Optimal. Leaf size=213 \[ -\frac {5 b \pi ^{5/2} x^2}{256 c}-\frac {59}{768} b c \pi ^{5/2} x^4-\frac {17}{288} b c^3 \pi ^{5/2} x^6-\frac {1}{64} b c^5 \pi ^{5/2} x^8+\frac {5 \pi ^{5/2} x \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{128 c^2}+\frac {5}{64} \pi ^2 x^3 \sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {5}{48} \pi x^3 \left (\pi +c^2 \pi x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{8} x^3 \left (\pi +c^2 \pi x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {5 \pi ^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{256 b c^3} \]
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Rubi [A]
time = 0.30, antiderivative size = 213, normalized size of antiderivative = 1.00, 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 = {5808, 5806,
5812, 5783, 30, 14, 272, 45} \begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 30
Rule 45
Rule 272
Rule 5783
Rule 5806
Rule 5808
Rule 5812
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 ) \text {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 ) \text {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 = 0.41, size = 196, normalized size = 0.92 \begin {gather*} \frac {\pi ^{5/2} \left (2880 a c x \sqrt {1+c^2 x^2}+22656 a c^3 x^3 \sqrt {1+c^2 x^2}+26112 a c^5 x^5 \sqrt {1+c^2 x^2}+9216 a c^7 x^7 \sqrt {1+c^2 x^2}-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 )-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 )\right )}{73728 c^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int x^{2} \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {5}{2}} \left (a +b \arcsinh \left (c x \right )\right )\, dx\]
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 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 63.80, size = 350, normalized size = 1.64 \begin {gather*} \begin {cases} \frac {\pi ^{\frac {5}{2}} a c^{4} x^{7} \sqrt {c^{2} x^{2} + 1}}{8} + \frac {17 \pi ^{\frac {5}{2}} a c^{2} x^{5} \sqrt {c^{2} x^{2} + 1}}{48} + \frac {59 \pi ^{\frac {5}{2}} a x^{3} \sqrt {c^{2} x^{2} + 1}}{192} + \frac {5 \pi ^{\frac {5}{2}} a x \sqrt {c^{2} x^{2} + 1}}{128 c^{2}} - \frac {5 \pi ^{\frac {5}{2}} a \operatorname {asinh}{\left (c x \right )}}{128 c^{3}} - \frac {\pi ^{\frac {5}{2}} b c^{5} x^{8}}{64} + \frac {\pi ^{\frac {5}{2}} b c^{4} x^{7} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{8} - \frac {17 \pi ^{\frac {5}{2}} b c^{3} x^{6}}{288} + \frac {17 \pi ^{\frac {5}{2}} b c^{2} x^{5} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{48} - \frac {59 \pi ^{\frac {5}{2}} b c x^{4}}{768} + \frac {59 \pi ^{\frac {5}{2}} b x^{3} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{192} - \frac {5 \pi ^{\frac {5}{2}} b x^{2}}{256 c} + \frac {5 \pi ^{\frac {5}{2}} b x \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{128 c^{2}} - \frac {5 \pi ^{\frac {5}{2}} b \operatorname {asinh}^{2}{\left (c x \right )}}{256 c^{3}} & \text {for}\: c \neq 0 \\\frac {\pi ^{\frac {5}{2}} a x^{3}}{3} & \text {otherwise} \end {cases} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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