Optimal. Leaf size=129 \[ \frac {4 \sqrt {3+5 x}}{77 \sqrt {1-2 x} \sqrt {2+3 x}}+\frac {186 \sqrt {1-2 x} \sqrt {3+5 x}}{539 \sqrt {2+3 x}}-\frac {62}{49} \sqrt {\frac {3}{11}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {8}{49} \sqrt {\frac {3}{11}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right ) \]
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Rubi [A]
time = 0.03, antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps
used = 5, 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 {8}{49} \sqrt {\frac {3}{11}} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {62}{49} \sqrt {\frac {3}{11}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {186 \sqrt {1-2 x} \sqrt {5 x+3}}{539 \sqrt {3 x+2}}+\frac {4 \sqrt {5 x+3}}{77 \sqrt {1-2 x} \sqrt {3 x+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}{(1-2 x)^{3/2} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx &=\frac {4 \sqrt {3+5 x}}{77 \sqrt {1-2 x} \sqrt {2+3 x}}-\frac {2}{77} \int \frac {-\frac {51}{2}-15 x}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx\\ &=\frac {4 \sqrt {3+5 x}}{77 \sqrt {1-2 x} \sqrt {2+3 x}}+\frac {186 \sqrt {1-2 x} \sqrt {3+5 x}}{539 \sqrt {2+3 x}}-\frac {4}{539} \int \frac {-\frac {345}{2}-\frac {465 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx\\ &=\frac {4 \sqrt {3+5 x}}{77 \sqrt {1-2 x} \sqrt {2+3 x}}+\frac {186 \sqrt {1-2 x} \sqrt {3+5 x}}{539 \sqrt {2+3 x}}+\frac {12}{49} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx+\frac {186}{539} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx\\ &=\frac {4 \sqrt {3+5 x}}{77 \sqrt {1-2 x} \sqrt {2+3 x}}+\frac {186 \sqrt {1-2 x} \sqrt {3+5 x}}{539 \sqrt {2+3 x}}-\frac {62}{49} \sqrt {\frac {3}{11}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {8}{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 = 5.97, size = 122, normalized size = 0.95 \begin {gather*} \frac {2 (107-186 x) \sqrt {2+3 x} \sqrt {3+5 x}+62 \sqrt {2-4 x} (2+3 x) E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )+70 \sqrt {2-4 x} (2+3 x) F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )}{539 \sqrt {1-2 x} (2+3 x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 133, normalized size = 1.03
method | result | size |
default | \(-\frac {2 \sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}\, \left (66 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-31 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-930 x^{2}-23 x +321\right )}{539 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\) | \(133\) |
elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {2 \left (-18-30 x \right ) \left (\frac {107}{3234}-\frac {31 x}{539}\right )}{\sqrt {\left (x^{2}+\frac {1}{6} x -\frac {1}{3}\right ) \left (-18-30 x \right )}}+\frac {230 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{3773 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {310 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{3773 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(201\) |
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 = 0.18, size = 38, normalized size = 0.29 \begin {gather*} \frac {2 \, {\left (186 \, x - 107\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{539 \, {\left (6 \, x^{2} + x - 2\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}{\left (1 - 2 x\right )^{\frac {3}{2}} \left (3 x + 2\right )^{\frac {3}{2}} \sqrt {5 x + 3}}\, 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}{{\left (1-2\,x\right )}^{3/2}\,{\left (3\,x+2\right )}^{3/2}\,\sqrt {5\,x+3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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