Optimal. Leaf size=222 \[ \frac {1353340 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{35721}-\frac {62596 \sqrt {1-2 x} (3+5 x)^{3/2}}{3969 \sqrt {2+3 x}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{21 (2+3 x)^{7/2}}+\frac {74 (1-2 x)^{3/2} (3+5 x)^{5/2}}{63 (2+3 x)^{5/2}}-\frac {1844 \sqrt {1-2 x} (3+5 x)^{5/2}}{567 (2+3 x)^{3/2}}-\frac {904798 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{35721}+\frac {270668 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{35721} \]
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Rubi [A]
time = 0.05, antiderivative size = 222, 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 = {99, 155, 159,
164, 114, 120} \begin {gather*} \frac {270668 \sqrt {\frac {11}{3}} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{35721}-\frac {904798 \sqrt {\frac {11}{3}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{35721}-\frac {1844 \sqrt {1-2 x} (5 x+3)^{5/2}}{567 (3 x+2)^{3/2}}+\frac {74 (1-2 x)^{3/2} (5 x+3)^{5/2}}{63 (3 x+2)^{5/2}}-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{21 (3 x+2)^{7/2}}-\frac {62596 \sqrt {1-2 x} (5 x+3)^{3/2}}{3969 \sqrt {3 x+2}}+\frac {1353340 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}{35721} \end {gather*}
Antiderivative was successfully verified.
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Rule 99
Rule 114
Rule 120
Rule 155
Rule 159
Rule 164
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{9/2}} \, dx &=-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{21 (2+3 x)^{7/2}}+\frac {2}{21} \int \frac {\left (-\frac {5}{2}-50 x\right ) (1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^{7/2}} \, dx\\ &=-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{21 (2+3 x)^{7/2}}+\frac {74 (1-2 x)^{3/2} (3+5 x)^{5/2}}{63 (2+3 x)^{5/2}}-\frac {4}{315} \int \frac {\left (-\frac {1415}{2}-\frac {3275 x}{2}\right ) \sqrt {1-2 x} (3+5 x)^{3/2}}{(2+3 x)^{5/2}} \, dx\\ &=-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{21 (2+3 x)^{7/2}}+\frac {74 (1-2 x)^{3/2} (3+5 x)^{5/2}}{63 (2+3 x)^{5/2}}-\frac {1844 \sqrt {1-2 x} (3+5 x)^{5/2}}{567 (2+3 x)^{3/2}}+\frac {8 \int \frac {\left (\frac {19045}{4}-22200 x\right ) (3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^{3/2}} \, dx}{2835}\\ &=-\frac {62596 \sqrt {1-2 x} (3+5 x)^{3/2}}{3969 \sqrt {2+3 x}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{21 (2+3 x)^{7/2}}+\frac {74 (1-2 x)^{3/2} (3+5 x)^{5/2}}{63 (2+3 x)^{5/2}}-\frac {1844 \sqrt {1-2 x} (3+5 x)^{5/2}}{567 (2+3 x)^{3/2}}+\frac {16 \int \frac {\left (\frac {1656225}{8}-\frac {5075025 x}{4}\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{59535}\\ &=\frac {1353340 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{35721}-\frac {62596 \sqrt {1-2 x} (3+5 x)^{3/2}}{3969 \sqrt {2+3 x}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{21 (2+3 x)^{7/2}}+\frac {74 (1-2 x)^{3/2} (3+5 x)^{5/2}}{63 (2+3 x)^{5/2}}-\frac {1844 \sqrt {1-2 x} (3+5 x)^{5/2}}{567 (2+3 x)^{3/2}}-\frac {16 \int \frac {-\frac {2298225}{2}-\frac {33929925 x}{8}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{535815}\\ &=\frac {1353340 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{35721}-\frac {62596 \sqrt {1-2 x} (3+5 x)^{3/2}}{3969 \sqrt {2+3 x}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{21 (2+3 x)^{7/2}}+\frac {74 (1-2 x)^{3/2} (3+5 x)^{5/2}}{63 (2+3 x)^{5/2}}-\frac {1844 \sqrt {1-2 x} (3+5 x)^{5/2}}{567 (2+3 x)^{3/2}}+\frac {904798 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{35721}-\frac {1488674 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{35721}\\ &=\frac {1353340 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{35721}-\frac {62596 \sqrt {1-2 x} (3+5 x)^{3/2}}{3969 \sqrt {2+3 x}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{21 (2+3 x)^{7/2}}+\frac {74 (1-2 x)^{3/2} (3+5 x)^{5/2}}{63 (2+3 x)^{5/2}}-\frac {1844 \sqrt {1-2 x} (3+5 x)^{5/2}}{567 (2+3 x)^{3/2}}-\frac {904798 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{35721}+\frac {270668 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{35721}\\ \end {align*}
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Mathematica [A]
time = 8.93, size = 109, normalized size = 0.49 \begin {gather*} \frac {2 \left (\frac {3 \sqrt {1-2 x} \sqrt {3+5 x} \left (2337569+11107911 x+17788023 x^2+9846603 x^3+396900 x^4\right )}{(2+3 x)^{7/2}}+\sqrt {2} \left (452399 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-2685410 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )\right )\right )}{107163} \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. \(405\) vs.
\(2(162)=324\).
time = 0.10, size = 406, normalized size = 1.83
method | result | size |
elliptic | \(-\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {74 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{2187 \left (\frac {2}{3}+x \right )^{3}}-\frac {26882 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{45927 \left (\frac {2}{3}+x \right )^{2}}+\frac {-\frac {6509780}{35721} x^{2}-\frac {650978}{35721} x +\frac {650978}{11907}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}+\frac {1225720 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{750141 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {4523990 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{750141 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {200 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{729}-\frac {14 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{19683 \left (\frac {2}{3}+x \right )^{4}}\right )}{\left (10 x^{2}+x -3\right ) \sqrt {2+3 x}}\) | \(303\) |
default | \(\frac {2 \left (12214773 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+60291297 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+24429546 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+120582594 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+16286364 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+80388396 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+11907000 x^{6}+3619192 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+17864088 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+296588790 x^{5}+559608399 x^{4}+297981972 x^{3}-56641404 x^{2}-92958492 x -21038121\right ) \sqrt {3+5 x}\, \sqrt {1-2 x}}{107163 \left (10 x^{2}+x -3\right ) \left (2+3 x \right )^{\frac {7}{2}}}\) | \(406\) |
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 = 0.35, size = 65, normalized size = 0.29 \begin {gather*} \frac {2 \, {\left (396900 \, x^{4} + 9846603 \, x^{3} + 17788023 \, x^{2} + 11107911 \, x + 2337569\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{35721 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\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 = -0.00 \begin {gather*} \int \frac {{\left (1-2\,x\right )}^{5/2}\,{\left (5\,x+3\right )}^{5/2}}{{\left (3\,x+2\right )}^{9/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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