Optimal. Leaf size=173 \[ \frac {11 (5 x+3)^{3/2}}{21 (1-2 x)^{3/2} (3 x+2)^3}+\frac {15755 \sqrt {5 x+3}}{86436 \sqrt {1-2 x}}-\frac {2365 \sqrt {5 x+3}}{8232 \sqrt {1-2 x} (3 x+2)}-\frac {187 \sqrt {5 x+3}}{588 \sqrt {1-2 x} (3 x+2)^2}+\frac {32 \sqrt {5 x+3}}{441 \sqrt {1-2 x} (3 x+2)^3}-\frac {2585 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{19208 \sqrt {7}} \]
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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 = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {98, 149, 151, 152, 12, 93, 204} \[ \frac {11 (5 x+3)^{3/2}}{21 (1-2 x)^{3/2} (3 x+2)^3}+\frac {15755 \sqrt {5 x+3}}{86436 \sqrt {1-2 x}}-\frac {2365 \sqrt {5 x+3}}{8232 \sqrt {1-2 x} (3 x+2)}-\frac {187 \sqrt {5 x+3}}{588 \sqrt {1-2 x} (3 x+2)^2}+\frac {32 \sqrt {5 x+3}}{441 \sqrt {1-2 x} (3 x+2)^3}-\frac {2585 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{19208 \sqrt {7}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 93
Rule 98
Rule 149
Rule 151
Rule 152
Rule 204
Rubi steps
\begin {align*} \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{5/2} (2+3 x)^4} \, dx &=\frac {11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^3}-\frac {1}{21} \int \frac {\left (-123-\frac {465 x}{2}\right ) \sqrt {3+5 x}}{(1-2 x)^{3/2} (2+3 x)^4} \, dx\\ &=\frac {32 \sqrt {3+5 x}}{441 \sqrt {1-2 x} (2+3 x)^3}+\frac {11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^3}-\frac {\int \frac {-\frac {24783}{2}-\frac {43065 x}{2}}{(1-2 x)^{3/2} (2+3 x)^3 \sqrt {3+5 x}} \, dx}{1323}\\ &=\frac {32 \sqrt {3+5 x}}{441 \sqrt {1-2 x} (2+3 x)^3}-\frac {187 \sqrt {3+5 x}}{588 \sqrt {1-2 x} (2+3 x)^2}+\frac {11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^3}-\frac {\int \frac {-\frac {264495}{4}-117810 x}{(1-2 x)^{3/2} (2+3 x)^2 \sqrt {3+5 x}} \, dx}{18522}\\ &=\frac {32 \sqrt {3+5 x}}{441 \sqrt {1-2 x} (2+3 x)^3}-\frac {187 \sqrt {3+5 x}}{588 \sqrt {1-2 x} (2+3 x)^2}-\frac {2365 \sqrt {3+5 x}}{8232 \sqrt {1-2 x} (2+3 x)}+\frac {11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^3}-\frac {\int \frac {-\frac {2149455}{8}-\frac {744975 x}{2}}{(1-2 x)^{3/2} (2+3 x) \sqrt {3+5 x}} \, dx}{129654}\\ &=\frac {15755 \sqrt {3+5 x}}{86436 \sqrt {1-2 x}}+\frac {32 \sqrt {3+5 x}}{441 \sqrt {1-2 x} (2+3 x)^3}-\frac {187 \sqrt {3+5 x}}{588 \sqrt {1-2 x} (2+3 x)^2}-\frac {2365 \sqrt {3+5 x}}{8232 \sqrt {1-2 x} (2+3 x)}+\frac {11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^3}+\frac {\int \frac {5374215}{16 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{4991679}\\ &=\frac {15755 \sqrt {3+5 x}}{86436 \sqrt {1-2 x}}+\frac {32 \sqrt {3+5 x}}{441 \sqrt {1-2 x} (2+3 x)^3}-\frac {187 \sqrt {3+5 x}}{588 \sqrt {1-2 x} (2+3 x)^2}-\frac {2365 \sqrt {3+5 x}}{8232 \sqrt {1-2 x} (2+3 x)}+\frac {11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^3}+\frac {2585 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{38416}\\ &=\frac {15755 \sqrt {3+5 x}}{86436 \sqrt {1-2 x}}+\frac {32 \sqrt {3+5 x}}{441 \sqrt {1-2 x} (2+3 x)^3}-\frac {187 \sqrt {3+5 x}}{588 \sqrt {1-2 x} (2+3 x)^2}-\frac {2365 \sqrt {3+5 x}}{8232 \sqrt {1-2 x} (2+3 x)}+\frac {11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^3}+\frac {2585 \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{19208}\\ &=\frac {15755 \sqrt {3+5 x}}{86436 \sqrt {1-2 x}}+\frac {32 \sqrt {3+5 x}}{441 \sqrt {1-2 x} (2+3 x)^3}-\frac {187 \sqrt {3+5 x}}{588 \sqrt {1-2 x} (2+3 x)^2}-\frac {2365 \sqrt {3+5 x}}{8232 \sqrt {1-2 x} (2+3 x)}+\frac {11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^3}-\frac {2585 \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 \[ -\frac {7 \sqrt {5 x+3} \left (567180 x^4+552780 x^3-169221 x^2-304730 x-75888\right )-7755 \sqrt {7-14 x} (2 x-1) (3 x+2)^3 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{403368 (1-2 x)^{3/2} (3 x+2)^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.19, size = 131, normalized size = 0.76 \[ -\frac {7755 \, \sqrt {7} {\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 14 \, {\left (567180 \, x^{4} + 552780 \, x^{3} - 169221 \, x^{2} - 304730 \, x - 75888\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{806736 \, {\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 4.15, size = 349, normalized size = 2.02 \[ \frac {517}{537824} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {88 \, {\left (151 \, \sqrt {5} {\left (5 \, x + 3\right )} - 1023 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{1260525 \, {\left (2 \, x - 1\right )}^{2}} - \frac {11 \, \sqrt {10} {\left (3629 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} + 2900800 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} + \frac {755384000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} - \frac {3021536000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{67228 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{3}} \]
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 \[ \frac {\left (837540 \sqrt {7}\, x^{5} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+837540 \sqrt {7}\, x^{4} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-7940520 \sqrt {-10 x^{2}-x +3}\, x^{4}-348975 \sqrt {7}\, x^{3} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-7738920 \sqrt {-10 x^{2}-x +3}\, x^{3}-449790 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+2369094 \sqrt {-10 x^{2}-x +3}\, x^{2}+31020 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+4266220 \sqrt {-10 x^{2}-x +3}\, x +62040 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+1062432 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {-2 x +1}\, \sqrt {5 x +3}}{806736 \left (3 x +2\right )^{3} \left (2 x -1\right )^{2} \sqrt {-10 x^{2}-x +3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.26, size = 240, normalized size = 1.39 \[ \frac {2585}{268912} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {78775 \, x}{86436 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {11755}{172872 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {17875 \, x}{12348 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} - \frac {1}{1701 \, {\left (27 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{3} + 54 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} + 36 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 8 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} + \frac {239}{15876 \, {\left (9 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} + 12 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 4 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} - \frac {4997}{31752 \, {\left (3 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 2 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} + \frac {901885}{666792 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} \]
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 \[ \int \frac {{\left (5\,x+3\right )}^{5/2}}{{\left (1-2\,x\right )}^{5/2}\,{\left (3\,x+2\right )}^4} \,d x \]
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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