Optimal. Leaf size=142 \[ -\frac {2 \sqrt {1-2 x} (2+3 x)^4}{15 (3+5 x)^{3/2}}-\frac {524 \sqrt {1-2 x} (2+3 x)^3}{825 \sqrt {3+5 x}}+\frac {623 \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}}{1375}+\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} (2563+8940 x)}{220000}+\frac {35511 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{20000 \sqrt {10}} \]
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Rubi [A]
time = 0.03, antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {99, 155, 158,
152, 56, 222} \begin {gather*} \frac {35511 \text {ArcSin}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{20000 \sqrt {10}}-\frac {2 \sqrt {1-2 x} (3 x+2)^4}{15 (5 x+3)^{3/2}}-\frac {524 \sqrt {1-2 x} (3 x+2)^3}{825 \sqrt {5 x+3}}+\frac {623 \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^2}{1375}+\frac {7 \sqrt {1-2 x} \sqrt {5 x+3} (8940 x+2563)}{220000} \end {gather*}
Antiderivative was successfully verified.
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Rule 56
Rule 99
Rule 152
Rule 155
Rule 158
Rule 222
Rubi steps
\begin {align*} \int \frac {\sqrt {1-2 x} (2+3 x)^4}{(3+5 x)^{5/2}} \, dx &=-\frac {2 \sqrt {1-2 x} (2+3 x)^4}{15 (3+5 x)^{3/2}}+\frac {2}{15} \int \frac {(10-27 x) (2+3 x)^3}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx\\ &=-\frac {2 \sqrt {1-2 x} (2+3 x)^4}{15 (3+5 x)^{3/2}}-\frac {524 \sqrt {1-2 x} (2+3 x)^3}{825 \sqrt {3+5 x}}+\frac {4}{825} \int \frac {\left (882-\frac {5607 x}{2}\right ) (2+3 x)^2}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {2 \sqrt {1-2 x} (2+3 x)^4}{15 (3+5 x)^{3/2}}-\frac {524 \sqrt {1-2 x} (2+3 x)^3}{825 \sqrt {3+5 x}}+\frac {623 \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}}{1375}-\frac {2 \int \frac {(2+3 x) \left (-\frac {10521}{2}+\frac {46935 x}{4}\right )}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{12375}\\ &=-\frac {2 \sqrt {1-2 x} (2+3 x)^4}{15 (3+5 x)^{3/2}}-\frac {524 \sqrt {1-2 x} (2+3 x)^3}{825 \sqrt {3+5 x}}+\frac {623 \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}}{1375}+\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} (2563+8940 x)}{220000}+\frac {35511 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{40000}\\ &=-\frac {2 \sqrt {1-2 x} (2+3 x)^4}{15 (3+5 x)^{3/2}}-\frac {524 \sqrt {1-2 x} (2+3 x)^3}{825 \sqrt {3+5 x}}+\frac {623 \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}}{1375}+\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} (2563+8940 x)}{220000}+\frac {35511 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{20000 \sqrt {5}}\\ &=-\frac {2 \sqrt {1-2 x} (2+3 x)^4}{15 (3+5 x)^{3/2}}-\frac {524 \sqrt {1-2 x} (2+3 x)^3}{825 \sqrt {3+5 x}}+\frac {623 \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}}{1375}+\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} (2563+8940 x)}{220000}+\frac {35511 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{20000 \sqrt {10}}\\ \end {align*}
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Mathematica [A]
time = 0.20, size = 74, normalized size = 0.52 \begin {gather*} \frac {\frac {10 \sqrt {1-2 x} \left (-218953+995870 x+6384015 x^2+8999100 x^3+3564000 x^4\right )}{(3+5 x)^{3/2}}-1171863 \sqrt {10} \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{6600000} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 147, normalized size = 1.04
method | result | size |
default | \(\frac {\left (71280000 x^{4} \sqrt {-10 x^{2}-x +3}+29296575 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{2}+179982000 x^{3} \sqrt {-10 x^{2}-x +3}+35155890 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x +127680300 x^{2} \sqrt {-10 x^{2}-x +3}+10546767 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+19917400 x \sqrt {-10 x^{2}-x +3}-4379060 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}}{13200000 \sqrt {-10 x^{2}-x +3}\, \left (3+5 x \right )^{\frac {3}{2}}}\) | \(147\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [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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Fricas [A]
time = 1.07, size = 101, normalized size = 0.71 \begin {gather*} -\frac {1171863 \, \sqrt {10} {\left (25 \, x^{2} + 30 \, x + 9\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 20 \, {\left (3564000 \, x^{4} + 8999100 \, x^{3} + 6384015 \, x^{2} + 995870 \, x - 218953\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{13200000 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {1 - 2 x} \left (3 x + 2\right )^{4}}{\left (5 x + 3\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.72, size = 184, normalized size = 1.30 \begin {gather*} \frac {27}{500000} \, {\left (4 \, {\left (8 \, \sqrt {5} {\left (5 \, x + 3\right )} + 5 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} - 475 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - \frac {1}{8250000} \, \sqrt {10} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}}} + \frac {3156 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}}\right )} + \frac {35511}{200000} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + \frac {\sqrt {10} {\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (\frac {789 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} + 4\right )}}{515625 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}} \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 (3\,x+2\right )}^4}{{\left (5\,x+3\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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