Optimal. Leaf size=48 \[ -\frac{b^2 x (a+b x)^{n+1} \, _2F_1\left (3,n+1;n+2;\frac{b x}{a}+1\right )}{a^3 (n+1) \sqrt{c x^2}} \]
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Rubi [A] time = 0.011783, antiderivative size = 48, 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} \[ -\frac{b^2 x (a+b x)^{n+1} \, _2F_1\left (3,n+1;n+2;\frac{b x}{a}+1\right )}{a^3 (n+1) \sqrt{c x^2}} \]
Antiderivative was successfully verified.
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Rule 15
Rule 65
Rubi steps
\begin{align*} \int \frac{(a+b x)^n}{x^2 \sqrt{c x^2}} \, dx &=\frac{x \int \frac{(a+b x)^n}{x^3} \, dx}{\sqrt{c x^2}}\\ &=-\frac{b^2 x (a+b x)^{1+n} \, _2F_1\left (3,1+n;2+n;1+\frac{b x}{a}\right )}{a^3 (1+n) \sqrt{c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0106825, size = 51, normalized size = 1.06 \[ -\frac{b^2 c x^3 (a+b x)^{n+1} \, _2F_1\left (3,n+1;n+2;\frac{b x}{a}+1\right )}{a^3 (n+1) \left (c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.026, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( bx+a \right ) ^{n}}{{x}^{2}}{\frac{1}{\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{{\left (b x + a\right )}^{n}}{\sqrt{c x^{2}} x^{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}}{\left (b x + a\right )}^{n}}{c x^{4}}, 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 (a + b x\right )^{n}}{x^{2} \sqrt{c x^{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 (b x + a\right )}^{n}}{\sqrt{c x^{2}} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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