Optimal. Leaf size=144 \[ \frac {19}{18} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{6 (2+3 x)^2}+\frac {5 (1-2 x)^{3/2} \sqrt {3+5 x}}{4 (2+3 x)}+\frac {118}{27} \sqrt {\frac {2}{5}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )-\frac {155}{108} \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right ) \]
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Rubi [A]
time = 0.04, antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 8, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {99, 154, 159,
163, 56, 222, 95, 210} \begin {gather*} \frac {118}{27} \sqrt {\frac {2}{5}} \text {ArcSin}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )-\frac {155}{108} \sqrt {7} \text {ArcTan}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )-\frac {\sqrt {5 x+3} (1-2 x)^{5/2}}{6 (3 x+2)^2}+\frac {5 \sqrt {5 x+3} (1-2 x)^{3/2}}{4 (3 x+2)}+\frac {19}{18} \sqrt {5 x+3} \sqrt {1-2 x} \end {gather*}
Antiderivative was successfully verified.
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Rule 56
Rule 95
Rule 99
Rule 154
Rule 159
Rule 163
Rule 210
Rule 222
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{(2+3 x)^3} \, dx &=-\frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{6 (2+3 x)^2}+\frac {1}{6} \int \frac {\left (-\frac {25}{2}-30 x\right ) (1-2 x)^{3/2}}{(2+3 x)^2 \sqrt {3+5 x}} \, dx\\ &=-\frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{6 (2+3 x)^2}+\frac {5 (1-2 x)^{3/2} \sqrt {3+5 x}}{4 (2+3 x)}-\frac {1}{18} \int \frac {\left (-\frac {915}{4}-285 x\right ) \sqrt {1-2 x}}{(2+3 x) \sqrt {3+5 x}} \, dx\\ &=\frac {19}{18} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{6 (2+3 x)^2}+\frac {5 (1-2 x)^{3/2} \sqrt {3+5 x}}{4 (2+3 x)}-\frac {1}{270} \int \frac {-\frac {14865}{4}-3540 x}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=\frac {19}{18} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{6 (2+3 x)^2}+\frac {5 (1-2 x)^{3/2} \sqrt {3+5 x}}{4 (2+3 x)}+\frac {118}{27} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx+\frac {1085}{216} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=\frac {19}{18} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{6 (2+3 x)^2}+\frac {5 (1-2 x)^{3/2} \sqrt {3+5 x}}{4 (2+3 x)}+\frac {1085}{108} \text {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )+\frac {236 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{27 \sqrt {5}}\\ &=\frac {19}{18} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{6 (2+3 x)^2}+\frac {5 (1-2 x)^{3/2} \sqrt {3+5 x}}{4 (2+3 x)}+\frac {118}{27} \sqrt {\frac {2}{5}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )-\frac {155}{108} \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.27, size = 108, normalized size = 0.75 \begin {gather*} \frac {1}{540} \left (\frac {15 \sqrt {1-2 x} \left (708+2485 x+2319 x^2+240 x^3\right )}{(2+3 x)^2 \sqrt {3+5 x}}-472 \sqrt {10} \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )-775 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 208, normalized size = 1.44
method | result | size |
risch | \(-\frac {\sqrt {3+5 x}\, \left (-1+2 x \right ) \left (48 x^{2}+435 x +236\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{36 \left (2+3 x \right )^{2} \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right )}\, \sqrt {1-2 x}}-\frac {\left (-\frac {59 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )}{135}-\frac {155 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right )}{216}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(138\) |
default | \(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (4248 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{2}+6975 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}+5664 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x +9300 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +1440 x^{2} \sqrt {-10 x^{2}-x +3}+1888 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+3100 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+13050 x \sqrt {-10 x^{2}-x +3}+7080 \sqrt {-10 x^{2}-x +3}\right )}{1080 \sqrt {-10 x^{2}-x +3}\, \left (2+3 x \right )^{2}}\) | \(208\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.69, size = 101, normalized size = 0.70 \begin {gather*} \frac {59}{135} \, \sqrt {10} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) + \frac {155}{216} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {13}{9} \, \sqrt {-10 \, x^{2} - x + 3} + \frac {7 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{6 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac {49 \, \sqrt {-10 \, x^{2} - x + 3}}{36 \, {\left (3 \, x + 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.51, size = 147, normalized size = 1.02 \begin {gather*} -\frac {472 \, \sqrt {5} \sqrt {2} {\left (9 \, x^{2} + 12 \, x + 4\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 ) + 775 \, \sqrt {7} {\left (9 \, x^{2} + 12 \, x + 4\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 30 \, {\left (48 \, x^{2} + 435 \, x + 236\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{1080 \, {\left (9 \, x^{2} + 12 \, x + 4\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 (1 - 2 x\right )^{\frac {5}{2}} \sqrt {5 x + 3}}{\left (3 x + 2\right )^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 338 vs.
\(2 (106) = 212\).
time = 0.78, size = 338, normalized size = 2.35 \begin {gather*} \frac {31}{432} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} + \frac {59}{135} \, \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} + \frac {4}{135} \, \sqrt {5} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - \frac {77 \, \sqrt {10} {\left (17 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} - \frac {13720 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {54880 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{54 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\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 (1-2\,x\right )}^{5/2}\,\sqrt {5\,x+3}}{{\left (3\,x+2\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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