Optimal. Leaf size=142 \[ \frac {76587 \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}}{17600}+\frac {939}{880} \sqrt {1-2 x} (2+3 x)^3 \sqrt {3+5 x}+\frac {7 (2+3 x)^4 \sqrt {3+5 x}}{11 \sqrt {1-2 x}}+\frac {21 \sqrt {1-2 x} \sqrt {3+5 x} (18424549+7645620 x)}{2816000}-\frac {291096141 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{256000 \sqrt {10}} \]
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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 = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {100, 158, 152,
56, 222} \begin {gather*} -\frac {291096141 \text {ArcSin}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{256000 \sqrt {10}}+\frac {7 \sqrt {5 x+3} (3 x+2)^4}{11 \sqrt {1-2 x}}+\frac {939}{880} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^3+\frac {76587 \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^2}{17600}+\frac {21 \sqrt {1-2 x} \sqrt {5 x+3} (7645620 x+18424549)}{2816000} \end {gather*}
Antiderivative was successfully verified.
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Rule 56
Rule 100
Rule 152
Rule 158
Rule 222
Rubi steps
\begin {align*} \int \frac {(2+3 x)^5}{(1-2 x)^{3/2} \sqrt {3+5 x}} \, dx &=\frac {7 (2+3 x)^4 \sqrt {3+5 x}}{11 \sqrt {1-2 x}}-\frac {1}{11} \int \frac {(2+3 x)^3 \left (285+\frac {939 x}{2}\right )}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=\frac {939}{880} \sqrt {1-2 x} (2+3 x)^3 \sqrt {3+5 x}+\frac {7 (2+3 x)^4 \sqrt {3+5 x}}{11 \sqrt {1-2 x}}+\frac {1}{440} \int \frac {\left (-35007-\frac {229761 x}{4}\right ) (2+3 x)^2}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=\frac {76587 \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}}{17600}+\frac {939}{880} \sqrt {1-2 x} (2+3 x)^3 \sqrt {3+5 x}+\frac {7 (2+3 x)^4 \sqrt {3+5 x}}{11 \sqrt {1-2 x}}-\frac {\int \frac {(2+3 x) \left (\frac {12307617}{4}+\frac {40139505 x}{8}\right )}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{13200}\\ &=\frac {76587 \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}}{17600}+\frac {939}{880} \sqrt {1-2 x} (2+3 x)^3 \sqrt {3+5 x}+\frac {7 (2+3 x)^4 \sqrt {3+5 x}}{11 \sqrt {1-2 x}}+\frac {21 \sqrt {1-2 x} \sqrt {3+5 x} (18424549+7645620 x)}{2816000}-\frac {291096141 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{512000}\\ &=\frac {76587 \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}}{17600}+\frac {939}{880} \sqrt {1-2 x} (2+3 x)^3 \sqrt {3+5 x}+\frac {7 (2+3 x)^4 \sqrt {3+5 x}}{11 \sqrt {1-2 x}}+\frac {21 \sqrt {1-2 x} \sqrt {3+5 x} (18424549+7645620 x)}{2816000}-\frac {291096141 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{256000 \sqrt {5}}\\ &=\frac {76587 \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}}{17600}+\frac {939}{880} \sqrt {1-2 x} (2+3 x)^3 \sqrt {3+5 x}+\frac {7 (2+3 x)^4 \sqrt {3+5 x}}{11 \sqrt {1-2 x}}+\frac {21 \sqrt {1-2 x} \sqrt {3+5 x} (18424549+7645620 x)}{2816000}-\frac {291096141 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{256000 \sqrt {10}}\\ \end {align*}
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Mathematica [A]
time = 0.20, size = 78, normalized size = 0.55 \begin {gather*} \frac {-10 \sqrt {3+5 x} \left (-488641609+332129358 x+171939240 x^2+76887360 x^3+17107200 x^4\right )+3202057551 \sqrt {10-20 x} \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{28160000 \sqrt {1-2 x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 140, normalized size = 0.99
method | result | size |
default | \(-\frac {\left (-342144000 x^{4} \sqrt {-10 x^{2}-x +3}-1537747200 x^{3} \sqrt {-10 x^{2}-x +3}+6404115102 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x -3438784800 x^{2} \sqrt {-10 x^{2}-x +3}-3202057551 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-6642587160 x \sqrt {-10 x^{2}-x +3}+9772832180 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {3+5 x}\, \sqrt {1-2 x}}{56320000 \left (-1+2 x \right ) \sqrt {-10 x^{2}-x +3}}\) | \(140\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.60, size = 99, normalized size = 0.70 \begin {gather*} \frac {243}{80} \, \sqrt {-10 \, x^{2} - x + 3} x^{3} + \frac {24273}{1600} \, \sqrt {-10 \, x^{2} - x + 3} x^{2} - \frac {291096141}{5120000} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) + \frac {487863}{12800} \, \sqrt {-10 \, x^{2} - x + 3} x + \frac {19975419}{256000} \, \sqrt {-10 \, x^{2} - x + 3} - \frac {16807 \, \sqrt {-10 \, x^{2} - x + 3}}{176 \, {\left (2 \, x - 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.78, size = 91, normalized size = 0.64 \begin {gather*} \frac {3202057551 \, \sqrt {10} {\left (2 \, x - 1\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 (17107200 \, x^{4} + 76887360 \, x^{3} + 171939240 \, x^{2} + 332129358 \, x - 488641609\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{56320000 \, {\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 )^{5}}{\left (1 - 2 x\right )^{\frac {3}{2}} \sqrt {5 x + 3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.96, size = 97, normalized size = 0.68 \begin {gather*} -\frac {291096141}{2560000} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + \frac {{\left (198 \, {\left (12 \, {\left (8 \, {\left (36 \, \sqrt {5} {\left (5 \, x + 3\right )} + 377 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 29669 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 4900505 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} - 16010291851 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{352000000 \, {\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 )}^5}{{\left (1-2\,x\right )}^{3/2}\,\sqrt {5\,x+3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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