Integrand size = 46, antiderivative size = 179 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2} (f+g x)} \, dx=-\frac {2 (c d f-a e g) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{g^2 \sqrt {d+e x}}+\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{3 g (d+e x)^{3/2}}+\frac {2 (c d f-a e g)^{3/2} \arctan \left (\frac {\sqrt {g} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d f-a e g} \sqrt {d+e x}}\right )}{g^{5/2}} \]
2/3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/g/(e*x+d)^(3/2)+2*(-a*e*g+c*d* f)^(3/2)*arctan(g^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(-a*e*g+c* d*f)^(1/2)/(e*x+d)^(1/2))/g^(5/2)-2*(-a*e*g+c*d*f)*(a*d*e+(a*e^2+c*d^2)*x+ c*d*e*x^2)^(1/2)/g^2/(e*x+d)^(1/2)
Time = 0.19 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.74 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2} (f+g x)} \, dx=\frac {2 \sqrt {a e+c d x} \sqrt {d+e x} \left (\sqrt {g} \sqrt {a e+c d x} (4 a e g+c d (-3 f+g x))+3 (c d f-a e g)^{3/2} \arctan \left (\frac {\sqrt {g} \sqrt {a e+c d x}}{\sqrt {c d f-a e g}}\right )\right )}{3 g^{5/2} \sqrt {(a e+c d x) (d+e x)}} \]
(2*Sqrt[a*e + c*d*x]*Sqrt[d + e*x]*(Sqrt[g]*Sqrt[a*e + c*d*x]*(4*a*e*g + c *d*(-3*f + g*x)) + 3*(c*d*f - a*e*g)^(3/2)*ArcTan[(Sqrt[g]*Sqrt[a*e + c*d* x])/Sqrt[c*d*f - a*e*g]]))/(3*g^(5/2)*Sqrt[(a*e + c*d*x)*(d + e*x)])
Time = 0.48 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.03, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {1250, 1250, 1255, 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.
| \(\displaystyle \int \frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{(d+e x)^{3/2} (f+g x)} \, dx\) |
| \(\Big \downarrow \) 1250 |
| \(\displaystyle \frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 g (d+e x)^{3/2}}-\frac {(c d f-a e g) \int \frac {\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{\sqrt {d+e x} (f+g x)}dx}{g}\) |
| \(\Big \downarrow \) 1250 |
| \(\displaystyle \frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 g (d+e x)^{3/2}}-\frac {(c d f-a e g) \left (\frac {2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{g \sqrt {d+e x}}-\frac {(c d f-a e g) \int \frac {\sqrt {d+e x}}{(f+g x) \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{g}\right )}{g}\) |
| \(\Big \downarrow \) 1255 |
| \(\displaystyle \frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 g (d+e x)^{3/2}}-\frac {(c d f-a e g) \left (\frac {2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{g \sqrt {d+e x}}-\frac {2 e^2 (c d f-a e g) \int \frac {1}{(c d f-a e g) e^2+\frac {g \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right ) e^2}{d+e x}}d\frac {\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{\sqrt {d+e x}}}{g}\right )}{g}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 g (d+e x)^{3/2}}-\frac {(c d f-a e g) \left (\frac {2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{g \sqrt {d+e x}}-\frac {2 \sqrt {c d f-a e g} \arctan \left (\frac {\sqrt {g} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} \sqrt {c d f-a e g}}\right )}{g^{3/2}}\right )}{g}\) |
(2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(3*g*(d + e*x)^(3/2)) - ((c*d*f - a*e*g)*((2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(g*Sqrt[ d + e*x]) - (2*Sqrt[c*d*f - a*e*g]*ArcTan[(Sqrt[g]*Sqrt[a*d*e + (c*d^2 + a *e^2)*x + c*d*e*x^2])/(Sqrt[c*d*f - a*e*g]*Sqrt[d + e*x])])/g^(3/2)))/g
3.7.94.3.1 Defintions of rubi rules used
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_)*((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[Sqrt[(d_) + (e_.)*(x_)]/(((f_.) + (g_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Simp[2*e^2 Subst[Int[1/(c*(e*f + d*g) - b*e *g + e^2*g*x^2), x], x, Sqrt[a + b*x + c*x^2]/Sqrt[d + e*x]], x] /; FreeQ[{ a, b, c, d, e, f, g}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0]
Time = 0.56 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.41
| method | result | size |
| default | \(-\frac {2 \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (3 \,\operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) a^{2} e^{2} g^{2}-6 \,\operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) a c d e f g +3 \,\operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{2} d^{2} f^{2}-\sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, c d g x -4 \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, a e g +3 \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, c d f \right )}{3 \sqrt {e x +d}\, \sqrt {c d x +a e}\, g^{2} \sqrt {\left (a e g -c d f \right ) g}}\) | \(253\) |
-2/3*((c*d*x+a*e)*(e*x+d))^(1/2)*(3*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c* d*f)*g)^(1/2))*a^2*e^2*g^2-6*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g) ^(1/2))*a*c*d*e*f*g+3*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2)) *c^2*d^2*f^2-(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)*c*d*g*x-4*(c*d*x+a* e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)*a*e*g+3*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)* g)^(1/2)*c*d*f)/(e*x+d)^(1/2)/(c*d*x+a*e)^(1/2)/g^2/((a*e*g-c*d*f)*g)^(1/2 )
Time = 0.35 (sec) , antiderivative size = 408, normalized size of antiderivative = 2.28 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2} (f+g x)} \, dx=\left [-\frac {3 \, {\left (c d^{2} f - a d e g + {\left (c d e f - a e^{2} g\right )} x\right )} \sqrt {-\frac {c d f - a e g}{g}} \log \left (-\frac {c d e g x^{2} - c d^{2} f + 2 \, a d e g - 2 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d} g \sqrt {-\frac {c d f - a e g}{g}} - {\left (c d e f - {\left (c d^{2} + 2 \, a e^{2}\right )} g\right )} x}{e g x^{2} + d f + {\left (e f + d g\right )} x}\right ) - 2 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (c d g x - 3 \, c d f + 4 \, a e g\right )} \sqrt {e x + d}}{3 \, {\left (e g^{2} x + d g^{2}\right )}}, -\frac {2 \, {\left (3 \, {\left (c d^{2} f - a d e g + {\left (c d e f - a e^{2} g\right )} x\right )} \sqrt {\frac {c d f - a e g}{g}} \arctan \left (\frac {\sqrt {e x + d} \sqrt {\frac {c d f - a e g}{g}}}{\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}\right ) - \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (c d g x - 3 \, c d f + 4 \, a e g\right )} \sqrt {e x + d}\right )}}{3 \, {\left (e g^{2} x + d g^{2}\right )}}\right ] \]
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d)^(3/2)/(g*x+f),x, algorithm="fricas")
[-1/3*(3*(c*d^2*f - a*d*e*g + (c*d*e*f - a*e^2*g)*x)*sqrt(-(c*d*f - a*e*g) /g)*log(-(c*d*e*g*x^2 - c*d^2*f + 2*a*d*e*g - 2*sqrt(c*d*e*x^2 + a*d*e + ( c*d^2 + a*e^2)*x)*sqrt(e*x + d)*g*sqrt(-(c*d*f - a*e*g)/g) - (c*d*e*f - (c *d^2 + 2*a*e^2)*g)*x)/(e*g*x^2 + d*f + (e*f + d*g)*x)) - 2*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(c*d*g*x - 3*c*d*f + 4*a*e*g)*sqrt(e*x + d))/ (e*g^2*x + d*g^2), -2/3*(3*(c*d^2*f - a*d*e*g + (c*d*e*f - a*e^2*g)*x)*sqr t((c*d*f - a*e*g)/g)*arctan(sqrt(e*x + d)*sqrt((c*d*f - a*e*g)/g)/sqrt(c*d *e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)) - sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a *e^2)*x)*(c*d*g*x - 3*c*d*f + 4*a*e*g)*sqrt(e*x + d))/(e*g^2*x + d*g^2)]
\[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2} (f+g x)} \, dx=\int \frac {\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {3}{2}}}{\left (d + e x\right )^{\frac {3}{2}} \left (f + g x\right )}\, dx \]
\[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2} (f+g x)} \, dx=\int { \frac {{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {3}{2}}}{{\left (e x + d\right )}^{\frac {3}{2}} {\left (g x + f\right )}} \,d x } \]
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d)^(3/2)/(g*x+f),x, algorithm="maxima")
Leaf count of result is larger than twice the leaf count of optimal. 506 vs. \(2 (157) = 314\).
Time = 0.47 (sec) , antiderivative size = 506, normalized size of antiderivative = 2.83 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2} (f+g x)} \, dx=\frac {2 \, {\left (c^{2} d^{2} f^{2} {\left | e \right |} - 2 \, a c d e f g {\left | e \right |} + a^{2} e^{2} g^{2} {\left | e \right |}\right )} \arctan \left (\frac {\sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} g}{\sqrt {c d f g - a e g^{2}} e}\right )}{\sqrt {c d f g - a e g^{2}} e g^{2}} - \frac {2 \, {\left (3 \, c^{2} d^{2} e^{2} f^{2} {\left | e \right |} \arctan \left (\frac {\sqrt {-c d^{2} e + a e^{3}} g}{\sqrt {c d f g - a e g^{2}} e}\right ) - 6 \, a c d e^{3} f g {\left | e \right |} \arctan \left (\frac {\sqrt {-c d^{2} e + a e^{3}} g}{\sqrt {c d f g - a e g^{2}} e}\right ) + 3 \, a^{2} e^{4} g^{2} {\left | e \right |} \arctan \left (\frac {\sqrt {-c d^{2} e + a e^{3}} g}{\sqrt {c d f g - a e g^{2}} e}\right ) - 3 \, \sqrt {-c d^{2} e + a e^{3}} \sqrt {c d f g - a e g^{2}} c d e f {\left | e \right |} - \sqrt {-c d^{2} e + a e^{3}} \sqrt {c d f g - a e g^{2}} c d^{2} g {\left | e \right |} + 4 \, \sqrt {-c d^{2} e + a e^{3}} \sqrt {c d f g - a e g^{2}} a e^{2} g {\left | e \right |}\right )}}{3 \, \sqrt {c d f g - a e g^{2}} e^{3} g^{2}} - \frac {2 \, {\left (3 \, \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} c d e^{10} f g {\left | e \right |} - 3 \, \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} a e^{11} g^{2} {\left | e \right |} - {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} e^{8} g^{2} {\left | e \right |}\right )}}{3 \, e^{12} g^{3}} \]
2*(c^2*d^2*f^2*abs(e) - 2*a*c*d*e*f*g*abs(e) + a^2*e^2*g^2*abs(e))*arctan( sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)*g/(sqrt(c*d*f*g - a*e*g^2)*e))/(sq rt(c*d*f*g - a*e*g^2)*e*g^2) - 2/3*(3*c^2*d^2*e^2*f^2*abs(e)*arctan(sqrt(- c*d^2*e + a*e^3)*g/(sqrt(c*d*f*g - a*e*g^2)*e)) - 6*a*c*d*e^3*f*g*abs(e)*a rctan(sqrt(-c*d^2*e + a*e^3)*g/(sqrt(c*d*f*g - a*e*g^2)*e)) + 3*a^2*e^4*g^ 2*abs(e)*arctan(sqrt(-c*d^2*e + a*e^3)*g/(sqrt(c*d*f*g - a*e*g^2)*e)) - 3* sqrt(-c*d^2*e + a*e^3)*sqrt(c*d*f*g - a*e*g^2)*c*d*e*f*abs(e) - sqrt(-c*d^ 2*e + a*e^3)*sqrt(c*d*f*g - a*e*g^2)*c*d^2*g*abs(e) + 4*sqrt(-c*d^2*e + a* e^3)*sqrt(c*d*f*g - a*e*g^2)*a*e^2*g*abs(e))/(sqrt(c*d*f*g - a*e*g^2)*e^3* g^2) - 2/3*(3*sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)*c*d*e^10*f*g*abs(e) - 3*sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)*a*e^11*g^2*abs(e) - ((e*x + d) *c*d*e - c*d^2*e + a*e^3)^(3/2)*e^8*g^2*abs(e))/(e^12*g^3)
Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2} (f+g x)} \, dx=\int \frac {{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{3/2}}{\left (f+g\,x\right )\,{\left (d+e\,x\right )}^{3/2}} \,d x \]