Integrand size = 24, antiderivative size = 144 \[ \int \frac {a+c x^2}{(d+e x)^2 (f+g x)^{3/2}} \, dx=-\frac {2 \left (c f^2+a g^2\right )}{g (e f-d g)^2 \sqrt {f+g x}}-\frac {\left (c d^2+a e^2\right ) \sqrt {f+g x}}{e (e f-d g)^2 (d+e x)}+\frac {\left (3 a e^2 g+c d (4 e f-d g)\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{e^{3/2} (e f-d g)^{5/2}} \]
(3*a*e^2*g+c*d*(-d*g+4*e*f))*arctanh(e^(1/2)*(g*x+f)^(1/2)/(-d*g+e*f)^(1/2 ))/e^(3/2)/(-d*g+e*f)^(5/2)-2*(a*g^2+c*f^2)/g/(-d*g+e*f)^2/(g*x+f)^(1/2)-( a*e^2+c*d^2)*(g*x+f)^(1/2)/e/(-d*g+e*f)^2/(e*x+d)
Time = 0.47 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.03 \[ \int \frac {a+c x^2}{(d+e x)^2 (f+g x)^{3/2}} \, dx=\frac {-c \left (2 d e f^2+2 e^2 f^2 x+d^2 g (f+g x)\right )-a e g (2 d g+e (f+3 g x))}{e g (e f-d g)^2 (d+e x) \sqrt {f+g x}}+\frac {\left (-3 a e^2 g+c d (-4 e f+d g)\right ) \arctan \left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {-e f+d g}}\right )}{e^{3/2} (-e f+d g)^{5/2}} \]
(-(c*(2*d*e*f^2 + 2*e^2*f^2*x + d^2*g*(f + g*x))) - a*e*g*(2*d*g + e*(f + 3*g*x)))/(e*g*(e*f - d*g)^2*(d + e*x)*Sqrt[f + g*x]) + ((-3*a*e^2*g + c*d* (-4*e*f + d*g))*ArcTan[(Sqrt[e]*Sqrt[f + g*x])/Sqrt[-(e*f) + d*g]])/(e^(3/ 2)*(-(e*f) + d*g)^(5/2))
Time = 0.39 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.19, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {649, 1582, 25, 27, 359, 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.
| \(\displaystyle \int \frac {a+c x^2}{(d+e x)^2 (f+g x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 649 |
| \(\displaystyle \frac {2 \int \frac {c f^2-2 c (f+g x) f+a g^2+c (f+g x)^2}{(f+g x) (e f-d g-e (f+g x))^2}d\sqrt {f+g x}}{g}\) |
| \(\Big \downarrow \) 1582 |
| \(\displaystyle \frac {2 \left (\frac {g^2 \sqrt {f+g x} \left (a e^2+c d^2\right )}{2 e (e f-d g)^2 (-d g-e (f+g x)+e f)}-\frac {\int -\frac {e \left (2 e (e f-d g) \left (c f^2+a g^2\right )+\left (a e^2 g^2-c \left (2 e^2 f^2-4 d e g f+d^2 g^2\right )\right ) (f+g x)\right )}{(f+g x) (e f-d g-e (f+g x))}d\sqrt {f+g x}}{2 e^2 (e f-d g)^2}\right )}{g}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 \left (\frac {\int \frac {e \left (2 e (e f-d g) \left (c f^2+a g^2\right )+\left (a e^2 g^2-c \left (2 e^2 f^2-4 d e g f+d^2 g^2\right )\right ) (f+g x)\right )}{(f+g x) (e f-d g-e (f+g x))}d\sqrt {f+g x}}{2 e^2 (e f-d g)^2}+\frac {g^2 \sqrt {f+g x} \left (a e^2+c d^2\right )}{2 e (e f-d g)^2 (-d g-e (f+g x)+e f)}\right )}{g}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \left (\frac {\int \frac {2 e (e f-d g) \left (c f^2+a g^2\right )+\left (a e^2 g^2-c \left (2 e^2 f^2-4 d e g f+d^2 g^2\right )\right ) (f+g x)}{(f+g x) (e f-d g-e (f+g x))}d\sqrt {f+g x}}{2 e (e f-d g)^2}+\frac {g^2 \sqrt {f+g x} \left (a e^2+c d^2\right )}{2 e (e f-d g)^2 (-d g-e (f+g x)+e f)}\right )}{g}\) |
| \(\Big \downarrow \) 359 |
| \(\displaystyle \frac {2 \left (\frac {g \left (3 a e^2 g+c d (4 e f-d g)\right ) \int \frac {1}{e f-d g-e (f+g x)}d\sqrt {f+g x}-\frac {2 e \left (a g^2+c f^2\right )}{\sqrt {f+g x}}}{2 e (e f-d g)^2}+\frac {g^2 \sqrt {f+g x} \left (a e^2+c d^2\right )}{2 e (e f-d g)^2 (-d g-e (f+g x)+e f)}\right )}{g}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {2 \left (\frac {\frac {g \left (3 a e^2 g+c d (4 e f-d g)\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{\sqrt {e} \sqrt {e f-d g}}-\frac {2 e \left (a g^2+c f^2\right )}{\sqrt {f+g x}}}{2 e (e f-d g)^2}+\frac {g^2 \sqrt {f+g x} \left (a e^2+c d^2\right )}{2 e (e f-d g)^2 (-d g-e (f+g x)+e f)}\right )}{g}\) |
(2*(((c*d^2 + a*e^2)*g^2*Sqrt[f + g*x])/(2*e*(e*f - d*g)^2*(e*f - d*g - e* (f + g*x))) + ((-2*e*(c*f^2 + a*g^2))/Sqrt[f + g*x] + (g*(3*a*e^2*g + c*d* (4*e*f - d*g))*ArcTanh[(Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g]])/(Sqrt[e]* Sqrt[e*f - d*g]))/(2*e*(e*f - d*g)^2)))/g
3.7.1.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x _Symbol] :> Simp[c*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*e*(m + 1))), x] + Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(a*e^2*(m + 1)) Int[(e*x)^(m + 2)* (a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] && !ILtQ[p, -1]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^ 2)^(p_.), x_Symbol] :> Simp[2/e^(n + 2*p + 1) Subst[Int[x^(2*m + 1)*(e*f - d*g + g*x^2)^n*(c*d^2 + a*e^2 - 2*c*d*x^2 + c*x^4)^p, x], x, Sqrt[d + e*x ]], x] /; FreeQ[{a, c, d, e, f, g}, x] && IGtQ[p, 0] && ILtQ[n, 0] && Integ erQ[m + 1/2]
Int[(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^ 4)^(p_.), x_Symbol] :> Simp[(-d)^(m/2 - 1)*(c*d^2 - b*d*e + a*e^2)^p*x*((d + e*x^2)^(q + 1)/(2*e^(2*p + m/2)*(q + 1))), x] + Simp[(-d)^(m/2 - 1)/(2*e^ (2*p)*(q + 1)) Int[x^m*(d + e*x^2)^(q + 1)*ExpandToSum[Together[(1/(d + e *x^2))*(2*(-d)^(-m/2 + 1)*e^(2*p)*(q + 1)*(a + b*x^2 + c*x^4)^p - ((c*d^2 - b*d*e + a*e^2)^p/(e^(m/2)*x^m))*(d + e*(2*q + 3)*x^2))], x], x], x] /; Fre eQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && ILtQ[q, -1] && ILtQ[m/2, 0]
Time = 0.48 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.06
| method | result | size |
| derivativedivides | \(\frac {-\frac {2 g \left (\frac {g \left (e^{2} a +c \,d^{2}\right ) \sqrt {g x +f}}{2 e \left (e \left (g x +f \right )+d g -e f \right )}+\frac {\left (3 a \,e^{2} g -c \,d^{2} g +4 c d e f \right ) \arctan \left (\frac {e \sqrt {g x +f}}{\sqrt {\left (d g -e f \right ) e}}\right )}{2 e \sqrt {\left (d g -e f \right ) e}}\right )}{\left (d g -e f \right )^{2}}-\frac {2 \left (a \,g^{2}+c \,f^{2}\right )}{\left (d g -e f \right )^{2} \sqrt {g x +f}}}{g}\) | \(152\) |
| default | \(\frac {-\frac {2 g \left (\frac {g \left (e^{2} a +c \,d^{2}\right ) \sqrt {g x +f}}{2 e \left (e \left (g x +f \right )+d g -e f \right )}+\frac {\left (3 a \,e^{2} g -c \,d^{2} g +4 c d e f \right ) \arctan \left (\frac {e \sqrt {g x +f}}{\sqrt {\left (d g -e f \right ) e}}\right )}{2 e \sqrt {\left (d g -e f \right ) e}}\right )}{\left (d g -e f \right )^{2}}-\frac {2 \left (a \,g^{2}+c \,f^{2}\right )}{\left (d g -e f \right )^{2} \sqrt {g x +f}}}{g}\) | \(152\) |
| pseudoelliptic | \(-\frac {2 \left (\frac {3 \sqrt {g x +f}\, \left (a \,e^{2} g -\frac {1}{3} c \,d^{2} g +\frac {4}{3} c d e f \right ) \left (e x +d \right ) g \arctan \left (\frac {e \sqrt {g x +f}}{\sqrt {\left (d g -e f \right ) e}}\right )}{2}+\sqrt {\left (d g -e f \right ) e}\, \left (\left (\frac {3}{2} a \,g^{2} x +c \,f^{2} x +\frac {1}{2} a f g \right ) e^{2}+d \left (a \,g^{2}+c \,f^{2}\right ) e +\frac {c \,d^{2} g \left (g x +f \right )}{2}\right )\right )}{\sqrt {\left (d g -e f \right ) e}\, \sqrt {g x +f}\, g \left (e x +d \right ) \left (d g -e f \right )^{2} e}\) | \(166\) |
2/g*(-g/(d*g-e*f)^2*(1/2*g*(a*e^2+c*d^2)/e*(g*x+f)^(1/2)/(e*(g*x+f)+d*g-e* f)+1/2*(3*a*e^2*g-c*d^2*g+4*c*d*e*f)/e/((d*g-e*f)*e)^(1/2)*arctan(e*(g*x+f )^(1/2)/((d*g-e*f)*e)^(1/2)))-(a*g^2+c*f^2)/(d*g-e*f)^2/(g*x+f)^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 446 vs. \(2 (130) = 260\).
Time = 0.32 (sec) , antiderivative size = 906, normalized size of antiderivative = 6.29 \[ \int \frac {a+c x^2}{(d+e x)^2 (f+g x)^{3/2}} \, dx=\left [\frac {{\left (4 \, c d^{2} e f^{2} g - {\left (c d^{3} - 3 \, a d e^{2}\right )} f g^{2} + {\left (4 \, c d e^{2} f g^{2} - {\left (c d^{2} e - 3 \, a e^{3}\right )} g^{3}\right )} x^{2} + {\left (4 \, c d e^{2} f^{2} g + 3 \, {\left (c d^{2} e + a e^{3}\right )} f g^{2} - {\left (c d^{3} - 3 \, a d e^{2}\right )} g^{3}\right )} x\right )} \sqrt {e^{2} f - d e g} \log \left (\frac {e g x + 2 \, e f - d g + 2 \, \sqrt {e^{2} f - d e g} \sqrt {g x + f}}{e x + d}\right ) - 2 \, {\left (2 \, c d e^{3} f^{3} - 2 \, a d^{2} e^{2} g^{3} - {\left (c d^{2} e^{2} - a e^{4}\right )} f^{2} g - {\left (c d^{3} e - a d e^{3}\right )} f g^{2} + {\left (2 \, c e^{4} f^{3} - 2 \, c d e^{3} f^{2} g + {\left (c d^{2} e^{2} + 3 \, a e^{4}\right )} f g^{2} - {\left (c d^{3} e + 3 \, a d e^{3}\right )} g^{3}\right )} x\right )} \sqrt {g x + f}}{2 \, {\left (d e^{5} f^{4} g - 3 \, d^{2} e^{4} f^{3} g^{2} + 3 \, d^{3} e^{3} f^{2} g^{3} - d^{4} e^{2} f g^{4} + {\left (e^{6} f^{3} g^{2} - 3 \, d e^{5} f^{2} g^{3} + 3 \, d^{2} e^{4} f g^{4} - d^{3} e^{3} g^{5}\right )} x^{2} + {\left (e^{6} f^{4} g - 2 \, d e^{5} f^{3} g^{2} + 2 \, d^{3} e^{3} f g^{4} - d^{4} e^{2} g^{5}\right )} x\right )}}, -\frac {{\left (4 \, c d^{2} e f^{2} g - {\left (c d^{3} - 3 \, a d e^{2}\right )} f g^{2} + {\left (4 \, c d e^{2} f g^{2} - {\left (c d^{2} e - 3 \, a e^{3}\right )} g^{3}\right )} x^{2} + {\left (4 \, c d e^{2} f^{2} g + 3 \, {\left (c d^{2} e + a e^{3}\right )} f g^{2} - {\left (c d^{3} - 3 \, a d e^{2}\right )} g^{3}\right )} x\right )} \sqrt {-e^{2} f + d e g} \arctan \left (\frac {\sqrt {-e^{2} f + d e g} \sqrt {g x + f}}{e g x + e f}\right ) + {\left (2 \, c d e^{3} f^{3} - 2 \, a d^{2} e^{2} g^{3} - {\left (c d^{2} e^{2} - a e^{4}\right )} f^{2} g - {\left (c d^{3} e - a d e^{3}\right )} f g^{2} + {\left (2 \, c e^{4} f^{3} - 2 \, c d e^{3} f^{2} g + {\left (c d^{2} e^{2} + 3 \, a e^{4}\right )} f g^{2} - {\left (c d^{3} e + 3 \, a d e^{3}\right )} g^{3}\right )} x\right )} \sqrt {g x + f}}{d e^{5} f^{4} g - 3 \, d^{2} e^{4} f^{3} g^{2} + 3 \, d^{3} e^{3} f^{2} g^{3} - d^{4} e^{2} f g^{4} + {\left (e^{6} f^{3} g^{2} - 3 \, d e^{5} f^{2} g^{3} + 3 \, d^{2} e^{4} f g^{4} - d^{3} e^{3} g^{5}\right )} x^{2} + {\left (e^{6} f^{4} g - 2 \, d e^{5} f^{3} g^{2} + 2 \, d^{3} e^{3} f g^{4} - d^{4} e^{2} g^{5}\right )} x}\right ] \]
[1/2*((4*c*d^2*e*f^2*g - (c*d^3 - 3*a*d*e^2)*f*g^2 + (4*c*d*e^2*f*g^2 - (c *d^2*e - 3*a*e^3)*g^3)*x^2 + (4*c*d*e^2*f^2*g + 3*(c*d^2*e + a*e^3)*f*g^2 - (c*d^3 - 3*a*d*e^2)*g^3)*x)*sqrt(e^2*f - d*e*g)*log((e*g*x + 2*e*f - d*g + 2*sqrt(e^2*f - d*e*g)*sqrt(g*x + f))/(e*x + d)) - 2*(2*c*d*e^3*f^3 - 2* a*d^2*e^2*g^3 - (c*d^2*e^2 - a*e^4)*f^2*g - (c*d^3*e - a*d*e^3)*f*g^2 + (2 *c*e^4*f^3 - 2*c*d*e^3*f^2*g + (c*d^2*e^2 + 3*a*e^4)*f*g^2 - (c*d^3*e + 3* a*d*e^3)*g^3)*x)*sqrt(g*x + f))/(d*e^5*f^4*g - 3*d^2*e^4*f^3*g^2 + 3*d^3*e ^3*f^2*g^3 - d^4*e^2*f*g^4 + (e^6*f^3*g^2 - 3*d*e^5*f^2*g^3 + 3*d^2*e^4*f* g^4 - d^3*e^3*g^5)*x^2 + (e^6*f^4*g - 2*d*e^5*f^3*g^2 + 2*d^3*e^3*f*g^4 - d^4*e^2*g^5)*x), -((4*c*d^2*e*f^2*g - (c*d^3 - 3*a*d*e^2)*f*g^2 + (4*c*d*e ^2*f*g^2 - (c*d^2*e - 3*a*e^3)*g^3)*x^2 + (4*c*d*e^2*f^2*g + 3*(c*d^2*e + a*e^3)*f*g^2 - (c*d^3 - 3*a*d*e^2)*g^3)*x)*sqrt(-e^2*f + d*e*g)*arctan(sqr t(-e^2*f + d*e*g)*sqrt(g*x + f)/(e*g*x + e*f)) + (2*c*d*e^3*f^3 - 2*a*d^2* e^2*g^3 - (c*d^2*e^2 - a*e^4)*f^2*g - (c*d^3*e - a*d*e^3)*f*g^2 + (2*c*e^4 *f^3 - 2*c*d*e^3*f^2*g + (c*d^2*e^2 + 3*a*e^4)*f*g^2 - (c*d^3*e + 3*a*d*e^ 3)*g^3)*x)*sqrt(g*x + f))/(d*e^5*f^4*g - 3*d^2*e^4*f^3*g^2 + 3*d^3*e^3*f^2 *g^3 - d^4*e^2*f*g^4 + (e^6*f^3*g^2 - 3*d*e^5*f^2*g^3 + 3*d^2*e^4*f*g^4 - d^3*e^3*g^5)*x^2 + (e^6*f^4*g - 2*d*e^5*f^3*g^2 + 2*d^3*e^3*f*g^4 - d^4*e^ 2*g^5)*x)]
Timed out. \[ \int \frac {a+c x^2}{(d+e x)^2 (f+g x)^{3/2}} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {a+c x^2}{(d+e x)^2 (f+g x)^{3/2}} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e*(d*g-e*f)>0)', see `assume?` f or more de
Time = 0.28 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.58 \[ \int \frac {a+c x^2}{(d+e x)^2 (f+g x)^{3/2}} \, dx=-\frac {{\left (4 \, c d e f - c d^{2} g + 3 \, a e^{2} g\right )} \arctan \left (\frac {\sqrt {g x + f} e}{\sqrt {-e^{2} f + d e g}}\right )}{{\left (e^{3} f^{2} - 2 \, d e^{2} f g + d^{2} e g^{2}\right )} \sqrt {-e^{2} f + d e g}} - \frac {2 \, {\left (g x + f\right )} c e^{2} f^{2} - 2 \, c e^{2} f^{3} + 2 \, c d e f^{2} g + {\left (g x + f\right )} c d^{2} g^{2} + 3 \, {\left (g x + f\right )} a e^{2} g^{2} - 2 \, a e^{2} f g^{2} + 2 \, a d e g^{3}}{{\left (e^{3} f^{2} g - 2 \, d e^{2} f g^{2} + d^{2} e g^{3}\right )} {\left ({\left (g x + f\right )}^{\frac {3}{2}} e - \sqrt {g x + f} e f + \sqrt {g x + f} d g\right )}} \]
-(4*c*d*e*f - c*d^2*g + 3*a*e^2*g)*arctan(sqrt(g*x + f)*e/sqrt(-e^2*f + d* e*g))/((e^3*f^2 - 2*d*e^2*f*g + d^2*e*g^2)*sqrt(-e^2*f + d*e*g)) - (2*(g*x + f)*c*e^2*f^2 - 2*c*e^2*f^3 + 2*c*d*e*f^2*g + (g*x + f)*c*d^2*g^2 + 3*(g *x + f)*a*e^2*g^2 - 2*a*e^2*f*g^2 + 2*a*d*e*g^3)/((e^3*f^2*g - 2*d*e^2*f*g ^2 + d^2*e*g^3)*((g*x + f)^(3/2)*e - sqrt(g*x + f)*e*f + sqrt(g*x + f)*d*g ))
Time = 12.67 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.30 \[ \int \frac {a+c x^2}{(d+e x)^2 (f+g x)^{3/2}} \, dx=-\frac {\frac {2\,\left (c\,f^2+a\,g^2\right )}{d\,g-e\,f}+\frac {\left (f+g\,x\right )\,\left (c\,d^2\,g^2+2\,c\,e^2\,f^2+3\,a\,e^2\,g^2\right )}{e\,{\left (d\,g-e\,f\right )}^2}}{\sqrt {f+g\,x}\,\left (d\,g^2-e\,f\,g\right )+e\,g\,{\left (f+g\,x\right )}^{3/2}}-\frac {\mathrm {atan}\left (\frac {\sqrt {f+g\,x}\,\left (d^2\,e\,g^2-2\,d\,e^2\,f\,g+e^3\,f^2\right )}{\sqrt {e}\,{\left (d\,g-e\,f\right )}^{5/2}}\right )\,\left (-c\,g\,d^2+4\,c\,f\,d\,e+3\,a\,g\,e^2\right )}{e^{3/2}\,{\left (d\,g-e\,f\right )}^{5/2}} \]
- ((2*(a*g^2 + c*f^2))/(d*g - e*f) + ((f + g*x)*(3*a*e^2*g^2 + c*d^2*g^2 + 2*c*e^2*f^2))/(e*(d*g - e*f)^2))/((f + g*x)^(1/2)*(d*g^2 - e*f*g) + e*g*( f + g*x)^(3/2)) - (atan(((f + g*x)^(1/2)*(e^3*f^2 + d^2*e*g^2 - 2*d*e^2*f* g))/(e^(1/2)*(d*g - e*f)^(5/2)))*(3*a*e^2*g - c*d^2*g + 4*c*d*e*f))/(e^(3/ 2)*(d*g - e*f)^(5/2))