Optimal. Leaf size=191 \[ \frac {676 \sqrt {1-2 x} \sqrt {3+5 x}}{15435 (2+3 x)^{5/2}}-\frac {101902 \sqrt {1-2 x} \sqrt {3+5 x}}{324135 (2+3 x)^{3/2}}+\frac {816622 \sqrt {1-2 x} \sqrt {3+5 x}}{2268945 \sqrt {2+3 x}}+\frac {2 \sqrt {1-2 x} (3+5 x)^{3/2}}{147 (2+3 x)^{7/2}}-\frac {816622 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{2268945}-\frac {265648 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{2268945} \]
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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 = {100, 155, 157,
164, 114, 120} \begin {gather*} -\frac {265648 \sqrt {\frac {11}{3}} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{2268945}-\frac {816622 \sqrt {\frac {11}{3}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{2268945}+\frac {2 \sqrt {1-2 x} (5 x+3)^{3/2}}{147 (3 x+2)^{7/2}}+\frac {816622 \sqrt {1-2 x} \sqrt {5 x+3}}{2268945 \sqrt {3 x+2}}-\frac {101902 \sqrt {1-2 x} \sqrt {5 x+3}}{324135 (3 x+2)^{3/2}}+\frac {676 \sqrt {1-2 x} \sqrt {5 x+3}}{15435 (3 x+2)^{5/2}} \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 {(3+5 x)^{5/2}}{\sqrt {1-2 x} (2+3 x)^{9/2}} \, dx &=\frac {2 \sqrt {1-2 x} (3+5 x)^{3/2}}{147 (2+3 x)^{7/2}}-\frac {2}{147} \int \frac {\left (-342-\frac {1195 x}{2}\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^{7/2}} \, dx\\ &=\frac {676 \sqrt {1-2 x} \sqrt {3+5 x}}{15435 (2+3 x)^{5/2}}+\frac {2 \sqrt {1-2 x} (3+5 x)^{3/2}}{147 (2+3 x)^{7/2}}-\frac {4 \int \frac {-\frac {115673}{4}-\frac {198985 x}{4}}{\sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx}{15435}\\ &=\frac {676 \sqrt {1-2 x} \sqrt {3+5 x}}{15435 (2+3 x)^{5/2}}-\frac {101902 \sqrt {1-2 x} \sqrt {3+5 x}}{324135 (2+3 x)^{3/2}}+\frac {2 \sqrt {1-2 x} (3+5 x)^{3/2}}{147 (2+3 x)^{7/2}}-\frac {8 \int \frac {-\frac {475777}{8}-\frac {254755 x}{4}}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx}{324135}\\ &=\frac {676 \sqrt {1-2 x} \sqrt {3+5 x}}{15435 (2+3 x)^{5/2}}-\frac {101902 \sqrt {1-2 x} \sqrt {3+5 x}}{324135 (2+3 x)^{3/2}}+\frac {816622 \sqrt {1-2 x} \sqrt {3+5 x}}{2268945 \sqrt {2+3 x}}+\frac {2 \sqrt {1-2 x} (3+5 x)^{3/2}}{147 (2+3 x)^{7/2}}-\frac {16 \int \frac {-\frac {1955465}{8}-\frac {2041555 x}{8}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{2268945}\\ &=\frac {676 \sqrt {1-2 x} \sqrt {3+5 x}}{15435 (2+3 x)^{5/2}}-\frac {101902 \sqrt {1-2 x} \sqrt {3+5 x}}{324135 (2+3 x)^{3/2}}+\frac {816622 \sqrt {1-2 x} \sqrt {3+5 x}}{2268945 \sqrt {2+3 x}}+\frac {2 \sqrt {1-2 x} (3+5 x)^{3/2}}{147 (2+3 x)^{7/2}}+\frac {816622 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{2268945}+\frac {1461064 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{2268945}\\ &=\frac {676 \sqrt {1-2 x} \sqrt {3+5 x}}{15435 (2+3 x)^{5/2}}-\frac {101902 \sqrt {1-2 x} \sqrt {3+5 x}}{324135 (2+3 x)^{3/2}}+\frac {816622 \sqrt {1-2 x} \sqrt {3+5 x}}{2268945 \sqrt {2+3 x}}+\frac {2 \sqrt {1-2 x} (3+5 x)^{3/2}}{147 (2+3 x)^{7/2}}-\frac {816622 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{2268945}-\frac {265648 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{2268945}\\ \end {align*}
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Mathematica [A]
time = 4.86, size = 104, normalized size = 0.54 \begin {gather*} \frac {2 \left (\frac {3 \sqrt {1-2 x} \sqrt {3+5 x} \left (1985537+10645545 x+18838881 x^2+11024397 x^3\right )}{(2+3 x)^{7/2}}+\sqrt {2} \left (408311 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )+1783285 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )\right )\right )}{6806835} \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 | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {38 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{15435 \left (\frac {2}{3}+x \right )^{3}}-\frac {101902 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{2917215 \left (\frac {2}{3}+x \right )^{2}}+\frac {-\frac {1633244}{453789} x^{2}-\frac {816622}{2268945} x +\frac {816622}{756315}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}+\frac {782186 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{9529569 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {816622 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{9529569 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {2 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{35721 \left (\frac {2}{3}+x \right )^{4}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(273\) |
default | \(-\frac {2 \left (59173092 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-11024397 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+118346184 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-22048794 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+78897456 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-14699196 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+17532768 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-3266488 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-330731910 x^{5}-598239621 x^{4}-276663420 x^{3}+78047184 x^{2}+89853294 x +17869833\right ) \sqrt {1-2 x}\, \sqrt {3+5 x}}{6806835 \left (10 x^{2}+x -3\right ) \left (2+3 x \right )^{\frac {7}{2}}}\) | \(401\) |
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 = 0.24, size = 60, normalized size = 0.31 \begin {gather*} \frac {2 \, {\left (11024397 \, x^{3} + 18838881 \, x^{2} + 10645545 \, x + 1985537\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{2268945 \, {\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 (5\,x+3\right )}^{5/2}}{\sqrt {1-2\,x}\,{\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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