Optimal. Leaf size=129 \[ -\frac {2 \sqrt {1-2 x} (2+3 x)^{3/2}}{55 \sqrt {3+5 x}}-\frac {27}{275} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {438}{125} \sqrt {\frac {3}{11}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {17}{125} \sqrt {\frac {3}{11}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right ) \]
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Rubi [A]
time = 0.03, antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps
used = 5, 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 {17}{125} \sqrt {\frac {3}{11}} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {438}{125} \sqrt {\frac {3}{11}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {2 \sqrt {1-2 x} (3 x+2)^{3/2}}{55 \sqrt {5 x+3}}-\frac {27}{275} \sqrt {1-2 x} \sqrt {5 x+3} \sqrt {3 x+2} \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)^{5/2}}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx &=-\frac {2 \sqrt {1-2 x} (2+3 x)^{3/2}}{55 \sqrt {3+5 x}}-\frac {2}{55} \int \frac {\left (-\frac {75}{2}-\frac {81 x}{2}\right ) \sqrt {2+3 x}}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {2 \sqrt {1-2 x} (2+3 x)^{3/2}}{55 \sqrt {3+5 x}}-\frac {27}{275} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {2}{825} \int \frac {\frac {5067}{4}+1971 x}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {2 \sqrt {1-2 x} (2+3 x)^{3/2}}{55 \sqrt {3+5 x}}-\frac {27}{275} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {51}{250} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx+\frac {1314 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{1375}\\ &=-\frac {2 \sqrt {1-2 x} (2+3 x)^{3/2}}{55 \sqrt {3+5 x}}-\frac {27}{275} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {438}{125} \sqrt {\frac {3}{11}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {17}{125} \sqrt {\frac {3}{11}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\\ \end {align*}
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Mathematica [A]
time = 3.85, size = 97, normalized size = 0.75 \begin {gather*} \frac {-\frac {10 \sqrt {1-2 x} \sqrt {2+3 x} (101+165 x)}{\sqrt {3+5 x}}+876 \sqrt {2} E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-315 \sqrt {2} F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )}{2750} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 138, normalized size = 1.07
method | result | size |
default | \(-\frac {\sqrt {2+3 x}\, \sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (561 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-876 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+9900 x^{3}+7710 x^{2}-2290 x -2020\right )}{2750 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\) | \(138\) |
elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {3 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{25}+\frac {563 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{3850 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {438 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{1925 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {2 \left (-30 x^{2}-5 x +10\right )}{1375 \sqrt {\left (x +\frac {3}{5}\right ) \left (-30 x^{2}-5 x +10\right )}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(220\) |
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 = 0.20, size = 28, normalized size = 0.22 \begin {gather*} -\frac {{\left (165 \, x + 101\right )} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{275 \, \sqrt {5 \, x + 3}} \end {gather*}
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 \begin {gather*} \int \frac {\left (3 x + 2\right )^{\frac {5}{2}}}{\sqrt {1 - 2 x} \left (5 x + 3\right )^{\frac {3}{2}}}\, dx \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 )}^{5/2}}{\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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