Optimal. Leaf size=79 \[ \frac{2 (b x)^{3/2} (c+d x)^n \left (\frac{d x}{c}+1\right )^{-n} (e+f x)^p \left (\frac{f x}{e}+1\right )^{-p} F_1\left (\frac{3}{2};-n,-p;\frac{5}{2};-\frac{d x}{c},-\frac{f x}{e}\right )}{3 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0462138, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {135, 133} \[ \frac{2 (b x)^{3/2} (c+d x)^n \left (\frac{d x}{c}+1\right )^{-n} (e+f x)^p \left (\frac{f x}{e}+1\right )^{-p} F_1\left (\frac{3}{2};-n,-p;\frac{5}{2};-\frac{d x}{c},-\frac{f x}{e}\right )}{3 b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 135
Rule 133
Rubi steps
\begin{align*} \int \sqrt{b x} (c+d x)^n (e+f x)^p \, dx &=\left ((c+d x)^n \left (1+\frac{d x}{c}\right )^{-n}\right ) \int \sqrt{b x} \left (1+\frac{d x}{c}\right )^n (e+f x)^p \, dx\\ &=\left ((c+d x)^n \left (1+\frac{d x}{c}\right )^{-n} (e+f x)^p \left (1+\frac{f x}{e}\right )^{-p}\right ) \int \sqrt{b x} \left (1+\frac{d x}{c}\right )^n \left (1+\frac{f x}{e}\right )^p \, dx\\ &=\frac{2 (b x)^{3/2} (c+d x)^n \left (1+\frac{d x}{c}\right )^{-n} (e+f x)^p \left (1+\frac{f x}{e}\right )^{-p} F_1\left (\frac{3}{2};-n,-p;\frac{5}{2};-\frac{d x}{c},-\frac{f x}{e}\right )}{3 b}\\ \end{align*}
Mathematica [A] time = 0.0537861, size = 79, normalized size = 1. \[ \frac{2}{3} x \sqrt{b x} (c+d x)^n \left (\frac{c+d x}{c}\right )^{-n} (e+f x)^p \left (\frac{e+f x}{e}\right )^{-p} F_1\left (\frac{3}{2};-n,-p;\frac{5}{2};-\frac{d x}{c},-\frac{f x}{e}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.039, size = 0, normalized size = 0. \begin{align*} \int \sqrt{bx} \left ( dx+c \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 \sqrt{b x}{\left (d x + c\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 (d x + c\right )}^{n}{\left (f x + e\right )}^{p}, 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 \sqrt{b x}{\left (d x + c\right )}^{n}{\left (f x + e\right )}^{p}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]