Optimal. Leaf size=646 \[ -\frac{x (1-a x)^{7/8} (a x+1)^{9/8}}{3 a^2}-\frac{(1-a x)^{7/8} (a x+1)^{9/8}}{24 a^3}-\frac{11 (1-a x)^{7/8} \sqrt [8]{a x+1}}{32 a^3}-\frac{11 \sqrt{2-\sqrt{2}} \log \left (\frac{\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}-\frac{\sqrt{2-\sqrt{2}} \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+1\right )}{256 a^3}+\frac{11 \sqrt{2-\sqrt{2}} \log \left (\frac{\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+\frac{\sqrt{2-\sqrt{2}} \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+1\right )}{256 a^3}-\frac{11 \sqrt{2+\sqrt{2}} \log \left (\frac{\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}-\frac{\sqrt{2+\sqrt{2}} \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+1\right )}{256 a^3}+\frac{11 \sqrt{2+\sqrt{2}} \log \left (\frac{\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+\frac{\sqrt{2+\sqrt{2}} \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+1\right )}{256 a^3}+\frac{11 \sqrt{2+\sqrt{2}} \tan ^{-1}\left (\frac{\sqrt{2-\sqrt{2}}-\frac{2 \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}}{\sqrt{2+\sqrt{2}}}\right )}{128 a^3}+\frac{11 \sqrt{2-\sqrt{2}} \tan ^{-1}\left (\frac{\sqrt{2+\sqrt{2}}-\frac{2 \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}}{\sqrt{2-\sqrt{2}}}\right )}{128 a^3}-\frac{11 \sqrt{2+\sqrt{2}} \tan ^{-1}\left (\frac{\frac{2 \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+\sqrt{2-\sqrt{2}}}{\sqrt{2+\sqrt{2}}}\right )}{128 a^3}-\frac{11 \sqrt{2-\sqrt{2}} \tan ^{-1}\left (\frac{\frac{2 \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+\sqrt{2+\sqrt{2}}}{\sqrt{2-\sqrt{2}}}\right )}{128 a^3} \]
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Rubi [A] time = 0.701821, antiderivative size = 646, normalized size of antiderivative = 1., number of steps used = 27, number of rules used = 13, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.929, Rules used = {6126, 90, 80, 50, 63, 331, 299, 1122, 1169, 634, 618, 204, 628} \[ -\frac{x (1-a x)^{7/8} (a x+1)^{9/8}}{3 a^2}-\frac{(1-a x)^{7/8} (a x+1)^{9/8}}{24 a^3}-\frac{11 (1-a x)^{7/8} \sqrt [8]{a x+1}}{32 a^3}-\frac{11 \sqrt{2-\sqrt{2}} \log \left (\frac{\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}-\frac{\sqrt{2-\sqrt{2}} \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+1\right )}{256 a^3}+\frac{11 \sqrt{2-\sqrt{2}} \log \left (\frac{\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+\frac{\sqrt{2-\sqrt{2}} \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+1\right )}{256 a^3}-\frac{11 \sqrt{2+\sqrt{2}} \log \left (\frac{\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}-\frac{\sqrt{2+\sqrt{2}} \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+1\right )}{256 a^3}+\frac{11 \sqrt{2+\sqrt{2}} \log \left (\frac{\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+\frac{\sqrt{2+\sqrt{2}} \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+1\right )}{256 a^3}+\frac{11 \sqrt{2+\sqrt{2}} \tan ^{-1}\left (\frac{\sqrt{2-\sqrt{2}}-\frac{2 \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}}{\sqrt{2+\sqrt{2}}}\right )}{128 a^3}+\frac{11 \sqrt{2-\sqrt{2}} \tan ^{-1}\left (\frac{\sqrt{2+\sqrt{2}}-\frac{2 \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}}{\sqrt{2-\sqrt{2}}}\right )}{128 a^3}-\frac{11 \sqrt{2+\sqrt{2}} \tan ^{-1}\left (\frac{\frac{2 \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+\sqrt{2-\sqrt{2}}}{\sqrt{2+\sqrt{2}}}\right )}{128 a^3}-\frac{11 \sqrt{2-\sqrt{2}} \tan ^{-1}\left (\frac{\frac{2 \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+\sqrt{2+\sqrt{2}}}{\sqrt{2-\sqrt{2}}}\right )}{128 a^3} \]
Antiderivative was successfully verified.
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Rule 6126
Rule 90
Rule 80
Rule 50
Rule 63
Rule 331
Rule 299
Rule 1122
Rule 1169
Rule 634
Rule 618
Rule 204
Rule 628
Rubi steps
\begin{align*} \int e^{\frac{1}{4} \tanh ^{-1}(a x)} x^2 \, dx &=\int \frac{x^2 \sqrt [8]{1+a x}}{\sqrt [8]{1-a x}} \, dx\\ &=-\frac{x (1-a x)^{7/8} (1+a x)^{9/8}}{3 a^2}-\frac{\int \frac{\left (-1-\frac{a x}{4}\right ) \sqrt [8]{1+a x}}{\sqrt [8]{1-a x}} \, dx}{3 a^2}\\ &=-\frac{(1-a x)^{7/8} (1+a x)^{9/8}}{24 a^3}-\frac{x (1-a x)^{7/8} (1+a x)^{9/8}}{3 a^2}+\frac{11 \int \frac{\sqrt [8]{1+a x}}{\sqrt [8]{1-a x}} \, dx}{32 a^2}\\ &=-\frac{11 (1-a x)^{7/8} \sqrt [8]{1+a x}}{32 a^3}-\frac{(1-a x)^{7/8} (1+a x)^{9/8}}{24 a^3}-\frac{x (1-a x)^{7/8} (1+a x)^{9/8}}{3 a^2}+\frac{11 \int \frac{1}{\sqrt [8]{1-a x} (1+a x)^{7/8}} \, dx}{128 a^2}\\ &=-\frac{11 (1-a x)^{7/8} \sqrt [8]{1+a x}}{32 a^3}-\frac{(1-a x)^{7/8} (1+a x)^{9/8}}{24 a^3}-\frac{x (1-a x)^{7/8} (1+a x)^{9/8}}{3 a^2}-\frac{11 \operatorname{Subst}\left (\int \frac{x^6}{\left (2-x^8\right )^{7/8}} \, dx,x,\sqrt [8]{1-a x}\right )}{16 a^3}\\ &=-\frac{11 (1-a x)^{7/8} \sqrt [8]{1+a x}}{32 a^3}-\frac{(1-a x)^{7/8} (1+a x)^{9/8}}{24 a^3}-\frac{x (1-a x)^{7/8} (1+a x)^{9/8}}{3 a^2}-\frac{11 \operatorname{Subst}\left (\int \frac{x^6}{1+x^8} \, dx,x,\frac{\sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}\right )}{16 a^3}\\ &=-\frac{11 (1-a x)^{7/8} \sqrt [8]{1+a x}}{32 a^3}-\frac{(1-a x)^{7/8} (1+a x)^{9/8}}{24 a^3}-\frac{x (1-a x)^{7/8} (1+a x)^{9/8}}{3 a^2}-\frac{11 \operatorname{Subst}\left (\int \frac{x^4}{1-\sqrt{2} x^2+x^4} \, dx,x,\frac{\sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}\right )}{32 \sqrt{2} a^3}+\frac{11 \operatorname{Subst}\left (\int \frac{x^4}{1+\sqrt{2} x^2+x^4} \, dx,x,\frac{\sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}\right )}{32 \sqrt{2} a^3}\\ &=-\frac{11 (1-a x)^{7/8} \sqrt [8]{1+a x}}{32 a^3}-\frac{(1-a x)^{7/8} (1+a x)^{9/8}}{24 a^3}-\frac{x (1-a x)^{7/8} (1+a x)^{9/8}}{3 a^2}+\frac{11 \operatorname{Subst}\left (\int \frac{1-\sqrt{2} x^2}{1-\sqrt{2} x^2+x^4} \, dx,x,\frac{\sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}\right )}{32 \sqrt{2} a^3}-\frac{11 \operatorname{Subst}\left (\int \frac{1+\sqrt{2} x^2}{1+\sqrt{2} x^2+x^4} \, dx,x,\frac{\sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}\right )}{32 \sqrt{2} a^3}\\ &=-\frac{11 (1-a x)^{7/8} \sqrt [8]{1+a x}}{32 a^3}-\frac{(1-a x)^{7/8} (1+a x)^{9/8}}{24 a^3}-\frac{x (1-a x)^{7/8} (1+a x)^{9/8}}{3 a^2}-\frac{11 \operatorname{Subst}\left (\int \frac{\sqrt{2-\sqrt{2}}-\left (1-\sqrt{2}\right ) x}{1-\sqrt{2-\sqrt{2}} x+x^2} \, dx,x,\frac{\sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}\right )}{64 \sqrt{2 \left (2-\sqrt{2}\right )} a^3}-\frac{11 \operatorname{Subst}\left (\int \frac{\sqrt{2-\sqrt{2}}+\left (1-\sqrt{2}\right ) x}{1+\sqrt{2-\sqrt{2}} x+x^2} \, dx,x,\frac{\sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}\right )}{64 \sqrt{2 \left (2-\sqrt{2}\right )} a^3}+\frac{11 \operatorname{Subst}\left (\int \frac{\sqrt{2+\sqrt{2}}-\left (1+\sqrt{2}\right ) x}{1-\sqrt{2+\sqrt{2}} x+x^2} \, dx,x,\frac{\sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}\right )}{64 \sqrt{2 \left (2+\sqrt{2}\right )} a^3}+\frac{11 \operatorname{Subst}\left (\int \frac{\sqrt{2+\sqrt{2}}+\left (1+\sqrt{2}\right ) x}{1+\sqrt{2+\sqrt{2}} x+x^2} \, dx,x,\frac{\sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}\right )}{64 \sqrt{2 \left (2+\sqrt{2}\right )} a^3}\\ &=-\frac{11 (1-a x)^{7/8} \sqrt [8]{1+a x}}{32 a^3}-\frac{(1-a x)^{7/8} (1+a x)^{9/8}}{24 a^3}-\frac{x (1-a x)^{7/8} (1+a x)^{9/8}}{3 a^2}-\frac{\left (11 \sqrt{\frac{1}{2} \left (3-2 \sqrt{2}\right )}\right ) \operatorname{Subst}\left (\int \frac{1}{1-\sqrt{2+\sqrt{2}} x+x^2} \, dx,x,\frac{\sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}\right )}{128 a^3}-\frac{\left (11 \sqrt{\frac{1}{2} \left (3-2 \sqrt{2}\right )}\right ) \operatorname{Subst}\left (\int \frac{1}{1+\sqrt{2+\sqrt{2}} x+x^2} \, dx,x,\frac{\sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}\right )}{128 a^3}-\frac{\left (11 \sqrt{2-\sqrt{2}}\right ) \operatorname{Subst}\left (\int \frac{-\sqrt{2-\sqrt{2}}+2 x}{1-\sqrt{2-\sqrt{2}} x+x^2} \, dx,x,\frac{\sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}\right )}{256 a^3}+\frac{\left (11 \sqrt{2-\sqrt{2}}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{2-\sqrt{2}}+2 x}{1+\sqrt{2-\sqrt{2}} x+x^2} \, dx,x,\frac{\sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}\right )}{256 a^3}-\frac{\left (11 \sqrt{2+\sqrt{2}}\right ) \operatorname{Subst}\left (\int \frac{-\sqrt{2+\sqrt{2}}+2 x}{1-\sqrt{2+\sqrt{2}} x+x^2} \, dx,x,\frac{\sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}\right )}{256 a^3}+\frac{\left (11 \sqrt{2+\sqrt{2}}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{2+\sqrt{2}}+2 x}{1+\sqrt{2+\sqrt{2}} x+x^2} \, dx,x,\frac{\sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}\right )}{256 a^3}-\frac{\left (11 \sqrt{\frac{1}{2} \left (3+2 \sqrt{2}\right )}\right ) \operatorname{Subst}\left (\int \frac{1}{1-\sqrt{2-\sqrt{2}} x+x^2} \, dx,x,\frac{\sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}\right )}{128 a^3}-\frac{\left (11 \sqrt{\frac{1}{2} \left (3+2 \sqrt{2}\right )}\right ) \operatorname{Subst}\left (\int \frac{1}{1+\sqrt{2-\sqrt{2}} x+x^2} \, dx,x,\frac{\sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}\right )}{128 a^3}\\ &=-\frac{11 (1-a x)^{7/8} \sqrt [8]{1+a x}}{32 a^3}-\frac{(1-a x)^{7/8} (1+a x)^{9/8}}{24 a^3}-\frac{x (1-a x)^{7/8} (1+a x)^{9/8}}{3 a^2}-\frac{11 \sqrt{2-\sqrt{2}} \log \left (1+\frac{\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}-\frac{\sqrt{2-\sqrt{2}} \sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}\right )}{256 a^3}+\frac{11 \sqrt{2-\sqrt{2}} \log \left (1+\frac{\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}+\frac{\sqrt{2-\sqrt{2}} \sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}\right )}{256 a^3}-\frac{11 \sqrt{2+\sqrt{2}} \log \left (1+\frac{\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}-\frac{\sqrt{2+\sqrt{2}} \sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}\right )}{256 a^3}+\frac{11 \sqrt{2+\sqrt{2}} \log \left (1+\frac{\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}+\frac{\sqrt{2+\sqrt{2}} \sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}\right )}{256 a^3}+\frac{\left (11 \sqrt{\frac{1}{2} \left (3-2 \sqrt{2}\right )}\right ) \operatorname{Subst}\left (\int \frac{1}{-2+\sqrt{2}-x^2} \, dx,x,-\sqrt{2+\sqrt{2}}+\frac{2 \sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}\right )}{64 a^3}+\frac{\left (11 \sqrt{\frac{1}{2} \left (3-2 \sqrt{2}\right )}\right ) \operatorname{Subst}\left (\int \frac{1}{-2+\sqrt{2}-x^2} \, dx,x,\sqrt{2+\sqrt{2}}+\frac{2 \sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}\right )}{64 a^3}+\frac{\left (11 \sqrt{\frac{1}{2} \left (3+2 \sqrt{2}\right )}\right ) \operatorname{Subst}\left (\int \frac{1}{-2-\sqrt{2}-x^2} \, dx,x,-\sqrt{2-\sqrt{2}}+\frac{2 \sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}\right )}{64 a^3}+\frac{\left (11 \sqrt{\frac{1}{2} \left (3+2 \sqrt{2}\right )}\right ) \operatorname{Subst}\left (\int \frac{1}{-2-\sqrt{2}-x^2} \, dx,x,\sqrt{2-\sqrt{2}}+\frac{2 \sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}\right )}{64 a^3}\\ &=-\frac{11 (1-a x)^{7/8} \sqrt [8]{1+a x}}{32 a^3}-\frac{(1-a x)^{7/8} (1+a x)^{9/8}}{24 a^3}-\frac{x (1-a x)^{7/8} (1+a x)^{9/8}}{3 a^2}+\frac{11 \sqrt{2+\sqrt{2}} \tan ^{-1}\left (\frac{\sqrt{2-\sqrt{2}}-\frac{2 \sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}}{\sqrt{2+\sqrt{2}}}\right )}{128 a^3}+\frac{11 \sqrt{2-\sqrt{2}} \tan ^{-1}\left (\frac{\sqrt{2+\sqrt{2}}-\frac{2 \sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}}{\sqrt{2-\sqrt{2}}}\right )}{128 a^3}-\frac{11 \sqrt{2+\sqrt{2}} \tan ^{-1}\left (\frac{\sqrt{2-\sqrt{2}}+\frac{2 \sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}}{\sqrt{2+\sqrt{2}}}\right )}{128 a^3}-\frac{11 \sqrt{2-\sqrt{2}} \tan ^{-1}\left (\frac{\sqrt{2+\sqrt{2}}+\frac{2 \sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}}{\sqrt{2-\sqrt{2}}}\right )}{128 a^3}-\frac{11 \sqrt{2-\sqrt{2}} \log \left (1+\frac{\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}-\frac{\sqrt{2-\sqrt{2}} \sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}\right )}{256 a^3}+\frac{11 \sqrt{2-\sqrt{2}} \log \left (1+\frac{\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}+\frac{\sqrt{2-\sqrt{2}} \sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}\right )}{256 a^3}-\frac{11 \sqrt{2+\sqrt{2}} \log \left (1+\frac{\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}-\frac{\sqrt{2+\sqrt{2}} \sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}\right )}{256 a^3}+\frac{11 \sqrt{2+\sqrt{2}} \log \left (1+\frac{\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}+\frac{\sqrt{2+\sqrt{2}} \sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}\right )}{256 a^3}\\ \end{align*}
Mathematica [C] time = 0.0353602, size = 70, normalized size = 0.11 \[ -\frac{(1-a x)^{7/8} \left (66 \sqrt [8]{2} \text{Hypergeometric2F1}\left (-\frac{1}{8},\frac{7}{8},\frac{15}{8},\frac{1}{2} (1-a x)\right )+7 \sqrt [8]{a x+1} \left (8 a^2 x^2+9 a x+1\right )\right )}{168 a^3} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.036, size = 0, normalized size = 0. \begin{align*} \int \sqrt [4]{{(ax+1){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}}{x}^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \left (\frac{a x + 1}{\sqrt{-a^{2} x^{2} + 1}}\right )^{\frac{1}{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.39818, size = 6882, normalized size = 10.65 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \left (\frac{a x + 1}{\sqrt{-a^{2} x^{2} + 1}}\right )^{\frac{1}{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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