3.1682 \(\int \frac{(c+d x)^{5/4}}{(a+b x)^{13/4}} \, dx\)

Optimal. Leaf size=32 \[ -\frac{4 (c+d x)^{9/4}}{9 (a+b x)^{9/4} (b c-a d)} \]

[Out]

(-4*(c + d*x)^(9/4))/(9*(b*c - a*d)*(a + b*x)^(9/4))

________________________________________________________________________________________

Rubi [A]  time = 0.002907, 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} \[ -\frac{4 (c+d x)^{9/4}}{9 (a+b x)^{9/4} (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^(5/4)/(a + b*x)^(13/4),x]

[Out]

(-4*(c + d*x)^(9/4))/(9*(b*c - a*d)*(a + b*x)^(9/4))

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{(c+d x)^{5/4}}{(a+b x)^{13/4}} \, dx &=-\frac{4 (c+d x)^{9/4}}{9 (b c-a d) (a+b x)^{9/4}}\\ \end{align*}

Mathematica [A]  time = 0.0149801, size = 32, normalized size = 1. \[ -\frac{4 (c+d x)^{9/4}}{9 (a+b x)^{9/4} (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^(5/4)/(a + b*x)^(13/4),x]

[Out]

(-4*(c + d*x)^(9/4))/(9*(b*c - a*d)*(a + b*x)^(9/4))

________________________________________________________________________________________

Maple [A]  time = 0.005, size = 27, normalized size = 0.8 \begin{align*}{\frac{4}{9\,ad-9\,bc} \left ( dx+c \right ) ^{{\frac{9}{4}}} \left ( bx+a \right ) ^{-{\frac{9}{4}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^(5/4)/(b*x+a)^(13/4),x)

[Out]

4/9/(b*x+a)^(9/4)*(d*x+c)^(9/4)/(a*d-b*c)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d x + c\right )}^{\frac{5}{4}}}{{\left (b x + a\right )}^{\frac{13}{4}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(5/4)/(b*x+a)^(13/4),x, algorithm="maxima")

[Out]

integrate((d*x + c)^(5/4)/(b*x + a)^(13/4), x)

________________________________________________________________________________________

Fricas [B]  time = 2.63168, size = 221, normalized size = 6.91 \begin{align*} -\frac{4 \,{\left (d^{2} x^{2} + 2 \, c d x + c^{2}\right )}{\left (b x + a\right )}^{\frac{3}{4}}{\left (d x + c\right )}^{\frac{1}{4}}}{9 \,{\left (a^{3} b c - a^{4} d +{\left (b^{4} c - a b^{3} d\right )} x^{3} + 3 \,{\left (a b^{3} c - a^{2} b^{2} d\right )} x^{2} + 3 \,{\left (a^{2} b^{2} c - a^{3} b d\right )} x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(5/4)/(b*x+a)^(13/4),x, algorithm="fricas")

[Out]

-4/9*(d^2*x^2 + 2*c*d*x + c^2)*(b*x + a)^(3/4)*(d*x + c)^(1/4)/(a^3*b*c - a^4*d + (b^4*c - a*b^3*d)*x^3 + 3*(a
*b^3*c - a^2*b^2*d)*x^2 + 3*(a^2*b^2*c - a^3*b*d)*x)

________________________________________________________________________________________

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((d*x+c)**(5/4)/(b*x+a)**(13/4),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d x + c\right )}^{\frac{5}{4}}}{{\left (b x + a\right )}^{\frac{13}{4}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(5/4)/(b*x+a)^(13/4),x, algorithm="giac")

[Out]

integrate((d*x + c)^(5/4)/(b*x + a)^(13/4), x)