Integrand size = 25, antiderivative size = 262 \[ \int \frac {(2+x)^{3/4} (3+5 x)}{4+7 x+2 x^2} \, dx=\frac {10}{3} (2+x)^{3/4}-\frac {\sqrt [4]{-861187+208885 \sqrt {17}} \arctan \left (\frac {\sqrt {2} \sqrt [4]{2+x}}{\sqrt [4]{1+\sqrt {17}}}\right )}{2 \sqrt {17}}+\frac {\sqrt [4]{861187+208885 \sqrt {17}} \arctan \left (1-\frac {2 \sqrt [4]{2+x}}{\sqrt [4]{-1+\sqrt {17}}}\right )}{2 \sqrt {34}}-\frac {\sqrt [4]{861187+208885 \sqrt {17}} \arctan \left (1+\frac {2 \sqrt [4]{2+x}}{\sqrt [4]{-1+\sqrt {17}}}\right )}{2 \sqrt {34}}+\frac {\sqrt [4]{-861187+208885 \sqrt {17}} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{2+x}}{\sqrt [4]{1+\sqrt {17}}}\right )}{2 \sqrt {17}}+\frac {\sqrt [4]{861187+208885 \sqrt {17}} \text {arctanh}\left (\frac {2 \sqrt [4]{-1+\sqrt {17}} \sqrt [4]{2+x}}{\sqrt {-1+\sqrt {17}}+2 \sqrt {2+x}}\right )}{2 \sqrt {34}} \] Output:
10/3*(2+x)^(3/4)-1/34*(-861187+208885*17^(1/2))^(1/4)*arctan(2^(1/2)*(2+x) ^(1/4)/(1+17^(1/2))^(1/4))*17^(1/2)-1/68*(861187+208885*17^(1/2))^(1/4)*ar ctan(-1+2*(2+x)^(1/4)/(-1+17^(1/2))^(1/4))*34^(1/2)-1/68*(861187+208885*17 ^(1/2))^(1/4)*arctan(1+2*(2+x)^(1/4)/(-1+17^(1/2))^(1/4))*34^(1/2)+1/34*(- 861187+208885*17^(1/2))^(1/4)*arctanh(2^(1/2)*(2+x)^(1/4)/(1+17^(1/2))^(1/ 4))*17^(1/2)+1/68*(861187+208885*17^(1/2))^(1/4)*arctanh(2*(-1+17^(1/2))^( 1/4)*(2+x)^(1/4)/((-1+17^(1/2))^(1/2)+2*(2+x)^(1/2)))*34^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.10 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.31 \[ \int \frac {(2+x)^{3/4} (3+5 x)}{4+7 x+2 x^2} \, dx=\frac {10}{3} (2+x)^{3/4}-\frac {1}{2} \text {RootSum}\left [-2-\text {$\#$1}^4+2 \text {$\#$1}^8\&,\frac {-10 \log \left (\sqrt [4]{2+x}-\text {$\#$1}\right )+9 \log \left (\sqrt [4]{2+x}-\text {$\#$1}\right ) \text {$\#$1}^4}{-\text {$\#$1}+4 \text {$\#$1}^5}\&\right ] \] Input:
Integrate[((2 + x)^(3/4)*(3 + 5*x))/(4 + 7*x + 2*x^2),x]
Output:
(10*(2 + x)^(3/4))/3 - RootSum[-2 - #1^4 + 2*#1^8 & , (-10*Log[(2 + x)^(1/ 4) - #1] + 9*Log[(2 + x)^(1/4) - #1]*#1^4)/(-#1 + 4*#1^5) & ]/2
Time = 0.78 (sec) , antiderivative size = 319, normalized size of antiderivative = 1.22, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {1196, 25, 1200, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {(x+2)^{3/4} (5 x+3)}{2 x^2+7 x+4} \, dx\) |
| \(\Big \downarrow \) 1196 |
| \(\displaystyle \frac {1}{2} \int -\frac {9 x+8}{\sqrt [4]{x+2} \left (2 x^2+7 x+4\right )}dx+\frac {10}{3} (x+2)^{3/4}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {10}{3} (x+2)^{3/4}-\frac {1}{2} \int \frac {9 x+8}{\sqrt [4]{x+2} \left (2 x^2+7 x+4\right )}dx\) |
| \(\Big \downarrow \) 1200 |
| \(\displaystyle \frac {10}{3} (x+2)^{3/4}-\frac {1}{2} \int \left (\frac {9-\frac {31}{\sqrt {17}}}{\left (4 x-\sqrt {17}+7\right ) \sqrt [4]{x+2}}+\frac {9+\frac {31}{\sqrt {17}}}{\left (4 x+\sqrt {17}+7\right ) \sqrt [4]{x+2}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} \left (-\frac {\sqrt [4]{835540 \sqrt {17}-3444748} \arctan \left (\frac {\sqrt {2} \sqrt [4]{x+2}}{\sqrt [4]{1+\sqrt {17}}}\right )}{\sqrt {34}}+\frac {\sqrt [4]{861187+208885 \sqrt {17}} \arctan \left (1-\frac {2 \sqrt [4]{x+2}}{\sqrt [4]{\sqrt {17}-1}}\right )}{\sqrt {34}}-\frac {\sqrt [4]{861187+208885 \sqrt {17}} \arctan \left (\frac {2 \sqrt [4]{x+2}}{\sqrt [4]{\sqrt {17}-1}}+1\right )}{\sqrt {34}}+\frac {\sqrt [4]{835540 \sqrt {17}-3444748} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{x+2}}{\sqrt [4]{1+\sqrt {17}}}\right )}{\sqrt {34}}-\frac {\sqrt [4]{861187+208885 \sqrt {17}} \log \left (2 \sqrt {x+2}-2 \sqrt [4]{\sqrt {17}-1} \sqrt [4]{x+2}+\sqrt {\sqrt {17}-1}\right )}{2 \sqrt {34}}+\frac {\sqrt [4]{861187+208885 \sqrt {17}} \log \left (2 \sqrt {x+2}+2 \sqrt [4]{\sqrt {17}-1} \sqrt [4]{x+2}+\sqrt {\sqrt {17}-1}\right )}{2 \sqrt {34}}\right )+\frac {10}{3} (x+2)^{3/4}\) |
Input:
Int[((2 + x)^(3/4)*(3 + 5*x))/(4 + 7*x + 2*x^2),x]
Output:
(10*(2 + x)^(3/4))/3 + (-(((-3444748 + 835540*Sqrt[17])^(1/4)*ArcTan[(Sqrt [2]*(2 + x)^(1/4))/(1 + Sqrt[17])^(1/4)])/Sqrt[34]) + ((861187 + 208885*Sq rt[17])^(1/4)*ArcTan[1 - (2*(2 + x)^(1/4))/(-1 + Sqrt[17])^(1/4)])/Sqrt[34 ] - ((861187 + 208885*Sqrt[17])^(1/4)*ArcTan[1 + (2*(2 + x)^(1/4))/(-1 + S qrt[17])^(1/4)])/Sqrt[34] + ((-3444748 + 835540*Sqrt[17])^(1/4)*ArcTanh[(S qrt[2]*(2 + x)^(1/4))/(1 + Sqrt[17])^(1/4)])/Sqrt[34] - ((861187 + 208885* Sqrt[17])^(1/4)*Log[Sqrt[-1 + Sqrt[17]] - 2*(-1 + Sqrt[17])^(1/4)*(2 + x)^ (1/4) + 2*Sqrt[2 + x]])/(2*Sqrt[34]) + ((861187 + 208885*Sqrt[17])^(1/4)*L og[Sqrt[-1 + Sqrt[17]] + 2*(-1 + Sqrt[17])^(1/4)*(2 + x)^(1/4) + 2*Sqrt[2 + x]])/(2*Sqrt[34]))/2
Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[g*((d + e*x)^m/(c*m)), x] + Simp[1/c Int [(d + e*x)^(m - 1)*(Simp[c*d*f - a*e*g + (g*c*d - b*e*g + c*e*f)*x, x]/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && FractionQ[m] & & GtQ[m, 0]
Int[(((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.))/((a_.) + (b_.)* (x_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*((f + g* x)^n/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && In tegersQ[n]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 36.16 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.24
| method | result | size |
| derivativedivides | \(\frac {10 \left (2+x \right )^{\frac {3}{4}}}{3}-\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (2 \textit {\_Z}^{8}-\textit {\_Z}^{4}-2\right )}{\sum }\frac {\left (9 \textit {\_R}^{6}-10 \textit {\_R}^{2}\right ) \ln \left (\left (2+x \right )^{\frac {1}{4}}-\textit {\_R} \right )}{4 \textit {\_R}^{7}-\textit {\_R}^{3}}\right )}{2}\) | \(62\) |
| default | \(\frac {10 \left (2+x \right )^{\frac {3}{4}}}{3}-\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (2 \textit {\_Z}^{8}-\textit {\_Z}^{4}-2\right )}{\sum }\frac {\left (9 \textit {\_R}^{6}-10 \textit {\_R}^{2}\right ) \ln \left (\left (2+x \right )^{\frac {1}{4}}-\textit {\_R} \right )}{4 \textit {\_R}^{7}-\textit {\_R}^{3}}\right )}{2}\) | \(62\) |
| risch | \(\frac {10 \left (2+x \right )^{\frac {3}{4}}}{3}-\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (2 \textit {\_Z}^{8}-\textit {\_Z}^{4}-2\right )}{\sum }\frac {\left (9 \textit {\_R}^{6}-10 \textit {\_R}^{2}\right ) \ln \left (\left (2+x \right )^{\frac {1}{4}}-\textit {\_R} \right )}{4 \textit {\_R}^{7}-\textit {\_R}^{3}}\right )}{2}\) | \(62\) |
| trager | \(\text {Expression too large to display}\) | \(3565\) |
Input:
int((2+x)^(3/4)*(5*x+3)/(2*x^2+7*x+4),x,method=_RETURNVERBOSE)
Output:
10/3*(2+x)^(3/4)-1/2*sum((9*_R^6-10*_R^2)/(4*_R^7-_R^3)*ln((2+x)^(1/4)-_R) ,_R=RootOf(2*_Z^8-_Z^4-2))
Leaf count of result is larger than twice the leaf count of optimal. 424 vs. \(2 (178) = 356\).
Time = 0.10 (sec) , antiderivative size = 424, normalized size of antiderivative = 1.62 \[ \int \frac {(2+x)^{3/4} (3+5 x)}{4+7 x+2 x^2} \, dx =\text {Too large to display} \] Input:
integrate((2+x)^(3/4)*(3+5*x)/(2*x^2+7*x+4),x, algorithm="fricas")
Output:
-1/4*sqrt(1/17)*sqrt(-sqrt(208885*sqrt(17) - 861187))*log(sqrt(1/17)*sqrt( 208885*sqrt(17) - 861187)*(25301*sqrt(17) + 104227)*sqrt(-sqrt(208885*sqrt (17) - 861187)) + 1124864*(x + 2)^(1/4)) + 1/4*sqrt(1/17)*sqrt(-sqrt(20888 5*sqrt(17) - 861187))*log(-sqrt(1/17)*sqrt(208885*sqrt(17) - 861187)*(2530 1*sqrt(17) + 104227)*sqrt(-sqrt(208885*sqrt(17) - 861187)) + 1124864*(x + 2)^(1/4)) + 1/4*sqrt(1/17)*sqrt(-sqrt(-208885*sqrt(17) - 861187))*log(sqrt (1/17)*(25301*sqrt(17) - 104227)*sqrt(-208885*sqrt(17) - 861187)*sqrt(-sqr t(-208885*sqrt(17) - 861187)) + 1124864*(x + 2)^(1/4)) - 1/4*sqrt(1/17)*sq rt(-sqrt(-208885*sqrt(17) - 861187))*log(-sqrt(1/17)*(25301*sqrt(17) - 104 227)*sqrt(-208885*sqrt(17) - 861187)*sqrt(-sqrt(-208885*sqrt(17) - 861187) ) + 1124864*(x + 2)^(1/4)) + 1/4*sqrt(1/17)*(208885*sqrt(17) - 861187)^(1/ 4)*log(sqrt(1/17)*(208885*sqrt(17) - 861187)^(3/4)*(25301*sqrt(17) + 10422 7) + 1124864*(x + 2)^(1/4)) - 1/4*sqrt(1/17)*(208885*sqrt(17) - 861187)^(1 /4)*log(-sqrt(1/17)*(208885*sqrt(17) - 861187)^(3/4)*(25301*sqrt(17) + 104 227) + 1124864*(x + 2)^(1/4)) - 1/4*sqrt(1/17)*(-208885*sqrt(17) - 861187) ^(1/4)*log(sqrt(1/17)*(25301*sqrt(17) - 104227)*(-208885*sqrt(17) - 861187 )^(3/4) + 1124864*(x + 2)^(1/4)) + 1/4*sqrt(1/17)*(-208885*sqrt(17) - 8611 87)^(1/4)*log(-sqrt(1/17)*(25301*sqrt(17) - 104227)*(-208885*sqrt(17) - 86 1187)^(3/4) + 1124864*(x + 2)^(1/4)) + 10/3*(x + 2)^(3/4)
\[ \int \frac {(2+x)^{3/4} (3+5 x)}{4+7 x+2 x^2} \, dx=\int \frac {\left (x + 2\right )^{\frac {3}{4}} \cdot \left (5 x + 3\right )}{2 x^{2} + 7 x + 4}\, dx \] Input:
integrate((2+x)**(3/4)*(3+5*x)/(2*x**2+7*x+4),x)
Output:
Integral((x + 2)**(3/4)*(5*x + 3)/(2*x**2 + 7*x + 4), x)
\[ \int \frac {(2+x)^{3/4} (3+5 x)}{4+7 x+2 x^2} \, dx=\int { \frac {{\left (5 \, x + 3\right )} {\left (x + 2\right )}^{\frac {3}{4}}}{2 \, x^{2} + 7 \, x + 4} \,d x } \] Input:
integrate((2+x)^(3/4)*(3+5*x)/(2*x^2+7*x+4),x, algorithm="maxima")
Output:
integrate((5*x + 3)*(x + 2)^(3/4)/(2*x^2 + 7*x + 4), x)
\[ \int \frac {(2+x)^{3/4} (3+5 x)}{4+7 x+2 x^2} \, dx=\int { \frac {{\left (5 \, x + 3\right )} {\left (x + 2\right )}^{\frac {3}{4}}}{2 \, x^{2} + 7 \, x + 4} \,d x } \] Input:
integrate((2+x)^(3/4)*(3+5*x)/(2*x^2+7*x+4),x, algorithm="giac")
Output:
integrate((5*x + 3)*(x + 2)^(3/4)/(2*x^2 + 7*x + 4), x)
Time = 11.55 (sec) , antiderivative size = 290, normalized size of antiderivative = 1.11 \[ \int \frac {(2+x)^{3/4} (3+5 x)}{4+7 x+2 x^2} \, dx=\frac {10\,{\left (x+2\right )}^{3/4}}{3}-\frac {\sqrt {17}\,\mathrm {atan}\left (\frac {99638145\,{\left (208885\,\sqrt {17}-861187\right )}^{1/4}\,{\left (x+2\right )}^{1/4}}{8\,\left (\frac {820709165\,\sqrt {17}}{8}-\frac {3382057035}{8}\right )}-\frac {24021775\,\sqrt {17}\,{\left (208885\,\sqrt {17}-861187\right )}^{1/4}\,{\left (x+2\right )}^{1/4}}{8\,\left (\frac {820709165\,\sqrt {17}}{8}-\frac {3382057035}{8}\right )}\right )\,{\left (208885\,\sqrt {17}-861187\right )}^{1/4}}{34}-\frac {\sqrt {17}\,\mathrm {atan}\left (\frac {99638145\,{\left (-208885\,\sqrt {17}-861187\right )}^{1/4}\,{\left (x+2\right )}^{1/4}}{8\,\left (\frac {820709165\,\sqrt {17}}{8}+\frac {3382057035}{8}\right )}+\frac {24021775\,\sqrt {17}\,{\left (-208885\,\sqrt {17}-861187\right )}^{1/4}\,{\left (x+2\right )}^{1/4}}{8\,\left (\frac {820709165\,\sqrt {17}}{8}+\frac {3382057035}{8}\right )}\right )\,{\left (-208885\,\sqrt {17}-861187\right )}^{1/4}}{34}-\frac {\sqrt {17}\,\mathrm {atan}\left (\frac {{\left (208885\,\sqrt {17}-861187\right )}^{1/4}\,{\left (x+2\right )}^{1/4}\,99638145{}\mathrm {i}}{8\,\left (\frac {820709165\,\sqrt {17}}{8}-\frac {3382057035}{8}\right )}-\frac {\sqrt {17}\,{\left (208885\,\sqrt {17}-861187\right )}^{1/4}\,{\left (x+2\right )}^{1/4}\,24021775{}\mathrm {i}}{8\,\left (\frac {820709165\,\sqrt {17}}{8}-\frac {3382057035}{8}\right )}\right )\,{\left (208885\,\sqrt {17}-861187\right )}^{1/4}\,1{}\mathrm {i}}{34}-\frac {\sqrt {17}\,\mathrm {atan}\left (\frac {{\left (-208885\,\sqrt {17}-861187\right )}^{1/4}\,{\left (x+2\right )}^{1/4}\,99638145{}\mathrm {i}}{8\,\left (\frac {820709165\,\sqrt {17}}{8}+\frac {3382057035}{8}\right )}+\frac {\sqrt {17}\,{\left (-208885\,\sqrt {17}-861187\right )}^{1/4}\,{\left (x+2\right )}^{1/4}\,24021775{}\mathrm {i}}{8\,\left (\frac {820709165\,\sqrt {17}}{8}+\frac {3382057035}{8}\right )}\right )\,{\left (-208885\,\sqrt {17}-861187\right )}^{1/4}\,1{}\mathrm {i}}{34} \] Input:
int(((5*x + 3)*(x + 2)^(3/4))/(7*x + 2*x^2 + 4),x)
Output:
(10*(x + 2)^(3/4))/3 - (17^(1/2)*atan((99638145*(208885*17^(1/2) - 861187) ^(1/4)*(x + 2)^(1/4))/(8*((820709165*17^(1/2))/8 - 3382057035/8)) - (24021 775*17^(1/2)*(208885*17^(1/2) - 861187)^(1/4)*(x + 2)^(1/4))/(8*((82070916 5*17^(1/2))/8 - 3382057035/8)))*(208885*17^(1/2) - 861187)^(1/4))/34 - (17 ^(1/2)*atan((99638145*(- 208885*17^(1/2) - 861187)^(1/4)*(x + 2)^(1/4))/(8 *((820709165*17^(1/2))/8 + 3382057035/8)) + (24021775*17^(1/2)*(- 208885*1 7^(1/2) - 861187)^(1/4)*(x + 2)^(1/4))/(8*((820709165*17^(1/2))/8 + 338205 7035/8)))*(- 208885*17^(1/2) - 861187)^(1/4))/34 - (17^(1/2)*atan(((208885 *17^(1/2) - 861187)^(1/4)*(x + 2)^(1/4)*99638145i)/(8*((820709165*17^(1/2) )/8 - 3382057035/8)) - (17^(1/2)*(208885*17^(1/2) - 861187)^(1/4)*(x + 2)^ (1/4)*24021775i)/(8*((820709165*17^(1/2))/8 - 3382057035/8)))*(208885*17^( 1/2) - 861187)^(1/4)*1i)/34 - (17^(1/2)*atan(((- 208885*17^(1/2) - 861187) ^(1/4)*(x + 2)^(1/4)*99638145i)/(8*((820709165*17^(1/2))/8 + 3382057035/8) ) + (17^(1/2)*(- 208885*17^(1/2) - 861187)^(1/4)*(x + 2)^(1/4)*24021775i)/ (8*((820709165*17^(1/2))/8 + 3382057035/8)))*(- 208885*17^(1/2) - 861187)^ (1/4)*1i)/34
\[ \int \frac {(2+x)^{3/4} (3+5 x)}{4+7 x+2 x^2} \, dx=\frac {52 \left (x +2\right )^{\frac {3}{4}}}{21}-\frac {10 \left (\int \frac {\left (x +2\right )^{\frac {3}{4}}}{2 x^{3}+11 x^{2}+18 x +8}d x \right )}{7}+\frac {9 \left (\int \frac {\left (x +2\right )^{\frac {3}{4}} x^{2}}{2 x^{3}+11 x^{2}+18 x +8}d x \right )}{7} \] Input:
int((2+x)^(3/4)*(3+5*x)/(2*x^2+7*x+4),x)
Output:
(52*(x + 2)**(3/4) - 30*int((x + 2)**(3/4)/(2*x**3 + 11*x**2 + 18*x + 8),x ) + 27*int(((x + 2)**(3/4)*x**2)/(2*x**3 + 11*x**2 + 18*x + 8),x))/21