Optimal. Leaf size=173 \[ \frac {15755 \sqrt {3+5 x}}{86436 \sqrt {1-2 x}}+\frac {32 \sqrt {3+5 x}}{441 \sqrt {1-2 x} (2+3 x)^3}-\frac {187 \sqrt {3+5 x}}{588 \sqrt {1-2 x} (2+3 x)^2}-\frac {2365 \sqrt {3+5 x}}{8232 \sqrt {1-2 x} (2+3 x)}+\frac {11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^3}-\frac {2585 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{19208 \sqrt {7}} \]
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Rubi [A]
time = 0.04, 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 = {100, 154, 156,
157, 12, 95, 210} \begin {gather*} -\frac {2585 \text {ArcTan}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{19208 \sqrt {7}}+\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 95
Rule 100
Rule 154
Rule 156
Rule 157
Rule 210
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 \text {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 = 2.51, size = 152, normalized size = 0.88 \begin {gather*} \frac {5 \left (-\frac {7 \sqrt {3+5 x} \left (-75888-304730 x-169221 x^2+552780 x^3+567180 x^4\right )}{5 (1-2 x)^{3/2} (2+3 x)^3}+1551 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {2 \left (34+\sqrt {1155}\right )} \sqrt {3+5 x}}{-\sqrt {11}+\sqrt {5-10 x}}\right )+1551 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {6+10 x}}{\sqrt {34+\sqrt {1155}} \left (-\sqrt {11}+\sqrt {5-10 x}\right )}\right )\right )}{403368} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(304\) vs.
\(2(134)=268\).
time = 0.08, size = 305, normalized size = 1.76
method | result | size |
default | \(\frac {\left (837540 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{5}+837540 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}-348975 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}-7940520 x^{4} \sqrt {-10 x^{2}-x +3}-449790 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}-7738920 x^{3} \sqrt {-10 x^{2}-x +3}+31020 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +2369094 x^{2} \sqrt {-10 x^{2}-x +3}+62040 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+4266220 x \sqrt {-10 x^{2}-x +3}+1062432 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}\, \sqrt {3+5 x}}{806736 \left (2+3 x \right )^{3} \left (-1+2 x \right )^{2} \sqrt {-10 x^{2}-x +3}}\) | \(305\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 240, normalized size = 1.39 \begin {gather*} \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}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.48, size = 131, normalized size = 0.76 \begin {gather*} -\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 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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. 349 vs.
\(2 (134) = 268\).
time = 3.69, size = 349, normalized size = 2.02 \begin {gather*} \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}} \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 (5\,x+3\right )}^{5/2}}{{\left (1-2\,x\right )}^{5/2}\,{\left (3\,x+2\right )}^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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