Optimal. Leaf size=160 \[ -\frac {37 \sqrt {1-2 x} (2+3 x)^{3/2}}{605 \sqrt {3+5 x}}+\frac {7 (2+3 x)^{5/2}}{11 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {2388 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{3025}+\frac {55019 \sqrt {\frac {3}{11}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{2750}+\frac {823 \sqrt {\frac {3}{11}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1375} \]
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Rubi [A]
time = 0.04, antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {100, 155, 159,
164, 114, 120} \begin {gather*} \frac {823 \sqrt {\frac {3}{11}} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1375}+\frac {55019 \sqrt {\frac {3}{11}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{2750}+\frac {7 (3 x+2)^{5/2}}{11 \sqrt {1-2 x} \sqrt {5 x+3}}-\frac {37 \sqrt {1-2 x} (3 x+2)^{3/2}}{605 \sqrt {5 x+3}}+\frac {2388 \sqrt {1-2 x} \sqrt {5 x+3} \sqrt {3 x+2}}{3025} \end {gather*}
Antiderivative was successfully verified.
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Rule 100
Rule 114
Rule 120
Rule 155
Rule 159
Rule 164
Rubi steps
\begin {align*} \int \frac {(2+3 x)^{7/2}}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx &=\frac {7 (2+3 x)^{5/2}}{11 \sqrt {1-2 x} \sqrt {3+5 x}}-\frac {1}{11} \int \frac {(2+3 x)^{3/2} \left (\frac {241}{2}+207 x\right )}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx\\ &=-\frac {37 \sqrt {1-2 x} (2+3 x)^{3/2}}{605 \sqrt {3+5 x}}+\frac {7 (2+3 x)^{5/2}}{11 \sqrt {1-2 x} \sqrt {3+5 x}}-\frac {2}{605} \int \frac {\sqrt {2+3 x} \left (\frac {8775}{4}+3582 x\right )}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {37 \sqrt {1-2 x} (2+3 x)^{3/2}}{605 \sqrt {3+5 x}}+\frac {7 (2+3 x)^{5/2}}{11 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {2388 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{3025}+\frac {2 \int \frac {-\frac {156699}{2}-\frac {495171 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{9075}\\ &=-\frac {37 \sqrt {1-2 x} (2+3 x)^{3/2}}{605 \sqrt {3+5 x}}+\frac {7 (2+3 x)^{5/2}}{11 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {2388 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{3025}-\frac {2469 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{2750}-\frac {165057 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{30250}\\ &=-\frac {37 \sqrt {1-2 x} (2+3 x)^{3/2}}{605 \sqrt {3+5 x}}+\frac {7 (2+3 x)^{5/2}}{11 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {2388 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{3025}+\frac {55019 \sqrt {\frac {3}{11}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{2750}+\frac {823 \sqrt {\frac {3}{11}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1375}\\ \end {align*}
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Mathematica [A]
time = 7.37, size = 127, normalized size = 0.79 \begin {gather*} \frac {10 \sqrt {2+3 x} \sqrt {3+5 x} \left (14494+20897 x-5445 x^2\right )-55019 \sqrt {2-4 x} (3+5 x) E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )+27860 \sqrt {2-4 x} (3+5 x) F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )}{30250 \sqrt {1-2 x} (3+5 x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 138, normalized size = 0.86
method | result | size |
default | \(\frac {\sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (27159 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-55019 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+163350 x^{3}-518010 x^{2}-852760 x -289880\right )}{907500 x^{3}+695750 x^{2}-211750 x -181500}\) | \(138\) |
elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {2 \left (-20-30 x \right ) \left (\frac {25721}{121000}+\frac {42883 x}{121000}\right )}{\sqrt {\left (x^{2}+\frac {1}{10} x -\frac {3}{10}\right ) \left (-20-30 x \right )}}+\frac {9 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{50}-\frac {17411 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{21175 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {55019 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{42350 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(220\) |
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 = 0.15, size = 43, normalized size = 0.27 \begin {gather*} \frac {{\left (5445 \, x^{2} - 20897 \, x - 14494\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{3025 \, {\left (10 \, x^{2} + x - 3\right )}} \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 {{\left (3\,x+2\right )}^{7/2}}{{\left (1-2\,x\right )}^{3/2}\,{\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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