Optimal. Leaf size=222 \[ -\frac {1241596 \sqrt {1-2 x} \sqrt {3+5 x}}{750141 \sqrt {2+3 x}}-\frac {13316 \sqrt {1-2 x} (3+5 x)^{3/2}}{35721 (2+3 x)^{3/2}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{27 (2+3 x)^{9/2}}+\frac {370 (1-2 x)^{3/2} (3+5 x)^{5/2}}{567 (2+3 x)^{7/2}}+\frac {2776 \sqrt {1-2 x} (3+5 x)^{5/2}}{1701 (2+3 x)^{5/2}}-\frac {100444 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{750141}-\frac {1241596 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{750141} \]
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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 = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {99, 155, 164,
114, 120} \begin {gather*} -\frac {1241596 \sqrt {\frac {11}{3}} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{750141}-\frac {100444 \sqrt {\frac {11}{3}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{750141}+\frac {2776 \sqrt {1-2 x} (5 x+3)^{5/2}}{1701 (3 x+2)^{5/2}}+\frac {370 (1-2 x)^{3/2} (5 x+3)^{5/2}}{567 (3 x+2)^{7/2}}-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{27 (3 x+2)^{9/2}}-\frac {13316 \sqrt {1-2 x} (5 x+3)^{3/2}}{35721 (3 x+2)^{3/2}}-\frac {1241596 \sqrt {1-2 x} \sqrt {5 x+3}}{750141 \sqrt {3 x+2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 99
Rule 114
Rule 120
Rule 155
Rule 164
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{11/2}} \, dx &=-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{27 (2+3 x)^{9/2}}+\frac {2}{27} \int \frac {\left (-\frac {5}{2}-50 x\right ) (1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^{9/2}} \, dx\\ &=-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{27 (2+3 x)^{9/2}}+\frac {370 (1-2 x)^{3/2} (3+5 x)^{5/2}}{567 (2+3 x)^{7/2}}-\frac {4}{567} \int \frac {\left (-1120-\frac {1625 x}{2}\right ) \sqrt {1-2 x} (3+5 x)^{3/2}}{(2+3 x)^{7/2}} \, dx\\ &=-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{27 (2+3 x)^{9/2}}+\frac {370 (1-2 x)^{3/2} (3+5 x)^{5/2}}{567 (2+3 x)^{7/2}}+\frac {2776 \sqrt {1-2 x} (3+5 x)^{5/2}}{1701 (2+3 x)^{5/2}}+\frac {8 \int \frac {(3+5 x)^{3/2} \left (\frac {28945}{4}+\frac {9225 x}{2}\right )}{\sqrt {1-2 x} (2+3 x)^{5/2}} \, dx}{8505}\\ &=-\frac {13316 \sqrt {1-2 x} (3+5 x)^{3/2}}{35721 (2+3 x)^{3/2}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{27 (2+3 x)^{9/2}}+\frac {370 (1-2 x)^{3/2} (3+5 x)^{5/2}}{567 (2+3 x)^{7/2}}+\frac {2776 \sqrt {1-2 x} (3+5 x)^{5/2}}{1701 (2+3 x)^{5/2}}+\frac {16 \int \frac {\sqrt {3+5 x} \left (\frac {2510595}{8}+\frac {359475 x}{2}\right )}{\sqrt {1-2 x} (2+3 x)^{3/2}} \, dx}{535815}\\ &=-\frac {1241596 \sqrt {1-2 x} \sqrt {3+5 x}}{750141 \sqrt {2+3 x}}-\frac {13316 \sqrt {1-2 x} (3+5 x)^{3/2}}{35721 (2+3 x)^{3/2}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{27 (2+3 x)^{9/2}}+\frac {370 (1-2 x)^{3/2} (3+5 x)^{5/2}}{567 (2+3 x)^{7/2}}+\frac {2776 \sqrt {1-2 x} (3+5 x)^{5/2}}{1701 (2+3 x)^{5/2}}+\frac {32 \int \frac {\frac {53475825}{16}+\frac {1883325 x}{8}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{11252115}\\ &=-\frac {1241596 \sqrt {1-2 x} \sqrt {3+5 x}}{750141 \sqrt {2+3 x}}-\frac {13316 \sqrt {1-2 x} (3+5 x)^{3/2}}{35721 (2+3 x)^{3/2}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{27 (2+3 x)^{9/2}}+\frac {370 (1-2 x)^{3/2} (3+5 x)^{5/2}}{567 (2+3 x)^{7/2}}+\frac {2776 \sqrt {1-2 x} (3+5 x)^{5/2}}{1701 (2+3 x)^{5/2}}+\frac {100444 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{750141}+\frac {6828778 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{750141}\\ &=-\frac {1241596 \sqrt {1-2 x} \sqrt {3+5 x}}{750141 \sqrt {2+3 x}}-\frac {13316 \sqrt {1-2 x} (3+5 x)^{3/2}}{35721 (2+3 x)^{3/2}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{27 (2+3 x)^{9/2}}+\frac {370 (1-2 x)^{3/2} (3+5 x)^{5/2}}{567 (2+3 x)^{7/2}}+\frac {2776 \sqrt {1-2 x} (3+5 x)^{5/2}}{1701 (2+3 x)^{5/2}}-\frac {100444 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{750141}-\frac {1241596 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{750141}\\ \end {align*}
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Mathematica [A]
time = 8.99, size = 109, normalized size = 0.49 \begin {gather*} \frac {2 \left (\frac {3 \sqrt {1-2 x} \sqrt {3+5 x} \left (12903031+71920155 x+142557831 x^2+115002639 x^3+29072682 x^4\right )}{(2+3 x)^{9/2}}+\sqrt {2} \left (50222 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )+10192945 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )\right )\right )}{2250423} \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. \(493\) vs.
\(2(162)=324\).
time = 0.10, size = 494, normalized size = 2.23
method | result | size |
elliptic | \(-\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {98 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{531441 \left (\frac {2}{3}+x \right )^{5}}-\frac {44990 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{413343 \left (\frac {2}{3}+x \right )^{3}}+\frac {1406 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{177147 \left (\frac {2}{3}+x \right )^{4}}+\frac {-\frac {7178440}{750141} x^{2}-\frac {717844}{750141} x +\frac {717844}{250047}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}+\frac {396566 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{964467 \left (\frac {2}{3}+x \right )^{2}}+\frac {7130110 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15752961 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {502220 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{15752961 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\left (10 x^{2}+x -3\right ) \sqrt {2+3 x}}\) | \(308\) |
default | \(-\frac {2 \left (829696527 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{4} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-4067982 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{4} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+2212524072 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-10847952 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+2212524072 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-10847952 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+983344032 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-4821312 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-872180460 x^{6}+163890672 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-803552 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-3537297216 x^{5}-4360088709 x^{4}-1550254392 x^{3}+680169084 x^{2}+608572302 x +116127279\right ) \sqrt {3+5 x}\, \sqrt {1-2 x}}{2250423 \left (10 x^{2}+x -3\right ) \left (2+3 x \right )^{\frac {9}{2}}}\) | \(494\) |
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 = 70, normalized size = 0.32 \begin {gather*} \frac {2 \, {\left (29072682 \, x^{4} + 115002639 \, x^{3} + 142557831 \, x^{2} + 71920155 \, x + 12903031\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{750141 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\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 )}^{11/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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