Optimal. Leaf size=165 \[ \frac {1}{6} x \left (\pi c^2 x^2+\pi \right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {5}{24} \pi x \left (\pi c^2 x^2+\pi \right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {5}{16} \pi ^2 x \sqrt {\pi c^2 x^2+\pi } \left (a+b \sinh ^{-1}(c x)\right )+\frac {5 \pi ^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{32 b c}-\frac {5}{96} \pi ^{5/2} b c^3 x^4-\frac {\pi ^{5/2} b \left (c^2 x^2+1\right )^3}{36 c}-\frac {25}{96} \pi ^{5/2} b c x^2 \]
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Rubi [A] time = 0.16, antiderivative size = 254, normalized size of antiderivative = 1.54, number of steps used = 8, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {5684, 5682, 5675, 30, 14, 261} \[ \frac {1}{6} x \left (\pi c^2 x^2+\pi \right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {5}{24} \pi x \left (\pi c^2 x^2+\pi \right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {5}{16} \pi ^2 x \sqrt {\pi c^2 x^2+\pi } \left (a+b \sinh ^{-1}(c x)\right )+\frac {5 \pi ^2 \sqrt {\pi c^2 x^2+\pi } \left (a+b \sinh ^{-1}(c x)\right )^2}{32 b c \sqrt {c^2 x^2+1}}-\frac {5 \pi ^2 b c^3 x^4 \sqrt {\pi c^2 x^2+\pi }}{96 \sqrt {c^2 x^2+1}}-\frac {25 \pi ^2 b c x^2 \sqrt {\pi c^2 x^2+\pi }}{96 \sqrt {c^2 x^2+1}}-\frac {\pi ^2 b \left (c^2 x^2+1\right )^{5/2} \sqrt {\pi c^2 x^2+\pi }}{36 c} \]
Antiderivative was successfully verified.
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Rule 14
Rule 30
Rule 261
Rule 5675
Rule 5682
Rule 5684
Rubi steps
\begin {align*} \int \left (\pi +c^2 \pi x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx &=\frac {1}{6} x \left (\pi +c^2 \pi x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{6} (5 \pi ) \int \left (\pi +c^2 \pi x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx-\frac {\left (b c \pi ^2 \sqrt {\pi +c^2 \pi x^2}\right ) \int x \left (1+c^2 x^2\right )^2 \, dx}{6 \sqrt {1+c^2 x^2}}\\ &=-\frac {b \pi ^2 \left (1+c^2 x^2\right )^{5/2} \sqrt {\pi +c^2 \pi x^2}}{36 c}+\frac {5}{24} \pi x \left (\pi +c^2 \pi x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{6} x \left (\pi +c^2 \pi x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{8} \left (5 \pi ^2\right ) \int \sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx-\frac {\left (5 b c \pi ^2 \sqrt {\pi +c^2 \pi x^2}\right ) \int x \left (1+c^2 x^2\right ) \, dx}{24 \sqrt {1+c^2 x^2}}\\ &=-\frac {b \pi ^2 \left (1+c^2 x^2\right )^{5/2} \sqrt {\pi +c^2 \pi x^2}}{36 c}+\frac {5}{16} \pi ^2 x \sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {5}{24} \pi x \left (\pi +c^2 \pi x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{6} x \left (\pi +c^2 \pi x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {\left (5 \pi ^2 \sqrt {\pi +c^2 \pi x^2}\right ) \int \frac {a+b \sinh ^{-1}(c x)}{\sqrt {1+c^2 x^2}} \, dx}{16 \sqrt {1+c^2 x^2}}-\frac {\left (5 b c \pi ^2 \sqrt {\pi +c^2 \pi x^2}\right ) \int \left (x+c^2 x^3\right ) \, dx}{24 \sqrt {1+c^2 x^2}}-\frac {\left (5 b c \pi ^2 \sqrt {\pi +c^2 \pi x^2}\right ) \int x \, dx}{16 \sqrt {1+c^2 x^2}}\\ &=-\frac {25 b c \pi ^2 x^2 \sqrt {\pi +c^2 \pi x^2}}{96 \sqrt {1+c^2 x^2}}-\frac {5 b c^3 \pi ^2 x^4 \sqrt {\pi +c^2 \pi x^2}}{96 \sqrt {1+c^2 x^2}}-\frac {b \pi ^2 \left (1+c^2 x^2\right )^{5/2} \sqrt {\pi +c^2 \pi x^2}}{36 c}+\frac {5}{16} \pi ^2 x \sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {5}{24} \pi x \left (\pi +c^2 \pi x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{6} x \left (\pi +c^2 \pi x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {5 \pi ^2 \sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{32 b c \sqrt {1+c^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.38, size = 153, normalized size = 0.93 \[ \frac {\pi ^{5/2} \left (12 \sinh ^{-1}(c x) \left (60 a+45 b \sinh \left (2 \sinh ^{-1}(c x)\right )+9 b \sinh \left (4 \sinh ^{-1}(c x)\right )+b \sinh \left (6 \sinh ^{-1}(c x)\right )\right )+1584 a c x \sqrt {c^2 x^2+1}+384 a c^5 x^5 \sqrt {c^2 x^2+1}+1248 a c^3 x^3 \sqrt {c^2 x^2+1}+360 b \sinh ^{-1}(c x)^2-270 b \cosh \left (2 \sinh ^{-1}(c x)\right )-27 b \cosh \left (4 \sinh ^{-1}(c x)\right )-2 b \cosh \left (6 \sinh ^{-1}(c x)\right )\right )}{2304 c} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.79, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {\pi + \pi c^{2} x^{2}} {\left (\pi ^{2} a c^{4} x^{4} + 2 \, \pi ^{2} a c^{2} x^{2} + \pi ^{2} a + {\left (\pi ^{2} b c^{4} x^{4} + 2 \, \pi ^{2} b c^{2} x^{2} + \pi ^{2} b\right )} \operatorname {arsinh}\left (c x\right )\right )}, x\right ) \]
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 [A] time = 0.07, size = 228, normalized size = 1.38 \[ \frac {x \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {5}{2}} a}{6}+\frac {5 a \pi x \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}{24}+\frac {5 a \,\pi ^{2} x \sqrt {\pi \,c^{2} x^{2}+\pi }}{16}+\frac {5 a \,\pi ^{3} \ln \left (\frac {\pi x \,c^{2}}{\sqrt {\pi \,c^{2}}}+\sqrt {\pi \,c^{2} x^{2}+\pi }\right )}{16 \sqrt {\pi \,c^{2}}}+\frac {b \,\pi ^{\frac {5}{2}} c^{4} \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{5}}{6}-\frac {b \,\pi ^{\frac {5}{2}} c^{5} x^{6}}{36}+\frac {13 b \,\pi ^{\frac {5}{2}} c^{2} \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{3}}{24}-\frac {13 b \,c^{3} \pi ^{\frac {5}{2}} x^{4}}{96}+\frac {11 b \,\pi ^{\frac {5}{2}} \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x}{16}-\frac {11 b c \,\pi ^{\frac {5}{2}} x^{2}}{32}+\frac {5 b \,\pi ^{\frac {5}{2}} \arcsinh \left (c x \right )^{2}}{32 c}-\frac {17 b \,\pi ^{\frac {5}{2}}}{72 c} \]
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 \[ \text {Exception raised: RuntimeError} \]
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 \[ \int \left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (\Pi \,c^2\,x^2+\Pi \right )}^{5/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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