Integrand size = 29, antiderivative size = 138 \[ \int \frac {d+e x+f x^2}{(g+h x) \left (a+c x^2\right )^{3/2}} \, dx=-\frac {a (c e g-c d h+a f h)-c (c d g-a f g+a e h) x}{a c \left (c g^2+a h^2\right ) \sqrt {a+c x^2}}-\frac {\left (f g^2-e g h+d h^2\right ) \text {arctanh}\left (\frac {a h-c g x}{\sqrt {c g^2+a h^2} \sqrt {a+c x^2}}\right )}{\left (c g^2+a h^2\right )^{3/2}} \]
-(d*h^2-e*g*h+f*g^2)*arctanh((-c*g*x+a*h)/(a*h^2+c*g^2)^(1/2)/(c*x^2+a)^(1 /2))/(a*h^2+c*g^2)^(3/2)+(-a*(a*f*h-c*d*h+c*e*g)+c*(a*e*h-a*f*g+c*d*g)*x)/ a/c/(a*h^2+c*g^2)/(c*x^2+a)^(1/2)
Time = 0.66 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.07 \[ \int \frac {d+e x+f x^2}{(g+h x) \left (a+c x^2\right )^{3/2}} \, dx=\frac {-a^2 f h+c^2 d g x+a c (-e g+d h-f g x+e h x)}{a c \left (c g^2+a h^2\right ) \sqrt {a+c x^2}}+\frac {2 \left (f g^2+h (-e g+d h)\right ) \arctan \left (\frac {\sqrt {c} (g+h x)-h \sqrt {a+c x^2}}{\sqrt {-c g^2-a h^2}}\right )}{\left (-c g^2-a h^2\right )^{3/2}} \]
(-(a^2*f*h) + c^2*d*g*x + a*c*(-(e*g) + d*h - f*g*x + e*h*x))/(a*c*(c*g^2 + a*h^2)*Sqrt[a + c*x^2]) + (2*(f*g^2 + h*(-(e*g) + d*h))*ArcTan[(Sqrt[c]* (g + h*x) - h*Sqrt[a + c*x^2])/Sqrt[-(c*g^2) - a*h^2]])/(-(c*g^2) - a*h^2) ^(3/2)
Time = 0.34 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {2178, 25, 27, 488, 219}
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.
| \(\displaystyle \int \frac {d+e x+f x^2}{\left (a+c x^2\right )^{3/2} (g+h x)} \, dx\) |
| \(\Big \downarrow \) 2178 |
| \(\displaystyle -\frac {\int -\frac {a c \left (f g^2-e h g+d h^2\right )}{\left (c g^2+a h^2\right ) (g+h x) \sqrt {c x^2+a}}dx}{a c}-\frac {a (a f h-c d h+c e g)-c x (a e h-a f g+c d g)}{a c \sqrt {a+c x^2} \left (a h^2+c g^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {a c \left (f g^2-e h g+d h^2\right )}{\left (c g^2+a h^2\right ) (g+h x) \sqrt {c x^2+a}}dx}{a c}-\frac {a (a f h-c d h+c e g)-c x (a e h-a f g+c d g)}{a c \sqrt {a+c x^2} \left (a h^2+c g^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\left (d h^2-e g h+f g^2\right ) \int \frac {1}{(g+h x) \sqrt {c x^2+a}}dx}{a h^2+c g^2}-\frac {a (a f h-c d h+c e g)-c x (a e h-a f g+c d g)}{a c \sqrt {a+c x^2} \left (a h^2+c g^2\right )}\) |
| \(\Big \downarrow \) 488 |
| \(\displaystyle -\frac {\left (d h^2-e g h+f g^2\right ) \int \frac {1}{c g^2+a h^2-\frac {(a h-c g x)^2}{c x^2+a}}d\frac {a h-c g x}{\sqrt {c x^2+a}}}{a h^2+c g^2}-\frac {a (a f h-c d h+c e g)-c x (a e h-a f g+c d g)}{a c \sqrt {a+c x^2} \left (a h^2+c g^2\right )}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {\left (d h^2-e g h+f g^2\right ) \text {arctanh}\left (\frac {a h-c g x}{\sqrt {a+c x^2} \sqrt {a h^2+c g^2}}\right )}{\left (a h^2+c g^2\right )^{3/2}}-\frac {a (a f h-c d h+c e g)-c x (a e h-a f g+c d g)}{a c \sqrt {a+c x^2} \left (a h^2+c g^2\right )}\) |
-((a*(c*e*g - c*d*h + a*f*h) - c*(c*d*g - a*f*g + a*e*h)*x)/(a*c*(c*g^2 + a*h^2)*Sqrt[a + c*x^2])) - ((f*g^2 - e*g*h + d*h^2)*ArcTanh[(a*h - c*g*x)/ (Sqrt[c*g^2 + a*h^2]*Sqrt[a + c*x^2])])/(c*g^2 + a*h^2)^(3/2)
3.2.12.3.1 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[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> -Subst[ Int[1/(b*c^2 + a*d^2 - x^2), x], x, (a*d - b*c*x)/Sqrt[a + b*x^2]] /; FreeQ [{a, b, c, d}, x]
Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] : > With[{Qx = PolynomialQuotient[(d + e*x)^m*Pq, a + b*x^2, x], R = Coeff[Po lynomialRemainder[(d + e*x)^m*Pq, a + b*x^2, x], x, 0], S = Coeff[Polynomia lRemainder[(d + e*x)^m*Pq, a + b*x^2, x], x, 1]}, Simp[(a*S - b*R*x)*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*b*(p + 1)) Int[(d + e*x )^m*(a + b*x^2)^(p + 1)*ExpandToSum[(2*a*b*(p + 1)*Qx)/(d + e*x)^m + (b*R*( 2*p + 3))/(d + e*x)^m, x], x], x]] /; FreeQ[{a, b, d, e}, x] && PolyQ[Pq, x ] && NeQ[b*d^2 + a*e^2, 0] && LtQ[p, -1] && ILtQ[m, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(385\) vs. \(2(129)=258\).
Time = 0.51 (sec) , antiderivative size = 386, normalized size of antiderivative = 2.80
| method | result | size |
| default | \(\frac {\frac {e h x}{a \sqrt {c \,x^{2}+a}}-\frac {f h}{c \sqrt {c \,x^{2}+a}}-\frac {f g x}{a \sqrt {c \,x^{2}+a}}}{h^{2}}+\frac {\left (d \,h^{2}-e g h +f \,g^{2}\right ) \left (\frac {h^{2}}{\left (a \,h^{2}+c \,g^{2}\right ) \sqrt {\left (x +\frac {g}{h}\right )^{2} c -\frac {2 c g \left (x +\frac {g}{h}\right )}{h}+\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}}+\frac {2 c g h \left (2 c \left (x +\frac {g}{h}\right )-\frac {2 c g}{h}\right )}{\left (a \,h^{2}+c \,g^{2}\right ) \left (\frac {4 c \left (a \,h^{2}+c \,g^{2}\right )}{h^{2}}-\frac {4 c^{2} g^{2}}{h^{2}}\right ) \sqrt {\left (x +\frac {g}{h}\right )^{2} c -\frac {2 c g \left (x +\frac {g}{h}\right )}{h}+\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}}-\frac {h^{2} \ln \left (\frac {\frac {2 a \,h^{2}+2 c \,g^{2}}{h^{2}}-\frac {2 c g \left (x +\frac {g}{h}\right )}{h}+2 \sqrt {\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}\, \sqrt {\left (x +\frac {g}{h}\right )^{2} c -\frac {2 c g \left (x +\frac {g}{h}\right )}{h}+\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}}{x +\frac {g}{h}}\right )}{\left (a \,h^{2}+c \,g^{2}\right ) \sqrt {\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}}\right )}{h^{3}}\) | \(386\) |
1/h^2*(e*h*x/a/(c*x^2+a)^(1/2)-f*h/c/(c*x^2+a)^(1/2)-f*g*x/a/(c*x^2+a)^(1/ 2))+(d*h^2-e*g*h+f*g^2)/h^3*(1/(a*h^2+c*g^2)*h^2/((x+1/h*g)^2*c-2*c*g/h*(x +1/h*g)+(a*h^2+c*g^2)/h^2)^(1/2)+2*c*g*h/(a*h^2+c*g^2)*(2*c*(x+1/h*g)-2*c* g/h)/(4*c*(a*h^2+c*g^2)/h^2-4*c^2*g^2/h^2)/((x+1/h*g)^2*c-2*c*g/h*(x+1/h*g )+(a*h^2+c*g^2)/h^2)^(1/2)-1/(a*h^2+c*g^2)*h^2/((a*h^2+c*g^2)/h^2)^(1/2)*l n((2*(a*h^2+c*g^2)/h^2-2*c*g/h*(x+1/h*g)+2*((a*h^2+c*g^2)/h^2)^(1/2)*((x+1 /h*g)^2*c-2*c*g/h*(x+1/h*g)+(a*h^2+c*g^2)/h^2)^(1/2))/(x+1/h*g)))
Leaf count of result is larger than twice the leaf count of optimal. 347 vs. \(2 (130) = 260\).
Time = 1.06 (sec) , antiderivative size = 721, normalized size of antiderivative = 5.22 \[ \int \frac {d+e x+f x^2}{(g+h x) \left (a+c x^2\right )^{3/2}} \, dx=\left [\frac {{\left (a^{2} c f g^{2} - a^{2} c e g h + a^{2} c d h^{2} + {\left (a c^{2} f g^{2} - a c^{2} e g h + a c^{2} d h^{2}\right )} x^{2}\right )} \sqrt {c g^{2} + a h^{2}} \log \left (\frac {2 \, a c g h x - a c g^{2} - 2 \, a^{2} h^{2} - {\left (2 \, c^{2} g^{2} + a c h^{2}\right )} x^{2} - 2 \, \sqrt {c g^{2} + a h^{2}} {\left (c g x - a h\right )} \sqrt {c x^{2} + a}}{h^{2} x^{2} + 2 \, g h x + g^{2}}\right ) - 2 \, {\left (a c^{2} e g^{3} + a^{2} c e g h^{2} - {\left (a c^{2} d - a^{2} c f\right )} g^{2} h - {\left (a^{2} c d - a^{3} f\right )} h^{3} - {\left (a c^{2} e g^{2} h + a^{2} c e h^{3} + {\left (c^{3} d - a c^{2} f\right )} g^{3} + {\left (a c^{2} d - a^{2} c f\right )} g h^{2}\right )} x\right )} \sqrt {c x^{2} + a}}{2 \, {\left (a^{2} c^{3} g^{4} + 2 \, a^{3} c^{2} g^{2} h^{2} + a^{4} c h^{4} + {\left (a c^{4} g^{4} + 2 \, a^{2} c^{3} g^{2} h^{2} + a^{3} c^{2} h^{4}\right )} x^{2}\right )}}, -\frac {{\left (a^{2} c f g^{2} - a^{2} c e g h + a^{2} c d h^{2} + {\left (a c^{2} f g^{2} - a c^{2} e g h + a c^{2} d h^{2}\right )} x^{2}\right )} \sqrt {-c g^{2} - a h^{2}} \arctan \left (\frac {\sqrt {-c g^{2} - a h^{2}} {\left (c g x - a h\right )} \sqrt {c x^{2} + a}}{a c g^{2} + a^{2} h^{2} + {\left (c^{2} g^{2} + a c h^{2}\right )} x^{2}}\right ) + {\left (a c^{2} e g^{3} + a^{2} c e g h^{2} - {\left (a c^{2} d - a^{2} c f\right )} g^{2} h - {\left (a^{2} c d - a^{3} f\right )} h^{3} - {\left (a c^{2} e g^{2} h + a^{2} c e h^{3} + {\left (c^{3} d - a c^{2} f\right )} g^{3} + {\left (a c^{2} d - a^{2} c f\right )} g h^{2}\right )} x\right )} \sqrt {c x^{2} + a}}{a^{2} c^{3} g^{4} + 2 \, a^{3} c^{2} g^{2} h^{2} + a^{4} c h^{4} + {\left (a c^{4} g^{4} + 2 \, a^{2} c^{3} g^{2} h^{2} + a^{3} c^{2} h^{4}\right )} x^{2}}\right ] \]
[1/2*((a^2*c*f*g^2 - a^2*c*e*g*h + a^2*c*d*h^2 + (a*c^2*f*g^2 - a*c^2*e*g* h + a*c^2*d*h^2)*x^2)*sqrt(c*g^2 + a*h^2)*log((2*a*c*g*h*x - a*c*g^2 - 2*a ^2*h^2 - (2*c^2*g^2 + a*c*h^2)*x^2 - 2*sqrt(c*g^2 + a*h^2)*(c*g*x - a*h)*s qrt(c*x^2 + a))/(h^2*x^2 + 2*g*h*x + g^2)) - 2*(a*c^2*e*g^3 + a^2*c*e*g*h^ 2 - (a*c^2*d - a^2*c*f)*g^2*h - (a^2*c*d - a^3*f)*h^3 - (a*c^2*e*g^2*h + a ^2*c*e*h^3 + (c^3*d - a*c^2*f)*g^3 + (a*c^2*d - a^2*c*f)*g*h^2)*x)*sqrt(c* x^2 + a))/(a^2*c^3*g^4 + 2*a^3*c^2*g^2*h^2 + a^4*c*h^4 + (a*c^4*g^4 + 2*a^ 2*c^3*g^2*h^2 + a^3*c^2*h^4)*x^2), -((a^2*c*f*g^2 - a^2*c*e*g*h + a^2*c*d* h^2 + (a*c^2*f*g^2 - a*c^2*e*g*h + a*c^2*d*h^2)*x^2)*sqrt(-c*g^2 - a*h^2)* arctan(sqrt(-c*g^2 - a*h^2)*(c*g*x - a*h)*sqrt(c*x^2 + a)/(a*c*g^2 + a^2*h ^2 + (c^2*g^2 + a*c*h^2)*x^2)) + (a*c^2*e*g^3 + a^2*c*e*g*h^2 - (a*c^2*d - a^2*c*f)*g^2*h - (a^2*c*d - a^3*f)*h^3 - (a*c^2*e*g^2*h + a^2*c*e*h^3 + ( c^3*d - a*c^2*f)*g^3 + (a*c^2*d - a^2*c*f)*g*h^2)*x)*sqrt(c*x^2 + a))/(a^2 *c^3*g^4 + 2*a^3*c^2*g^2*h^2 + a^4*c*h^4 + (a*c^4*g^4 + 2*a^2*c^3*g^2*h^2 + a^3*c^2*h^4)*x^2)]
\[ \int \frac {d+e x+f x^2}{(g+h x) \left (a+c x^2\right )^{3/2}} \, dx=\int \frac {d + e x + f x^{2}}{\left (a + c x^{2}\right )^{\frac {3}{2}} \left (g + h x\right )}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 453 vs. \(2 (130) = 260\).
Time = 0.24 (sec) , antiderivative size = 453, normalized size of antiderivative = 3.28 \[ \int \frac {d+e x+f x^2}{(g+h x) \left (a+c x^2\right )^{3/2}} \, dx=\frac {c f g^{3} x}{\sqrt {c x^{2} + a} a c g^{2} h^{2} + \sqrt {c x^{2} + a} a^{2} h^{4}} - \frac {c e g^{2} x}{\sqrt {c x^{2} + a} a c g^{2} h + \sqrt {c x^{2} + a} a^{2} h^{3}} + \frac {c d g x}{\sqrt {c x^{2} + a} a c g^{2} + \sqrt {c x^{2} + a} a^{2} h^{2}} + \frac {f g^{2}}{\sqrt {c x^{2} + a} c g^{2} h + \sqrt {c x^{2} + a} a h^{3}} - \frac {e g}{\sqrt {c x^{2} + a} c g^{2} + \sqrt {c x^{2} + a} a h^{2}} + \frac {d}{\frac {\sqrt {c x^{2} + a} c g^{2}}{h} + \sqrt {c x^{2} + a} a h} - \frac {f g x}{\sqrt {c x^{2} + a} a h^{2}} + \frac {e x}{\sqrt {c x^{2} + a} a h} + \frac {f g^{2} \operatorname {arsinh}\left (\frac {c g x}{\sqrt {a c} {\left | h x + g \right |}} - \frac {a h}{\sqrt {a c} {\left | h x + g \right |}}\right )}{{\left (a + \frac {c g^{2}}{h^{2}}\right )}^{\frac {3}{2}} h^{3}} - \frac {e g \operatorname {arsinh}\left (\frac {c g x}{\sqrt {a c} {\left | h x + g \right |}} - \frac {a h}{\sqrt {a c} {\left | h x + g \right |}}\right )}{{\left (a + \frac {c g^{2}}{h^{2}}\right )}^{\frac {3}{2}} h^{2}} + \frac {d \operatorname {arsinh}\left (\frac {c g x}{\sqrt {a c} {\left | h x + g \right |}} - \frac {a h}{\sqrt {a c} {\left | h x + g \right |}}\right )}{{\left (a + \frac {c g^{2}}{h^{2}}\right )}^{\frac {3}{2}} h} - \frac {f}{\sqrt {c x^{2} + a} c h} \]
c*f*g^3*x/(sqrt(c*x^2 + a)*a*c*g^2*h^2 + sqrt(c*x^2 + a)*a^2*h^4) - c*e*g^ 2*x/(sqrt(c*x^2 + a)*a*c*g^2*h + sqrt(c*x^2 + a)*a^2*h^3) + c*d*g*x/(sqrt( c*x^2 + a)*a*c*g^2 + sqrt(c*x^2 + a)*a^2*h^2) + f*g^2/(sqrt(c*x^2 + a)*c*g ^2*h + sqrt(c*x^2 + a)*a*h^3) - e*g/(sqrt(c*x^2 + a)*c*g^2 + sqrt(c*x^2 + a)*a*h^2) + d/(sqrt(c*x^2 + a)*c*g^2/h + sqrt(c*x^2 + a)*a*h) - f*g*x/(sqr t(c*x^2 + a)*a*h^2) + e*x/(sqrt(c*x^2 + a)*a*h) + f*g^2*arcsinh(c*g*x/(sqr t(a*c)*abs(h*x + g)) - a*h/(sqrt(a*c)*abs(h*x + g)))/((a + c*g^2/h^2)^(3/2 )*h^3) - e*g*arcsinh(c*g*x/(sqrt(a*c)*abs(h*x + g)) - a*h/(sqrt(a*c)*abs(h *x + g)))/((a + c*g^2/h^2)^(3/2)*h^2) + d*arcsinh(c*g*x/(sqrt(a*c)*abs(h*x + g)) - a*h/(sqrt(a*c)*abs(h*x + g)))/((a + c*g^2/h^2)^(3/2)*h) - f/(sqrt (c*x^2 + a)*c*h)
Leaf count of result is larger than twice the leaf count of optimal. 288 vs. \(2 (130) = 260\).
Time = 0.29 (sec) , antiderivative size = 288, normalized size of antiderivative = 2.09 \[ \int \frac {d+e x+f x^2}{(g+h x) \left (a+c x^2\right )^{3/2}} \, dx=\frac {\frac {{\left (c^{3} d g^{3} - a c^{2} f g^{3} + a c^{2} e g^{2} h + a c^{2} d g h^{2} - a^{2} c f g h^{2} + a^{2} c e h^{3}\right )} x}{a c^{3} g^{4} + 2 \, a^{2} c^{2} g^{2} h^{2} + a^{3} c h^{4}} - \frac {a c^{2} e g^{3} - a c^{2} d g^{2} h + a^{2} c f g^{2} h + a^{2} c e g h^{2} - a^{2} c d h^{3} + a^{3} f h^{3}}{a c^{3} g^{4} + 2 \, a^{2} c^{2} g^{2} h^{2} + a^{3} c h^{4}}}{\sqrt {c x^{2} + a}} - \frac {2 \, {\left (f g^{2} - e g h + d h^{2}\right )} \arctan \left (\frac {{\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} h + \sqrt {c} g}{\sqrt {-c g^{2} - a h^{2}}}\right )}{{\left (c g^{2} + a h^{2}\right )} \sqrt {-c g^{2} - a h^{2}}} \]
((c^3*d*g^3 - a*c^2*f*g^3 + a*c^2*e*g^2*h + a*c^2*d*g*h^2 - a^2*c*f*g*h^2 + a^2*c*e*h^3)*x/(a*c^3*g^4 + 2*a^2*c^2*g^2*h^2 + a^3*c*h^4) - (a*c^2*e*g^ 3 - a*c^2*d*g^2*h + a^2*c*f*g^2*h + a^2*c*e*g*h^2 - a^2*c*d*h^3 + a^3*f*h^ 3)/(a*c^3*g^4 + 2*a^2*c^2*g^2*h^2 + a^3*c*h^4))/sqrt(c*x^2 + a) - 2*(f*g^2 - e*g*h + d*h^2)*arctan(((sqrt(c)*x - sqrt(c*x^2 + a))*h + sqrt(c)*g)/sqr t(-c*g^2 - a*h^2))/((c*g^2 + a*h^2)*sqrt(-c*g^2 - a*h^2))
Timed out. \[ \int \frac {d+e x+f x^2}{(g+h x) \left (a+c x^2\right )^{3/2}} \, dx=\int \frac {f\,x^2+e\,x+d}{\left (g+h\,x\right )\,{\left (c\,x^2+a\right )}^{3/2}} \,d x \]