Integrand size = 46, antiderivative size = 274 \[ \int \frac {(d+e x)^{3/2}}{(f+g x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=-\frac {2 \sqrt {d+e x}}{(c d f-a e g) (f+g x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {5 g \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{2 (c d f-a e g)^2 \sqrt {d+e x} (f+g x)^2}-\frac {15 c d g \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 (c d f-a e g)^3 \sqrt {d+e x} (f+g x)}-\frac {15 c^2 d^2 \sqrt {g} \arctan \left (\frac {\sqrt {g} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d f-a e g} \sqrt {d+e x}}\right )}{4 (c d f-a e g)^{7/2}} \]
-15/4*c^2*d^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^(1/2)/(-a*e*g+c*d*f)^(7/2)-2*(e*x+d)^(1/2 )/(-a*e*g+c*d*f)/(g*x+f)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-5/2*g*( a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(-a*e*g+c*d*f)^2/(g*x+f)^2/(e*x+d)^ (1/2)-15/4*c*d*g*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(-a*e*g+c*d*f)^3/ (g*x+f)/(e*x+d)^(1/2)
Time = 0.53 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.68 \[ \int \frac {(d+e x)^{3/2}}{(f+g x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=-\frac {\sqrt {d+e x} \left (\sqrt {c d f-a e g} \left (-2 a^2 e^2 g^2+a c d e g (9 f+5 g x)+c^2 d^2 \left (8 f^2+25 f g x+15 g^2 x^2\right )\right )+15 c^2 d^2 \sqrt {g} \sqrt {a e+c d x} (f+g x)^2 \arctan \left (\frac {\sqrt {g} \sqrt {a e+c d x}}{\sqrt {c d f-a e g}}\right )\right )}{4 (c d f-a e g)^{7/2} \sqrt {(a e+c d x) (d+e x)} (f+g x)^2} \]
-1/4*(Sqrt[d + e*x]*(Sqrt[c*d*f - a*e*g]*(-2*a^2*e^2*g^2 + a*c*d*e*g*(9*f + 5*g*x) + c^2*d^2*(8*f^2 + 25*f*g*x + 15*g^2*x^2)) + 15*c^2*d^2*Sqrt[g]*S qrt[a*e + c*d*x]*(f + g*x)^2*ArcTan[(Sqrt[g]*Sqrt[a*e + c*d*x])/Sqrt[c*d*f - a*e*g]]))/((c*d*f - a*e*g)^(7/2)*Sqrt[(a*e + c*d*x)*(d + e*x)]*(f + g*x )^2)
Time = 0.60 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.08, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.109, Rules used = {1252, 1254, 1254, 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.
| \(\displaystyle \int \frac {(d+e x)^{3/2}}{(f+g x)^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1252 |
| \(\displaystyle -\frac {5 g \int \frac {\sqrt {d+e x}}{(f+g x)^3 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{c d f-a e g}-\frac {2 \sqrt {d+e x}}{(f+g x)^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)}\) |
| \(\Big \downarrow \) 1254 |
| \(\displaystyle -\frac {5 g \left (\frac {3 c d \int \frac {\sqrt {d+e x}}{(f+g x)^2 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{4 (c d f-a e g)}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 \sqrt {d+e x} (f+g x)^2 (c d f-a e g)}\right )}{c d f-a e g}-\frac {2 \sqrt {d+e x}}{(f+g x)^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)}\) |
| \(\Big \downarrow \) 1254 |
| \(\displaystyle -\frac {5 g \left (\frac {3 c d \left (\frac {c d \int \frac {\sqrt {d+e x}}{(f+g x) \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{2 (c d f-a e g)}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} (f+g x) (c d f-a e g)}\right )}{4 (c d f-a e g)}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 \sqrt {d+e x} (f+g x)^2 (c d f-a e g)}\right )}{c d f-a e g}-\frac {2 \sqrt {d+e x}}{(f+g x)^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)}\) |
| \(\Big \downarrow \) 1255 |
| \(\displaystyle -\frac {5 g \left (\frac {3 c d \left (\frac {c d e^2 \int \frac {1}{(c d f-a e g) e^2+\frac {g \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right ) e^2}{d+e x}}d\frac {\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{\sqrt {d+e x}}}{c d f-a e g}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} (f+g x) (c d f-a e g)}\right )}{4 (c d f-a e g)}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 \sqrt {d+e x} (f+g x)^2 (c d f-a e g)}\right )}{c d f-a e g}-\frac {2 \sqrt {d+e x}}{(f+g x)^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle -\frac {5 g \left (\frac {3 c d \left (\frac {c d \arctan \left (\frac {\sqrt {g} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} \sqrt {c d f-a e g}}\right )}{\sqrt {g} (c d f-a e g)^{3/2}}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} (f+g x) (c d f-a e g)}\right )}{4 (c d f-a e g)}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 \sqrt {d+e x} (f+g x)^2 (c d f-a e g)}\right )}{c d f-a e g}-\frac {2 \sqrt {d+e x}}{(f+g x)^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)}\) |
(-2*Sqrt[d + e*x])/((c*d*f - a*e*g)*(f + g*x)^2*Sqrt[a*d*e + (c*d^2 + a*e^ 2)*x + c*d*e*x^2]) - (5*g*(Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(2* (c*d*f - a*e*g)*Sqrt[d + e*x]*(f + g*x)^2) + (3*c*d*(Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/((c*d*f - a*e*g)*Sqrt[d + e*x]*(f + g*x)) + (c*d*Ar cTan[(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])])/(Sqrt[g]*(c*d*f - a*e*g)^(3/2))))/(4*(c*d*f - a*e*g ))))/(c*d*f - a*e*g)
3.7.71.3.1 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[e^2*(d + e*x)^(m - 1)*(f + g*x)^(n + 1)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(c*e*f + c*d*g - b*e*g))), x] + Si mp[e^2*g*((m - n - 2)/((p + 1)*(c*e*f + c*d*g - b*e*g))) 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] && LtQ[p, -1] && RationalQ[n]
Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-e^2)*(d + e*x)^(m - 1)*(f + g*x)^ (n + 1)*((a + b*x + c*x^2)^(p + 1)/((n + 1)*(c*e*f + c*d*g - b*e*g))), x] - Simp[c*e*((m - n - 2)/((n + 1)*(c*e*f + c*d*g - b*e*g))) Int[(d + e*x)^m *(f + g*x)^(n + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[m + p, 0] && LtQ[n, -1 ] && IntegerQ[2*p]
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 = 0.55 (sec) , antiderivative size = 369, normalized size of antiderivative = 1.35
| method | result | size |
| default | \(-\frac {\sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (15 \,\operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) \sqrt {c d x +a e}\, c^{2} d^{2} g^{3} x^{2}+30 \,\operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) \sqrt {c d x +a e}\, c^{2} d^{2} f \,g^{2} x -15 \sqrt {\left (a e g -c d f \right ) g}\, c^{2} d^{2} g^{2} x^{2}+15 \,\operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) \sqrt {c d x +a e}\, c^{2} d^{2} f^{2} g -5 \sqrt {\left (a e g -c d f \right ) g}\, a c d e \,g^{2} x -25 \sqrt {\left (a e g -c d f \right ) g}\, c^{2} d^{2} f g x +2 \sqrt {\left (a e g -c d f \right ) g}\, a^{2} e^{2} g^{2}-9 \sqrt {\left (a e g -c d f \right ) g}\, a c d e f g -8 \sqrt {\left (a e g -c d f \right ) g}\, c^{2} d^{2} f^{2}\right )}{4 \sqrt {e x +d}\, \left (c d x +a e \right ) \left (a e g -c d f \right )^{3} \left (g x +f \right )^{2} \sqrt {\left (a e g -c d f \right ) g}}\) | \(369\) |
-1/4/(e*x+d)^(1/2)*((c*d*x+a*e)*(e*x+d))^(1/2)*(15*arctanh(g*(c*d*x+a*e)^( 1/2)/((a*e*g-c*d*f)*g)^(1/2))*(c*d*x+a*e)^(1/2)*c^2*d^2*g^3*x^2+30*arctanh (g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2))*(c*d*x+a*e)^(1/2)*c^2*d^2*f* g^2*x-15*((a*e*g-c*d*f)*g)^(1/2)*c^2*d^2*g^2*x^2+15*arctanh(g*(c*d*x+a*e)^ (1/2)/((a*e*g-c*d*f)*g)^(1/2))*(c*d*x+a*e)^(1/2)*c^2*d^2*f^2*g-5*((a*e*g-c *d*f)*g)^(1/2)*a*c*d*e*g^2*x-25*((a*e*g-c*d*f)*g)^(1/2)*c^2*d^2*f*g*x+2*(( a*e*g-c*d*f)*g)^(1/2)*a^2*e^2*g^2-9*((a*e*g-c*d*f)*g)^(1/2)*a*c*d*e*f*g-8* ((a*e*g-c*d*f)*g)^(1/2)*c^2*d^2*f^2)/(c*d*x+a*e)/(a*e*g-c*d*f)^3/(g*x+f)^2 /((a*e*g-c*d*f)*g)^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 910 vs. \(2 (244) = 488\).
Time = 0.49 (sec) , antiderivative size = 1863, normalized size of antiderivative = 6.80 \[ \int \frac {(d+e x)^{3/2}}{(f+g x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\text {Too large to display} \]
integrate((e*x+d)^(3/2)/(g*x+f)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2), x, algorithm="fricas")
[-1/8*(15*(c^3*d^3*e*g^2*x^4 + a*c^2*d^3*e*f^2 + (2*c^3*d^3*e*f*g + (c^3*d ^4 + a*c^2*d^2*e^2)*g^2)*x^3 + (c^3*d^3*e*f^2 + a*c^2*d^3*e*g^2 + 2*(c^3*d ^4 + a*c^2*d^2*e^2)*f*g)*x^2 + (2*a*c^2*d^3*e*f*g + (c^3*d^4 + a*c^2*d^2*e ^2)*f^2)*x)*sqrt(-g/(c*d*f - a*e*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)*(c*d*f - a*e*g)*sqrt(e* x + d)*sqrt(-g/(c*d*f - a*e*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*(15*c^2*d^2*g^2*x^2 + 8*c^2*d^2*f^2 + 9*a* c*d*e*f*g - 2*a^2*e^2*g^2 + 5*(5*c^2*d^2*f*g + a*c*d*e*g^2)*x)*sqrt(c*d*e* x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d))/(a*c^3*d^4*e*f^5 - 3*a^2*c ^2*d^3*e^2*f^4*g + 3*a^3*c*d^2*e^3*f^3*g^2 - a^4*d*e^4*f^2*g^3 + (c^4*d^4* e*f^3*g^2 - 3*a*c^3*d^3*e^2*f^2*g^3 + 3*a^2*c^2*d^2*e^3*f*g^4 - a^3*c*d*e^ 4*g^5)*x^4 + (2*c^4*d^4*e*f^4*g + (c^4*d^5 - 5*a*c^3*d^3*e^2)*f^3*g^2 - 3* (a*c^3*d^4*e - a^2*c^2*d^2*e^3)*f^2*g^3 + (3*a^2*c^2*d^3*e^2 + a^3*c*d*e^4 )*f*g^4 - (a^3*c*d^2*e^3 + a^4*e^5)*g^5)*x^3 + (c^4*d^4*e*f^5 - a^4*d*e^4* g^5 + (2*c^4*d^5 - a*c^3*d^3*e^2)*f^4*g - (5*a*c^3*d^4*e + 3*a^2*c^2*d^2*e ^3)*f^3*g^2 + (3*a^2*c^2*d^3*e^2 + 5*a^3*c*d*e^4)*f^2*g^3 + (a^3*c*d^2*e^3 - 2*a^4*e^5)*f*g^4)*x^2 - (2*a^4*d*e^4*f*g^4 - (c^4*d^5 + a*c^3*d^3*e^2)* f^5 + (a*c^3*d^4*e + 3*a^2*c^2*d^2*e^3)*f^4*g + 3*(a^2*c^2*d^3*e^2 - a^3*c *d*e^4)*f^3*g^2 - (5*a^3*c*d^2*e^3 - a^4*e^5)*f^2*g^3)*x), -1/4*(15*(c^3*d ^3*e*g^2*x^4 + a*c^2*d^3*e*f^2 + (2*c^3*d^3*e*f*g + (c^3*d^4 + a*c^2*d^...
Timed out. \[ \int \frac {(d+e x)^{3/2}}{(f+g x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\text {Timed out} \]
\[ \int \frac {(d+e x)^{3/2}}{(f+g x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {3}{2}}}{{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {3}{2}} {\left (g x + f\right )}^{3}} \,d x } \]
integrate((e*x+d)^(3/2)/(g*x+f)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2), x, algorithm="maxima")
Leaf count of result is larger than twice the leaf count of optimal. 1390 vs. \(2 (244) = 488\).
Time = 0.70 (sec) , antiderivative size = 1390, normalized size of antiderivative = 5.07 \[ \int \frac {(d+e x)^{3/2}}{(f+g x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\text {Too large to display} \]
integrate((e*x+d)^(3/2)/(g*x+f)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2), x, algorithm="giac")
-1/4*(15*c^2*d^2*g*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))/((c^3*d^3*e^2*f^3*abs(e) - 3*a*c^2*d^2*e^3*f^2*g*ab s(e) + 3*a^2*c*d*e^4*f*g^2*abs(e) - a^3*e^5*g^3*abs(e))*sqrt(c*d*f*g - a*e *g^2)*e) + 8*c^2*d^2/((c^3*d^3*e^2*f^3*abs(e) - 3*a*c^2*d^2*e^3*f^2*g*abs( e) + 3*a^2*c*d*e^4*f*g^2*abs(e) - a^3*e^5*g^3*abs(e))*sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)) + (9*sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)*c^3*d^3* e^2*f*g - 9*sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)*a*c^2*d^2*e^3*g^2 + 7* ((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*c^2*d^2*g^2)/((c^3*d^3*e^2*f^3*a bs(e) - 3*a*c^2*d^2*e^3*f^2*g*abs(e) + 3*a^2*c*d*e^4*f*g^2*abs(e) - a^3*e^ 5*g^3*abs(e))*(c*d*e^2*f - a*e^3*g + ((e*x + d)*c*d*e - c*d^2*e + a*e^3)*g )^2))*e^4 + 1/4*(15*sqrt(-c*d^2*e + a*e^3)*c^2*d^2*e^3*f^2*g*arctan(sqrt(- c*d^2*e + a*e^3)*g/(sqrt(c*d*f*g - a*e*g^2)*e)) - 30*sqrt(-c*d^2*e + a*e^3 )*c^2*d^3*e^2*f*g^2*arctan(sqrt(-c*d^2*e + a*e^3)*g/(sqrt(c*d*f*g - a*e*g^ 2)*e)) + 15*sqrt(-c*d^2*e + a*e^3)*c^2*d^4*e*g^3*arctan(sqrt(-c*d^2*e + a* e^3)*g/(sqrt(c*d*f*g - a*e*g^2)*e)) + 8*sqrt(c*d*f*g - a*e*g^2)*c^2*d^2*e^ 4*f^2 - 25*sqrt(c*d*f*g - a*e*g^2)*c^2*d^3*e^3*f*g + 9*sqrt(c*d*f*g - a*e* g^2)*a*c*d*e^5*f*g + 15*sqrt(c*d*f*g - a*e*g^2)*c^2*d^4*e^2*g^2 - 5*sqrt(c *d*f*g - a*e*g^2)*a*c*d^2*e^4*g^2 - 2*sqrt(c*d*f*g - a*e*g^2)*a^2*e^6*g^2) /(sqrt(-c*d^2*e + a*e^3)*sqrt(c*d*f*g - a*e*g^2)*c^3*d^3*e^2*f^5*abs(e) - 2*sqrt(-c*d^2*e + a*e^3)*sqrt(c*d*f*g - a*e*g^2)*c^3*d^4*e*f^4*g*abs(e)...
Timed out. \[ \int \frac {(d+e x)^{3/2}}{(f+g x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\int \frac {{\left (d+e\,x\right )}^{3/2}}{{\left (f+g\,x\right )}^3\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{3/2}} \,d x \]