Optimal. Leaf size=156 \[ \frac {6 \sqrt {1-2 x}}{7 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {1340 \sqrt {1-2 x} \sqrt {2+3 x}}{231 (3+5 x)^{3/2}}+\frac {89020 \sqrt {1-2 x} \sqrt {2+3 x}}{2541 \sqrt {3+5 x}}-\frac {17804 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{77 \sqrt {33}}-\frac {536 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{77 \sqrt {33}} \]
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Rubi [A]
time = 0.04, antiderivative size = 156, 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 {536 F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{77 \sqrt {33}}-\frac {17804 E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{77 \sqrt {33}}+\frac {89020 \sqrt {1-2 x} \sqrt {3 x+2}}{2541 \sqrt {5 x+3}}-\frac {1340 \sqrt {1-2 x} \sqrt {3 x+2}}{231 (5 x+3)^{3/2}}+\frac {6 \sqrt {1-2 x}}{7 \sqrt {3 x+2} (5 x+3)^{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}{\sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}} \, dx &=\frac {6 \sqrt {1-2 x}}{7 \sqrt {2+3 x} (3+5 x)^{3/2}}+\frac {2}{7} \int \frac {40-45 x}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}} \, dx\\ &=\frac {6 \sqrt {1-2 x}}{7 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {1340 \sqrt {1-2 x} \sqrt {2+3 x}}{231 (3+5 x)^{3/2}}-\frac {4}{231} \int \frac {\frac {3245}{2}-1005 x}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}} \, dx\\ &=\frac {6 \sqrt {1-2 x}}{7 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {1340 \sqrt {1-2 x} \sqrt {2+3 x}}{231 (3+5 x)^{3/2}}+\frac {89020 \sqrt {1-2 x} \sqrt {2+3 x}}{2541 \sqrt {3+5 x}}+\frac {8 \int \frac {21135+\frac {66765 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{2541}\\ &=\frac {6 \sqrt {1-2 x}}{7 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {1340 \sqrt {1-2 x} \sqrt {2+3 x}}{231 (3+5 x)^{3/2}}+\frac {89020 \sqrt {1-2 x} \sqrt {2+3 x}}{2541 \sqrt {3+5 x}}+\frac {268}{77} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx+\frac {17804}{847} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx\\ &=\frac {6 \sqrt {1-2 x}}{7 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {1340 \sqrt {1-2 x} \sqrt {2+3 x}}{231 (3+5 x)^{3/2}}+\frac {89020 \sqrt {1-2 x} \sqrt {2+3 x}}{2541 \sqrt {3+5 x}}-\frac {17804 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{77 \sqrt {33}}-\frac {536 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{77 \sqrt {33}}\\ \end {align*}
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Mathematica [A]
time = 3.29, size = 99, normalized size = 0.63 \begin {gather*} \frac {2 \left (\frac {\sqrt {1-2 x} \left (253409+823580 x+667650 x^2\right )}{\sqrt {2+3 x} (3+5 x)^{3/2}}+2 \sqrt {2} \left (4451 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-2240 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )\right )\right )}{2541} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 215, normalized size = 1.38
method | result | size |
default | \(-\frac {2 \sqrt {2+3 x}\, \sqrt {1-2 x}\, \left (22110 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-44510 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+13266 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-26706 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-1335300 x^{3}-979510 x^{2}+316762 x +253409\right )}{2541 \left (3+5 x \right )^{\frac {3}{2}} \left (6 x^{2}+x -2\right )}\) | \(215\) |
elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {-\frac {540}{7} x^{2}-\frac {54}{7} x +\frac {162}{7}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}+\frac {56360 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{17787 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {89020 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{17787 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {2 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{33 \left (x +\frac {3}{5}\right )^{2}}+\frac {-\frac {16100}{121} x^{2}-\frac {8050}{363} x +\frac {16100}{363}}{\sqrt {\left (x +\frac {3}{5}\right ) \left (-30 x^{2}-5 x +10\right )}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(253\) |
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 = 0.16, size = 50, normalized size = 0.32 \begin {gather*} \frac {2 \, {\left (667650 \, x^{2} + 823580 \, x + 253409\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{2541 \, {\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\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}{\sqrt {1 - 2 x} \left (3 x + 2\right )^{\frac {3}{2}} \left (5 x + 3\right )^{\frac {5}{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 {1}{\sqrt {1-2\,x}\,{\left (3\,x+2\right )}^{3/2}\,{\left (5\,x+3\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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