Optimal. Leaf size=121 \[ -\frac {3}{40} \sqrt {1-2 x} (3 x+2) (5 x+3)^{5/2}-\frac {251}{800} \sqrt {1-2 x} (5 x+3)^{5/2}-\frac {14529 \sqrt {1-2 x} (5 x+3)^{3/2}}{6400}-\frac {479457 \sqrt {1-2 x} \sqrt {5 x+3}}{25600}+\frac {5274027 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{25600 \sqrt {10}} \]
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Rubi [A] time = 0.03, antiderivative size = 121, 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 = {90, 80, 50, 54, 216} \begin {gather*} -\frac {3}{40} \sqrt {1-2 x} (3 x+2) (5 x+3)^{5/2}-\frac {251}{800} \sqrt {1-2 x} (5 x+3)^{5/2}-\frac {14529 \sqrt {1-2 x} (5 x+3)^{3/2}}{6400}-\frac {479457 \sqrt {1-2 x} \sqrt {5 x+3}}{25600}+\frac {5274027 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{25600 \sqrt {10}} \end {gather*}
Antiderivative was successfully verified.
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Rule 50
Rule 54
Rule 80
Rule 90
Rule 216
Rubi steps
\begin {align*} \int \frac {(2+3 x)^2 (3+5 x)^{3/2}}{\sqrt {1-2 x}} \, dx &=-\frac {3}{40} \sqrt {1-2 x} (2+3 x) (3+5 x)^{5/2}-\frac {1}{40} \int \frac {\left (-244-\frac {753 x}{2}\right ) (3+5 x)^{3/2}}{\sqrt {1-2 x}} \, dx\\ &=-\frac {251}{800} \sqrt {1-2 x} (3+5 x)^{5/2}-\frac {3}{40} \sqrt {1-2 x} (2+3 x) (3+5 x)^{5/2}+\frac {14529 \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x}} \, dx}{1600}\\ &=-\frac {14529 \sqrt {1-2 x} (3+5 x)^{3/2}}{6400}-\frac {251}{800} \sqrt {1-2 x} (3+5 x)^{5/2}-\frac {3}{40} \sqrt {1-2 x} (2+3 x) (3+5 x)^{5/2}+\frac {479457 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x}} \, dx}{12800}\\ &=-\frac {479457 \sqrt {1-2 x} \sqrt {3+5 x}}{25600}-\frac {14529 \sqrt {1-2 x} (3+5 x)^{3/2}}{6400}-\frac {251}{800} \sqrt {1-2 x} (3+5 x)^{5/2}-\frac {3}{40} \sqrt {1-2 x} (2+3 x) (3+5 x)^{5/2}+\frac {5274027 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{51200}\\ &=-\frac {479457 \sqrt {1-2 x} \sqrt {3+5 x}}{25600}-\frac {14529 \sqrt {1-2 x} (3+5 x)^{3/2}}{6400}-\frac {251}{800} \sqrt {1-2 x} (3+5 x)^{5/2}-\frac {3}{40} \sqrt {1-2 x} (2+3 x) (3+5 x)^{5/2}+\frac {5274027 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{25600 \sqrt {5}}\\ &=-\frac {479457 \sqrt {1-2 x} \sqrt {3+5 x}}{25600}-\frac {14529 \sqrt {1-2 x} (3+5 x)^{3/2}}{6400}-\frac {251}{800} \sqrt {1-2 x} (3+5 x)^{5/2}-\frac {3}{40} \sqrt {1-2 x} (2+3 x) (3+5 x)^{5/2}+\frac {5274027 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{25600 \sqrt {10}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 83, normalized size = 0.69 \begin {gather*} -\frac {\sqrt {1-2 x} \left (10 \sqrt {2 x-1} \sqrt {5 x+3} \left (144000 x^3+469600 x^2+698580 x+760653\right )+5274027 \sqrt {10} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )\right )}{256000 \sqrt {2 x-1}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.18, size = 125, normalized size = 1.03 \begin {gather*} -\frac {121 \sqrt {1-2 x} \left (\frac {5448375 (1-2 x)^3}{(5 x+3)^3}+\frac {7990950 (1-2 x)^2}{(5 x+3)^2}+\frac {4240420 (1-2 x)}{5 x+3}+905704\right )}{25600 \sqrt {5 x+3} \left (\frac {5 (1-2 x)}{5 x+3}+2\right )^4}-\frac {5274027 \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}} \sqrt {1-2 x}}{\sqrt {5 x+3}}\right )}{25600 \sqrt {10}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.53, size = 72, normalized size = 0.60 \begin {gather*} -\frac {1}{25600} \, {\left (144000 \, x^{3} + 469600 \, x^{2} + 698580 \, x + 760653\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {5274027}{512000} \, \sqrt {10} \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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.25, size = 63, normalized size = 0.52 \begin {gather*} -\frac {1}{256000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (180 \, x + 371\right )} {\left (5 \, x + 3\right )} + 14529\right )} {\left (5 \, x + 3\right )} + 479457\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 5274027 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 104, normalized size = 0.86 \begin {gather*} \frac {\sqrt {5 x +3}\, \sqrt {-2 x +1}\, \left (-2880000 \sqrt {-10 x^{2}-x +3}\, x^{3}-9392000 \sqrt {-10 x^{2}-x +3}\, x^{2}-13971600 \sqrt {-10 x^{2}-x +3}\, x +5274027 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-15213060 \sqrt {-10 x^{2}-x +3}\right )}{512000 \sqrt {-10 x^{2}-x +3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.39, size = 75, normalized size = 0.62 \begin {gather*} -\frac {45}{8} \, \sqrt {-10 \, x^{2} - x + 3} x^{3} - \frac {587}{32} \, \sqrt {-10 \, x^{2} - x + 3} x^{2} - \frac {34929}{1280} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {5274027}{512000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) - \frac {760653}{25600} \, \sqrt {-10 \, x^{2} - x + 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 {{\left (3\,x+2\right )}^2\,{\left (5\,x+3\right )}^{3/2}}{\sqrt {1-2\,x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 165.34, size = 398, normalized size = 3.29 \begin {gather*} \frac {2 \sqrt {5} \left (\begin {cases} \frac {121 \sqrt {2} \left (\frac {\sqrt {2} \sqrt {5 - 10 x} \left (- 20 x - 1\right ) \sqrt {5 x + 3}}{968} - \frac {\sqrt {2} \sqrt {5 - 10 x} \sqrt {5 x + 3}}{22} + \frac {3 \operatorname {asin}{\left (\frac {\sqrt {22} \sqrt {5 x + 3}}{11} \right )}}{8}\right )}{8} & \text {for}\: x \geq - \frac {3}{5} \wedge x < \frac {1}{2} \end {cases}\right )}{125} + \frac {12 \sqrt {5} \left (\begin {cases} \frac {1331 \sqrt {2} \left (\frac {\sqrt {2} \left (5 - 10 x\right )^{\frac {3}{2}} \left (5 x + 3\right )^{\frac {3}{2}}}{3993} + \frac {3 \sqrt {2} \sqrt {5 - 10 x} \left (- 20 x - 1\right ) \sqrt {5 x + 3}}{1936} - \frac {\sqrt {2} \sqrt {5 - 10 x} \sqrt {5 x + 3}}{22} + \frac {5 \operatorname {asin}{\left (\frac {\sqrt {22} \sqrt {5 x + 3}}{11} \right )}}{16}\right )}{16} & \text {for}\: x \geq - \frac {3}{5} \wedge x < \frac {1}{2} \end {cases}\right )}{125} + \frac {18 \sqrt {5} \left (\begin {cases} \frac {14641 \sqrt {2} \left (\frac {2 \sqrt {2} \left (5 - 10 x\right )^{\frac {3}{2}} \left (5 x + 3\right )^{\frac {3}{2}}}{3993} + \frac {7 \sqrt {2} \sqrt {5 - 10 x} \left (- 20 x - 1\right ) \sqrt {5 x + 3}}{3872} + \frac {\sqrt {2} \sqrt {5 - 10 x} \sqrt {5 x + 3} \left (- 12100 x - 128 \left (5 x + 3\right )^{3} + 1056 \left (5 x + 3\right )^{2} - 5929\right )}{1874048} - \frac {\sqrt {2} \sqrt {5 - 10 x} \sqrt {5 x + 3}}{22} + \frac {35 \operatorname {asin}{\left (\frac {\sqrt {22} \sqrt {5 x + 3}}{11} \right )}}{128}\right )}{32} & \text {for}\: x \geq - \frac {3}{5} \wedge x < \frac {1}{2} \end {cases}\right )}{125} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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