Optimal. Leaf size=222 \[ -\frac {2 (1-2 x)^{3/2} \sqrt {3+5 x}}{27 (2+3 x)^{9/2}}+\frac {82 \sqrt {1-2 x} \sqrt {3+5 x}}{567 (2+3 x)^{7/2}}+\frac {13136 \sqrt {1-2 x} \sqrt {3+5 x}}{19845 (2+3 x)^{5/2}}+\frac {613276 \sqrt {1-2 x} \sqrt {3+5 x}}{138915 (2+3 x)^{3/2}}+\frac {42623864 \sqrt {1-2 x} \sqrt {3+5 x}}{972405 \sqrt {2+3 x}}-\frac {42623864 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{972405}-\frac {1282376 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{972405} \]
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Rubi [A]
time = 0.06, 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 {1282376 \sqrt {\frac {11}{3}} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{972405}-\frac {42623864 \sqrt {\frac {11}{3}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{972405}-\frac {2 \sqrt {5 x+3} (1-2 x)^{3/2}}{27 (3 x+2)^{9/2}}+\frac {42623864 \sqrt {5 x+3} \sqrt {1-2 x}}{972405 \sqrt {3 x+2}}+\frac {613276 \sqrt {5 x+3} \sqrt {1-2 x}}{138915 (3 x+2)^{3/2}}+\frac {13136 \sqrt {5 x+3} \sqrt {1-2 x}}{19845 (3 x+2)^{5/2}}+\frac {82 \sqrt {5 x+3} \sqrt {1-2 x}}{567 (3 x+2)^{7/2}} \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} \sqrt {3+5 x}}{(2+3 x)^{11/2}} \, dx &=-\frac {2 (1-2 x)^{3/2} \sqrt {3+5 x}}{27 (2+3 x)^{9/2}}+\frac {2}{27} \int \frac {\left (-\frac {13}{2}-20 x\right ) \sqrt {1-2 x}}{(2+3 x)^{9/2} \sqrt {3+5 x}} \, dx\\ &=-\frac {2 (1-2 x)^{3/2} \sqrt {3+5 x}}{27 (2+3 x)^{9/2}}+\frac {82 \sqrt {1-2 x} \sqrt {3+5 x}}{567 (2+3 x)^{7/2}}-\frac {4}{567} \int \frac {-299+\frac {745 x}{2}}{\sqrt {1-2 x} (2+3 x)^{7/2} \sqrt {3+5 x}} \, dx\\ &=-\frac {2 (1-2 x)^{3/2} \sqrt {3+5 x}}{27 (2+3 x)^{9/2}}+\frac {82 \sqrt {1-2 x} \sqrt {3+5 x}}{567 (2+3 x)^{7/2}}+\frac {13136 \sqrt {1-2 x} \sqrt {3+5 x}}{19845 (2+3 x)^{5/2}}-\frac {8 \int \frac {-\frac {87639}{4}+24630 x}{\sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx}{19845}\\ &=-\frac {2 (1-2 x)^{3/2} \sqrt {3+5 x}}{27 (2+3 x)^{9/2}}+\frac {82 \sqrt {1-2 x} \sqrt {3+5 x}}{567 (2+3 x)^{7/2}}+\frac {13136 \sqrt {1-2 x} \sqrt {3+5 x}}{19845 (2+3 x)^{5/2}}+\frac {613276 \sqrt {1-2 x} \sqrt {3+5 x}}{138915 (2+3 x)^{3/2}}-\frac {16 \int \frac {-\frac {3794793}{4}+\frac {2299785 x}{4}}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx}{416745}\\ &=-\frac {2 (1-2 x)^{3/2} \sqrt {3+5 x}}{27 (2+3 x)^{9/2}}+\frac {82 \sqrt {1-2 x} \sqrt {3+5 x}}{567 (2+3 x)^{7/2}}+\frac {13136 \sqrt {1-2 x} \sqrt {3+5 x}}{19845 (2+3 x)^{5/2}}+\frac {613276 \sqrt {1-2 x} \sqrt {3+5 x}}{138915 (2+3 x)^{3/2}}+\frac {42623864 \sqrt {1-2 x} \sqrt {3+5 x}}{972405 \sqrt {2+3 x}}-\frac {32 \int \frac {-\frac {101193495}{8}-\frac {79919745 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{2917215}\\ &=-\frac {2 (1-2 x)^{3/2} \sqrt {3+5 x}}{27 (2+3 x)^{9/2}}+\frac {82 \sqrt {1-2 x} \sqrt {3+5 x}}{567 (2+3 x)^{7/2}}+\frac {13136 \sqrt {1-2 x} \sqrt {3+5 x}}{19845 (2+3 x)^{5/2}}+\frac {613276 \sqrt {1-2 x} \sqrt {3+5 x}}{138915 (2+3 x)^{3/2}}+\frac {42623864 \sqrt {1-2 x} \sqrt {3+5 x}}{972405 \sqrt {2+3 x}}+\frac {7053068 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{972405}+\frac {42623864 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{972405}\\ &=-\frac {2 (1-2 x)^{3/2} \sqrt {3+5 x}}{27 (2+3 x)^{9/2}}+\frac {82 \sqrt {1-2 x} \sqrt {3+5 x}}{567 (2+3 x)^{7/2}}+\frac {13136 \sqrt {1-2 x} \sqrt {3+5 x}}{19845 (2+3 x)^{5/2}}+\frac {613276 \sqrt {1-2 x} \sqrt {3+5 x}}{138915 (2+3 x)^{3/2}}+\frac {42623864 \sqrt {1-2 x} \sqrt {3+5 x}}{972405 \sqrt {2+3 x}}-\frac {42623864 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{972405}-\frac {1282376 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{972405}\\ \end {align*}
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Mathematica [A]
time = 7.03, size = 111, normalized size = 0.50 \begin {gather*} \frac {4 \left (\frac {3 \sqrt {1-2 x} \sqrt {3+5 x} \left (359554583+2127363207 x+4722182964 x^2+4661331894 x^3+1726266492 x^4\right )}{2 (2+3 x)^{9/2}}+\sqrt {2} \left (10655966 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-5366165 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )\right )\right )}{2917215} \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 | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {14 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{19683 \left (\frac {2}{3}+x \right )^{5}}+\frac {110 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{45927 \left (\frac {2}{3}+x \right )^{4}}+\frac {13136 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{535815 \left (\frac {2}{3}+x \right )^{3}}+\frac {613276 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{1250235 \left (\frac {2}{3}+x \right )^{2}}+\frac {-\frac {85247728}{194481} x^{2}-\frac {42623864}{972405} x +\frac {42623864}{324135}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}+\frac {26984932 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{4084101 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {42623864 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{4084101 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(297\) |
default | \(-\frac {2 \left (856947762 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{4} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-1726266492 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{4} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+2285194032 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-4603377312 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+2285194032 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-4603377312 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+1015641792 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-2045945472 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-51787994760 x^{6}+169273632 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-340990912 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-145018756296 x^{5}-140113086174 x^{4}-36035458056 x^{3}+25330919565 x^{2}+18067605114 x +3235991247\right ) \sqrt {1-2 x}\, \sqrt {3+5 x}}{2917215 \left (10 x^{2}+x -3\right ) \left (2+3 x \right )^{\frac {9}{2}}}\) | \(494\) |
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 = 0.34, size = 70, normalized size = 0.32 \begin {gather*} \frac {2 \, {\left (1726266492 \, x^{4} + 4661331894 \, x^{3} + 4722182964 \, x^{2} + 2127363207 \, x + 359554583\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{972405 \, {\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}\,\sqrt {5\,x+3}}{{\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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