Integrand size = 29, antiderivative size = 168 \[ \int \frac {d+e x+f x^2}{(g+h x)^2 \sqrt {a+c x^2}} \, dx=-\frac {\left (f g^2-e g h+d h^2\right ) \sqrt {a+c x^2}}{h \left (c g^2+a h^2\right ) (g+h x)}+\frac {f \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{\sqrt {c} h^2}+\frac {\left (a h^2 (2 f g-e h)+c \left (f g^3-d g h^2\right )\right ) \text {arctanh}\left (\frac {a h-c g x}{\sqrt {c g^2+a h^2} \sqrt {a+c x^2}}\right )}{h^2 \left (c g^2+a h^2\right )^{3/2}} \]
(a*h^2*(-e*h+2*f*g)+c*(-d*g*h^2+f*g^3))*arctanh((-c*g*x+a*h)/(a*h^2+c*g^2) ^(1/2)/(c*x^2+a)^(1/2))/h^2/(a*h^2+c*g^2)^(3/2)+f*arctanh(x*c^(1/2)/(c*x^2 +a)^(1/2))/h^2/c^(1/2)-(d*h^2-e*g*h+f*g^2)*(c*x^2+a)^(1/2)/h/(a*h^2+c*g^2) /(h*x+g)
Time = 0.85 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.05 \[ \int \frac {d+e x+f x^2}{(g+h x)^2 \sqrt {a+c x^2}} \, dx=-\frac {\frac {h \left (f g^2+h (-e g+d h)\right ) \sqrt {a+c x^2}}{\left (c g^2+a h^2\right ) (g+h x)}+\frac {2 \left (a h^2 (2 f g-e h)+c \left (f g^3-d g h^2\right )\right ) \arctan \left (\frac {\sqrt {c} (g+h x)-h \sqrt {a+c x^2}}{\sqrt {-c g^2-a h^2}}\right )}{\left (-c g^2-a h^2\right )^{3/2}}+\frac {f \log \left (-\sqrt {c} x+\sqrt {a+c x^2}\right )}{\sqrt {c}}}{h^2} \]
-(((h*(f*g^2 + h*(-(e*g) + d*h))*Sqrt[a + c*x^2])/((c*g^2 + a*h^2)*(g + h* x)) + (2*(a*h^2*(2*f*g - e*h) + c*(f*g^3 - d*g*h^2))*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 ) + (f*Log[-(Sqrt[c]*x) + Sqrt[a + c*x^2]])/Sqrt[c])/h^2)
Time = 0.42 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.11, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2182, 25, 719, 224, 219, 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.
| \(\displaystyle \int \frac {d+e x+f x^2}{\sqrt {a+c x^2} (g+h x)^2} \, dx\) |
| \(\Big \downarrow \) 2182 |
| \(\displaystyle -\frac {\int -\frac {c d g-a f g+a e h+f \left (\frac {c g^2}{h}+a h\right ) x}{(g+h x) \sqrt {c x^2+a}}dx}{a h^2+c g^2}-\frac {\sqrt {a+c x^2} \left (d h^2-e g h+f g^2\right )}{h (g+h x) \left (a h^2+c g^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {c d g-a f g+a e h+f \left (\frac {c g^2}{h}+a h\right ) x}{(g+h x) \sqrt {c x^2+a}}dx}{a h^2+c g^2}-\frac {\sqrt {a+c x^2} \left (d h^2-e g h+f g^2\right )}{h (g+h x) \left (a h^2+c g^2\right )}\) |
| \(\Big \downarrow \) 719 |
| \(\displaystyle \frac {\left (a e h-2 a f g+c d g-\frac {c f g^3}{h^2}\right ) \int \frac {1}{(g+h x) \sqrt {c x^2+a}}dx+\frac {f \left (a h^2+c g^2\right ) \int \frac {1}{\sqrt {c x^2+a}}dx}{h^2}}{a h^2+c g^2}-\frac {\sqrt {a+c x^2} \left (d h^2-e g h+f g^2\right )}{h (g+h x) \left (a h^2+c g^2\right )}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {\left (a e h-2 a f g+c d g-\frac {c f g^3}{h^2}\right ) \int \frac {1}{(g+h x) \sqrt {c x^2+a}}dx+\frac {f \left (a h^2+c g^2\right ) \int \frac {1}{1-\frac {c x^2}{c x^2+a}}d\frac {x}{\sqrt {c x^2+a}}}{h^2}}{a h^2+c g^2}-\frac {\sqrt {a+c x^2} \left (d h^2-e g h+f g^2\right )}{h (g+h x) \left (a h^2+c g^2\right )}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\left (a e h-2 a f g+c d g-\frac {c f g^3}{h^2}\right ) \int \frac {1}{(g+h x) \sqrt {c x^2+a}}dx+\frac {f \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right ) \left (a h^2+c g^2\right )}{\sqrt {c} h^2}}{a h^2+c g^2}-\frac {\sqrt {a+c x^2} \left (d h^2-e g h+f g^2\right )}{h (g+h x) \left (a h^2+c g^2\right )}\) |
| \(\Big \downarrow \) 488 |
| \(\displaystyle \frac {\frac {f \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right ) \left (a h^2+c g^2\right )}{\sqrt {c} h^2}-\left (a e h-2 a f g+c d g-\frac {c f g^3}{h^2}\right ) \int \frac {1}{c g^2+a h^2-\frac {(a h-c g x)^2}{c x^2+a}}d\frac {a h-c g x}{\sqrt {c x^2+a}}}{a h^2+c g^2}-\frac {\sqrt {a+c x^2} \left (d h^2-e g h+f g^2\right )}{h (g+h x) \left (a h^2+c g^2\right )}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {f \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right ) \left (a h^2+c g^2\right )}{\sqrt {c} h^2}-\frac {\text {arctanh}\left (\frac {a h-c g x}{\sqrt {a+c x^2} \sqrt {a h^2+c g^2}}\right ) \left (a e h-2 a f g+c d g-\frac {c f g^3}{h^2}\right )}{\sqrt {a h^2+c g^2}}}{a h^2+c g^2}-\frac {\sqrt {a+c x^2} \left (d h^2-e g h+f g^2\right )}{h (g+h x) \left (a h^2+c g^2\right )}\) |
-(((f*g^2 - e*g*h + d*h^2)*Sqrt[a + c*x^2])/(h*(c*g^2 + a*h^2)*(g + h*x))) + ((f*(c*g^2 + a*h^2)*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/(Sqrt[c]*h^2) - ((c*d*g - 2*a*f*g - (c*f*g^3)/h^2 + a*e*h)*ArcTanh[(a*h - c*g*x)/(Sqrt[ c*g^2 + a*h^2]*Sqrt[a + c*x^2])])/Sqrt[c*g^2 + a*h^2])/(c*g^2 + a*h^2)
3.2.6.3.1 Defintions of rubi rules used
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/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 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[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[g/e Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Simp[(e*f - d*g)/e Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && !IGtQ[m, 0]
Int[(Pq_)*((d_) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Qx = PolynomialQuotient[Pq, d + e*x, x], R = PolynomialRemainder[Pq, d + e*x, x]}, Simp[e*R*(d + e*x)^(m + 1)*((a + b*x^2)^(p + 1)/((m + 1)*(b* d^2 + a*e^2))), x] + Simp[1/((m + 1)*(b*d^2 + a*e^2)) Int[(d + e*x)^(m + 1)*(a + b*x^2)^p*ExpandToSum[(m + 1)*(b*d^2 + a*e^2)*Qx + b*d*R*(m + 1) - b *e*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, d, e, p}, x] && PolyQ[Pq, x] && NeQ[b*d^2 + a*e^2, 0] && LtQ[m, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(389\) vs. \(2(154)=308\).
Time = 0.60 (sec) , antiderivative size = 390, normalized size of antiderivative = 2.32
| method | result | size |
| default | \(\frac {f \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )}{h^{2} \sqrt {c}}-\frac {\left (e h -2 f g \right ) \ln \left (\frac {\frac {2 a \,h^{2}+2 c \,g^{2}}{h^{2}}-\frac {2 c g \left (x +\frac {g}{h}\right )}{h}+2 \sqrt {\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}\, \sqrt {\left (x +\frac {g}{h}\right )^{2} c -\frac {2 c g \left (x +\frac {g}{h}\right )}{h}+\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}}{x +\frac {g}{h}}\right )}{h^{3} \sqrt {\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}}+\frac {\left (d \,h^{2}-e g h +f \,g^{2}\right ) \left (-\frac {h^{2} \sqrt {\left (x +\frac {g}{h}\right )^{2} c -\frac {2 c g \left (x +\frac {g}{h}\right )}{h}+\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}}{\left (a \,h^{2}+c \,g^{2}\right ) \left (x +\frac {g}{h}\right )}-\frac {c g h \ln \left (\frac {\frac {2 a \,h^{2}+2 c \,g^{2}}{h^{2}}-\frac {2 c g \left (x +\frac {g}{h}\right )}{h}+2 \sqrt {\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}\, \sqrt {\left (x +\frac {g}{h}\right )^{2} c -\frac {2 c g \left (x +\frac {g}{h}\right )}{h}+\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}}{x +\frac {g}{h}}\right )}{\left (a \,h^{2}+c \,g^{2}\right ) \sqrt {\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}}\right )}{h^{4}}\) | \(390\) |
f/h^2*ln(x*c^(1/2)+(c*x^2+a)^(1/2))/c^(1/2)-1/h^3*(e*h-2*f*g)/((a*h^2+c*g^ 2)/h^2)^(1/2)*ln((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))+1/h^4*(d*h^2-e*g*h+f*g^2)*(-1/(a*h^2+c*g^2)*h^2/(x+1/h*g)*((x+1/h*g) ^2*c-2*c*g/h*(x+1/h*g)+(a*h^2+c*g^2)/h^2)^(1/2)-c*g*h/(a*h^2+c*g^2)/((a*h^ 2+c*g^2)/h^2)^(1/2)*ln((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)))
Timed out. \[ \int \frac {d+e x+f x^2}{(g+h x)^2 \sqrt {a+c x^2}} \, dx=\text {Timed out} \]
\[ \int \frac {d+e x+f x^2}{(g+h x)^2 \sqrt {a+c x^2}} \, dx=\int \frac {d + e x + f x^{2}}{\sqrt {a + c x^{2}} \left (g + h x\right )^{2}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 419 vs. \(2 (155) = 310\).
Time = 0.23 (sec) , antiderivative size = 419, normalized size of antiderivative = 2.49 \[ \int \frac {d+e x+f x^2}{(g+h x)^2 \sqrt {a+c x^2}} \, dx=-\frac {\sqrt {c x^{2} + a} f g^{2}}{c g^{2} h^{2} x + a h^{4} x + c g^{3} h + a g h^{3}} + \frac {\sqrt {c x^{2} + a} e g}{c g^{2} h x + a h^{3} x + c g^{3} + a g h^{2}} - \frac {\sqrt {c x^{2} + a} d}{c g^{2} x + a h^{2} x + \frac {c g^{3}}{h} + a g h} + \frac {f \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {c} h^{2}} + \frac {c f g^{3} \operatorname {arsinh}\left (\frac {c g x}{\sqrt {a c} {\left | h x + g \right |}} - \frac {a h}{\sqrt {a c} {\left | h x + g \right |}}\right )}{{\left (a + \frac {c g^{2}}{h^{2}}\right )}^{\frac {3}{2}} h^{5}} - \frac {c e g^{2} \operatorname {arsinh}\left (\frac {c g x}{\sqrt {a c} {\left | h x + g \right |}} - \frac {a h}{\sqrt {a c} {\left | h x + g \right |}}\right )}{{\left (a + \frac {c g^{2}}{h^{2}}\right )}^{\frac {3}{2}} h^{4}} + \frac {c d g \operatorname {arsinh}\left (\frac {c g x}{\sqrt {a c} {\left | h x + g \right |}} - \frac {a h}{\sqrt {a c} {\left | h x + g \right |}}\right )}{{\left (a + \frac {c g^{2}}{h^{2}}\right )}^{\frac {3}{2}} h^{3}} - \frac {2 \, f g \operatorname {arsinh}\left (\frac {c g x}{\sqrt {a c} {\left | h x + g \right |}} - \frac {a h}{\sqrt {a c} {\left | h x + g \right |}}\right )}{\sqrt {a + \frac {c g^{2}}{h^{2}}} h^{3}} + \frac {e \operatorname {arsinh}\left (\frac {c g x}{\sqrt {a c} {\left | h x + g \right |}} - \frac {a h}{\sqrt {a c} {\left | h x + g \right |}}\right )}{\sqrt {a + \frac {c g^{2}}{h^{2}}} h^{2}} \]
-sqrt(c*x^2 + a)*f*g^2/(c*g^2*h^2*x + a*h^4*x + c*g^3*h + a*g*h^3) + sqrt( c*x^2 + a)*e*g/(c*g^2*h*x + a*h^3*x + c*g^3 + a*g*h^2) - sqrt(c*x^2 + a)*d /(c*g^2*x + a*h^2*x + c*g^3/h + a*g*h) + f*arcsinh(c*x/sqrt(a*c))/(sqrt(c) *h^2) + c*f*g^3*arcsinh(c*g*x/(sqrt(a*c)*abs(h*x + g)) - a*h/(sqrt(a*c)*ab s(h*x + g)))/((a + c*g^2/h^2)^(3/2)*h^5) - c*e*g^2*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^4 ) + c*d*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^3) - 2*f*g*arcsinh(c*g*x/(sqrt(a*c)*abs(h* x + g)) - a*h/(sqrt(a*c)*abs(h*x + g)))/(sqrt(a + c*g^2/h^2)*h^3) + e*arcs inh(c*g*x/(sqrt(a*c)*abs(h*x + g)) - a*h/(sqrt(a*c)*abs(h*x + g)))/(sqrt(a + c*g^2/h^2)*h^2)
Exception generated. \[ \int \frac {d+e x+f x^2}{(g+h x)^2 \sqrt {a+c x^2}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
Timed out. \[ \int \frac {d+e x+f x^2}{(g+h x)^2 \sqrt {a+c x^2}} \, dx=\int \frac {f\,x^2+e\,x+d}{{\left (g+h\,x\right )}^2\,\sqrt {c\,x^2+a}} \,d x \]