Optimal. Leaf size=197 \[ -\frac {2}{27} (2 x+3)^{3/2} \left (3 x^2+5 x+2\right )^{3/2}+\frac {202}{189} \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{3/2}+\frac {\sqrt {2 x+3} (30033 x+27914) \sqrt {3 x^2+5 x+2}}{8505}+\frac {5773 \sqrt {-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{3402 \sqrt {3} \sqrt {3 x^2+5 x+2}}-\frac {4729 \sqrt {-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{2430 \sqrt {3} \sqrt {3 x^2+5 x+2}} \]
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Rubi [A] time = 0.13, antiderivative size = 197, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {832, 814, 843, 718, 424, 419} \[ -\frac {2}{27} (2 x+3)^{3/2} \left (3 x^2+5 x+2\right )^{3/2}+\frac {202}{189} \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{3/2}+\frac {\sqrt {2 x+3} (30033 x+27914) \sqrt {3 x^2+5 x+2}}{8505}+\frac {5773 \sqrt {-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{3402 \sqrt {3} \sqrt {3 x^2+5 x+2}}-\frac {4729 \sqrt {-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{2430 \sqrt {3} \sqrt {3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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Rule 419
Rule 424
Rule 718
Rule 814
Rule 832
Rule 843
Rubi steps
\begin {align*} \int (5-x) (3+2 x)^{3/2} \sqrt {2+5 x+3 x^2} \, dx &=-\frac {2}{27} (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{3/2}+\frac {2}{27} \int \sqrt {3+2 x} \left (231+\frac {303 x}{2}\right ) \sqrt {2+5 x+3 x^2} \, dx\\ &=\frac {202}{189} \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{3/2}-\frac {2}{27} (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{3/2}+\frac {4}{567} \int \frac {\left (\frac {14259}{4}+\frac {10011 x}{4}\right ) \sqrt {2+5 x+3 x^2}}{\sqrt {3+2 x}} \, dx\\ &=\frac {\sqrt {3+2 x} (27914+30033 x) \sqrt {2+5 x+3 x^2}}{8505}+\frac {202}{189} \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{3/2}-\frac {2}{27} (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{3/2}-\frac {2 \int \frac {\frac {52833}{2}+\frac {99309 x}{4}}{\sqrt {3+2 x} \sqrt {2+5 x+3 x^2}} \, dx}{25515}\\ &=\frac {\sqrt {3+2 x} (27914+30033 x) \sqrt {2+5 x+3 x^2}}{8505}+\frac {202}{189} \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{3/2}-\frac {2}{27} (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{3/2}+\frac {5773 \int \frac {1}{\sqrt {3+2 x} \sqrt {2+5 x+3 x^2}} \, dx}{6804}-\frac {4729 \int \frac {\sqrt {3+2 x}}{\sqrt {2+5 x+3 x^2}} \, dx}{4860}\\ &=\frac {\sqrt {3+2 x} (27914+30033 x) \sqrt {2+5 x+3 x^2}}{8505}+\frac {202}{189} \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{3/2}-\frac {2}{27} (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{3/2}+\frac {\left (5773 \sqrt {-2-5 x-3 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {2 x^2}{3}}} \, dx,x,\frac {\sqrt {6+6 x}}{\sqrt {2}}\right )}{3402 \sqrt {3} \sqrt {2+5 x+3 x^2}}-\frac {\left (4729 \sqrt {-2-5 x-3 x^2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+\frac {2 x^2}{3}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {6+6 x}}{\sqrt {2}}\right )}{2430 \sqrt {3} \sqrt {2+5 x+3 x^2}}\\ &=\frac {\sqrt {3+2 x} (27914+30033 x) \sqrt {2+5 x+3 x^2}}{8505}+\frac {202}{189} \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{3/2}-\frac {2}{27} (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{3/2}-\frac {4729 \sqrt {-2-5 x-3 x^2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{2430 \sqrt {3} \sqrt {2+5 x+3 x^2}}+\frac {5773 \sqrt {-2-5 x-3 x^2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{3402 \sqrt {3} \sqrt {2+5 x+3 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.32, size = 203, normalized size = 1.03 \[ -\frac {2 \left (68040 x^6-59940 x^5-1799874 x^4-5185953 x^3-6208230 x^2-3389617 x-695446\right ) \sqrt {2 x+3}-15784 \sqrt {5} \sqrt {\frac {x+1}{2 x+3}} \sqrt {\frac {3 x+2}{2 x+3}} (2 x+3)^2 F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {2 x+3}}\right )|\frac {3}{5}\right )+33103 \sqrt {5} \sqrt {\frac {x+1}{2 x+3}} \sqrt {\frac {3 x+2}{2 x+3}} (2 x+3)^2 E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {2 x+3}}\right )|\frac {3}{5}\right )}{51030 (2 x+3) \sqrt {3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.68, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\sqrt {3 \, x^{2} + 5 \, x + 2} {\left (2 \, x^{2} - 7 \, x - 15\right )} \sqrt {2 \, x + 3}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int -\sqrt {3 \, x^{2} + 5 \, x + 2} {\left (2 \, x + 3\right )}^{\frac {3}{2}} {\left (x - 5\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 151, normalized size = 0.77 \[ -\frac {\sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}\, \left (1360800 x^{6}-1198800 x^{5}-35997480 x^{4}-103719060 x^{3}-126150780 x^{2}-71102640 x -33103 \sqrt {2 x +3}\, \sqrt {15}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, \EllipticE \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+4238 \sqrt {2 x +3}\, \sqrt {15}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, \EllipticF \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )-15233040\right )}{510300 \left (6 x^{3}+19 x^{2}+19 x +6\right )} \]
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 \[ -\int \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (2 \, x + 3\right )}^{\frac {3}{2}} {\left (x - 5\right )}\,{d x} \]
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 \[ -\int {\left (2\,x+3\right )}^{3/2}\,\left (x-5\right )\,\sqrt {3\,x^2+5\,x+2} \,d x \]
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 \[ - \int \left (- 15 \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 7 x \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int 2 x^{2} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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