Optimal. Leaf size=187 \[ \frac {2 \sqrt {1-2 x}}{7 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {428 \sqrt {1-2 x}}{49 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {94420 \sqrt {1-2 x} \sqrt {2+3 x}}{1617 (3+5 x)^{3/2}}+\frac {6277760 \sqrt {1-2 x} \sqrt {2+3 x}}{17787 \sqrt {3+5 x}}-\frac {1255552 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{539 \sqrt {33}}-\frac {37768 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{539 \sqrt {33}} \]
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Rubi [A]
time = 0.05, antiderivative size = 187, 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 {37768 F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{539 \sqrt {33}}-\frac {1255552 E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{539 \sqrt {33}}+\frac {6277760 \sqrt {1-2 x} \sqrt {3 x+2}}{17787 \sqrt {5 x+3}}-\frac {94420 \sqrt {1-2 x} \sqrt {3 x+2}}{1617 (5 x+3)^{3/2}}+\frac {428 \sqrt {1-2 x}}{49 \sqrt {3 x+2} (5 x+3)^{3/2}}+\frac {2 \sqrt {1-2 x}}{7 (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}{\sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}} \, dx &=\frac {2 \sqrt {1-2 x}}{7 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {2}{21} \int \frac {57-75 x}{\sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}} \, dx\\ &=\frac {2 \sqrt {1-2 x}}{7 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {428 \sqrt {1-2 x}}{49 \sqrt {2+3 x} (3+5 x)^{3/2}}+\frac {4}{147} \int \frac {\frac {8385}{2}-4815 x}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}} \, dx\\ &=\frac {2 \sqrt {1-2 x}}{7 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {428 \sqrt {1-2 x}}{49 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {94420 \sqrt {1-2 x} \sqrt {2+3 x}}{1617 (3+5 x)^{3/2}}-\frac {8 \int \frac {\frac {343365}{2}-\frac {212445 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}} \, dx}{4851}\\ &=\frac {2 \sqrt {1-2 x}}{7 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {428 \sqrt {1-2 x}}{49 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {94420 \sqrt {1-2 x} \sqrt {2+3 x}}{1617 (3+5 x)^{3/2}}+\frac {6277760 \sqrt {1-2 x} \sqrt {2+3 x}}{17787 \sqrt {3+5 x}}+\frac {16 \int \frac {\frac {8942355}{4}+3531240 x}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{53361}\\ &=\frac {2 \sqrt {1-2 x}}{7 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {428 \sqrt {1-2 x}}{49 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {94420 \sqrt {1-2 x} \sqrt {2+3 x}}{1617 (3+5 x)^{3/2}}+\frac {6277760 \sqrt {1-2 x} \sqrt {2+3 x}}{17787 \sqrt {3+5 x}}+\frac {18884}{539} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx+\frac {1255552 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{5929}\\ &=\frac {2 \sqrt {1-2 x}}{7 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {428 \sqrt {1-2 x}}{49 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {94420 \sqrt {1-2 x} \sqrt {2+3 x}}{1617 (3+5 x)^{3/2}}+\frac {6277760 \sqrt {1-2 x} \sqrt {2+3 x}}{17787 \sqrt {3+5 x}}-\frac {1255552 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{539 \sqrt {33}}-\frac {37768 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{539 \sqrt {33}}\\ \end {align*}
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Mathematica [A]
time = 4.40, size = 104, normalized size = 0.56 \begin {gather*} \frac {2 \left (\frac {\sqrt {1-2 x} \left (35747225+169778606 x+268408770 x^2+141249600 x^3\right )}{(2+3 x)^{3/2} (3+5 x)^{3/2}}+2 \sqrt {2} \left (313888 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-158095 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )\right )\right )}{17787} \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. \(304\) vs.
\(2(139)=278\).
time = 0.13, size = 305, normalized size = 1.63
method | result | size |
elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {\left (-\frac {1294}{51975}-\frac {136 x}{3465}\right ) \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{\left (x^{2}+\frac {19}{15} x +\frac {2}{5}\right )^{2}}-\frac {2 \left (15-30 x \right ) \left (-\frac {5966174}{266805}-\frac {627776 x}{17787}\right )}{\sqrt {\left (x^{2}+\frac {19}{15} x +\frac {2}{5}\right ) \left (15-30 x \right )}}+\frac {3974380 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{124509 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {6277760 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{124509 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(234\) |
default | \(-\frac {2 \sqrt {1-2 x}\, \left (4673790 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-9416640 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+5920134 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-11927744 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+1869516 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-3766656 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-282499200 x^{4}-395567940 x^{3}-71148442 x^{2}+98284156 x +35747225\right )}{17787 \left (2+3 x \right )^{\frac {3}{2}} \left (3+5 x \right )^{\frac {3}{2}} \left (-1+2 x \right )}\) | \(305\) |
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 = 0.19, size = 60, normalized size = 0.32 \begin {gather*} \frac {2 \, {\left (141249600 \, x^{3} + 268408770 \, x^{2} + 169778606 \, x + 35747225\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{17787 \, {\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\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}{\sqrt {1-2\,x}\,{\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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