Optimal. Leaf size=222 \[ -\frac {558524 \sqrt {1-2 x} \sqrt {3+5 x}}{1250235 (2+3 x)^{3/2}}+\frac {17830424 \sqrt {1-2 x} \sqrt {3+5 x}}{8751645 \sqrt {2+3 x}}-\frac {1864 \sqrt {1-2 x} (3+5 x)^{3/2}}{6615 (2+3 x)^{5/2}}-\frac {2 (1-2 x)^{3/2} (3+5 x)^{5/2}}{27 (2+3 x)^{9/2}}+\frac {362 \sqrt {1-2 x} (3+5 x)^{5/2}}{567 (2+3 x)^{7/2}}-\frac {17830424 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{8751645}-\frac {1717916 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{8751645} \]
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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, 157,
164, 114, 120} \begin {gather*} -\frac {1717916 \sqrt {\frac {11}{3}} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{8751645}-\frac {17830424 \sqrt {\frac {11}{3}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{8751645}+\frac {362 \sqrt {1-2 x} (5 x+3)^{5/2}}{567 (3 x+2)^{7/2}}-\frac {2 (1-2 x)^{3/2} (5 x+3)^{5/2}}{27 (3 x+2)^{9/2}}-\frac {1864 \sqrt {1-2 x} (5 x+3)^{3/2}}{6615 (3 x+2)^{5/2}}+\frac {17830424 \sqrt {1-2 x} \sqrt {5 x+3}}{8751645 \sqrt {3 x+2}}-\frac {558524 \sqrt {1-2 x} \sqrt {5 x+3}}{1250235 (3 x+2)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 99
Rule 114
Rule 120
Rule 155
Rule 157
Rule 164
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^{11/2}} \, dx &=-\frac {2 (1-2 x)^{3/2} (3+5 x)^{5/2}}{27 (2+3 x)^{9/2}}+\frac {2}{27} \int \frac {\left (\frac {7}{2}-40 x\right ) \sqrt {1-2 x} (3+5 x)^{3/2}}{(2+3 x)^{9/2}} \, dx\\ &=-\frac {2 (1-2 x)^{3/2} (3+5 x)^{5/2}}{27 (2+3 x)^{9/2}}+\frac {362 \sqrt {1-2 x} (3+5 x)^{5/2}}{567 (2+3 x)^{7/2}}-\frac {4}{567} \int \frac {(3+5 x)^{3/2} \left (-584+\frac {345 x}{2}\right )}{\sqrt {1-2 x} (2+3 x)^{7/2}} \, dx\\ &=-\frac {1864 \sqrt {1-2 x} (3+5 x)^{3/2}}{6615 (2+3 x)^{5/2}}-\frac {2 (1-2 x)^{3/2} (3+5 x)^{5/2}}{27 (2+3 x)^{9/2}}+\frac {362 \sqrt {1-2 x} (3+5 x)^{5/2}}{567 (2+3 x)^{7/2}}-\frac {8 \int \frac {\sqrt {3+5 x} \left (-\frac {127341}{4}+\frac {18435 x}{4}\right )}{\sqrt {1-2 x} (2+3 x)^{5/2}} \, dx}{59535}\\ &=-\frac {558524 \sqrt {1-2 x} \sqrt {3+5 x}}{1250235 (2+3 x)^{3/2}}-\frac {1864 \sqrt {1-2 x} (3+5 x)^{3/2}}{6615 (2+3 x)^{5/2}}-\frac {2 (1-2 x)^{3/2} (3+5 x)^{5/2}}{27 (2+3 x)^{9/2}}+\frac {362 \sqrt {1-2 x} (3+5 x)^{5/2}}{567 (2+3 x)^{7/2}}-\frac {16 \int \frac {-744972-\frac {2253255 x}{8}}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx}{3750705}\\ &=-\frac {558524 \sqrt {1-2 x} \sqrt {3+5 x}}{1250235 (2+3 x)^{3/2}}+\frac {17830424 \sqrt {1-2 x} \sqrt {3+5 x}}{8751645 \sqrt {2+3 x}}-\frac {1864 \sqrt {1-2 x} (3+5 x)^{3/2}}{6615 (2+3 x)^{5/2}}-\frac {2 (1-2 x)^{3/2} (3+5 x)^{5/2}}{27 (2+3 x)^{9/2}}+\frac {362 \sqrt {1-2 x} (3+5 x)^{5/2}}{567 (2+3 x)^{7/2}}-\frac {32 \int \frac {-\frac {94409715}{16}-\frac {33432045 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{26254935}\\ &=-\frac {558524 \sqrt {1-2 x} \sqrt {3+5 x}}{1250235 (2+3 x)^{3/2}}+\frac {17830424 \sqrt {1-2 x} \sqrt {3+5 x}}{8751645 \sqrt {2+3 x}}-\frac {1864 \sqrt {1-2 x} (3+5 x)^{3/2}}{6615 (2+3 x)^{5/2}}-\frac {2 (1-2 x)^{3/2} (3+5 x)^{5/2}}{27 (2+3 x)^{9/2}}+\frac {362 \sqrt {1-2 x} (3+5 x)^{5/2}}{567 (2+3 x)^{7/2}}+\frac {9448538 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{8751645}+\frac {17830424 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{8751645}\\ &=-\frac {558524 \sqrt {1-2 x} \sqrt {3+5 x}}{1250235 (2+3 x)^{3/2}}+\frac {17830424 \sqrt {1-2 x} \sqrt {3+5 x}}{8751645 \sqrt {2+3 x}}-\frac {1864 \sqrt {1-2 x} (3+5 x)^{3/2}}{6615 (2+3 x)^{5/2}}-\frac {2 (1-2 x)^{3/2} (3+5 x)^{5/2}}{27 (2+3 x)^{9/2}}+\frac {362 \sqrt {1-2 x} (3+5 x)^{5/2}}{567 (2+3 x)^{7/2}}-\frac {17830424 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{8751645}-\frac {1717916 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{8751645}\\ \end {align*}
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Mathematica [A]
time = 8.32, size = 110, normalized size = 0.50 \begin {gather*} \frac {\frac {24 \sqrt {1-2 x} \sqrt {3+5 x} \left (159578303+955601637 x+2115318249 x^2+2043155529 x^3+722132172 x^4\right )}{(2+3 x)^{9/2}}+8 \sqrt {2} \left (8915212 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )+5257595 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )\right )}{105019740} \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 = 0.12, size = 494, normalized size = 2.23
method | result | size |
elliptic | \(-\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {205474 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{4822335 \left (\frac {2}{3}+x \right )^{3}}+\frac {1243066 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{11252115 \left (\frac {2}{3}+x \right )^{2}}+\frac {-\frac {35660848}{1750329} x^{2}-\frac {17830424}{8751645} x +\frac {17830424}{2917215}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}+\frac {12587962 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{36756909 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {17830424 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{36756909 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {14 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{177147 \left (\frac {2}{3}+x \right )^{5}}+\frac {1370 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{413343 \left (\frac {2}{3}+x \right )^{4}}\right )}{\left (10 x^{2}+x -3\right ) \sqrt {2+3 x}}\) | \(308\) |
default | \(-\frac {2 \left (1147997367 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{4} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-722132172 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{4} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+3061326312 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-1925685792 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+3061326312 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-1925685792 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+1360589472 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-855860352 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-21663965160 x^{6}+226764912 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-142643392 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-63461062386 x^{5}-63089824509 x^{4}-16625604096 x^{3}+11383710240 x^{2}+8121679824 x +1436204727\right ) \sqrt {3+5 x}\, \sqrt {1-2 x}}{26254935 \left (10 x^{2}+x -3\right ) \left (2+3 x \right )^{\frac {9}{2}}}\) | \(494\) |
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 = 0.48, size = 70, normalized size = 0.32 \begin {gather*} \frac {2 \, {\left (722132172 \, x^{4} + 2043155529 \, x^{3} + 2115318249 \, x^{2} + 955601637 \, x + 159578303\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{8751645 \, {\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 )}^{3/2}\,{\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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