Optimal. Leaf size=95 \[ -\frac{(a x+1)^{5/4}}{x \sqrt [4]{1-a x}}+\frac{10 a \sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}-5 a \tan ^{-1}\left (\frac{\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-5 a \tanh ^{-1}\left (\frac{\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0386483, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {6126, 94, 93, 212, 206, 203} \[ -\frac{(a x+1)^{5/4}}{x \sqrt [4]{1-a x}}+\frac{10 a \sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}-5 a \tan ^{-1}\left (\frac{\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-5 a \tanh ^{-1}\left (\frac{\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6126
Rule 94
Rule 93
Rule 212
Rule 206
Rule 203
Rubi steps
\begin{align*} \int \frac{e^{\frac{5}{2} \tanh ^{-1}(a x)}}{x^2} \, dx &=\int \frac{(1+a x)^{5/4}}{x^2 (1-a x)^{5/4}} \, dx\\ &=-\frac{(1+a x)^{5/4}}{x \sqrt [4]{1-a x}}+\frac{1}{2} (5 a) \int \frac{\sqrt [4]{1+a x}}{x (1-a x)^{5/4}} \, dx\\ &=\frac{10 a \sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}-\frac{(1+a x)^{5/4}}{x \sqrt [4]{1-a x}}+\frac{1}{2} (5 a) \int \frac{1}{x \sqrt [4]{1-a x} (1+a x)^{3/4}} \, dx\\ &=\frac{10 a \sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}-\frac{(1+a x)^{5/4}}{x \sqrt [4]{1-a x}}+(10 a) \operatorname{Subst}\left (\int \frac{1}{-1+x^4} \, dx,x,\frac{\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )\\ &=\frac{10 a \sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}-\frac{(1+a x)^{5/4}}{x \sqrt [4]{1-a x}}-(5 a) \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\frac{\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )-(5 a) \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\frac{\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )\\ &=\frac{10 a \sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}-\frac{(1+a x)^{5/4}}{x \sqrt [4]{1-a x}}-5 a \tan ^{-1}\left (\frac{\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )-5 a \tanh ^{-1}\left (\frac{\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )\\ \end{align*}
Mathematica [C] time = 0.018959, size = 74, normalized size = 0.78 \[ \frac{10 a x (a x-1) \text{Hypergeometric2F1}\left (\frac{3}{4},1,\frac{7}{4},\frac{1-a x}{a x+1}\right )+3 \left (9 a^2 x^2+8 a x-1\right )}{3 x \sqrt [4]{1-a x} (a x+1)^{3/4}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.106, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}} \left ({(ax+1){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) ^{{\frac{5}{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 (\frac{a x + 1}{\sqrt{-a^{2} x^{2} + 1}}\right )^{\frac{5}{2}}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.69999, size = 296, normalized size = 3.12 \begin{align*} -\frac{10 \, a x \arctan \left (\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}\right ) + 5 \, a x \log \left (\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} + 1\right ) - 5 \, a x \log \left (\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} - 1\right ) - 2 \,{\left (9 \, a x - 1\right )} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}}{2 \, x} \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{\left (\frac{a x + 1}{\sqrt{-a^{2} x^{2} + 1}}\right )^{\frac{5}{2}}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]