Optimal. Leaf size=377 \[ \frac{\left (i \sqrt{14} \left (6565 d^2-2 d e (1313-3206 m)+e^2 (3939-98 m)\right )+80360 d^2-5922 d e m-32144 d e+19138 e^2 m+48216 e^2\right ) (d+e x)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{5 (d+e x)}{5 d+i \sqrt{14} e-e}\right )}{19600 (m+1) \left (5 d+i \left (\sqrt{14}+i\right ) e\right ) \left (5 d^2-2 d e+3 e^2\right )}+\frac{\left (-i \sqrt{14} \left (6565 d^2-2 d e (1313-3206 m)+e^2 (3939-98 m)\right )+80360 d^2-5922 d e m-32144 d e+19138 e^2 m+48216 e^2\right ) (d+e x)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{5 (d+e x)}{5 d-\left (1+i \sqrt{14}\right ) e}\right )}{19600 (m+1) \left (5 d-\left (1+i \sqrt{14}\right ) e\right ) \left (5 d^2-2 d e+3 e^2\right )}-\frac{(x (423 d-1367 e)+1367 d-293 e) (d+e x)^{m+1}}{700 \left (5 x^2+2 x+3\right ) \left (5 d^2-2 d e+3 e^2\right )}+\frac{4 (d+e x)^{m+1}}{25 e (m+1)} \]
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Rubi [A] time = 0.899862, antiderivative size = 377, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.079, Rules used = {1648, 1628, 68} \[ \frac{\left (i \sqrt{14} \left (6565 d^2-2 d e (1313-3206 m)+e^2 (3939-98 m)\right )+80360 d^2-5922 d e m-32144 d e+19138 e^2 m+48216 e^2\right ) (d+e x)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{5 (d+e x)}{5 d+i \sqrt{14} e-e}\right )}{19600 (m+1) \left (5 d+i \left (\sqrt{14}+i\right ) e\right ) \left (5 d^2-2 d e+3 e^2\right )}+\frac{\left (-i \sqrt{14} \left (6565 d^2-2 d e (1313-3206 m)+e^2 (3939-98 m)\right )+80360 d^2-5922 d e m-32144 d e+19138 e^2 m+48216 e^2\right ) (d+e x)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{5 (d+e x)}{5 d-\left (1+i \sqrt{14}\right ) e}\right )}{19600 (m+1) \left (5 d-\left (1+i \sqrt{14}\right ) e\right ) \left (5 d^2-2 d e+3 e^2\right )}-\frac{(x (423 d-1367 e)+1367 d-293 e) (d+e x)^{m+1}}{700 \left (5 x^2+2 x+3\right ) \left (5 d^2-2 d e+3 e^2\right )}+\frac{4 (d+e x)^{m+1}}{25 e (m+1)} \]
Antiderivative was successfully verified.
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Rule 1648
Rule 1628
Rule 68
Rubi steps
\begin{align*} \int \frac{(d+e x)^m \left (2+x+3 x^2-5 x^3+4 x^4\right )}{\left (3+2 x+5 x^2\right )^2} \, dx &=-\frac{(1367 d-293 e+(423 d-1367 e) x) (d+e x)^{1+m}}{700 \left (5 d^2-2 d e+3 e^2\right ) \left (3+2 x+5 x^2\right )}+\frac{\int \frac{(d+e x)^m \left (\frac{2}{25} \left (1845 d^2-d e (738-1367 m)+e^2 (1107-293 m)\right )-\frac{2}{25} \left (4620 d^2-3 d e (616+141 m)+e^2 (2772+1367 m)\right ) x+\frac{224}{5} \left (5 d^2-2 d e+3 e^2\right ) x^2\right )}{3+2 x+5 x^2} \, dx}{56 \left (5 d^2-2 d e+3 e^2\right )}\\ &=-\frac{(1367 d-293 e+(423 d-1367 e) x) (d+e x)^{1+m}}{700 \left (5 d^2-2 d e+3 e^2\right ) \left (3+2 x+5 x^2\right )}+\frac{\int \left (\frac{224}{25} \left (5 d^2-2 d e+3 e^2\right ) (d+e x)^m+\frac{\left (-\frac{2296 d^2}{5}+\frac{4592 d e}{25}-\frac{6888 e^2}{25}+\frac{846 d e m}{25}-\frac{2734 e^2 m}{25}-\frac{1}{25} i \sqrt{\frac{2}{7}} \left (6565 d^2-2626 d e+3939 e^2+6412 d e m-98 e^2 m\right )\right ) (d+e x)^m}{2-2 i \sqrt{14}+10 x}+\frac{\left (-\frac{2296 d^2}{5}+\frac{4592 d e}{25}-\frac{6888 e^2}{25}+\frac{846 d e m}{25}-\frac{2734 e^2 m}{25}+\frac{1}{25} i \sqrt{\frac{2}{7}} \left (6565 d^2-2626 d e+3939 e^2+6412 d e m-98 e^2 m\right )\right ) (d+e x)^m}{2+2 i \sqrt{14}+10 x}\right ) \, dx}{56 \left (5 d^2-2 d e+3 e^2\right )}\\ &=\frac{4 (d+e x)^{1+m}}{25 e (1+m)}-\frac{(1367 d-293 e+(423 d-1367 e) x) (d+e x)^{1+m}}{700 \left (5 d^2-2 d e+3 e^2\right ) \left (3+2 x+5 x^2\right )}-\frac{\left (80360 d^2-32144 d e+48216 e^2-i \sqrt{14} \left (6565 d^2-2 d e (1313-3206 m)+e^2 (3939-98 m)\right )-5922 d e m+19138 e^2 m\right ) \int \frac{(d+e x)^m}{2+2 i \sqrt{14}+10 x} \, dx}{9800 \left (5 d^2-2 d e+3 e^2\right )}-\frac{\left (80360 d^2-32144 d e+48216 e^2+i \sqrt{14} \left (6565 d^2-2 d e (1313-3206 m)+e^2 (3939-98 m)\right )-5922 d e m+19138 e^2 m\right ) \int \frac{(d+e x)^m}{2-2 i \sqrt{14}+10 x} \, dx}{9800 \left (5 d^2-2 d e+3 e^2\right )}\\ &=\frac{4 (d+e x)^{1+m}}{25 e (1+m)}-\frac{(1367 d-293 e+(423 d-1367 e) x) (d+e x)^{1+m}}{700 \left (5 d^2-2 d e+3 e^2\right ) \left (3+2 x+5 x^2\right )}+\frac{\left (80360 d^2-32144 d e+48216 e^2+i \sqrt{14} \left (6565 d^2-2 d e (1313-3206 m)+e^2 (3939-98 m)\right )-5922 d e m+19138 e^2 m\right ) (d+e x)^{1+m} \, _2F_1\left (1,1+m;2+m;\frac{5 (d+e x)}{5 d-e+i \sqrt{14} e}\right )}{19600 \left (5 d+i \left (i+\sqrt{14}\right ) e\right ) \left (5 d^2-2 d e+3 e^2\right ) (1+m)}+\frac{\left (80360 d^2-32144 d e+48216 e^2-i \sqrt{14} \left (6565 d^2-2 d e (1313-3206 m)+e^2 (3939-98 m)\right )-5922 d e m+19138 e^2 m\right ) (d+e x)^{1+m} \, _2F_1\left (1,1+m;2+m;\frac{5 (d+e x)}{5 d-\left (1+i \sqrt{14}\right ) e}\right )}{19600 \left (5 d-\left (1+i \sqrt{14}\right ) e\right ) \left (5 d^2-2 d e+3 e^2\right ) (1+m)}\\ \end{align*}
Mathematica [A] time = 1.75703, size = 441, normalized size = 1.17 \[ \frac{(d+e x)^{m+1} \left (-\frac{\sqrt{14} \left (\frac{\left (2115 d^2+d e \left (-846+\left (-6412+423 i \sqrt{14}\right ) m\right )+e^2 \left (1269+\left (98-1367 i \sqrt{14}\right ) m\right )\right ) \, _2F_1\left (1,m+1;m+2;\frac{5 (d+e x)}{5 d+\left (-1-i \sqrt{14}\right ) e}\right )}{5 i d+\left (\sqrt{14}-i\right ) e}-\frac{\left (2115 d^2-d e \left (846+\left (6412+423 i \sqrt{14}\right ) m\right )+e^2 \left (1269+\left (98+1367 i \sqrt{14}\right ) m\right )\right ) \, _2F_1\left (1,m+1;m+2;\frac{5 (d+e x)}{5 d+i \left (i+\sqrt{14}\right ) e}\right )}{5 i d-\left (\sqrt{14}+i\right ) e}\right )}{(m+1) \left (5 d^2-2 d e+3 e^2\right )}-\frac{28 (d (423 x+1367)-e (1367 x+293))}{\left (5 x^2+2 x+3\right ) \left (5 d^2-2 d e+3 e^2\right )}+\frac{56 \left (31 \sqrt{14}+287 i\right ) \, _2F_1\left (1,m+1;m+2;\frac{5 (d+e x)}{5 d+\left (-1-i \sqrt{14}\right ) e}\right )}{(m+1) \left (5 i d+\left (\sqrt{14}-i\right ) e\right )}+\frac{56 \left (31 \sqrt{14}-287 i\right ) \, _2F_1\left (1,m+1;m+2;\frac{5 (d+e x)}{5 d+i \left (i+\sqrt{14}\right ) e}\right )}{(m+1) \left (\left (\sqrt{14}+i\right ) e-5 i d\right )}+\frac{3136}{e m+e}\right )}{19600} \]
Antiderivative was successfully verified.
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Maple [F] time = 2.375, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( ex+d \right ) ^{m} \left ( 4\,{x}^{4}-5\,{x}^{3}+3\,{x}^{2}+x+2 \right ) }{ \left ( 5\,{x}^{2}+2\,x+3 \right ) ^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (4 \, x^{4} - 5 \, x^{3} + 3 \, x^{2} + x + 2\right )}{\left (e x + d\right )}^{m}}{{\left (5 \, x^{2} + 2 \, x + 3\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (4 \, x^{4} - 5 \, x^{3} + 3 \, x^{2} + x + 2\right )}{\left (e x + d\right )}^{m}}{25 \, x^{4} + 20 \, x^{3} + 34 \, x^{2} + 12 \, x + 9}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (4 \, x^{4} - 5 \, x^{3} + 3 \, x^{2} + x + 2\right )}{\left (e x + d\right )}^{m}}{{\left (5 \, x^{2} + 2 \, x + 3\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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