3.1775 \(\int \frac{\sqrt [6]{a+b x}}{(c+d x)^{13/6}} \, dx\)

Optimal. Leaf size=32 \[ \frac{6 (a+b x)^{7/6}}{7 (c+d x)^{7/6} (b c-a d)} \]

[Out]

(6*(a + b*x)^(7/6))/(7*(b*c - a*d)*(c + d*x)^(7/6))

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Rubi [A]  time = 0.0032505, 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} \[ \frac{6 (a+b x)^{7/6}}{7 (c+d x)^{7/6} (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^(1/6)/(c + d*x)^(13/6),x]

[Out]

(6*(a + b*x)^(7/6))/(7*(b*c - a*d)*(c + d*x)^(7/6))

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{\sqrt [6]{a+b x}}{(c+d x)^{13/6}} \, dx &=\frac{6 (a+b x)^{7/6}}{7 (b c-a d) (c+d x)^{7/6}}\\ \end{align*}

Mathematica [A]  time = 0.0095994, size = 32, normalized size = 1. \[ \frac{6 (a+b x)^{7/6}}{7 (c+d x)^{7/6} (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^(1/6)/(c + d*x)^(13/6),x]

[Out]

(6*(a + b*x)^(7/6))/(7*(b*c - a*d)*(c + d*x)^(7/6))

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Maple [A]  time = 0.003, size = 27, normalized size = 0.8 \begin{align*} -{\frac{6}{7\,ad-7\,bc} \left ( bx+a \right ) ^{{\frac{7}{6}}} \left ( dx+c \right ) ^{-{\frac{7}{6}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^(1/6)/(d*x+c)^(13/6),x)

[Out]

-6/7*(b*x+a)^(7/6)/(d*x+c)^(7/6)/(a*d-b*c)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x + a\right )}^{\frac{1}{6}}}{{\left (d x + c\right )}^{\frac{13}{6}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(1/6)/(d*x+c)^(13/6),x, algorithm="maxima")

[Out]

integrate((b*x + a)^(1/6)/(d*x + c)^(13/6), x)

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Fricas [B]  time = 1.49147, size = 142, normalized size = 4.44 \begin{align*} \frac{6 \,{\left (b x + a\right )}^{\frac{7}{6}}{\left (d x + c\right )}^{\frac{5}{6}}}{7 \,{\left (b c^{3} - a c^{2} d +{\left (b c d^{2} - a d^{3}\right )} x^{2} + 2 \,{\left (b c^{2} d - a c d^{2}\right )} x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(1/6)/(d*x+c)^(13/6),x, algorithm="fricas")

[Out]

6/7*(b*x + a)^(7/6)*(d*x + c)^(5/6)/(b*c^3 - a*c^2*d + (b*c*d^2 - a*d^3)*x^2 + 2*(b*c^2*d - a*c*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((b*x+a)**(1/6)/(d*x+c)**(13/6),x)

[Out]

Timed out

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Giac [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((b*x+a)^(1/6)/(d*x+c)^(13/6),x, algorithm="giac")

[Out]

Timed out