Optimal. Leaf size=116 \[ -\frac {3}{40} \sqrt {1-2 x} (5 x+3)^{7/2}-\frac {247}{480} \sqrt {1-2 x} (5 x+3)^{5/2}-\frac {2717}{768} \sqrt {1-2 x} (5 x+3)^{3/2}-\frac {29887 \sqrt {1-2 x} \sqrt {5 x+3}}{1024}+\frac {328757 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{1024 \sqrt {10}} \]
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Rubi [A] time = 0.03, antiderivative size = 116, 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 = {80, 50, 54, 216} \[ -\frac {3}{40} \sqrt {1-2 x} (5 x+3)^{7/2}-\frac {247}{480} \sqrt {1-2 x} (5 x+3)^{5/2}-\frac {2717}{768} \sqrt {1-2 x} (5 x+3)^{3/2}-\frac {29887 \sqrt {1-2 x} \sqrt {5 x+3}}{1024}+\frac {328757 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{1024 \sqrt {10}} \]
Antiderivative was successfully verified.
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Rule 50
Rule 54
Rule 80
Rule 216
Rubi steps
\begin {align*} \int \frac {(2+3 x) (3+5 x)^{5/2}}{\sqrt {1-2 x}} \, dx &=-\frac {3}{40} \sqrt {1-2 x} (3+5 x)^{7/2}+\frac {247}{80} \int \frac {(3+5 x)^{5/2}}{\sqrt {1-2 x}} \, dx\\ &=-\frac {247}{480} \sqrt {1-2 x} (3+5 x)^{5/2}-\frac {3}{40} \sqrt {1-2 x} (3+5 x)^{7/2}+\frac {2717}{192} \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x}} \, dx\\ &=-\frac {2717}{768} \sqrt {1-2 x} (3+5 x)^{3/2}-\frac {247}{480} \sqrt {1-2 x} (3+5 x)^{5/2}-\frac {3}{40} \sqrt {1-2 x} (3+5 x)^{7/2}+\frac {29887}{512} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x}} \, dx\\ &=-\frac {29887 \sqrt {1-2 x} \sqrt {3+5 x}}{1024}-\frac {2717}{768} \sqrt {1-2 x} (3+5 x)^{3/2}-\frac {247}{480} \sqrt {1-2 x} (3+5 x)^{5/2}-\frac {3}{40} \sqrt {1-2 x} (3+5 x)^{7/2}+\frac {328757 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{2048}\\ &=-\frac {29887 \sqrt {1-2 x} \sqrt {3+5 x}}{1024}-\frac {2717}{768} \sqrt {1-2 x} (3+5 x)^{3/2}-\frac {247}{480} \sqrt {1-2 x} (3+5 x)^{5/2}-\frac {3}{40} \sqrt {1-2 x} (3+5 x)^{7/2}+\frac {328757 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{1024 \sqrt {5}}\\ &=-\frac {29887 \sqrt {1-2 x} \sqrt {3+5 x}}{1024}-\frac {2717}{768} \sqrt {1-2 x} (3+5 x)^{3/2}-\frac {247}{480} \sqrt {1-2 x} (3+5 x)^{5/2}-\frac {3}{40} \sqrt {1-2 x} (3+5 x)^{7/2}+\frac {328757 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{1024 \sqrt {10}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 83, normalized size = 0.72 \[ -\frac {\sqrt {1-2 x} \left (10 \sqrt {2 x-1} \sqrt {5 x+3} \left (28800 x^3+91360 x^2+132868 x+142713\right )+986271 \sqrt {10} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )\right )}{30720 \sqrt {2 x-1}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.05, size = 72, normalized size = 0.62 \[ -\frac {1}{3072} \, {\left (28800 \, x^{3} + 91360 \, x^{2} + 132868 \, x + 142713\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {328757}{20480} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.11, size = 63, normalized size = 0.54 \[ -\frac {1}{30720} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (36 \, x + 71\right )} {\left (5 \, x + 3\right )} + 2717\right )} {\left (5 \, x + 3\right )} + 89661\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 986271 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 104, normalized size = 0.90 \[ \frac {\sqrt {5 x +3}\, \sqrt {-2 x +1}\, \left (-576000 \sqrt {-10 x^{2}-x +3}\, x^{3}-1827200 \sqrt {-10 x^{2}-x +3}\, x^{2}-2657360 \sqrt {-10 x^{2}-x +3}\, x +986271 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-2854260 \sqrt {-10 x^{2}-x +3}\right )}{61440 \sqrt {-10 x^{2}-x +3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.44, size = 75, normalized size = 0.65 \[ -\frac {75}{8} \, \sqrt {-10 \, x^{2} - x + 3} x^{3} - \frac {2855}{96} \, \sqrt {-10 \, x^{2} - x + 3} x^{2} - \frac {33217}{768} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {328757}{20480} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) - \frac {47571}{1024} \, \sqrt {-10 \, x^{2} - x + 3} \]
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 \[ \int \frac {\left (3\,x+2\right )\,{\left (5\,x+3\right )}^{5/2}}{\sqrt {1-2\,x}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 166.18, size = 298, normalized size = 2.57 \[ \frac {2 \sqrt {5} \left (\begin {cases} \frac {1331 \sqrt {2} \left (\frac {\sqrt {2} \left (5 - 10 x\right )^{\frac {3}{2}} \left (5 x + 3\right )^{\frac {3}{2}}}{3993} + \frac {3 \sqrt {2} \sqrt {5 - 10 x} \left (- 20 x - 1\right ) \sqrt {5 x + 3}}{1936} - \frac {\sqrt {2} \sqrt {5 - 10 x} \sqrt {5 x + 3}}{22} + \frac {5 \operatorname {asin}{\left (\frac {\sqrt {22} \sqrt {5 x + 3}}{11} \right )}}{16}\right )}{16} & \text {for}\: x \geq - \frac {3}{5} \wedge x < \frac {1}{2} \end {cases}\right )}{25} + \frac {6 \sqrt {5} \left (\begin {cases} \frac {14641 \sqrt {2} \left (\frac {2 \sqrt {2} \left (5 - 10 x\right )^{\frac {3}{2}} \left (5 x + 3\right )^{\frac {3}{2}}}{3993} + \frac {7 \sqrt {2} \sqrt {5 - 10 x} \left (- 20 x - 1\right ) \sqrt {5 x + 3}}{3872} + \frac {\sqrt {2} \sqrt {5 - 10 x} \sqrt {5 x + 3} \left (- 12100 x - 128 \left (5 x + 3\right )^{3} + 1056 \left (5 x + 3\right )^{2} - 5929\right )}{1874048} - \frac {\sqrt {2} \sqrt {5 - 10 x} \sqrt {5 x + 3}}{22} + \frac {35 \operatorname {asin}{\left (\frac {\sqrt {22} \sqrt {5 x + 3}}{11} \right )}}{128}\right )}{32} & \text {for}\: x \geq - \frac {3}{5} \wedge x < \frac {1}{2} \end {cases}\right )}{25} \]
Verification of antiderivative is not currently implemented for this CAS.
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