Optimal. Leaf size=115 \[ -\frac {2 (139 x+121) (2 x+3)^3}{9 \left (3 x^2+5 x+2\right )^{3/2}}+\frac {4 (7976 x+6809) (2 x+3)}{27 \sqrt {3 x^2+5 x+2}}-\frac {6848}{9} \sqrt {3 x^2+5 x+2}+\frac {152 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{27 \sqrt {3}} \]
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Rubi [A] time = 0.06, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {818, 640, 621, 206} \begin {gather*} -\frac {2 (139 x+121) (2 x+3)^3}{9 \left (3 x^2+5 x+2\right )^{3/2}}+\frac {4 (7976 x+6809) (2 x+3)}{27 \sqrt {3 x^2+5 x+2}}-\frac {6848}{9} \sqrt {3 x^2+5 x+2}+\frac {152 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{27 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 621
Rule 640
Rule 818
Rubi steps
\begin {align*} \int \frac {(5-x) (3+2 x)^4}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx &=-\frac {2 (3+2 x)^3 (121+139 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}+\frac {2}{9} \int \frac {(3+2 x)^2 (-117+272 x)}{\left (2+5 x+3 x^2\right )^{3/2}} \, dx\\ &=-\frac {2 (3+2 x)^3 (121+139 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}+\frac {4 (3+2 x) (6809+7976 x)}{27 \sqrt {2+5 x+3 x^2}}+\frac {4}{27} \int \frac {-12802-15408 x}{\sqrt {2+5 x+3 x^2}} \, dx\\ &=-\frac {2 (3+2 x)^3 (121+139 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}+\frac {4 (3+2 x) (6809+7976 x)}{27 \sqrt {2+5 x+3 x^2}}-\frac {6848}{9} \sqrt {2+5 x+3 x^2}+\frac {152}{27} \int \frac {1}{\sqrt {2+5 x+3 x^2}} \, dx\\ &=-\frac {2 (3+2 x)^3 (121+139 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}+\frac {4 (3+2 x) (6809+7976 x)}{27 \sqrt {2+5 x+3 x^2}}-\frac {6848}{9} \sqrt {2+5 x+3 x^2}+\frac {304}{27} \operatorname {Subst}\left (\int \frac {1}{12-x^2} \, dx,x,\frac {5+6 x}{\sqrt {2+5 x+3 x^2}}\right )\\ &=-\frac {2 (3+2 x)^3 (121+139 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}+\frac {4 (3+2 x) (6809+7976 x)}{27 \sqrt {2+5 x+3 x^2}}-\frac {6848}{9} \sqrt {2+5 x+3 x^2}+\frac {152 \tanh ^{-1}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{27 \sqrt {3}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 83, normalized size = 0.72 \begin {gather*} \frac {2 \left (-216 x^4+176160 x^3+438540 x^2+76 \sqrt {3} \left (3 x^2+5 x+2\right )^{3/2} \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {9 x^2+15 x+6}}\right )+354459 x+92457\right )}{81 \left (3 x^2+5 x+2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.54, size = 86, normalized size = 0.75 \begin {gather*} \frac {304 \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {3} (x+1)}\right )}{27 \sqrt {3}}-\frac {2 \sqrt {3 x^2+5 x+2} \left (72 x^4-58720 x^3-146180 x^2-118153 x-30819\right )}{27 (x+1)^2 (3 x+2)^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 117, normalized size = 1.02 \begin {gather*} \frac {2 \, {\left (38 \, \sqrt {3} {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \log \left (4 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\right ) - 3 \, {\left (72 \, x^{4} - 58720 \, x^{3} - 146180 \, x^{2} - 118153 \, x - 30819\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}}{81 \, {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 68, normalized size = 0.59 \begin {gather*} -\frac {152}{81} \, \sqrt {3} \log \left ({\left | -2 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) - \frac {2 \, {\left ({\left (4 \, {\left (2 \, {\left (9 \, x - 7340\right )} x - 36545\right )} x - 118153\right )} x - 30819\right )}}{27 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 178, normalized size = 1.55 \begin {gather*} -\frac {16 x^{4}}{3 \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}-\frac {152 x^{3}}{27 \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}-\frac {2380 x^{2}}{27 \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}-\frac {152 x}{27 \sqrt {3 x^{2}+5 x +2}}-\frac {14639 x}{81 \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}+\frac {152 \sqrt {3}\, \ln \left (\frac {\left (3 x +\frac {5}{2}\right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right )}{81}-\frac {145763}{1458 \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}+\frac {380}{81 \sqrt {3 x^{2}+5 x +2}}-\frac {16181 \left (6 x +5\right )}{1458 \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}+\frac {\frac {118048 x}{81}+\frac {295120}{243}}{\sqrt {3 x^{2}+5 x +2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.16, size = 214, normalized size = 1.86 \begin {gather*} -\frac {16 \, x^{4}}{3 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}} - \frac {152}{81} \, x {\left (\frac {1410 \, x}{\sqrt {3 \, x^{2} + 5 \, x + 2}} + \frac {9 \, x^{2}}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}} + \frac {1175}{\sqrt {3 \, x^{2} + 5 \, x + 2}} - \frac {55 \, x}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}} - \frac {46}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}}\right )} + \frac {152}{81} \, \sqrt {3} \log \left (2 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) + \frac {71440}{81} \, \sqrt {3 \, x^{2} + 5 \, x + 2} - \frac {60704 \, x}{81 \, \sqrt {3 \, x^{2} + 5 \, x + 2}} - \frac {920 \, x^{2}}{9 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}} - \frac {15680}{27 \, \sqrt {3 \, x^{2} + 5 \, x + 2}} - \frac {13066 \, x}{81 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}} - \frac {6766}{81 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{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 (2\,x+3\right )}^4\,\left (x-5\right )}{{\left (3\,x^2+5\,x+2\right )}^{5/2}} \,d 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 \left (- \frac {999 x}{9 x^{4} \sqrt {3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 20 x \sqrt {3 x^{2} + 5 x + 2} + 4 \sqrt {3 x^{2} + 5 x + 2}}\right )\, dx - \int \left (- \frac {864 x^{2}}{9 x^{4} \sqrt {3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 20 x \sqrt {3 x^{2} + 5 x + 2} + 4 \sqrt {3 x^{2} + 5 x + 2}}\right )\, dx - \int \left (- \frac {264 x^{3}}{9 x^{4} \sqrt {3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 20 x \sqrt {3 x^{2} + 5 x + 2} + 4 \sqrt {3 x^{2} + 5 x + 2}}\right )\, dx - \int \frac {16 x^{4}}{9 x^{4} \sqrt {3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 20 x \sqrt {3 x^{2} + 5 x + 2} + 4 \sqrt {3 x^{2} + 5 x + 2}}\, dx - \int \frac {16 x^{5}}{9 x^{4} \sqrt {3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 20 x \sqrt {3 x^{2} + 5 x + 2} + 4 \sqrt {3 x^{2} + 5 x + 2}}\, dx - \int \left (- \frac {405}{9 x^{4} \sqrt {3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 20 x \sqrt {3 x^{2} + 5 x + 2} + 4 \sqrt {3 x^{2} + 5 x + 2}}\right )\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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