Optimal. Leaf size=98 \[ \frac {2 b x}{3 c^3 \sqrt {\pi }}-\frac {b x^3}{9 c \sqrt {\pi }}-\frac {2 \sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c^4 \pi }+\frac {x^2 \sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c^2 \pi } \]
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Rubi [A]
time = 0.11, antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {5812, 5798, 8,
30} \begin {gather*} \frac {x^2 \sqrt {\pi c^2 x^2+\pi } \left (a+b \sinh ^{-1}(c x)\right )}{3 \pi c^2}-\frac {2 \sqrt {\pi c^2 x^2+\pi } \left (a+b \sinh ^{-1}(c x)\right )}{3 \pi c^4}+\frac {2 b x}{3 \sqrt {\pi } c^3}-\frac {b x^3}{9 \sqrt {\pi } c} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 30
Rule 5798
Rule 5812
Rubi steps
\begin {align*} \int \frac {x^3 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {\pi +c^2 \pi x^2}} \, dx &=\frac {x^2 \sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c^2 \pi }-\frac {2 \int \frac {x \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {\pi +c^2 \pi x^2}} \, dx}{3 c^2}-\frac {\left (b \sqrt {1+c^2 x^2}\right ) \int x^2 \, dx}{3 c \sqrt {\pi +c^2 \pi x^2}}\\ &=-\frac {b x^3 \sqrt {1+c^2 x^2}}{9 c \sqrt {\pi +c^2 \pi x^2}}-\frac {2 \sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c^4 \pi }+\frac {x^2 \sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c^2 \pi }+\frac {\left (2 b \sqrt {1+c^2 x^2}\right ) \int 1 \, dx}{3 c^3 \sqrt {\pi +c^2 \pi x^2}}\\ &=\frac {2 b x \sqrt {1+c^2 x^2}}{3 c^3 \sqrt {\pi +c^2 \pi x^2}}-\frac {b x^3 \sqrt {1+c^2 x^2}}{9 c \sqrt {\pi +c^2 \pi x^2}}-\frac {2 \sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c^4 \pi }+\frac {x^2 \sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c^2 \pi }\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 82, normalized size = 0.84 \begin {gather*} \frac {3 a \left (-2+c^2 x^2\right ) \sqrt {1+c^2 x^2}+b \left (6 c x-c^3 x^3\right )+3 b \left (-2+c^2 x^2\right ) \sqrt {1+c^2 x^2} \sinh ^{-1}(c x)}{9 c^4 \sqrt {\pi }} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {x^{3} \left (a +b \arcsinh \left (c x \right )\right )}{\sqrt {\pi \,c^{2} x^{2}+\pi }}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 117, normalized size = 1.19 \begin {gather*} \frac {1}{3} \, b {\left (\frac {\sqrt {\pi + \pi c^{2} x^{2}} x^{2}}{\pi c^{2}} - \frac {2 \, \sqrt {\pi + \pi c^{2} x^{2}}}{\pi c^{4}}\right )} \operatorname {arsinh}\left (c x\right ) + \frac {1}{3} \, a {\left (\frac {\sqrt {\pi + \pi c^{2} x^{2}} x^{2}}{\pi c^{2}} - \frac {2 \, \sqrt {\pi + \pi c^{2} x^{2}}}{\pi c^{4}}\right )} - \frac {{\left (c^{2} x^{3} - 6 \, x\right )} b}{9 \, \sqrt {\pi } c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 132, normalized size = 1.35 \begin {gather*} \frac {3 \, \sqrt {\pi + \pi c^{2} x^{2}} {\left (b c^{4} x^{4} - b c^{2} x^{2} - 2 \, b\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + \sqrt {\pi + \pi c^{2} x^{2}} {\left (3 \, a c^{4} x^{4} - 3 \, a c^{2} x^{2} - {\left (b c^{3} x^{3} - 6 \, b c x\right )} \sqrt {c^{2} x^{2} + 1} - 6 \, a\right )}}{9 \, {\left (\pi c^{6} x^{2} + \pi c^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 2.01, size = 122, normalized size = 1.24 \begin {gather*} \frac {a \left (\begin {cases} \frac {x^{2} \sqrt {c^{2} x^{2} + 1}}{3 c^{2}} - \frac {2 \sqrt {c^{2} x^{2} + 1}}{3 c^{4}} & \text {for}\: c \neq 0 \\\frac {x^{4}}{4} & \text {otherwise} \end {cases}\right )}{\sqrt {\pi }} + \frac {b \left (\begin {cases} - \frac {x^{3}}{9 c} + \frac {x^{2} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{3 c^{2}} + \frac {2 x}{3 c^{3}} - \frac {2 \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{3 c^{4}} & \text {for}\: c \neq 0 \\0 & \text {otherwise} \end {cases}\right )}{\sqrt {\pi }} \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 = -0.01 \begin {gather*} \int \frac {x^3\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}{\sqrt {\Pi \,c^2\,x^2+\Pi }} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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