Optimal. Leaf size=61 \[ \frac {\left (\pi c^2 x^2+\pi \right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 \pi c^2}-\frac {1}{9} \sqrt {\pi } b c x^3-\frac {\sqrt {\pi } b x}{3 c} \]
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Rubi [A] time = 0.07, antiderivative size = 105, normalized size of antiderivative = 1.72, number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {5717} \[ \frac {\left (\pi c^2 x^2+\pi \right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 \pi c^2}-\frac {b c x^3 \sqrt {\pi c^2 x^2+\pi }}{9 \sqrt {c^2 x^2+1}}-\frac {b x \sqrt {\pi c^2 x^2+\pi }}{3 c \sqrt {c^2 x^2+1}} \]
Antiderivative was successfully verified.
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Rule 5717
Rubi steps
\begin {align*} \int x \sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx &=\frac {\left (\pi +c^2 \pi x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c^2 \pi }-\frac {\left (b \sqrt {\pi +c^2 \pi x^2}\right ) \int \left (1+c^2 x^2\right ) \, dx}{3 c \sqrt {1+c^2 x^2}}\\ &=-\frac {b x \sqrt {\pi +c^2 \pi x^2}}{3 c \sqrt {1+c^2 x^2}}-\frac {b c x^3 \sqrt {\pi +c^2 \pi x^2}}{9 \sqrt {1+c^2 x^2}}+\frac {\left (\pi +c^2 \pi x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c^2 \pi }\\ \end {align*}
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Mathematica [A] time = 0.12, size = 63, normalized size = 1.03 \[ \frac {\sqrt {\pi } \left (3 a \left (c^2 x^2+1\right )^{3/2}-b c x \left (c^2 x^2+3\right )+3 b \left (c^2 x^2+1\right )^{3/2} \sinh ^{-1}(c x)\right )}{9 c^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.69, size = 127, normalized size = 2.08 \[ \frac {3 \, \sqrt {\pi + \pi c^{2} x^{2}} {\left (b c^{4} x^{4} + 2 \, b c^{2} x^{2} + b\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + \sqrt {\pi + \pi c^{2} x^{2}} {\left (3 \, a c^{4} x^{4} + 6 \, a c^{2} x^{2} - {\left (b c^{3} x^{3} + 3 \, b c x\right )} \sqrt {c^{2} x^{2} + 1} + 3 \, a\right )}}{9 \, {\left (c^{4} x^{2} + c^{2}\right )}} \]
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 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 108, normalized size = 1.77 \[ \frac {a \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}{3 \pi \,c^{2}}+\frac {b \sqrt {\pi }\, \left (3 \arcsinh \left (c x \right ) c^{4} x^{4}+6 \arcsinh \left (c x \right ) c^{2} x^{2}-c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+3 \arcsinh \left (c x \right )-3 c x \sqrt {c^{2} x^{2}+1}\right )}{9 c^{2} \sqrt {c^{2} x^{2}+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.50, size = 73, normalized size = 1.20 \[ \frac {{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {3}{2}} b \operatorname {arsinh}\left (c x\right )}{3 \, \pi c^{2}} - \frac {{\left (\pi ^{\frac {3}{2}} c^{2} x^{3} + 3 \, \pi ^{\frac {3}{2}} x\right )} b}{9 \, \pi c} + \frac {{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {3}{2}} a}{3 \, \pi c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int x\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,\sqrt {\Pi \,c^2\,x^2+\Pi } \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.20, size = 141, normalized size = 2.31 \[ \begin {cases} \frac {\sqrt {\pi } a x^{2} \sqrt {c^{2} x^{2} + 1}}{3} + \frac {\sqrt {\pi } a \sqrt {c^{2} x^{2} + 1}}{3 c^{2}} - \frac {\sqrt {\pi } b c x^{3}}{9} + \frac {\sqrt {\pi } b x^{2} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{3} - \frac {\sqrt {\pi } b x}{3 c} + \frac {\sqrt {\pi } b \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{3 c^{2}} & \text {for}\: c \neq 0 \\\frac {\sqrt {\pi } a x^{2}}{2} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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