Optimal. Leaf size=165 \[ \frac{1}{6} x \left (\pi c^2 x^2+\pi \right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{5}{24} \pi x \left (\pi c^2 x^2+\pi \right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{5}{16} \pi ^2 x \sqrt{\pi c^2 x^2+\pi } \left (a+b \sinh ^{-1}(c x)\right )+\frac{5 \pi ^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{32 b c}-\frac{5}{96} \pi ^{5/2} b c^3 x^4-\frac{\pi ^{5/2} b \left (c^2 x^2+1\right )^3}{36 c}-\frac{25}{96} \pi ^{5/2} b c x^2 \]
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Rubi [A] time = 0.164508, antiderivative size = 254, normalized size of antiderivative = 1.54, number of steps used = 8, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {5684, 5682, 5675, 30, 14, 261} \[ \frac{1}{6} x \left (\pi c^2 x^2+\pi \right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{5}{24} \pi x \left (\pi c^2 x^2+\pi \right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{5}{16} \pi ^2 x \sqrt{\pi c^2 x^2+\pi } \left (a+b \sinh ^{-1}(c x)\right )+\frac{5 \pi ^2 \sqrt{\pi c^2 x^2+\pi } \left (a+b \sinh ^{-1}(c x)\right )^2}{32 b c \sqrt{c^2 x^2+1}}-\frac{5 \pi ^2 b c^3 x^4 \sqrt{\pi c^2 x^2+\pi }}{96 \sqrt{c^2 x^2+1}}-\frac{25 \pi ^2 b c x^2 \sqrt{\pi c^2 x^2+\pi }}{96 \sqrt{c^2 x^2+1}}-\frac{\pi ^2 b \left (c^2 x^2+1\right )^{5/2} \sqrt{\pi c^2 x^2+\pi }}{36 c} \]
Antiderivative was successfully verified.
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Rule 5684
Rule 5682
Rule 5675
Rule 30
Rule 14
Rule 261
Rubi steps
\begin{align*} \int \left (\pi +c^2 \pi x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx &=\frac{1}{6} x \left (\pi +c^2 \pi x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{6} (5 \pi ) \int \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 \left (1+c^2 x^2\right )^2 \, dx}{6 \sqrt{1+c^2 x^2}}\\ &=-\frac{b \pi ^2 \left (1+c^2 x^2\right )^{5/2} \sqrt{\pi +c^2 \pi x^2}}{36 c}+\frac{5}{24} \pi x \left (\pi +c^2 \pi x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{6} x \left (\pi +c^2 \pi x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{8} \left (5 \pi ^2\right ) \int \sqrt{\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx-\frac{\left (5 b c \pi ^2 \sqrt{\pi +c^2 \pi x^2}\right ) \int x \left (1+c^2 x^2\right ) \, dx}{24 \sqrt{1+c^2 x^2}}\\ &=-\frac{b \pi ^2 \left (1+c^2 x^2\right )^{5/2} \sqrt{\pi +c^2 \pi x^2}}{36 c}+\frac{5}{16} \pi ^2 x \sqrt{\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{5}{24} \pi x \left (\pi +c^2 \pi x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{6} x \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}{16 \sqrt{1+c^2 x^2}}-\frac{\left (5 b c \pi ^2 \sqrt{\pi +c^2 \pi x^2}\right ) \int \left (x+c^2 x^3\right ) \, dx}{24 \sqrt{1+c^2 x^2}}-\frac{\left (5 b c \pi ^2 \sqrt{\pi +c^2 \pi x^2}\right ) \int x \, dx}{16 \sqrt{1+c^2 x^2}}\\ &=-\frac{25 b c \pi ^2 x^2 \sqrt{\pi +c^2 \pi x^2}}{96 \sqrt{1+c^2 x^2}}-\frac{5 b c^3 \pi ^2 x^4 \sqrt{\pi +c^2 \pi x^2}}{96 \sqrt{1+c^2 x^2}}-\frac{b \pi ^2 \left (1+c^2 x^2\right )^{5/2} \sqrt{\pi +c^2 \pi x^2}}{36 c}+\frac{5}{16} \pi ^2 x \sqrt{\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{5}{24} \pi x \left (\pi +c^2 \pi x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{6} x \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}{32 b c \sqrt{1+c^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.379247, size = 153, normalized size = 0.93 \[ \frac{\pi ^{5/2} \left (12 \sinh ^{-1}(c x) \left (60 a+45 b \sinh \left (2 \sinh ^{-1}(c x)\right )+9 b \sinh \left (4 \sinh ^{-1}(c x)\right )+b \sinh \left (6 \sinh ^{-1}(c x)\right )\right )+384 a c^5 x^5 \sqrt{c^2 x^2+1}+1248 a c^3 x^3 \sqrt{c^2 x^2+1}+1584 a c x \sqrt{c^2 x^2+1}+360 b \sinh ^{-1}(c x)^2-270 b \cosh \left (2 \sinh ^{-1}(c x)\right )-27 b \cosh \left (4 \sinh ^{-1}(c x)\right )-2 b \cosh \left (6 \sinh ^{-1}(c x)\right )\right )}{2304 c} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.052, size = 228, normalized size = 1.4 \begin{align*}{\frac{ax}{6} \left ( \pi \,{c}^{2}{x}^{2}+\pi \right ) ^{{\frac{5}{2}}}}+{\frac{5\,a\pi \,x}{24} \left ( \pi \,{c}^{2}{x}^{2}+\pi \right ) ^{{\frac{3}{2}}}}+{\frac{5\,a{\pi }^{2}x}{16}\sqrt{\pi \,{c}^{2}{x}^{2}+\pi }}+{\frac{5\,a{\pi }^{3}}{16}\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}^{5}}{6}\sqrt{{c}^{2}{x}^{2}+1}}-{\frac{b{\pi }^{{\frac{5}{2}}}{c}^{5}{x}^{6}}{36}}+{\frac{13\,b{\pi }^{5/2}{c}^{2}{\it Arcsinh} \left ( cx \right ){x}^{3}}{24}\sqrt{{c}^{2}{x}^{2}+1}}-{\frac{13\,b{c}^{3}{\pi }^{5/2}{x}^{4}}{96}}+{\frac{11\,b{\pi }^{5/2}{\it Arcsinh} \left ( cx \right ) x}{16}\sqrt{{c}^{2}{x}^{2}+1}}-{\frac{11\,bc{\pi }^{5/2}{x}^{2}}{32}}+{\frac{5\,b{\pi }^{5/2} \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{2}}{32\,c}}-{\frac{17\,b{\pi }^{5/2}}{72\,c}} \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^{4} + 2 \, \pi ^{2} a c^{2} x^{2} + \pi ^{2} a +{\left (\pi ^{2} b c^{4} x^{4} + 2 \, \pi ^{2} b c^{2} x^{2} + \pi ^{2} b\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 )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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