Integrand size = 35, antiderivative size = 116 \[ \int \frac {d+e x}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{c d}+\frac {\left (c d^2-a e^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} (d+e x)}{\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{c^{3/2} d^{3/2} \sqrt {e}} \] Output:
(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c/d+(-a*e^2+c*d^2)*arctanh(c^(1/2) *d^(1/2)*(e*x+d)/e^(1/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/c^(3/2)/ d^(3/2)/e^(1/2)
Time = 0.28 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.17 \[ \int \frac {d+e x}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {\sqrt {c} \sqrt {d} \sqrt {e} (a e+c d x) (d+e x)+\left (c d^2-a e^2\right ) \sqrt {a e+c d x} \sqrt {d+e x} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {e} \sqrt {a e+c d x}}\right )}{c^{3/2} d^{3/2} \sqrt {e} \sqrt {(a e+c d x) (d+e x)}} \] Input:
Integrate[(d + e*x)/Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2],x]
Output:
(Sqrt[c]*Sqrt[d]*Sqrt[e]*(a*e + c*d*x)*(d + e*x) + (c*d^2 - a*e^2)*Sqrt[a* e + c*d*x]*Sqrt[d + e*x]*ArcTanh[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/(Sqrt[e]* Sqrt[a*e + c*d*x])])/(c^(3/2)*d^(3/2)*Sqrt[e]*Sqrt[(a*e + c*d*x)*(d + e*x) ])
Time = 0.39 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.16, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {1160, 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.
\(\displaystyle \int \frac {d+e x}{\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \, dx\) |
\(\Big \downarrow \) 1160 |
\(\displaystyle \frac {\left (d^2-\frac {a e^2}{c}\right ) \int \frac {1}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{2 d}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c d}\) |
\(\Big \downarrow \) 1092 |
\(\displaystyle \frac {\left (d^2-\frac {a e^2}{c}\right ) \int \frac {1}{4 c d e-\frac {\left (c d^2+2 c e x d+a e^2\right )^2}{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}d\frac {c d^2+2 c e x d+a e^2}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}}{d}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c d}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {\left (d^2-\frac {a e^2}{c}\right ) \text {arctanh}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2 \sqrt {c} d^{3/2} \sqrt {e}}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c d}\) |
Input:
Int[(d + e*x)/Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2],x]
Output:
Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(c*d) + ((d^2 - (a*e^2)/c)*Arc Tanh[(c*d^2 + a*e^2 + 2*c*d*e*x)/(2*Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*d*e + ( c*d^2 + a*e^2)*x + c*d*e*x^2])])/(2*Sqrt[c]*d^(3/2)*Sqrt[e])
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_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol ] :> Simp[e*((a + b*x + c*x^2)^(p + 1)/(2*c*(p + 1))), x] + Simp[(2*c*d - b *e)/(2*c) Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[p, -1]
Time = 0.99 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.60
method | result | size |
default | \(\frac {d \ln \left (\frac {\frac {1}{2} a \,e^{2}+\frac {1}{2} c \,d^{2}+c d x e}{\sqrt {d e c}}+\sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}\right )}{\sqrt {d e c}}+e \left (\frac {\sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}{d e c}-\frac {\left (a \,e^{2}+c \,d^{2}\right ) \ln \left (\frac {\frac {1}{2} a \,e^{2}+\frac {1}{2} c \,d^{2}+c d x e}{\sqrt {d e c}}+\sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}\right )}{2 d e c \sqrt {d e c}}\right )\) | \(186\) |
Input:
int((e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2),x,method=_RETURNVERBOS E)
Output:
d*ln((1/2*a*e^2+1/2*c*d^2+c*d*x*e)/(d*e*c)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+c* d*x^2*e)^(1/2))/(d*e*c)^(1/2)+e*(1/d/e/c*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e) ^(1/2)-1/2*(a*e^2+c*d^2)/d/e/c*ln((1/2*a*e^2+1/2*c*d^2+c*d*x*e)/(d*e*c)^(1 /2)+(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2))/(d*e*c)^(1/2))
Time = 0.11 (sec) , antiderivative size = 338, normalized size of antiderivative = 2.91 \[ \int \frac {d+e x}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\left [\frac {4 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} c d e - {\left (c d^{2} - a e^{2}\right )} \sqrt {c d e} \log \left (8 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4} - 4 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt {c d e} + 8 \, {\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right )}{4 \, c^{2} d^{2} e}, \frac {2 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} c d e - {\left (c d^{2} - a e^{2}\right )} \sqrt {-c d e} \arctan \left (\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt {-c d e}}{2 \, {\left (c^{2} d^{2} e^{2} x^{2} + a c d^{2} e^{2} + {\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right )}}\right )}{2 \, c^{2} d^{2} e}\right ] \] Input:
integrate((e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x, algorithm="fr icas")
Output:
[1/4*(4*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*c*d*e - (c*d^2 - a*e^2 )*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*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) + 8*(c^2*d^3*e + a*c*d*e^3)*x))/(c^2*d^2*e), 1/2*(2*sqrt(c*d*e *x^2 + a*d*e + (c*d^2 + a*e^2)*x)*c*d*e - (c*d^2 - a*e^2)*sqrt(-c*d*e)*arc tan(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^2*x^2 + a*c*d^2*e^2 + (c^2*d^3*e + a*c*d*e^3 )*x)))/(c^2*d^2*e)]
Leaf count of result is larger than twice the leaf count of optimal. 335 vs. \(2 (107) = 214\).
Time = 0.68 (sec) , antiderivative size = 335, normalized size of antiderivative = 2.89 \[ \int \frac {d+e x}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\begin {cases} \left (d - \frac {a e^{2} + c d^{2}}{2 c d}\right ) \left (\begin {cases} \frac {\log {\left (a e^{2} + c d^{2} + 2 c d e x + 2 \sqrt {c d e} \sqrt {a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )} \right )}}{\sqrt {c d e}} & \text {for}\: a d e - \frac {\left (a e^{2} + c d^{2}\right )^{2}}{4 c d e} \neq 0 \\\frac {\left (x - \frac {- a e^{2} - c d^{2}}{2 c d e}\right ) \log {\left (x - \frac {- a e^{2} - c d^{2}}{2 c d e} \right )}}{\sqrt {c d e \left (x - \frac {- a e^{2} - c d^{2}}{2 c d e}\right )^{2}}} & \text {otherwise} \end {cases}\right ) + \frac {\sqrt {a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{c d} & \text {for}\: c d e \neq 0 \\\frac {2 d \sqrt {a d e + x \left (a e^{2} + c d^{2}\right )} + \frac {2 e \left (- a d e \sqrt {a d e + x \left (a e^{2} + c d^{2}\right )} + \frac {\left (a d e + x \left (a e^{2} + c d^{2}\right )\right )^{\frac {3}{2}}}{3}\right )}{a e^{2} + c d^{2}}}{a e^{2} + c d^{2}} & \text {for}\: a e^{2} + c d^{2} \neq 0 \\\frac {d x + \frac {e x^{2}}{2}}{\sqrt {a d e}} & \text {otherwise} \end {cases} \] Input:
integrate((e*x+d)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2),x)
Output:
Piecewise(((d - (a*e**2 + c*d**2)/(2*c*d))*Piecewise((log(a*e**2 + c*d**2 + 2*c*d*e*x + 2*sqrt(c*d*e)*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2)) )/sqrt(c*d*e), Ne(a*d*e - (a*e**2 + c*d**2)**2/(4*c*d*e), 0)), ((x - (-a*e **2 - c*d**2)/(2*c*d*e))*log(x - (-a*e**2 - c*d**2)/(2*c*d*e))/sqrt(c*d*e* (x - (-a*e**2 - c*d**2)/(2*c*d*e))**2), True)) + sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))/(c*d), Ne(c*d*e, 0)), ((2*d*sqrt(a*d*e + x*(a*e**2 + c*d**2)) + 2*e*(-a*d*e*sqrt(a*d*e + x*(a*e**2 + c*d**2)) + (a*d*e + x*(a* e**2 + c*d**2))**(3/2)/3)/(a*e**2 + c*d**2))/(a*e**2 + c*d**2), Ne(a*e**2 + c*d**2, 0)), ((d*x + e*x**2/2)/sqrt(a*d*e), True))
Exception generated. \[ \int \frac {d+e x}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x, algorithm="ma xima")
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(a*e^2-c*d^2>0)', see `assume?` f or more de
Time = 0.18 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.03 \[ \int \frac {d+e x}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=-\frac {{\left (c d^{2} - a e^{2}\right )} \log \left ({\left | -c d^{2} - a e^{2} - 2 \, \sqrt {c d e} {\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )} \right |}\right )}{2 \, \sqrt {c d e} c d} + \frac {\sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}}{c d} \] Input:
integrate((e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x, algorithm="gi ac")
Output:
-1/2*(c*d^2 - a*e^2)*log(abs(-c*d^2 - a*e^2 - 2*sqrt(c*d*e)*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))))/(sqrt(c*d*e)*c*d) + sqrt (c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e)/(c*d)
Time = 5.90 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.24 \[ \int \frac {d+e x}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{c\,d}-\frac {a\,e^3\,\ln \left (2\,\sqrt {\left (a\,e+c\,d\,x\right )\,\left (d+e\,x\right )}\,\sqrt {c\,d\,e}+a\,e^2+c\,d^2+2\,c\,d\,e\,x\right )}{2\,{\left (c\,d\,e\right )}^{3/2}}+\frac {c\,d^2\,e\,\ln \left (2\,\sqrt {\left (a\,e+c\,d\,x\right )\,\left (d+e\,x\right )}\,\sqrt {c\,d\,e}+a\,e^2+c\,d^2+2\,c\,d\,e\,x\right )}{2\,{\left (c\,d\,e\right )}^{3/2}} \] Input:
int((d + e*x)/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2),x)
Output:
(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)/(c*d) - (a*e^3*log(2*((a*e + c*d*x)*(d + e*x))^(1/2)*(c*d*e)^(1/2) + a*e^2 + c*d^2 + 2*c*d*e*x))/(2*(c *d*e)^(3/2)) + (c*d^2*e*log(2*((a*e + c*d*x)*(d + e*x))^(1/2)*(c*d*e)^(1/2 ) + a*e^2 + c*d^2 + 2*c*d*e*x))/(2*(c*d*e)^(3/2))
Time = 0.24 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.16 \[ \int \frac {d+e x}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {\sqrt {e x +d}\, \sqrt {c d x +a e}\, c d e -\sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) a \,e^{2}+\sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) c \,d^{2}}{c^{2} d^{2} e} \] Input:
int((e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x)
Output:
(sqrt(d + e*x)*sqrt(a*e + c*d*x)*c*d*e - 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))*a*e**2 + 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))*c*d**2)/(c**2*d**2*e)