Optimal. Leaf size=94 \[ -\frac {2 (1-2 x)^{5/2}}{5 \sqrt {3+5 x}}-\frac {33}{50} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {1}{5} (1-2 x)^{3/2} \sqrt {3+5 x}-\frac {363 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{50 \sqrt {10}} \]
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Rubi [A]
time = 0.02, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {49, 52, 56, 222}
\begin {gather*} -\frac {363 \text {ArcSin}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{50 \sqrt {10}}-\frac {2 (1-2 x)^{5/2}}{5 \sqrt {5 x+3}}-\frac {1}{5} \sqrt {5 x+3} (1-2 x)^{3/2}-\frac {33}{50} \sqrt {5 x+3} \sqrt {1-2 x} \end {gather*}
Antiderivative was successfully verified.
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Rule 49
Rule 52
Rule 56
Rule 222
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{5/2}}{(3+5 x)^{3/2}} \, dx &=-\frac {2 (1-2 x)^{5/2}}{5 \sqrt {3+5 x}}-2 \int \frac {(1-2 x)^{3/2}}{\sqrt {3+5 x}} \, dx\\ &=-\frac {2 (1-2 x)^{5/2}}{5 \sqrt {3+5 x}}-\frac {1}{5} (1-2 x)^{3/2} \sqrt {3+5 x}-\frac {33}{10} \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx\\ &=-\frac {2 (1-2 x)^{5/2}}{5 \sqrt {3+5 x}}-\frac {33}{50} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {1}{5} (1-2 x)^{3/2} \sqrt {3+5 x}-\frac {363}{100} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {2 (1-2 x)^{5/2}}{5 \sqrt {3+5 x}}-\frac {33}{50} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {1}{5} (1-2 x)^{3/2} \sqrt {3+5 x}-\frac {363 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{50 \sqrt {5}}\\ &=-\frac {2 (1-2 x)^{5/2}}{5 \sqrt {3+5 x}}-\frac {33}{50} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {1}{5} (1-2 x)^{3/2} \sqrt {3+5 x}-\frac {363 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{50 \sqrt {10}}\\ \end {align*}
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Mathematica [A]
time = 0.14, size = 68, normalized size = 0.72 \begin {gather*} \frac {10 \sqrt {1-2 x} \left (-149-75 x+20 x^2\right )+363 \sqrt {30+50 x} \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{500 \sqrt {3+5 x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (1-2 x \right )^{\frac {5}{2}}}{\left (3+5 x \right )^{\frac {3}{2}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.55, size = 75, normalized size = 0.80 \begin {gather*} -\frac {4 \, x^{3}}{5 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {17 \, x^{2}}{5 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {363}{1000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {223 \, x}{50 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {149}{50 \, \sqrt {-10 \, x^{2} - x + 3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.51, size = 81, normalized size = 0.86 \begin {gather*} \frac {363 \, \sqrt {10} {\left (5 \, x + 3\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 20 \, {\left (20 \, x^{2} - 75 \, x - 149\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{1000 \, {\left (5 \, x + 3\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 4.44, size = 228, normalized size = 2.43 \begin {gather*} \begin {cases} \frac {4 i \left (x + \frac {3}{5}\right )^{\frac {5}{2}}}{5 \sqrt {10 x - 5}} - \frac {121 i \left (x + \frac {3}{5}\right )^{\frac {3}{2}}}{25 \sqrt {10 x - 5}} + \frac {121 i \sqrt {x + \frac {3}{5}}}{250 \sqrt {10 x - 5}} + \frac {363 \sqrt {10} i \operatorname {acosh}{\left (\frac {\sqrt {110} \sqrt {x + \frac {3}{5}}}{11} \right )}}{500} + \frac {2662 i}{625 \sqrt {x + \frac {3}{5}} \sqrt {10 x - 5}} & \text {for}\: \left |{x + \frac {3}{5}}\right | > \frac {11}{10} \\- \frac {363 \sqrt {10} \operatorname {asin}{\left (\frac {\sqrt {110} \sqrt {x + \frac {3}{5}}}{11} \right )}}{500} - \frac {4 \left (x + \frac {3}{5}\right )^{\frac {5}{2}}}{5 \sqrt {5 - 10 x}} + \frac {121 \left (x + \frac {3}{5}\right )^{\frac {3}{2}}}{25 \sqrt {5 - 10 x}} - \frac {121 \sqrt {x + \frac {3}{5}}}{250 \sqrt {5 - 10 x}} - \frac {2662}{625 \sqrt {5 - 10 x} \sqrt {x + \frac {3}{5}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.50, size = 111, normalized size = 1.18 \begin {gather*} \frac {1}{1250} \, {\left (4 \, \sqrt {5} {\left (5 \, x + 3\right )} - 99 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - \frac {363}{500} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) - \frac {121 \, \sqrt {10} {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{1250 \, \sqrt {5 \, x + 3}} + \frac {242 \, \sqrt {10} \sqrt {5 \, x + 3}}{625 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}} \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}}{{\left (5\,x+3\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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