Optimal. Leaf size=100 \[ \frac {11 (5 x+3)^2}{7 \sqrt {1-2 x} (3 x+2)^3}+\frac {2 \sqrt {1-2 x} (470 x+297)}{441 (3 x+2)^3}-\frac {4660 \sqrt {1-2 x}}{3087 (3 x+2)}-\frac {9320 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{3087 \sqrt {21}} \]
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Rubi [A] time = 0.03, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {98, 145, 51, 63, 206} \[ \frac {11 (5 x+3)^2}{7 \sqrt {1-2 x} (3 x+2)^3}+\frac {2 \sqrt {1-2 x} (470 x+297)}{441 (3 x+2)^3}-\frac {4660 \sqrt {1-2 x}}{3087 (3 x+2)}-\frac {9320 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{3087 \sqrt {21}} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 98
Rule 145
Rule 206
Rubi steps
\begin {align*} \int \frac {(3+5 x)^3}{(1-2 x)^{3/2} (2+3 x)^4} \, dx &=\frac {11 (3+5 x)^2}{7 \sqrt {1-2 x} (2+3 x)^3}-\frac {1}{7} \int \frac {(-74-160 x) (3+5 x)}{\sqrt {1-2 x} (2+3 x)^4} \, dx\\ &=\frac {11 (3+5 x)^2}{7 \sqrt {1-2 x} (2+3 x)^3}+\frac {2 \sqrt {1-2 x} (297+470 x)}{441 (2+3 x)^3}+\frac {4660}{441} \int \frac {1}{\sqrt {1-2 x} (2+3 x)^2} \, dx\\ &=-\frac {4660 \sqrt {1-2 x}}{3087 (2+3 x)}+\frac {11 (3+5 x)^2}{7 \sqrt {1-2 x} (2+3 x)^3}+\frac {2 \sqrt {1-2 x} (297+470 x)}{441 (2+3 x)^3}+\frac {4660 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx}{3087}\\ &=-\frac {4660 \sqrt {1-2 x}}{3087 (2+3 x)}+\frac {11 (3+5 x)^2}{7 \sqrt {1-2 x} (2+3 x)^3}+\frac {2 \sqrt {1-2 x} (297+470 x)}{441 (2+3 x)^3}-\frac {4660 \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{3087}\\ &=-\frac {4660 \sqrt {1-2 x}}{3087 (2+3 x)}+\frac {11 (3+5 x)^2}{7 \sqrt {1-2 x} (2+3 x)^3}+\frac {2 \sqrt {1-2 x} (297+470 x)}{441 (2+3 x)^3}-\frac {9320 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{3087 \sqrt {21}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 59, normalized size = 0.59 \[ \frac {18640 (3 x+2)^3 \, _2F_1\left (-\frac {1}{2},2;\frac {1}{2};\frac {3}{7}-\frac {6 x}{7}\right )+49 \left (7875 x^2+10434 x+3457\right )}{27783 \sqrt {1-2 x} (3 x+2)^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.95, size = 99, normalized size = 0.99 \[ \frac {4660 \, \sqrt {21} {\left (54 \, x^{4} + 81 \, x^{3} + 18 \, x^{2} - 20 \, x - 8\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \, {\left (83880 \, x^{3} + 178015 \, x^{2} + 125154 \, x + 29177\right )} \sqrt {-2 \, x + 1}}{64827 \, {\left (54 \, x^{4} + 81 \, x^{3} + 18 \, x^{2} - 20 \, x - 8\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.27, size = 93, normalized size = 0.93 \[ \frac {4660}{64827} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {2662}{2401 \, \sqrt {-2 \, x + 1}} + \frac {29853 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 137186 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 157633 \, \sqrt {-2 \, x + 1}}{86436 \, {\left (3 \, x + 2\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 66, normalized size = 0.66 \[ -\frac {9320 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{64827}+\frac {2662}{2401 \sqrt {-2 x +1}}+\frac {-\frac {6634 \left (-2 x +1\right )^{\frac {5}{2}}}{2401}+\frac {39196 \left (-2 x +1\right )^{\frac {3}{2}}}{3087}-\frac {6434 \sqrt {-2 x +1}}{441}}{\left (-6 x -4\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.46, size = 101, normalized size = 1.01 \[ \frac {4660}{64827} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {2 \, {\left (41940 \, {\left (2 \, x - 1\right )}^{3} + 303835 \, {\left (2 \, x - 1\right )}^{2} + 1464316 \, x - 145187\right )}}{3087 \, {\left (27 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 189 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 441 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 343 \, \sqrt {-2 \, x + 1}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.24, size = 82, normalized size = 0.82 \[ \frac {\frac {59768\,x}{1701}+\frac {86810\,{\left (2\,x-1\right )}^2}{11907}+\frac {9320\,{\left (2\,x-1\right )}^3}{9261}-\frac {5926}{1701}}{\frac {343\,\sqrt {1-2\,x}}{27}-\frac {49\,{\left (1-2\,x\right )}^{3/2}}{3}+7\,{\left (1-2\,x\right )}^{5/2}-{\left (1-2\,x\right )}^{7/2}}-\frac {9320\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{64827} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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