Optimal. Leaf size=207 \[ \frac {(115 x+114) \left (3 x^2+5 x+2\right )^{5/2}}{99 (2 x+3)^{11/2}}+\frac {(18699 x+24161) \left (3 x^2+5 x+2\right )^{3/2}}{34650 (2 x+3)^{7/2}}+\frac {(948443 x+1301762) \sqrt {3 x^2+5 x+2}}{346500 (2 x+3)^{3/2}}+\frac {198109 \sqrt {-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{46200 \sqrt {3} \sqrt {3 x^2+5 x+2}}-\frac {107857 \sqrt {-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{33000 \sqrt {3} \sqrt {3 x^2+5 x+2}} \]
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Rubi [A] time = 0.13, antiderivative size = 207, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {810, 843, 718, 424, 419} \[ \frac {(115 x+114) \left (3 x^2+5 x+2\right )^{5/2}}{99 (2 x+3)^{11/2}}+\frac {(18699 x+24161) \left (3 x^2+5 x+2\right )^{3/2}}{34650 (2 x+3)^{7/2}}+\frac {(948443 x+1301762) \sqrt {3 x^2+5 x+2}}{346500 (2 x+3)^{3/2}}+\frac {198109 \sqrt {-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{46200 \sqrt {3} \sqrt {3 x^2+5 x+2}}-\frac {107857 \sqrt {-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{33000 \sqrt {3} \sqrt {3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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Rule 419
Rule 424
Rule 718
Rule 810
Rule 843
Rubi steps
\begin {align*} \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^{13/2}} \, dx &=\frac {(114+115 x) \left (2+5 x+3 x^2\right )^{5/2}}{99 (3+2 x)^{11/2}}-\frac {1}{198} \int \frac {(326+321 x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^{9/2}} \, dx\\ &=\frac {(24161+18699 x) \left (2+5 x+3 x^2\right )^{3/2}}{34650 (3+2 x)^{7/2}}+\frac {(114+115 x) \left (2+5 x+3 x^2\right )^{5/2}}{99 (3+2 x)^{11/2}}+\frac {\int \frac {(-31975-37437 x) \sqrt {2+5 x+3 x^2}}{(3+2 x)^{5/2}} \, dx}{23100}\\ &=\frac {(1301762+948443 x) \sqrt {2+5 x+3 x^2}}{346500 (3+2 x)^{3/2}}+\frac {(24161+18699 x) \left (2+5 x+3 x^2\right )^{3/2}}{34650 (3+2 x)^{7/2}}+\frac {(114+115 x) \left (2+5 x+3 x^2\right )^{5/2}}{99 (3+2 x)^{11/2}}-\frac {\int \frac {1911678+2264997 x}{\sqrt {3+2 x} \sqrt {2+5 x+3 x^2}} \, dx}{693000}\\ &=\frac {(1301762+948443 x) \sqrt {2+5 x+3 x^2}}{346500 (3+2 x)^{3/2}}+\frac {(24161+18699 x) \left (2+5 x+3 x^2\right )^{3/2}}{34650 (3+2 x)^{7/2}}+\frac {(114+115 x) \left (2+5 x+3 x^2\right )^{5/2}}{99 (3+2 x)^{11/2}}-\frac {107857 \int \frac {\sqrt {3+2 x}}{\sqrt {2+5 x+3 x^2}} \, dx}{66000}+\frac {198109 \int \frac {1}{\sqrt {3+2 x} \sqrt {2+5 x+3 x^2}} \, dx}{92400}\\ &=\frac {(1301762+948443 x) \sqrt {2+5 x+3 x^2}}{346500 (3+2 x)^{3/2}}+\frac {(24161+18699 x) \left (2+5 x+3 x^2\right )^{3/2}}{34650 (3+2 x)^{7/2}}+\frac {(114+115 x) \left (2+5 x+3 x^2\right )^{5/2}}{99 (3+2 x)^{11/2}}-\frac {\left (107857 \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 )}{33000 \sqrt {3} \sqrt {2+5 x+3 x^2}}+\frac {\left (198109 \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 )}{46200 \sqrt {3} \sqrt {2+5 x+3 x^2}}\\ &=\frac {(1301762+948443 x) \sqrt {2+5 x+3 x^2}}{346500 (3+2 x)^{3/2}}+\frac {(24161+18699 x) \left (2+5 x+3 x^2\right )^{3/2}}{34650 (3+2 x)^{7/2}}+\frac {(114+115 x) \left (2+5 x+3 x^2\right )^{5/2}}{99 (3+2 x)^{11/2}}-\frac {107857 \sqrt {-2-5 x-3 x^2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{33000 \sqrt {3} \sqrt {2+5 x+3 x^2}}+\frac {198109 \sqrt {-2-5 x-3 x^2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{46200 \sqrt {3} \sqrt {2+5 x+3 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.45, size = 227, normalized size = 1.10 \[ -\frac {4 (2 x+3)^5 \left (1509998 \left (3 x^2+5 x+2\right )-160672 \sqrt {5} \sqrt {\frac {x+1}{2 x+3}} \sqrt {\frac {3 x+2}{2 x+3}} (2 x+3)^{3/2} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {2 x+3}}\right )|\frac {3}{5}\right )+754999 \sqrt {5} \sqrt {\frac {x+1}{2 x+3}} \sqrt {\frac {3 x+2}{2 x+3}} (2 x+3)^{3/2} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {2 x+3}}\right )|\frac {3}{5}\right )\right )-8 \left (3 x^2+5 x+2\right ) \left (21041468 x^5+140915480 x^4+387989550 x^3+544712540 x^2+387631385 x+111387702\right )}{2772000 (2 x+3)^{11/2} \sqrt {3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (9 \, x^{5} - 15 \, x^{4} - 113 \, x^{3} - 165 \, x^{2} - 96 \, x - 20\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3}}{128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187}, x\right ) \]
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 \[ \int -\frac {{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} {\left (x - 5\right )}}{{\left (2 \, x + 3\right )}^{\frac {13}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 575, normalized size = 2.78 \[ \frac {1262488080 x^{7}+10559075600 x^{6}+24159968 \sqrt {15}\, \sqrt {2 x +3}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, x^{5} \EllipticE \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+7537472 \sqrt {15}\, \sqrt {2 x +3}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, x^{5} \EllipticF \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+38212579720 x^{5}+181199760 \sqrt {15}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, \sqrt {2 x +3}\, x^{4} \EllipticE \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+56531040 \sqrt {15}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, \sqrt {2 x +3}\, x^{4} \EllipticF \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+77118326600 x^{4}+543599280 \sqrt {15}\, \sqrt {2 x +3}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, x^{3} \EllipticE \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+169593120 \sqrt {15}\, \sqrt {2 x +3}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, x^{3} \EllipticF \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+93248719100 x^{3}+815398920 \sqrt {15}\, \sqrt {2 x +3}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, x^{2} \EllipticE \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+254389680 \sqrt {15}\, \sqrt {2 x +3}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, x^{2} \EllipticF \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+67234902220 x^{2}+611549190 \sqrt {15}\, \sqrt {2 x +3}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, x \EllipticE \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+190792260 \sqrt {15}\, \sqrt {2 x +3}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, x \EllipticF \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+26644025600 x +183464757 \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 )+57237678 \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 )+4455508080}{6930000 \sqrt {3 x^{2}+5 x +2}\, \left (2 x +3\right )^{\frac {11}{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 \[ -\int \frac {{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} {\left (x - 5\right )}}{{\left (2 \, x + 3\right )}^{\frac {13}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ -\int \frac {\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{5/2}}{{\left (2\,x+3\right )}^{13/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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