Optimal. Leaf size=166 \[ \frac {185}{27} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {35}{4} \sqrt {1-2 x} (3+5 x)^{3/2}-\frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{6 (2+3 x)^2}+\frac {181 \sqrt {1-2 x} (3+5 x)^{5/2}}{36 (2+3 x)}+\frac {1945}{324} \sqrt {\frac {5}{2}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )+\frac {6829 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{324 \sqrt {7}} \]
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Rubi [A]
time = 0.05, antiderivative size = 166, normalized size of antiderivative = 1.00, number of steps
used = 9, 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 {1945}{324} \sqrt {\frac {5}{2}} \text {ArcSin}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )+\frac {6829 \text {ArcTan}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{324 \sqrt {7}}+\frac {181 \sqrt {1-2 x} (5 x+3)^{5/2}}{36 (3 x+2)}-\frac {(1-2 x)^{3/2} (5 x+3)^{5/2}}{6 (3 x+2)^2}-\frac {35}{4} \sqrt {1-2 x} (5 x+3)^{3/2}+\frac {185}{27} \sqrt {1-2 x} \sqrt {5 x+3} \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)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^3} \, dx &=-\frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{6 (2+3 x)^2}+\frac {1}{6} \int \frac {\left (\frac {7}{2}-40 x\right ) \sqrt {1-2 x} (3+5 x)^{3/2}}{(2+3 x)^2} \, dx\\ &=-\frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{6 (2+3 x)^2}+\frac {181 \sqrt {1-2 x} (3+5 x)^{5/2}}{36 (2+3 x)}-\frac {1}{18} \int \frac {\left (\frac {1789}{4}-1890 x\right ) (3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)} \, dx\\ &=-\frac {35}{4} \sqrt {1-2 x} (3+5 x)^{3/2}-\frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{6 (2+3 x)^2}+\frac {181 \sqrt {1-2 x} (3+5 x)^{5/2}}{36 (2+3 x)}+\frac {1}{216} \int \frac {(909-8880 x) \sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)} \, dx\\ &=\frac {185}{27} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {35}{4} \sqrt {1-2 x} (3+5 x)^{3/2}-\frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{6 (2+3 x)^2}+\frac {181 \sqrt {1-2 x} (3+5 x)^{5/2}}{36 (2+3 x)}-\frac {\int \frac {-25242-58350 x}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{1296}\\ &=\frac {185}{27} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {35}{4} \sqrt {1-2 x} (3+5 x)^{3/2}-\frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{6 (2+3 x)^2}+\frac {181 \sqrt {1-2 x} (3+5 x)^{5/2}}{36 (2+3 x)}-\frac {6829}{648} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx+\frac {9725}{648} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=\frac {185}{27} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {35}{4} \sqrt {1-2 x} (3+5 x)^{3/2}-\frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{6 (2+3 x)^2}+\frac {181 \sqrt {1-2 x} (3+5 x)^{5/2}}{36 (2+3 x)}-\frac {6829}{324} \text {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )+\frac {1}{324} \left (1945 \sqrt {5}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )\\ &=\frac {185}{27} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {35}{4} \sqrt {1-2 x} (3+5 x)^{3/2}-\frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{6 (2+3 x)^2}+\frac {181 \sqrt {1-2 x} (3+5 x)^{5/2}}{36 (2+3 x)}+\frac {1945}{324} \sqrt {\frac {5}{2}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )+\frac {6829 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{324 \sqrt {7}}\\ \end {align*}
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Mathematica [A]
time = 0.31, size = 113, normalized size = 0.68 \begin {gather*} \frac {\frac {42 \sqrt {1-2 x} \left (3696+15115 x+18210 x^2+2775 x^3-4500 x^4\right )}{(2+3 x)^2 \sqrt {3+5 x}}-13615 \sqrt {10} \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )+13658 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{4536} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 225, normalized size = 1.36
method | result | size |
risch | \(\frac {\sqrt {3+5 x}\, \left (-1+2 x \right ) \left (900 x^{3}-1095 x^{2}-2985 x -1232\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{108 \left (2+3 x \right )^{2} \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right )}\, \sqrt {1-2 x}}+\frac {\left (\frac {1945 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )}{1296}-\frac {6829 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right )}{4536}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(142\) |
default | \(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (122535 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{2}-122922 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}-75600 x^{3} \sqrt {-10 x^{2}-x +3}+163380 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x -163896 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +91980 x^{2} \sqrt {-10 x^{2}-x +3}+54460 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-54632 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+250740 x \sqrt {-10 x^{2}-x +3}+103488 \sqrt {-10 x^{2}-x +3}\right )}{9072 \sqrt {-10 x^{2}-x +3}\, \left (2+3 x \right )^{2}}\) | \(225\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 130, normalized size = 0.78 \begin {gather*} -\frac {5}{63} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} - \frac {{\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{14 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac {535}{126} \, \sqrt {-10 \, x^{2} - x + 3} x + \frac {1945}{1296} \, \sqrt {10} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) - \frac {6829}{4536} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {1627}{378} \, \sqrt {-10 \, x^{2} - x + 3} - \frac {59 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{84 \, {\left (3 \, x + 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.42, size = 152, normalized size = 0.92 \begin {gather*} -\frac {13615 \, \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 ) - 13658 \, \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 ) + 84 \, {\left (900 \, x^{3} - 1095 \, x^{2} - 2985 \, x - 1232\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{9072 \, {\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(-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. 351 vs.
\(2 (122) = 244\).
time = 1.28, size = 351, normalized size = 2.11 \begin {gather*} -\frac {6829}{45360} \, \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 {1}{108} \, {\left (4 \, \sqrt {5} {\left (5 \, x + 3\right )} - 63 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + \frac {1945}{1296} \, \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 {55 \, \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 {5992 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} - \frac {23968 \, \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 )}^{3/2}\,{\left (5\,x+3\right )}^{5/2}}{{\left (3\,x+2\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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