Optimal. Leaf size=192 \[ -\frac {5 \left (a+b \sinh ^{-1}(c x)\right )^2}{4 \pi ^{5/2} b c^7}-\frac {x^5 \left (a+b \sinh ^{-1}(c x)\right )}{3 \pi c^2 \left (\pi c^2 x^2+\pi \right )^{3/2}}+\frac {5 x \sqrt {\pi c^2 x^2+\pi } \left (a+b \sinh ^{-1}(c x)\right )}{2 \pi ^3 c^6}-\frac {5 x^3 \left (a+b \sinh ^{-1}(c x)\right )}{3 \pi ^2 c^4 \sqrt {\pi c^2 x^2+\pi }}-\frac {b x^2}{4 \pi ^{5/2} c^5}-\frac {b}{6 \pi ^{5/2} c^7 \left (c^2 x^2+1\right )}-\frac {7 b \log \left (c^2 x^2+1\right )}{6 \pi ^{5/2} c^7} \]
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Rubi [A] time = 0.43, antiderivative size = 256, normalized size of antiderivative = 1.33, number of steps used = 11, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {5751, 5758, 5675, 30, 266, 43} \[ -\frac {x^5 \left (a+b \sinh ^{-1}(c x)\right )}{3 \pi c^2 \left (\pi c^2 x^2+\pi \right )^{3/2}}-\frac {5 x^3 \left (a+b \sinh ^{-1}(c x)\right )}{3 \pi ^2 c^4 \sqrt {\pi c^2 x^2+\pi }}+\frac {5 x \sqrt {\pi c^2 x^2+\pi } \left (a+b \sinh ^{-1}(c x)\right )}{2 \pi ^3 c^6}-\frac {5 \left (a+b \sinh ^{-1}(c x)\right )^2}{4 \pi ^{5/2} b c^7}-\frac {b x^2 \sqrt {c^2 x^2+1}}{4 \pi ^2 c^5 \sqrt {\pi c^2 x^2+\pi }}-\frac {b}{6 \pi ^2 c^7 \sqrt {c^2 x^2+1} \sqrt {\pi c^2 x^2+\pi }}-\frac {7 b \sqrt {c^2 x^2+1} \log \left (c^2 x^2+1\right )}{6 \pi ^2 c^7 \sqrt {\pi c^2 x^2+\pi }} \]
Antiderivative was successfully verified.
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Rule 30
Rule 43
Rule 266
Rule 5675
Rule 5751
Rule 5758
Rubi steps
\begin {align*} \int \frac {x^6 \left (a+b \sinh ^{-1}(c x)\right )}{\left (\pi +c^2 \pi x^2\right )^{5/2}} \, dx &=-\frac {x^5 \left (a+b \sinh ^{-1}(c x)\right )}{3 c^2 \pi \left (\pi +c^2 \pi x^2\right )^{3/2}}+\frac {5 \int \frac {x^4 \left (a+b \sinh ^{-1}(c x)\right )}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx}{3 c^2 \pi }+\frac {\left (b \sqrt {1+c^2 x^2}\right ) \int \frac {x^5}{\left (1+c^2 x^2\right )^2} \, dx}{3 c \pi ^2 \sqrt {\pi +c^2 \pi x^2}}\\ &=-\frac {x^5 \left (a+b \sinh ^{-1}(c x)\right )}{3 c^2 \pi \left (\pi +c^2 \pi x^2\right )^{3/2}}-\frac {5 x^3 \left (a+b \sinh ^{-1}(c x)\right )}{3 c^4 \pi ^2 \sqrt {\pi +c^2 \pi x^2}}+\frac {5 \int \frac {x^2 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {\pi +c^2 \pi x^2}} \, dx}{c^4 \pi ^2}+\frac {\left (5 b \sqrt {1+c^2 x^2}\right ) \int \frac {x^3}{1+c^2 x^2} \, dx}{3 c^3 \pi ^2 \sqrt {\pi +c^2 \pi x^2}}+\frac {\left (b \sqrt {1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (1+c^2 x\right )^2} \, dx,x,x^2\right )}{6 c \pi ^2 \sqrt {\pi +c^2 \pi x^2}}\\ &=-\frac {x^5 \left (a+b \sinh ^{-1}(c x)\right )}{3 c^2 \pi \left (\pi +c^2 \pi x^2\right )^{3/2}}-\frac {5 x^3 \left (a+b \sinh ^{-1}(c x)\right )}{3 c^4 \pi ^2 \sqrt {\pi +c^2 \pi x^2}}+\frac {5 x \sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 c^6 \pi ^3}-\frac {5 \int \frac {a+b \sinh ^{-1}(c x)}{\sqrt {\pi +c^2 \pi x^2}} \, dx}{2 c^6 \pi ^2}-\frac {\left (5 b \sqrt {1+c^2 x^2}\right ) \int x \, dx}{2 c^5 \pi ^2 \sqrt {\pi +c^2 \pi x^2}}+\frac {\left (5 b \sqrt {1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {x}{1+c^2 x} \, dx,x,x^2\right )}{6 c^3 \pi ^2 \sqrt {\pi +c^2 \pi x^2}}+\frac {\left (b \sqrt {1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{c^4}+\frac {1}{c^4 \left (1+c^2 x\right )^2}-\frac {2}{c^4 \left (1+c^2 x\right )}\right ) \, dx,x,x^2\right )}{6 c \pi ^2 \sqrt {\pi +c^2 \pi x^2}}\\ &=-\frac {b}{6 c^7 \pi ^2 \sqrt {1+c^2 x^2} \sqrt {\pi +c^2 \pi x^2}}-\frac {13 b x^2 \sqrt {1+c^2 x^2}}{12 c^5 \pi ^2 \sqrt {\pi +c^2 \pi x^2}}-\frac {x^5 \left (a+b \sinh ^{-1}(c x)\right )}{3 c^2 \pi \left (\pi +c^2 \pi x^2\right )^{3/2}}-\frac {5 x^3 \left (a+b \sinh ^{-1}(c x)\right )}{3 c^4 \pi ^2 \sqrt {\pi +c^2 \pi x^2}}+\frac {5 x \sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 c^6 \pi ^3}-\frac {5 \left (a+b \sinh ^{-1}(c x)\right )^2}{4 b c^7 \pi ^{5/2}}-\frac {b \sqrt {1+c^2 x^2} \log \left (1+c^2 x^2\right )}{3 c^7 \pi ^2 \sqrt {\pi +c^2 \pi x^2}}+\frac {\left (5 b \sqrt {1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{c^2}-\frac {1}{c^2 \left (1+c^2 x\right )}\right ) \, dx,x,x^2\right )}{6 c^3 \pi ^2 \sqrt {\pi +c^2 \pi x^2}}\\ &=-\frac {b}{6 c^7 \pi ^2 \sqrt {1+c^2 x^2} \sqrt {\pi +c^2 \pi x^2}}-\frac {b x^2 \sqrt {1+c^2 x^2}}{4 c^5 \pi ^2 \sqrt {\pi +c^2 \pi x^2}}-\frac {x^5 \left (a+b \sinh ^{-1}(c x)\right )}{3 c^2 \pi \left (\pi +c^2 \pi x^2\right )^{3/2}}-\frac {5 x^3 \left (a+b \sinh ^{-1}(c x)\right )}{3 c^4 \pi ^2 \sqrt {\pi +c^2 \pi x^2}}+\frac {5 x \sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 c^6 \pi ^3}-\frac {5 \left (a+b \sinh ^{-1}(c x)\right )^2}{4 b c^7 \pi ^{5/2}}-\frac {7 b \sqrt {1+c^2 x^2} \log \left (1+c^2 x^2\right )}{6 c^7 \pi ^2 \sqrt {\pi +c^2 \pi x^2}}\\ \end {align*}
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Mathematica [A] time = 0.47, size = 202, normalized size = 1.05 \[ \frac {4 \sinh ^{-1}(c x) \left (b c x \left (3 c^4 x^4+20 c^2 x^2+15\right )-15 a \left (c^2 x^2+1\right )^{3/2}\right )+12 a c^5 x^5+80 a c^3 x^3+60 a c x-9 b c^2 x^2 \sqrt {c^2 x^2+1}-7 b \sqrt {c^2 x^2+1}-28 b \left (c^2 x^2+1\right )^{3/2} \log \left (c^2 x^2+1\right )-30 b \left (c^2 x^2+1\right )^{3/2} \sinh ^{-1}(c x)^2-6 b c^4 x^4 \sqrt {c^2 x^2+1}}{24 \pi ^{5/2} c^7 \left (c^2 x^2+1\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.55, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {\pi + \pi c^{2} x^{2}} {\left (b x^{6} \operatorname {arsinh}\left (c x\right ) + a x^{6}\right )}}{\pi ^{3} c^{6} x^{6} + 3 \, \pi ^{3} c^{4} x^{4} + 3 \, \pi ^{3} c^{2} x^{2} + \pi ^{3}}, x\right ) \]
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 [B] time = 0.51, size = 970, normalized size = 5.05 \[ \text {result too large to display} \]
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 \[ \frac {1}{6} \, a {\left (\frac {3 \, x^{5}}{\pi {\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {3}{2}} c^{2}} + \frac {5 \, x {\left (\frac {3 \, x^{2}}{\pi {\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {3}{2}} c^{2}} + \frac {2}{\pi {\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {3}{2}} c^{4}}\right )}}{c^{2}} + \frac {5 \, x}{\pi ^{2} \sqrt {\pi + \pi c^{2} x^{2}} c^{6}} - \frac {15 \, \operatorname {arsinh}\left (c x\right )}{\pi ^{\frac {5}{2}} c^{7}}\right )} + b \int \frac {x^{6} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )}{{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {x^6\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}{{\left (\Pi \,c^2\,x^2+\Pi \right )}^{5/2}} \,d x \]
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 \[ \frac {\int \frac {a x^{6}}{c^{4} x^{4} \sqrt {c^{2} x^{2} + 1} + 2 c^{2} x^{2} \sqrt {c^{2} x^{2} + 1} + \sqrt {c^{2} x^{2} + 1}}\, dx + \int \frac {b x^{6} \operatorname {asinh}{\left (c x \right )}}{c^{4} x^{4} \sqrt {c^{2} x^{2} + 1} + 2 c^{2} x^{2} \sqrt {c^{2} x^{2} + 1} + \sqrt {c^{2} x^{2} + 1}}\, dx}{\pi ^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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