Optimal. Leaf size=67 \[ \frac{2 (b+c x) \sqrt{b x+c x^2} (d x)^m \left (-\frac{c x}{b}\right )^{-m-\frac{1}{2}} \, _2F_1\left (\frac{3}{2},-m-\frac{1}{2};\frac{5}{2};\frac{c x}{b}+1\right )}{3 c} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0251237, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {674, 67, 65} \[ \frac{2 (b+c x) \sqrt{b x+c x^2} (d x)^m \left (-\frac{c x}{b}\right )^{-m-\frac{1}{2}} \, _2F_1\left (\frac{3}{2},-m-\frac{1}{2};\frac{5}{2};\frac{c x}{b}+1\right )}{3 c} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 674
Rule 67
Rule 65
Rubi steps
\begin{align*} \int (d x)^m \sqrt{b x+c x^2} \, dx &=\frac{\left (x^{-\frac{1}{2}-m} (d x)^m \sqrt{b x+c x^2}\right ) \int x^{\frac{1}{2}+m} \sqrt{b+c x} \, dx}{\sqrt{b+c x}}\\ &=\frac{\left (\left (-\frac{c x}{b}\right )^{-\frac{1}{2}-m} (d x)^m \sqrt{b x+c x^2}\right ) \int \left (-\frac{c x}{b}\right )^{\frac{1}{2}+m} \sqrt{b+c x} \, dx}{\sqrt{b+c x}}\\ &=\frac{2 \left (-\frac{c x}{b}\right )^{-\frac{1}{2}-m} (d x)^m (b+c x) \sqrt{b x+c x^2} \, _2F_1\left (\frac{3}{2},-\frac{1}{2}-m;\frac{5}{2};1+\frac{c x}{b}\right )}{3 c}\\ \end{align*}
Mathematica [A] time = 0.0681754, size = 60, normalized size = 0.9 \[ -\frac{2 (x (b+c x))^{3/2} (d x)^m \left (-\frac{c x}{b}\right )^{-m-\frac{3}{2}} \, _2F_1\left (\frac{3}{2},-m-\frac{1}{2};\frac{5}{2};\frac{c x}{b}+1\right )}{3 b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.427, size = 0, normalized size = 0. \begin{align*} \int \left ( dx \right ) ^{m}\sqrt{c{x}^{2}+bx}\, 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{c x^{2} + b x} \left (d x\right )^{m}\,{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{c x^{2} + b x} \left (d x\right )^{m}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d x\right )^{m} \sqrt{x \left (b + c x\right )}\, dx \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{c x^{2} + b x} \left (d x\right )^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]