Optimal. Leaf size=359 \[ -\frac{(a x+1)^{11/4} (1-a x)^{13/4}}{6 a}-\frac{11 (a x+1)^{7/4} (1-a x)^{13/4}}{60 a}-\frac{77 (a x+1)^{3/4} (1-a x)^{13/4}}{480 a}+\frac{77 (a x+1)^{3/4} (1-a x)^{9/4}}{960 a}+\frac{231 (a x+1)^{3/4} (1-a x)^{5/4}}{1280 a}+\frac{231 (a x+1)^{3/4} \sqrt [4]{1-a x}}{512 a}+\frac{231 \log \left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}-\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{1024 \sqrt{2} a}-\frac{231 \log \left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}+\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{1024 \sqrt{2} a}+\frac{231 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{512 \sqrt{2} a}-\frac{231 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{512 \sqrt{2} a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.306347, antiderivative size = 359, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 10, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {6140, 50, 63, 240, 211, 1165, 628, 1162, 617, 204} \[ -\frac{(a x+1)^{11/4} (1-a x)^{13/4}}{6 a}-\frac{11 (a x+1)^{7/4} (1-a x)^{13/4}}{60 a}-\frac{77 (a x+1)^{3/4} (1-a x)^{13/4}}{480 a}+\frac{77 (a x+1)^{3/4} (1-a x)^{9/4}}{960 a}+\frac{231 (a x+1)^{3/4} (1-a x)^{5/4}}{1280 a}+\frac{231 (a x+1)^{3/4} \sqrt [4]{1-a x}}{512 a}+\frac{231 \log \left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}-\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{1024 \sqrt{2} a}-\frac{231 \log \left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}+\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{1024 \sqrt{2} a}+\frac{231 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{512 \sqrt{2} a}-\frac{231 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{512 \sqrt{2} a} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6140
Rule 50
Rule 63
Rule 240
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int e^{\frac{1}{2} \tanh ^{-1}(a x)} \left (1-a^2 x^2\right )^{5/2} \, dx &=\int (1-a x)^{9/4} (1+a x)^{11/4} \, dx\\ &=-\frac{(1-a x)^{13/4} (1+a x)^{11/4}}{6 a}+\frac{11}{12} \int (1-a x)^{9/4} (1+a x)^{7/4} \, dx\\ &=-\frac{11 (1-a x)^{13/4} (1+a x)^{7/4}}{60 a}-\frac{(1-a x)^{13/4} (1+a x)^{11/4}}{6 a}+\frac{77}{120} \int (1-a x)^{9/4} (1+a x)^{3/4} \, dx\\ &=-\frac{77 (1-a x)^{13/4} (1+a x)^{3/4}}{480 a}-\frac{11 (1-a x)^{13/4} (1+a x)^{7/4}}{60 a}-\frac{(1-a x)^{13/4} (1+a x)^{11/4}}{6 a}+\frac{77}{320} \int \frac{(1-a x)^{9/4}}{\sqrt [4]{1+a x}} \, dx\\ &=\frac{77 (1-a x)^{9/4} (1+a x)^{3/4}}{960 a}-\frac{77 (1-a x)^{13/4} (1+a x)^{3/4}}{480 a}-\frac{11 (1-a x)^{13/4} (1+a x)^{7/4}}{60 a}-\frac{(1-a x)^{13/4} (1+a x)^{11/4}}{6 a}+\frac{231}{640} \int \frac{(1-a x)^{5/4}}{\sqrt [4]{1+a x}} \, dx\\ &=\frac{231 (1-a x)^{5/4} (1+a x)^{3/4}}{1280 a}+\frac{77 (1-a x)^{9/4} (1+a x)^{3/4}}{960 a}-\frac{77 (1-a x)^{13/4} (1+a x)^{3/4}}{480 a}-\frac{11 (1-a x)^{13/4} (1+a x)^{7/4}}{60 a}-\frac{(1-a x)^{13/4} (1+a x)^{11/4}}{6 a}+\frac{231}{512} \int \frac{\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}} \, dx\\ &=\frac{231 \sqrt [4]{1-a x} (1+a x)^{3/4}}{512 a}+\frac{231 (1-a x)^{5/4} (1+a x)^{3/4}}{1280 a}+\frac{77 (1-a x)^{9/4} (1+a x)^{3/4}}{960 a}-\frac{77 (1-a x)^{13/4} (1+a x)^{3/4}}{480 a}-\frac{11 (1-a x)^{13/4} (1+a x)^{7/4}}{60 a}-\frac{(1-a x)^{13/4} (1+a x)^{11/4}}{6 a}+\frac{231 \int \frac{1}{(1-a x)^{3/4} \sqrt [4]{1+a x}} \, dx}{1024}\\ &=\frac{231 \sqrt [4]{1-a x} (1+a x)^{3/4}}{512 a}+\frac{231 (1-a x)^{5/4} (1+a x)^{3/4}}{1280 a}+\frac{77 (1-a x)^{9/4} (1+a x)^{3/4}}{960 a}-\frac{77 (1-a x)^{13/4} (1+a x)^{3/4}}{480 a}-\frac{11 (1-a x)^{13/4} (1+a x)^{7/4}}{60 a}-\frac{(1-a x)^{13/4} (1+a x)^{11/4}}{6 a}-\frac{231 \operatorname{Subst}\left (\int \frac{1}{\sqrt [4]{2-x^4}} \, dx,x,\sqrt [4]{1-a x}\right )}{256 a}\\ &=\frac{231 \sqrt [4]{1-a x} (1+a x)^{3/4}}{512 a}+\frac{231 (1-a x)^{5/4} (1+a x)^{3/4}}{1280 a}+\frac{77 (1-a x)^{9/4} (1+a x)^{3/4}}{960 a}-\frac{77 (1-a x)^{13/4} (1+a x)^{3/4}}{480 a}-\frac{11 (1-a x)^{13/4} (1+a x)^{7/4}}{60 a}-\frac{(1-a x)^{13/4} (1+a x)^{11/4}}{6 a}-\frac{231 \operatorname{Subst}\left (\int \frac{1}{1+x^4} \, dx,x,\frac{\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{256 a}\\ &=\frac{231 \sqrt [4]{1-a x} (1+a x)^{3/4}}{512 a}+\frac{231 (1-a x)^{5/4} (1+a x)^{3/4}}{1280 a}+\frac{77 (1-a x)^{9/4} (1+a x)^{3/4}}{960 a}-\frac{77 (1-a x)^{13/4} (1+a x)^{3/4}}{480 a}-\frac{11 (1-a x)^{13/4} (1+a x)^{7/4}}{60 a}-\frac{(1-a x)^{13/4} (1+a x)^{11/4}}{6 a}-\frac{231 \operatorname{Subst}\left (\int \frac{1-x^2}{1+x^4} \, dx,x,\frac{\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{512 a}-\frac{231 \operatorname{Subst}\left (\int \frac{1+x^2}{1+x^4} \, dx,x,\frac{\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{512 a}\\ &=\frac{231 \sqrt [4]{1-a x} (1+a x)^{3/4}}{512 a}+\frac{231 (1-a x)^{5/4} (1+a x)^{3/4}}{1280 a}+\frac{77 (1-a x)^{9/4} (1+a x)^{3/4}}{960 a}-\frac{77 (1-a x)^{13/4} (1+a x)^{3/4}}{480 a}-\frac{11 (1-a x)^{13/4} (1+a x)^{7/4}}{60 a}-\frac{(1-a x)^{13/4} (1+a x)^{11/4}}{6 a}-\frac{231 \operatorname{Subst}\left (\int \frac{1}{1-\sqrt{2} x+x^2} \, dx,x,\frac{\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{1024 a}-\frac{231 \operatorname{Subst}\left (\int \frac{1}{1+\sqrt{2} x+x^2} \, dx,x,\frac{\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{1024 a}+\frac{231 \operatorname{Subst}\left (\int \frac{\sqrt{2}+2 x}{-1-\sqrt{2} x-x^2} \, dx,x,\frac{\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{1024 \sqrt{2} a}+\frac{231 \operatorname{Subst}\left (\int \frac{\sqrt{2}-2 x}{-1+\sqrt{2} x-x^2} \, dx,x,\frac{\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{1024 \sqrt{2} a}\\ &=\frac{231 \sqrt [4]{1-a x} (1+a x)^{3/4}}{512 a}+\frac{231 (1-a x)^{5/4} (1+a x)^{3/4}}{1280 a}+\frac{77 (1-a x)^{9/4} (1+a x)^{3/4}}{960 a}-\frac{77 (1-a x)^{13/4} (1+a x)^{3/4}}{480 a}-\frac{11 (1-a x)^{13/4} (1+a x)^{7/4}}{60 a}-\frac{(1-a x)^{13/4} (1+a x)^{11/4}}{6 a}+\frac{231 \log \left (1+\frac{\sqrt{1-a x}}{\sqrt{1+a x}}-\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{1024 \sqrt{2} a}-\frac{231 \log \left (1+\frac{\sqrt{1-a x}}{\sqrt{1+a x}}+\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{1024 \sqrt{2} a}-\frac{231 \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{512 \sqrt{2} a}+\frac{231 \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{512 \sqrt{2} a}\\ &=\frac{231 \sqrt [4]{1-a x} (1+a x)^{3/4}}{512 a}+\frac{231 (1-a x)^{5/4} (1+a x)^{3/4}}{1280 a}+\frac{77 (1-a x)^{9/4} (1+a x)^{3/4}}{960 a}-\frac{77 (1-a x)^{13/4} (1+a x)^{3/4}}{480 a}-\frac{11 (1-a x)^{13/4} (1+a x)^{7/4}}{60 a}-\frac{(1-a x)^{13/4} (1+a x)^{11/4}}{6 a}+\frac{231 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{512 \sqrt{2} a}-\frac{231 \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{512 \sqrt{2} a}+\frac{231 \log \left (1+\frac{\sqrt{1-a x}}{\sqrt{1+a x}}-\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{1024 \sqrt{2} a}-\frac{231 \log \left (1+\frac{\sqrt{1-a x}}{\sqrt{1+a x}}+\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{1024 \sqrt{2} a}\\ \end{align*}
Mathematica [C] time = 0.0185667, size = 42, normalized size = 0.12 \[ -\frac{16\ 2^{3/4} (1-a x)^{13/4} \text{Hypergeometric2F1}\left (-\frac{11}{4},\frac{13}{4},\frac{17}{4},\frac{1}{2} (1-a x)\right )}{13 a} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.201, size = 0, normalized size = 0. \begin{align*} \int \sqrt{{(ax+1){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}} \left ( -{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{\left (-a^{2} x^{2} + 1\right )}^{\frac{5}{2}} \sqrt{\frac{a x + 1}{\sqrt{-a^{2} x^{2} + 1}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 2.96471, size = 1405, normalized size = 3.91 \begin{align*} -\frac{13860 \, \sqrt{2} a \frac{1}{a^{4}}^{\frac{1}{4}} \arctan \left (\sqrt{2} a \sqrt{\frac{\sqrt{2}{\left (a^{4} x - a^{3}\right )} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{4}}^{\frac{3}{4}} +{\left (a^{3} x - a^{2}\right )} \sqrt{\frac{1}{a^{4}}} - \sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{4}}^{\frac{1}{4}} - \sqrt{2} a \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{4}}^{\frac{1}{4}} - 1\right ) + 13860 \, \sqrt{2} a \frac{1}{a^{4}}^{\frac{1}{4}} \arctan \left (\sqrt{2} a \sqrt{-\frac{\sqrt{2}{\left (a^{4} x - a^{3}\right )} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{4}}^{\frac{3}{4}} -{\left (a^{3} x - a^{2}\right )} \sqrt{\frac{1}{a^{4}}} + \sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{4}}^{\frac{1}{4}} - \sqrt{2} a \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{4}}^{\frac{1}{4}} + 1\right ) + 3465 \, \sqrt{2} a \frac{1}{a^{4}}^{\frac{1}{4}} \log \left (\frac{\sqrt{2}{\left (a^{4} x - a^{3}\right )} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{4}}^{\frac{3}{4}} +{\left (a^{3} x - a^{2}\right )} \sqrt{\frac{1}{a^{4}}} - \sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right ) - 3465 \, \sqrt{2} a \frac{1}{a^{4}}^{\frac{1}{4}} \log \left (-\frac{\sqrt{2}{\left (a^{4} x - a^{3}\right )} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{4}}^{\frac{3}{4}} -{\left (a^{3} x - a^{2}\right )} \sqrt{\frac{1}{a^{4}}} + \sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right ) - 4 \,{\left (1280 \, a^{5} x^{5} + 128 \, a^{4} x^{4} - 4144 \, a^{3} x^{3} - 520 \, a^{2} x^{2} + 5174 \, a x + 1547\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}}{30720 \, a} \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{\left (-a^{2} x^{2} + 1\right )}^{\frac{5}{2}} \sqrt{\frac{a x + 1}{\sqrt{-a^{2} x^{2} + 1}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]