Optimal. Leaf size=77 \[ -\frac {b \pi ^{3/2} x}{5 c}-\frac {2}{15} b c \pi ^{3/2} x^3-\frac {1}{25} b c^3 \pi ^{3/2} x^5+\frac {\left (\pi +c^2 \pi x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{5 c^2 \pi } \]
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Rubi [A]
time = 0.05, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {5798, 200}
\begin {gather*} \frac {\left (\pi c^2 x^2+\pi \right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{5 \pi c^2}-\frac {1}{25} \pi ^{3/2} b c^3 x^5-\frac {2}{15} \pi ^{3/2} b c x^3-\frac {\pi ^{3/2} b x}{5 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 200
Rule 5798
Rubi steps
\begin {align*} \int x \left (\pi +c^2 \pi x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx &=\frac {\left (\pi +c^2 \pi x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{5 c^2 \pi }-\frac {\left (b \pi \sqrt {\pi +c^2 \pi x^2}\right ) \int \left (1+c^2 x^2\right )^2 \, dx}{5 c \sqrt {1+c^2 x^2}}\\ &=\frac {\left (\pi +c^2 \pi x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{5 c^2 \pi }-\frac {\left (b \pi \sqrt {\pi +c^2 \pi x^2}\right ) \int \left (1+2 c^2 x^2+c^4 x^4\right ) \, dx}{5 c \sqrt {1+c^2 x^2}}\\ &=-\frac {b \pi x \sqrt {\pi +c^2 \pi x^2}}{5 c \sqrt {1+c^2 x^2}}-\frac {2 b c \pi x^3 \sqrt {\pi +c^2 \pi x^2}}{15 \sqrt {1+c^2 x^2}}-\frac {b c^3 \pi x^5 \sqrt {\pi +c^2 \pi x^2}}{25 \sqrt {1+c^2 x^2}}+\frac {\left (\pi +c^2 \pi x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{5 c^2 \pi }\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 72, normalized size = 0.94 \begin {gather*} \frac {\pi ^{3/2} \left (15 a \left (1+c^2 x^2\right )^{5/2}-b c x \left (15+10 c^2 x^2+3 c^4 x^4\right )+15 b \left (1+c^2 x^2\right )^{5/2} \sinh ^{-1}(c x)\right )}{75 c^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int x \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}} \left (a +b \arcsinh \left (c x \right )\right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 85, normalized size = 1.10 \begin {gather*} \frac {{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {5}{2}} b \operatorname {arsinh}\left (c x\right )}{5 \, \pi c^{2}} + \frac {{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {5}{2}} a}{5 \, \pi c^{2}} - \frac {{\left (3 \, \pi ^{\frac {5}{2}} c^{4} x^{5} + 10 \, \pi ^{\frac {5}{2}} c^{2} x^{3} + 15 \, \pi ^{\frac {5}{2}} x\right )} b}{75 \, \pi c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 167 vs.
\(2 (61) = 122\).
time = 0.38, size = 167, normalized size = 2.17 \begin {gather*} \frac {15 \, \sqrt {\pi + \pi c^{2} x^{2}} {\left (\pi b c^{6} x^{6} + 3 \, \pi b c^{4} x^{4} + 3 \, \pi b c^{2} x^{2} + \pi b\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + \sqrt {\pi + \pi c^{2} x^{2}} {\left (15 \, \pi a c^{6} x^{6} + 45 \, \pi a c^{4} x^{4} + 45 \, \pi a c^{2} x^{2} + 15 \, \pi a - {\left (3 \, \pi b c^{5} x^{5} + 10 \, \pi b c^{3} x^{3} + 15 \, \pi b c x\right )} \sqrt {c^{2} x^{2} + 1}\right )}}{75 \, {\left (c^{4} x^{2} + c^{2}\right )}} \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. 221 vs.
\(2 (70) = 140\).
time = 3.96, size = 221, normalized size = 2.87 \begin {gather*} \begin {cases} \frac {\pi ^{\frac {3}{2}} a c^{2} x^{4} \sqrt {c^{2} x^{2} + 1}}{5} + \frac {2 \pi ^{\frac {3}{2}} a x^{2} \sqrt {c^{2} x^{2} + 1}}{5} + \frac {\pi ^{\frac {3}{2}} a \sqrt {c^{2} x^{2} + 1}}{5 c^{2}} - \frac {\pi ^{\frac {3}{2}} b c^{3} x^{5}}{25} + \frac {\pi ^{\frac {3}{2}} b c^{2} x^{4} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{5} - \frac {2 \pi ^{\frac {3}{2}} b c x^{3}}{15} + \frac {2 \pi ^{\frac {3}{2}} b x^{2} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{5} - \frac {\pi ^{\frac {3}{2}} b x}{5 c} + \frac {\pi ^{\frac {3}{2}} b \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{5 c^{2}} & \text {for}\: c \neq 0 \\\frac {\pi ^{\frac {3}{2}} a x^{2}}{2} & \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 = -0.01 \begin {gather*} \int x\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\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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