Optimal. Leaf size=210 \[ \frac {15}{64} b^2 \pi ^{3/2} x \sqrt {1+c^2 x^2}+\frac {1}{32} b^2 \pi ^{3/2} x \left (1+c^2 x^2\right )^{3/2}-\frac {9 b^2 \pi ^{3/2} \sinh ^{-1}(c x)}{64 c}-\frac {3}{8} b c \pi ^{3/2} x^2 \left (a+b \sinh ^{-1}(c x)\right )-\frac {b \pi ^{3/2} \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )}{8 c}+\frac {3}{8} \pi x \sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{4} x \left (\pi +c^2 \pi x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {\pi ^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^3}{8 b c} \]
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Rubi [A]
time = 0.15, antiderivative size = 210, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 8, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {5786, 5785,
5783, 5776, 327, 221, 5798, 201} \begin {gather*} \frac {1}{4} x \left (\pi c^2 x^2+\pi \right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {3}{8} \pi x \sqrt {\pi c^2 x^2+\pi } \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {\pi ^{3/2} b \left (c^2 x^2+1\right )^2 \left (a+b \sinh ^{-1}(c x)\right )}{8 c}-\frac {3}{8} \pi ^{3/2} b c x^2 \left (a+b \sinh ^{-1}(c x)\right )+\frac {\pi ^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^3}{8 b c}+\frac {1}{32} \pi ^{3/2} b^2 x \left (c^2 x^2+1\right )^{3/2}+\frac {15}{64} \pi ^{3/2} b^2 x \sqrt {c^2 x^2+1}-\frac {9 \pi ^{3/2} b^2 \sinh ^{-1}(c x)}{64 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 201
Rule 221
Rule 327
Rule 5776
Rule 5783
Rule 5785
Rule 5786
Rule 5798
Rubi steps
\begin {align*} \int \left (\pi +c^2 \pi x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx &=\frac {1}{4} x \left (\pi +c^2 \pi x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{4} (3 \pi ) \int \sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx-\frac {\left (b c \pi \sqrt {\pi +c^2 \pi x^2}\right ) \int x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{2 \sqrt {1+c^2 x^2}}\\ &=-\frac {b \pi \left (1+c^2 x^2\right )^{3/2} \sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )}{8 c}+\frac {3}{8} \pi x \sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{4} x \left (\pi +c^2 \pi x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {\left (3 \pi \sqrt {\pi +c^2 \pi x^2}\right ) \int \frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt {1+c^2 x^2}} \, dx}{8 \sqrt {1+c^2 x^2}}+\frac {\left (b^2 \pi \sqrt {\pi +c^2 \pi x^2}\right ) \int \left (1+c^2 x^2\right )^{3/2} \, dx}{8 \sqrt {1+c^2 x^2}}-\frac {\left (3 b c \pi \sqrt {\pi +c^2 \pi x^2}\right ) \int x \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{4 \sqrt {1+c^2 x^2}}\\ &=\frac {1}{32} b^2 \pi x \left (1+c^2 x^2\right ) \sqrt {\pi +c^2 \pi x^2}-\frac {3 b c \pi x^2 \sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )}{8 \sqrt {1+c^2 x^2}}-\frac {b \pi \left (1+c^2 x^2\right )^{3/2} \sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )}{8 c}+\frac {3}{8} \pi x \sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{4} x \left (\pi +c^2 \pi x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {\pi \sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{8 b c \sqrt {1+c^2 x^2}}+\frac {\left (3 b^2 \pi \sqrt {\pi +c^2 \pi x^2}\right ) \int \sqrt {1+c^2 x^2} \, dx}{32 \sqrt {1+c^2 x^2}}+\frac {\left (3 b^2 c^2 \pi \sqrt {\pi +c^2 \pi x^2}\right ) \int \frac {x^2}{\sqrt {1+c^2 x^2}} \, dx}{8 \sqrt {1+c^2 x^2}}\\ &=\frac {15}{64} b^2 \pi x \sqrt {\pi +c^2 \pi x^2}+\frac {1}{32} b^2 \pi x \left (1+c^2 x^2\right ) \sqrt {\pi +c^2 \pi x^2}-\frac {3 b c \pi x^2 \sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )}{8 \sqrt {1+c^2 x^2}}-\frac {b \pi \left (1+c^2 x^2\right )^{3/2} \sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )}{8 c}+\frac {3}{8} \pi x \sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{4} x \left (\pi +c^2 \pi x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {\pi \sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{8 b c \sqrt {1+c^2 x^2}}+\frac {\left (3 b^2 \pi \sqrt {\pi +c^2 \pi x^2}\right ) \int \frac {1}{\sqrt {1+c^2 x^2}} \, dx}{64 \sqrt {1+c^2 x^2}}-\frac {\left (3 b^2 \pi \sqrt {\pi +c^2 \pi x^2}\right ) \int \frac {1}{\sqrt {1+c^2 x^2}} \, dx}{16 \sqrt {1+c^2 x^2}}\\ &=\frac {15}{64} b^2 \pi x \sqrt {\pi +c^2 \pi x^2}+\frac {1}{32} b^2 \pi x \left (1+c^2 x^2\right ) \sqrt {\pi +c^2 \pi x^2}-\frac {9 b^2 \pi \sqrt {\pi +c^2 \pi x^2} \sinh ^{-1}(c x)}{64 c \sqrt {1+c^2 x^2}}-\frac {3 b c \pi x^2 \sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )}{8 \sqrt {1+c^2 x^2}}-\frac {b \pi \left (1+c^2 x^2\right )^{3/2} \sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )}{8 c}+\frac {3}{8} \pi x \sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{4} x \left (\pi +c^2 \pi x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {\pi \sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{8 b c \sqrt {1+c^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.39, size = 202, normalized size = 0.96 \begin {gather*} \frac {\pi ^{3/2} \left (160 a^2 c x \sqrt {1+c^2 x^2}+64 a^2 c^3 x^3 \sqrt {1+c^2 x^2}+32 b^2 \sinh ^{-1}(c x)^3-64 a b \cosh \left (2 \sinh ^{-1}(c x)\right )-4 a b \cosh \left (4 \sinh ^{-1}(c x)\right )+32 b^2 \sinh \left (2 \sinh ^{-1}(c x)\right )+b^2 \sinh \left (4 \sinh ^{-1}(c x)\right )+8 b \sinh ^{-1}(c x)^2 \left (12 a+8 b \sinh \left (2 \sinh ^{-1}(c x)\right )+b \sinh \left (4 \sinh ^{-1}(c x)\right )\right )+4 \sinh ^{-1}(c x) \left (-16 b^2 \cosh \left (2 \sinh ^{-1}(c x)\right )-b^2 \cosh \left (4 \sinh ^{-1}(c x)\right )+4 a \left (6 a+8 b \sinh \left (2 \sinh ^{-1}(c x)\right )+b \sinh \left (4 \sinh ^{-1}(c x)\right )\right )\right )\right )}{256 c} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}} \left (a +b \arcsinh \left (c x \right )\right )^{2}\, dx\]
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 405 vs.
\(2 (197) = 394\).
time = 3.70, size = 405, normalized size = 1.93 \begin {gather*} \begin {cases} \frac {\pi ^{\frac {3}{2}} a^{2} c^{2} x^{3} \sqrt {c^{2} x^{2} + 1}}{4} + \frac {5 \pi ^{\frac {3}{2}} a^{2} x \sqrt {c^{2} x^{2} + 1}}{8} + \frac {3 \pi ^{\frac {3}{2}} a^{2} \operatorname {asinh}{\left (c x \right )}}{8 c} - \frac {\pi ^{\frac {3}{2}} a b c^{3} x^{4}}{8} + \frac {\pi ^{\frac {3}{2}} a b c^{2} x^{3} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{2} - \frac {5 \pi ^{\frac {3}{2}} a b c x^{2}}{8} + \frac {5 \pi ^{\frac {3}{2}} a b x \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{4} + \frac {3 \pi ^{\frac {3}{2}} a b \operatorname {asinh}^{2}{\left (c x \right )}}{8 c} - \frac {\pi ^{\frac {3}{2}} b^{2} c^{3} x^{4} \operatorname {asinh}{\left (c x \right )}}{8} + \frac {\pi ^{\frac {3}{2}} b^{2} c^{2} x^{3} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (c x \right )}}{4} + \frac {\pi ^{\frac {3}{2}} b^{2} c^{2} x^{3} \sqrt {c^{2} x^{2} + 1}}{32} - \frac {5 \pi ^{\frac {3}{2}} b^{2} c x^{2} \operatorname {asinh}{\left (c x \right )}}{8} + \frac {5 \pi ^{\frac {3}{2}} b^{2} x \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (c x \right )}}{8} + \frac {17 \pi ^{\frac {3}{2}} b^{2} x \sqrt {c^{2} x^{2} + 1}}{64} + \frac {\pi ^{\frac {3}{2}} b^{2} \operatorname {asinh}^{3}{\left (c x \right )}}{8 c} - \frac {17 \pi ^{\frac {3}{2}} b^{2} \operatorname {asinh}{\left (c x \right )}}{64 c} & \text {for}\: c \neq 0 \\\pi ^{\frac {3}{2}} a^{2} x & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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 {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (\Pi \,c^2\,x^2+\Pi \right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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