Optimal. Leaf size=173 \[ \frac {465 \sqrt {5 x+3}}{9604 \sqrt {1-2 x}}-\frac {85 \sqrt {5 x+3}}{2744 \sqrt {1-2 x} (3 x+2)}-\frac {23 \sqrt {5 x+3}}{196 \sqrt {1-2 x} (3 x+2)^2}-\frac {32 \sqrt {5 x+3}}{147 \sqrt {1-2 x} (3 x+2)^3}+\frac {11 \sqrt {5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)^3}-\frac {9395 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{19208 \sqrt {7}} \]
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Rubi [A] time = 0.06, antiderivative size = 173, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {98, 151, 152, 12, 93, 204} \[ \frac {465 \sqrt {5 x+3}}{9604 \sqrt {1-2 x}}-\frac {85 \sqrt {5 x+3}}{2744 \sqrt {1-2 x} (3 x+2)}-\frac {23 \sqrt {5 x+3}}{196 \sqrt {1-2 x} (3 x+2)^2}-\frac {32 \sqrt {5 x+3}}{147 \sqrt {1-2 x} (3 x+2)^3}+\frac {11 \sqrt {5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)^3}-\frac {9395 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{19208 \sqrt {7}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 93
Rule 98
Rule 151
Rule 152
Rule 204
Rubi steps
\begin {align*} \int \frac {(3+5 x)^{3/2}}{(1-2 x)^{5/2} (2+3 x)^4} \, dx &=\frac {11 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^3}-\frac {1}{21} \int \frac {-233-\frac {795 x}{2}}{(1-2 x)^{3/2} (2+3 x)^4 \sqrt {3+5 x}} \, dx\\ &=\frac {11 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^3}-\frac {32 \sqrt {3+5 x}}{147 \sqrt {1-2 x} (2+3 x)^3}-\frac {1}{441} \int \frac {-\frac {3357}{2}-2880 x}{(1-2 x)^{3/2} (2+3 x)^3 \sqrt {3+5 x}} \, dx\\ &=\frac {11 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^3}-\frac {32 \sqrt {3+5 x}}{147 \sqrt {1-2 x} (2+3 x)^3}-\frac {23 \sqrt {3+5 x}}{196 \sqrt {1-2 x} (2+3 x)^2}-\frac {\int \frac {-\frac {36855}{4}-14490 x}{(1-2 x)^{3/2} (2+3 x)^2 \sqrt {3+5 x}} \, dx}{6174}\\ &=\frac {11 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^3}-\frac {32 \sqrt {3+5 x}}{147 \sqrt {1-2 x} (2+3 x)^3}-\frac {23 \sqrt {3+5 x}}{196 \sqrt {1-2 x} (2+3 x)^2}-\frac {85 \sqrt {3+5 x}}{2744 \sqrt {1-2 x} (2+3 x)}-\frac {\int \frac {-\frac {268695}{8}-\frac {26775 x}{2}}{(1-2 x)^{3/2} (2+3 x) \sqrt {3+5 x}} \, dx}{43218}\\ &=\frac {465 \sqrt {3+5 x}}{9604 \sqrt {1-2 x}}+\frac {11 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^3}-\frac {32 \sqrt {3+5 x}}{147 \sqrt {1-2 x} (2+3 x)^3}-\frac {23 \sqrt {3+5 x}}{196 \sqrt {1-2 x} (2+3 x)^2}-\frac {85 \sqrt {3+5 x}}{2744 \sqrt {1-2 x} (2+3 x)}+\frac {\int \frac {6510735}{16 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{1663893}\\ &=\frac {465 \sqrt {3+5 x}}{9604 \sqrt {1-2 x}}+\frac {11 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^3}-\frac {32 \sqrt {3+5 x}}{147 \sqrt {1-2 x} (2+3 x)^3}-\frac {23 \sqrt {3+5 x}}{196 \sqrt {1-2 x} (2+3 x)^2}-\frac {85 \sqrt {3+5 x}}{2744 \sqrt {1-2 x} (2+3 x)}+\frac {9395 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{38416}\\ &=\frac {465 \sqrt {3+5 x}}{9604 \sqrt {1-2 x}}+\frac {11 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^3}-\frac {32 \sqrt {3+5 x}}{147 \sqrt {1-2 x} (2+3 x)^3}-\frac {23 \sqrt {3+5 x}}{196 \sqrt {1-2 x} (2+3 x)^2}-\frac {85 \sqrt {3+5 x}}{2744 \sqrt {1-2 x} (2+3 x)}+\frac {9395 \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{19208}\\ &=\frac {465 \sqrt {3+5 x}}{9604 \sqrt {1-2 x}}+\frac {11 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^3}-\frac {32 \sqrt {3+5 x}}{147 \sqrt {1-2 x} (2+3 x)^3}-\frac {23 \sqrt {3+5 x}}{196 \sqrt {1-2 x} (2+3 x)^2}-\frac {85 \sqrt {3+5 x}}{2744 \sqrt {1-2 x} (2+3 x)}-\frac {9395 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{19208 \sqrt {7}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 100, normalized size = 0.58 \[ -\frac {7 \sqrt {5 x+3} \left (150660 x^4+193860 x^3-17127 x^2-80510 x-19296\right )-28185 \sqrt {7-14 x} (2 x-1) (3 x+2)^3 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{403368 (1-2 x)^{3/2} (3 x+2)^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.15, size = 131, normalized size = 0.76 \[ -\frac {28185 \, \sqrt {7} {\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 14 \, {\left (150660 \, x^{4} + 193860 \, x^{3} - 17127 \, x^{2} - 80510 \, x - 19296\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{806736 \, {\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 3.25, size = 349, normalized size = 2.02 \[ \frac {1879}{537824} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {8 \, {\left (512 \, \sqrt {5} {\left (5 \, x + 3\right )} - 3201 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{1260525 \, {\left (2 \, x - 1\right )}^{2}} - \frac {99 \, \sqrt {10} {\left (727 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} + 548800 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} + \frac {20776000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} - \frac {83104000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{67228 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 305, normalized size = 1.76 \[ \frac {\left (3043980 \sqrt {7}\, x^{5} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+3043980 \sqrt {7}\, x^{4} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-2109240 \sqrt {-10 x^{2}-x +3}\, x^{4}-1268325 \sqrt {7}\, x^{3} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-2714040 \sqrt {-10 x^{2}-x +3}\, x^{3}-1634730 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+239778 \sqrt {-10 x^{2}-x +3}\, x^{2}+112740 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+1127140 \sqrt {-10 x^{2}-x +3}\, x +225480 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+270144 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {-2 x +1}\, \sqrt {5 x +3}}{806736 \left (3 x +2\right )^{3} \left (2 x -1\right )^{2} \sqrt {-10 x^{2}-x +3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.53, size = 240, normalized size = 1.39 \[ \frac {9395}{268912} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {2325 \, x}{9604 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {5395}{57624 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {7625 \, x}{12348 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} + \frac {1}{567 \, {\left (27 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{3} + 54 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} + 36 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 8 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} - \frac {169}{5292 \, {\left (9 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} + 12 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 4 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} + \frac {1987}{10584 \, {\left (3 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 2 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} + \frac {2165}{222264 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} \]
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 \[ \int \frac {{\left (5\,x+3\right )}^{3/2}}{{\left (1-2\,x\right )}^{5/2}\,{\left (3\,x+2\right )}^4} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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