Optimal. Leaf size=156 \[ -\frac {107 \sqrt {1-2 x} \sqrt {2+3 x}}{1815 (3+5 x)^{3/2}}+\frac {7 (2+3 x)^{3/2}}{11 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {4157 \sqrt {1-2 x} \sqrt {2+3 x}}{19965 \sqrt {3+5 x}}+\frac {4157 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3025 \sqrt {33}}-\frac {412 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3025 \sqrt {33}} \]
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Rubi [A]
time = 0.04, antiderivative size = 156, 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, 157,
164, 114, 120} \begin {gather*} -\frac {412 F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3025 \sqrt {33}}+\frac {4157 E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3025 \sqrt {33}}+\frac {7 (3 x+2)^{3/2}}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}-\frac {4157 \sqrt {1-2 x} \sqrt {3 x+2}}{19965 \sqrt {5 x+3}}-\frac {107 \sqrt {1-2 x} \sqrt {3 x+2}}{1815 (5 x+3)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 100
Rule 114
Rule 120
Rule 155
Rule 157
Rule 164
Rubi steps
\begin {align*} \int \frac {(2+3 x)^{5/2}}{(1-2 x)^{3/2} (3+5 x)^{5/2}} \, dx &=\frac {7 (2+3 x)^{3/2}}{11 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {1}{11} \int \frac {\left (-\frac {25}{2}-3 x\right ) \sqrt {2+3 x}}{\sqrt {1-2 x} (3+5 x)^{5/2}} \, dx\\ &=-\frac {107 \sqrt {1-2 x} \sqrt {2+3 x}}{1815 (3+5 x)^{3/2}}+\frac {7 (2+3 x)^{3/2}}{11 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {2 \int \frac {-\frac {1573}{4}-309 x}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}} \, dx}{1815}\\ &=-\frac {107 \sqrt {1-2 x} \sqrt {2+3 x}}{1815 (3+5 x)^{3/2}}+\frac {7 (2+3 x)^{3/2}}{11 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {4157 \sqrt {1-2 x} \sqrt {2+3 x}}{19965 \sqrt {3+5 x}}+\frac {4 \int \frac {-\frac {6123}{4}-\frac {12471 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{19965}\\ &=-\frac {107 \sqrt {1-2 x} \sqrt {2+3 x}}{1815 (3+5 x)^{3/2}}+\frac {7 (2+3 x)^{3/2}}{11 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {4157 \sqrt {1-2 x} \sqrt {2+3 x}}{19965 \sqrt {3+5 x}}+\frac {206 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{3025}-\frac {4157 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{33275}\\ &=-\frac {107 \sqrt {1-2 x} \sqrt {2+3 x}}{1815 (3+5 x)^{3/2}}+\frac {7 (2+3 x)^{3/2}}{11 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {4157 \sqrt {1-2 x} \sqrt {2+3 x}}{19965 \sqrt {3+5 x}}+\frac {4157 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3025 \sqrt {33}}-\frac {412 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3025 \sqrt {33}}\\ \end {align*}
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Mathematica [A]
time = 7.42, size = 97, normalized size = 0.62 \begin {gather*} \frac {\sqrt {2} \left (\frac {5 \sqrt {4+6 x} \left (5881+22313 x+20785 x^2\right )}{\sqrt {1-2 x} (3+5 x)^{3/2}}-4157 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )+10955 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )\right )}{99825} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 215, normalized size = 1.38
method | result | size |
default | \(-\frac {\sqrt {2+3 x}\, \sqrt {1-2 x}\, \left (33990 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+20785 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+20394 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+12471 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+623550 x^{3}+1085090 x^{2}+622690 x +117620\right )}{99825 \left (3+5 x \right )^{\frac {3}{2}} \left (6 x^{2}+x -2\right )}\) | \(215\) |
elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {49 \left (-30 x^{2}-38 x -12\right )}{1331 \sqrt {\left (-\frac {1}{2}+x \right ) \left (-30 x^{2}-38 x -12\right )}}-\frac {2041 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{139755 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {4157 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{139755 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {2 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{45375 \left (x +\frac {3}{5}\right )^{2}}-\frac {482 \left (-30 x^{2}-5 x +10\right )}{99825 \sqrt {\left (x +\frac {3}{5}\right ) \left (-30 x^{2}-5 x +10\right )}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(253\) |
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 = 0.27, size = 50, normalized size = 0.32 \begin {gather*} -\frac {2 \, {\left (20785 \, x^{2} + 22313 \, x + 5881\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{19965 \, {\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\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 )}^{5/2}}{{\left (1-2\,x\right )}^{3/2}\,{\left (5\,x+3\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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