Optimal. Leaf size=175 \[ \frac {(337 x+408) \left (3 x^2+5 x+2\right )^{3/2}}{75 (2 x+3)^{5/2}}-\frac {(181 x+614) \sqrt {3 x^2+5 x+2}}{50 \sqrt {2 x+3}}-\frac {243 \sqrt {3} \sqrt {-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{20 \sqrt {3 x^2+5 x+2}}+\frac {2779 \sqrt {-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{100 \sqrt {3} \sqrt {3 x^2+5 x+2}} \]
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Rubi [A] time = 0.10, antiderivative size = 175, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {810, 812, 843, 718, 424, 419} \[ \frac {(337 x+408) \left (3 x^2+5 x+2\right )^{3/2}}{75 (2 x+3)^{5/2}}-\frac {(181 x+614) \sqrt {3 x^2+5 x+2}}{50 \sqrt {2 x+3}}-\frac {243 \sqrt {3} \sqrt {-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{20 \sqrt {3 x^2+5 x+2}}+\frac {2779 \sqrt {-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{100 \sqrt {3} \sqrt {3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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Rule 419
Rule 424
Rule 718
Rule 810
Rule 812
Rule 843
Rubi steps
\begin {align*} \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^{7/2}} \, dx &=\frac {(408+337 x) \left (2+5 x+3 x^2\right )^{3/2}}{75 (3+2 x)^{5/2}}-\frac {1}{50} \int \frac {(472+543 x) \sqrt {2+5 x+3 x^2}}{(3+2 x)^{3/2}} \, dx\\ &=-\frac {(614+181 x) \sqrt {2+5 x+3 x^2}}{50 \sqrt {3+2 x}}+\frac {(408+337 x) \left (2+5 x+3 x^2\right )^{3/2}}{75 (3+2 x)^{5/2}}+\frac {1}{300} \int \frac {7038+8337 x}{\sqrt {3+2 x} \sqrt {2+5 x+3 x^2}} \, dx\\ &=-\frac {(614+181 x) \sqrt {2+5 x+3 x^2}}{50 \sqrt {3+2 x}}+\frac {(408+337 x) \left (2+5 x+3 x^2\right )^{3/2}}{75 (3+2 x)^{5/2}}+\frac {2779}{200} \int \frac {\sqrt {3+2 x}}{\sqrt {2+5 x+3 x^2}} \, dx-\frac {729}{40} \int \frac {1}{\sqrt {3+2 x} \sqrt {2+5 x+3 x^2}} \, dx\\ &=-\frac {(614+181 x) \sqrt {2+5 x+3 x^2}}{50 \sqrt {3+2 x}}+\frac {(408+337 x) \left (2+5 x+3 x^2\right )^{3/2}}{75 (3+2 x)^{5/2}}+\frac {\left (2779 \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 )}{100 \sqrt {3} \sqrt {2+5 x+3 x^2}}-\frac {\left (243 \sqrt {3} \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 )}{20 \sqrt {2+5 x+3 x^2}}\\ &=-\frac {(614+181 x) \sqrt {2+5 x+3 x^2}}{50 \sqrt {3+2 x}}+\frac {(408+337 x) \left (2+5 x+3 x^2\right )^{3/2}}{75 (3+2 x)^{5/2}}+\frac {2779 \sqrt {-2-5 x-3 x^2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{100 \sqrt {3} \sqrt {2+5 x+3 x^2}}-\frac {243 \sqrt {3} \sqrt {-2-5 x-3 x^2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{20 \sqrt {2+5 x+3 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.34, size = 192, normalized size = 1.10 \[ \frac {-900 x^5+16800 x^4+100610 x^3+190440 x^2+147790 x-592 \sqrt {5} \sqrt {\frac {x+1}{2 x+3}} (2 x+3)^{7/2} \sqrt {\frac {3 x+2}{2 x+3}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {2 x+3}}\right )|\frac {3}{5}\right )+2779 \sqrt {5} \sqrt {\frac {x+1}{2 x+3}} (2 x+3)^{7/2} \sqrt {\frac {3 x+2}{2 x+3}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {2 x+3}}\right )|\frac {3}{5}\right )+40260}{300 (2 x+3)^{5/2} \sqrt {3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (3 \, x^{3} - 10 \, x^{2} - 23 \, x - 10\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3}}{16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81}, x\right ) \]
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 \[ \int -\frac {{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} {\left (x - 5\right )}}{{\left (2 \, x + 3\right )}^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 301, normalized size = 1.72 \[ -\frac {9000 x^{5}+498960 x^{4}+2106380 x^{3}+11116 \sqrt {15}\, \sqrt {2 x +3}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, x^{2} \EllipticE \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+3464 \sqrt {15}\, \sqrt {2 x +3}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, x^{2} \EllipticF \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+3375700 x^{2}+33348 \sqrt {15}\, \sqrt {2 x +3}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, x \EllipticE \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+10392 \sqrt {15}\, \sqrt {2 x +3}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, x \EllipticF \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+2357120 x +25011 \sqrt {2 x +3}\, \sqrt {15}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, \EllipticE \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+7794 \sqrt {2 x +3}\, \sqrt {15}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, \EllipticF \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+597840}{3000 \sqrt {3 x^{2}+5 x +2}\, \left (2 x +3\right )^{\frac {5}{2}}} \]
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 \[ -\int \frac {{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} {\left (x - 5\right )}}{{\left (2 \, x + 3\right )}^{\frac {7}{2}}}\,{d x} \]
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 \[ -\int \frac {\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{3/2}}{{\left (2\,x+3\right )}^{7/2}} \,d x \]
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 \[ - \int \left (- \frac {10 \sqrt {3 x^{2} + 5 x + 2}}{8 x^{3} \sqrt {2 x + 3} + 36 x^{2} \sqrt {2 x + 3} + 54 x \sqrt {2 x + 3} + 27 \sqrt {2 x + 3}}\right )\, dx - \int \left (- \frac {23 x \sqrt {3 x^{2} + 5 x + 2}}{8 x^{3} \sqrt {2 x + 3} + 36 x^{2} \sqrt {2 x + 3} + 54 x \sqrt {2 x + 3} + 27 \sqrt {2 x + 3}}\right )\, dx - \int \left (- \frac {10 x^{2} \sqrt {3 x^{2} + 5 x + 2}}{8 x^{3} \sqrt {2 x + 3} + 36 x^{2} \sqrt {2 x + 3} + 54 x \sqrt {2 x + 3} + 27 \sqrt {2 x + 3}}\right )\, dx - \int \frac {3 x^{3} \sqrt {3 x^{2} + 5 x + 2}}{8 x^{3} \sqrt {2 x + 3} + 36 x^{2} \sqrt {2 x + 3} + 54 x \sqrt {2 x + 3} + 27 \sqrt {2 x + 3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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