Integrand size = 48, antiderivative size = 167 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x} \sqrt {f+g x}} \, dx=\frac {\sqrt {f+g x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{g \sqrt {d+e x}}-\frac {(c d f-a e g) \sqrt {a e+c d x} \sqrt {d+e x} \text {arctanh}\left (\frac {\sqrt {g} \sqrt {a e+c d x}}{\sqrt {c} \sqrt {d} \sqrt {f+g x}}\right )}{\sqrt {c} \sqrt {d} g^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \]
-(-a*e*g+c*d*f)*arctanh(g^(1/2)*(c*d*x+a*e)^(1/2)/c^(1/2)/d^(1/2)/(g*x+f)^ (1/2))*(c*d*x+a*e)^(1/2)*(e*x+d)^(1/2)/g^(3/2)/c^(1/2)/d^(1/2)/(a*d*e+(a*e ^2+c*d^2)*x+c*d*e*x^2)^(1/2)+(g*x+f)^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^ 2)^(1/2)/g/(e*x+d)^(1/2)
Time = 0.26 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.86 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x} \sqrt {f+g x}} \, dx=\frac {\sqrt {(a e+c d x) (d+e x)} \sqrt {f+g x}}{g \sqrt {d+e x}}+\frac {(-c d f+a e g) \sqrt {(a e+c d x) (d+e x)} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {f+g x}}{\sqrt {g} \sqrt {a e+c d x}}\right )}{\sqrt {c} \sqrt {d} g^{3/2} \sqrt {a e+c d x} \sqrt {d+e x}} \]
(Sqrt[(a*e + c*d*x)*(d + e*x)]*Sqrt[f + g*x])/(g*Sqrt[d + e*x]) + ((-(c*d* f) + a*e*g)*Sqrt[(a*e + c*d*x)*(d + e*x)]*ArcTanh[(Sqrt[c]*Sqrt[d]*Sqrt[f + g*x])/(Sqrt[g]*Sqrt[a*e + c*d*x])])/(Sqrt[c]*Sqrt[d]*g^(3/2)*Sqrt[a*e + c*d*x]*Sqrt[d + e*x])
Time = 0.36 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {1250, 1268, 66, 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.
| \(\displaystyle \int \frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} \sqrt {f+g x}} \, dx\) |
| \(\Big \downarrow \) 1250 |
| \(\displaystyle \frac {\sqrt {f+g x} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{g \sqrt {d+e x}}-\frac {(c d f-a e g) \int \frac {\sqrt {d+e x}}{\sqrt {f+g x} \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{2 g}\) |
| \(\Big \downarrow \) 1268 |
| \(\displaystyle \frac {\sqrt {f+g x} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{g \sqrt {d+e x}}-\frac {\sqrt {d+e x} \sqrt {a e+c d x} (c d f-a e g) \int \frac {1}{\sqrt {a e+c d x} \sqrt {f+g x}}dx}{2 g \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\) |
| \(\Big \downarrow \) 66 |
| \(\displaystyle \frac {\sqrt {f+g x} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{g \sqrt {d+e x}}-\frac {\sqrt {d+e x} \sqrt {a e+c d x} (c d f-a e g) \int \frac {1}{c d-\frac {g (a e+c d x)}{f+g x}}d\frac {\sqrt {a e+c d x}}{\sqrt {f+g x}}}{g \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\sqrt {f+g x} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{g \sqrt {d+e x}}-\frac {\sqrt {d+e x} \sqrt {a e+c d x} (c d f-a e g) \text {arctanh}\left (\frac {\sqrt {g} \sqrt {a e+c d x}}{\sqrt {c} \sqrt {d} \sqrt {f+g x}}\right )}{\sqrt {c} \sqrt {d} g^{3/2} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\) |
(Sqrt[f + g*x]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(g*Sqrt[d + e* x]) - ((c*d*f - a*e*g)*Sqrt[a*e + c*d*x]*Sqrt[d + e*x]*ArcTanh[(Sqrt[g]*Sq rt[a*e + c*d*x])/(Sqrt[c]*Sqrt[d]*Sqrt[f + g*x])])/(Sqrt[c]*Sqrt[d]*g^(3/2 )*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])
3.8.36.3.1 Defintions of rubi rules used
Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ 2 Subst[Int[1/(b - d*x^2), x], x, Sqrt[a + b*x]/Sqrt[c + d*x]], x] /; Fre eQ[{a, b, c, d}, x] && !GtQ[c - a*(d/b), 0]
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_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(d + e*x)^m)*(f + g*x)^(n + 1)*(( a + b*x + c*x^2)^p/(g*(m - n - 1))), x] - Simp[m*((c*e*f + c*d*g - b*e*g)/( e^2*g*(m - n - 1))) Int[(d + e*x)^(m + 1)*(f + g*x)^n*(a + b*x + c*x^2)^( p - 1), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[m + p, 0] && GtQ[p, 0] && NeQ[m - n - 1, 0] && !IGtQ[n, 0] && !(IntegerQ[n + p] && LtQ[n + p + 2, 0]) && RationalQ[n]
Int[((d_) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x_ ) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a + b*x + c*x^2)^FracPart[p]/((d + e*x)^FracPart[p]*(a/d + (c*x)/e)^FracPart[p]) Int[(d + e*x)^(m + p)*(f + g*x)^n*(a/d + (c/e)*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x ] && EqQ[c*d^2 - b*d*e + a*e^2, 0]
Time = 0.55 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.13
| method | result | size |
| default | \(\frac {\sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \sqrt {g x +f}\, \left (\ln \left (\frac {2 c d g x +a e g +c d f +2 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {c d g}}{2 \sqrt {c d g}}\right ) a e g -\ln \left (\frac {2 c d g x +a e g +c d f +2 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {c d g}}{2 \sqrt {c d g}}\right ) c d f +2 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {c d g}\right )}{2 \sqrt {e x +d}\, \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, g \sqrt {c d g}}\) | \(188\) |
int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(g*x+f)^(1/2)/(e*x+d)^(1/2),x, method=_RETURNVERBOSE)
1/2*((c*d*x+a*e)*(e*x+d))^(1/2)*(g*x+f)^(1/2)/(e*x+d)^(1/2)*(ln(1/2*(2*c*d *g*x+a*e*g+c*d*f+2*((g*x+f)*(c*d*x+a*e))^(1/2)*(c*d*g)^(1/2))/(c*d*g)^(1/2 ))*a*e*g-ln(1/2*(2*c*d*g*x+a*e*g+c*d*f+2*((g*x+f)*(c*d*x+a*e))^(1/2)*(c*d* g)^(1/2))/(c*d*g)^(1/2))*c*d*f+2*((g*x+f)*(c*d*x+a*e))^(1/2)*(c*d*g)^(1/2) )/((g*x+f)*(c*d*x+a*e))^(1/2)/g/(c*d*g)^(1/2)
Time = 0.76 (sec) , antiderivative size = 516, normalized size of antiderivative = 3.09 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x} \sqrt {f+g x}} \, dx=\left [\frac {4 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d} \sqrt {g x + f} c d g - {\left (c d^{2} f - a d e g + {\left (c d e f - a e^{2} g\right )} x\right )} \sqrt {c d g} \log \left (-\frac {8 \, c^{2} d^{2} e g^{2} x^{3} + c^{2} d^{3} f^{2} + 6 \, a c d^{2} e f g + a^{2} d e^{2} g^{2} + 4 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d g x + c d f + a e g\right )} \sqrt {c d g} \sqrt {e x + d} \sqrt {g x + f} + 8 \, {\left (c^{2} d^{2} e f g + {\left (c^{2} d^{3} + a c d e^{2}\right )} g^{2}\right )} x^{2} + {\left (c^{2} d^{2} e f^{2} + 2 \, {\left (4 \, c^{2} d^{3} + 3 \, a c d e^{2}\right )} f g + {\left (8 \, a c d^{2} e + a^{2} e^{3}\right )} g^{2}\right )} x}{e x + d}\right )}{4 \, {\left (c d e g^{2} x + c d^{2} g^{2}\right )}}, \frac {2 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d} \sqrt {g x + f} c d g + {\left (c d^{2} f - a d e g + {\left (c d e f - a e^{2} g\right )} x\right )} \sqrt {-c d g} \arctan \left (\frac {2 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {-c d g} \sqrt {e x + d} \sqrt {g x + f}}{2 \, c d e g x^{2} + c d^{2} f + a d e g + {\left (c d e f + {\left (2 \, c d^{2} + a e^{2}\right )} g\right )} x}\right )}{2 \, {\left (c d e g^{2} x + c d^{2} g^{2}\right )}}\right ] \]
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(g*x+f)^(1/2)/(e*x+d)^(1 /2),x, algorithm="fricas")
[1/4*(4*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d)*sqrt(g*x + f)*c*d*g - (c*d^2*f - a*d*e*g + (c*d*e*f - a*e^2*g)*x)*sqrt(c*d*g)*log( -(8*c^2*d^2*e*g^2*x^3 + c^2*d^3*f^2 + 6*a*c*d^2*e*f*g + a^2*d*e^2*g^2 + 4* sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(2*c*d*g*x + c*d*f + a*e*g)*sq rt(c*d*g)*sqrt(e*x + d)*sqrt(g*x + f) + 8*(c^2*d^2*e*f*g + (c^2*d^3 + a*c* d*e^2)*g^2)*x^2 + (c^2*d^2*e*f^2 + 2*(4*c^2*d^3 + 3*a*c*d*e^2)*f*g + (8*a* c*d^2*e + a^2*e^3)*g^2)*x)/(e*x + d)))/(c*d*e*g^2*x + c*d^2*g^2), 1/2*(2*s qrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d)*sqrt(g*x + f)*c*d *g + (c*d^2*f - a*d*e*g + (c*d*e*f - a*e^2*g)*x)*sqrt(-c*d*g)*arctan(2*sqr t(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(-c*d*g)*sqrt(e*x + d)*sqrt(g *x + f)/(2*c*d*e*g*x^2 + c*d^2*f + a*d*e*g + (c*d*e*f + (2*c*d^2 + a*e^2)* g)*x)))/(c*d*e*g^2*x + c*d^2*g^2)]
\[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x} \sqrt {f+g x}} \, dx=\int \frac {\sqrt {\left (d + e x\right ) \left (a e + c d x\right )}}{\sqrt {d + e x} \sqrt {f + g x}}\, dx \]
\[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x} \sqrt {f+g x}} \, dx=\int { \frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{\sqrt {e x + d} \sqrt {g x + f}} \,d x } \]
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(g*x+f)^(1/2)/(e*x+d)^(1 /2),x, algorithm="maxima")
Leaf count of result is larger than twice the leaf count of optimal. 409 vs. \(2 (139) = 278\).
Time = 0.46 (sec) , antiderivative size = 409, normalized size of antiderivative = 2.45 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x} \sqrt {f+g x}} \, dx=\frac {\frac {{\left (\frac {{\left (c d e f - a e^{2} g\right )} \log \left ({\left | -\sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} \sqrt {c d g} + \sqrt {c^{2} d^{2} e^{2} f - a c d e^{3} g + {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )} c d g} \right |}\right )}{\sqrt {c d g} g} + \frac {\sqrt {c^{2} d^{2} e^{2} f - a c d e^{3} g + {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )} c d g} \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}}}{c d e g}\right )} {\left | c \right |} {\left | d \right |}}{c d} - \frac {c^{2} d^{2} e^{2} f {\left | c \right |} {\left | d \right |} \log \left ({\left | -\sqrt {-c d^{2} e + a e^{3}} \sqrt {c d g} + \sqrt {c^{2} d^{2} e^{2} f - c^{2} d^{3} e g} \right |}\right ) - a c d e^{3} g {\left | c \right |} {\left | d \right |} \log \left ({\left | -\sqrt {-c d^{2} e + a e^{3}} \sqrt {c d g} + \sqrt {c^{2} d^{2} e^{2} f - c^{2} d^{3} e g} \right |}\right ) + \sqrt {c^{2} d^{2} e^{2} f - c^{2} d^{3} e g} \sqrt {-c d^{2} e + a e^{3}} \sqrt {c d g} {\left | c \right |} {\left | d \right |}}{\sqrt {c d g} c^{2} d^{2} e g}}{e} \]
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(g*x+f)^(1/2)/(e*x+d)^(1 /2),x, algorithm="giac")
(((c*d*e*f - a*e^2*g)*log(abs(-sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)*sqr t(c*d*g) + sqrt(c^2*d^2*e^2*f - a*c*d*e^3*g + ((e*x + d)*c*d*e - c*d^2*e + a*e^3)*c*d*g)))/(sqrt(c*d*g)*g) + sqrt(c^2*d^2*e^2*f - a*c*d*e^3*g + ((e* x + d)*c*d*e - c*d^2*e + a*e^3)*c*d*g)*sqrt((e*x + d)*c*d*e - c*d^2*e + a* e^3)/(c*d*e*g))*abs(c)*abs(d)/(c*d) - (c^2*d^2*e^2*f*abs(c)*abs(d)*log(abs (-sqrt(-c*d^2*e + a*e^3)*sqrt(c*d*g) + sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g))) - a*c*d*e^3*g*abs(c)*abs(d)*log(abs(-sqrt(-c*d^2*e + a*e^3)*sqrt(c*d*g) + sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g))) + sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g)*s qrt(-c*d^2*e + a*e^3)*sqrt(c*d*g)*abs(c)*abs(d))/(sqrt(c*d*g)*c^2*d^2*e*g) )/e
Timed out. \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x} \sqrt {f+g x}} \, dx=\int \frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{\sqrt {f+g\,x}\,\sqrt {d+e\,x}} \,d x \]