Optimal. Leaf size=205 \[ -\frac {(x+73) \left (3 x^2+5 x+2\right )^{5/2}}{11 \sqrt {2 x+3}}+\frac {5 \sqrt {2 x+3} (3031 x+218) \left (3 x^2+5 x+2\right )^{3/2}}{1386}-\frac {(21871-471213 x) \sqrt {2 x+3} \sqrt {3 x^2+5 x+2}}{24948}-\frac {4145485 \sqrt {-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{49896 \sqrt {3} \sqrt {3 x^2+5 x+2}}+\frac {451331 \sqrt {-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{7128 \sqrt {3} \sqrt {3 x^2+5 x+2}} \]
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Rubi [A] time = 0.13, antiderivative size = 205, 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 = {812, 814, 843, 718, 424, 419} \[ -\frac {(x+73) \left (3 x^2+5 x+2\right )^{5/2}}{11 \sqrt {2 x+3}}+\frac {5 \sqrt {2 x+3} (3031 x+218) \left (3 x^2+5 x+2\right )^{3/2}}{1386}-\frac {(21871-471213 x) \sqrt {2 x+3} \sqrt {3 x^2+5 x+2}}{24948}-\frac {4145485 \sqrt {-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{49896 \sqrt {3} \sqrt {3 x^2+5 x+2}}+\frac {451331 \sqrt {-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{7128 \sqrt {3} \sqrt {3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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Rule 419
Rule 424
Rule 718
Rule 812
Rule 814
Rule 843
Rubi steps
\begin {align*} \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^{3/2}} \, dx &=-\frac {(73+x) \left (2+5 x+3 x^2\right )^{5/2}}{11 \sqrt {3+2 x}}-\frac {5}{22} \int \frac {(-361-433 x) \left (2+5 x+3 x^2\right )^{3/2}}{\sqrt {3+2 x}} \, dx\\ &=\frac {5 \sqrt {3+2 x} (218+3031 x) \left (2+5 x+3 x^2\right )^{3/2}}{1386}-\frac {(73+x) \left (2+5 x+3 x^2\right )^{5/2}}{11 \sqrt {3+2 x}}+\frac {5 \int \frac {(43918+52357 x) \sqrt {2+5 x+3 x^2}}{\sqrt {3+2 x}} \, dx}{2772}\\ &=-\frac {(21871-471213 x) \sqrt {3+2 x} \sqrt {2+5 x+3 x^2}}{24948}+\frac {5 \sqrt {3+2 x} (218+3031 x) \left (2+5 x+3 x^2\right )^{3/2}}{1386}-\frac {(73+x) \left (2+5 x+3 x^2\right )^{5/2}}{11 \sqrt {3+2 x}}-\frac {\int \frac {-2666233-3159317 x}{\sqrt {3+2 x} \sqrt {2+5 x+3 x^2}} \, dx}{49896}\\ &=-\frac {(21871-471213 x) \sqrt {3+2 x} \sqrt {2+5 x+3 x^2}}{24948}+\frac {5 \sqrt {3+2 x} (218+3031 x) \left (2+5 x+3 x^2\right )^{3/2}}{1386}-\frac {(73+x) \left (2+5 x+3 x^2\right )^{5/2}}{11 \sqrt {3+2 x}}+\frac {451331 \int \frac {\sqrt {3+2 x}}{\sqrt {2+5 x+3 x^2}} \, dx}{14256}-\frac {4145485 \int \frac {1}{\sqrt {3+2 x} \sqrt {2+5 x+3 x^2}} \, dx}{99792}\\ &=-\frac {(21871-471213 x) \sqrt {3+2 x} \sqrt {2+5 x+3 x^2}}{24948}+\frac {5 \sqrt {3+2 x} (218+3031 x) \left (2+5 x+3 x^2\right )^{3/2}}{1386}-\frac {(73+x) \left (2+5 x+3 x^2\right )^{5/2}}{11 \sqrt {3+2 x}}+\frac {\left (451331 \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 )}{7128 \sqrt {3} \sqrt {2+5 x+3 x^2}}-\frac {\left (4145485 \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 )}{49896 \sqrt {3} \sqrt {2+5 x+3 x^2}}\\ &=-\frac {(21871-471213 x) \sqrt {3+2 x} \sqrt {2+5 x+3 x^2}}{24948}+\frac {5 \sqrt {3+2 x} (218+3031 x) \left (2+5 x+3 x^2\right )^{3/2}}{1386}-\frac {(73+x) \left (2+5 x+3 x^2\right )^{5/2}}{11 \sqrt {3+2 x}}+\frac {451331 \sqrt {-2-5 x-3 x^2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{7128 \sqrt {3} \sqrt {2+5 x+3 x^2}}-\frac {4145485 \sqrt {-2-5 x-3 x^2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{49896 \sqrt {3} \sqrt {2+5 x+3 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.40, size = 205, normalized size = 1.00 \[ \frac {2 \left (-183708 x^7+401436 x^6+3305934 x^5+7163046 x^4+6935769 x^3+6834513 x^2+6998740 x-336013 \sqrt {5} \sqrt {\frac {x+1}{2 x+3}} (2 x+3)^{3/2} \sqrt {\frac {3 x+2}{2 x+3}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {2 x+3}}\right )|\frac {3}{5}\right )+2657740\right )+3159317 \sqrt {5} \sqrt {\frac {x+1}{2 x+3}} \sqrt {\frac {3 x+2}{2 x+3}} (2 x+3)^{3/2} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {2 x+3}}\right )|\frac {3}{5}\right )}{149688 \sqrt {2 x+3} \sqrt {3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.65, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (9 \, x^{5} - 15 \, x^{4} - 113 \, x^{3} - 165 \, x^{2} - 96 \, x - 20\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3}}{4 \, x^{2} + 12 \, x + 9}, x\right ) \]
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 \[ \int -\frac {{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} {\left (x - 5\right )}}{{\left (2 \, x + 3\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 156, normalized size = 0.76 \[ -\frac {\sqrt {3 x^{2}+5 x +2}\, \sqrt {2 x +3}\, \left (3674160 x^{7}-8028720 x^{6}-66118680 x^{5}-143260920 x^{4}-138715380 x^{3}+52868760 x^{2}+175956900 x +3159317 \sqrt {2 x +3}\, \sqrt {15}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, \EllipticE \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+986168 \sqrt {2 x +3}\, \sqrt {15}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, \EllipticF \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+73217880\right )}{1496880 \left (6 x^{3}+19 x^{2}+19 x +6\right )} \]
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 \[ -\int \frac {{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} {\left (x - 5\right )}}{{\left (2 \, x + 3\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ -\int \frac {\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{5/2}}{{\left (2\,x+3\right )}^{3/2}} \,d x \]
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 \[ - \int \left (- \frac {20 \sqrt {3 x^{2} + 5 x + 2}}{2 x \sqrt {2 x + 3} + 3 \sqrt {2 x + 3}}\right )\, dx - \int \left (- \frac {96 x \sqrt {3 x^{2} + 5 x + 2}}{2 x \sqrt {2 x + 3} + 3 \sqrt {2 x + 3}}\right )\, dx - \int \left (- \frac {165 x^{2} \sqrt {3 x^{2} + 5 x + 2}}{2 x \sqrt {2 x + 3} + 3 \sqrt {2 x + 3}}\right )\, dx - \int \left (- \frac {113 x^{3} \sqrt {3 x^{2} + 5 x + 2}}{2 x \sqrt {2 x + 3} + 3 \sqrt {2 x + 3}}\right )\, dx - \int \left (- \frac {15 x^{4} \sqrt {3 x^{2} + 5 x + 2}}{2 x \sqrt {2 x + 3} + 3 \sqrt {2 x + 3}}\right )\, dx - \int \frac {9 x^{5} \sqrt {3 x^{2} + 5 x + 2}}{2 x \sqrt {2 x + 3} + 3 \sqrt {2 x + 3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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