Optimal. Leaf size=45 \[ \frac {2}{11 \sqrt {1-2 x} \sqrt {5 x+3}}-\frac {20 \sqrt {1-2 x}}{121 \sqrt {5 x+3}} \]
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Rubi [A] time = 0.01, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {45, 37} \begin {gather*} \frac {2}{11 \sqrt {1-2 x} \sqrt {5 x+3}}-\frac {20 \sqrt {1-2 x}}{121 \sqrt {5 x+3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 45
Rubi steps
\begin {align*} \int \frac {1}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx &=\frac {2}{11 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {10}{11} \int \frac {1}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx\\ &=\frac {2}{11 \sqrt {1-2 x} \sqrt {3+5 x}}-\frac {20 \sqrt {1-2 x}}{121 \sqrt {3+5 x}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 27, normalized size = 0.60 \begin {gather*} \frac {2 (20 x+1)}{121 \sqrt {1-2 x} \sqrt {5 x+3}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.08, size = 38, normalized size = 0.84 \begin {gather*} -\frac {2 \sqrt {5 x+3} \left (\frac {5 (1-2 x)}{5 x+3}-2\right )}{121 \sqrt {1-2 x}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.39, size = 31, normalized size = 0.69 \begin {gather*} -\frac {2 \, {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{121 \, {\left (10 \, x^{2} + x - 3\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.15, size = 87, normalized size = 1.93 \begin {gather*} -\frac {\sqrt {10} {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{242 \, \sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{605 \, {\left (2 \, x - 1\right )}} + \frac {2 \, \sqrt {10} \sqrt {5 \, x + 3}}{121 \, {\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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maple [A] time = 0.00, size = 22, normalized size = 0.49 \begin {gather*} \frac {\frac {40 x}{121}+\frac {2}{121}}{\sqrt {-2 x +1}\, \sqrt {5 x +3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.62, size = 30, normalized size = 0.67 \begin {gather*} \frac {40 \, x}{121 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {2}{121 \, \sqrt {-10 \, x^{2} - x + 3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.40, size = 34, normalized size = 0.76 \begin {gather*} \frac {\sqrt {5\,x+3}\,\left (\frac {8\,x}{121}+\frac {2}{605}\right )}{x\,\sqrt {1-2\,x}+\frac {3\,\sqrt {1-2\,x}}{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.79, size = 116, normalized size = 2.58 \begin {gather*} \begin {cases} \frac {40 \sqrt {10} \sqrt {-1 + \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )}{605 - 1210 x} - \frac {22 \sqrt {10} \sqrt {-1 + \frac {11}{10 \left (x + \frac {3}{5}\right )}}}{605 - 1210 x} & \text {for}\: \frac {11}{10 \left |{x + \frac {3}{5}}\right |} > 1 \\\frac {40 \sqrt {10} i \sqrt {1 - \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )}{605 - 1210 x} - \frac {22 \sqrt {10} i \sqrt {1 - \frac {11}{10 \left (x + \frac {3}{5}\right )}}}{605 - 1210 x} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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