Optimal. Leaf size=160 \[ -\frac {2 \sqrt {1-2 x} (2+3 x)^{5/2}}{55 \sqrt {3+5 x}}-\frac {2577 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{6875}-\frac {69 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{1375}-\frac {61151 \sqrt {\frac {3}{11}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{6250}-\frac {942 \sqrt {\frac {3}{11}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3125} \]
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Rubi [A]
time = 0.04, antiderivative size = 160, 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 = {100, 159, 164,
114, 120} \begin {gather*} -\frac {942 \sqrt {\frac {3}{11}} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3125}-\frac {61151 \sqrt {\frac {3}{11}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{6250}-\frac {2 \sqrt {1-2 x} (3 x+2)^{5/2}}{55 \sqrt {5 x+3}}-\frac {69 \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{3/2}}{1375}-\frac {2577 \sqrt {1-2 x} \sqrt {5 x+3} \sqrt {3 x+2}}{6875} \end {gather*}
Antiderivative was successfully verified.
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Rule 100
Rule 114
Rule 120
Rule 159
Rule 164
Rubi steps
\begin {align*} \int \frac {(2+3 x)^{7/2}}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx &=-\frac {2 \sqrt {1-2 x} (2+3 x)^{5/2}}{55 \sqrt {3+5 x}}-\frac {2}{55} \int \frac {\left (-\frac {81}{2}-\frac {69 x}{2}\right ) (2+3 x)^{3/2}}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {2 \sqrt {1-2 x} (2+3 x)^{5/2}}{55 \sqrt {3+5 x}}-\frac {69 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{1375}+\frac {2 \int \frac {\sqrt {2+3 x} \left (\frac {9825}{4}+\frac {7731 x}{2}\right )}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{1375}\\ &=-\frac {2 \sqrt {1-2 x} (2+3 x)^{5/2}}{55 \sqrt {3+5 x}}-\frac {2577 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{6875}-\frac {69 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{1375}-\frac {2 \int \frac {-\frac {348867}{4}-\frac {550359 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{20625}\\ &=-\frac {2 \sqrt {1-2 x} (2+3 x)^{5/2}}{55 \sqrt {3+5 x}}-\frac {2577 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{6875}-\frac {69 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{1375}+\frac {1413 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{3125}+\frac {183453 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{68750}\\ &=-\frac {2 \sqrt {1-2 x} (2+3 x)^{5/2}}{55 \sqrt {3+5 x}}-\frac {2577 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{6875}-\frac {69 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{1375}-\frac {61151 \sqrt {\frac {3}{11}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{6250}-\frac {942 \sqrt {\frac {3}{11}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3125}\\ \end {align*}
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Mathematica [A]
time = 4.46, size = 122, normalized size = 0.76 \begin {gather*} \frac {61151 \sqrt {2} (3+5 x) E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-5 \left (2 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x} \left (10801+22440 x+7425 x^2\right )+6013 \sqrt {2} (3+5 x) F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )\right )}{68750 (3+5 x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 143, normalized size = 0.89
method | result | size |
default | \(-\frac {\sqrt {2+3 x}\, \sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (31086 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-61151 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+445500 x^{4}+1420650 x^{3}+723960 x^{2}-340790 x -216020\right )}{68750 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\) | \(143\) |
elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {27 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{125}-\frac {327 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{625}+\frac {38763 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{96250 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {61151 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{96250 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {2 \left (-30 x^{2}-5 x +10\right )}{6875 \sqrt {\left (x +\frac {3}{5}\right ) \left (-30 x^{2}-5 x +10\right )}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(240\) |
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 = 0.25, size = 33, normalized size = 0.21 \begin {gather*} -\frac {{\left (7425 \, x^{2} + 22440 \, x + 10801\right )} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{6875 \, \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 = -0.01 \begin {gather*} \int \frac {{\left (3\,x+2\right )}^{7/2}}{\sqrt {1-2\,x}\,{\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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