Integrand size = 28, antiderivative size = 166 \[ \int \frac {b+2 c x}{(d+e x) \left (a+b x+c x^2\right )^{3/2}} \, dx=-\frac {2 \left (\left (b^2-4 a c\right ) (c d-b e)-c \left (b^2-4 a c\right ) e x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \sqrt {a+b x+c x^2}}-\frac {e (2 c d-b e) \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{\left (c d^2-b d e+a e^2\right )^{3/2}} \]
-e*(-b*e+2*c*d)*arctanh(1/2*(b*d-2*a*e+(-b*e+2*c*d)*x)/(a*e^2-b*d*e+c*d^2) ^(1/2)/(c*x^2+b*x+a)^(1/2))/(a*e^2-b*d*e+c*d^2)^(3/2)-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)^ (1/2)
Time = 1.14 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.80 \[ \int \frac {b+2 c x}{(d+e x) \left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {2 \left (\frac {-c d+b e+c e x}{\sqrt {a+x (b+c x)}}+\frac {e (2 c d-b e) \arctan \left (\frac {\sqrt {-c d^2+e (b d-a e)} x}{\sqrt {a} (d+e x)-d \sqrt {a+x (b+c x)}}\right )}{\sqrt {-c d^2+e (b d-a e)}}\right )}{c d^2+e (-b d+a e)} \]
(2*((-(c*d) + b*e + c*e*x)/Sqrt[a + x*(b + c*x)] + (e*(2*c*d - b*e)*ArcTan [(Sqrt[-(c*d^2) + e*(b*d - a*e)]*x)/(Sqrt[a]*(d + e*x) - d*Sqrt[a + x*(b + c*x)])])/Sqrt[-(c*d^2) + e*(b*d - a*e)]))/(c*d^2 + e*(-(b*d) + a*e))
Time = 0.34 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {1235, 27, 1154, 219}
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.
| \(\displaystyle \int \frac {b+2 c x}{(d+e x) \left (a+b x+c x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1235 |
| \(\displaystyle -\frac {2 \int \frac {\left (b^2-4 a c\right ) e (2 c d-b e)}{2 (d+e x) \sqrt {c x^2+b x+a}}dx}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (\left (b^2-4 a c\right ) (c d-b e)-c e x \left (b^2-4 a c\right )\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {e (2 c d-b e) \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{a e^2-b d e+c d^2}-\frac {2 \left (\left (b^2-4 a c\right ) (c d-b e)-c e x \left (b^2-4 a c\right )\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {2 e (2 c d-b e) \int \frac {1}{4 \left (c d^2-b e d+a e^2\right )-\frac {(b d-2 a e+(2 c d-b e) x)^2}{c x^2+b x+a}}d\left (-\frac {b d-2 a e+(2 c d-b e) x}{\sqrt {c x^2+b x+a}}\right )}{a e^2-b d e+c d^2}-\frac {2 \left (\left (b^2-4 a c\right ) (c d-b e)-c e x \left (b^2-4 a c\right )\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {e (2 c d-b e) \text {arctanh}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{\left (a e^2-b d e+c d^2\right )^{3/2}}-\frac {2 \left (\left (b^2-4 a c\right ) (c d-b e)-c e x \left (b^2-4 a c\right )\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}\) |
(-2*((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[a + b*x + c*x^2]) - (e*(2*c*d - b*e)*ArcTanh[(b*d - 2*a*e + (2*c*d - b*e)*x)/(2*Sqrt[c*d^2 - b*d*e + a*e^2]*Sqrt[a + b*x + c*x^2])])/(c*d^2 - b*d*e + a*e^2)^(3/2)
3.16.86.3.1 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[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym bol] :> Simp[-2 Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c , d, e}, x]
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] )
Leaf count of result is larger than twice the leaf count of optimal. \(444\) vs. \(2(156)=312\).
Time = 0.71 (sec) , antiderivative size = 445, normalized size of antiderivative = 2.68
| method | result | size |
| default | \(\frac {4 c \left (2 c x +b \right )}{e \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}+\frac {\left (b e -2 c d \right ) \left (\frac {e^{2}}{\left (e^{2} a -b d e +c \,d^{2}\right ) \sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}}-\frac {\left (b e -2 c d \right ) e \left (2 c \left (x +\frac {d}{e}\right )+\frac {b e -2 c d}{e}\right )}{\left (e^{2} a -b d e +c \,d^{2}\right ) \left (\frac {4 c \left (e^{2} a -b d e +c \,d^{2}\right )}{e^{2}}-\frac {\left (b e -2 c d \right )^{2}}{e^{2}}\right ) \sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}}-\frac {e^{2} \ln \left (\frac {\frac {2 e^{2} a -2 b d e +2 c \,d^{2}}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{\left (e^{2} a -b d e +c \,d^{2}\right ) \sqrt {\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}}\right )}{e^{2}}\) | \(445\) |
4*c/e*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)+(b*e-2*c*d)/e^2*(1/(a*e^2- b*d*e+c*d^2)*e^2/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^ 2)^(1/2)-(b*e-2*c*d)*e/(a*e^2-b*d*e+c*d^2)*(2*c*(x+d/e)+(b*e-2*c*d)/e)/(4* c*(a*e^2-b*d*e+c*d^2)/e^2-(b*e-2*c*d)^2/e^2)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x +d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)-1/(a*e^2-b*d*e+c*d^2)*e^2/((a*e^2-b*d *e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(x+d/e)+2 *((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2 -b*d*e+c*d^2)/e^2)^(1/2))/(x+d/e)))
Leaf count of result is larger than twice the leaf count of optimal. 438 vs. \(2 (156) = 312\).
Time = 0.84 (sec) , antiderivative size = 918, normalized size of antiderivative = 5.53 \[ \int \frac {b+2 c x}{(d+e x) \left (a+b x+c x^2\right )^{3/2}} \, dx=\left [-\frac {{\left (2 \, a c d e - a b e^{2} + {\left (2 \, c^{2} d e - b c e^{2}\right )} x^{2} + {\left (2 \, b c d e - b^{2} e^{2}\right )} x\right )} \sqrt {c d^{2} - b d e + a e^{2}} \log \left (\frac {8 \, a b d e - 8 \, a^{2} e^{2} - {\left (b^{2} + 4 \, a c\right )} d^{2} - {\left (8 \, c^{2} d^{2} - 8 \, b c d e + {\left (b^{2} + 4 \, a c\right )} e^{2}\right )} x^{2} - 4 \, \sqrt {c d^{2} - b d e + a e^{2}} \sqrt {c x^{2} + b x + a} {\left (b d - 2 \, a e + {\left (2 \, c d - b e\right )} x\right )} - 2 \, {\left (4 \, b c d^{2} + 4 \, a b e^{2} - {\left (3 \, b^{2} + 4 \, a c\right )} d e\right )} x}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) + 4 \, {\left (c^{2} d^{3} - 2 \, b c d^{2} e - a b e^{3} + {\left (b^{2} + a c\right )} d e^{2} - {\left (c^{2} d^{2} e - b c d e^{2} + a c e^{3}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{2 \, {\left (a c^{2} d^{4} - 2 \, a b c d^{3} e - 2 \, a^{2} b d e^{3} + a^{3} e^{4} + {\left (a b^{2} + 2 \, a^{2} c\right )} d^{2} e^{2} + {\left (c^{3} d^{4} - 2 \, b c^{2} d^{3} e - 2 \, a b c d e^{3} + a^{2} c e^{4} + {\left (b^{2} c + 2 \, a c^{2}\right )} d^{2} e^{2}\right )} x^{2} + {\left (b c^{2} d^{4} - 2 \, b^{2} c d^{3} e - 2 \, a b^{2} d e^{3} + a^{2} b e^{4} + {\left (b^{3} + 2 \, a b c\right )} d^{2} e^{2}\right )} x\right )}}, -\frac {{\left (2 \, a c d e - a b e^{2} + {\left (2 \, c^{2} d e - b c e^{2}\right )} x^{2} + {\left (2 \, b c d e - b^{2} e^{2}\right )} x\right )} \sqrt {-c d^{2} + b d e - a e^{2}} \arctan \left (-\frac {\sqrt {-c d^{2} + b d e - a e^{2}} \sqrt {c x^{2} + b x + a} {\left (b d - 2 \, a e + {\left (2 \, c d - b e\right )} x\right )}}{2 \, {\left (a c d^{2} - a b d e + a^{2} e^{2} + {\left (c^{2} d^{2} - b c d e + a c e^{2}\right )} x^{2} + {\left (b c d^{2} - b^{2} d e + a b e^{2}\right )} x\right )}}\right ) + 2 \, {\left (c^{2} d^{3} - 2 \, b c d^{2} e - a b e^{3} + {\left (b^{2} + a c\right )} d e^{2} - {\left (c^{2} d^{2} e - b c d e^{2} + a c e^{3}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{a c^{2} d^{4} - 2 \, a b c d^{3} e - 2 \, a^{2} b d e^{3} + a^{3} e^{4} + {\left (a b^{2} + 2 \, a^{2} c\right )} d^{2} e^{2} + {\left (c^{3} d^{4} - 2 \, b c^{2} d^{3} e - 2 \, a b c d e^{3} + a^{2} c e^{4} + {\left (b^{2} c + 2 \, a c^{2}\right )} d^{2} e^{2}\right )} x^{2} + {\left (b c^{2} d^{4} - 2 \, b^{2} c d^{3} e - 2 \, a b^{2} d e^{3} + a^{2} b e^{4} + {\left (b^{3} + 2 \, a b c\right )} d^{2} e^{2}\right )} x}\right ] \]
[-1/2*((2*a*c*d*e - a*b*e^2 + (2*c^2*d*e - b*c*e^2)*x^2 + (2*b*c*d*e - b^2 *e^2)*x)*sqrt(c*d^2 - b*d*e + a*e^2)*log((8*a*b*d*e - 8*a^2*e^2 - (b^2 + 4 *a*c)*d^2 - (8*c^2*d^2 - 8*b*c*d*e + (b^2 + 4*a*c)*e^2)*x^2 - 4*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(c*x^2 + b*x + a)*(b*d - 2*a*e + (2*c*d - b*e)*x) - 2*(4*b*c*d^2 + 4*a*b*e^2 - (3*b^2 + 4*a*c)*d*e)*x)/(e^2*x^2 + 2*d*e*x + d^ 2)) + 4*(c^2*d^3 - 2*b*c*d^2*e - a*b*e^3 + (b^2 + a*c)*d*e^2 - (c^2*d^2*e - b*c*d*e^2 + a*c*e^3)*x)*sqrt(c*x^2 + b*x + a))/(a*c^2*d^4 - 2*a*b*c*d^3* e - 2*a^2*b*d*e^3 + a^3*e^4 + (a*b^2 + 2*a^2*c)*d^2*e^2 + (c^3*d^4 - 2*b*c ^2*d^3*e - 2*a*b*c*d*e^3 + a^2*c*e^4 + (b^2*c + 2*a*c^2)*d^2*e^2)*x^2 + (b *c^2*d^4 - 2*b^2*c*d^3*e - 2*a*b^2*d*e^3 + a^2*b*e^4 + (b^3 + 2*a*b*c)*d^2 *e^2)*x), -((2*a*c*d*e - a*b*e^2 + (2*c^2*d*e - b*c*e^2)*x^2 + (2*b*c*d*e - b^2*e^2)*x)*sqrt(-c*d^2 + b*d*e - a*e^2)*arctan(-1/2*sqrt(-c*d^2 + b*d*e - a*e^2)*sqrt(c*x^2 + b*x + a)*(b*d - 2*a*e + (2*c*d - b*e)*x)/(a*c*d^2 - a*b*d*e + a^2*e^2 + (c^2*d^2 - b*c*d*e + a*c*e^2)*x^2 + (b*c*d^2 - b^2*d* e + a*b*e^2)*x)) + 2*(c^2*d^3 - 2*b*c*d^2*e - a*b*e^3 + (b^2 + a*c)*d*e^2 - (c^2*d^2*e - b*c*d*e^2 + a*c*e^3)*x)*sqrt(c*x^2 + b*x + a))/(a*c^2*d^4 - 2*a*b*c*d^3*e - 2*a^2*b*d*e^3 + a^3*e^4 + (a*b^2 + 2*a^2*c)*d^2*e^2 + (c^ 3*d^4 - 2*b*c^2*d^3*e - 2*a*b*c*d*e^3 + a^2*c*e^4 + (b^2*c + 2*a*c^2)*d^2* e^2)*x^2 + (b*c^2*d^4 - 2*b^2*c*d^3*e - 2*a*b^2*d*e^3 + a^2*b*e^4 + (b^3 + 2*a*b*c)*d^2*e^2)*x)]
\[ \int \frac {b+2 c x}{(d+e x) \left (a+b x+c x^2\right )^{3/2}} \, dx=\int \frac {b + 2 c x}{\left (d + e x\right ) \left (a + b x + c x^{2}\right )^{\frac {3}{2}}}\, dx \]
Exception generated. \[ \int \frac {b+2 c x}{(d+e x) \left (a+b x+c x^2\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume((b/e-(2*c*d)/e^2)^2>0)', see `as sume?` for
Leaf count of result is larger than twice the leaf count of optimal. 515 vs. \(2 (156) = 312\).
Time = 0.28 (sec) , antiderivative size = 515, normalized size of antiderivative = 3.10 \[ \int \frac {b+2 c x}{(d+e x) \left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {2 \, {\left (\frac {{\left (b^{2} c^{2} d^{2} e - 4 \, a c^{3} d^{2} e - b^{3} c d e^{2} + 4 \, a b c^{2} d e^{2} + a b^{2} c e^{3} - 4 \, a^{2} c^{2} e^{3}\right )} x}{b^{2} c^{2} d^{4} - 4 \, a c^{3} d^{4} - 2 \, b^{3} c d^{3} e + 8 \, a b c^{2} d^{3} e + b^{4} d^{2} e^{2} - 2 \, a b^{2} c d^{2} e^{2} - 8 \, a^{2} c^{2} d^{2} e^{2} - 2 \, a b^{3} d e^{3} + 8 \, a^{2} b c d e^{3} + a^{2} b^{2} e^{4} - 4 \, a^{3} c e^{4}} - \frac {b^{2} c^{2} d^{3} - 4 \, a c^{3} d^{3} - 2 \, b^{3} c d^{2} e + 8 \, a b c^{2} d^{2} e + b^{4} d e^{2} - 3 \, a b^{2} c d e^{2} - 4 \, a^{2} c^{2} d e^{2} - a b^{3} e^{3} + 4 \, a^{2} b c e^{3}}{b^{2} c^{2} d^{4} - 4 \, a c^{3} d^{4} - 2 \, b^{3} c d^{3} e + 8 \, a b c^{2} d^{3} e + b^{4} d^{2} e^{2} - 2 \, a b^{2} c d^{2} e^{2} - 8 \, a^{2} c^{2} d^{2} e^{2} - 2 \, a b^{3} d e^{3} + 8 \, a^{2} b c d e^{3} + a^{2} b^{2} e^{4} - 4 \, a^{3} c e^{4}}\right )}}{\sqrt {c x^{2} + b x + a}} - \frac {2 \, {\left (2 \, c d e - b e^{2}\right )} \arctan \left (-\frac {{\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} e + \sqrt {c} d}{\sqrt {-c d^{2} + b d e - a e^{2}}}\right )}{{\left (c d^{2} - b d e + a e^{2}\right )} \sqrt {-c d^{2} + b d e - a e^{2}}} \]
2*((b^2*c^2*d^2*e - 4*a*c^3*d^2*e - b^3*c*d*e^2 + 4*a*b*c^2*d*e^2 + a*b^2* c*e^3 - 4*a^2*c^2*e^3)*x/(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) - (b^2*c^2*d^3 - 4* a*c^3*d^3 - 2*b^3*c*d^2*e + 8*a*b*c^2*d^2*e + b^4*d*e^2 - 3*a*b^2*c*d*e^2 - 4*a^2*c^2*d*e^2 - a*b^3*e^3 + 4*a^2*b*c*e^3)/(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))/sqrt(c*x^2 + b*x + a) - 2*(2*c*d*e - b*e^2)*arctan(-((sqrt(c)*x - sqrt (c*x^2 + b*x + a))*e + sqrt(c)*d)/sqrt(-c*d^2 + b*d*e - a*e^2))/((c*d^2 - b*d*e + a*e^2)*sqrt(-c*d^2 + b*d*e - a*e^2))
Timed out. \[ \int \frac {b+2 c x}{(d+e x) \left (a+b x+c x^2\right )^{3/2}} \, dx=\int \frac {b+2\,c\,x}{\left (d+e\,x\right )\,{\left (c\,x^2+b\,x+a\right )}^{3/2}} \,d x \]