Optimal. Leaf size=191 \[ -\frac {3632 \sqrt {1-2 x} \sqrt {3+5 x}}{6615 (2+3 x)^{3/2}}+\frac {119732 \sqrt {1-2 x} \sqrt {3+5 x}}{46305 \sqrt {2+3 x}}-\frac {2 (1-2 x)^{3/2} (3+5 x)^{3/2}}{21 (2+3 x)^{7/2}}+\frac {74 \sqrt {1-2 x} (3+5 x)^{3/2}}{105 (2+3 x)^{5/2}}-\frac {119732 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{46305}-\frac {7388 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{46305} \]
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Rubi [A]
time = 0.04, antiderivative size = 191, normalized size of antiderivative = 1.00, number of steps
used = 7, 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 {7388 \sqrt {\frac {11}{3}} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{46305}-\frac {119732 \sqrt {\frac {11}{3}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{46305}+\frac {74 \sqrt {1-2 x} (5 x+3)^{3/2}}{105 (3 x+2)^{5/2}}-\frac {2 (1-2 x)^{3/2} (5 x+3)^{3/2}}{21 (3 x+2)^{7/2}}+\frac {119732 \sqrt {1-2 x} \sqrt {5 x+3}}{46305 \sqrt {3 x+2}}-\frac {3632 \sqrt {1-2 x} \sqrt {5 x+3}}{6615 (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)^{3/2}}{(2+3 x)^{9/2}} \, dx &=-\frac {2 (1-2 x)^{3/2} (3+5 x)^{3/2}}{21 (2+3 x)^{7/2}}+\frac {2}{21} \int \frac {\left (-\frac {3}{2}-30 x\right ) \sqrt {1-2 x} \sqrt {3+5 x}}{(2+3 x)^{7/2}} \, dx\\ &=-\frac {2 (1-2 x)^{3/2} (3+5 x)^{3/2}}{21 (2+3 x)^{7/2}}+\frac {74 \sqrt {1-2 x} (3+5 x)^{3/2}}{105 (2+3 x)^{5/2}}-\frac {4}{315} \int \frac {\sqrt {3+5 x} \left (-369+\frac {255 x}{2}\right )}{\sqrt {1-2 x} (2+3 x)^{5/2}} \, dx\\ &=-\frac {3632 \sqrt {1-2 x} \sqrt {3+5 x}}{6615 (2+3 x)^{3/2}}-\frac {2 (1-2 x)^{3/2} (3+5 x)^{3/2}}{21 (2+3 x)^{7/2}}+\frac {74 \sqrt {1-2 x} (3+5 x)^{3/2}}{105 (2+3 x)^{5/2}}-\frac {8 \int \frac {-\frac {30243}{4}-\frac {465 x}{4}}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx}{19845}\\ &=-\frac {3632 \sqrt {1-2 x} \sqrt {3+5 x}}{6615 (2+3 x)^{3/2}}+\frac {119732 \sqrt {1-2 x} \sqrt {3+5 x}}{46305 \sqrt {2+3 x}}-\frac {2 (1-2 x)^{3/2} (3+5 x)^{3/2}}{21 (2+3 x)^{7/2}}+\frac {74 \sqrt {1-2 x} (3+5 x)^{3/2}}{105 (2+3 x)^{5/2}}-\frac {16 \int \frac {-\frac {599745}{8}-\frac {448995 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{138915}\\ &=-\frac {3632 \sqrt {1-2 x} \sqrt {3+5 x}}{6615 (2+3 x)^{3/2}}+\frac {119732 \sqrt {1-2 x} \sqrt {3+5 x}}{46305 \sqrt {2+3 x}}-\frac {2 (1-2 x)^{3/2} (3+5 x)^{3/2}}{21 (2+3 x)^{7/2}}+\frac {74 \sqrt {1-2 x} (3+5 x)^{3/2}}{105 (2+3 x)^{5/2}}+\frac {40634 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{46305}+\frac {119732 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{46305}\\ &=-\frac {3632 \sqrt {1-2 x} \sqrt {3+5 x}}{6615 (2+3 x)^{3/2}}+\frac {119732 \sqrt {1-2 x} \sqrt {3+5 x}}{46305 \sqrt {2+3 x}}-\frac {2 (1-2 x)^{3/2} (3+5 x)^{3/2}}{21 (2+3 x)^{7/2}}+\frac {74 \sqrt {1-2 x} (3+5 x)^{3/2}}{105 (2+3 x)^{5/2}}-\frac {119732 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{46305}-\frac {7388 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{46305}\\ \end {align*}
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Mathematica [A]
time = 7.80, size = 104, normalized size = 0.54 \begin {gather*} \frac {2 \left (\frac {3 \sqrt {1-2 x} \sqrt {3+5 x} \left (519367+2314860 x+3385161 x^2+1616382 x^3\right )}{(2+3 x)^{7/2}}+\sqrt {2} \left (59866 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )+1085 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )\right )\right )}{138915} \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. \(400\) vs.
\(2(139)=278\).
time = 0.10, size = 401, normalized size = 2.10
method | result | size |
elliptic | \(-\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {592 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{25515 \left (\frac {2}{3}+x \right )^{3}}+\frac {4838 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{59535 \left (\frac {2}{3}+x \right )^{2}}+\frac {-\frac {239464}{9261} x^{2}-\frac {119732}{46305} x +\frac {119732}{15435}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}+\frac {79966 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{194481 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {119732 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{194481 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {2 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{2187 \left (\frac {2}{3}+x \right )^{4}}\right )}{\left (10 x^{2}+x -3\right ) \sqrt {2+3 x}}\) | \(284\) |
default | \(-\frac {2 \left (1645677 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-1616382 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+3291354 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-3232764 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+2194236 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-2155176 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+487608 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-478928 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-48491460 x^{5}-106403976 x^{4}-65053845 x^{3}+7940859 x^{2}+19275639 x +4674303\right ) \sqrt {3+5 x}\, \sqrt {1-2 x}}{138915 \left (10 x^{2}+x -3\right ) \left (2+3 x \right )^{\frac {7}{2}}}\) | \(401\) |
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 = 0.17, size = 60, normalized size = 0.31 \begin {gather*} \frac {2 \, {\left (1616382 \, x^{3} + 3385161 \, x^{2} + 2314860 \, x + 519367\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{46305 \, {\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 = -0.01 \begin {gather*} \int \frac {{\left (1-2\,x\right )}^{3/2}\,{\left (5\,x+3\right )}^{3/2}}{{\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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