Optimal. Leaf size=65 \[ \frac{2 (b+c x) (d x)^m \left (-\frac{c x}{b}\right )^{\frac{1}{2}-m} \, _2F_1\left (\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{c x}{b}+1\right )}{c \sqrt{b x+c x^2}} \]
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Rubi [A] time = 0.0253687, antiderivative size = 65, 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} \[ \frac{2 (b+c x) (d x)^m \left (-\frac{c x}{b}\right )^{\frac{1}{2}-m} \, _2F_1\left (\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{c x}{b}+1\right )}{c \sqrt{b x+c x^2}} \]
Antiderivative was successfully verified.
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Rule 674
Rule 67
Rule 65
Rubi steps
\begin{align*} \int \frac{(d x)^m}{\sqrt{b x+c x^2}} \, dx &=\frac{\left (x^{\frac{1}{2}-m} (d x)^m \sqrt{b+c x}\right ) \int \frac{x^{-\frac{1}{2}+m}}{\sqrt{b+c x}} \, dx}{\sqrt{b x+c x^2}}\\ &=\frac{\left (\left (-\frac{c x}{b}\right )^{\frac{1}{2}-m} (d x)^m \sqrt{b+c x}\right ) \int \frac{\left (-\frac{c x}{b}\right )^{-\frac{1}{2}+m}}{\sqrt{b+c x}} \, dx}{\sqrt{b x+c x^2}}\\ &=\frac{2 \left (-\frac{c x}{b}\right )^{\frac{1}{2}-m} (d x)^m (b+c x) \, _2F_1\left (\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};1+\frac{c x}{b}\right )}{c \sqrt{b x+c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0577265, size = 58, normalized size = 0.89 \[ -\frac{2 \sqrt{x (b+c x)} (d x)^m \left (-\frac{c x}{b}\right )^{-m-\frac{1}{2}} \, _2F_1\left (\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{c x}{b}+1\right )}{b} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.4, size = 0, normalized size = 0. \begin{align*} \int{ \left ( dx \right ) ^{m}{\frac{1}{\sqrt{c{x}^{2}+bx}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (d x\right )^{m}}{\sqrt{c x^{2} + b x}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\left (d x\right )^{m}}{\sqrt{c x^{2} + b x}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\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.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (d x\right )^{m}}{\sqrt{c x^{2} + b x}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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