Integrand size = 28, antiderivative size = 469 \[ \int \frac {b+2 c x}{(d+e x)^{3/2} \left (a+b x+c x^2\right )^2} \, dx=\frac {3 e^2 (2 c d-b e)}{\left (c d^2-b d e+a e^2\right )^2 \sqrt {d+e x}}-\frac {\left (b^2-4 a c\right ) (c d-b e)-c \left (b^2-4 a c\right ) e x}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x} \left (a+b x+c x^2\right )}+\frac {3 \sqrt {c} e \left (2 c^2 d^2+b \left (b+\sqrt {b^2-4 a c}\right ) e^2-2 c e \left (b d+\sqrt {b^2-4 a c} d+a e\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {2} \sqrt {b^2-4 a c} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} \left (c d^2-b d e+a e^2\right )^2}-\frac {3 \sqrt {c} e \left (2 c^2 d^2+b \left (b-\sqrt {b^2-4 a c}\right ) e^2-2 c e \left (b d-\sqrt {b^2-4 a c} d+a e\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {2} \sqrt {b^2-4 a c} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} \left (c d^2-b d e+a e^2\right )^2} \]
3*e^2*(-b*e+2*c*d)/(a*e^2-b*d*e+c*d^2)^2/(e*x+d)^(1/2)+(-(-4*a*c+b^2)*(-b* e+c*d)+c*(-4*a*c+b^2)*e*x)/(-4*a*c+b^2)/(a*e^2-b*d*e+c*d^2)/(c*x^2+b*x+a)/ (e*x+d)^(1/2)+3/2*e*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))*c^(1/2)*(2*c^2*d^2+b*e^2*(b+(-4*a*c+b^2)^(1/2))-2* c*e*(b*d+a*e+d*(-4*a*c+b^2)^(1/2)))/(a*e^2-b*d*e+c*d^2)^2*2^(1/2)/(-4*a*c+ b^2)^(1/2)/(2*c*d-e*(b-(-4*a*c+b^2)^(1/2)))^(1/2)-3/2*e*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))*c^(1/2)*(2*c^2 *d^2+b*e^2*(b-(-4*a*c+b^2)^(1/2))-2*c*e*(b*d+a*e-d*(-4*a*c+b^2)^(1/2)))/(a *e^2-b*d*e+c*d^2)^2*2^(1/2)/(-4*a*c+b^2)^(1/2)/(2*c*d-e*(b+(-4*a*c+b^2)^(1 /2)))^(1/2)
Time = 3.47 (sec) , antiderivative size = 429, normalized size of antiderivative = 0.91 \[ \int \frac {b+2 c x}{(d+e x)^{3/2} \left (a+b x+c x^2\right )^2} \, dx=\frac {e \left (-\frac {2 \left (c^2 d \left (d^2-d e x-6 e^2 x^2\right )+b e^2 (2 a e+b (d+3 e x))-c e \left (a e (5 d+e x)+b \left (2 d^2+5 d e x-3 e^2 x^2\right )\right )\right )}{e \sqrt {d+e x} (a+x (b+c x))}-\frac {3 \sqrt {2} \sqrt {c} \left (2 c^2 d^2+b \left (b+\sqrt {b^2-4 a c}\right ) e^2-2 c e \left (b d+\sqrt {b^2-4 a c} d+a e\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e-\sqrt {b^2-4 a c} e}}\right )}{\sqrt {b^2-4 a c} \sqrt {-2 c d+\left (b-\sqrt {b^2-4 a c}\right ) e}}+\frac {3 \sqrt {2} \sqrt {c} \left (2 c^2 d^2+b \left (b-\sqrt {b^2-4 a c}\right ) e^2-2 c e \left (b d-\sqrt {b^2-4 a c} d+a e\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {b^2-4 a c} \sqrt {-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{2 \left (c d^2+e (-b d+a e)\right )^2} \]
(e*((-2*(c^2*d*(d^2 - d*e*x - 6*e^2*x^2) + b*e^2*(2*a*e + b*(d + 3*e*x)) - c*e*(a*e*(5*d + e*x) + b*(2*d^2 + 5*d*e*x - 3*e^2*x^2))))/(e*Sqrt[d + e*x ]*(a + x*(b + c*x))) - (3*Sqrt[2]*Sqrt[c]*(2*c^2*d^2 + b*(b + Sqrt[b^2 - 4 *a*c])*e^2 - 2*c*e*(b*d + Sqrt[b^2 - 4*a*c]*d + a*e))*ArcTan[(Sqrt[2]*Sqrt [c]*Sqrt[d + e*x])/Sqrt[-2*c*d + b*e - Sqrt[b^2 - 4*a*c]*e]])/(Sqrt[b^2 - 4*a*c]*Sqrt[-2*c*d + (b - Sqrt[b^2 - 4*a*c])*e]) + (3*Sqrt[2]*Sqrt[c]*(2*c ^2*d^2 + b*(b - Sqrt[b^2 - 4*a*c])*e^2 - 2*c*e*(b*d - Sqrt[b^2 - 4*a*c]*d + a*e))*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e]])/(Sqrt[b^2 - 4*a*c]*Sqrt[-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e ])))/(2*(c*d^2 + e*(-(b*d) + a*e))^2)
Time = 0.90 (sec) , antiderivative size = 480, normalized size of antiderivative = 1.02, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1235, 27, 1198, 1197, 27, 1480, 221}
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.
| \(\displaystyle \int \frac {b+2 c x}{(d+e x)^{3/2} \left (a+b x+c x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 1235 |
| \(\displaystyle -\frac {\int \frac {3 \left (b^2-4 a c\right ) e (c d-b e-c e x)}{2 (d+e x)^{3/2} \left (c x^2+b x+a\right )}dx}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {\left (b^2-4 a c\right ) (c d-b e)-c e x \left (b^2-4 a c\right )}{\left (b^2-4 a c\right ) \sqrt {d+e x} \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {3 e \int \frac {c d-b e-c e x}{(d+e x)^{3/2} \left (c x^2+b x+a\right )}dx}{2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (b^2-4 a c\right ) (c d-b e)-c e x \left (b^2-4 a c\right )}{\left (b^2-4 a c\right ) \sqrt {d+e x} \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 1198 |
| \(\displaystyle -\frac {3 e \left (\frac {\int \frac {c^2 d^2+b^2 e^2-c e (2 b d+a e)-c e (2 c d-b e) x}{\sqrt {d+e x} \left (c x^2+b x+a\right )}dx}{a e^2-b d e+c d^2}-\frac {2 e (2 c d-b e)}{\sqrt {d+e x} \left (a e^2-b d e+c d^2\right )}\right )}{2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (b^2-4 a c\right ) (c d-b e)-c e x \left (b^2-4 a c\right )}{\left (b^2-4 a c\right ) \sqrt {d+e x} \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 1197 |
| \(\displaystyle -\frac {3 e \left (\frac {2 \int \frac {e \left (3 c^2 d^2+b^2 e^2-c e (3 b d+a e)-c (2 c d-b e) (d+e x)\right )}{c d^2-b e d+a e^2+c (d+e x)^2-(2 c d-b e) (d+e x)}d\sqrt {d+e x}}{a e^2-b d e+c d^2}-\frac {2 e (2 c d-b e)}{\sqrt {d+e x} \left (a e^2-b d e+c d^2\right )}\right )}{2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (b^2-4 a c\right ) (c d-b e)-c e x \left (b^2-4 a c\right )}{\left (b^2-4 a c\right ) \sqrt {d+e x} \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {3 e \left (\frac {2 e \int \frac {3 c^2 d^2+b^2 e^2-c e (3 b d+a e)-c (2 c d-b e) (d+e x)}{c d^2-b e d+a e^2+c (d+e x)^2-(2 c d-b e) (d+e x)}d\sqrt {d+e x}}{a e^2-b d e+c d^2}-\frac {2 e (2 c d-b e)}{\sqrt {d+e x} \left (a e^2-b d e+c d^2\right )}\right )}{2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (b^2-4 a c\right ) (c d-b e)-c e x \left (b^2-4 a c\right )}{\left (b^2-4 a c\right ) \sqrt {d+e x} \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle -\frac {3 e \left (\frac {2 e \left (\frac {c \left (-2 c e \left (d \sqrt {b^2-4 a c}+a e+b d\right )+b e^2 \left (\sqrt {b^2-4 a c}+b\right )+2 c^2 d^2\right ) \int \frac {1}{\frac {1}{2} \left (\left (b-\sqrt {b^2-4 a c}\right ) e-2 c d\right )+c (d+e x)}d\sqrt {d+e x}}{2 e \sqrt {b^2-4 a c}}-\frac {c \left (-2 c e \left (-d \sqrt {b^2-4 a c}+a e+b d\right )+b e^2 \left (b-\sqrt {b^2-4 a c}\right )+2 c^2 d^2\right ) \int \frac {1}{\frac {1}{2} \left (\left (b+\sqrt {b^2-4 a c}\right ) e-2 c d\right )+c (d+e x)}d\sqrt {d+e x}}{2 e \sqrt {b^2-4 a c}}\right )}{a e^2-b d e+c d^2}-\frac {2 e (2 c d-b e)}{\sqrt {d+e x} \left (a e^2-b d e+c d^2\right )}\right )}{2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (b^2-4 a c\right ) (c d-b e)-c e x \left (b^2-4 a c\right )}{\left (b^2-4 a c\right ) \sqrt {d+e x} \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {3 e \left (\frac {2 e \left (\frac {\sqrt {c} \left (-2 c e \left (-d \sqrt {b^2-4 a c}+a e+b d\right )+b e^2 \left (b-\sqrt {b^2-4 a c}\right )+2 c^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{\sqrt {2} e \sqrt {b^2-4 a c} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}-\frac {\sqrt {c} \left (-2 c e \left (d \sqrt {b^2-4 a c}+a e+b d\right )+b e^2 \left (\sqrt {b^2-4 a c}+b\right )+2 c^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{\sqrt {2} e \sqrt {b^2-4 a c} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{a e^2-b d e+c d^2}-\frac {2 e (2 c d-b e)}{\sqrt {d+e x} \left (a e^2-b d e+c d^2\right )}\right )}{2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (b^2-4 a c\right ) (c d-b e)-c e x \left (b^2-4 a c\right )}{\left (b^2-4 a c\right ) \sqrt {d+e x} \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )}\) |
-(((b^2 - 4*a*c)*(c*d - b*e) - c*(b^2 - 4*a*c)*e*x)/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*Sqrt[d + e*x]*(a + b*x + c*x^2))) - (3*e*((-2*e*(2*c*d - b*e))/((c*d^2 - b*d*e + a*e^2)*Sqrt[d + e*x]) + (2*e*(-((Sqrt[c]*(2*c^2*d^ 2 + b*(b + Sqrt[b^2 - 4*a*c])*e^2 - 2*c*e*(b*d + Sqrt[b^2 - 4*a*c]*d + a*e ))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b - Sqrt[b^2 - 4* a*c])*e]])/(Sqrt[2]*Sqrt[b^2 - 4*a*c]*e*Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c ])*e])) + (Sqrt[c]*(2*c^2*d^2 + b*(b - Sqrt[b^2 - 4*a*c])*e^2 - 2*c*e*(b*d - Sqrt[b^2 - 4*a*c]*d + a*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqr t[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/(Sqrt[2]*Sqrt[b^2 - 4*a*c]*e*Sqrt[2 *c*d - (b + Sqrt[b^2 - 4*a*c])*e])))/(c*d^2 - b*d*e + a*e^2)))/(2*(c*d^2 - b*d*e + a*e^2))
3.17.22.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
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_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(e*f - d*g)*((d + e*x)^(m + 1)/((m + 1)*(c *d^2 - b*d*e + a*e^2))), x] + Simp[1/(c*d^2 - b*d*e + a*e^2) Int[(d + e*x )^(m + 1)*(Simp[c*d*f - f*b*e + a*e*g - c*(e*f - d*g)*x, x]/(a + b*x + c*x^ 2)), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && FractionQ[m] && LtQ[m, -1 ]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2 *a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^m *(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2*(p + m + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d* m + b*e*m) - b*d*(3*c*d - b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p] )
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 = 0.67 (sec) , antiderivative size = 500, normalized size of antiderivative = 1.07
| method | result | size |
| derivativedivides | \(2 e^{2} \left (-\frac {b e -2 c d}{\left (e^{2} a -b d e +c \,d^{2}\right )^{2} \sqrt {e x +d}}+\frac {\frac {\left (-\frac {1}{2} b c e +c^{2} d \right ) \left (e x +d \right )^{\frac {3}{2}}+\left (\frac {1}{2} a c \,e^{2}-\frac {1}{2} b^{2} e^{2}+\frac {3}{2} b c d e -\frac {3}{2} c^{2} d^{2}\right ) \sqrt {e x +d}}{c \left (e x +d \right )^{2}+b e \left (e x +d \right )-2 c d \left (e x +d \right )+e^{2} a -b d e +c \,d^{2}}+6 c \left (\frac {\left (-2 a c \,e^{2}+b^{2} e^{2}-2 b c d e +2 c^{2} d^{2}-\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b e +2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c d \right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}-\frac {\left (2 a c \,e^{2}-b^{2} e^{2}+2 b c d e -2 c^{2} d^{2}-\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b e +2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c d \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{\left (e^{2} a -b d e +c \,d^{2}\right )^{2}}\right )\) | \(500\) |
| default | \(2 e^{2} \left (-\frac {b e -2 c d}{\left (e^{2} a -b d e +c \,d^{2}\right )^{2} \sqrt {e x +d}}+\frac {\frac {\left (-\frac {1}{2} b c e +c^{2} d \right ) \left (e x +d \right )^{\frac {3}{2}}+\left (\frac {1}{2} a c \,e^{2}-\frac {1}{2} b^{2} e^{2}+\frac {3}{2} b c d e -\frac {3}{2} c^{2} d^{2}\right ) \sqrt {e x +d}}{c \left (e x +d \right )^{2}+b e \left (e x +d \right )-2 c d \left (e x +d \right )+e^{2} a -b d e +c \,d^{2}}+6 c \left (\frac {\left (-2 a c \,e^{2}+b^{2} e^{2}-2 b c d e +2 c^{2} d^{2}-\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b e +2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c d \right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}-\frac {\left (2 a c \,e^{2}-b^{2} e^{2}+2 b c d e -2 c^{2} d^{2}-\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b e +2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c d \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{\left (e^{2} a -b d e +c \,d^{2}\right )^{2}}\right )\) | \(500\) |
| pseudoelliptic | \(-\frac {3 \left (\sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {2}\, \sqrt {e x +d}\, e^{2} \left (c \,x^{2}+b x +a \right ) \left (\left (-\frac {b e}{2}+c d \right ) \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}+\left (a c -\frac {b^{2}}{2}\right ) e^{2}+b c d e -c^{2} d^{2}\right ) c \,\operatorname {arctanh}\left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}}\right )+\left (\left (\left (\frac {b e}{2}-c d \right ) \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}+\left (a c -\frac {b^{2}}{2}\right ) e^{2}+b c d e -c^{2} d^{2}\right ) \sqrt {2}\, \sqrt {e x +d}\, e^{2} \left (c \,x^{2}+b x +a \right ) c \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}}\right )+\frac {2 \left (\left (-\frac {x \left (-3 b x +a \right ) c}{2}+b \left (\frac {3 b x}{2}+a \right )\right ) e^{3}-\frac {5 d \left (\frac {6 c^{2} x^{2}}{5}+\left (b x +a \right ) c -\frac {b^{2}}{5}\right ) e^{2}}{2}-d^{2} \left (\frac {c x}{2}+b \right ) c e +\frac {c^{2} d^{3}}{2}\right ) \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}}{3}\right ) \sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\right )}{\sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {e x +d}\, \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\, \left (c \,x^{2}+b x +a \right ) \left (e^{2} a -b d e +c \,d^{2}\right )^{2}}\) | \(529\) |
2*e^2*(-1/(a*e^2-b*d*e+c*d^2)^2*(b*e-2*c*d)/(e*x+d)^(1/2)+1/(a*e^2-b*d*e+c *d^2)^2*(((-1/2*b*c*e+c^2*d)*(e*x+d)^(3/2)+(1/2*a*c*e^2-1/2*b^2*e^2+3/2*b* c*d*e-3/2*c^2*d^2)*(e*x+d)^(1/2))/(c*(e*x+d)^2+b*e*(e*x+d)-2*c*d*(e*x+d)+e ^2*a-b*d*e+c*d^2)+6*c*(1/8*(-2*a*c*e^2+b^2*e^2-2*b*c*d*e+2*c^2*d^2-(-e^2*( 4*a*c-b^2))^(1/2)*b*e+2*(-e^2*(4*a*c-b^2))^(1/2)*c*d)/(-e^2*(4*a*c-b^2))^( 1/2)*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))-1/8*(2*a* c*e^2-b^2*e^2+2*b*c*d*e-2*c^2*d^2-(-e^2*(4*a*c-b^2))^(1/2)*b*e+2*(-e^2*(4* a*c-b^2))^(1/2)*c*d)/(-e^2*(4*a*c-b^2))^(1/2)*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. 26583 vs. \(2 (416) = 832\).
Time = 6.68 (sec) , antiderivative size = 26583, normalized size of antiderivative = 56.68 \[ \int \frac {b+2 c x}{(d+e x)^{3/2} \left (a+b x+c x^2\right )^2} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {b+2 c x}{(d+e x)^{3/2} \left (a+b x+c x^2\right )^2} \, dx=\text {Timed out} \]
\[ \int \frac {b+2 c x}{(d+e x)^{3/2} \left (a+b x+c x^2\right )^2} \, dx=\int { \frac {2 \, c x + b}{{\left (c x^{2} + b x + a\right )}^{2} {\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 3464 vs. \(2 (416) = 832\).
Time = 1.13 (sec) , antiderivative size = 3464, normalized size of antiderivative = 7.39 \[ \int \frac {b+2 c x}{(d+e x)^{3/2} \left (a+b x+c x^2\right )^2} \, dx=\text {Too large to display} \]
(6*(e*x + d)^2*c^2*d*e^2 - 11*(e*x + d)*c^2*d^2*e^2 + 4*c^2*d^3*e^2 - 3*(e *x + d)^2*b*c*e^3 + 11*(e*x + d)*b*c*d*e^3 - 6*b*c*d^2*e^3 - 3*(e*x + d)*b ^2*e^4 + (e*x + d)*a*c*e^4 + 2*b^2*d*e^4 + 4*a*c*d*e^4 - 2*a*b*e^5)/((c^2* d^4 - 2*b*c*d^3*e + b^2*d^2*e^2 + 2*a*c*d^2*e^2 - 2*a*b*d*e^3 + a^2*e^4)*( (e*x + d)^(5/2)*c - 2*(e*x + d)^(3/2)*c*d + sqrt(e*x + d)*c*d^2 + (e*x + d )^(3/2)*b*e - sqrt(e*x + d)*b*d*e + sqrt(e*x + d)*a*e^2)) + 3/8*((c^2*d^4* e - 2*b*c*d^3*e^2 + b^2*d^2*e^3 + 2*a*c*d^2*e^3 - 2*a*b*d*e^4 + a^2*e^5)^2 *(2*(b^2*c - 4*a*c^2)*d*e^2 - (b^3 - 4*a*b*c)*e^3)*sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e) - 2*(3*sqrt(b^2 - 4*a*c)*c^4*d^6*e^2 - 9*sqrt(b^ 2 - 4*a*c)*b*c^3*d^5*e^3 + 5*(2*b^2*c^2 + a*c^3)*sqrt(b^2 - 4*a*c)*d^4*e^4 - 5*(b^3*c + 2*a*b*c^2)*sqrt(b^2 - 4*a*c)*d^3*e^5 + (b^4 + 7*a*b^2*c + a^ 2*c^2)*sqrt(b^2 - 4*a*c)*d^2*e^6 - (2*a*b^3 + a^2*b*c)*sqrt(b^2 - 4*a*c)*d *e^7 + (a^2*b^2 - a^3*c)*sqrt(b^2 - 4*a*c)*e^8)*sqrt(-4*c^2*d + 2*(b*c - s qrt(b^2 - 4*a*c)*c)*e)*abs(c^2*d^4*e - 2*b*c*d^3*e^2 + b^2*d^2*e^3 + 2*a*c *d^2*e^3 - 2*a*b*d*e^4 + a^2*e^5) + (4*c^7*d^11*e^2 - 22*b*c^6*d^10*e^3 + 4*(13*b^2*c^5 + 3*a*c^6)*d^9*e^4 - 3*(23*b^3*c^4 + 18*a*b*c^5)*d^8*e^5 + 8 *(7*b^4*c^3 + 13*a*b^2*c^4 + a^2*c^5)*d^7*e^6 - 28*(b^5*c^2 + 4*a*b^3*c^3 + a^2*b*c^4)*d^6*e^7 + 8*(b^6*c + 9*a*b^4*c^2 + 6*a^2*b^2*c^3 - a^3*c^4)*d ^5*e^8 - (b^7 + 26*a*b^5*c + 50*a^2*b^3*c^2 - 20*a^3*b*c^3)*d^4*e^9 + 4*(a *b^6 + 7*a^2*b^4*c - 2*a^3*b^2*c^2 - 3*a^4*c^3)*d^3*e^10 - 2*(3*a^2*b^5...
Time = 19.99 (sec) , antiderivative size = 58573, normalized size of antiderivative = 124.89 \[ \int \frac {b+2 c x}{(d+e x)^{3/2} \left (a+b x+c x^2\right )^2} \, dx=\text {Too large to display} \]
- atan((((-(9*(b^7*e^7 + b^4*e^7*(-(4*a*c - b^2)^3)^(1/2) - 20*a^3*b*c^3*e ^7 + 8*a*c^6*d^5*e^2 + 40*a^3*c^4*d*e^6 + 25*a^2*b^3*c^2*e^7 + a^2*c^2*e^7 *(-(4*a*c - b^2)^3)^(1/2) - 80*a^2*c^5*d^3*e^4 - 2*b^2*c^5*d^5*e^2 + 5*b^3 *c^4*d^4*e^3 - 10*b^4*c^3*d^3*e^4 + 10*b^5*c^2*d^2*e^5 + 5*c^4*d^4*e^3*(-( 4*a*c - b^2)^3)^(1/2) - 9*a*b^5*c*e^7 - 5*b^6*c*d*e^6 + 10*b^2*c^2*d^2*e^5 *(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^2*c*e^7*(-(4*a*c - b^2)^3)^(1/2) - 20*a* b*c^5*d^4*e^3 + 40*a*b^4*c^2*d*e^6 - 5*b^3*c*d*e^6*(-(4*a*c - b^2)^3)^(1/2 ) + 60*a*b^2*c^4*d^3*e^4 - 70*a*b^3*c^3*d^2*e^5 + 120*a^2*b*c^4*d^2*e^5 - 90*a^2*b^2*c^3*d*e^6 - 10*a*c^3*d^2*e^5*(-(4*a*c - b^2)^3)^(1/2) - 10*b*c^ 3*d^3*e^4*(-(4*a*c - b^2)^3)^(1/2) + 10*a*b*c^2*d*e^6*(-(4*a*c - b^2)^3)^( 1/2)))/(8*(16*a^2*c^7*d^10 + a^5*b^4*e^10 + 16*a^7*c^2*e^10 + b^4*c^5*d^10 - b^9*d^5*e^5 - 8*a*b^2*c^6*d^10 - 8*a^6*b^2*c*e^10 + 5*a*b^8*d^4*e^6 - 5 *a^4*b^5*d*e^9 - 5*b^5*c^4*d^9*e + 5*b^8*c*d^6*e^4 - 10*a^2*b^7*d^3*e^7 + 10*a^3*b^6*d^2*e^8 + 80*a^3*c^6*d^8*e^2 + 160*a^4*c^5*d^6*e^4 + 160*a^5*c^ 4*d^4*e^6 + 80*a^6*c^3*d^2*e^8 + 10*b^6*c^3*d^8*e^2 - 10*b^7*c^2*d^7*e^3 + 120*a^2*b^2*c^5*d^8*e^2 - 150*a^2*b^4*c^3*d^6*e^4 + 114*a^2*b^5*c^2*d^5*e ^5 + 400*a^3*b^2*c^4*d^6*e^4 - 80*a^3*b^3*c^3*d^5*e^5 - 150*a^3*b^4*c^2*d^ 4*e^6 + 400*a^4*b^2*c^3*d^4*e^6 + 120*a^5*b^2*c^2*d^2*e^8 + 40*a*b^3*c^5*d ^9*e - 12*a*b^7*c*d^5*e^5 - 80*a^2*b*c^6*d^9*e + 40*a^5*b^3*c*d*e^9 - 80*a ^6*b*c^2*d*e^9 - 75*a*b^4*c^4*d^8*e^2 + 60*a*b^5*c^3*d^7*e^3 - 10*a*b^6...