3.1782 \(\int \frac{(a+b x)^{5/6}}{(c+d x)^{17/6}} \, dx\)

Optimal. Leaf size=32 \[ \frac{6 (a+b x)^{11/6}}{11 (c+d x)^{11/6} (b c-a d)} \]

[Out]

(6*(a + b*x)^(11/6))/(11*(b*c - a*d)*(c + d*x)^(11/6))

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Rubi [A]  time = 0.0032335, 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} \[ \frac{6 (a+b x)^{11/6}}{11 (c+d x)^{11/6} (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^(5/6)/(c + d*x)^(17/6),x]

[Out]

(6*(a + b*x)^(11/6))/(11*(b*c - a*d)*(c + d*x)^(11/6))

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{(a+b x)^{5/6}}{(c+d x)^{17/6}} \, dx &=\frac{6 (a+b x)^{11/6}}{11 (b c-a d) (c+d x)^{11/6}}\\ \end{align*}

Mathematica [A]  time = 0.0107152, size = 32, normalized size = 1. \[ \frac{6 (a+b x)^{11/6}}{11 (c+d x)^{11/6} (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^(5/6)/(c + d*x)^(17/6),x]

[Out]

(6*(a + b*x)^(11/6))/(11*(b*c - a*d)*(c + d*x)^(11/6))

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Maple [A]  time = 0.005, size = 27, normalized size = 0.8 \begin{align*} -{\frac{6}{11\,ad-11\,bc} \left ( bx+a \right ) ^{{\frac{11}{6}}} \left ( dx+c \right ) ^{-{\frac{11}{6}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^(5/6)/(d*x+c)^(17/6),x)

[Out]

-6/11*(b*x+a)^(11/6)/(d*x+c)^(11/6)/(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{5}{6}}}{{\left (d x + c\right )}^{\frac{17}{6}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(5/6)/(d*x+c)^(17/6),x, algorithm="maxima")

[Out]

integrate((b*x + a)^(5/6)/(d*x + c)^(17/6), x)

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Fricas [B]  time = 1.81663, size = 144, normalized size = 4.5 \begin{align*} \frac{6 \,{\left (b x + a\right )}^{\frac{11}{6}}{\left (d x + c\right )}^{\frac{1}{6}}}{11 \,{\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)^(5/6)/(d*x+c)^(17/6),x, algorithm="fricas")

[Out]

6/11*(b*x + a)^(11/6)*(d*x + c)^(1/6)/(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)**(5/6)/(d*x+c)**(17/6),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x + a\right )}^{\frac{5}{6}}}{{\left (d x + c\right )}^{\frac{17}{6}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(5/6)/(d*x+c)^(17/6),x, algorithm="giac")

[Out]

integrate((b*x + a)^(5/6)/(d*x + c)^(17/6), x)