Optimal. Leaf size=218 \[ \frac {4439 \sqrt {1-2 x} (2+3 x)^{5/2}}{19965 \sqrt {3+5 x}}-\frac {896 (2+3 x)^{7/2}}{363 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {7 (2+3 x)^{9/2}}{33 (1-2 x)^{3/2} \sqrt {3+5 x}}-\frac {21713939 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{1663750}-\frac {932783 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{332750}-\frac {1508889271 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1512500 \sqrt {33}}-\frac {11346991 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{378125 \sqrt {33}} \]
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Rubi [A]
time = 0.06, antiderivative size = 218, normalized size of antiderivative = 1.00, number of steps
used = 8, 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 {11346991 F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{378125 \sqrt {33}}-\frac {1508889271 E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1512500 \sqrt {33}}+\frac {7 (3 x+2)^{9/2}}{33 (1-2 x)^{3/2} \sqrt {5 x+3}}-\frac {896 (3 x+2)^{7/2}}{363 \sqrt {1-2 x} \sqrt {5 x+3}}+\frac {4439 \sqrt {1-2 x} (3 x+2)^{5/2}}{19965 \sqrt {5 x+3}}-\frac {932783 \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{3/2}}{332750}-\frac {21713939 \sqrt {1-2 x} \sqrt {5 x+3} \sqrt {3 x+2}}{1663750} \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)^{11/2}}{(1-2 x)^{5/2} (3+5 x)^{3/2}} \, dx &=\frac {7 (2+3 x)^{9/2}}{33 (1-2 x)^{3/2} \sqrt {3+5 x}}-\frac {1}{33} \int \frac {(2+3 x)^{7/2} \left (\frac {485}{2}+411 x\right )}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx\\ &=-\frac {896 (2+3 x)^{7/2}}{363 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {7 (2+3 x)^{9/2}}{33 (1-2 x)^{3/2} \sqrt {3+5 x}}-\frac {1}{363} \int \frac {\left (-23785-\frac {80763 x}{2}\right ) (2+3 x)^{5/2}}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx\\ &=\frac {4439 \sqrt {1-2 x} (2+3 x)^{5/2}}{19965 \sqrt {3+5 x}}-\frac {896 (2+3 x)^{7/2}}{363 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {7 (2+3 x)^{9/2}}{33 (1-2 x)^{3/2} \sqrt {3+5 x}}-\frac {2 \int \frac {\left (-\frac {1710201}{4}-\frac {2798349 x}{4}\right ) (2+3 x)^{3/2}}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{19965}\\ &=\frac {4439 \sqrt {1-2 x} (2+3 x)^{5/2}}{19965 \sqrt {3+5 x}}-\frac {896 (2+3 x)^{7/2}}{363 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {7 (2+3 x)^{9/2}}{33 (1-2 x)^{3/2} \sqrt {3+5 x}}-\frac {932783 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{332750}+\frac {2 \int \frac {\sqrt {2+3 x} \left (\frac {240978825}{8}+\frac {195425451 x}{4}\right )}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{499125}\\ &=\frac {4439 \sqrt {1-2 x} (2+3 x)^{5/2}}{19965 \sqrt {3+5 x}}-\frac {896 (2+3 x)^{7/2}}{363 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {7 (2+3 x)^{9/2}}{33 (1-2 x)^{3/2} \sqrt {3+5 x}}-\frac {21713939 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{1663750}-\frac {932783 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{332750}-\frac {2 \int \frac {-\frac {8597342907}{8}-\frac {13580003439 x}{8}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{7486875}\\ &=\frac {4439 \sqrt {1-2 x} (2+3 x)^{5/2}}{19965 \sqrt {3+5 x}}-\frac {896 (2+3 x)^{7/2}}{363 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {7 (2+3 x)^{9/2}}{33 (1-2 x)^{3/2} \sqrt {3+5 x}}-\frac {21713939 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{1663750}-\frac {932783 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{332750}+\frac {11346991 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{756250}+\frac {1508889271 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{16637500}\\ &=\frac {4439 \sqrt {1-2 x} (2+3 x)^{5/2}}{19965 \sqrt {3+5 x}}-\frac {896 (2+3 x)^{7/2}}{363 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {7 (2+3 x)^{9/2}}{33 (1-2 x)^{3/2} \sqrt {3+5 x}}-\frac {21713939 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{1663750}-\frac {932783 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{332750}-\frac {1508889271 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1512500 \sqrt {33}}-\frac {11346991 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{378125 \sqrt {33}}\\ \end {align*}
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Mathematica [A]
time = 8.91, size = 107, normalized size = 0.49 \begin {gather*} \frac {-\frac {5 \sqrt {4+6 x} \left (356556921-376752444 x-1463754851 x^2+286777260 x^3+48514950 x^4\right )}{(1-2 x)^{3/2} \sqrt {3+5 x}}+1508889271 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-759987865 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )}{24956250 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 234, normalized size = 1.07
method | result | size |
default | \(-\frac {\sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (1497802812 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-3017778542 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-748901406 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+1508889271 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+1455448500 x^{5}+9573616800 x^{4}-38177100330 x^{3}-40577670340 x^{2}+3161658750 x +7131138420\right )}{49912500 \left (15 x^{2}+19 x +6\right ) \left (-1+2 x \right )^{2}}\) | \(234\) |
elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {243 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{500}-\frac {1917 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{625}+\frac {955260323 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{69877500 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {1508889271 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{69877500 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {-\frac {4237765}{10648} x^{2}-\frac {16103507}{31944} x -\frac {847553}{5324}}{\sqrt {\left (-\frac {1}{2}+x \right ) \left (-30 x^{2}-38 x -12\right )}}+\frac {16807 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{11616 \left (-\frac {1}{2}+x \right )^{2}}-\frac {2 \left (-30 x^{2}-5 x +10\right )}{831875 \sqrt {\left (x +\frac {3}{5}\right ) \left (-30 x^{2}-5 x +10\right )}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(292\) |
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 = 60, normalized size = 0.28 \begin {gather*} -\frac {{\left (48514950 \, x^{4} + 286777260 \, x^{3} - 1463754851 \, x^{2} - 376752444 \, x + 356556921\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{4991250 \, {\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\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 = -0.00 \begin {gather*} \int \frac {{\left (3\,x+2\right )}^{11/2}}{{\left (1-2\,x\right )}^{5/2}\,{\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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