Optimal. Leaf size=93 \[ -\frac {b \pi ^{5/2} x}{7 c}-\frac {1}{7} b c \pi ^{5/2} x^3-\frac {3}{35} b c^3 \pi ^{5/2} x^5-\frac {1}{49} b c^5 \pi ^{5/2} x^7+\frac {\left (\pi +c^2 \pi x^2\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{7 c^2 \pi } \]
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Rubi [A]
time = 0.06, antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {5798, 200}
\begin {gather*} \frac {\left (\pi c^2 x^2+\pi \right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{7 \pi c^2}-\frac {1}{49} \pi ^{5/2} b c^5 x^7-\frac {3}{35} \pi ^{5/2} b c^3 x^5-\frac {1}{7} \pi ^{5/2} b c x^3-\frac {\pi ^{5/2} b x}{7 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 200
Rule 5798
Rubi steps
\begin {align*} \int x \left (\pi +c^2 \pi x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx &=\frac {\left (\pi +c^2 \pi x^2\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{7 c^2 \pi }-\frac {\left (b \pi ^2 \sqrt {\pi +c^2 \pi x^2}\right ) \int \left (1+c^2 x^2\right )^3 \, dx}{7 c \sqrt {1+c^2 x^2}}\\ &=\frac {\left (\pi +c^2 \pi x^2\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{7 c^2 \pi }-\frac {\left (b \pi ^2 \sqrt {\pi +c^2 \pi x^2}\right ) \int \left (1+3 c^2 x^2+3 c^4 x^4+c^6 x^6\right ) \, dx}{7 c \sqrt {1+c^2 x^2}}\\ &=-\frac {b \pi ^2 x \sqrt {\pi +c^2 \pi x^2}}{7 c \sqrt {1+c^2 x^2}}-\frac {b c \pi ^2 x^3 \sqrt {\pi +c^2 \pi x^2}}{7 \sqrt {1+c^2 x^2}}-\frac {3 b c^3 \pi ^2 x^5 \sqrt {\pi +c^2 \pi x^2}}{35 \sqrt {1+c^2 x^2}}-\frac {b c^5 \pi ^2 x^7 \sqrt {\pi +c^2 \pi x^2}}{49 \sqrt {1+c^2 x^2}}+\frac {\left (\pi +c^2 \pi x^2\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{7 c^2 \pi }\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 80, normalized size = 0.86 \begin {gather*} \frac {\pi ^{5/2} \left (35 a \left (1+c^2 x^2\right )^{7/2}-b c x \left (35+35 c^2 x^2+21 c^4 x^4+5 c^6 x^6\right )+35 b \left (1+c^2 x^2\right )^{7/2} \sinh ^{-1}(c x)\right )}{245 c^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int x \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {5}{2}} \left (a +b \arcsinh \left (c x \right )\right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 96, normalized size = 1.03 \begin {gather*} \frac {{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {7}{2}} b \operatorname {arsinh}\left (c x\right )}{7 \, \pi c^{2}} + \frac {{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {7}{2}} a}{7 \, \pi c^{2}} - \frac {{\left (5 \, \pi ^{\frac {7}{2}} c^{6} x^{7} + 21 \, \pi ^{\frac {7}{2}} c^{4} x^{5} + 35 \, \pi ^{\frac {7}{2}} c^{2} x^{3} + 35 \, \pi ^{\frac {7}{2}} x\right )} b}{245 \, \pi c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 225 vs.
\(2 (73) = 146\).
time = 0.37, size = 225, normalized size = 2.42 \begin {gather*} \frac {35 \, \sqrt {\pi + \pi c^{2} x^{2}} {\left (\pi ^{2} b c^{8} x^{8} + 4 \, \pi ^{2} b c^{6} x^{6} + 6 \, \pi ^{2} b c^{4} x^{4} + 4 \, \pi ^{2} b c^{2} x^{2} + \pi ^{2} b\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + \sqrt {\pi + \pi c^{2} x^{2}} {\left (35 \, \pi ^{2} a c^{8} x^{8} + 140 \, \pi ^{2} a c^{6} x^{6} + 210 \, \pi ^{2} a c^{4} x^{4} + 140 \, \pi ^{2} a c^{2} x^{2} + 35 \, \pi ^{2} a - {\left (5 \, \pi ^{2} b c^{7} x^{7} + 21 \, \pi ^{2} b c^{5} x^{5} + 35 \, \pi ^{2} b c^{3} x^{3} + 35 \, \pi ^{2} b c x\right )} \sqrt {c^{2} x^{2} + 1}\right )}}{245 \, {\left (c^{4} x^{2} + c^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 299 vs.
\(2 (85) = 170\).
time = 37.57, size = 299, normalized size = 3.22 \begin {gather*} \begin {cases} \frac {\pi ^{\frac {5}{2}} a c^{4} x^{6} \sqrt {c^{2} x^{2} + 1}}{7} + \frac {3 \pi ^{\frac {5}{2}} a c^{2} x^{4} \sqrt {c^{2} x^{2} + 1}}{7} + \frac {3 \pi ^{\frac {5}{2}} a x^{2} \sqrt {c^{2} x^{2} + 1}}{7} + \frac {\pi ^{\frac {5}{2}} a \sqrt {c^{2} x^{2} + 1}}{7 c^{2}} - \frac {\pi ^{\frac {5}{2}} b c^{5} x^{7}}{49} + \frac {\pi ^{\frac {5}{2}} b c^{4} x^{6} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{7} - \frac {3 \pi ^{\frac {5}{2}} b c^{3} x^{5}}{35} + \frac {3 \pi ^{\frac {5}{2}} b c^{2} x^{4} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{7} - \frac {\pi ^{\frac {5}{2}} b c x^{3}}{7} + \frac {3 \pi ^{\frac {5}{2}} b x^{2} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{7} - \frac {\pi ^{\frac {5}{2}} b x}{7 c} + \frac {\pi ^{\frac {5}{2}} b \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{7 c^{2}} & \text {for}\: c \neq 0 \\\frac {\pi ^{\frac {5}{2}} a x^{2}}{2} & \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 x\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (\Pi \,c^2\,x^2+\Pi \right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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