Optimal. Leaf size=191 \[ -\frac {4282 \sqrt {1-2 x} \sqrt {3+5 x}}{15435 (2+3 x)^{3/2}}+\frac {173482 \sqrt {1-2 x} \sqrt {3+5 x}}{108045 \sqrt {2+3 x}}-\frac {118 \sqrt {1-2 x} (3+5 x)^{3/2}}{735 (2+3 x)^{5/2}}-\frac {2 \sqrt {1-2 x} (3+5 x)^{5/2}}{21 (2+3 x)^{7/2}}-\frac {173482 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{108045}+\frac {23612 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{108045} \]
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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 {23612 \sqrt {\frac {11}{3}} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{108045}-\frac {173482 \sqrt {\frac {11}{3}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{108045}-\frac {2 \sqrt {1-2 x} (5 x+3)^{5/2}}{21 (3 x+2)^{7/2}}-\frac {118 \sqrt {1-2 x} (5 x+3)^{3/2}}{735 (3 x+2)^{5/2}}+\frac {173482 \sqrt {1-2 x} \sqrt {5 x+3}}{108045 \sqrt {3 x+2}}-\frac {4282 \sqrt {1-2 x} \sqrt {5 x+3}}{15435 (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 {\sqrt {1-2 x} (3+5 x)^{5/2}}{(2+3 x)^{9/2}} \, dx &=-\frac {2 \sqrt {1-2 x} (3+5 x)^{5/2}}{21 (2+3 x)^{7/2}}+\frac {2}{21} \int \frac {\left (\frac {19}{2}-30 x\right ) (3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^{7/2}} \, dx\\ &=-\frac {118 \sqrt {1-2 x} (3+5 x)^{3/2}}{735 (2+3 x)^{5/2}}-\frac {2 \sqrt {1-2 x} (3+5 x)^{5/2}}{21 (2+3 x)^{7/2}}+\frac {4 \int \frac {\left (\frac {603}{4}-\frac {4365 x}{2}\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^{5/2}} \, dx}{2205}\\ &=-\frac {4282 \sqrt {1-2 x} \sqrt {3+5 x}}{15435 (2+3 x)^{3/2}}-\frac {118 \sqrt {1-2 x} (3+5 x)^{3/2}}{735 (2+3 x)^{5/2}}-\frac {2 \sqrt {1-2 x} (3+5 x)^{5/2}}{21 (2+3 x)^{7/2}}+\frac {8 \int \frac {-\frac {222417}{8}-90495 x}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx}{138915}\\ &=-\frac {4282 \sqrt {1-2 x} \sqrt {3+5 x}}{15435 (2+3 x)^{3/2}}+\frac {173482 \sqrt {1-2 x} \sqrt {3+5 x}}{108045 \sqrt {2+3 x}}-\frac {118 \sqrt {1-2 x} (3+5 x)^{3/2}}{735 (2+3 x)^{5/2}}-\frac {2 \sqrt {1-2 x} (3+5 x)^{5/2}}{21 (2+3 x)^{7/2}}+\frac {16 \int \frac {\frac {878805}{4}+\frac {3903345 x}{8}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{972405}\\ &=-\frac {4282 \sqrt {1-2 x} \sqrt {3+5 x}}{15435 (2+3 x)^{3/2}}+\frac {173482 \sqrt {1-2 x} \sqrt {3+5 x}}{108045 \sqrt {2+3 x}}-\frac {118 \sqrt {1-2 x} (3+5 x)^{3/2}}{735 (2+3 x)^{5/2}}-\frac {2 \sqrt {1-2 x} (3+5 x)^{5/2}}{21 (2+3 x)^{7/2}}-\frac {129866 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{108045}+\frac {173482 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{108045}\\ &=-\frac {4282 \sqrt {1-2 x} \sqrt {3+5 x}}{15435 (2+3 x)^{3/2}}+\frac {173482 \sqrt {1-2 x} \sqrt {3+5 x}}{108045 \sqrt {2+3 x}}-\frac {118 \sqrt {1-2 x} (3+5 x)^{3/2}}{735 (2+3 x)^{5/2}}-\frac {2 \sqrt {1-2 x} (3+5 x)^{5/2}}{21 (2+3 x)^{7/2}}-\frac {173482 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{108045}+\frac {23612 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{108045}\\ \end {align*}
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Mathematica [A]
time = 4.98, size = 104, normalized size = 0.54 \begin {gather*} \frac {2 \left (\frac {3 \sqrt {1-2 x} \sqrt {3+5 x} \left (535637+2623695 x+4290411 x^2+2342007 x^3\right )}{(2+3 x)^{7/2}}+\sqrt {2} \left (86741 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-281540 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )\right )\right )}{324135} \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 | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {1054 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{178605 \left (\frac {2}{3}+x \right )^{3}}-\frac {37486 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{416745 \left (\frac {2}{3}+x \right )^{2}}+\frac {-\frac {346964}{21609} x^{2}-\frac {173482}{108045} x +\frac {173482}{36015}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}+\frac {78116 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{453789 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {173482 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{453789 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {2 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{15309 \left (\frac {2}{3}+x \right )^{4}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(273\) |
default | \(\frac {2 \left (5259573 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+2342007 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+10519146 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+4684014 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+7012764 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+3122676 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+1558392 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+693928 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+70260210 x^{5}+135738351 x^{4}+70504020 x^{3}-14673504 x^{2}-22006344 x -4820733\right ) \sqrt {1-2 x}\, \sqrt {3+5 x}}{324135 \left (10 x^{2}+x -3\right ) \left (2+3 x \right )^{\frac {7}{2}}}\) | \(401\) |
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 = 0.27, size = 60, normalized size = 0.31 \begin {gather*} \frac {2 \, {\left (2342007 \, x^{3} + 4290411 \, x^{2} + 2623695 \, x + 535637\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{108045 \, {\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 {\sqrt {1-2\,x}\,{\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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