Optimal. Leaf size=249 \[ -\frac {2 (1-2 x)^{5/2} (2+3 x)^{7/2}}{5 \sqrt {3+5 x}}-\frac {703672 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{32484375}+\frac {2020841 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{6496875}+\frac {346636 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}{259875}-\frac {2972 \sqrt {1-2 x} (2+3 x)^{7/2} \sqrt {3+5 x}}{7425}-\frac {48}{275} (1-2 x)^{3/2} (2+3 x)^{7/2} \sqrt {3+5 x}-\frac {264260033 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{29531250 \sqrt {33}}-\frac {7261561 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{14765625 \sqrt {33}} \]
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Rubi [A]
time = 0.07, antiderivative size = 249, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {99, 159, 164,
114, 120} \begin {gather*} -\frac {7261561 F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{14765625 \sqrt {33}}-\frac {264260033 E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{29531250 \sqrt {33}}-\frac {48}{275} (1-2 x)^{3/2} \sqrt {5 x+3} (3 x+2)^{7/2}-\frac {2972 \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{7/2}}{7425}-\frac {2 (1-2 x)^{5/2} (3 x+2)^{7/2}}{5 \sqrt {5 x+3}}+\frac {346636 \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{5/2}}{259875}+\frac {2020841 \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{3/2}}{6496875}-\frac {703672 \sqrt {1-2 x} \sqrt {5 x+3} \sqrt {3 x+2}}{32484375} \end {gather*}
Antiderivative was successfully verified.
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Rule 99
Rule 114
Rule 120
Rule 159
Rule 164
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{5/2} (2+3 x)^{7/2}}{(3+5 x)^{3/2}} \, dx &=-\frac {2 (1-2 x)^{5/2} (2+3 x)^{7/2}}{5 \sqrt {3+5 x}}+\frac {2}{5} \int \frac {\left (\frac {1}{2}-36 x\right ) (1-2 x)^{3/2} (2+3 x)^{5/2}}{\sqrt {3+5 x}} \, dx\\ &=-\frac {2 (1-2 x)^{5/2} (2+3 x)^{7/2}}{5 \sqrt {3+5 x}}-\frac {48}{275} (1-2 x)^{3/2} (2+3 x)^{7/2} \sqrt {3+5 x}+\frac {4}{825} \int \frac {\left (\frac {2829}{4}-\frac {11145 x}{2}\right ) \sqrt {1-2 x} (2+3 x)^{5/2}}{\sqrt {3+5 x}} \, dx\\ &=-\frac {2 (1-2 x)^{5/2} (2+3 x)^{7/2}}{5 \sqrt {3+5 x}}-\frac {2972 \sqrt {1-2 x} (2+3 x)^{7/2} \sqrt {3+5 x}}{7425}-\frac {48}{275} (1-2 x)^{3/2} (2+3 x)^{7/2} \sqrt {3+5 x}+\frac {8 \int \frac {\left (\frac {1741605}{8}-\frac {1299885 x}{2}\right ) (2+3 x)^{5/2}}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{111375}\\ &=-\frac {2 (1-2 x)^{5/2} (2+3 x)^{7/2}}{5 \sqrt {3+5 x}}+\frac {346636 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}{259875}-\frac {2972 \sqrt {1-2 x} (2+3 x)^{7/2} \sqrt {3+5 x}}{7425}-\frac {48}{275} (1-2 x)^{3/2} (2+3 x)^{7/2} \sqrt {3+5 x}-\frac {8 \int \frac {(2+3 x)^{3/2} \left (-1265280+\frac {30312615 x}{8}\right )}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{3898125}\\ &=-\frac {2 (1-2 x)^{5/2} (2+3 x)^{7/2}}{5 \sqrt {3+5 x}}+\frac {2020841 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{6496875}+\frac {346636 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}{259875}-\frac {2972 \sqrt {1-2 x} (2+3 x)^{7/2} \sqrt {3+5 x}}{7425}-\frac {48}{275} (1-2 x)^{3/2} (2+3 x)^{7/2} \sqrt {3+5 x}+\frac {8 \int \frac {\sqrt {2+3 x} \left (\frac {254408625}{16}+3958155 x\right )}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{97453125}\\ &=-\frac {2 (1-2 x)^{5/2} (2+3 x)^{7/2}}{5 \sqrt {3+5 x}}-\frac {703672 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{32484375}+\frac {2020841 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{6496875}+\frac {346636 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}{259875}-\frac {2972 \sqrt {1-2 x} (2+3 x)^{7/2} \sqrt {3+5 x}}{7425}-\frac {48}{275} (1-2 x)^{3/2} (2+3 x)^{7/2} \sqrt {3+5 x}-\frac {8 \int \frac {-\frac {3926957715}{8}-\frac {11891701485 x}{16}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{1461796875}\\ &=-\frac {2 (1-2 x)^{5/2} (2+3 x)^{7/2}}{5 \sqrt {3+5 x}}-\frac {703672 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{32484375}+\frac {2020841 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{6496875}+\frac {346636 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}{259875}-\frac {2972 \sqrt {1-2 x} (2+3 x)^{7/2} \sqrt {3+5 x}}{7425}-\frac {48}{275} (1-2 x)^{3/2} (2+3 x)^{7/2} \sqrt {3+5 x}+\frac {7261561 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{29531250}+\frac {264260033 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{324843750}\\ &=-\frac {2 (1-2 x)^{5/2} (2+3 x)^{7/2}}{5 \sqrt {3+5 x}}-\frac {703672 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{32484375}+\frac {2020841 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{6496875}+\frac {346636 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}{259875}-\frac {2972 \sqrt {1-2 x} (2+3 x)^{7/2} \sqrt {3+5 x}}{7425}-\frac {48}{275} (1-2 x)^{3/2} (2+3 x)^{7/2} \sqrt {3+5 x}-\frac {264260033 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{29531250 \sqrt {33}}-\frac {7261561 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{14765625 \sqrt {33}}\\ \end {align*}
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Mathematica [A]
time = 9.10, size = 125, normalized size = 0.50 \begin {gather*} \frac {30 \sqrt {1-2 x} \sqrt {2+3 x} \left (26378214+71568535 x-32807925 x^2-141221250 x^3+56227500 x^4+127575000 x^5\right )+264260033 \sqrt {6+10 x} E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-24628520 \sqrt {6+10 x} F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )}{974531250 \sqrt {3+5 x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 158, normalized size = 0.63
method | result | size |
default | \(-\frac {\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}\, \left (-22963500000 x^{7}-13948200000 x^{6}+239631513 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-264260033 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+31387500000 x^{5}+13515714000 x^{4}-20371373550 x^{3}-8863610070 x^{2}+3502765680 x +1582692840\right )}{974531250 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\) | \(158\) |
elliptic | \(-\frac {\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {216 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}\, x^{4}}{275}-\frac {242 \left (-30 x^{2}-5 x +10\right )}{78125 \sqrt {\left (x +\frac {3}{5}\right ) \left (-30 x^{2}-5 x +10\right )}}-\frac {172 x^{3} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{1375}-\frac {114694 x^{2} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{144375}+\frac {198269 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{721875}+\frac {8960444 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{32484375}+\frac {87265727 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{682171875 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {264260033 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{1364343750 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\left (6 x^{2}+x -2\right ) \sqrt {3+5 x}}\) | \(317\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.26, size = 48, normalized size = 0.19 \begin {gather*} \frac {{\left (127575000 \, x^{5} + 56227500 \, x^{4} - 141221250 \, x^{3} - 32807925 \, x^{2} + 71568535 \, x + 26378214\right )} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{32484375 \, \sqrt {5 \, x + 3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {{\left (1-2\,x\right )}^{5/2}\,{\left (3\,x+2\right )}^{7/2}}{{\left (5\,x+3\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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