Optimal. Leaf size=189 \[ -\frac {2 \sqrt {1-2 x} (2+3 x)^{7/2}}{5 \sqrt {3+5 x}}-\frac {2486 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{21875}+\frac {183 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{4375}+\frac {48}{175} \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}-\frac {203179 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{218750}-\frac {38723 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{109375 \sqrt {33}} \]
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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 = {99, 159, 164,
114, 120} \begin {gather*} -\frac {38723 F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{109375 \sqrt {33}}-\frac {203179 \sqrt {\frac {11}{3}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{218750}-\frac {2 \sqrt {1-2 x} (3 x+2)^{7/2}}{5 \sqrt {5 x+3}}+\frac {48}{175} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{5/2}+\frac {183 \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{3/2}}{4375}-\frac {2486 \sqrt {1-2 x} \sqrt {5 x+3} \sqrt {3 x+2}}{21875} \end {gather*}
Antiderivative was successfully verified.
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Rule 99
Rule 114
Rule 120
Rule 159
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} (2+3 x)^{7/2}}{5 \sqrt {3+5 x}}+\frac {2}{5} \int \frac {\left (\frac {17}{2}-24 x\right ) (2+3 x)^{5/2}}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {2 \sqrt {1-2 x} (2+3 x)^{7/2}}{5 \sqrt {3+5 x}}+\frac {48}{175} \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}-\frac {2}{175} \int \frac {(2+3 x)^{3/2} \left (-79+\frac {183 x}{2}\right )}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {2 \sqrt {1-2 x} (2+3 x)^{7/2}}{5 \sqrt {3+5 x}}+\frac {183 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{4375}+\frac {48}{175} \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}+\frac {2 \int \frac {\sqrt {2+3 x} \left (\frac {11225}{4}+3729 x\right )}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{4375}\\ &=-\frac {2 \sqrt {1-2 x} (2+3 x)^{7/2}}{5 \sqrt {3+5 x}}-\frac {2486 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{21875}+\frac {183 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{4375}+\frac {48}{175} \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}-\frac {2 \int \frac {-97239-\frac {609537 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{65625}\\ &=-\frac {2 \sqrt {1-2 x} (2+3 x)^{7/2}}{5 \sqrt {3+5 x}}-\frac {2486 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{21875}+\frac {183 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{4375}+\frac {48}{175} \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}+\frac {38723 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{218750}+\frac {203179 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{218750}\\ &=-\frac {2 \sqrt {1-2 x} (2+3 x)^{7/2}}{5 \sqrt {3+5 x}}-\frac {2486 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{21875}+\frac {183 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{4375}+\frac {48}{175} \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}-\frac {203179 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{218750}-\frac {38723 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{109375 \sqrt {33}}\\ \end {align*}
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Mathematica [A]
time = 4.35, size = 107, normalized size = 0.57 \begin {gather*} \frac {\frac {30 \sqrt {1-2 x} \sqrt {2+3 x} \left (32+25955 x+63225 x^2+33750 x^3\right )}{\sqrt {3+5 x}}+203179 \sqrt {2} E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-87010 \sqrt {2} F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )}{656250} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 148, normalized size = 0.78
method | result | size |
default | \(-\frac {\sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (116169 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-203179 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-6075000 x^{5}-12393000 x^{4}-4543650 x^{3}+3009090 x^{2}+1556340 x +1920\right )}{656250 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\) | \(148\) |
elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {54 x^{2} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{175}+\frac {1719 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{4375}+\frac {34 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{21875}+\frac {64826 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{459375 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {203179 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{918750 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {2 \left (-30 x^{2}-5 x +10\right )}{3125 \sqrt {\left (x +\frac {3}{5}\right ) \left (-30 x^{2}-5 x +10\right )}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(262\) |
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 = 0.14, size = 38, normalized size = 0.20 \begin {gather*} \frac {{\left (33750 \, x^{3} + 63225 \, x^{2} + 25955 \, x + 32\right )} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{21875 \, \sqrt {5 \, x + 3}} \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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