Optimal. Leaf size=137 \[ \frac {5 b x}{3 c^5 \pi ^{3/2}}-\frac {b x^3}{9 c^3 \pi ^{3/2}}-\frac {a+b \sinh ^{-1}(c x)}{c^6 \pi \sqrt {\pi +c^2 \pi x^2}}-\frac {2 \sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )}{c^6 \pi ^2}+\frac {\left (\pi +c^2 \pi x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c^6 \pi ^3}+\frac {b \text {ArcTan}(c x)}{c^6 \pi ^{3/2}} \]
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Rubi [A]
time = 0.13, antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {272, 45, 5804,
12, 1167, 209} \begin {gather*} \frac {\left (\pi c^2 x^2+\pi \right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 \pi ^3 c^6}-\frac {2 \sqrt {\pi c^2 x^2+\pi } \left (a+b \sinh ^{-1}(c x)\right )}{\pi ^2 c^6}-\frac {a+b \sinh ^{-1}(c x)}{\pi c^6 \sqrt {\pi c^2 x^2+\pi }}+\frac {b \text {ArcTan}(c x)}{\pi ^{3/2} c^6}+\frac {5 b x}{3 \pi ^{3/2} c^5}-\frac {b x^3}{9 \pi ^{3/2} c^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 45
Rule 209
Rule 272
Rule 1167
Rule 5804
Rubi steps
\begin {align*} \int \frac {x^5 \left (a+b \sinh ^{-1}(c x)\right )}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx &=-\frac {a+b \sinh ^{-1}(c x)}{c^6 \pi ^{3/2} \sqrt {1+c^2 x^2}}-\frac {2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{c^6 \pi ^{3/2}}+\frac {\left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c^6 \pi ^{3/2}}-\frac {(b c) \int \frac {-8-4 c^2 x^2+c^4 x^4}{3 c^6+3 c^8 x^2} \, dx}{\pi ^{3/2}}\\ &=-\frac {a+b \sinh ^{-1}(c x)}{c^6 \pi ^{3/2} \sqrt {1+c^2 x^2}}-\frac {2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{c^6 \pi ^{3/2}}+\frac {\left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c^6 \pi ^{3/2}}-\frac {(b c) \int \left (-\frac {5}{3 c^6}+\frac {x^2}{3 c^4}-\frac {3}{3 c^6+3 c^8 x^2}\right ) \, dx}{\pi ^{3/2}}\\ &=\frac {5 b x}{3 c^5 \pi ^{3/2}}-\frac {b x^3}{9 c^3 \pi ^{3/2}}-\frac {a+b \sinh ^{-1}(c x)}{c^6 \pi ^{3/2} \sqrt {1+c^2 x^2}}-\frac {2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{c^6 \pi ^{3/2}}+\frac {\left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c^6 \pi ^{3/2}}+\frac {(3 b c) \int \frac {1}{3 c^6+3 c^8 x^2} \, dx}{\pi ^{3/2}}\\ &=\frac {5 b x}{3 c^5 \pi ^{3/2}}-\frac {b x^3}{9 c^3 \pi ^{3/2}}-\frac {a+b \sinh ^{-1}(c x)}{c^6 \pi ^{3/2} \sqrt {1+c^2 x^2}}-\frac {2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{c^6 \pi ^{3/2}}+\frac {\left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c^6 \pi ^{3/2}}+\frac {b \tan ^{-1}(c x)}{c^6 \pi ^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 131, normalized size = 0.96 \begin {gather*} \frac {-24 a-12 a c^2 x^2+3 a c^4 x^4+15 b c x \sqrt {1+c^2 x^2}-b c^3 x^3 \sqrt {1+c^2 x^2}+3 b \left (-8-4 c^2 x^2+c^4 x^4\right ) \sinh ^{-1}(c x)+9 b \sqrt {1+c^2 x^2} \text {ArcTan}(c x)}{9 c^6 \pi ^{3/2} \sqrt {1+c^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 5.19, size = 229, normalized size = 1.67
method | result | size |
default | \(a \left (\frac {x^{4}}{3 \pi \,c^{2} \sqrt {\pi \,c^{2} x^{2}+\pi }}-\frac {4 \left (\frac {x^{2}}{\pi \,c^{2} \sqrt {\pi \,c^{2} x^{2}+\pi }}+\frac {2}{\pi \,c^{4} \sqrt {\pi \,c^{2} x^{2}+\pi }}\right )}{3 c^{2}}\right )-\frac {5 b \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}}{3 \pi ^{\frac {3}{2}} c^{6}}-\frac {i b \ln \left (c x +\sqrt {c^{2} x^{2}+1}-i\right )}{c^{6} \pi ^{\frac {3}{2}}}+\frac {i b \ln \left (c x +\sqrt {c^{2} x^{2}+1}+i\right )}{c^{6} \pi ^{\frac {3}{2}}}-\frac {b \,x^{3}}{9 c^{3} \pi ^{\frac {3}{2}}}+\frac {5 b x}{3 c^{5} \pi ^{\frac {3}{2}}}+\frac {b \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{2}}{3 \pi ^{\frac {3}{2}} c^{4}}-\frac {b \arcsinh \left (c x \right )}{\pi ^{\frac {3}{2}} \sqrt {c^{2} x^{2}+1}\, c^{6}}\) | \(229\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.43, size = 196, normalized size = 1.43 \begin {gather*} -\frac {9 \, \sqrt {\pi } {\left (b c^{2} x^{2} + b\right )} \arctan \left (-\frac {2 \, \sqrt {\pi } \sqrt {\pi + \pi c^{2} x^{2}} \sqrt {c^{2} x^{2} + 1} c x}{\pi - \pi c^{4} x^{4}}\right ) - 6 \, \sqrt {\pi + \pi c^{2} x^{2}} {\left (b c^{4} x^{4} - 4 \, b c^{2} x^{2} - 8 \, b\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - 2 \, \sqrt {\pi + \pi c^{2} x^{2}} {\left (3 \, a c^{4} x^{4} - 12 \, a c^{2} x^{2} - {\left (b c^{3} x^{3} - 15 \, b c x\right )} \sqrt {c^{2} x^{2} + 1} - 24 \, a\right )}}{18 \, {\left (\pi ^{2} c^{8} x^{2} + \pi ^{2} c^{6}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {a x^{5}}{c^{2} x^{2} \sqrt {c^{2} x^{2} + 1} + \sqrt {c^{2} x^{2} + 1}}\, dx + \int \frac {b x^{5} \operatorname {asinh}{\left (c x \right )}}{c^{2} x^{2} \sqrt {c^{2} x^{2} + 1} + \sqrt {c^{2} x^{2} + 1}}\, dx}{\pi ^{\frac {3}{2}}} \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^5\,\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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