Optimal. Leaf size=156 \[ \frac {2657 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{15125 \sqrt {33}}+\frac {7 (3 x+2)^{5/2}}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}-\frac {107 \sqrt {1-2 x} (3 x+2)^{3/2}}{1815 (5 x+3)^{3/2}}-\frac {4289 \sqrt {1-2 x} \sqrt {3 x+2}}{99825 \sqrt {5 x+3}}+\frac {118898 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{15125 \sqrt {33}} \]
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Rubi [A] time = 0.05, antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {98, 150, 158, 113, 119} \[ \frac {7 (3 x+2)^{5/2}}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}-\frac {107 \sqrt {1-2 x} (3 x+2)^{3/2}}{1815 (5 x+3)^{3/2}}-\frac {4289 \sqrt {1-2 x} \sqrt {3 x+2}}{99825 \sqrt {5 x+3}}+\frac {2657 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{15125 \sqrt {33}}+\frac {118898 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{15125 \sqrt {33}} \]
Antiderivative was successfully verified.
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Rule 98
Rule 113
Rule 119
Rule 150
Rule 158
Rubi steps
\begin {align*} \int \frac {(2+3 x)^{7/2}}{(1-2 x)^{3/2} (3+5 x)^{5/2}} \, dx &=\frac {7 (2+3 x)^{5/2}}{11 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {1}{11} \int \frac {(2+3 x)^{3/2} \left (\frac {101}{2}+102 x\right )}{\sqrt {1-2 x} (3+5 x)^{5/2}} \, dx\\ &=-\frac {107 \sqrt {1-2 x} (2+3 x)^{3/2}}{1815 (3+5 x)^{3/2}}+\frac {7 (2+3 x)^{5/2}}{11 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {2 \int \frac {\sqrt {2+3 x} \left (\frac {11645}{4}+\frac {10419 x}{2}\right )}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx}{1815}\\ &=-\frac {107 \sqrt {1-2 x} (2+3 x)^{3/2}}{1815 (3+5 x)^{3/2}}+\frac {7 (2+3 x)^{5/2}}{11 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {4289 \sqrt {1-2 x} \sqrt {2+3 x}}{99825 \sqrt {3+5 x}}-\frac {4 \int \frac {\frac {445569}{8}+\frac {178347 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{99825}\\ &=-\frac {107 \sqrt {1-2 x} (2+3 x)^{3/2}}{1815 (3+5 x)^{3/2}}+\frac {7 (2+3 x)^{5/2}}{11 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {4289 \sqrt {1-2 x} \sqrt {2+3 x}}{99825 \sqrt {3+5 x}}-\frac {2657 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{30250}-\frac {118898 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{166375}\\ &=-\frac {107 \sqrt {1-2 x} (2+3 x)^{3/2}}{1815 (3+5 x)^{3/2}}+\frac {7 (2+3 x)^{5/2}}{11 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {4289 \sqrt {1-2 x} \sqrt {2+3 x}}{99825 \sqrt {3+5 x}}+\frac {118898 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{15125 \sqrt {33}}+\frac {2657 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{15125 \sqrt {33}}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 102, normalized size = 0.65 \[ \frac {150115 \sqrt {2} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )+\frac {10 \sqrt {3 x+2} \left (649925 x^2+772474 x+229463\right )}{\sqrt {1-2 x} (5 x+3)^{3/2}}-237796 \sqrt {2} E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )}{998250} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.69, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{500 \, x^{5} + 400 \, x^{4} - 235 \, x^{3} - 207 \, x^{2} + 27 \, x + 27}, x\right ) \]
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 \[ \int \frac {{\left (3 \, x + 2\right )}^{\frac {7}{2}}}{{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.03, size = 219, normalized size = 1.40 \[ -\frac {\sqrt {3 x +2}\, \sqrt {-2 x +1}\, \left (19497750 x^{3}+36172720 x^{2}-1188980 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+750575 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+22333370 x -713388 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+450345 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+4589260\right )}{998250 \left (5 x +3\right )^{\frac {3}{2}} \left (6 x^{2}+x -2\right )} \]
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 \[ \int \frac {{\left (3 \, x + 2\right )}^{\frac {7}{2}}}{{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (3\,x+2\right )}^{7/2}}{{\left (1-2\,x\right )}^{3/2}\,{\left (5\,x+3\right )}^{5/2}} \,d x \]
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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