Optimal. Leaf size=63 \[ \frac{2 \pi ^n (b x)^{7/2} (e+f x)^p \left (\frac{f x}{e}+1\right )^{-p} F_1\left (\frac{7}{2};-n,-p;\frac{9}{2};-\frac{d x}{\pi },-\frac{f x}{e}\right )}{7 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0293217, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {135, 133} \[ \frac{2 \pi ^n (b x)^{7/2} (e+f x)^p \left (\frac{f x}{e}+1\right )^{-p} F_1\left (\frac{7}{2};-n,-p;\frac{9}{2};-\frac{d x}{\pi },-\frac{f x}{e}\right )}{7 b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 135
Rule 133
Rubi steps
\begin{align*} \int (b x)^{5/2} (\pi +d x)^n (e+f x)^p \, dx &=\left ((e+f x)^p \left (1+\frac{f x}{e}\right )^{-p}\right ) \int (b x)^{5/2} (\pi +d x)^n \left (1+\frac{f x}{e}\right )^p \, dx\\ &=\frac{2 \pi ^n (b x)^{7/2} (e+f x)^p \left (1+\frac{f x}{e}\right )^{-p} F_1\left (\frac{7}{2};-n,-p;\frac{9}{2};-\frac{d x}{\pi },-\frac{f x}{e}\right )}{7 b}\\ \end{align*}
Mathematica [A] time = 0.0709525, size = 62, normalized size = 0.98 \[ \frac{2}{7} \pi ^n x (b x)^{5/2} (e+f x)^p \left (\frac{e+f x}{e}\right )^{-p} F_1\left (\frac{7}{2};-n,-p;\frac{9}{2};-\frac{d x}{\pi },-\frac{f x}{e}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.043, size = 0, normalized size = 0. \begin{align*} \int \left ( bx \right ) ^{{\frac{5}{2}}} \left ( dx+\pi \right ) ^{n} \left ( fx+e \right ) ^{p}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b x\right )^{\frac{5}{2}}{\left (\pi + d x\right )}^{n}{\left (f x + e\right )}^{p}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sqrt{b x}{\left (\pi + d x\right )}^{n}{\left (f x + e\right )}^{p} b^{2} x^{2}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b x\right )^{\frac{5}{2}}{\left (\pi + d x\right )}^{n}{\left (f x + e\right )}^{p}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]