Optimal. Leaf size=194 \[ -\frac{113 a^2 \sqrt [4]{a x+1}}{96 x^2 \sqrt [4]{1-a x}}+\frac{2467 a^4 \sqrt [4]{a x+1}}{192 \sqrt [4]{1-a x}}-\frac{521 a^3 \sqrt [4]{a x+1}}{192 x \sqrt [4]{1-a x}}-\frac{475}{64} a^4 \tan ^{-1}\left (\frac{\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\frac{475}{64} a^4 \tanh ^{-1}\left (\frac{\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\frac{17 a \sqrt [4]{a x+1}}{24 x^3 \sqrt [4]{1-a x}}-\frac{\sqrt [4]{a x+1}}{4 x^4 \sqrt [4]{1-a x}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0940809, antiderivative size = 194, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 9, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.643, Rules used = {6126, 98, 151, 155, 12, 93, 212, 206, 203} \[ -\frac{113 a^2 \sqrt [4]{a x+1}}{96 x^2 \sqrt [4]{1-a x}}+\frac{2467 a^4 \sqrt [4]{a x+1}}{192 \sqrt [4]{1-a x}}-\frac{521 a^3 \sqrt [4]{a x+1}}{192 x \sqrt [4]{1-a x}}-\frac{475}{64} a^4 \tan ^{-1}\left (\frac{\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\frac{475}{64} a^4 \tanh ^{-1}\left (\frac{\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\frac{17 a \sqrt [4]{a x+1}}{24 x^3 \sqrt [4]{1-a x}}-\frac{\sqrt [4]{a x+1}}{4 x^4 \sqrt [4]{1-a x}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6126
Rule 98
Rule 151
Rule 155
Rule 12
Rule 93
Rule 212
Rule 206
Rule 203
Rubi steps
\begin{align*} \int \frac{e^{\frac{5}{2} \tanh ^{-1}(a x)}}{x^5} \, dx &=\int \frac{(1+a x)^{5/4}}{x^5 (1-a x)^{5/4}} \, dx\\ &=-\frac{\sqrt [4]{1+a x}}{4 x^4 \sqrt [4]{1-a x}}-\frac{1}{4} \int \frac{-\frac{17 a}{2}-8 a^2 x}{x^4 (1-a x)^{5/4} (1+a x)^{3/4}} \, dx\\ &=-\frac{\sqrt [4]{1+a x}}{4 x^4 \sqrt [4]{1-a x}}-\frac{17 a \sqrt [4]{1+a x}}{24 x^3 \sqrt [4]{1-a x}}+\frac{1}{12} \int \frac{\frac{113 a^2}{4}+\frac{51 a^3 x}{2}}{x^3 (1-a x)^{5/4} (1+a x)^{3/4}} \, dx\\ &=-\frac{\sqrt [4]{1+a x}}{4 x^4 \sqrt [4]{1-a x}}-\frac{17 a \sqrt [4]{1+a x}}{24 x^3 \sqrt [4]{1-a x}}-\frac{113 a^2 \sqrt [4]{1+a x}}{96 x^2 \sqrt [4]{1-a x}}-\frac{1}{24} \int \frac{-\frac{521 a^3}{8}-\frac{113 a^4 x}{2}}{x^2 (1-a x)^{5/4} (1+a x)^{3/4}} \, dx\\ &=-\frac{\sqrt [4]{1+a x}}{4 x^4 \sqrt [4]{1-a x}}-\frac{17 a \sqrt [4]{1+a x}}{24 x^3 \sqrt [4]{1-a x}}-\frac{113 a^2 \sqrt [4]{1+a x}}{96 x^2 \sqrt [4]{1-a x}}-\frac{521 a^3 \sqrt [4]{1+a x}}{192 x \sqrt [4]{1-a x}}+\frac{1}{24} \int \frac{\frac{1425 a^4}{16}+\frac{521 a^5 x}{8}}{x (1-a x)^{5/4} (1+a x)^{3/4}} \, dx\\ &=\frac{2467 a^4 \sqrt [4]{1+a x}}{192 \sqrt [4]{1-a x}}-\frac{\sqrt [4]{1+a x}}{4 x^4 \sqrt [4]{1-a x}}-\frac{17 a \sqrt [4]{1+a x}}{24 x^3 \sqrt [4]{1-a x}}-\frac{113 a^2 \sqrt [4]{1+a x}}{96 x^2 \sqrt [4]{1-a x}}-\frac{521 a^3 \sqrt [4]{1+a x}}{192 x \sqrt [4]{1-a x}}-\frac{\int -\frac{1425 a^5}{32 x \sqrt [4]{1-a x} (1+a x)^{3/4}} \, dx}{12 a}\\ &=\frac{2467 a^4 \sqrt [4]{1+a x}}{192 \sqrt [4]{1-a x}}-\frac{\sqrt [4]{1+a x}}{4 x^4 \sqrt [4]{1-a x}}-\frac{17 a \sqrt [4]{1+a x}}{24 x^3 \sqrt [4]{1-a x}}-\frac{113 a^2 \sqrt [4]{1+a x}}{96 x^2 \sqrt [4]{1-a x}}-\frac{521 a^3 \sqrt [4]{1+a x}}{192 x \sqrt [4]{1-a x}}+\frac{1}{128} \left (475 a^4\right ) \int \frac{1}{x \sqrt [4]{1-a x} (1+a x)^{3/4}} \, dx\\ &=\frac{2467 a^4 \sqrt [4]{1+a x}}{192 \sqrt [4]{1-a x}}-\frac{\sqrt [4]{1+a x}}{4 x^4 \sqrt [4]{1-a x}}-\frac{17 a \sqrt [4]{1+a x}}{24 x^3 \sqrt [4]{1-a x}}-\frac{113 a^2 \sqrt [4]{1+a x}}{96 x^2 \sqrt [4]{1-a x}}-\frac{521 a^3 \sqrt [4]{1+a x}}{192 x \sqrt [4]{1-a x}}+\frac{1}{32} \left (475 a^4\right ) \operatorname{Subst}\left (\int \frac{1}{-1+x^4} \, dx,x,\frac{\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )\\ &=\frac{2467 a^4 \sqrt [4]{1+a x}}{192 \sqrt [4]{1-a x}}-\frac{\sqrt [4]{1+a x}}{4 x^4 \sqrt [4]{1-a x}}-\frac{17 a \sqrt [4]{1+a x}}{24 x^3 \sqrt [4]{1-a x}}-\frac{113 a^2 \sqrt [4]{1+a x}}{96 x^2 \sqrt [4]{1-a x}}-\frac{521 a^3 \sqrt [4]{1+a x}}{192 x \sqrt [4]{1-a x}}-\frac{1}{64} \left (475 a^4\right ) \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\frac{\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )-\frac{1}{64} \left (475 a^4\right ) \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\frac{\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )\\ &=\frac{2467 a^4 \sqrt [4]{1+a x}}{192 \sqrt [4]{1-a x}}-\frac{\sqrt [4]{1+a x}}{4 x^4 \sqrt [4]{1-a x}}-\frac{17 a \sqrt [4]{1+a x}}{24 x^3 \sqrt [4]{1-a x}}-\frac{113 a^2 \sqrt [4]{1+a x}}{96 x^2 \sqrt [4]{1-a x}}-\frac{521 a^3 \sqrt [4]{1+a x}}{192 x \sqrt [4]{1-a x}}-\frac{475}{64} a^4 \tan ^{-1}\left (\frac{\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )-\frac{475}{64} a^4 \tanh ^{-1}\left (\frac{\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )\\ \end{align*}
Mathematica [C] time = 0.0322788, size = 99, normalized size = 0.51 \[ \frac{950 a^4 x^4 (a x-1) \text{Hypergeometric2F1}\left (\frac{3}{4},1,\frac{7}{4},\frac{1-a x}{a x+1}\right )+2467 a^5 x^5+1946 a^4 x^4-747 a^3 x^3-362 a^2 x^2-184 a x-48}{192 x^4 \sqrt [4]{1-a x} (a x+1)^{3/4}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.112, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{5}} \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^{5}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.00803, size = 390, normalized size = 2.01 \begin{align*} -\frac{2850 \, a^{4} x^{4} \arctan \left (\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}\right ) + 1425 \, a^{4} x^{4} \log \left (\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} + 1\right ) - 1425 \, a^{4} x^{4} \log \left (\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} - 1\right ) - 2 \,{\left (2467 \, a^{4} x^{4} - 521 \, a^{3} x^{3} - 226 \, a^{2} x^{2} - 136 \, a x - 48\right )} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}}{384 \, x^{4}} \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^{5}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]