Optimal. Leaf size=187 \[ \frac {2063 \sqrt {1-2 x} (2+3 x)^{3/2}}{19965 (3+5 x)^{3/2}}-\frac {140 (2+3 x)^{5/2}}{121 \sqrt {1-2 x} (3+5 x)^{3/2}}+\frac {7 (2+3 x)^{7/2}}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac {70226 \sqrt {1-2 x} \sqrt {2+3 x}}{1098075 \sqrt {3+5 x}}-\frac {4971289 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{332750 \sqrt {33}}-\frac {76163 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{166375 \sqrt {33}} \]
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Rubi [A]
time = 0.04, antiderivative size = 187, 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 = {100, 155, 164,
114, 120} \begin {gather*} -\frac {76163 F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{166375 \sqrt {33}}-\frac {4971289 E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{332750 \sqrt {33}}+\frac {7 (3 x+2)^{7/2}}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}-\frac {140 (3 x+2)^{5/2}}{121 \sqrt {1-2 x} (5 x+3)^{3/2}}+\frac {2063 \sqrt {1-2 x} (3 x+2)^{3/2}}{19965 (5 x+3)^{3/2}}+\frac {70226 \sqrt {1-2 x} \sqrt {3 x+2}}{1098075 \sqrt {5 x+3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 100
Rule 114
Rule 120
Rule 155
Rule 164
Rubi steps
\begin {align*} \int \frac {(2+3 x)^{9/2}}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx &=\frac {7 (2+3 x)^{7/2}}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac {1}{33} \int \frac {(2+3 x)^{5/2} \left (\frac {219}{2}+201 x\right )}{(1-2 x)^{3/2} (3+5 x)^{5/2}} \, dx\\ &=-\frac {140 (2+3 x)^{5/2}}{121 \sqrt {1-2 x} (3+5 x)^{3/2}}+\frac {7 (2+3 x)^{7/2}}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac {1}{363} \int \frac {\left (-3261-\frac {12933 x}{2}\right ) (2+3 x)^{3/2}}{\sqrt {1-2 x} (3+5 x)^{5/2}} \, dx\\ &=\frac {2063 \sqrt {1-2 x} (2+3 x)^{3/2}}{19965 (3+5 x)^{3/2}}-\frac {140 (2+3 x)^{5/2}}{121 \sqrt {1-2 x} (3+5 x)^{3/2}}+\frac {7 (2+3 x)^{7/2}}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac {2 \int \frac {\left (-\frac {748365}{4}-\frac {1317501 x}{4}\right ) \sqrt {2+3 x}}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx}{59895}\\ &=\frac {2063 \sqrt {1-2 x} (2+3 x)^{3/2}}{19965 (3+5 x)^{3/2}}-\frac {140 (2+3 x)^{5/2}}{121 \sqrt {1-2 x} (3+5 x)^{3/2}}+\frac {7 (2+3 x)^{7/2}}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac {70226 \sqrt {1-2 x} \sqrt {2+3 x}}{1098075 \sqrt {3+5 x}}-\frac {4 \int \frac {-\frac {7088247}{2}-\frac {44741601 x}{8}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{3294225}\\ &=\frac {2063 \sqrt {1-2 x} (2+3 x)^{3/2}}{19965 (3+5 x)^{3/2}}-\frac {140 (2+3 x)^{5/2}}{121 \sqrt {1-2 x} (3+5 x)^{3/2}}+\frac {7 (2+3 x)^{7/2}}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac {70226 \sqrt {1-2 x} \sqrt {2+3 x}}{1098075 \sqrt {3+5 x}}+\frac {76163 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{332750}+\frac {4971289 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{3660250}\\ &=\frac {2063 \sqrt {1-2 x} (2+3 x)^{3/2}}{19965 (3+5 x)^{3/2}}-\frac {140 (2+3 x)^{5/2}}{121 \sqrt {1-2 x} (3+5 x)^{3/2}}+\frac {7 (2+3 x)^{7/2}}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac {70226 \sqrt {1-2 x} \sqrt {2+3 x}}{1098075 \sqrt {3+5 x}}-\frac {4971289 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{332750 \sqrt {33}}-\frac {76163 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{166375 \sqrt {33}}\\ \end {align*}
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Mathematica [A]
time = 8.73, size = 107, normalized size = 0.57 \begin {gather*} \frac {\frac {10 \sqrt {2+3 x} \left (-2780992+2244393 x+30619782 x^2+31924075 x^3\right )}{(1-2 x)^{3/2} (3+5 x)^{3/2}}+4971289 \sqrt {2} E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-2457910 \sqrt {2} F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )}{10980750} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(304\) vs.
\(2(139)=278\).
time = 0.11, size = 305, normalized size = 1.63
method | result | size |
elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {\left (\frac {900367}{18150000}+\frac {1500641 x}{18150000}\right ) \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{\left (x^{2}+\frac {1}{10} x -\frac {3}{10}\right )^{2}}-\frac {2 \left (-20-30 x \right ) \left (-\frac {54854749}{439230000}-\frac {1276963 x}{8784600}\right )}{\sqrt {\left (x^{2}+\frac {1}{10} x -\frac {3}{10}\right ) \left (-20-30 x \right )}}+\frac {1575166 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{7686525 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {4971289 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{15373050 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(234\) |
default | \(-\frac {\sqrt {1-2 x}\, \left (25133790 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-49712890 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+2513379 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-4971289 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-7540137 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+14913867 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-957722250 x^{4}-1557074960 x^{3}-679727430 x^{2}+38541900 x +55619840\right )}{10980750 \left (3+5 x \right )^{\frac {3}{2}} \left (-1+2 x \right )^{2} \sqrt {2+3 x}}\) | \(305\) |
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 = 0.16, size = 60, normalized size = 0.32 \begin {gather*} \frac {{\left (31924075 \, x^{3} + 30619782 \, x^{2} + 2244393 \, x - 2780992\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{1098075 \, {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\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 )}^{9/2}}{{\left (1-2\,x\right )}^{5/2}\,{\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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