Optimal. Leaf size=229 \[ -\frac {2}{39} (2 x+3)^{3/2} \left (3 x^2+5 x+2\right )^{5/2}+\frac {886 \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{5/2}}{1287}+\frac {\sqrt {2 x+3} (50771 x+43822) \left (3 x^2+5 x+2\right )^{3/2}}{27027}-\frac {\sqrt {2 x+3} (783711 x+486863) \sqrt {3 x^2+5 x+2}}{2432430}+\frac {332459 \sqrt {-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{972972 \sqrt {3} \sqrt {3 x^2+5 x+2}}-\frac {152657 \sqrt {-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{694980 \sqrt {3} \sqrt {3 x^2+5 x+2}} \]
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Rubi [A] time = 0.16, antiderivative size = 229, normalized size of antiderivative = 1.00, number of steps used = 9, 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} \[ -\frac {2}{39} (2 x+3)^{3/2} \left (3 x^2+5 x+2\right )^{5/2}+\frac {886 \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{5/2}}{1287}+\frac {\sqrt {2 x+3} (50771 x+43822) \left (3 x^2+5 x+2\right )^{3/2}}{27027}-\frac {\sqrt {2 x+3} (783711 x+486863) \sqrt {3 x^2+5 x+2}}{2432430}+\frac {332459 \sqrt {-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{972972 \sqrt {3} \sqrt {3 x^2+5 x+2}}-\frac {152657 \sqrt {-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{694980 \sqrt {3} \sqrt {3 x^2+5 x+2}} \]
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} \left (2+5 x+3 x^2\right )^{3/2} \, dx &=-\frac {2}{39} (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{5/2}+\frac {2}{39} \int \sqrt {3+2 x} \left (336+\frac {443 x}{2}\right ) \left (2+5 x+3 x^2\right )^{3/2} \, dx\\ &=\frac {886 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{5/2}}{1287}-\frac {2}{39} (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{5/2}+\frac {4 \int \frac {\left (\frac {31531}{4}+\frac {21759 x}{4}\right ) \left (2+5 x+3 x^2\right )^{3/2}}{\sqrt {3+2 x}} \, dx}{1287}\\ &=\frac {\sqrt {3+2 x} (43822+50771 x) \left (2+5 x+3 x^2\right )^{3/2}}{27027}+\frac {886 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{5/2}}{1287}-\frac {2}{39} (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{5/2}-\frac {2 \int \frac {\left (\frac {161889}{2}+\frac {261237 x}{4}\right ) \sqrt {2+5 x+3 x^2}}{\sqrt {3+2 x}} \, dx}{81081}\\ &=-\frac {\sqrt {3+2 x} (486863+783711 x) \sqrt {2+5 x+3 x^2}}{2432430}+\frac {\sqrt {3+2 x} (43822+50771 x) \left (2+5 x+3 x^2\right )^{3/2}}{27027}+\frac {886 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{5/2}}{1287}-\frac {2}{39} (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{5/2}+\frac {\int \frac {-\frac {2315253}{4}-\frac {3205797 x}{4}}{\sqrt {3+2 x} \sqrt {2+5 x+3 x^2}} \, dx}{3648645}\\ &=-\frac {\sqrt {3+2 x} (486863+783711 x) \sqrt {2+5 x+3 x^2}}{2432430}+\frac {\sqrt {3+2 x} (43822+50771 x) \left (2+5 x+3 x^2\right )^{3/2}}{27027}+\frac {886 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{5/2}}{1287}-\frac {2}{39} (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{5/2}-\frac {152657 \int \frac {\sqrt {3+2 x}}{\sqrt {2+5 x+3 x^2}} \, dx}{1389960}+\frac {332459 \int \frac {1}{\sqrt {3+2 x} \sqrt {2+5 x+3 x^2}} \, dx}{1945944}\\ &=-\frac {\sqrt {3+2 x} (486863+783711 x) \sqrt {2+5 x+3 x^2}}{2432430}+\frac {\sqrt {3+2 x} (43822+50771 x) \left (2+5 x+3 x^2\right )^{3/2}}{27027}+\frac {886 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{5/2}}{1287}-\frac {2}{39} (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{5/2}-\frac {\left (152657 \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 )}{694980 \sqrt {3} \sqrt {2+5 x+3 x^2}}+\frac {\left (332459 \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 )}{972972 \sqrt {3} \sqrt {2+5 x+3 x^2}}\\ &=-\frac {\sqrt {3+2 x} (486863+783711 x) \sqrt {2+5 x+3 x^2}}{2432430}+\frac {\sqrt {3+2 x} (43822+50771 x) \left (2+5 x+3 x^2\right )^{3/2}}{27027}+\frac {886 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{5/2}}{1287}-\frac {2}{39} (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{5/2}-\frac {152657 \sqrt {-2-5 x-3 x^2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{694980 \sqrt {3} \sqrt {2+5 x+3 x^2}}+\frac {332459 \sqrt {-2-5 x-3 x^2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{972972 \sqrt {3} \sqrt {2+5 x+3 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.33, size = 213, normalized size = 0.93 \[ -\frac {2 \left (40415760 x^8+52050600 x^7-895236300 x^6-4079217510 x^5-7944858702 x^4-8470029969 x^3-5141306625 x^2-1668494576 x-224705588\right ) \sqrt {2 x+3}-71222 \sqrt {5} \sqrt {\frac {x+1}{2 x+3}} \sqrt {\frac {3 x+2}{2 x+3}} (2 x+3)^2 F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {2 x+3}}\right )|\frac {3}{5}\right )+1068599 \sqrt {5} \sqrt {\frac {x+1}{2 x+3}} \sqrt {\frac {3 x+2}{2 x+3}} (2 x+3)^2 E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {2 x+3}}\right )|\frac {3}{5}\right )}{14594580 (2 x+3) \sqrt {3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.70, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (6 \, x^{4} - 11 \, x^{3} - 76 \, x^{2} - 89 \, x - 30\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3}, x\right ) \]
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 \[ \int -{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} {\left (2 \, x + 3\right )}^{\frac {3}{2}} {\left (x - 5\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 161, normalized size = 0.70 \[ \frac {\sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}\, \left (-808315200 x^{8}-1041012000 x^{7}+17904726000 x^{6}+81584350200 x^{5}+158897174040 x^{4}+169400599380 x^{3}+102890248440 x^{2}+33476751420 x +1068599 \sqrt {2 x +3}\, \sqrt {15}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, \EllipticE \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+593696 \sqrt {2 x +3}\, \sqrt {15}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, \EllipticF \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+4536855720\right )}{875674800 x^{3}+2772970200 x^{2}+2772970200 x +875674800} \]
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 \[ -\int {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} {\left (2 \, x + 3\right )}^{\frac {3}{2}} {\left (x - 5\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ -\int {\left (2\,x+3\right )}^{3/2}\,\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{3/2} \,d x \]
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 \[ - \int \left (- 30 \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 89 x \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 76 x^{2} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 11 x^{3} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int 6 x^{4} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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