Optimal. Leaf size=160 \[ \frac {2 \sqrt {1-2 x}}{7 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {288 \sqrt {1-2 x}}{49 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {31940 \sqrt {1-2 x} \sqrt {2+3 x}}{539 \sqrt {3+5 x}}+\frac {6388}{49} \sqrt {\frac {3}{11}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {192}{49} \sqrt {\frac {3}{11}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right ) \]
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Rubi [A]
time = 0.03, antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps
used = 6, 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 {192}{49} \sqrt {\frac {3}{11}} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {6388}{49} \sqrt {\frac {3}{11}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {31940 \sqrt {1-2 x} \sqrt {3 x+2}}{539 \sqrt {5 x+3}}+\frac {288 \sqrt {1-2 x}}{49 \sqrt {3 x+2} \sqrt {5 x+3}}+\frac {2 \sqrt {1-2 x}}{7 (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}{\sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}} \, dx &=\frac {2 \sqrt {1-2 x}}{7 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {2}{21} \int \frac {42-45 x}{\sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}} \, dx\\ &=\frac {2 \sqrt {1-2 x}}{7 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {288 \sqrt {1-2 x}}{49 \sqrt {2+3 x} \sqrt {3+5 x}}+\frac {4}{147} \int \frac {\frac {3495}{2}-1080 x}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}} \, dx\\ &=\frac {2 \sqrt {1-2 x}}{7 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {288 \sqrt {1-2 x}}{49 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {31940 \sqrt {1-2 x} \sqrt {2+3 x}}{539 \sqrt {3+5 x}}-\frac {8 \int \frac {\frac {45495}{2}+\frac {71865 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{1617}\\ &=\frac {2 \sqrt {1-2 x}}{7 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {288 \sqrt {1-2 x}}{49 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {31940 \sqrt {1-2 x} \sqrt {2+3 x}}{539 \sqrt {3+5 x}}-\frac {288}{49} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx-\frac {19164}{539} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx\\ &=\frac {2 \sqrt {1-2 x}}{7 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {288 \sqrt {1-2 x}}{49 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {31940 \sqrt {1-2 x} \sqrt {2+3 x}}{539 \sqrt {3+5 x}}+\frac {6388}{49} \sqrt {\frac {3}{11}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {192}{49} \sqrt {\frac {3}{11}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\\ \end {align*}
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Mathematica [A]
time = 4.00, size = 100, normalized size = 0.62 \begin {gather*} \frac {2}{539} \left (-\frac {\sqrt {1-2 x} \left (60635+186888 x+143730 x^2\right )}{(2+3 x)^{3/2} \sqrt {3+5 x}}-2 \sqrt {2} \left (1597 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-805 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 [A]
time = 0.11, size = 215, normalized size = 1.34
method | result | size |
default | \(\frac {2 \sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (4752 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-9582 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+3168 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-6388 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-287460 x^{3}-230046 x^{2}+65618 x +60635\right )}{539 \left (2+3 x \right )^{\frac {3}{2}} \left (10 x^{2}+x -3\right )}\) | \(215\) |
elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {358 \left (-30 x^{2}-3 x +9\right )}{49 \sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}-\frac {20220 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{3773 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {31940 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{3773 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {50 \left (-30 x^{2}-5 x +10\right )}{11 \sqrt {\left (x +\frac {3}{5}\right ) \left (-30 x^{2}-5 x +10\right )}}-\frac {2 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{21 \left (\frac {2}{3}+x \right )^{2}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(253\) |
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 = 0.26, size = 50, normalized size = 0.31 \begin {gather*} -\frac {2 \, {\left (143730 \, x^{2} + 186888 \, x + 60635\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{539 \, {\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {1 - 2 x} \left (3 x + 2\right )^{\frac {5}{2}} \left (5 x + 3\right )^{\frac {3}{2}}}\, dx \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 )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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