Integrand size = 28, antiderivative size = 175 \[ \int \frac {b+2 c x}{\sqrt {d+e x} \left (a+b x+c x^2\right )} \, dx=-\frac {2 \sqrt {2} \sqrt {c} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}-\frac {2 \sqrt {2} \sqrt {c} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \]
-2*arctanh(2^(1/2)*c^(1/2)*(e*x+d)^(1/2)/(2*c*d-e*(b-(-4*a*c+b^2)^(1/2)))^ (1/2))*2^(1/2)*c^(1/2)/(2*c*d-e*(b-(-4*a*c+b^2)^(1/2)))^(1/2)-2*arctanh(2^ (1/2)*c^(1/2)*(e*x+d)^(1/2)/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2)))^(1/2))*2^(1/2 )*c^(1/2)/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2)))^(1/2)
Time = 0.49 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.92 \[ \int \frac {b+2 c x}{\sqrt {d+e x} \left (a+b x+c x^2\right )} \, dx=2 \sqrt {2} \sqrt {c} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e-\sqrt {b^2-4 a c} e}}\right )}{\sqrt {-2 c d+\left (b-\sqrt {b^2-4 a c}\right ) e}}+\frac {\arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e}}\right ) \]
2*Sqrt[2]*Sqrt[c]*(ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[-2*c*d + b* e - Sqrt[b^2 - 4*a*c]*e]]/Sqrt[-2*c*d + (b - Sqrt[b^2 - 4*a*c])*e] + ArcTa n[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e] ]/Sqrt[-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e])
Time = 0.29 (sec) , antiderivative size = 1, normalized size of antiderivative = 0.01, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {1197, 25, 1480}
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.
| \(\displaystyle \int \frac {b+2 c x}{\sqrt {d+e x} \left (a+b x+c x^2\right )} \, dx\) |
| \(\Big \downarrow \) 1197 |
| \(\displaystyle 2 \int -\frac {2 c d-b e-2 c (d+e x)}{c d^2-b e d+a e^2+c (d+e x)^2-(2 c d-b e) (d+e x)}d\sqrt {d+e x}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -2 \int \frac {2 c d-b e-2 c (d+e x)}{c d^2-b e d+a e^2+c (d+e x)^2-(2 c d-b e) (d+e x)}d\sqrt {d+e x}\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle \text {Indeterminate}\) |
3.17.15.3.1 Defintions of rubi rules used
Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)), x_Symbol] :> Simp[2 Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; Fr eeQ[{a, b, c, d, e, f, g}, x]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : > With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q)) Int[1/( b/2 - q/2 + c*x^2), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q)) Int[1/(b/2 + q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^2 - 4*a*c]
Time = 0.48 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.91
| method | result | size |
| derivativedivides | \(8 c \left (\frac {\sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{4 \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}-\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{4 \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )\) | \(159\) |
| default | \(8 c \left (\frac {\sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{4 \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}-\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{4 \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )\) | \(159\) |
| pseudoelliptic | \(\frac {2 \sqrt {2}\, c \left (\sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}}\right )-\operatorname {arctanh}\left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}}\right ) \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\right )}{\sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}}\) | \(208\) |
8*c*(1/4*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctan(c*( e*x+d)^(1/2)*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))-1/4*2 ^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctanh(c*(e*x+d)^( 1/2)*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 1317 vs. \(2 (139) = 278\).
Time = 0.32 (sec) , antiderivative size = 1317, normalized size of antiderivative = 7.53 \[ \int \frac {b+2 c x}{\sqrt {d+e x} \left (a+b x+c x^2\right )} \, dx=\text {Too large to display} \]
-1/2*sqrt(2)*sqrt((2*c*d - b*e + (c*d^2 - b*d*e + a*e^2)*sqrt((b^2 - 4*a*c )*e^2/(c^2*d^4 - 2*b*c*d^3*e - 2*a*b*d*e^3 + a^2*e^4 + (b^2 + 2*a*c)*d^2*e ^2)))/(c*d^2 - b*d*e + a*e^2))*log(sqrt(2)*(2*c*d - b*e - (c*d^2 - b*d*e + a*e^2)*sqrt((b^2 - 4*a*c)*e^2/(c^2*d^4 - 2*b*c*d^3*e - 2*a*b*d*e^3 + a^2* e^4 + (b^2 + 2*a*c)*d^2*e^2)))*sqrt((2*c*d - b*e + (c*d^2 - b*d*e + a*e^2) *sqrt((b^2 - 4*a*c)*e^2/(c^2*d^4 - 2*b*c*d^3*e - 2*a*b*d*e^3 + a^2*e^4 + ( b^2 + 2*a*c)*d^2*e^2)))/(c*d^2 - b*d*e + a*e^2)) + 4*sqrt(e*x + d)*c) + 1/ 2*sqrt(2)*sqrt((2*c*d - b*e + (c*d^2 - b*d*e + a*e^2)*sqrt((b^2 - 4*a*c)*e ^2/(c^2*d^4 - 2*b*c*d^3*e - 2*a*b*d*e^3 + a^2*e^4 + (b^2 + 2*a*c)*d^2*e^2) ))/(c*d^2 - b*d*e + a*e^2))*log(-sqrt(2)*(2*c*d - b*e - (c*d^2 - b*d*e + a *e^2)*sqrt((b^2 - 4*a*c)*e^2/(c^2*d^4 - 2*b*c*d^3*e - 2*a*b*d*e^3 + a^2*e^ 4 + (b^2 + 2*a*c)*d^2*e^2)))*sqrt((2*c*d - b*e + (c*d^2 - b*d*e + a*e^2)*s qrt((b^2 - 4*a*c)*e^2/(c^2*d^4 - 2*b*c*d^3*e - 2*a*b*d*e^3 + a^2*e^4 + (b^ 2 + 2*a*c)*d^2*e^2)))/(c*d^2 - b*d*e + a*e^2)) + 4*sqrt(e*x + d)*c) - 1/2* sqrt(2)*sqrt((2*c*d - b*e - (c*d^2 - b*d*e + a*e^2)*sqrt((b^2 - 4*a*c)*e^2 /(c^2*d^4 - 2*b*c*d^3*e - 2*a*b*d*e^3 + a^2*e^4 + (b^2 + 2*a*c)*d^2*e^2))) /(c*d^2 - b*d*e + a*e^2))*log(sqrt(2)*(2*c*d - b*e + (c*d^2 - b*d*e + a*e^ 2)*sqrt((b^2 - 4*a*c)*e^2/(c^2*d^4 - 2*b*c*d^3*e - 2*a*b*d*e^3 + a^2*e^4 + (b^2 + 2*a*c)*d^2*e^2)))*sqrt((2*c*d - b*e - (c*d^2 - b*d*e + a*e^2)*sqrt ((b^2 - 4*a*c)*e^2/(c^2*d^4 - 2*b*c*d^3*e - 2*a*b*d*e^3 + a^2*e^4 + (b^...
\[ \int \frac {b+2 c x}{\sqrt {d+e x} \left (a+b x+c x^2\right )} \, dx=\int \frac {b + 2 c x}{\sqrt {d + e x} \left (a + b x + c x^{2}\right )}\, dx \]
\[ \int \frac {b+2 c x}{\sqrt {d+e x} \left (a+b x+c x^2\right )} \, dx=\int { \frac {2 \, c x + b}{{\left (c x^{2} + b x + a\right )} \sqrt {e x + d}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 338 vs. \(2 (139) = 278\).
Time = 0.30 (sec) , antiderivative size = 338, normalized size of antiderivative = 1.93 \[ \int \frac {b+2 c x}{\sqrt {d+e x} \left (a+b x+c x^2\right )} \, dx=\frac {{\left (\sqrt {-4 \, c^{2} d + 2 \, {\left (b c - \sqrt {b^{2} - 4 \, a c} c\right )} e} \sqrt {b^{2} - 4 \, a c} {\left | e \right |} - \sqrt {-4 \, c^{2} d + 2 \, {\left (b c - \sqrt {b^{2} - 4 \, a c} c\right )} e} {\left (2 \, c d - b e\right )}\right )} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {e x + d}}{\sqrt {-\frac {2 \, c d - b e + \sqrt {{\left (2 \, c d - b e\right )}^{2} - 4 \, {\left (c d^{2} - b d e + a e^{2}\right )} c}}{c}}}\right )}{2 \, {\left (c d^{2} - b d e + a e^{2}\right )} {\left | c \right |}} - \frac {{\left (\sqrt {-4 \, c^{2} d + 2 \, {\left (b c + \sqrt {b^{2} - 4 \, a c} c\right )} e} \sqrt {b^{2} - 4 \, a c} {\left | e \right |} + \sqrt {-4 \, c^{2} d + 2 \, {\left (b c + \sqrt {b^{2} - 4 \, a c} c\right )} e} {\left (2 \, c d - b e\right )}\right )} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {e x + d}}{\sqrt {-\frac {2 \, c d - b e - \sqrt {{\left (2 \, c d - b e\right )}^{2} - 4 \, {\left (c d^{2} - b d e + a e^{2}\right )} c}}{c}}}\right )}{2 \, {\left (c d^{2} - b d e + a e^{2}\right )} {\left | c \right |}} \]
1/2*(sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e)*sqrt(b^2 - 4*a*c)*ab s(e) - sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e)*(2*c*d - b*e))*arc tan(2*sqrt(1/2)*sqrt(e*x + d)/sqrt(-(2*c*d - b*e + sqrt((2*c*d - b*e)^2 - 4*(c*d^2 - b*d*e + a*e^2)*c))/c))/((c*d^2 - b*d*e + a*e^2)*abs(c)) - 1/2*( sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e)*sqrt(b^2 - 4*a*c)*abs(e) + sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e)*(2*c*d - b*e))*arctan(2 *sqrt(1/2)*sqrt(e*x + d)/sqrt(-(2*c*d - b*e - sqrt((2*c*d - b*e)^2 - 4*(c* d^2 - b*d*e + a*e^2)*c))/c))/((c*d^2 - b*d*e + a*e^2)*abs(c))
Time = 11.01 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.17 \[ \int \frac {b+2 c x}{\sqrt {d+e x} \left (a+b x+c x^2\right )} \, dx=\mathrm {atan}\left (\sqrt {\frac {2\,c\,d-b\,e+e\,\sqrt {b^2-4\,a\,c}}{2\,c\,d^2-2\,b\,d\,e+2\,a\,e^2}}\,\sqrt {d+e\,x}\,1{}\mathrm {i}\right )\,\sqrt {\frac {2\,c\,d-b\,e+e\,\sqrt {b^2-4\,a\,c}}{2\,c\,d^2-2\,b\,d\,e+2\,a\,e^2}}\,2{}\mathrm {i}+\mathrm {atan}\left (\sqrt {-\frac {b\,e-2\,c\,d+e\,\sqrt {b^2-4\,a\,c}}{2\,c\,d^2-2\,b\,d\,e+2\,a\,e^2}}\,\sqrt {d+e\,x}\,1{}\mathrm {i}\right )\,\sqrt {-\frac {b\,e-2\,c\,d+e\,\sqrt {b^2-4\,a\,c}}{2\,c\,d^2-2\,b\,d\,e+2\,a\,e^2}}\,2{}\mathrm {i} \]
atan(((2*c*d - b*e + e*(b^2 - 4*a*c)^(1/2))/(2*a*e^2 + 2*c*d^2 - 2*b*d*e)) ^(1/2)*(d + e*x)^(1/2)*1i)*((2*c*d - b*e + e*(b^2 - 4*a*c)^(1/2))/(2*a*e^2 + 2*c*d^2 - 2*b*d*e))^(1/2)*2i + atan((-(b*e - 2*c*d + e*(b^2 - 4*a*c)^(1 /2))/(2*a*e^2 + 2*c*d^2 - 2*b*d*e))^(1/2)*(d + e*x)^(1/2)*1i)*(-(b*e - 2*c *d + e*(b^2 - 4*a*c)^(1/2))/(2*a*e^2 + 2*c*d^2 - 2*b*d*e))^(1/2)*2i