Optimal. Leaf size=51 \[ -\frac{b^2 x (a+b x)^{n+1} \, _2F_1\left (3,n+1;n+2;\frac{b x}{a}+1\right )}{a^3 c (n+1) \sqrt{c x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0136584, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {15, 65} \[ -\frac{b^2 x (a+b x)^{n+1} \, _2F_1\left (3,n+1;n+2;\frac{b x}{a}+1\right )}{a^3 c (n+1) \sqrt{c x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 15
Rule 65
Rubi steps
\begin{align*} \int \frac{(a+b x)^n}{\left (c x^2\right )^{3/2}} \, dx &=\frac{x \int \frac{(a+b x)^n}{x^3} \, dx}{c \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 c (1+n) \sqrt{c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0025603, size = 50, normalized size = 0.98 \[ -\frac{b^2 x^3 (a+b x)^{n+1} \, _2F_1\left (3,n+1;n+2;\frac{b x}{a}+1\right )}{a^3 (n+1) \left (c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.026, size = 0, normalized size = 0. \begin{align*} \int{ \left ( bx+a \right ) ^{n} \left ( c{x}^{2} \right ) ^{-{\frac{3}{2}}}}\, 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 \frac{{\left (b x + a\right )}^{n}}{\left (c x^{2}\right )^{\frac{3}{2}}}\,{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 (\frac{\sqrt{c x^{2}}{\left (b x + a\right )}^{n}}{c^{2} x^{4}}, 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 \frac{\left (a + b x\right )^{n}}{\left (c x^{2}\right )^{\frac{3}{2}}}\, 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 \frac{{\left (b x + a\right )}^{n}}{\left (c x^{2}\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]