Optimal. Leaf size=146 \[ \frac {\sqrt {\pi c^2 x^2+\pi } \left (a+b \sinh ^{-1}(c x)\right )}{\pi ^3 c^6}+\frac {2 \left (a+b \sinh ^{-1}(c x)\right )}{\pi ^2 c^6 \sqrt {\pi c^2 x^2+\pi }}-\frac {a+b \sinh ^{-1}(c x)}{3 \pi c^6 \left (\pi c^2 x^2+\pi \right )^{3/2}}-\frac {11 b \tan ^{-1}(c x)}{6 \pi ^{5/2} c^6}-\frac {b x}{\pi ^{5/2} c^5}+\frac {b x}{6 \pi ^{5/2} c^5 \left (c^2 x^2+1\right )} \]
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Rubi [A] time = 0.18, antiderivative size = 149, normalized size of antiderivative = 1.02, number of steps used = 5, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {266, 43, 5732, 12, 1157, 388, 203} \[ \frac {\sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{\pi ^{5/2} c^6}+\frac {2 \left (a+b \sinh ^{-1}(c x)\right )}{\pi ^{5/2} c^6 \sqrt {c^2 x^2+1}}-\frac {a+b \sinh ^{-1}(c x)}{3 \pi ^{5/2} c^6 \left (c^2 x^2+1\right )^{3/2}}+\frac {b x}{6 \pi ^{5/2} c^5 \left (c^2 x^2+1\right )}-\frac {b x}{\pi ^{5/2} c^5}-\frac {11 b \tan ^{-1}(c x)}{6 \pi ^{5/2} c^6} \]
Antiderivative was successfully verified.
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Rule 12
Rule 43
Rule 203
Rule 266
Rule 388
Rule 1157
Rule 5732
Rubi steps
\begin {align*} \int \frac {x^5 \left (a+b \sinh ^{-1}(c x)\right )}{\left (\pi +c^2 \pi x^2\right )^{5/2}} \, dx &=-\frac {a+b \sinh ^{-1}(c x)}{3 c^6 \pi ^{5/2} \left (1+c^2 x^2\right )^{3/2}}+\frac {2 \left (a+b \sinh ^{-1}(c x)\right )}{c^6 \pi ^{5/2} \sqrt {1+c^2 x^2}}+\frac {\sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{c^6 \pi ^{5/2}}-\frac {(b c) \int \frac {8+12 c^2 x^2+3 c^4 x^4}{3 c^6 \left (1+c^2 x^2\right )^2} \, dx}{\pi ^{5/2}}\\ &=-\frac {a+b \sinh ^{-1}(c x)}{3 c^6 \pi ^{5/2} \left (1+c^2 x^2\right )^{3/2}}+\frac {2 \left (a+b \sinh ^{-1}(c x)\right )}{c^6 \pi ^{5/2} \sqrt {1+c^2 x^2}}+\frac {\sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{c^6 \pi ^{5/2}}-\frac {b \int \frac {8+12 c^2 x^2+3 c^4 x^4}{\left (1+c^2 x^2\right )^2} \, dx}{3 c^5 \pi ^{5/2}}\\ &=\frac {b x}{6 c^5 \pi ^{5/2} \left (1+c^2 x^2\right )}-\frac {a+b \sinh ^{-1}(c x)}{3 c^6 \pi ^{5/2} \left (1+c^2 x^2\right )^{3/2}}+\frac {2 \left (a+b \sinh ^{-1}(c x)\right )}{c^6 \pi ^{5/2} \sqrt {1+c^2 x^2}}+\frac {\sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{c^6 \pi ^{5/2}}+\frac {b \int \frac {-17-6 c^2 x^2}{1+c^2 x^2} \, dx}{6 c^5 \pi ^{5/2}}\\ &=-\frac {b x}{c^5 \pi ^{5/2}}+\frac {b x}{6 c^5 \pi ^{5/2} \left (1+c^2 x^2\right )}-\frac {a+b \sinh ^{-1}(c x)}{3 c^6 \pi ^{5/2} \left (1+c^2 x^2\right )^{3/2}}+\frac {2 \left (a+b \sinh ^{-1}(c x)\right )}{c^6 \pi ^{5/2} \sqrt {1+c^2 x^2}}+\frac {\sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{c^6 \pi ^{5/2}}-\frac {(11 b) \int \frac {1}{1+c^2 x^2} \, dx}{6 c^5 \pi ^{5/2}}\\ &=-\frac {b x}{c^5 \pi ^{5/2}}+\frac {b x}{6 c^5 \pi ^{5/2} \left (1+c^2 x^2\right )}-\frac {a+b \sinh ^{-1}(c x)}{3 c^6 \pi ^{5/2} \left (1+c^2 x^2\right )^{3/2}}+\frac {2 \left (a+b \sinh ^{-1}(c x)\right )}{c^6 \pi ^{5/2} \sqrt {1+c^2 x^2}}+\frac {\sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{c^6 \pi ^{5/2}}-\frac {11 b \tan ^{-1}(c x)}{6 c^6 \pi ^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.21, size = 132, normalized size = 0.90 \[ \frac {6 a c^4 x^4+24 a c^2 x^2+16 a-5 b c x \sqrt {c^2 x^2+1}-11 b \left (c^2 x^2+1\right )^{3/2} \tan ^{-1}(c x)+2 b \left (3 c^4 x^4+12 c^2 x^2+8\right ) \sinh ^{-1}(c x)-6 b c^3 x^3 \sqrt {c^2 x^2+1}}{6 \pi ^{5/2} c^6 \left (c^2 x^2+1\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.89, size = 218, normalized size = 1.49 \[ \frac {11 \, \sqrt {\pi } {\left (b c^{4} x^{4} + 2 \, b c^{2} x^{2} + b\right )} \arctan \left (-\frac {2 \, \sqrt {\pi } \sqrt {\pi + \pi c^{2} x^{2}} \sqrt {c^{2} x^{2} + 1} c x}{\pi - \pi c^{4} x^{4}}\right ) + 4 \, \sqrt {\pi + \pi c^{2} x^{2}} {\left (3 \, b c^{4} x^{4} + 12 \, b c^{2} x^{2} + 8 \, b\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + 2 \, \sqrt {\pi + \pi c^{2} x^{2}} {\left (6 \, a c^{4} x^{4} + 24 \, a c^{2} x^{2} - {\left (6 \, b c^{3} x^{3} + 5 \, b c x\right )} \sqrt {c^{2} x^{2} + 1} + 16 \, a\right )}}{12 \, {\left (\pi ^{3} c^{10} x^{4} + 2 \, \pi ^{3} c^{8} x^{2} + \pi ^{3} c^{6}\right )}} \]
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 [C] time = 0.48, size = 231, normalized size = 1.58 \[ \frac {a \,x^{4}}{\pi \,c^{2} \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}+\frac {4 a \,x^{2}}{c^{4} \pi \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}+\frac {8 a}{3 c^{6} \pi \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}+\frac {b \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}}{\pi ^{\frac {5}{2}} c^{6}}-\frac {b x}{c^{5} \pi ^{\frac {5}{2}}}+\frac {2 b \arcsinh \left (c x \right ) x^{2}}{\pi ^{\frac {5}{2}} \left (c^{2} x^{2}+1\right )^{\frac {3}{2}} c^{4}}+\frac {b x}{6 c^{5} \pi ^{\frac {5}{2}} \left (c^{2} x^{2}+1\right )}+\frac {5 b \arcsinh \left (c x \right )}{3 \pi ^{\frac {5}{2}} \left (c^{2} x^{2}+1\right )^{\frac {3}{2}} c^{6}}+\frac {11 i b \ln \left (c x +\sqrt {c^{2} x^{2}+1}-i\right )}{6 c^{6} \pi ^{\frac {5}{2}}}-\frac {11 i b \ln \left (c x +\sqrt {c^{2} x^{2}+1}+i\right )}{6 c^{6} \pi ^{\frac {5}{2}}} \]
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 \[ \frac {1}{3} \, b {\left (\frac {{\left (3 \, \sqrt {\pi } c^{4} x^{4} + 12 \, \sqrt {\pi } c^{2} x^{2} + 8 \, \sqrt {\pi }\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )}{{\left (\pi ^{3} c^{8} x^{2} + \pi ^{3} c^{6}\right )} \sqrt {c^{2} x^{2} + 1}} + 3 \, \int \frac {3 \, \sqrt {\pi } c^{4} x^{4} + 12 \, \sqrt {\pi } c^{2} x^{2} + 8 \, \sqrt {\pi }}{3 \, {\left (\pi ^{3} c^{11} x^{6} + 2 \, \pi ^{3} c^{9} x^{4} + \pi ^{3} c^{7} x^{2} + {\left (\pi ^{3} c^{10} x^{5} + 2 \, \pi ^{3} c^{8} x^{3} + \pi ^{3} c^{6} x\right )} \sqrt {c^{2} x^{2} + 1}\right )}}\,{d x} - 3 \, \int \frac {3 \, \sqrt {\pi } c^{4} x^{4} + 12 \, \sqrt {\pi } c^{2} x^{2} + 8 \, \sqrt {\pi }}{3 \, {\left (\pi ^{3} c^{8} x^{3} + \pi ^{3} c^{6} x\right )} \sqrt {c^{2} x^{2} + 1}}\,{d x}\right )} + \frac {1}{3} \, a {\left (\frac {3 \, x^{4}}{\pi {\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {3}{2}} c^{2}} + \frac {12 \, x^{2}}{\pi {\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {3}{2}} c^{4}} + \frac {8}{\pi {\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {3}{2}} c^{6}}\right )} \]
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 \[ \int \frac {x^5\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}{{\left (\Pi \,c^2\,x^2+\Pi \right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {a x^{5}}{c^{4} x^{4} \sqrt {c^{2} x^{2} + 1} + 2 c^{2} x^{2} \sqrt {c^{2} x^{2} + 1} + \sqrt {c^{2} x^{2} + 1}}\, dx + \int \frac {b x^{5} \operatorname {asinh}{\left (c x \right )}}{c^{4} x^{4} \sqrt {c^{2} x^{2} + 1} + 2 c^{2} x^{2} \sqrt {c^{2} x^{2} + 1} + \sqrt {c^{2} x^{2} + 1}}\, dx}{\pi ^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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