Optimal. Leaf size=218 \[ \frac {4}{77 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {54 \sqrt {1-2 x}}{539 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {14496 \sqrt {1-2 x}}{3773 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {3205940 \sqrt {1-2 x} \sqrt {2+3 x}}{124509 (3+5 x)^{3/2}}+\frac {213119320 \sqrt {1-2 x} \sqrt {2+3 x}}{1369599 \sqrt {3+5 x}}-\frac {42623864 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{41503 \sqrt {33}}-\frac {1282376 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{41503 \sqrt {33}} \]
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Rubi [A]
time = 0.06, antiderivative size = 218, normalized size of antiderivative = 1.00, number of steps
used = 8, 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 {1282376 F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{41503 \sqrt {33}}-\frac {42623864 E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{41503 \sqrt {33}}+\frac {213119320 \sqrt {1-2 x} \sqrt {3 x+2}}{1369599 \sqrt {5 x+3}}-\frac {3205940 \sqrt {1-2 x} \sqrt {3 x+2}}{124509 (5 x+3)^{3/2}}+\frac {14496 \sqrt {1-2 x}}{3773 \sqrt {3 x+2} (5 x+3)^{3/2}}+\frac {54 \sqrt {1-2 x}}{539 (3 x+2)^{3/2} (5 x+3)^{3/2}}+\frac {4}{77 \sqrt {1-2 x} (3 x+2)^{3/2} (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}{(1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{5/2}} \, dx &=\frac {4}{77 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}}-\frac {2}{77} \int \frac {-\frac {167}{2}-105 x}{\sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}} \, dx\\ &=\frac {4}{77 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {54 \sqrt {1-2 x}}{539 (2+3 x)^{3/2} (3+5 x)^{3/2}}-\frac {4 \int \frac {-1137+\frac {2025 x}{2}}{\sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}} \, dx}{1617}\\ &=\frac {4}{77 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {54 \sqrt {1-2 x}}{539 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {14496 \sqrt {1-2 x}}{3773 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {8 \int \frac {-\frac {285195}{4}+81540 x}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}} \, dx}{11319}\\ &=\frac {4}{77 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {54 \sqrt {1-2 x}}{539 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {14496 \sqrt {1-2 x}}{3773 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {3205940 \sqrt {1-2 x} \sqrt {2+3 x}}{124509 (3+5 x)^{3/2}}+\frac {16 \int \frac {-\frac {5827965}{2}+\frac {7213365 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}} \, dx}{373527}\\ &=\frac {4}{77 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {54 \sqrt {1-2 x}}{539 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {14496 \sqrt {1-2 x}}{3773 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {3205940 \sqrt {1-2 x} \sqrt {2+3 x}}{124509 (3+5 x)^{3/2}}+\frac {213119320 \sqrt {1-2 x} \sqrt {2+3 x}}{1369599 \sqrt {3+5 x}}-\frac {32 \int \frac {-\frac {303580485}{8}-\frac {239759235 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{4108797}\\ &=\frac {4}{77 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {54 \sqrt {1-2 x}}{539 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {14496 \sqrt {1-2 x}}{3773 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {3205940 \sqrt {1-2 x} \sqrt {2+3 x}}{124509 (3+5 x)^{3/2}}+\frac {213119320 \sqrt {1-2 x} \sqrt {2+3 x}}{1369599 \sqrt {3+5 x}}+\frac {641188 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{41503}+\frac {42623864 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{456533}\\ &=\frac {4}{77 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {54 \sqrt {1-2 x}}{539 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {14496 \sqrt {1-2 x}}{3773 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {3205940 \sqrt {1-2 x} \sqrt {2+3 x}}{124509 (3+5 x)^{3/2}}+\frac {213119320 \sqrt {1-2 x} \sqrt {2+3 x}}{1369599 \sqrt {3+5 x}}-\frac {42623864 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{41503 \sqrt {33}}-\frac {1282376 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{41503 \sqrt {33}}\\ \end {align*}
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Mathematica [A]
time = 7.69, size = 109, normalized size = 0.50 \begin {gather*} \frac {2 \left (\frac {1213551469+3336610202 x-2415287594 x^2-13428808080 x^3-9590369400 x^4}{\sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}}+2 \sqrt {2} \left (10655966 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-5366165 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )\right )\right )}{1369599} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 305, normalized size = 1.40
method | result | size |
elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {32 \left (-30 x^{2}-38 x -12\right )}{456533 \sqrt {\left (-\frac {1}{2}+x \right ) \left (-30 x^{2}-38 x -12\right )}}+\frac {\left (-\frac {44102}{4002075}-\frac {4628 x}{266805}\right ) \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{\left (x^{2}+\frac {19}{15} x +\frac {2}{5}\right )^{2}}-\frac {2 \left (15-30 x \right ) \left (-\frac {40508198}{4108797}-\frac {21311980 x}{1369599}\right )}{\sqrt {\left (x^{2}+\frac {19}{15} x +\frac {2}{5}\right ) \left (15-30 x \right )}}+\frac {134924660 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{9587193 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {213119320 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{9587193 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(262\) |
default | \(-\frac {2 \sqrt {1-2 x}\, \left (158694030 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-319678980 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+201012438 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-404926708 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+63477612 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-127871592 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-9590369400 x^{4}-13428808080 x^{3}-2415287594 x^{2}+3336610202 x +1213551469\right )}{1369599 \left (2+3 x \right )^{\frac {3}{2}} \left (3+5 x \right )^{\frac {3}{2}} \left (-1+2 x \right )}\) | \(305\) |
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 = 0.17, size = 70, normalized size = 0.32 \begin {gather*} \frac {2 \, {\left (9590369400 \, x^{4} + 13428808080 \, x^{3} + 2415287594 \, x^{2} - 3336610202 \, x - 1213551469\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{1369599 \, {\left (450 \, x^{5} + 915 \, x^{4} + 512 \, x^{3} - 85 \, x^{2} - 156 \, x - 36\right )}} \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 = -0.00 \begin {gather*} \int \frac {1}{{\left (1-2\,x\right )}^{3/2}\,{\left (3\,x+2\right )}^{5/2}\,{\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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