3.1801 \(\int \frac{(a+b x)^{7/6}}{(c+d x)^{19/6}} \, dx\)

Optimal. Leaf size=32 \[ \frac{6 (a+b x)^{13/6}}{13 (c+d x)^{13/6} (b c-a d)} \]

[Out]

(6*(a + b*x)^(13/6))/(13*(b*c - a*d)*(c + d*x)^(13/6))

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Rubi [A]  time = 0.0030103, 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} \[ \frac{6 (a+b x)^{13/6}}{13 (c+d x)^{13/6} (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^(7/6)/(c + d*x)^(19/6),x]

[Out]

(6*(a + b*x)^(13/6))/(13*(b*c - a*d)*(c + d*x)^(13/6))

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)^{7/6}}{(c+d x)^{19/6}} \, dx &=\frac{6 (a+b x)^{13/6}}{13 (b c-a d) (c+d x)^{13/6}}\\ \end{align*}

Mathematica [A]  time = 0.0129569, size = 32, normalized size = 1. \[ \frac{6 (a+b x)^{13/6}}{13 (c+d x)^{13/6} (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^(7/6)/(c + d*x)^(19/6),x]

[Out]

(6*(a + b*x)^(13/6))/(13*(b*c - a*d)*(c + d*x)^(13/6))

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Maple [A]  time = 0.004, size = 27, normalized size = 0.8 \begin{align*} -{\frac{6}{13\,ad-13\,bc} \left ( bx+a \right ) ^{{\frac{13}{6}}} \left ( dx+c \right ) ^{-{\frac{13}{6}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^(7/6)/(d*x+c)^(19/6),x)

[Out]

-6/13*(b*x+a)^(13/6)/(d*x+c)^(13/6)/(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{7}{6}}}{{\left (d x + c\right )}^{\frac{19}{6}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(7/6)/(d*x+c)^(19/6),x, algorithm="maxima")

[Out]

integrate((b*x + a)^(7/6)/(d*x + c)^(19/6), x)

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Fricas [B]  time = 1.76069, size = 221, normalized size = 6.91 \begin{align*} \frac{6 \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}{\left (b x + a\right )}^{\frac{1}{6}}{\left (d x + c\right )}^{\frac{5}{6}}}{13 \,{\left (b c^{4} - a c^{3} d +{\left (b c d^{3} - a d^{4}\right )} x^{3} + 3 \,{\left (b c^{2} d^{2} - a c d^{3}\right )} x^{2} + 3 \,{\left (b c^{3} d - a c^{2} d^{2}\right )} x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(7/6)/(d*x+c)^(19/6),x, algorithm="fricas")

[Out]

6/13*(b^2*x^2 + 2*a*b*x + a^2)*(b*x + a)^(1/6)*(d*x + c)^(5/6)/(b*c^4 - a*c^3*d + (b*c*d^3 - a*d^4)*x^3 + 3*(b
*c^2*d^2 - a*c*d^3)*x^2 + 3*(b*c^3*d - a*c^2*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)**(7/6)/(d*x+c)**(19/6),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)^(7/6)/(d*x+c)^(19/6),x, algorithm="giac")

[Out]

Timed out