Optimal. Leaf size=141 \[ \frac{\left (\pi c^2 x^2+\pi \right )^{9/2} \left (a+b \sinh ^{-1}(c x)\right )}{9 \pi ^2 c^4}-\frac{\left (\pi c^2 x^2+\pi \right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{7 \pi c^4}-\frac{1}{81} \pi ^{5/2} b c^5 x^9-\frac{19}{441} \pi ^{5/2} b c^3 x^7+\frac{2 \pi ^{5/2} b x}{63 c^3}-\frac{1}{21} \pi ^{5/2} b c x^5-\frac{\pi ^{5/2} b x^3}{189 c} \]
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Rubi [A] time = 0.152781, antiderivative size = 143, normalized size of antiderivative = 1.01, number of steps used = 4, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {266, 43, 5732, 12, 373} \[ \frac{\pi ^{5/2} \left (c^2 x^2+1\right )^{9/2} \left (a+b \sinh ^{-1}(c x)\right )}{9 c^4}-\frac{\pi ^{5/2} \left (c^2 x^2+1\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{7 c^4}-\frac{1}{81} \pi ^{5/2} b c^5 x^9-\frac{19}{441} \pi ^{5/2} b c^3 x^7+\frac{2 \pi ^{5/2} b x}{63 c^3}-\frac{1}{21} \pi ^{5/2} b c x^5-\frac{\pi ^{5/2} b x^3}{189 c} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rule 5732
Rule 12
Rule 373
Rubi steps
\begin{align*} \int x^3 \left (\pi +c^2 \pi x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx &=-\frac{\pi ^{5/2} \left (1+c^2 x^2\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{7 c^4}+\frac{\pi ^{5/2} \left (1+c^2 x^2\right )^{9/2} \left (a+b \sinh ^{-1}(c x)\right )}{9 c^4}-\left (b c \pi ^{5/2}\right ) \int \frac{\left (1+c^2 x^2\right )^3 \left (-2+7 c^2 x^2\right )}{63 c^4} \, dx\\ &=-\frac{\pi ^{5/2} \left (1+c^2 x^2\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{7 c^4}+\frac{\pi ^{5/2} \left (1+c^2 x^2\right )^{9/2} \left (a+b \sinh ^{-1}(c x)\right )}{9 c^4}-\frac{\left (b \pi ^{5/2}\right ) \int \left (1+c^2 x^2\right )^3 \left (-2+7 c^2 x^2\right ) \, dx}{63 c^3}\\ &=-\frac{\pi ^{5/2} \left (1+c^2 x^2\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{7 c^4}+\frac{\pi ^{5/2} \left (1+c^2 x^2\right )^{9/2} \left (a+b \sinh ^{-1}(c x)\right )}{9 c^4}-\frac{\left (b \pi ^{5/2}\right ) \int \left (-2+c^2 x^2+15 c^4 x^4+19 c^6 x^6+7 c^8 x^8\right ) \, dx}{63 c^3}\\ &=\frac{2 b \pi ^{5/2} x}{63 c^3}-\frac{b \pi ^{5/2} x^3}{189 c}-\frac{1}{21} b c \pi ^{5/2} x^5-\frac{19}{441} b c^3 \pi ^{5/2} x^7-\frac{1}{81} b c^5 \pi ^{5/2} x^9-\frac{\pi ^{5/2} \left (1+c^2 x^2\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{7 c^4}+\frac{\pi ^{5/2} \left (1+c^2 x^2\right )^{9/2} \left (a+b \sinh ^{-1}(c x)\right )}{9 c^4}\\ \end{align*}
Mathematica [A] time = 0.198417, size = 108, normalized size = 0.77 \[ \frac{\pi ^{5/2} \left (63 a \left (7 c^2 x^2-2\right ) \left (c^2 x^2+1\right )^{7/2}-b c x \left (49 c^8 x^8+171 c^6 x^6+189 c^4 x^4+21 c^2 x^2-126\right )+63 b \left (7 c^2 x^2-2\right ) \left (c^2 x^2+1\right )^{7/2} \sinh ^{-1}(c x)\right )}{3969 c^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.1, size = 226, normalized size = 1.6 \begin{align*} a \left ({\frac{{x}^{2}}{9\,\pi \,{c}^{2}} \left ( \pi \,{c}^{2}{x}^{2}+\pi \right ) ^{{\frac{7}{2}}}}-{\frac{2}{63\,\pi \,{c}^{4}} \left ( \pi \,{c}^{2}{x}^{2}+\pi \right ) ^{{\frac{7}{2}}}} \right ) +{\frac{b{\pi }^{{\frac{5}{2}}}}{3969\,{c}^{4}} \left ( 441\,{\it Arcsinh} \left ( cx \right ){c}^{10}{x}^{10}+1638\,{\it Arcsinh} \left ( cx \right ){c}^{8}{x}^{8}-49\,{c}^{9}{x}^{9}\sqrt{{c}^{2}{x}^{2}+1}+2142\,{\it Arcsinh} \left ( cx \right ){c}^{6}{x}^{6}-171\,{c}^{7}{x}^{7}\sqrt{{c}^{2}{x}^{2}+1}+1008\,{\it Arcsinh} \left ( cx \right ){c}^{4}{x}^{4}-189\,{c}^{5}{x}^{5}\sqrt{{c}^{2}{x}^{2}+1}-63\,{\it Arcsinh} \left ( cx \right ){c}^{2}{x}^{2}-21\,{c}^{3}{x}^{3}\sqrt{{c}^{2}{x}^{2}+1}-126\,{\it Arcsinh} \left ( cx \right ) +126\,cx\sqrt{{c}^{2}{x}^{2}+1} \right ){\frac{1}{\sqrt{{c}^{2}{x}^{2}+1}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.20911, size = 211, normalized size = 1.5 \begin{align*} \frac{1}{63} \,{\left (\frac{7 \,{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac{7}{2}} x^{2}}{\pi c^{2}} - \frac{2 \,{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac{7}{2}}}{\pi c^{4}}\right )} b \operatorname{arsinh}\left (c x\right ) + \frac{1}{63} \,{\left (\frac{7 \,{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac{7}{2}} x^{2}}{\pi c^{2}} - \frac{2 \,{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac{7}{2}}}{\pi c^{4}}\right )} a - \frac{{\left (49 \, \pi ^{\frac{5}{2}} c^{8} x^{9} + 171 \, \pi ^{\frac{5}{2}} c^{6} x^{7} + 189 \, \pi ^{\frac{5}{2}} c^{4} x^{5} + 21 \, \pi ^{\frac{5}{2}} c^{2} x^{3} - 126 \, \pi ^{\frac{5}{2}} x\right )} b}{3969 \, c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.47961, size = 613, normalized size = 4.35 \begin{align*} \frac{63 \, \sqrt{\pi + \pi c^{2} x^{2}}{\left (7 \, \pi ^{2} b c^{10} x^{10} + 26 \, \pi ^{2} b c^{8} x^{8} + 34 \, \pi ^{2} b c^{6} x^{6} + 16 \, \pi ^{2} b c^{4} x^{4} - \pi ^{2} b c^{2} x^{2} - 2 \, \pi ^{2} b\right )} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) + \sqrt{\pi + \pi c^{2} x^{2}}{\left (441 \, \pi ^{2} a c^{10} x^{10} + 1638 \, \pi ^{2} a c^{8} x^{8} + 2142 \, \pi ^{2} a c^{6} x^{6} + 1008 \, \pi ^{2} a c^{4} x^{4} - 63 \, \pi ^{2} a c^{2} x^{2} - 126 \, \pi ^{2} a -{\left (49 \, \pi ^{2} b c^{9} x^{9} + 171 \, \pi ^{2} b c^{7} x^{7} + 189 \, \pi ^{2} b c^{5} x^{5} + 21 \, \pi ^{2} b c^{3} x^{3} - 126 \, \pi ^{2} b c x\right )} \sqrt{c^{2} x^{2} + 1}\right )}}{3969 \,{\left (c^{6} x^{2} + c^{4}\right )}} \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(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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