Integrand size = 37, antiderivative size = 112 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^2} \, dx=-\frac {2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{e (d+e x)}+\frac {2 \sqrt {c} \sqrt {d} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c} \sqrt {d} (d+e x)}\right )}{e^{3/2}} \] Output:
-2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/e/(e*x+d)+2*c^(1/2)*d^(1/2)*arc tanh(e^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c^(1/2)/d^(1/2)/(e*x+ d))/e^(3/2)
Time = 0.16 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.98 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^2} \, dx=\frac {2 \sqrt {(a e+c d x) (d+e x)} \left (-\frac {\sqrt {e}}{d+e x}+\frac {\sqrt {c} \sqrt {d} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {e} \sqrt {a e+c d x}}\right )}{\sqrt {a e+c d x} \sqrt {d+e x}}\right )}{e^{3/2}} \] Input:
Integrate[Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(d + e*x)^2,x]
Output:
(2*Sqrt[(a*e + c*d*x)*(d + e*x)]*(-(Sqrt[e]/(d + e*x)) + (Sqrt[c]*Sqrt[d]* ArcTanh[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/(Sqrt[e]*Sqrt[a*e + c*d*x])])/(Sqr t[a*e + c*d*x]*Sqrt[d + e*x])))/e^(3/2)
Time = 0.40 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.11, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.135, Rules used = {1125, 25, 27, 1092, 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.
\(\displaystyle \int \frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{(d+e x)^2} \, dx\) |
\(\Big \downarrow \) 1125 |
\(\displaystyle -\frac {\int -\frac {c d e}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{e^2}-\frac {2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e (d+e x)}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\int \frac {c d e}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{e^2}-\frac {2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e (d+e x)}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {c d \int \frac {1}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{e}-\frac {2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e (d+e x)}\) |
\(\Big \downarrow \) 1092 |
\(\displaystyle \frac {2 c d \int \frac {1}{4 c d e-\frac {\left (c d^2+2 c e x d+a e^2\right )^2}{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}d\frac {c d^2+2 c e x d+a e^2}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}}{e}-\frac {2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e (d+e x)}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {\sqrt {c} \sqrt {d} \text {arctanh}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{e^{3/2}}-\frac {2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e (d+e x)}\) |
Input:
Int[Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(d + e*x)^2,x]
Output:
(-2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(e*(d + e*x)) + (Sqrt[c]* Sqrt[d]*ArcTanh[(c*d^2 + a*e^2 + 2*c*d*e*x)/(2*Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqr t[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])])/e^(3/2)
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/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2 Subst[I nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a , b, c}, x]
Int[((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_ Symbol] :> Simp[-2*e^(2*m + 3)*(Sqrt[a + b*x + c*x^2]/((-2*c*d + b*e)^(m + 2)*(d + e*x))), x] - Simp[e^(2*m + 2) Int[(1/Sqrt[a + b*x + c*x^2])*Expan dToSum[((-2*c*d + b*e)^(-m - 1) - ((-c)*d + b*e + c*e*x)^(-m - 1))/(d + e*x ), x], x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && ILtQ[m, 0] && EqQ[m + p, -3/2]
Leaf count of result is larger than twice the leaf count of optimal. \(211\) vs. \(2(96)=192\).
Time = 1.87 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.89
method | result | size |
default | \(\frac {-\frac {2 \left (d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{2}}+\frac {2 d e c \left (\sqrt {d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}+\frac {\left (a \,e^{2}-c \,d^{2}\right ) \ln \left (\frac {\frac {a \,e^{2}}{2}-\frac {c \,d^{2}}{2}+d e c \left (x +\frac {d}{e}\right )}{\sqrt {d e c}}+\sqrt {d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}\right )}{2 \sqrt {d e c}}\right )}{a \,e^{2}-c \,d^{2}}}{e^{2}}\) | \(212\) |
Input:
int((a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2)/(e*x+d)^2,x,method=_RETURNVERB OSE)
Output:
1/e^2*(-2/(a*e^2-c*d^2)/(x+d/e)^2*(d*e*c*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^ (3/2)+2*d*e*c/(a*e^2-c*d^2)*((d*e*c*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2) +1/2*(a*e^2-c*d^2)*ln((1/2*a*e^2-1/2*c*d^2+d*e*c*(x+d/e))/(d*e*c)^(1/2)+(d *e*c*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2))/(d*e*c)^(1/2)))
Time = 0.13 (sec) , antiderivative size = 326, normalized size of antiderivative = 2.91 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^2} \, dx=\left [\frac {{\left (e x + d\right )} \sqrt {\frac {c d}{e}} \log \left (8 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4} + 4 \, {\left (2 \, c d e^{2} x + c d^{2} e + a e^{3}\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {\frac {c d}{e}} + 8 \, {\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right ) - 4 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{2 \, {\left (e^{2} x + d e\right )}}, -\frac {{\left (e x + d\right )} \sqrt {-\frac {c d}{e}} \arctan \left (\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt {-\frac {c d}{e}}}{2 \, {\left (c^{2} d^{2} e x^{2} + a c d^{2} e + {\left (c^{2} d^{3} + a c d e^{2}\right )} x\right )}}\right ) + 2 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{e^{2} x + d e}\right ] \] Input:
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d)^2,x, algorithm=" fricas")
Output:
[1/2*((e*x + d)*sqrt(c*d/e)*log(8*c^2*d^2*e^2*x^2 + c^2*d^4 + 6*a*c*d^2*e^ 2 + a^2*e^4 + 4*(2*c*d*e^2*x + c*d^2*e + a*e^3)*sqrt(c*d*e*x^2 + a*d*e + ( c*d^2 + a*e^2)*x)*sqrt(c*d/e) + 8*(c^2*d^3*e + a*c*d*e^3)*x) - 4*sqrt(c*d* e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(e^2*x + d*e), -((e*x + d)*sqrt(-c*d/e )*arctan(1/2*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(2*c*d*e*x + c*d^ 2 + a*e^2)*sqrt(-c*d/e)/(c^2*d^2*e*x^2 + a*c*d^2*e + (c^2*d^3 + a*c*d*e^2) *x)) + 2*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(e^2*x + d*e)]
\[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^2} \, dx=\int \frac {\sqrt {\left (d + e x\right ) \left (a e + c d x\right )}}{\left (d + e x\right )^{2}}\, dx \] Input:
integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2)/(e*x+d)**2,x)
Output:
Integral(sqrt((d + e*x)*(a*e + c*d*x))/(d + e*x)**2, x)
Exception generated. \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^2} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d)^2,x, algorithm=" maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e*(a*e^2-c*d^2)>0)', see `assume ?` for mor
Time = 0.20 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.65 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^2} \, dx=-2 \, {\left (\frac {c d \arctan \left (\frac {\sqrt {c d e - \frac {c d^{2} e}{e x + d} + \frac {a e^{3}}{e x + d}}}{\sqrt {-c d e}}\right ) \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )}{\sqrt {-c d e} e {\left | e \right |}} + \frac {\sqrt {c d e - \frac {c d^{2} e}{e x + d} + \frac {a e^{3}}{e x + d}} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )}{e^{2} {\left | e \right |}} - \frac {{\left (c d e \arctan \left (\frac {\sqrt {c d e}}{\sqrt {-c d e}}\right ) + \sqrt {c d e} \sqrt {-c d e}\right )} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )}{\sqrt {-c d e} e^{2} {\left | e \right |}}\right )} {\left | e \right |} \] Input:
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d)^2,x, algorithm=" giac")
Output:
-2*(c*d*arctan(sqrt(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))/sqrt(-c*d *e))*sgn(1/(e*x + d))*sgn(e)/(sqrt(-c*d*e)*e*abs(e)) + sqrt(c*d*e - c*d^2* e/(e*x + d) + a*e^3/(e*x + d))*sgn(1/(e*x + d))*sgn(e)/(e^2*abs(e)) - (c*d *e*arctan(sqrt(c*d*e)/sqrt(-c*d*e)) + sqrt(c*d*e)*sqrt(-c*d*e))*sgn(1/(e*x + d))*sgn(e)/(sqrt(-c*d*e)*e^2*abs(e)))*abs(e)
Timed out. \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^2} \, dx=\int \frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{{\left (d+e\,x\right )}^2} \,d x \] Input:
int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)/(d + e*x)^2,x)
Output:
int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)/(d + e*x)^2, x)
Time = 0.24 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.33 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^2} \, dx=\frac {-2 \sqrt {e x +d}\, \sqrt {c d x +a e}\, e +2 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) d +2 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) e x -2 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, d -2 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, e x}{e^{2} \left (e x +d \right )} \] Input:
int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d)^2,x)
Output:
(2*( - sqrt(d + e*x)*sqrt(a*e + c*d*x)*e + sqrt(e)*sqrt(d)*sqrt(c)*log((sq rt(e)*sqrt(a*e + c*d*x) + sqrt(d)*sqrt(c)*sqrt(d + e*x))/sqrt(a*e**2 - c*d **2))*d + sqrt(e)*sqrt(d)*sqrt(c)*log((sqrt(e)*sqrt(a*e + c*d*x) + sqrt(d) *sqrt(c)*sqrt(d + e*x))/sqrt(a*e**2 - c*d**2))*e*x - sqrt(e)*sqrt(d)*sqrt( c)*d - sqrt(e)*sqrt(d)*sqrt(c)*e*x))/(e**2*(d + e*x))