Optimal. Leaf size=127 \[ -\frac {2 \sqrt {1-2 x} (2+3 x)^{3/2}}{5 \sqrt {3+5 x}}+\frac {8}{25} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {19}{125} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {106 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{125 \sqrt {33}} \]
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Rubi [A]
time = 0.03, antiderivative size = 127, 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 = {99, 159, 164,
114, 120} \begin {gather*} -\frac {106 F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{125 \sqrt {33}}-\frac {19}{125} \sqrt {\frac {11}{3}} 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}}{5 \sqrt {5 x+3}}+\frac {8}{25} \sqrt {1-2 x} \sqrt {5 x+3} \sqrt {3 x+2} \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 {\sqrt {1-2 x} (2+3 x)^{3/2}}{(3+5 x)^{3/2}} \, dx &=-\frac {2 \sqrt {1-2 x} (2+3 x)^{3/2}}{5 \sqrt {3+5 x}}+\frac {2}{5} \int \frac {\left (\frac {5}{2}-12 x\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}}{5 \sqrt {3+5 x}}+\frac {8}{25} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {2}{75} \int \frac {-33-\frac {57 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {2 \sqrt {1-2 x} (2+3 x)^{3/2}}{5 \sqrt {3+5 x}}+\frac {8}{25} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {19}{125} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx+\frac {53}{125} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {2 \sqrt {1-2 x} (2+3 x)^{3/2}}{5 \sqrt {3+5 x}}+\frac {8}{25} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {19}{125} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {106 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{125 \sqrt {33}}\\ \end {align*}
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Mathematica [A]
time = 3.34, size = 97, normalized size = 0.76 \begin {gather*} \frac {1}{375} \left (\frac {30 \sqrt {1-2 x} \sqrt {2+3 x} (2+5 x)}{\sqrt {3+5 x}}+19 \sqrt {2} E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )+140 \sqrt {2} F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 138, normalized size = 1.09
method | result | size |
default | \(-\frac {\sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (159 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-19 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-900 x^{3}-510 x^{2}+240 x +120\right )}{375 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\) | \(138\) |
elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {2 \left (-30 x^{2}-5 x +10\right )}{125 \sqrt {\left (x +\frac {3}{5}\right ) \left (-30 x^{2}-5 x +10\right )}}+\frac {22 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{525 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {19 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{525 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {2 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{25}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(220\) |
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 = 0.12, size = 28, normalized size = 0.22 \begin {gather*} \frac {2 \, {\left (5 \, x + 2\right )} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{25 \, \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 {\sqrt {1 - 2 x} \left (3 x + 2\right )^{\frac {3}{2}}}{\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 {\sqrt {1-2\,x}\,{\left (3\,x+2\right )}^{3/2}}{{\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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