Optimal. Leaf size=111 \[ -\frac {5}{16} b c \pi ^{3/2} x^2-\frac {1}{16} b c^3 \pi ^{3/2} x^4+\frac {3}{8} \pi x \sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{4} x \left (\pi +c^2 \pi x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {3 \pi ^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{16 b c} \]
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Rubi [A]
time = 0.07, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {5786, 5785,
5783, 30, 14} \begin {gather*} \frac {1}{4} x \left (\pi c^2 x^2+\pi \right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {3}{8} \pi x \sqrt {\pi c^2 x^2+\pi } \left (a+b \sinh ^{-1}(c x)\right )+\frac {3 \pi ^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{16 b c}-\frac {1}{16} \pi ^{3/2} b c^3 x^4-\frac {5}{16} \pi ^{3/2} b c x^2 \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 30
Rule 5783
Rule 5785
Rule 5786
Rubi steps
\begin {align*} \int \left (\pi +c^2 \pi x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx &=\frac {1}{4} x \left (\pi +c^2 \pi x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{4} (3 \pi ) \int \sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx-\frac {\left (b c \pi \sqrt {\pi +c^2 \pi x^2}\right ) \int x \left (1+c^2 x^2\right ) \, dx}{4 \sqrt {1+c^2 x^2}}\\ &=\frac {3}{8} \pi x \sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{4} x \left (\pi +c^2 \pi x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {\left (3 \pi \sqrt {\pi +c^2 \pi x^2}\right ) \int \frac {a+b \sinh ^{-1}(c x)}{\sqrt {1+c^2 x^2}} \, dx}{8 \sqrt {1+c^2 x^2}}-\frac {\left (b c \pi \sqrt {\pi +c^2 \pi x^2}\right ) \int \left (x+c^2 x^3\right ) \, dx}{4 \sqrt {1+c^2 x^2}}-\frac {\left (3 b c \pi \sqrt {\pi +c^2 \pi x^2}\right ) \int x \, dx}{8 \sqrt {1+c^2 x^2}}\\ &=-\frac {5 b c \pi x^2 \sqrt {\pi +c^2 \pi x^2}}{16 \sqrt {1+c^2 x^2}}-\frac {b c^3 \pi x^4 \sqrt {\pi +c^2 \pi x^2}}{16 \sqrt {1+c^2 x^2}}+\frac {3}{8} \pi x \sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{4} x \left (\pi +c^2 \pi x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {3 \pi \sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{16 b c \sqrt {1+c^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.16, size = 111, normalized size = 1.00 \begin {gather*} \frac {\pi ^{3/2} \left (80 a c x \sqrt {1+c^2 x^2}+32 a c^3 x^3 \sqrt {1+c^2 x^2}+24 b \sinh ^{-1}(c x)^2-16 b \cosh \left (2 \sinh ^{-1}(c x)\right )-b \cosh \left (4 \sinh ^{-1}(c x)\right )+4 \sinh ^{-1}(c x) \left (12 a+8 b \sinh \left (2 \sinh ^{-1}(c x)\right )+b \sinh \left (4 \sinh ^{-1}(c x)\right )\right )\right )}{128 c} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}} \left (a +b \arcsinh \left (c x \right )\right )\, dx\]
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 = 1.93, size = 185, normalized size = 1.67 \begin {gather*} \begin {cases} \frac {\pi ^{\frac {3}{2}} a c^{2} x^{3} \sqrt {c^{2} x^{2} + 1}}{4} + \frac {5 \pi ^{\frac {3}{2}} a x \sqrt {c^{2} x^{2} + 1}}{8} + \frac {3 \pi ^{\frac {3}{2}} a \operatorname {asinh}{\left (c x \right )}}{8 c} - \frac {\pi ^{\frac {3}{2}} b c^{3} x^{4}}{16} + \frac {\pi ^{\frac {3}{2}} b c^{2} x^{3} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{4} - \frac {5 \pi ^{\frac {3}{2}} b c x^{2}}{16} + \frac {5 \pi ^{\frac {3}{2}} b x \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{8} + \frac {3 \pi ^{\frac {3}{2}} b \operatorname {asinh}^{2}{\left (c x \right )}}{16 c} & \text {for}\: c \neq 0 \\\pi ^{\frac {3}{2}} a 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 = -0.01 \begin {gather*} \int \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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