Optimal. Leaf size=191 \[ \frac {6 \sqrt {1-2 x}}{35 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {436 \sqrt {1-2 x}}{245 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {60684 \sqrt {1-2 x}}{1715 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {1344984 \sqrt {1-2 x} \sqrt {2+3 x}}{3773 \sqrt {3+5 x}}+\frac {1344984 \sqrt {\frac {3}{11}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1715}+\frac {40456 \sqrt {\frac {3}{11}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1715} \]
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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 {40456 \sqrt {\frac {3}{11}} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1715}+\frac {1344984 \sqrt {\frac {3}{11}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1715}-\frac {1344984 \sqrt {1-2 x} \sqrt {3 x+2}}{3773 \sqrt {5 x+3}}+\frac {60684 \sqrt {1-2 x}}{1715 \sqrt {3 x+2} \sqrt {5 x+3}}+\frac {436 \sqrt {1-2 x}}{245 (3 x+2)^{3/2} \sqrt {5 x+3}}+\frac {6 \sqrt {1-2 x}}{35 (3 x+2)^{5/2} \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}{\sqrt {1-2 x} (2+3 x)^{7/2} (3+5 x)^{3/2}} \, dx &=\frac {6 \sqrt {1-2 x}}{35 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {2}{35} \int \frac {59-75 x}{\sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}} \, dx\\ &=\frac {6 \sqrt {1-2 x}}{35 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {436 \sqrt {1-2 x}}{245 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {4}{735} \int \frac {\frac {8631}{2}-4905 x}{\sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}} \, dx\\ &=\frac {6 \sqrt {1-2 x}}{35 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {436 \sqrt {1-2 x}}{245 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {60684 \sqrt {1-2 x}}{1715 \sqrt {2+3 x} \sqrt {3+5 x}}+\frac {8 \int \frac {183915-\frac {227565 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}} \, dx}{5145}\\ &=\frac {6 \sqrt {1-2 x}}{35 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {436 \sqrt {1-2 x}}{245 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {60684 \sqrt {1-2 x}}{1715 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {1344984 \sqrt {1-2 x} \sqrt {2+3 x}}{3773 \sqrt {3+5 x}}-\frac {16 \int \frac {\frac {9579285}{4}+\frac {7565535 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{56595}\\ &=\frac {6 \sqrt {1-2 x}}{35 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {436 \sqrt {1-2 x}}{245 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {60684 \sqrt {1-2 x}}{1715 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {1344984 \sqrt {1-2 x} \sqrt {2+3 x}}{3773 \sqrt {3+5 x}}-\frac {60684 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{1715}-\frac {4034952 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{18865}\\ &=\frac {6 \sqrt {1-2 x}}{35 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {436 \sqrt {1-2 x}}{245 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {60684 \sqrt {1-2 x}}{1715 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {1344984 \sqrt {1-2 x} \sqrt {2+3 x}}{3773 \sqrt {3+5 x}}+\frac {1344984 \sqrt {\frac {3}{11}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1715}+\frac {40456 \sqrt {\frac {3}{11}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1715}\\ \end {align*}
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Mathematica [A]
time = 4.29, size = 105, normalized size = 0.55 \begin {gather*} \frac {2 \left (-\frac {\sqrt {1-2 x} \left (25529443+116993058 x+178568982 x^2+90786420 x^3\right )}{(2+3 x)^{5/2} \sqrt {3+5 x}}-6 \sqrt {2} \left (112082 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-56455 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )\right )\right )}{18865} \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 = 0.11, size = 308, normalized size = 1.61
method | result | size |
elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {2 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{105 \left (\frac {2}{3}+x \right )^{3}}-\frac {646 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{735 \left (\frac {2}{3}+x \right )^{2}}-\frac {83294 \left (-30 x^{2}-3 x +9\right )}{1715 \sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}-\frac {851492 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{26411 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {1344984 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{26411 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {250 \left (-30 x^{2}-5 x +10\right )}{11 \sqrt {\left (x +\frac {3}{5}\right ) \left (-30 x^{2}-5 x +10\right )}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(277\) |
default | \(\frac {2 \sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (3003858 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-6052428 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+4005144 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-8069904 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+1335048 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-2689968 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-181572840 x^{4}-266351544 x^{3}-55417134 x^{2}+65934172 x +25529443\right )}{18865 \left (2+3 x \right )^{\frac {5}{2}} \left (10 x^{2}+x -3\right )}\) | \(308\) |
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 = 0.17, size = 60, normalized size = 0.31 \begin {gather*} -\frac {2 \, {\left (90786420 \, x^{3} + 178568982 \, x^{2} + 116993058 \, x + 25529443\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{18865 \, {\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 {1}{\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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