Optimal. Leaf size=191 \[ \frac {14 (1-2 x)^{3/2}}{15 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {2716 \sqrt {1-2 x}}{135 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {17468 \sqrt {1-2 x}}{45 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {105584 \sqrt {1-2 x} \sqrt {2+3 x}}{27 \sqrt {3+5 x}}+\frac {105584}{45} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {3176}{45} \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right ) \]
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Rubi [A]
time = 0.05, 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 = {100, 155, 157,
164, 114, 120} \begin {gather*} \frac {3176}{45} \sqrt {\frac {11}{3}} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {105584}{45} \sqrt {\frac {11}{3}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {14 (1-2 x)^{3/2}}{15 (3 x+2)^{5/2} \sqrt {5 x+3}}-\frac {105584 \sqrt {3 x+2} \sqrt {1-2 x}}{27 \sqrt {5 x+3}}+\frac {17468 \sqrt {1-2 x}}{45 \sqrt {3 x+2} \sqrt {5 x+3}}+\frac {2716 \sqrt {1-2 x}}{135 (3 x+2)^{3/2} \sqrt {5 x+3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 100
Rule 114
Rule 120
Rule 155
Rule 157
Rule 164
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{5/2}}{(2+3 x)^{7/2} (3+5 x)^{3/2}} \, dx &=\frac {14 (1-2 x)^{3/2}}{15 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {2}{15} \int \frac {(163-95 x) \sqrt {1-2 x}}{(2+3 x)^{5/2} (3+5 x)^{3/2}} \, dx\\ &=\frac {14 (1-2 x)^{3/2}}{15 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {2716 \sqrt {1-2 x}}{135 (2+3 x)^{3/2} \sqrt {3+5 x}}-\frac {4}{135} \int \frac {-\frac {17369}{2}+9900 x}{\sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}} \, dx\\ &=\frac {14 (1-2 x)^{3/2}}{15 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {2716 \sqrt {1-2 x}}{135 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {17468 \sqrt {1-2 x}}{45 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {8}{945} \int \frac {-\frac {741125}{2}+\frac {458535 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}} \, dx\\ &=\frac {14 (1-2 x)^{3/2}}{15 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {2716 \sqrt {1-2 x}}{135 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {17468 \sqrt {1-2 x}}{45 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {105584 \sqrt {1-2 x} \sqrt {2+3 x}}{27 \sqrt {3+5 x}}+\frac {16 \int \frac {-\frac {19301205}{4}-7621845 x}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{10395}\\ &=\frac {14 (1-2 x)^{3/2}}{15 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {2716 \sqrt {1-2 x}}{135 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {17468 \sqrt {1-2 x}}{45 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {105584 \sqrt {1-2 x} \sqrt {2+3 x}}{27 \sqrt {3+5 x}}-\frac {17468}{45} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx-\frac {105584}{45} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx\\ &=\frac {14 (1-2 x)^{3/2}}{15 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {2716 \sqrt {1-2 x}}{135 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {17468 \sqrt {1-2 x}}{45 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {105584 \sqrt {1-2 x} \sqrt {2+3 x}}{27 \sqrt {3+5 x}}+\frac {105584}{45} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {3176}{45} \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\\ \end {align*}
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Mathematica [A]
time = 8.51, size = 105, normalized size = 0.55 \begin {gather*} \frac {2}{135} \left (-\frac {3 \sqrt {1-2 x} \left (668031+3061396 x+4672674 x^2+2375640 x^3\right )}{(2+3 x)^{5/2} \sqrt {3+5 x}}-2 \sqrt {2} \left (26396 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-13295 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )\right )\right ) \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. \(307\) vs.
\(2(139)=278\).
time = 0.11, size = 308, normalized size = 1.61
method | result | size |
elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {98 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{405 \left (\frac {2}{3}+x \right )^{3}}-\frac {4102 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{405 \left (\frac {2}{3}+x \right )^{2}}-\frac {72914 \left (-30 x^{2}-3 x +9\right )}{135 \sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}-\frac {66844 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{189 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {105584 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{189 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {242 \left (-30 x^{2}-5 x +10\right )}{\sqrt {\left (x +\frac {3}{5}\right ) \left (-30 x^{2}-5 x +10\right )}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(277\) |
default | \(-\frac {2 \sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (475128 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-235818 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+633504 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-314424 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+211168 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-104808 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+14253840 x^{4}+20909124 x^{3}+4350354 x^{2}-5176002 x -2004093\right )}{135 \left (2+3 x \right )^{\frac {5}{2}} \left (10 x^{2}+x -3\right )}\) | \(308\) |
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 = 0.20, size = 60, normalized size = 0.31 \begin {gather*} -\frac {2 \, {\left (2375640 \, x^{3} + 4672674 \, x^{2} + 3061396 \, x + 668031\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{45 \, {\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\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 )}^{5/2}}{{\left (3\,x+2\right )}^{7/2}\,{\left (5\,x+3\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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