Optimal. Leaf size=160 \[ \frac {4 \sqrt {3+5 x}}{77 \sqrt {1-2 x} (2+3 x)^{3/2}}+\frac {54 \sqrt {1-2 x} \sqrt {3+5 x}}{539 (2+3 x)^{3/2}}+\frac {5256 \sqrt {1-2 x} \sqrt {3+5 x}}{3773 \sqrt {2+3 x}}-\frac {1752}{343} \sqrt {\frac {3}{11}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {68}{343} \sqrt {\frac {3}{11}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right ) \]
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Rubi [A]
time = 0.04, 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 = {106, 157, 164,
114, 120} \begin {gather*} -\frac {68}{343} \sqrt {\frac {3}{11}} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {1752}{343} \sqrt {\frac {3}{11}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {5256 \sqrt {1-2 x} \sqrt {5 x+3}}{3773 \sqrt {3 x+2}}+\frac {54 \sqrt {1-2 x} \sqrt {5 x+3}}{539 (3 x+2)^{3/2}}+\frac {4 \sqrt {5 x+3}}{77 \sqrt {1-2 x} (3 x+2)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 106
Rule 114
Rule 120
Rule 157
Rule 164
Rubi steps
\begin {align*} \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx &=\frac {4 \sqrt {3+5 x}}{77 \sqrt {1-2 x} (2+3 x)^{3/2}}-\frac {2}{77} \int \frac {-\frac {87}{2}-45 x}{\sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx\\ &=\frac {4 \sqrt {3+5 x}}{77 \sqrt {1-2 x} (2+3 x)^{3/2}}+\frac {54 \sqrt {1-2 x} \sqrt {3+5 x}}{539 (2+3 x)^{3/2}}-\frac {4 \int \frac {-522+\frac {405 x}{2}}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx}{1617}\\ &=\frac {4 \sqrt {3+5 x}}{77 \sqrt {1-2 x} (2+3 x)^{3/2}}+\frac {54 \sqrt {1-2 x} \sqrt {3+5 x}}{539 (2+3 x)^{3/2}}+\frac {5256 \sqrt {1-2 x} \sqrt {3+5 x}}{3773 \sqrt {2+3 x}}-\frac {8 \int \frac {-\frac {25335}{4}-9855 x}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{11319}\\ &=\frac {4 \sqrt {3+5 x}}{77 \sqrt {1-2 x} (2+3 x)^{3/2}}+\frac {54 \sqrt {1-2 x} \sqrt {3+5 x}}{539 (2+3 x)^{3/2}}+\frac {5256 \sqrt {1-2 x} \sqrt {3+5 x}}{3773 \sqrt {2+3 x}}+\frac {102}{343} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx+\frac {5256 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{3773}\\ &=\frac {4 \sqrt {3+5 x}}{77 \sqrt {1-2 x} (2+3 x)^{3/2}}+\frac {54 \sqrt {1-2 x} \sqrt {3+5 x}}{539 (2+3 x)^{3/2}}+\frac {5256 \sqrt {1-2 x} \sqrt {3+5 x}}{3773 \sqrt {2+3 x}}-\frac {1752}{343} \sqrt {\frac {3}{11}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {68}{343} \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 = 6.55, size = 99, normalized size = 0.62 \begin {gather*} \frac {2 \left (\frac {\sqrt {3+5 x} \left (5543-3006 x-15768 x^2\right )}{\sqrt {1-2 x} (2+3 x)^{3/2}}+3 \sqrt {2} \left (292 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-105 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )\right )\right )}{3773} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 215, normalized size = 1.34
method | result | size |
default | \(\frac {2 \sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (2628 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-1683 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+1752 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-1122 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+78840 x^{3}+62334 x^{2}-18697 x -16629\right )}{3773 \left (2+3 x \right )^{\frac {3}{2}} \left (10 x^{2}+x -3\right )}\) | \(215\) |
elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {8 \left (-30 x^{2}-38 x -12\right )}{3773 \sqrt {\left (-\frac {1}{2}+x \right ) \left (-30 x^{2}-38 x -12\right )}}+\frac {5630 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{26411 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {8760 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{26411 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {-\frac {4800}{343} x^{2}-\frac {480}{343} x +\frac {1440}{343}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}+\frac {2 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{147 \left (\frac {2}{3}+x \right )^{2}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(253\) |
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 = 0.18, size = 50, normalized size = 0.31 \begin {gather*} \frac {2 \, {\left (15768 \, x^{2} + 3006 \, x - 5543\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{3773 \, {\left (18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4\right )}} \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 {1}{\left (1 - 2 x\right )^{\frac {3}{2}} \left (3 x + 2\right )^{\frac {5}{2}} \sqrt {5 x + 3}}\, 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 {1}{{\left (1-2\,x\right )}^{3/2}\,{\left (3\,x+2\right )}^{5/2}\,\sqrt {5\,x+3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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