + b c + a d\right )}^{\frac{4}{3}}{\left (d x + c\right )}^{\frac{1}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)/((2*b*d*x + b*c + a*d)^(4/3)*(d*x + c)^(1/3)),x, algorithm="giac")

[Out]

integrate((b*x + a)/((2*b*d*x + b*c + a*d)^(4/3)*(d*x + c)^(1/3)), x)