3.1590 \(\int \frac{1}{(a+b x)^{5/3} \sqrt [3]{c+d x}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0029671, antiderivative size = 32, 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, Rules used = {37} \[ -\frac{3 (c+d x)^{2/3}}{2 (a+b x)^{2/3} (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b*x)^(5/3)*(c + d*x)^(1/3)),x]

[Out]

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

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

\begin{align*} \int \frac{1}{(a+b x)^{5/3} \sqrt [3]{c+d x}} \, dx &=-\frac{3 (c+d x)^{2/3}}{2 (b c-a d) (a+b x)^{2/3}}\\ \end{align*}

Mathematica [A]  time = 0.0121046, size = 32, normalized size = 1. \[ -\frac{3 (c+d x)^{2/3}}{2 (a+b x)^{2/3} (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b*x)^(5/3)*(c + d*x)^(1/3)),x]

[Out]

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

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Maple [A]  time = 0.005, size = 27, normalized size = 0.8 \begin{align*}{\frac{3}{2\,ad-2\,bc} \left ( dx+c \right ) ^{{\frac{2}{3}}} \left ( bx+a \right ) ^{-{\frac{2}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x+a)^(5/3)/(d*x+c)^(1/3),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x + a\right )}^{\frac{5}{3}}{\left (d x + c\right )}^{\frac{1}{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^(5/3)/(d*x+c)^(1/3),x, algorithm="maxima")

[Out]

integrate(1/((b*x + a)^(5/3)*(d*x + c)^(1/3)), x)

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Fricas [A]  time = 1.76267, size = 100, normalized size = 3.12 \begin{align*} -\frac{3 \,{\left (b x + a\right )}^{\frac{1}{3}}{\left (d x + c\right )}^{\frac{2}{3}}}{2 \,{\left (a b c - a^{2} d +{\left (b^{2} c - a b d\right )} x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^(5/3)/(d*x+c)^(1/3),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a + b x\right )^{\frac{5}{3}} \sqrt [3]{c + d x}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)**(5/3)/(d*x+c)**(1/3),x)

[Out]

Integral(1/((a + b*x)**(5/3)*(c + d*x)**(1/3)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x + a\right )}^{\frac{5}{3}}{\left (d x + c\right )}^{\frac{1}{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^(5/3)/(d*x+c)^(1/3),x, algorithm="giac")

[Out]

integrate(1/((b*x + a)^(5/3)*(d*x + c)^(1/3)), x)