Optimal. Leaf size=249 \[ -\frac {2 (1-2 x)^{5/2} \sqrt {3+5 x}}{33 (2+3 x)^{11/2}}+\frac {10 (1-2 x)^{3/2} \sqrt {3+5 x}}{99 (2+3 x)^{9/2}}+\frac {1900 \sqrt {1-2 x} \sqrt {3+5 x}}{2079 (2+3 x)^{7/2}}+\frac {76492 \sqrt {1-2 x} \sqrt {3+5 x}}{14553 (2+3 x)^{5/2}}+\frac {3560432 \sqrt {1-2 x} \sqrt {3+5 x}}{101871 (2+3 x)^{3/2}}+\frac {247408648 \sqrt {1-2 x} \sqrt {3+5 x}}{713097 \sqrt {2+3 x}}-\frac {247408648 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{64827 \sqrt {33}}-\frac {7442032 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{64827 \sqrt {33}} \]
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Rubi [A]
time = 0.06, antiderivative size = 249, normalized size of antiderivative = 1.00, number of steps
used = 9, 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 {7442032 F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{64827 \sqrt {33}}-\frac {247408648 E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{64827 \sqrt {33}}-\frac {2 \sqrt {5 x+3} (1-2 x)^{5/2}}{33 (3 x+2)^{11/2}}+\frac {10 \sqrt {5 x+3} (1-2 x)^{3/2}}{99 (3 x+2)^{9/2}}+\frac {247408648 \sqrt {5 x+3} \sqrt {1-2 x}}{713097 \sqrt {3 x+2}}+\frac {3560432 \sqrt {5 x+3} \sqrt {1-2 x}}{101871 (3 x+2)^{3/2}}+\frac {76492 \sqrt {5 x+3} \sqrt {1-2 x}}{14553 (3 x+2)^{5/2}}+\frac {1900 \sqrt {5 x+3} \sqrt {1-2 x}}{2079 (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)^{5/2} \sqrt {3+5 x}}{(2+3 x)^{13/2}} \, dx &=-\frac {2 (1-2 x)^{5/2} \sqrt {3+5 x}}{33 (2+3 x)^{11/2}}+\frac {2}{33} \int \frac {\left (-\frac {25}{2}-30 x\right ) (1-2 x)^{3/2}}{(2+3 x)^{11/2} \sqrt {3+5 x}} \, dx\\ &=-\frac {2 (1-2 x)^{5/2} \sqrt {3+5 x}}{33 (2+3 x)^{11/2}}+\frac {10 (1-2 x)^{3/2} \sqrt {3+5 x}}{99 (2+3 x)^{9/2}}-\frac {4}{891} \int \frac {\sqrt {1-2 x} \left (-\frac {1035}{2}+\frac {585 x}{2}\right )}{(2+3 x)^{9/2} \sqrt {3+5 x}} \, dx\\ &=-\frac {2 (1-2 x)^{5/2} \sqrt {3+5 x}}{33 (2+3 x)^{11/2}}+\frac {10 (1-2 x)^{3/2} \sqrt {3+5 x}}{99 (2+3 x)^{9/2}}+\frac {1900 \sqrt {1-2 x} \sqrt {3+5 x}}{2079 (2+3 x)^{7/2}}+\frac {8 \int \frac {\frac {149805}{4}-51390 x}{\sqrt {1-2 x} (2+3 x)^{7/2} \sqrt {3+5 x}} \, dx}{18711}\\ &=-\frac {2 (1-2 x)^{5/2} \sqrt {3+5 x}}{33 (2+3 x)^{11/2}}+\frac {10 (1-2 x)^{3/2} \sqrt {3+5 x}}{99 (2+3 x)^{9/2}}+\frac {1900 \sqrt {1-2 x} \sqrt {3+5 x}}{2079 (2+3 x)^{7/2}}+\frac {76492 \sqrt {1-2 x} \sqrt {3+5 x}}{14553 (2+3 x)^{5/2}}+\frac {16 \int \frac {2855520-\frac {12908025 x}{4}}{\sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx}{654885}\\ &=-\frac {2 (1-2 x)^{5/2} \sqrt {3+5 x}}{33 (2+3 x)^{11/2}}+\frac {10 (1-2 x)^{3/2} \sqrt {3+5 x}}{99 (2+3 x)^{9/2}}+\frac {1900 \sqrt {1-2 x} \sqrt {3+5 x}}{2079 (2+3 x)^{7/2}}+\frac {76492 \sqrt {1-2 x} \sqrt {3+5 x}}{14553 (2+3 x)^{5/2}}+\frac {3560432 \sqrt {1-2 x} \sqrt {3+5 x}}{101871 (2+3 x)^{3/2}}+\frac {32 \int \frac {\frac {991125045}{8}-\frac {150205725 x}{2}}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx}{13752585}\\ &=-\frac {2 (1-2 x)^{5/2} \sqrt {3+5 x}}{33 (2+3 x)^{11/2}}+\frac {10 (1-2 x)^{3/2} \sqrt {3+5 x}}{99 (2+3 x)^{9/2}}+\frac {1900 \sqrt {1-2 x} \sqrt {3+5 x}}{2079 (2+3 x)^{7/2}}+\frac {76492 \sqrt {1-2 x} \sqrt {3+5 x}}{14553 (2+3 x)^{5/2}}+\frac {3560432 \sqrt {1-2 x} \sqrt {3+5 x}}{101871 (2+3 x)^{3/2}}+\frac {247408648 \sqrt {1-2 x} \sqrt {3+5 x}}{713097 \sqrt {2+3 x}}+\frac {64 \int \frac {1651972050+\frac {20875104675 x}{8}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{96268095}\\ &=-\frac {2 (1-2 x)^{5/2} \sqrt {3+5 x}}{33 (2+3 x)^{11/2}}+\frac {10 (1-2 x)^{3/2} \sqrt {3+5 x}}{99 (2+3 x)^{9/2}}+\frac {1900 \sqrt {1-2 x} \sqrt {3+5 x}}{2079 (2+3 x)^{7/2}}+\frac {76492 \sqrt {1-2 x} \sqrt {3+5 x}}{14553 (2+3 x)^{5/2}}+\frac {3560432 \sqrt {1-2 x} \sqrt {3+5 x}}{101871 (2+3 x)^{3/2}}+\frac {247408648 \sqrt {1-2 x} \sqrt {3+5 x}}{713097 \sqrt {2+3 x}}+\frac {3721016 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{64827}+\frac {247408648 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{713097}\\ &=-\frac {2 (1-2 x)^{5/2} \sqrt {3+5 x}}{33 (2+3 x)^{11/2}}+\frac {10 (1-2 x)^{3/2} \sqrt {3+5 x}}{99 (2+3 x)^{9/2}}+\frac {1900 \sqrt {1-2 x} \sqrt {3+5 x}}{2079 (2+3 x)^{7/2}}+\frac {76492 \sqrt {1-2 x} \sqrt {3+5 x}}{14553 (2+3 x)^{5/2}}+\frac {3560432 \sqrt {1-2 x} \sqrt {3+5 x}}{101871 (2+3 x)^{3/2}}+\frac {247408648 \sqrt {1-2 x} \sqrt {3+5 x}}{713097 \sqrt {2+3 x}}-\frac {247408648 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{64827 \sqrt {33}}-\frac {7442032 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{64827 \sqrt {33}}\\ \end {align*}
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Mathematica [A]
time = 8.12, size = 115, normalized size = 0.46 \begin {gather*} \frac {\frac {24 \sqrt {1-2 x} \sqrt {3+5 x} \left (4174268813+30956769477 x+91862628912 x^2+136342955970 x^3+101209884912 x^4+30060150732 x^5\right )}{(2+3 x)^{11/2}}+32 \sqrt {2} \left (30926081 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-15576890 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )\right )}{8557164} \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. \(586\) vs.
\(2(185)=370\).
time = 0.10, size = 587, normalized size = 2.36
method | result | size |
elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {14 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{8019 \left (\frac {2}{3}+x \right )^{5}}+\frac {568 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{56133 \left (\frac {2}{3}+x \right )^{4}}+\frac {76492 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{392931 \left (\frac {2}{3}+x \right )^{3}}+\frac {3560432 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{916839 \left (\frac {2}{3}+x \right )^{2}}+\frac {-\frac {2474086480}{713097} x^{2}-\frac {247408648}{713097} x +\frac {247408648}{237699}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}+\frac {783157120 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{14975037 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {1237043240 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{14975037 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {98 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{216513 \left (\frac {2}{3}+x \right )^{6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(321\) |
default | \(-\frac {2 \left (14919413652 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{5} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-30060150732 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{5} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+49731378840 \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}-100200502440 \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}+66308505120 \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}-133600669920 \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}+44205670080 \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}-89067113280 \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}-901804521960 x^{7}+14735223360 \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}-29689037760 \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}-3126476999556 x^{6}+1964696448 \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 )-3958538368 \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 )-4123376977248 x^{5}-2254018771062 x^{4}+22795632684 x^{3}+608665287387 x^{2}+266088118854 x +37568419317\right ) \sqrt {3+5 x}\, \sqrt {1-2 x}}{2139291 \left (10 x^{2}+x -3\right ) \left (2+3 x \right )^{\frac {11}{2}}}\) | \(587\) |
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 = 80, normalized size = 0.32 \begin {gather*} \frac {2 \, {\left (30060150732 \, x^{5} + 101209884912 \, x^{4} + 136342955970 \, x^{3} + 91862628912 \, x^{2} + 30956769477 \, x + 4174268813\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{713097 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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}\,\sqrt {5\,x+3}}{{\left (3\,x+2\right )}^{13/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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