Optimal. Leaf size=136 \[ \frac{4 x^{m+1} \sqrt{c-a^2 c x^2} \text{Hypergeometric2F1}(1,m+1,m+2,-a x)}{(m+1) \sqrt{1-a^2 x^2}}-\frac{3 x^{m+1} \sqrt{c-a^2 c x^2}}{(m+1) \sqrt{1-a^2 x^2}}+\frac{a x^{m+2} \sqrt{c-a^2 c x^2}}{(m+2) \sqrt{1-a^2 x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.201518, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148, Rules used = {6153, 6150, 88, 64} \[ \frac{4 x^{m+1} \sqrt{c-a^2 c x^2} \, _2F_1(1,m+1;m+2;-a x)}{(m+1) \sqrt{1-a^2 x^2}}-\frac{3 x^{m+1} \sqrt{c-a^2 c x^2}}{(m+1) \sqrt{1-a^2 x^2}}+\frac{a x^{m+2} \sqrt{c-a^2 c x^2}}{(m+2) \sqrt{1-a^2 x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6153
Rule 6150
Rule 88
Rule 64
Rubi steps
\begin{align*} \int e^{-3 \tanh ^{-1}(a x)} x^m \sqrt{c-a^2 c x^2} \, dx &=\frac{\sqrt{c-a^2 c x^2} \int e^{-3 \tanh ^{-1}(a x)} x^m \sqrt{1-a^2 x^2} \, dx}{\sqrt{1-a^2 x^2}}\\ &=\frac{\sqrt{c-a^2 c x^2} \int \frac{x^m (1-a x)^2}{1+a x} \, dx}{\sqrt{1-a^2 x^2}}\\ &=\frac{\sqrt{c-a^2 c x^2} \int \left (-3 x^m+a x^{1+m}+\frac{4 x^m}{1+a x}\right ) \, dx}{\sqrt{1-a^2 x^2}}\\ &=-\frac{3 x^{1+m} \sqrt{c-a^2 c x^2}}{(1+m) \sqrt{1-a^2 x^2}}+\frac{a x^{2+m} \sqrt{c-a^2 c x^2}}{(2+m) \sqrt{1-a^2 x^2}}+\frac{\left (4 \sqrt{c-a^2 c x^2}\right ) \int \frac{x^m}{1+a x} \, dx}{\sqrt{1-a^2 x^2}}\\ &=-\frac{3 x^{1+m} \sqrt{c-a^2 c x^2}}{(1+m) \sqrt{1-a^2 x^2}}+\frac{a x^{2+m} \sqrt{c-a^2 c x^2}}{(2+m) \sqrt{1-a^2 x^2}}+\frac{4 x^{1+m} \sqrt{c-a^2 c x^2} \, _2F_1(1,1+m;2+m;-a x)}{(1+m) \sqrt{1-a^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0500204, size = 74, normalized size = 0.54 \[ \frac{x^{m+1} \sqrt{c-a^2 c x^2} (4 (m+2) \text{Hypergeometric2F1}(1,m+1,m+2,-a x)+m (a x-3)+a x-6)}{(m+1) (m+2) \sqrt{1-a^2 x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.435, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{m}}{ \left ( ax+1 \right ) ^{3}}\sqrt{-{a}^{2}c{x}^{2}+c} \left ( -{a}^{2}{x}^{2}+1 \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{\sqrt{-a^{2} c x^{2} + c}{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} x^{m}}{{\left (a x + 1\right )}^{3}}\,{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{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}{\left (a x - 1\right )} x^{m}}{a^{2} x^{2} + 2 \, a x + 1}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{\sqrt{-a^{2} c x^{2} + c}{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} x^{m}}{{\left (a x + 1\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]