Optimal. Leaf size=118 \[ \frac {13365}{128} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {405}{32} \sqrt {1-2 x} (3+5 x)^{3/2}+\frac {81}{44} \sqrt {1-2 x} (3+5 x)^{5/2}+\frac {7 (3+5 x)^{7/2}}{11 \sqrt {1-2 x}}-\frac {29403}{128} \sqrt {\frac {5}{2}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right ) \]
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Rubi [A]
time = 0.02, antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {79, 52, 56, 222}
\begin {gather*} -\frac {29403}{128} \sqrt {\frac {5}{2}} \text {ArcSin}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )+\frac {7 (5 x+3)^{7/2}}{11 \sqrt {1-2 x}}+\frac {81}{44} \sqrt {1-2 x} (5 x+3)^{5/2}+\frac {405}{32} \sqrt {1-2 x} (5 x+3)^{3/2}+\frac {13365}{128} \sqrt {1-2 x} \sqrt {5 x+3} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 56
Rule 79
Rule 222
Rubi steps
\begin {align*} \int \frac {(2+3 x) (3+5 x)^{5/2}}{(1-2 x)^{3/2}} \, dx &=\frac {7 (3+5 x)^{7/2}}{11 \sqrt {1-2 x}}-\frac {243}{22} \int \frac {(3+5 x)^{5/2}}{\sqrt {1-2 x}} \, dx\\ &=\frac {81}{44} \sqrt {1-2 x} (3+5 x)^{5/2}+\frac {7 (3+5 x)^{7/2}}{11 \sqrt {1-2 x}}-\frac {405}{8} \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x}} \, dx\\ &=\frac {405}{32} \sqrt {1-2 x} (3+5 x)^{3/2}+\frac {81}{44} \sqrt {1-2 x} (3+5 x)^{5/2}+\frac {7 (3+5 x)^{7/2}}{11 \sqrt {1-2 x}}-\frac {13365}{64} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x}} \, dx\\ &=\frac {13365}{128} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {405}{32} \sqrt {1-2 x} (3+5 x)^{3/2}+\frac {81}{44} \sqrt {1-2 x} (3+5 x)^{5/2}+\frac {7 (3+5 x)^{7/2}}{11 \sqrt {1-2 x}}-\frac {147015}{256} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=\frac {13365}{128} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {405}{32} \sqrt {1-2 x} (3+5 x)^{3/2}+\frac {81}{44} \sqrt {1-2 x} (3+5 x)^{5/2}+\frac {7 (3+5 x)^{7/2}}{11 \sqrt {1-2 x}}-\frac {1}{128} \left (29403 \sqrt {5}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )\\ &=\frac {13365}{128} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {405}{32} \sqrt {1-2 x} (3+5 x)^{3/2}+\frac {81}{44} \sqrt {1-2 x} (3+5 x)^{5/2}+\frac {7 (3+5 x)^{7/2}}{11 \sqrt {1-2 x}}-\frac {29403}{128} \sqrt {\frac {5}{2}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )\\ \end {align*}
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Mathematica [A]
time = 0.16, size = 73, normalized size = 0.62 \begin {gather*} \frac {-2 \sqrt {3+5 x} \left (-22545+14526 x+6120 x^2+1600 x^3\right )+29403 \sqrt {10-20 x} \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{256 \sqrt {1-2 x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 123, normalized size = 1.04
method | result | size |
default | \(-\frac {\left (-6400 x^{3} \sqrt {-10 x^{2}-x +3}+58806 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x -24480 x^{2} \sqrt {-10 x^{2}-x +3}-29403 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-58104 x \sqrt {-10 x^{2}-x +3}+90180 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}\, \sqrt {3+5 x}}{512 \left (-1+2 x \right ) \sqrt {-10 x^{2}-x +3}}\) | \(123\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.55, size = 92, normalized size = 0.78 \begin {gather*} -\frac {125 \, x^{4}}{2 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {4425 \, x^{3}}{16 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {45495 \, x^{2}}{64 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {29403}{512} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {69147 \, x}{128 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {67635}{128 \, \sqrt {-10 \, x^{2} - x + 3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.92, size = 92, normalized size = 0.78 \begin {gather*} \frac {29403 \, \sqrt {5} \sqrt {2} {\left (2 \, x - 1\right )} \arctan \left (\frac {\sqrt {5} \sqrt {2} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 4 \, {\left (1600 \, x^{3} + 6120 \, x^{2} + 14526 \, x - 22545\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{512 \, {\left (2 \, x - 1\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 {\left (3 x + 2\right ) \left (5 x + 3\right )^{\frac {5}{2}}}{\left (1 - 2 x\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.99, size = 84, normalized size = 0.71 \begin {gather*} -\frac {29403}{256} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + \frac {{\left (2 \, {\left (4 \, {\left (8 \, \sqrt {5} {\left (5 \, x + 3\right )} + 81 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 4455 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} - 147015 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{3200 \, {\left (2 \, x - 1\right )}} \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 (3\,x+2\right )\,{\left (5\,x+3\right )}^{5/2}}{{\left (1-2\,x\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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