Optimal. Leaf size=47 \[ \frac{b \sqrt{c x^2} (a+b x)^{n+1} \, _2F_1\left (2,n+1;n+2;\frac{b x}{a}+1\right )}{a^2 (n+1) x} \]
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Rubi [A] time = 0.0106019, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {15, 65} \[ \frac{b \sqrt{c x^2} (a+b x)^{n+1} \, _2F_1\left (2,n+1;n+2;\frac{b x}{a}+1\right )}{a^2 (n+1) x} \]
Antiderivative was successfully verified.
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Rule 15
Rule 65
Rubi steps
\begin{align*} \int \frac{\sqrt{c x^2} (a+b x)^n}{x^3} \, dx &=\frac{\sqrt{c x^2} \int \frac{(a+b x)^n}{x^2} \, dx}{x}\\ &=\frac{b \sqrt{c x^2} (a+b x)^{1+n} \, _2F_1\left (2,1+n;2+n;1+\frac{b x}{a}\right )}{a^2 (1+n) x}\\ \end{align*}
Mathematica [A] time = 0.0141412, size = 47, normalized size = 1. \[ \frac{b \sqrt{c x^2} (a+b x)^{n+1} \, _2F_1\left (2,n+1;n+2;\frac{b x}{a}+1\right )}{a^2 (n+1) x} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.022, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( bx+a \right ) ^{n}}{{x}^{3}}\sqrt{c{x}^{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{\sqrt{c x^{2}}{\left (b x + a\right )}^{n}}{x^{3}}\,{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}}{\left (b x + a\right )}^{n}}{x^{3}}, 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{\sqrt{c x^{2}} \left (a + b x\right )^{n}}{x^{3}}\, 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{\sqrt{c x^{2}}{\left (b x + a\right )}^{n}}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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