Optimal. Leaf size=213 \[ -\frac {5 \pi ^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{256 b c^3}+\frac {5 \pi ^{5/2} x \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{128 c^2}+\frac {1}{8} x^3 \left (\pi c^2 x^2+\pi \right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {5}{48} \pi x^3 \left (\pi c^2 x^2+\pi \right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {5}{64} \pi ^2 x^3 \sqrt {\pi c^2 x^2+\pi } \left (a+b \sinh ^{-1}(c x)\right )-\frac {1}{64} \pi ^{5/2} b c^5 x^8-\frac {17}{288} \pi ^{5/2} b c^3 x^6-\frac {59}{768} \pi ^{5/2} b c x^4-\frac {5 \pi ^{5/2} b x^2}{256 c} \]
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Rubi [A] time = 0.46, 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 14
Rule 30
Rule 43
Rule 266
Rule 5675
Rule 5742
Rule 5744
Rule 5758
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*}
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Mathematica [A] time = 0.61, size = 196, normalized size = 0.92 \[ \frac {\pi ^{5/2} \left (-24 \sinh ^{-1}(c x) \left (120 a+48 b \sinh \left (2 \sinh ^{-1}(c x)\right )-24 b \sinh \left (4 \sinh ^{-1}(c x)\right )-16 b \sinh \left (6 \sinh ^{-1}(c x)\right )-3 b \sinh \left (8 \sinh ^{-1}(c x)\right )\right )+2880 a c x \sqrt {c^2 x^2+1}+9216 a c^7 x^7 \sqrt {c^2 x^2+1}+26112 a c^5 x^5 \sqrt {c^2 x^2+1}+22656 a c^3 x^3 \sqrt {c^2 x^2+1}-1440 b \sinh ^{-1}(c x)^2+576 b \cosh \left (2 \sinh ^{-1}(c x)\right )-144 b \cosh \left (4 \sinh ^{-1}(c x)\right )-64 b \cosh \left (6 \sinh ^{-1}(c x)\right )-9 b \cosh \left (8 \sinh ^{-1}(c x)\right )\right )}{73728 c^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.75, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {\pi + \pi c^{2} x^{2}} {\left (\pi ^{2} a c^{4} x^{6} + 2 \, \pi ^{2} a c^{2} x^{4} + \pi ^{2} a x^{2} + {\left (\pi ^{2} b c^{4} x^{6} + 2 \, \pi ^{2} b c^{2} x^{4} + \pi ^{2} b x^{2}\right )} \operatorname {arsinh}\left (c x\right )\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {5}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.11, size = 301, normalized size = 1.41 \[ \frac {a x \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {7}{2}}}{8 \pi \,c^{2}}-\frac {a x \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {5}{2}}}{48 c^{2}}-\frac {5 a \pi x \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}{192 c^{2}}-\frac {5 a \,\pi ^{2} x \sqrt {\pi \,c^{2} x^{2}+\pi }}{128 c^{2}}-\frac {5 a \,\pi ^{3} \ln \left (\frac {\pi x \,c^{2}}{\sqrt {\pi \,c^{2}}}+\sqrt {\pi \,c^{2} x^{2}+\pi }\right )}{128 c^{2} \sqrt {\pi \,c^{2}}}+\frac {b \,\pi ^{\frac {5}{2}} c^{4} \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{7}}{8}-\frac {b \,c^{5} \pi ^{\frac {5}{2}} x^{8}}{64}+\frac {17 b \,\pi ^{\frac {5}{2}} c^{2} \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{5}}{48}-\frac {17 b \,c^{3} \pi ^{\frac {5}{2}} x^{6}}{288}+\frac {59 b \,\pi ^{\frac {5}{2}} \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{3}}{192}-\frac {59 b c \,\pi ^{\frac {5}{2}} x^{4}}{768}+\frac {5 b \,\pi ^{\frac {5}{2}} \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x}{128 c^{2}}-\frac {5 b \,\pi ^{\frac {5}{2}} x^{2}}{256 c}-\frac {5 b \,\pi ^{\frac {5}{2}} \arcsinh \left (c x \right )^{2}}{256 c^{3}}+\frac {b \,\pi ^{\frac {5}{2}}}{72 c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x^2\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (\Pi \,c^2\,x^2+\Pi \right )}^{5/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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