3.1808 \(\int \frac{1}{\sqrt [6]{a+b x} (c+d x)^{11/6}} \, dx\)

Optimal. Leaf size=32 \[ \frac{6 (a+b x)^{5/6}}{5 (c+d x)^{5/6} (b c-a d)} \]

[Out]

(6*(a + b*x)^(5/6))/(5*(b*c - a*d)*(c + d*x)^(5/6))

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Rubi [A]  time = 0.0031016, 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} \[ \frac{6 (a+b x)^{5/6}}{5 (c+d x)^{5/6} (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b*x)^(1/6)*(c + d*x)^(11/6)),x]

[Out]

(6*(a + b*x)^(5/6))/(5*(b*c - a*d)*(c + d*x)^(5/6))

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}{\sqrt [6]{a+b x} (c+d x)^{11/6}} \, dx &=\frac{6 (a+b x)^{5/6}}{5 (b c-a d) (c+d x)^{5/6}}\\ \end{align*}

Mathematica [A]  time = 0.0107222, size = 32, normalized size = 1. \[ \frac{6 (a+b x)^{5/6}}{5 (c+d x)^{5/6} (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b*x)^(1/6)*(c + d*x)^(11/6)),x]

[Out]

(6*(a + b*x)^(5/6))/(5*(b*c - a*d)*(c + d*x)^(5/6))

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Maple [A]  time = 0.004, size = 27, normalized size = 0.8 \begin{align*} -{\frac{6}{5\,ad-5\,bc} \left ( bx+a \right ) ^{{\frac{5}{6}}} \left ( dx+c \right ) ^{-{\frac{5}{6}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x+a)^(1/6)/(d*x+c)^(11/6),x)

[Out]

-6/5*(b*x+a)^(5/6)/(d*x+c)^(5/6)/(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{1}{6}}{\left (d x + c\right )}^{\frac{11}{6}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^(1/6)/(d*x+c)^(11/6),x, algorithm="maxima")

[Out]

integrate(1/((b*x + a)^(1/6)*(d*x + c)^(11/6)), x)

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Fricas [A]  time = 1.86014, size = 99, normalized size = 3.09 \begin{align*} \frac{6 \,{\left (b x + a\right )}^{\frac{5}{6}}{\left (d x + c\right )}^{\frac{1}{6}}}{5 \,{\left (b c^{2} - a c d +{\left (b c d - a d^{2}\right )} x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^(1/6)/(d*x+c)^(11/6),x, algorithm="fricas")

[Out]

6/5*(b*x + a)^(5/6)*(d*x + c)^(1/6)/(b*c^2 - a*c*d + (b*c*d - a*d^2)*x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)**(1/6)/(d*x+c)**(11/6),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x + a\right )}^{\frac{1}{6}}{\left (d x + c\right )}^{\frac{11}{6}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^(1/6)/(d*x+c)^(11/6),x, algorithm="giac")

[Out]

integrate(1/((b*x + a)^(1/6)*(d*x + c)^(11/6)), x)