Optimal. Leaf size=88 \[ \frac {1}{12} (4-x) \left (3 x^2+2\right )^{7/2}+\frac {91}{36} x \left (3 x^2+2\right )^{5/2}+\frac {455}{72} x \left (3 x^2+2\right )^{3/2}+\frac {455}{24} x \sqrt {3 x^2+2}+\frac {455 \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{12 \sqrt {3}} \]
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Rubi [A] time = 0.03, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {780, 195, 215} \[ \frac {1}{12} (4-x) \left (3 x^2+2\right )^{7/2}+\frac {91}{36} x \left (3 x^2+2\right )^{5/2}+\frac {455}{72} x \left (3 x^2+2\right )^{3/2}+\frac {455}{24} x \sqrt {3 x^2+2}+\frac {455 \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{12 \sqrt {3}} \]
Antiderivative was successfully verified.
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Rule 195
Rule 215
Rule 780
Rubi steps
\begin {align*} \int (5-x) (3+2 x) \left (2+3 x^2\right )^{5/2} \, dx &=\frac {1}{12} (4-x) \left (2+3 x^2\right )^{7/2}+\frac {91}{6} \int \left (2+3 x^2\right )^{5/2} \, dx\\ &=\frac {91}{36} x \left (2+3 x^2\right )^{5/2}+\frac {1}{12} (4-x) \left (2+3 x^2\right )^{7/2}+\frac {455}{18} \int \left (2+3 x^2\right )^{3/2} \, dx\\ &=\frac {455}{72} x \left (2+3 x^2\right )^{3/2}+\frac {91}{36} x \left (2+3 x^2\right )^{5/2}+\frac {1}{12} (4-x) \left (2+3 x^2\right )^{7/2}+\frac {455}{12} \int \sqrt {2+3 x^2} \, dx\\ &=\frac {455}{24} x \sqrt {2+3 x^2}+\frac {455}{72} x \left (2+3 x^2\right )^{3/2}+\frac {91}{36} x \left (2+3 x^2\right )^{5/2}+\frac {1}{12} (4-x) \left (2+3 x^2\right )^{7/2}+\frac {455}{12} \int \frac {1}{\sqrt {2+3 x^2}} \, dx\\ &=\frac {455}{24} x \sqrt {2+3 x^2}+\frac {455}{72} x \left (2+3 x^2\right )^{3/2}+\frac {91}{36} x \left (2+3 x^2\right )^{5/2}+\frac {1}{12} (4-x) \left (2+3 x^2\right )^{7/2}+\frac {455 \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{12 \sqrt {3}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 70, normalized size = 0.80 \[ \frac {1}{72} \left (910 \sqrt {3} \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )-3 \sqrt {3 x^2+2} \left (54 x^7-216 x^6-438 x^5-432 x^4-1111 x^3-288 x^2-985 x-64\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.55, size = 75, normalized size = 0.85 \[ -\frac {1}{24} \, {\left (54 \, x^{7} - 216 \, x^{6} - 438 \, x^{5} - 432 \, x^{4} - 1111 \, x^{3} - 288 \, x^{2} - 985 \, x - 64\right )} \sqrt {3 \, x^{2} + 2} + \frac {455}{72} \, \sqrt {3} \log \left (-\sqrt {3} \sqrt {3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 63, normalized size = 0.72 \[ -\frac {1}{24} \, {\left ({\left ({\left ({\left (6 \, {\left ({\left (9 \, {\left (x - 4\right )} x - 73\right )} x - 72\right )} x - 1111\right )} x - 288\right )} x - 985\right )} x - 64\right )} \sqrt {3 \, x^{2} + 2} - \frac {455}{36} \, \sqrt {3} \log \left (-\sqrt {3} x + \sqrt {3 \, x^{2} + 2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 73, normalized size = 0.83 \[ -\frac {\left (3 x^{2}+2\right )^{\frac {7}{2}} x}{12}+\frac {91 \left (3 x^{2}+2\right )^{\frac {5}{2}} x}{36}+\frac {455 \left (3 x^{2}+2\right )^{\frac {3}{2}} x}{72}+\frac {455 \sqrt {3 x^{2}+2}\, x}{24}+\frac {455 \sqrt {3}\, \arcsinh \left (\frac {\sqrt {6}\, x}{2}\right )}{36}+\frac {\left (3 x^{2}+2\right )^{\frac {7}{2}}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.16, size = 72, normalized size = 0.82 \[ -\frac {1}{12} \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}} x + \frac {1}{3} \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}} + \frac {91}{36} \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}} x + \frac {455}{72} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} x + \frac {455}{24} \, \sqrt {3 \, x^{2} + 2} x + \frac {455}{36} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{2} \, \sqrt {6} x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.86, size = 60, normalized size = 0.68 \[ \frac {455\,\sqrt {3}\,\mathrm {asinh}\left (\frac {\sqrt {6}\,x}{2}\right )}{36}+\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (-\frac {27\,x^7}{4}+27\,x^6+\frac {219\,x^5}{4}+54\,x^4+\frac {1111\,x^3}{8}+36\,x^2+\frac {985\,x}{8}+8\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 34.01, size = 143, normalized size = 1.62 \[ - \frac {9 x^{7} \sqrt {3 x^{2} + 2}}{4} + 9 x^{6} \sqrt {3 x^{2} + 2} + \frac {73 x^{5} \sqrt {3 x^{2} + 2}}{4} + 18 x^{4} \sqrt {3 x^{2} + 2} + \frac {1111 x^{3} \sqrt {3 x^{2} + 2}}{24} + 12 x^{2} \sqrt {3 x^{2} + 2} + \frac {985 x \sqrt {3 x^{2} + 2}}{24} + \frac {8 \sqrt {3 x^{2} + 2}}{3} + \frac {455 \sqrt {3} \operatorname {asinh}{\left (\frac {\sqrt {6} x}{2} \right )}}{36} \]
Verification of antiderivative is not currently implemented for this CAS.
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