Optimal. Leaf size=188 \[ \frac {3683}{42} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {167}{14} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}+\frac {12}{7} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {(2+3 x)^{3/2} (3+5 x)^{5/2}}{\sqrt {1-2 x}}+\frac {244879}{420} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {3683}{210} \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right ) \]
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Rubi [A]
time = 0.04, antiderivative size = 188, 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 {3683}{210} \sqrt {\frac {11}{3}} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {244879}{420} \sqrt {\frac {11}{3}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {(3 x+2)^{3/2} (5 x+3)^{5/2}}{\sqrt {1-2 x}}+\frac {12}{7} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}+\frac {167}{14} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}+\frac {3683}{42} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3} \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 {(2+3 x)^{3/2} (3+5 x)^{5/2}}{(1-2 x)^{3/2}} \, dx &=\frac {(2+3 x)^{3/2} (3+5 x)^{5/2}}{\sqrt {1-2 x}}-\int \frac {\sqrt {2+3 x} (3+5 x)^{3/2} \left (\frac {77}{2}+60 x\right )}{\sqrt {1-2 x}} \, dx\\ &=\frac {12}{7} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {(2+3 x)^{3/2} (3+5 x)^{5/2}}{\sqrt {1-2 x}}+\frac {1}{35} \int \frac {\left (-4105-\frac {12525 x}{2}\right ) (3+5 x)^{3/2}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx\\ &=\frac {167}{14} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}+\frac {12}{7} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {(2+3 x)^{3/2} (3+5 x)^{5/2}}{\sqrt {1-2 x}}-\frac {1}{525} \int \frac {\sqrt {3+5 x} \left (\frac {1077075}{4}+\frac {828675 x}{2}\right )}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx\\ &=\frac {3683}{42} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {167}{14} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}+\frac {12}{7} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {(2+3 x)^{3/2} (3+5 x)^{5/2}}{\sqrt {1-2 x}}+\frac {\int \frac {-\frac {17440875}{2}-\frac {55097775 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{4725}\\ &=\frac {3683}{42} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {167}{14} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}+\frac {12}{7} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {(2+3 x)^{3/2} (3+5 x)^{5/2}}{\sqrt {1-2 x}}-\frac {40513}{420} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx-\frac {244879}{420} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx\\ &=\frac {3683}{42} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {167}{14} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}+\frac {12}{7} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {(2+3 x)^{3/2} (3+5 x)^{5/2}}{\sqrt {1-2 x}}+\frac {244879}{420} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {3683}{210} \sqrt {\frac {11}{3}} 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 = 6.83, size = 115, normalized size = 0.61 \begin {gather*} \frac {-30 \sqrt {2+3 x} \sqrt {3+5 x} \left (-6590+3349 x+1650 x^2+450 x^3\right )-244879 \sqrt {2-4 x} E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )+123340 \sqrt {2-4 x} F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )}{1260 \sqrt {1-2 x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 148, normalized size = 0.79
method | result | size |
default | \(\frac {\sqrt {2+3 x}\, \sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (121539 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-244879 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+202500 x^{5}+999000 x^{4}+2528550 x^{3}-759570 x^{2}-3153480 x -1186200\right )}{37800 x^{3}+28980 x^{2}-8820 x -7560}\) | \(148\) |
elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \sqrt {3+5 x}\, \sqrt {2+3 x}\, \left (\frac {75 x^{2} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{14}+\frac {625 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{28}+\frac {8573 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{168}-\frac {77515 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{882 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {244879 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{1764 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {847 \left (-30 x^{2}-38 x -12\right )}{16 \sqrt {\left (-\frac {1}{2}+x \right ) \left (-30 x^{2}-38 x -12\right )}}\right )}{\sqrt {1-2 x}\, \left (15 x^{2}+19 x +6\right )}\) | \(274\) |
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 = 0.24, size = 45, normalized size = 0.24 \begin {gather*} \frac {{\left (450 \, x^{3} + 1650 \, x^{2} + 3349 \, x - 6590\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{42 \, {\left (2 \, x - 1\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 {{\left (3\,x+2\right )}^{3/2}\,{\left (5\,x+3\right )}^{5/2}}{{\left (1-2\,x\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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