Optimal. Leaf size=213 \[ \frac{1}{8} x^3 \left (\pi c^2 x^2+\pi \right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{5}{48} \pi x^3 \left (\pi c^2 x^2+\pi \right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{5}{64} \pi ^2 x^3 \sqrt{\pi c^2 x^2+\pi } \left (a+b \sinh ^{-1}(c x)\right )+\frac{5 \pi ^{5/2} x \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{128 c^2}-\frac{5 \pi ^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{256 b c^3}-\frac{1}{64} \pi ^{5/2} b c^5 x^8-\frac{17}{288} \pi ^{5/2} b c^3 x^6-\frac{59}{768} \pi ^{5/2} b c x^4-\frac{5 \pi ^{5/2} b x^2}{256 c} \]
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Rubi [A] time = 0.457729, antiderivative size = 337, normalized size of antiderivative = 1.58, number of steps used = 12, number of rules used = 8, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {5744, 5742, 5758, 5675, 30, 14, 266, 43} \[ \frac{1}{8} x^3 \left (\pi c^2 x^2+\pi \right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{5}{48} \pi x^3 \left (\pi c^2 x^2+\pi \right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{5}{64} \pi ^2 x^3 \sqrt{\pi c^2 x^2+\pi } \left (a+b \sinh ^{-1}(c x)\right )+\frac{5 \pi ^2 x \sqrt{\pi c^2 x^2+\pi } \left (a+b \sinh ^{-1}(c x)\right )}{128 c^2}-\frac{5 \pi ^2 \sqrt{\pi c^2 x^2+\pi } \left (a+b \sinh ^{-1}(c x)\right )^2}{256 b c^3 \sqrt{c^2 x^2+1}}-\frac{\pi ^2 b c^5 x^8 \sqrt{\pi c^2 x^2+\pi }}{64 \sqrt{c^2 x^2+1}}-\frac{17 \pi ^2 b c^3 x^6 \sqrt{\pi c^2 x^2+\pi }}{288 \sqrt{c^2 x^2+1}}-\frac{59 \pi ^2 b c x^4 \sqrt{\pi c^2 x^2+\pi }}{768 \sqrt{c^2 x^2+1}}-\frac{5 \pi ^2 b x^2 \sqrt{\pi c^2 x^2+\pi }}{256 c \sqrt{c^2 x^2+1}} \]
Antiderivative was successfully verified.
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Rule 5744
Rule 5742
Rule 5758
Rule 5675
Rule 30
Rule 14
Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^2 \left (\pi +c^2 \pi x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx &=\frac{1}{8} x^3 \left (\pi +c^2 \pi x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{8} (5 \pi ) \int x^2 \left (\pi +c^2 \pi x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx-\frac{\left (b c \pi ^2 \sqrt{\pi +c^2 \pi x^2}\right ) \int x^3 \left (1+c^2 x^2\right )^2 \, dx}{8 \sqrt{1+c^2 x^2}}\\ &=\frac{5}{48} \pi x^3 \left (\pi +c^2 \pi x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{8} x^3 \left (\pi +c^2 \pi x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{16} \left (5 \pi ^2\right ) \int x^2 \sqrt{\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx-\frac{\left (b c \pi ^2 \sqrt{\pi +c^2 \pi x^2}\right ) \operatorname{Subst}\left (\int x \left (1+c^2 x\right )^2 \, dx,x,x^2\right )}{16 \sqrt{1+c^2 x^2}}-\frac{\left (5 b c \pi ^2 \sqrt{\pi +c^2 \pi x^2}\right ) \int x^3 \left (1+c^2 x^2\right ) \, dx}{48 \sqrt{1+c^2 x^2}}\\ &=\frac{5}{64} \pi ^2 x^3 \sqrt{\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{5}{48} \pi x^3 \left (\pi +c^2 \pi x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{8} x^3 \left (\pi +c^2 \pi x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{\left (5 \pi ^2 \sqrt{\pi +c^2 \pi x^2}\right ) \int \frac{x^2 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{1+c^2 x^2}} \, dx}{64 \sqrt{1+c^2 x^2}}-\frac{\left (b c \pi ^2 \sqrt{\pi +c^2 \pi x^2}\right ) \operatorname{Subst}\left (\int \left (x+2 c^2 x^2+c^4 x^3\right ) \, dx,x,x^2\right )}{16 \sqrt{1+c^2 x^2}}-\frac{\left (5 b c \pi ^2 \sqrt{\pi +c^2 \pi x^2}\right ) \int x^3 \, dx}{64 \sqrt{1+c^2 x^2}}-\frac{\left (5 b c \pi ^2 \sqrt{\pi +c^2 \pi x^2}\right ) \int \left (x^3+c^2 x^5\right ) \, dx}{48 \sqrt{1+c^2 x^2}}\\ &=-\frac{59 b c \pi ^2 x^4 \sqrt{\pi +c^2 \pi x^2}}{768 \sqrt{1+c^2 x^2}}-\frac{17 b c^3 \pi ^2 x^6 \sqrt{\pi +c^2 \pi x^2}}{288 \sqrt{1+c^2 x^2}}-\frac{b c^5 \pi ^2 x^8 \sqrt{\pi +c^2 \pi x^2}}{64 \sqrt{1+c^2 x^2}}+\frac{5 \pi ^2 x \sqrt{\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )}{128 c^2}+\frac{5}{64} \pi ^2 x^3 \sqrt{\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{5}{48} \pi x^3 \left (\pi +c^2 \pi x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{8} x^3 \left (\pi +c^2 \pi x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{\left (5 \pi ^2 \sqrt{\pi +c^2 \pi x^2}\right ) \int \frac{a+b \sinh ^{-1}(c x)}{\sqrt{1+c^2 x^2}} \, dx}{128 c^2 \sqrt{1+c^2 x^2}}-\frac{\left (5 b \pi ^2 \sqrt{\pi +c^2 \pi x^2}\right ) \int x \, dx}{128 c \sqrt{1+c^2 x^2}}\\ &=-\frac{5 b \pi ^2 x^2 \sqrt{\pi +c^2 \pi x^2}}{256 c \sqrt{1+c^2 x^2}}-\frac{59 b c \pi ^2 x^4 \sqrt{\pi +c^2 \pi x^2}}{768 \sqrt{1+c^2 x^2}}-\frac{17 b c^3 \pi ^2 x^6 \sqrt{\pi +c^2 \pi x^2}}{288 \sqrt{1+c^2 x^2}}-\frac{b c^5 \pi ^2 x^8 \sqrt{\pi +c^2 \pi x^2}}{64 \sqrt{1+c^2 x^2}}+\frac{5 \pi ^2 x \sqrt{\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )}{128 c^2}+\frac{5}{64} \pi ^2 x^3 \sqrt{\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{5}{48} \pi x^3 \left (\pi +c^2 \pi x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{8} x^3 \left (\pi +c^2 \pi x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{5 \pi ^2 \sqrt{\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{256 b c^3 \sqrt{1+c^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.565704, size = 196, normalized size = 0.92 \[ \frac{\pi ^{5/2} \left (-24 \sinh ^{-1}(c x) \left (120 a+48 b \sinh \left (2 \sinh ^{-1}(c x)\right )-24 b \sinh \left (4 \sinh ^{-1}(c x)\right )-16 b \sinh \left (6 \sinh ^{-1}(c x)\right )-3 b \sinh \left (8 \sinh ^{-1}(c x)\right )\right )+9216 a c^7 x^7 \sqrt{c^2 x^2+1}+26112 a c^5 x^5 \sqrt{c^2 x^2+1}+22656 a c^3 x^3 \sqrt{c^2 x^2+1}+2880 a c x \sqrt{c^2 x^2+1}-1440 b \sinh ^{-1}(c x)^2+576 b \cosh \left (2 \sinh ^{-1}(c x)\right )-144 b \cosh \left (4 \sinh ^{-1}(c x)\right )-64 b \cosh \left (6 \sinh ^{-1}(c x)\right )-9 b \cosh \left (8 \sinh ^{-1}(c x)\right )\right )}{73728 c^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.093, size = 301, normalized size = 1.4 \begin{align*}{\frac{ax}{8\,\pi \,{c}^{2}} \left ( \pi \,{c}^{2}{x}^{2}+\pi \right ) ^{{\frac{7}{2}}}}-{\frac{ax}{48\,{c}^{2}} \left ( \pi \,{c}^{2}{x}^{2}+\pi \right ) ^{{\frac{5}{2}}}}-{\frac{5\,a\pi \,x}{192\,{c}^{2}} \left ( \pi \,{c}^{2}{x}^{2}+\pi \right ) ^{{\frac{3}{2}}}}-{\frac{5\,a{\pi }^{2}x}{128\,{c}^{2}}\sqrt{\pi \,{c}^{2}{x}^{2}+\pi }}-{\frac{5\,a{\pi }^{3}}{128\,{c}^{2}}\ln \left ({\pi \,{c}^{2}x{\frac{1}{\sqrt{\pi \,{c}^{2}}}}}+\sqrt{\pi \,{c}^{2}{x}^{2}+\pi } \right ){\frac{1}{\sqrt{\pi \,{c}^{2}}}}}+{\frac{b{\pi }^{{\frac{5}{2}}}{c}^{4}{\it Arcsinh} \left ( cx \right ){x}^{7}}{8}\sqrt{{c}^{2}{x}^{2}+1}}-{\frac{b{c}^{5}{\pi }^{{\frac{5}{2}}}{x}^{8}}{64}}+{\frac{17\,b{\pi }^{5/2}{c}^{2}{\it Arcsinh} \left ( cx \right ){x}^{5}}{48}\sqrt{{c}^{2}{x}^{2}+1}}-{\frac{17\,b{c}^{3}{\pi }^{5/2}{x}^{6}}{288}}+{\frac{59\,b{\pi }^{5/2}{\it Arcsinh} \left ( cx \right ){x}^{3}}{192}\sqrt{{c}^{2}{x}^{2}+1}}-{\frac{59\,bc{\pi }^{5/2}{x}^{4}}{768}}+{\frac{5\,b{\pi }^{5/2}{\it Arcsinh} \left ( cx \right ) x}{128\,{c}^{2}}\sqrt{{c}^{2}{x}^{2}+1}}-{\frac{5\,b{\pi }^{5/2}{x}^{2}}{256\,c}}-{\frac{5\,b{\pi }^{5/2} \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{2}}{256\,{c}^{3}}}+{\frac{b{\pi }^{{\frac{5}{2}}}}{72\,{c}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sqrt{\pi + \pi c^{2} x^{2}}{\left (\pi ^{2} a c^{4} x^{6} + 2 \, \pi ^{2} a c^{2} x^{4} + \pi ^{2} a x^{2} +{\left (\pi ^{2} b c^{4} x^{6} + 2 \, \pi ^{2} b c^{2} x^{4} + \pi ^{2} b x^{2}\right )} \operatorname{arsinh}\left (c x\right )\right )}, x\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] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac{5}{2}}{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )} x^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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