Optimal. Leaf size=66 \[ \frac{2 x (b+c x) (d x)^m \left (-\frac{c x}{b}\right )^{\frac{1}{2}-m} \, _2F_1\left (-\frac{1}{2},\frac{3}{2}-m;\frac{1}{2};\frac{c x}{b}+1\right )}{b \left (b x+c x^2\right )^{3/2}} \]
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Rubi [A] time = 0.0265839, antiderivative size = 66, 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} \[ \frac{2 x (b+c x) (d x)^m \left (-\frac{c x}{b}\right )^{\frac{1}{2}-m} \, _2F_1\left (-\frac{1}{2},\frac{3}{2}-m;\frac{1}{2};\frac{c x}{b}+1\right )}{b \left (b x+c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 674
Rule 67
Rule 65
Rubi steps
\begin{align*} \int \frac{(d x)^m}{\left (b x+c x^2\right )^{3/2}} \, dx &=\frac{\left (x^{\frac{3}{2}-m} (d x)^m (b+c x)^{3/2}\right ) \int \frac{x^{-\frac{3}{2}+m}}{(b+c x)^{3/2}} \, dx}{\left (b x+c x^2\right )^{3/2}}\\ &=-\frac{\left (c x \left (-\frac{c x}{b}\right )^{\frac{1}{2}-m} (d x)^m (b+c x)^{3/2}\right ) \int \frac{\left (-\frac{c x}{b}\right )^{-\frac{3}{2}+m}}{(b+c x)^{3/2}} \, dx}{b \left (b x+c x^2\right )^{3/2}}\\ &=\frac{2 x \left (-\frac{c x}{b}\right )^{\frac{1}{2}-m} (d x)^m (b+c x) \, _2F_1\left (-\frac{1}{2},\frac{3}{2}-m;\frac{1}{2};1+\frac{c x}{b}\right )}{b \left (b x+c x^2\right )^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0931142, size = 58, normalized size = 0.88 \[ \frac{2 (d x)^m \left (-\frac{c x}{b}\right )^{\frac{1}{2}-m} \, _2F_1\left (-\frac{1}{2},\frac{3}{2}-m;\frac{1}{2};\frac{c x}{b}+1\right )}{b \sqrt{x (b+c x)}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.438, size = 0, normalized size = 0. \begin{align*} \int{ \left ( dx \right ) ^{m} \left ( c{x}^{2}+bx \right ) ^{-{\frac{3}{2}}}}\, 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}}{{\left (c x^{2} + b x\right )}^{\frac{3}{2}}}\,{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{\sqrt{c x^{2} + b x} \left (d x\right )^{m}}{c^{2} x^{4} + 2 \, b c x^{3} + b^{2} x^{2}}, 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}}{\left (x \left (b + c x\right )\right )^{\frac{3}{2}}}\, 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}}{{\left (c x^{2} + b x\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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