Optimal. Leaf size=191 \[ \frac {4}{77 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {54 \sqrt {1-2 x}}{539 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {9876 \sqrt {1-2 x}}{3773 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {1100380 \sqrt {1-2 x} \sqrt {2+3 x}}{41503 \sqrt {3+5 x}}+\frac {220076 \sqrt {\frac {3}{11}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3773}+\frac {6584 \sqrt {\frac {3}{11}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3773} \]
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Rubi [A]
time = 0.05, antiderivative size = 191, normalized size of antiderivative = 1.00, number of steps
used = 7, 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 {6584 \sqrt {\frac {3}{11}} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3773}+\frac {220076 \sqrt {\frac {3}{11}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3773}-\frac {1100380 \sqrt {1-2 x} \sqrt {3 x+2}}{41503 \sqrt {5 x+3}}+\frac {9876 \sqrt {1-2 x}}{3773 \sqrt {3 x+2} \sqrt {5 x+3}}+\frac {54 \sqrt {1-2 x}}{539 (3 x+2)^{3/2} \sqrt {5 x+3}}+\frac {4}{77 \sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}} \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)^{3/2}} \, dx &=\frac {4}{77 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}-\frac {2}{77} \int \frac {-\frac {127}{2}-75 x}{\sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}} \, dx\\ &=\frac {4}{77 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {54 \sqrt {1-2 x}}{539 (2+3 x)^{3/2} \sqrt {3+5 x}}-\frac {4 \int \frac {-\frac {1659}{2}+\frac {1215 x}{2}}{\sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}} \, dx}{1617}\\ &=\frac {4}{77 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {54 \sqrt {1-2 x}}{539 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {9876 \sqrt {1-2 x}}{3773 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {8 \int \frac {-\frac {120615}{4}+\frac {37035 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}} \, dx}{11319}\\ &=\frac {4}{77 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {54 \sqrt {1-2 x}}{539 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {9876 \sqrt {1-2 x}}{3773 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {1100380 \sqrt {1-2 x} \sqrt {2+3 x}}{41503 \sqrt {3+5 x}}+\frac {16 \int \frac {-\frac {783495}{2}-\frac {2475855 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{124509}\\ &=\frac {4}{77 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {54 \sqrt {1-2 x}}{539 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {9876 \sqrt {1-2 x}}{3773 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {1100380 \sqrt {1-2 x} \sqrt {2+3 x}}{41503 \sqrt {3+5 x}}-\frac {9876 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{3773}-\frac {660228 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{41503}\\ &=\frac {4}{77 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {54 \sqrt {1-2 x}}{539 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {9876 \sqrt {1-2 x}}{3773 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {1100380 \sqrt {1-2 x} \sqrt {2+3 x}}{41503 \sqrt {3+5 x}}+\frac {220076 \sqrt {\frac {3}{11}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3773}+\frac {6584 \sqrt {\frac {3}{11}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3773}\\ \end {align*}
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Mathematica [A]
time = 7.45, size = 104, normalized size = 0.54 \begin {gather*} \frac {2 \left (\frac {-2088967-2259236 x+7926942 x^2+9903420 x^3}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}-2 \sqrt {2} \left (55019 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-27860 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )\right )\right )}{41503} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 215, normalized size = 1.13
method | result | size |
default | \(\frac {2 \sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (162954 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-330114 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+108636 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-220076 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-9903420 x^{3}-7926942 x^{2}+2259236 x +2088967\right )}{41503 \left (2+3 x \right )^{\frac {3}{2}} \left (10 x^{2}+x -3\right )}\) | \(215\) |
elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {2 \left (-20-30 x \right ) \left (-\frac {214327}{415030}+\frac {42883 x}{41503}\right )}{\sqrt {\left (x^{2}+\frac {1}{10} x -\frac {3}{10}\right ) \left (-20-30 x \right )}}-\frac {2 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{49 \left (\frac {2}{3}+x \right )^{2}}-\frac {1110 \left (-30 x^{2}-3 x +9\right )}{343 \sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}-\frac {696440 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{290521 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {1100380 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{290521 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(253\) |
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 = 0.45, size = 60, normalized size = 0.31 \begin {gather*} -\frac {2 \, {\left (9903420 \, x^{3} + 7926942 \, x^{2} - 2259236 \, x - 2088967\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{41503 \, {\left (90 \, x^{4} + 129 \, x^{3} + 25 \, x^{2} - 32 \, x - 12\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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}\,{\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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