Optimal. Leaf size=142 \[ -\frac {697}{88} \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}-\frac {299 (2+3 x)^3 \sqrt {3+5 x}}{66 \sqrt {1-2 x}}+\frac {(2+3 x)^4 \sqrt {3+5 x}}{3 (1-2 x)^{3/2}}-\frac {\sqrt {1-2 x} \sqrt {3+5 x} (17606479+7306140 x)}{70400}+\frac {13246251 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{6400 \sqrt {10}} \]
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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 {13246251 \text {ArcSin}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{6400 \sqrt {10}}+\frac {\sqrt {5 x+3} (3 x+2)^4}{3 (1-2 x)^{3/2}}-\frac {299 \sqrt {5 x+3} (3 x+2)^3}{66 \sqrt {1-2 x}}-\frac {697}{88} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^2-\frac {\sqrt {1-2 x} \sqrt {5 x+3} (7306140 x+17606479)}{70400} \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 {(2+3 x)^4 \sqrt {3+5 x}}{(1-2 x)^{5/2}} \, dx &=\frac {(2+3 x)^4 \sqrt {3+5 x}}{3 (1-2 x)^{3/2}}-\frac {1}{3} \int \frac {(2+3 x)^3 \left (41+\frac {135 x}{2}\right )}{(1-2 x)^{3/2} \sqrt {3+5 x}} \, dx\\ &=-\frac {299 (2+3 x)^3 \sqrt {3+5 x}}{66 \sqrt {1-2 x}}+\frac {(2+3 x)^4 \sqrt {3+5 x}}{3 (1-2 x)^{3/2}}-\frac {1}{33} \int \frac {\left (-4779-\frac {31365 x}{4}\right ) (2+3 x)^2}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {697}{88} \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}-\frac {299 (2+3 x)^3 \sqrt {3+5 x}}{66 \sqrt {1-2 x}}+\frac {(2+3 x)^4 \sqrt {3+5 x}}{3 (1-2 x)^{3/2}}+\frac {1}{990} \int \frac {(2+3 x) \left (\frac {1680165}{4}+\frac {5479605 x}{8}\right )}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {697}{88} \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}-\frac {299 (2+3 x)^3 \sqrt {3+5 x}}{66 \sqrt {1-2 x}}+\frac {(2+3 x)^4 \sqrt {3+5 x}}{3 (1-2 x)^{3/2}}-\frac {\sqrt {1-2 x} \sqrt {3+5 x} (17606479+7306140 x)}{70400}+\frac {13246251 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{12800}\\ &=-\frac {697}{88} \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}-\frac {299 (2+3 x)^3 \sqrt {3+5 x}}{66 \sqrt {1-2 x}}+\frac {(2+3 x)^4 \sqrt {3+5 x}}{3 (1-2 x)^{3/2}}-\frac {\sqrt {1-2 x} \sqrt {3+5 x} (17606479+7306140 x)}{70400}+\frac {13246251 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{6400 \sqrt {5}}\\ &=-\frac {697}{88} \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}-\frac {299 (2+3 x)^3 \sqrt {3+5 x}}{66 \sqrt {1-2 x}}+\frac {(2+3 x)^4 \sqrt {3+5 x}}{3 (1-2 x)^{3/2}}-\frac {\sqrt {1-2 x} \sqrt {3+5 x} (17606479+7306140 x)}{70400}+\frac {13246251 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{6400 \sqrt {10}}\\ \end {align*}
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Mathematica [A]
time = 0.20, size = 83, normalized size = 0.58 \begin {gather*} \frac {-10 \sqrt {3+5 x} \left (66038637-183672928 x+52700868 x^2+15040080 x^3+2851200 x^4\right )+437126283 \sqrt {10-20 x} (-1+2 x) \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{2112000 (1-2 x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 154, normalized size = 1.08
method | result | size |
default | \(\frac {\left (-57024000 x^{4} \sqrt {-10 x^{2}-x +3}+1748505132 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{2}-300801600 x^{3} \sqrt {-10 x^{2}-x +3}-1748505132 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x -1054017360 x^{2} \sqrt {-10 x^{2}-x +3}+437126283 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+3673458560 x \sqrt {-10 x^{2}-x +3}-1320772740 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}\, \sqrt {3+5 x}}{4224000 \left (-1+2 x \right )^{2} \sqrt {-10 x^{2}-x +3}}\) | \(154\) |
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 = 1.25, size = 101, normalized size = 0.71 \begin {gather*} -\frac {437126283 \, \sqrt {10} {\left (4 \, x^{2} - 4 \, 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 (2851200 \, x^{4} + 15040080 \, x^{3} + 52700868 \, x^{2} - 183672928 \, x + 66038637\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{4224000 \, {\left (4 \, x^{2} - 4 \, 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 )^{4} \sqrt {5 x + 3}}{\left (1 - 2 x\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.58, size = 97, normalized size = 0.68 \begin {gather*} \frac {13246251}{64000} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) - \frac {{\left (4 \, {\left (891 \, {\left (4 \, {\left (8 \, \sqrt {5} {\left (5 \, x + 3\right )} + 115 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 8919 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} - 291417650 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 4808389113 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{26400000 \, {\left (2 \, x - 1\right )}^{2}} \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 )}^4\,\sqrt {5\,x+3}}{{\left (1-2\,x\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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