Integrand size = 22, antiderivative size = 604 \[ \int \frac {1}{(d+e x)^{3/2} \left (a+b x+c x^2\right )^2} \, dx=-\frac {e \left (2 c^2 d^2+3 b^2 e^2-2 c e (b d+5 a e)\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 \sqrt {d+e x}}-\frac {b c d-b^2 e+2 a c e+c (2 c d-b e) x}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x} \left (a+b x+c x^2\right )}+\frac {\sqrt {c} \left (8 c^3 d^3+3 b^2 \left (b+\sqrt {b^2-4 a c}\right ) e^3-2 c^2 d e \left (6 b d-\sqrt {b^2-4 a c} d-16 a e\right )-2 c e^2 \left (b^2 d+b \sqrt {b^2-4 a c} d+8 a b e+5 a \sqrt {b^2-4 a c} e\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {2} \left (b^2-4 a c\right )^{3/2} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} \left (c d^2-b d e+a e^2\right )^2}-\frac {\sqrt {c} \left (8 c^3 d^3+3 b^2 \left (b-\sqrt {b^2-4 a c}\right ) e^3-2 c^2 d e \left (6 b d+\sqrt {b^2-4 a c} d-16 a e\right )-2 c e^2 \left (b^2 d-b \sqrt {b^2-4 a c} d+8 a b e-5 a \sqrt {b^2-4 a c} e\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {2} \left (b^2-4 a c\right )^{3/2} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} \left (c d^2-b d e+a e^2\right )^2} \]
-e*(2*c^2*d^2+3*b^2*e^2-2*c*e*(5*a*e+b*d))/(-4*a*c+b^2)/(a*e^2-b*d*e+c*d^2 )^2/(e*x+d)^(1/2)+(-b*c*d+b^2*e-2*a*c*e-c*(-b*e+2*c*d)*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)+1/2*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))*c^(1/2)*(8*c^3*d^3+3* b^2*e^3*(b+(-4*a*c+b^2)^(1/2))-2*c^2*d*e*(6*b*d-16*a*e-d*(-4*a*c+b^2)^(1/2 ))-2*c*e^2*(b^2*d+8*a*b*e+b*d*(-4*a*c+b^2)^(1/2)+5*a*e*(-4*a*c+b^2)^(1/2)) )/(-4*a*c+b^2)^(3/2)/(a*e^2-b*d*e+c*d^2)^2*2^(1/2)/(2*c*d-e*(b-(-4*a*c+b^2 )^(1/2)))^(1/2)-1/2*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))*c^(1/2)*(8*c^3*d^3+3*b^2*e^3*(b-(-4*a*c+b^2)^(1/2) )-2*c^2*d*e*(6*b*d-16*a*e+d*(-4*a*c+b^2)^(1/2))-2*c*e^2*(b^2*d+8*a*b*e-b*d *(-4*a*c+b^2)^(1/2)-5*a*e*(-4*a*c+b^2)^(1/2)))/(-4*a*c+b^2)^(3/2)/(a*e^2-b *d*e+c*d^2)^2*2^(1/2)/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2)))^(1/2)
Time = 7.44 (sec) , antiderivative size = 601, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(d+e x)^{3/2} \left (a+b x+c x^2\right )^2} \, dx=\frac {\frac {2 \left (-b^3 e^2 (d+3 e x)-2 c \left (-4 a^2 e^3+c^2 d^2 x (d+e x)+a c e \left (2 d^2+d e x-5 e^2 x^2\right )\right )+b^2 \left (-2 a e^3+c e \left (2 d^2+d e x-3 e^2 x^2\right )\right )+b c \left (a e^2 (3 d+11 e x)+c d \left (-d^2+d e x+2 e^2 x^2\right )\right )\right )}{\left (b^2-4 a c\right ) \sqrt {d+e x} (a+x (b+c x))}-\frac {\sqrt {2} \sqrt {c} \left (8 c^3 d^3+3 b^2 \left (b+\sqrt {b^2-4 a c}\right ) e^3+2 c^2 d e \left (-6 b d+\sqrt {b^2-4 a c} d+16 a e\right )-2 c e^2 \left (b^2 d+b \sqrt {b^2-4 a c} d+8 a b e+5 a \sqrt {b^2-4 a c} e\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e-\sqrt {b^2-4 a c} e}}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt {-2 c d+\left (b-\sqrt {b^2-4 a c}\right ) e}}+\frac {\sqrt {2} \sqrt {c} \left (8 c^3 d^3+3 b^2 \left (b-\sqrt {b^2-4 a c}\right ) e^3-2 c^2 d e \left (6 b d+\sqrt {b^2-4 a c} d-16 a e\right )+2 c e^2 \left (-b^2 d+b \sqrt {b^2-4 a c} d-8 a b e+5 a \sqrt {b^2-4 a c} e\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt {-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e}}}{2 \left (c d^2+e (-b d+a e)\right )^2} \]
((2*(-(b^3*e^2*(d + 3*e*x)) - 2*c*(-4*a^2*e^3 + c^2*d^2*x*(d + e*x) + a*c* e*(2*d^2 + d*e*x - 5*e^2*x^2)) + b^2*(-2*a*e^3 + c*e*(2*d^2 + d*e*x - 3*e^ 2*x^2)) + b*c*(a*e^2*(3*d + 11*e*x) + c*d*(-d^2 + d*e*x + 2*e^2*x^2))))/(( b^2 - 4*a*c)*Sqrt[d + e*x]*(a + x*(b + c*x))) - (Sqrt[2]*Sqrt[c]*(8*c^3*d^ 3 + 3*b^2*(b + Sqrt[b^2 - 4*a*c])*e^3 + 2*c^2*d*e*(-6*b*d + Sqrt[b^2 - 4*a *c]*d + 16*a*e) - 2*c*e^2*(b^2*d + b*Sqrt[b^2 - 4*a*c]*d + 8*a*b*e + 5*a*S qrt[b^2 - 4*a*c]*e))*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[-2*c*d + b*e - Sqrt[b^2 - 4*a*c]*e]])/((b^2 - 4*a*c)^(3/2)*Sqrt[-2*c*d + (b - Sqrt[ b^2 - 4*a*c])*e]) + (Sqrt[2]*Sqrt[c]*(8*c^3*d^3 + 3*b^2*(b - Sqrt[b^2 - 4* a*c])*e^3 - 2*c^2*d*e*(6*b*d + Sqrt[b^2 - 4*a*c]*d - 16*a*e) + 2*c*e^2*(-( b^2*d) + b*Sqrt[b^2 - 4*a*c]*d - 8*a*b*e + 5*a*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] ])/((b^2 - 4*a*c)^(3/2)*Sqrt[-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e]))/(2*(c*d ^2 + e*(-(b*d) + a*e))^2)
Time = 1.34 (sec) , antiderivative size = 577, normalized size of antiderivative = 0.96, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {1165, 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.
| \(\displaystyle \int \frac {1}{(d+e x)^{3/2} \left (a+b x+c x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 1165 |
| \(\displaystyle -\frac {\int \frac {4 c^2 d^2-b c e d-3 b^2 e^2+10 a c e^2+3 c e (2 c d-b e) x}{2 (d+e x)^{3/2} \left (c x^2+b x+a\right )}dx}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d}{\left (b^2-4 a c\right ) \sqrt {d+e x} \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {4 c^2 d^2-3 b^2 e^2-c e (b d-10 a e)+3 c e (2 c d-b e) x}{(d+e x)^{3/2} \left (c x^2+b x+a\right )}dx}{2 \left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d}{\left (b^2-4 a c\right ) \sqrt {d+e x} \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 1198 |
| \(\displaystyle -\frac {\frac {\int \frac {4 c^3 d^3-c^2 e (5 b d-16 a e) d+3 b^3 e^3-b c e^2 (2 b d+13 a e)+c e \left (2 c^2 d^2+3 b^2 e^2-2 c e (b d+5 a e)\right ) x}{\sqrt {d+e x} \left (c x^2+b x+a\right )}dx}{a e^2-b d e+c d^2}+\frac {2 e \left (-2 c e (5 a e+b d)+3 b^2 e^2+2 c^2 d^2\right )}{\sqrt {d+e x} \left (a e^2-b d e+c d^2\right )}}{2 \left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d}{\left (b^2-4 a c\right ) \sqrt {d+e x} \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 1197 |
| \(\displaystyle -\frac {\frac {2 \int \frac {e \left ((2 c d-b e) \left (c^2 d^2-3 b^2 e^2-c e (b d-13 a e)\right )+c \left (2 c^2 d^2+3 b^2 e^2-2 c e (b d+5 a e)\right ) (d+e x)\right )}{c d^2-b e d+a e^2+c (d+e x)^2-(2 c d-b e) (d+e x)}d\sqrt {d+e x}}{a e^2-b d e+c d^2}+\frac {2 e \left (-2 c e (5 a e+b d)+3 b^2 e^2+2 c^2 d^2\right )}{\sqrt {d+e x} \left (a e^2-b d e+c d^2\right )}}{2 \left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d}{\left (b^2-4 a c\right ) \sqrt {d+e x} \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {2 e \int \frac {(2 c d-b e) \left (c^2 d^2-3 b^2 e^2-c e (b d-13 a e)\right )+c \left (2 c^2 d^2+3 b^2 e^2-2 c e (b d+5 a e)\right ) (d+e x)}{c d^2-b e d+a e^2+c (d+e x)^2-(2 c d-b e) (d+e x)}d\sqrt {d+e x}}{a e^2-b d e+c d^2}+\frac {2 e \left (-2 c e (5 a e+b d)+3 b^2 e^2+2 c^2 d^2\right )}{\sqrt {d+e x} \left (a e^2-b d e+c d^2\right )}}{2 \left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d}{\left (b^2-4 a c\right ) \sqrt {d+e x} \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle -\frac {\frac {2 e \left (\frac {1}{2} c \left (-\frac {(2 c d-b e) \left (-4 c e (b d-4 a e)-3 b^2 e^2+4 c^2 d^2\right )}{e \sqrt {b^2-4 a c}}-2 c e (5 a e+b d)+3 b^2 e^2+2 c^2 d^2\right ) \int \frac {1}{\frac {1}{2} \left (\left (b+\sqrt {b^2-4 a c}\right ) e-2 c d\right )+c (d+e x)}d\sqrt {d+e x}+\frac {c \left (-2 c^2 d e \left (-d \sqrt {b^2-4 a c}-16 a e+6 b d\right )-2 c e^2 \left (b d \sqrt {b^2-4 a c}+5 a e \sqrt {b^2-4 a c}+8 a b e+b^2 d\right )+3 b^2 e^3 \left (\sqrt {b^2-4 a c}+b\right )+8 c^3 d^3\right ) \int \frac {1}{\frac {1}{2} \left (\left (b-\sqrt {b^2-4 a c}\right ) e-2 c d\right )+c (d+e x)}d\sqrt {d+e x}}{2 e \sqrt {b^2-4 a c}}\right )}{a e^2-b d e+c d^2}+\frac {2 e \left (-2 c e (5 a e+b d)+3 b^2 e^2+2 c^2 d^2\right )}{\sqrt {d+e x} \left (a e^2-b d e+c d^2\right )}}{2 \left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d}{\left (b^2-4 a c\right ) \sqrt {d+e x} \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {\frac {2 e \left (-\frac {\sqrt {c} \left (-\frac {(2 c d-b e) \left (-4 c e (b d-4 a e)-3 b^2 e^2+4 c^2 d^2\right )}{e \sqrt {b^2-4 a c}}-2 c e (5 a e+b d)+3 b^2 e^2+2 c^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{\sqrt {2} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}-\frac {\sqrt {c} \left (-2 c^2 d e \left (-d \sqrt {b^2-4 a c}-16 a e+6 b d\right )-2 c e^2 \left (b d \sqrt {b^2-4 a c}+5 a e \sqrt {b^2-4 a c}+8 a b e+b^2 d\right )+3 b^2 e^3 \left (\sqrt {b^2-4 a c}+b\right )+8 c^3 d^3\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{\sqrt {2} e \sqrt {b^2-4 a c} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{a e^2-b d e+c d^2}+\frac {2 e \left (-2 c e (5 a e+b d)+3 b^2 e^2+2 c^2 d^2\right )}{\sqrt {d+e x} \left (a e^2-b d e+c d^2\right )}}{2 \left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d}{\left (b^2-4 a c\right ) \sqrt {d+e x} \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )}\) |
-((b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*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))) - ((2*e*(2*c^2*d^2 + 3*b^2* e^2 - 2*c*e*(b*d + 5*a*e)))/((c*d^2 - b*d*e + a*e^2)*Sqrt[d + e*x]) + (2*e *(-((Sqrt[c]*(8*c^3*d^3 + 3*b^2*(b + Sqrt[b^2 - 4*a*c])*e^3 - 2*c^2*d*e*(6 *b*d - Sqrt[b^2 - 4*a*c]*d - 16*a*e) - 2*c*e^2*(b^2*d + b*Sqrt[b^2 - 4*a*c ]*d + 8*a*b*e + 5*a*Sqrt[b^2 - 4*a*c]*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 + 3*b ^2*e^2 - 2*c*e*(b*d + 5*a*e) - ((2*c*d - b*e)*(4*c^2*d^2 - 3*b^2*e^2 - 4*c *e*(b*d - 4*a*e)))/(Sqrt[b^2 - 4*a*c]*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[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e])))/(c*d^2 - b*d*e + a*e^2))/(2*(b^2 - 4*a*c)*(c*d ^2 - b*d*e + a*e^2))
3.23.100.3.1 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[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(d + e*x)^(m + 1)*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*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*Simp[b*c*d*e*(2*p - m + 2) + b^2*e^2*(m + p + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3) - c*e*(2*c*d - b*e)*(m + 2*p + 4)*x, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && LtQ[p, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]
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_)^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 = 1.04 (sec) , antiderivative size = 653, normalized size of antiderivative = 1.08
| method | result | size |
| pseudoelliptic | \(-\frac {2 \left (-\frac {5 \sqrt {e x +d}\, \left (\left (-\frac {c^{2} d^{2}}{5}+e \left (a e +\frac {b d}{5}\right ) c -\frac {3 b^{2} e^{2}}{10}\right ) \sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}+\frac {8 \left (b e -2 c d \right ) \left (\frac {c^{2} d^{2}}{4}+e \left (a e -\frac {b d}{4}\right ) c -\frac {3 b^{2} e^{2}}{16}\right )}{5}\right ) \sqrt {2}\, \left (c \,x^{2}+b x +a \right ) e c \sqrt {\left (b e -2 c d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\right ) c}\, \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\right ) c}}\right )}{8}+\sqrt {\left (-b e +2 c d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\right ) c}\, \left (\frac {5 \sqrt {e x +d}\, \sqrt {2}\, \left (c \,x^{2}+b x +a \right ) e c \left (\left (-\frac {c^{2} d^{2}}{5}+e \left (a e +\frac {b d}{5}\right ) c -\frac {3 b^{2} e^{2}}{10}\right ) \sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}-\frac {8 \left (b e -2 c d \right ) \left (\frac {c^{2} d^{2}}{4}+e \left (a e -\frac {b d}{4}\right ) c -\frac {3 b^{2} e^{2}}{16}\right )}{5}\right ) \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\right ) c}}\right )}{8}+\left (-\frac {d^{2} x \left (e x +d \right ) c^{3}}{4}+\left (\frac {5 e^{3} x^{2} a}{4}-\frac {d x \left (-b x +a \right ) e^{2}}{4}-\frac {\left (-\frac {b x}{4}+a \right ) d^{2} e}{2}-\frac {b \,d^{3}}{8}\right ) c^{2}+e \left (\left (a^{2}-\frac {3}{8} b^{2} x^{2}+\frac {11}{8} a b x \right ) e^{2}+\frac {3 \left (\frac {b x}{3}+a \right ) b d e}{8}+\frac {b^{2} d^{2}}{4}\right ) c -\frac {e^{2} \left (\left (\frac {3 b x}{2}+a \right ) e +\frac {b d}{2}\right ) b^{2}}{4}\right ) \sqrt {\left (b e -2 c d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\right ) c}\, \sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\right )\right )}{\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\, \sqrt {\left (b e -2 c d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\right ) c}\, \sqrt {e x +d}\, \sqrt {\left (-b e +2 c d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\right ) c}\, \left (a c -\frac {b^{2}}{4}\right ) \left (a \,e^{2}-b d e +c \,d^{2}\right )^{2} \left (c \,x^{2}+b x +a \right )}\) | \(653\) |
| derivativedivides | \(2 e^{3} \left (-\frac {\frac {\frac {c \left (2 a c \,e^{2}-b^{2} e^{2}+2 b c d e -2 c^{2} d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}}{2 e^{2} \left (4 a c -b^{2}\right )}+\frac {\left (3 a b c \,e^{3}-6 a \,c^{2} d \,e^{2}-b^{3} e^{3}+3 b^{2} c d \,e^{2}-3 b \,c^{2} d^{2} e +2 c^{3} d^{3}\right ) \sqrt {e x +d}}{2 e^{2} \left (4 a c -b^{2}\right )}}{c \left (e x +d \right )^{2}+b e \left (e x +d \right )-2 c d \left (e x +d \right )+a \,e^{2}-b d e +c \,d^{2}}+\frac {2 c \left (\frac {\left (-16 a b c \,e^{3}+32 a \,c^{2} d \,e^{2}+3 b^{3} e^{3}-2 b^{2} c d \,e^{2}-12 b \,c^{2} d^{2} e +8 c^{3} d^{3}+10 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a c \,e^{2}-3 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} e^{2}+2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b c d e -2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c^{2} d^{2}\right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}-\frac {\left (16 a b c \,e^{3}-32 a \,c^{2} d \,e^{2}-3 b^{3} e^{3}+2 b^{2} c d \,e^{2}+12 b \,c^{2} d^{2} e -8 c^{3} d^{3}+10 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a c \,e^{2}-3 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} e^{2}+2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b c d e -2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c^{2} d^{2}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{e^{2} \left (4 a c -b^{2}\right )}}{\left (a \,e^{2}-b d e +c \,d^{2}\right )^{2}}-\frac {1}{\left (a \,e^{2}-b d e +c \,d^{2}\right )^{2} \sqrt {e x +d}}\right )\) | \(746\) |
| default | \(2 e^{3} \left (-\frac {\frac {\frac {c \left (2 a c \,e^{2}-b^{2} e^{2}+2 b c d e -2 c^{2} d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}}{2 e^{2} \left (4 a c -b^{2}\right )}+\frac {\left (3 a b c \,e^{3}-6 a \,c^{2} d \,e^{2}-b^{3} e^{3}+3 b^{2} c d \,e^{2}-3 b \,c^{2} d^{2} e +2 c^{3} d^{3}\right ) \sqrt {e x +d}}{2 e^{2} \left (4 a c -b^{2}\right )}}{c \left (e x +d \right )^{2}+b e \left (e x +d \right )-2 c d \left (e x +d \right )+a \,e^{2}-b d e +c \,d^{2}}+\frac {2 c \left (\frac {\left (-16 a b c \,e^{3}+32 a \,c^{2} d \,e^{2}+3 b^{3} e^{3}-2 b^{2} c d \,e^{2}-12 b \,c^{2} d^{2} e +8 c^{3} d^{3}+10 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a c \,e^{2}-3 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} e^{2}+2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b c d e -2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c^{2} d^{2}\right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}-\frac {\left (16 a b c \,e^{3}-32 a \,c^{2} d \,e^{2}-3 b^{3} e^{3}+2 b^{2} c d \,e^{2}+12 b \,c^{2} d^{2} e -8 c^{3} d^{3}+10 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a c \,e^{2}-3 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} e^{2}+2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b c d e -2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c^{2} d^{2}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{e^{2} \left (4 a c -b^{2}\right )}}{\left (a \,e^{2}-b d e +c \,d^{2}\right )^{2}}-\frac {1}{\left (a \,e^{2}-b d e +c \,d^{2}\right )^{2} \sqrt {e x +d}}\right )\) | \(746\) |
-2/(-4*(a*c-1/4*b^2)*e^2)^(1/2)/((b*e-2*c*d+(-4*(a*c-1/4*b^2)*e^2)^(1/2))* c)^(1/2)*(-5/8*(e*x+d)^(1/2)*((-1/5*c^2*d^2+e*(a*e+1/5*b*d)*c-3/10*b^2*e^2 )*(-4*(a*c-1/4*b^2)*e^2)^(1/2)+8/5*(b*e-2*c*d)*(1/4*c^2*d^2+e*(a*e-1/4*b*d )*c-3/16*b^2*e^2))*2^(1/2)*(c*x^2+b*x+a)*e*c*((b*e-2*c*d+(-4*(a*c-1/4*b^2) *e^2)^(1/2))*c)^(1/2)*arctanh(c*(e*x+d)^(1/2)*2^(1/2)/((-b*e+2*c*d+(-4*(a* c-1/4*b^2)*e^2)^(1/2))*c)^(1/2))+((-b*e+2*c*d+(-4*(a*c-1/4*b^2)*e^2)^(1/2) )*c)^(1/2)*(5/8*(e*x+d)^(1/2)*2^(1/2)*(c*x^2+b*x+a)*e*c*((-1/5*c^2*d^2+e*( a*e+1/5*b*d)*c-3/10*b^2*e^2)*(-4*(a*c-1/4*b^2)*e^2)^(1/2)-8/5*(b*e-2*c*d)* (1/4*c^2*d^2+e*(a*e-1/4*b*d)*c-3/16*b^2*e^2))*arctan(c*(e*x+d)^(1/2)*2^(1/ 2)/((b*e-2*c*d+(-4*(a*c-1/4*b^2)*e^2)^(1/2))*c)^(1/2))+(-1/4*d^2*x*(e*x+d) *c^3+(5/4*e^3*x^2*a-1/4*d*x*(-b*x+a)*e^2-1/2*(-1/4*b*x+a)*d^2*e-1/8*b*d^3) *c^2+e*((a^2-3/8*b^2*x^2+11/8*a*b*x)*e^2+3/8*(1/3*b*x+a)*b*d*e+1/4*b^2*d^2 )*c-1/4*e^2*((3/2*b*x+a)*e+1/2*b*d)*b^2)*((b*e-2*c*d+(-4*(a*c-1/4*b^2)*e^2 )^(1/2))*c)^(1/2)*(-4*(a*c-1/4*b^2)*e^2)^(1/2)))/(e*x+d)^(1/2)/((-b*e+2*c* d+(-4*(a*c-1/4*b^2)*e^2)^(1/2))*c)^(1/2)/(a*c-1/4*b^2)/(a*e^2-b*d*e+c*d^2) ^2/(c*x^2+b*x+a)
Leaf count of result is larger than twice the leaf count of optimal. 40675 vs. \(2 (545) = 1090\).
Time = 35.76 (sec) , antiderivative size = 40675, normalized size of antiderivative = 67.34 \[ \int \frac {1}{(d+e x)^{3/2} \left (a+b x+c x^2\right )^2} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {1}{(d+e x)^{3/2} \left (a+b x+c x^2\right )^2} \, dx=\text {Timed out} \]
\[ \int \frac {1}{(d+e x)^{3/2} \left (a+b x+c x^2\right )^2} \, dx=\int { \frac {1}{{\left (c x^{2} + b x + a\right )}^{2} {\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 5642 vs. \(2 (545) = 1090\).
Time = 1.39 (sec) , antiderivative size = 5642, normalized size of antiderivative = 9.34 \[ \int \frac {1}{(d+e x)^{3/2} \left (a+b x+c x^2\right )^2} \, dx=\text {Too large to display} \]
-(2*(e*x + d)^2*c^3*d^2*e - 2*(e*x + d)*c^3*d^3*e - 2*(e*x + d)^2*b*c^2*d* e^2 + 3*(e*x + d)*b*c^2*d^2*e^2 + 3*(e*x + d)^2*b^2*c*e^3 - 10*(e*x + d)^2 *a*c^2*e^3 - 7*(e*x + d)*b^2*c*d*e^3 + 22*(e*x + d)*a*c^2*d*e^3 + 2*b^2*c* d^2*e^3 - 8*a*c^2*d^2*e^3 + 3*(e*x + d)*b^3*e^4 - 11*(e*x + d)*a*b*c*e^4 - 2*b^3*d*e^4 + 8*a*b*c*d*e^4 + 2*a*b^2*e^5 - 8*a^2*c*e^5)/((b^2*c^2*d^4 - 4*a*c^3*d^4 - 2*b^3*c*d^3*e + 8*a*b*c^2*d^3*e + b^4*d^2*e^2 - 2*a*b^2*c*d^ 2*e^2 - 8*a^2*c^2*d^2*e^2 - 2*a*b^3*d*e^3 + 8*a^2*b*c*d*e^3 + a^2*b^2*e^4 - 4*a^3*c*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)) - 1/8*((b^2*c^2*d^4*e - 4*a*c^3*d^4*e - 2*b^3*c*d^3*e^2 + 8*a*b*c^2*d^3*e^ 2 + b^4*d^2*e^3 - 2*a*b^2*c*d^2*e^3 - 8*a^2*c^2*d^2*e^3 - 2*a*b^3*d*e^4 + 8*a^2*b*c*d*e^4 + a^2*b^2*e^5 - 4*a^3*c*e^5)^2*(2*c^2*d^2*e - 2*b*c*d*e^2 + (3*b^2 - 10*a*c)*e^3)*sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e) + 2*(2*sqrt(b^2 - 4*a*c)*c^5*d^7*e - 7*sqrt(b^2 - 4*a*c)*b*c^4*d^6*e^2 + 3* (b^2*c^3 + 10*a*c^4)*sqrt(b^2 - 4*a*c)*d^5*e^3 + 5*(2*b^3*c^2 - 15*a*b*c^3 )*sqrt(b^2 - 4*a*c)*d^4*e^4 - (11*b^4*c - 48*a*b^2*c^2 - 54*a^2*c^3)*sqrt( b^2 - 4*a*c)*d^3*e^5 + 3*(b^5 + a*b^3*c - 27*a^2*b*c^2)*sqrt(b^2 - 4*a*c)* d^2*e^6 - (6*a*b^4 - 21*a^2*b^2*c - 26*a^3*c^2)*sqrt(b^2 - 4*a*c)*d*e^7 + (3*a^2*b^3 - 13*a^3*b*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(b^2*c^2*d^4*e - 4*a*c^3*d^4*e - 2*b^3*c*d^3*...
Time = 28.92 (sec) , antiderivative size = 126405, normalized size of antiderivative = 209.28 \[ \int \frac {1}{(d+e x)^{3/2} \left (a+b x+c x^2\right )^2} \, dx=\text {Too large to display} \]
((e*(d + e*x)^2*(2*c^3*d^2 - 10*a*c^2*e^2 + 3*b^2*c*e^2 - 2*b*c^2*d*e))/(( 4*a*c - b^2)*(a*e^2 + c*d^2 - b*d*e)^2) - (2*e^3)/(a*e^2 + c*d^2 - b*d*e) + (e*(b*e - 2*c*d)*(d + e*x)*(3*b^2*e^2 + c^2*d^2 - 11*a*c*e^2 - b*c*d*e)) /((4*a*c - b^2)*(a*e^2 + c*d^2 - b*d*e)^2))/(c*(d + e*x)^(5/2) + (b*e - 2* c*d)*(d + e*x)^(3/2) + (d + e*x)^(1/2)*(a*e^2 + c*d^2 - b*d*e)) - atan(((( d + e*x)^(1/2)*(25600*a^12*c^9*e^20 + 18*a^6*b^12*c^3*e^20 - 408*a^7*b^10* c^4*e^20 + 3764*a^8*b^8*c^5*e^20 - 17920*a^9*b^6*c^6*e^20 + 45696*a^10*b^4 *c^7*e^20 - 57344*a^11*b^2*c^8*e^20 - 4096*a^3*c^18*d^18*e^2 - 56320*a^4*c ^17*d^16*e^4 - 327680*a^5*c^16*d^14*e^6 - 987136*a^6*c^15*d^12*e^8 - 16793 60*a^7*c^14*d^10*e^10 - 1632256*a^8*c^13*d^8*e^12 - 819200*a^9*c^12*d^6*e^ 14 - 102400*a^10*c^11*d^4*e^16 + 77824*a^11*c^10*d^2*e^18 + 64*b^6*c^15*d^ 18*e^2 - 576*b^7*c^14*d^17*e^3 + 2228*b^8*c^13*d^16*e^4 - 4768*b^9*c^12*d^ 15*e^5 + 5960*b^10*c^11*d^14*e^6 - 3976*b^11*c^10*d^13*e^7 + 578*b^12*c^9* d^12*e^8 + 1004*b^13*c^8*d^11*e^9 - 442*b^14*c^7*d^10*e^10 - 320*b^15*c^6* d^9*e^11 + 362*b^16*c^5*d^8*e^12 - 132*b^17*c^4*d^7*e^13 + 18*b^18*c^3*d^6 *e^14 + 3072*a^2*b^2*c^17*d^18*e^2 - 27648*a^2*b^3*c^16*d^17*e^3 + 96384*a ^2*b^4*c^15*d^16*e^4 - 144384*a^2*b^5*c^14*d^15*e^5 + 5120*a^2*b^6*c^13*d^ 14*e^6 + 297472*a^2*b^7*c^12*d^13*e^7 - 404368*a^2*b^8*c^11*d^12*e^8 + 136 544*a^2*b^9*c^10*d^11*e^9 + 165736*a^2*b^10*c^9*d^10*e^10 - 188920*a^2*b^1 1*c^8*d^9*e^11 + 58766*a^2*b^12*c^7*d^8*e^12 + 10936*a^2*b^13*c^6*d^7*e...