Integrand size = 39, antiderivative size = 129 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^{5/2}} \, dx=-\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{e (d+e x)^{3/2}}+\frac {c d \arctan \left (\frac {\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d^2-a e^2} \sqrt {d+e x}}\right )}{e^{3/2} \sqrt {c d^2-a e^2}} \] Output:
-(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/e/(e*x+d)^(3/2)+c*d*arctan(e^(1/2 )*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(-a*e^2+c*d^2)^(1/2)/(e*x+d)^(1/ 2))/e^(3/2)/(-a*e^2+c*d^2)^(1/2)
Time = 0.25 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.87 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^{5/2}} \, dx=\frac {\sqrt {(a e+c d x) (d+e x)} \left (-\sqrt {e}+\frac {c d (d+e x) \arctan \left (\frac {\sqrt {e} \sqrt {a e+c d x}}{\sqrt {c d^2-a e^2}}\right )}{\sqrt {c d^2-a e^2} \sqrt {a e+c d x}}\right )}{e^{3/2} (d+e x)^{3/2}} \] Input:
Integrate[Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(d + e*x)^(5/2),x]
Output:
(Sqrt[(a*e + c*d*x)*(d + e*x)]*(-Sqrt[e] + (c*d*(d + e*x)*ArcTan[(Sqrt[e]* Sqrt[a*e + c*d*x])/Sqrt[c*d^2 - a*e^2]])/(Sqrt[c*d^2 - a*e^2]*Sqrt[a*e + c *d*x])))/(e^(3/2)*(d + e*x)^(3/2))
Time = 0.43 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {1130, 1136, 218}
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.
\(\displaystyle \int \frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{(d+e x)^{5/2}} \, dx\) |
\(\Big \downarrow \) 1130 |
\(\displaystyle \frac {c d \int \frac {1}{\sqrt {d+e x} \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{2 e}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e (d+e x)^{3/2}}\) |
\(\Big \downarrow \) 1136 |
\(\displaystyle c d \int \frac {1}{\frac {\left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right ) e^2}{d+e x}+\left (c d^2-a e^2\right ) e}d\frac {\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{\sqrt {d+e x}}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e (d+e x)^{3/2}}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {c d \arctan \left (\frac {\sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} \sqrt {c d^2-a e^2}}\right )}{e^{3/2} \sqrt {c d^2-a e^2}}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e (d+e x)^{3/2}}\) |
Input:
Int[Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(d + e*x)^(5/2),x]
Output:
-(Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(e*(d + e*x)^(3/2))) + (c*d* ArcTan[(Sqrt[e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(Sqrt[c*d^2 - a*e^2]*Sqrt[d + e*x])])/(e^(3/2)*Sqrt[c*d^2 - a*e^2])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(d + e*x)^(m + 1)*((a + b*x + c*x^2)^p/(e*(m + p + 1))), x] - Simp[c*(p/(e^2*(m + p + 1))) Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && GtQ[p, 0] && (LtQ[m, -2] || EqQ[m + 2*p + 1, 0]) && NeQ[m + p + 1, 0] & & IntegerQ[2*p]
Int[1/(Sqrt[(d_.) + (e_.)*(x_)]*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x _Symbol] :> Simp[2*e Subst[Int[1/(2*c*d - b*e + e^2*x^2), x], x, Sqrt[a + b*x + c*x^2]/Sqrt[d + e*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0]
Time = 1.08 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.19
method | result | size |
default | \(\frac {\left (-\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}}\right ) c d e x -\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}}\right ) c \,d^{2}-\sqrt {c d x +a e}\, \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\right ) \sqrt {\left (e x +d \right ) \left (c d x +a e \right )}}{\left (e x +d \right )^{\frac {3}{2}} \sqrt {c d x +a e}\, e \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}}\) | \(153\) |
Input:
int((a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2)/(e*x+d)^(5/2),x,method=_RETURN VERBOSE)
Output:
(-arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*c*d*e*x-arctanh(e*( c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*c*d^2-(c*d*x+a*e)^(1/2)*((a*e^2- c*d^2)*e)^(1/2))*((e*x+d)*(c*d*x+a*e))^(1/2)/(e*x+d)^(3/2)/(c*d*x+a*e)^(1/ 2)/e/((a*e^2-c*d^2)*e)^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 230 vs. \(2 (113) = 226\).
Time = 0.10 (sec) , antiderivative size = 482, normalized size of antiderivative = 3.74 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^{5/2}} \, dx=\left [-\frac {{\left (c d e^{2} x^{2} + 2 \, c d^{2} e x + c d^{3}\right )} \sqrt {-c d^{2} e + a e^{3}} \log \left (-\frac {c d e^{2} x^{2} + 2 \, a e^{3} x - c d^{3} + 2 \, a d e^{2} - 2 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {-c d^{2} e + a e^{3}} \sqrt {e x + d}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) + 2 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (c d^{2} e - a e^{3}\right )} \sqrt {e x + d}}{2 \, {\left (c d^{4} e^{2} - a d^{2} e^{4} + {\left (c d^{2} e^{4} - a e^{6}\right )} x^{2} + 2 \, {\left (c d^{3} e^{3} - a d e^{5}\right )} x\right )}}, -\frac {{\left (c d e^{2} x^{2} + 2 \, c d^{2} e x + c d^{3}\right )} \sqrt {c d^{2} e - a e^{3}} \arctan \left (-\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {c d^{2} e - a e^{3}} \sqrt {e x + d}}{c d^{3} - a d e^{2} + {\left (c d^{2} e - a e^{3}\right )} x}\right ) + \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (c d^{2} e - a e^{3}\right )} \sqrt {e x + d}}{c d^{4} e^{2} - a d^{2} e^{4} + {\left (c d^{2} e^{4} - a e^{6}\right )} x^{2} + 2 \, {\left (c d^{3} e^{3} - a d e^{5}\right )} x}\right ] \] Input:
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d)^(5/2),x, algorit hm="fricas")
Output:
[-1/2*((c*d*e^2*x^2 + 2*c*d^2*e*x + c*d^3)*sqrt(-c*d^2*e + a*e^3)*log(-(c* d*e^2*x^2 + 2*a*e^3*x - c*d^3 + 2*a*d*e^2 - 2*sqrt(c*d*e*x^2 + a*d*e + (c* d^2 + a*e^2)*x)*sqrt(-c*d^2*e + a*e^3)*sqrt(e*x + d))/(e^2*x^2 + 2*d*e*x + d^2)) + 2*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(c*d^2*e - a*e^3)*s qrt(e*x + d))/(c*d^4*e^2 - a*d^2*e^4 + (c*d^2*e^4 - a*e^6)*x^2 + 2*(c*d^3* e^3 - a*d*e^5)*x), -((c*d*e^2*x^2 + 2*c*d^2*e*x + c*d^3)*sqrt(c*d^2*e - a* e^3)*arctan(-sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(c*d^2*e - a* e^3)*sqrt(e*x + d)/(c*d^3 - a*d*e^2 + (c*d^2*e - a*e^3)*x)) + sqrt(c*d*e*x ^2 + a*d*e + (c*d^2 + a*e^2)*x)*(c*d^2*e - a*e^3)*sqrt(e*x + d))/(c*d^4*e^ 2 - a*d^2*e^4 + (c*d^2*e^4 - a*e^6)*x^2 + 2*(c*d^3*e^3 - a*d*e^5)*x)]
\[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^{5/2}} \, dx=\int \frac {\sqrt {\left (d + e x\right ) \left (a e + c d x\right )}}{\left (d + e x\right )^{\frac {5}{2}}}\, dx \] Input:
integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2)/(e*x+d)**(5/2),x)
Output:
Integral(sqrt((d + e*x)*(a*e + c*d*x))/(d + e*x)**(5/2), x)
\[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^{5/2}} \, dx=\int { \frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{{\left (e x + d\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d)^(5/2),x, algorit hm="maxima")
Output:
integrate(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)/(e*x + d)^(5/2), x)
Time = 0.16 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.86 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^{5/2}} \, dx=\frac {c d {\left (\frac {\arctan \left (\frac {\sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}}}{\sqrt {c d^{2} e - a e^{3}}}\right )}{\sqrt {c d^{2} e - a e^{3}} e} - \frac {\sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}}}{{\left (e x + d\right )} c d e^{2}}\right )} {\left | e \right |}}{e} \] Input:
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d)^(5/2),x, algorit hm="giac")
Output:
c*d*(arctan(sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)/sqrt(c*d^2*e - a*e^3)) /(sqrt(c*d^2*e - a*e^3)*e) - sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)/((e*x + d)*c*d*e^2))*abs(e)/e
Timed out. \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^{5/2}} \, dx=\int \frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{{\left (d+e\,x\right )}^{5/2}} \,d x \] Input:
int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)/(d + e*x)^(5/2),x)
Output:
int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)/(d + e*x)^(5/2), x)
Time = 0.24 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.30 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^{5/2}} \, dx=\frac {-\sqrt {e}\, \sqrt {-a \,e^{2}+c \,d^{2}}\, \mathit {atan} \left (\frac {\sqrt {c d x +a e}\, e}{\sqrt {e}\, \sqrt {-a \,e^{2}+c \,d^{2}}}\right ) c \,d^{2}-\sqrt {e}\, \sqrt {-a \,e^{2}+c \,d^{2}}\, \mathit {atan} \left (\frac {\sqrt {c d x +a e}\, e}{\sqrt {e}\, \sqrt {-a \,e^{2}+c \,d^{2}}}\right ) c d e x -\sqrt {c d x +a e}\, a \,e^{3}+\sqrt {c d x +a e}\, c \,d^{2} e}{e^{2} \left (a \,e^{3} x -c \,d^{2} e x +a d \,e^{2}-c \,d^{3}\right )} \] Input:
int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d)^(5/2),x)
Output:
( - sqrt(e)*sqrt( - a*e**2 + c*d**2)*atan((sqrt(a*e + c*d*x)*e)/(sqrt(e)*s qrt( - a*e**2 + c*d**2)))*c*d**2 - sqrt(e)*sqrt( - a*e**2 + c*d**2)*atan(( sqrt(a*e + c*d*x)*e)/(sqrt(e)*sqrt( - a*e**2 + c*d**2)))*c*d*e*x - sqrt(a* e + c*d*x)*a*e**3 + sqrt(a*e + c*d*x)*c*d**2*e)/(e**2*(a*d*e**2 + a*e**3*x - c*d**3 - c*d**2*e*x))