Optimal. Leaf size=189 \[ \frac {2 \sqrt {1-2 x}}{5 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {416 \sqrt {1-2 x}}{105 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {19268 \sqrt {1-2 x}}{245 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {116464 \sqrt {1-2 x} \sqrt {2+3 x}}{147 \sqrt {3+5 x}}+\frac {116464}{245} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {38536 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{245 \sqrt {33}} \]
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Rubi [A]
time = 0.05, antiderivative size = 189, 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 = {101, 157, 164,
114, 120} \begin {gather*} \frac {38536 F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{245 \sqrt {33}}+\frac {116464}{245} \sqrt {\frac {11}{3}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {116464 \sqrt {1-2 x} \sqrt {3 x+2}}{147 \sqrt {5 x+3}}+\frac {19268 \sqrt {1-2 x}}{245 \sqrt {3 x+2} \sqrt {5 x+3}}+\frac {416 \sqrt {1-2 x}}{105 (3 x+2)^{3/2} \sqrt {5 x+3}}+\frac {2 \sqrt {1-2 x}}{5 (3 x+2)^{5/2} \sqrt {5 x+3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 101
Rule 114
Rule 120
Rule 157
Rule 164
Rubi steps
\begin {align*} \int \frac {\sqrt {1-2 x}}{(2+3 x)^{7/2} (3+5 x)^{3/2}} \, dx &=\frac {2 \sqrt {1-2 x}}{5 (2+3 x)^{5/2} \sqrt {3+5 x}}-\frac {2}{5} \int \frac {-18+25 x}{\sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}} \, dx\\ &=\frac {2 \sqrt {1-2 x}}{5 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {416 \sqrt {1-2 x}}{105 (2+3 x)^{3/2} \sqrt {3+5 x}}-\frac {4}{105} \int \frac {-\frac {2737}{2}+1560 x}{\sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}} \, dx\\ &=\frac {2 \sqrt {1-2 x}}{5 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {416 \sqrt {1-2 x}}{105 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {19268 \sqrt {1-2 x}}{245 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {8}{735} \int \frac {-\frac {116785}{2}+\frac {72255 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}} \, dx\\ &=\frac {2 \sqrt {1-2 x}}{5 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {416 \sqrt {1-2 x}}{105 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {19268 \sqrt {1-2 x}}{245 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {116464 \sqrt {1-2 x} \sqrt {2+3 x}}{147 \sqrt {3+5 x}}+\frac {16 \int \frac {-\frac {3041445}{4}-1201035 x}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{8085}\\ &=\frac {2 \sqrt {1-2 x}}{5 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {416 \sqrt {1-2 x}}{105 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {19268 \sqrt {1-2 x}}{245 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {116464 \sqrt {1-2 x} \sqrt {2+3 x}}{147 \sqrt {3+5 x}}-\frac {19268}{245} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx-\frac {116464}{245} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx\\ &=\frac {2 \sqrt {1-2 x}}{5 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {416 \sqrt {1-2 x}}{105 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {19268 \sqrt {1-2 x}}{245 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {116464 \sqrt {1-2 x} \sqrt {2+3 x}}{147 \sqrt {3+5 x}}+\frac {116464}{245} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {38536 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{245 \sqrt {33}}\\ \end {align*}
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Mathematica [A]
time = 4.22, size = 105, normalized size = 0.56 \begin {gather*} \frac {2}{735} \left (-\frac {3 \sqrt {1-2 x} \left (736871+3376856 x+5154174 x^2+2620440 x^3\right )}{(2+3 x)^{5/2} \sqrt {3+5 x}}-2 \sqrt {2} \left (29116 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-14665 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(307\) vs.
\(2(139)=278\).
time = 0.10, size = 308, normalized size = 1.63
method | result | size |
elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {2 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{45 \left (\frac {2}{3}+x \right )^{3}}-\frac {626 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{315 \left (\frac {2}{3}+x \right )^{2}}-\frac {79714 \left (-30 x^{2}-3 x +9\right )}{735 \sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}-\frac {73732 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{1029 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {116464 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{1029 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {50 \left (-30 x^{2}-5 x +10\right )}{\sqrt {\left (x +\frac {3}{5}\right ) \left (-30 x^{2}-5 x +10\right )}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(277\) |
default | \(\frac {2 \sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (260118 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-524088 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+346824 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-698784 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+115608 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-232928 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-15722640 x^{4}-23063724 x^{3}-4798614 x^{2}+5709342 x +2210613\right )}{735 \left (2+3 x \right )^{\frac {5}{2}} \left (10 x^{2}+x -3\right )}\) | \(308\) |
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 = 0.11, size = 60, normalized size = 0.32 \begin {gather*} -\frac {2 \, {\left (2620440 \, x^{3} + 5154174 \, x^{2} + 3376856 \, x + 736871\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{245 \, {\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\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 = -0.01 \begin {gather*} \int \frac {\sqrt {1-2\,x}}{{\left (3\,x+2\right )}^{7/2}\,{\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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