Optimal. Leaf size=67 \[ -\frac {793 \sqrt {5 x+3}}{19965 \sqrt {1-2 x}}-\frac {1237}{3630 \sqrt {1-2 x} \sqrt {5 x+3}}+\frac {49}{66 (1-2 x)^{3/2} \sqrt {5 x+3}} \]
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Rubi [A] time = 0.01, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {89, 78, 37} \begin {gather*} -\frac {793 \sqrt {5 x+3}}{19965 \sqrt {1-2 x}}-\frac {1237}{3630 \sqrt {1-2 x} \sqrt {5 x+3}}+\frac {49}{66 (1-2 x)^{3/2} \sqrt {5 x+3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 78
Rule 89
Rubi steps
\begin {align*} \int \frac {(2+3 x)^2}{(1-2 x)^{5/2} (3+5 x)^{3/2}} \, dx &=\frac {49}{66 (1-2 x)^{3/2} \sqrt {3+5 x}}-\frac {1}{66} \int \frac {\frac {109}{2}+297 x}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx\\ &=\frac {49}{66 (1-2 x)^{3/2} \sqrt {3+5 x}}-\frac {1237}{3630 \sqrt {1-2 x} \sqrt {3+5 x}}-\frac {793 \int \frac {1}{(1-2 x)^{3/2} \sqrt {3+5 x}} \, dx}{3630}\\ &=\frac {49}{66 (1-2 x)^{3/2} \sqrt {3+5 x}}-\frac {1237}{3630 \sqrt {1-2 x} \sqrt {3+5 x}}-\frac {793 \sqrt {3+5 x}}{19965 \sqrt {1-2 x}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 32, normalized size = 0.48 \begin {gather*} \frac {2 \left (793 x^2+1440 x+564\right )}{3993 (1-2 x)^{3/2} \sqrt {5 x+3}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.10, size = 54, normalized size = 0.81 \begin {gather*} -\frac {2 (5 x+3)^{3/2} \left (\frac {3 (1-2 x)^2}{(5 x+3)^2}-\frac {42 (1-2 x)}{5 x+3}-49\right )}{3993 (1-2 x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.99, size = 43, normalized size = 0.64 \begin {gather*} \frac {2 \, {\left (793 \, x^{2} + 1440 \, x + 564\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{3993 \, {\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.27, size = 100, normalized size = 1.49 \begin {gather*} -\frac {\sqrt {10} {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{13310 \, \sqrt {5 \, x + 3}} + \frac {14 \, {\left (23 \, \sqrt {5} {\left (5 \, x + 3\right )} + 66 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{99825 \, {\left (2 \, x - 1\right )}^{2}} + \frac {2 \, \sqrt {10} \sqrt {5 \, x + 3}}{6655 \, {\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 = 27, normalized size = 0.40 \begin {gather*} \frac {\frac {1586}{3993} x^{2}+\frac {960}{1331} x +\frac {376}{1331}}{\left (-2 x +1\right )^{\frac {3}{2}} \sqrt {5 x +3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.48, size = 64, normalized size = 0.96 \begin {gather*} -\frac {793 \, x}{3993 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {3673}{7986 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {49}{66 \, {\left (2 \, \sqrt {-10 \, x^{2} - x + 3} x - \sqrt {-10 \, x^{2} - x + 3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.28, size = 52, normalized size = 0.78 \begin {gather*} -\frac {\sqrt {5\,x+3}\,\left (\frac {793\,x^2}{19965}+\frac {96\,x}{1331}+\frac {188}{6655}\right )}{\frac {x\,\sqrt {1-2\,x}}{10}-\frac {3\,\sqrt {1-2\,x}}{10}+x^2\,\sqrt {1-2\,x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (3 x + 2\right )^{2}}{\left (1 - 2 x\right )^{\frac {5}{2}} \left (5 x + 3\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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