Integrand size = 29, antiderivative size = 308 \[ \int \frac {\sqrt {a+c x^2} \left (d+e x+f x^2\right )}{(g+h x)^2} \, dx=-\frac {\left (2 \left (a h^2 (2 f g-e h)+c g \left (3 f g^2-h (2 e g-d h)\right )\right )-h \left (a f h^2+c \left (3 f g^2-2 h (e g-d h)\right )\right ) x\right ) \sqrt {a+c x^2}}{2 h^3 \left (c g^2+a h^2\right )}-\frac {\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{3/2}}{h \left (c g^2+a h^2\right ) (g+h x)}+\frac {\left (a f h^2+2 c \left (3 f g^2-h (2 e g-d h)\right )\right ) \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{2 \sqrt {c} h^4}+\frac {\left (a h^2 (2 f g-e h)+c g \left (3 f g^2-h (2 e g-d h)\right )\right ) \text {arctanh}\left (\frac {a h-c g x}{\sqrt {c g^2+a h^2} \sqrt {a+c x^2}}\right )}{h^4 \sqrt {c g^2+a h^2}} \]
-(d*h^2-e*g*h+f*g^2)*(c*x^2+a)^(3/2)/h/(a*h^2+c*g^2)/(h*x+g)+1/2*(a*f*h^2+ 2*c*(3*f*g^2-h*(-d*h+2*e*g)))*arctanh(x*c^(1/2)/(c*x^2+a)^(1/2))/h^4/c^(1/ 2)+(a*h^2*(-e*h+2*f*g)+c*g*(3*f*g^2-h*(-d*h+2*e*g)))*arctanh((-c*g*x+a*h)/ (a*h^2+c*g^2)^(1/2)/(c*x^2+a)^(1/2))/h^4/(a*h^2+c*g^2)^(1/2)-1/2*(2*a*h^2* (-e*h+2*f*g)+2*c*g*(3*f*g^2-h*(-d*h+2*e*g))-h*(a*f*h^2+c*(3*f*g^2-2*h*(-d* h+e*g)))*x)*(c*x^2+a)^(1/2)/h^3/(a*h^2+c*g^2)
Time = 0.97 (sec) , antiderivative size = 217, normalized size of antiderivative = 0.70 \[ \int \frac {\sqrt {a+c x^2} \left (d+e x+f x^2\right )}{(g+h x)^2} \, dx=\frac {\frac {h \sqrt {a+c x^2} \left (2 h (2 e g-d h+e h x)+f \left (-6 g^2-3 g h x+h^2 x^2\right )\right )}{g+h x}+\frac {4 \left (3 c f g^3+c g h (-2 e g+d h)+a h^2 (2 f g-e h)\right ) \arctan \left (\frac {\sqrt {c} (g+h x)-h \sqrt {a+c x^2}}{\sqrt {-c g^2-a h^2}}\right )}{\sqrt {-c g^2-a h^2}}-\frac {\left (6 c f g^2+a f h^2+2 c h (-2 e g+d h)\right ) \log \left (-\sqrt {c} x+\sqrt {a+c x^2}\right )}{\sqrt {c}}}{2 h^4} \]
((h*Sqrt[a + c*x^2]*(2*h*(2*e*g - d*h + e*h*x) + f*(-6*g^2 - 3*g*h*x + h^2 *x^2)))/(g + h*x) + (4*(3*c*f*g^3 + c*g*h*(-2*e*g + d*h) + a*h^2*(2*f*g - e*h))*ArcTan[(Sqrt[c]*(g + h*x) - h*Sqrt[a + c*x^2])/Sqrt[-(c*g^2) - a*h^2 ]])/Sqrt[-(c*g^2) - a*h^2] - ((6*c*f*g^2 + a*f*h^2 + 2*c*h*(-2*e*g + d*h)) *Log[-(Sqrt[c]*x) + Sqrt[a + c*x^2]])/Sqrt[c])/(2*h^4)
Time = 0.67 (sec) , antiderivative size = 323, normalized size of antiderivative = 1.05, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.345, Rules used = {2182, 25, 682, 25, 27, 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.
| \(\displaystyle \int \frac {\sqrt {a+c x^2} \left (d+e x+f x^2\right )}{(g+h x)^2} \, dx\) |
| \(\Big \downarrow \) 2182 |
| \(\displaystyle -\frac {\int -\frac {\left (c d g-a f g+a e h+\left (a f h-c \left (-\frac {3 f g^2}{h}+2 e g-2 d h\right )\right ) x\right ) \sqrt {c x^2+a}}{g+h x}dx}{a h^2+c g^2}-\frac {\left (a+c x^2\right )^{3/2} \left (d h^2-e g h+f g^2\right )}{h (g+h x) \left (a h^2+c g^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {\left (c d g-a f g+a e h+\left (a f h-c \left (-\frac {3 f g^2}{h}+2 e g-2 d h\right )\right ) x\right ) \sqrt {c x^2+a}}{g+h x}dx}{a h^2+c g^2}-\frac {\left (a+c x^2\right )^{3/2} \left (d h^2-e g h+f g^2\right )}{h (g+h x) \left (a h^2+c g^2\right )}\) |
| \(\Big \downarrow \) 682 |
| \(\displaystyle \frac {\frac {\int -\frac {c \left (c g^2+a h^2\right ) \left (a h (3 f g-2 e h)-\left (6 c f g^2+a f h^2-2 c h (2 e g-d h)\right ) x\right )}{h (g+h x) \sqrt {c x^2+a}}dx}{2 c h^2}-\frac {\sqrt {a+c x^2} \left (2 \left (a h^2 (2 f g-e h)-c g h (2 e g-d h)+3 c f g^3\right )-h x \left (a f h^2-2 c h (e g-d h)+3 c f g^2\right )\right )}{2 h^3}}{a h^2+c g^2}-\frac {\left (a+c x^2\right )^{3/2} \left (d h^2-e g h+f g^2\right )}{h (g+h x) \left (a h^2+c g^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {\int \frac {c \left (c g^2+a h^2\right ) \left (a h (3 f g-2 e h)-\left (6 c f g^2+a f h^2-2 c h (2 e g-d h)\right ) x\right )}{h (g+h x) \sqrt {c x^2+a}}dx}{2 c h^2}-\frac {\sqrt {a+c x^2} \left (2 \left (a h^2 (2 f g-e h)-c g h (2 e g-d h)+3 c f g^3\right )-h x \left (a f h^2-2 c h (e g-d h)+3 c f g^2\right )\right )}{2 h^3}}{a h^2+c g^2}-\frac {\left (a+c x^2\right )^{3/2} \left (d h^2-e g h+f g^2\right )}{h (g+h x) \left (a h^2+c g^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {\left (a h^2+c g^2\right ) \int \frac {a h (3 f g-2 e h)-\left (6 c f g^2+a f h^2-2 c h (2 e g-d h)\right ) x}{(g+h x) \sqrt {c x^2+a}}dx}{2 h^3}-\frac {\sqrt {a+c x^2} \left (2 \left (a h^2 (2 f g-e h)-c g h (2 e g-d h)+3 c f g^3\right )-h x \left (a f h^2-2 c h (e g-d h)+3 c f g^2\right )\right )}{2 h^3}}{a h^2+c g^2}-\frac {\left (a+c x^2\right )^{3/2} \left (d h^2-e g h+f g^2\right )}{h (g+h x) \left (a h^2+c g^2\right )}\) |
| \(\Big \downarrow \) 719 |
| \(\displaystyle \frac {-\frac {\left (a h^2+c g^2\right ) \left (\frac {2 \left (a h^2 (2 f g-e h)-c g h (2 e g-d h)+3 c f g^3\right ) \int \frac {1}{(g+h x) \sqrt {c x^2+a}}dx}{h}-\frac {\left (a f h^2-2 c h (2 e g-d h)+6 c f g^2\right ) \int \frac {1}{\sqrt {c x^2+a}}dx}{h}\right )}{2 h^3}-\frac {\sqrt {a+c x^2} \left (2 \left (a h^2 (2 f g-e h)-c g h (2 e g-d h)+3 c f g^3\right )-h x \left (a f h^2-2 c h (e g-d h)+3 c f g^2\right )\right )}{2 h^3}}{a h^2+c g^2}-\frac {\left (a+c x^2\right )^{3/2} \left (d h^2-e g h+f g^2\right )}{h (g+h x) \left (a h^2+c g^2\right )}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {-\frac {\left (a h^2+c g^2\right ) \left (\frac {2 \left (a h^2 (2 f g-e h)-c g h (2 e g-d h)+3 c f g^3\right ) \int \frac {1}{(g+h x) \sqrt {c x^2+a}}dx}{h}-\frac {\left (a f h^2-2 c h (2 e g-d h)+6 c f g^2\right ) \int \frac {1}{1-\frac {c x^2}{c x^2+a}}d\frac {x}{\sqrt {c x^2+a}}}{h}\right )}{2 h^3}-\frac {\sqrt {a+c x^2} \left (2 \left (a h^2 (2 f g-e h)-c g h (2 e g-d h)+3 c f g^3\right )-h x \left (a f h^2-2 c h (e g-d h)+3 c f g^2\right )\right )}{2 h^3}}{a h^2+c g^2}-\frac {\left (a+c x^2\right )^{3/2} \left (d h^2-e g h+f g^2\right )}{h (g+h x) \left (a h^2+c g^2\right )}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {-\frac {\left (a h^2+c g^2\right ) \left (\frac {2 \left (a h^2 (2 f g-e h)-c g h (2 e g-d h)+3 c f g^3\right ) \int \frac {1}{(g+h x) \sqrt {c x^2+a}}dx}{h}-\frac {\text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right ) \left (a f h^2-2 c h (2 e g-d h)+6 c f g^2\right )}{\sqrt {c} h}\right )}{2 h^3}-\frac {\sqrt {a+c x^2} \left (2 \left (a h^2 (2 f g-e h)-c g h (2 e g-d h)+3 c f g^3\right )-h x \left (a f h^2-2 c h (e g-d h)+3 c f g^2\right )\right )}{2 h^3}}{a h^2+c g^2}-\frac {\left (a+c x^2\right )^{3/2} \left (d h^2-e g h+f g^2\right )}{h (g+h x) \left (a h^2+c g^2\right )}\) |
| \(\Big \downarrow \) 488 |
| \(\displaystyle \frac {-\frac {\left (a h^2+c g^2\right ) \left (-\frac {2 \left (a h^2 (2 f g-e h)-c g h (2 e g-d h)+3 c f g^3\right ) \int \frac {1}{c g^2+a h^2-\frac {(a h-c g x)^2}{c x^2+a}}d\frac {a h-c g x}{\sqrt {c x^2+a}}}{h}-\frac {\text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right ) \left (a f h^2-2 c h (2 e g-d h)+6 c f g^2\right )}{\sqrt {c} h}\right )}{2 h^3}-\frac {\sqrt {a+c x^2} \left (2 \left (a h^2 (2 f g-e h)-c g h (2 e g-d h)+3 c f g^3\right )-h x \left (a f h^2-2 c h (e g-d h)+3 c f g^2\right )\right )}{2 h^3}}{a h^2+c g^2}-\frac {\left (a+c x^2\right )^{3/2} \left (d h^2-e g h+f g^2\right )}{h (g+h x) \left (a h^2+c g^2\right )}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {-\frac {\left (a h^2+c g^2\right ) \left (-\frac {\text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right ) \left (a f h^2-2 c h (2 e g-d h)+6 c f g^2\right )}{\sqrt {c} h}-\frac {2 \text {arctanh}\left (\frac {a h-c g x}{\sqrt {a+c x^2} \sqrt {a h^2+c g^2}}\right ) \left (a h^2 (2 f g-e h)-c g h (2 e g-d h)+3 c f g^3\right )}{h \sqrt {a h^2+c g^2}}\right )}{2 h^3}-\frac {\sqrt {a+c x^2} \left (2 \left (a h^2 (2 f g-e h)-c g h (2 e g-d h)+3 c f g^3\right )-h x \left (a f h^2-2 c h (e g-d h)+3 c f g^2\right )\right )}{2 h^3}}{a h^2+c g^2}-\frac {\left (a+c x^2\right )^{3/2} \left (d h^2-e g h+f g^2\right )}{h (g+h x) \left (a h^2+c g^2\right )}\) |
-(((f*g^2 - e*g*h + d*h^2)*(a + c*x^2)^(3/2))/(h*(c*g^2 + a*h^2)*(g + h*x) )) + (-1/2*((2*(3*c*f*g^3 - c*g*h*(2*e*g - d*h) + a*h^2*(2*f*g - e*h)) - h *(3*c*f*g^2 + a*f*h^2 - 2*c*h*(e*g - d*h))*x)*Sqrt[a + c*x^2])/h^3 - ((c*g ^2 + a*h^2)*(-(((6*c*f*g^2 + a*f*h^2 - 2*c*h*(2*e*g - d*h))*ArcTanh[(Sqrt[ c]*x)/Sqrt[a + c*x^2]])/(Sqrt[c]*h)) - (2*(3*c*f*g^3 - c*g*h*(2*e*g - d*h) + a*h^2*(2*f*g - e*h))*ArcTanh[(a*h - c*g*x)/(Sqrt[c*g^2 + a*h^2]*Sqrt[a + c*x^2])])/(h*Sqrt[c*g^2 + a*h^2])))/(2*h^3))/(c*g^2 + a*h^2)
3.1.83.3.1 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/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[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*c*d*(2*p + 1) + g*c*e*(m + 2*p + 1)*x)*((a + c*x^2)^p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), x] + Simp[2*(p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))) Int[(d + e*x) ^m*(a + c*x^2)^(p - 1)*Simp[f*a*c*e^2*(m + 2*p + 2) + a*c*d*e*g*m - (c^2*f* d*e*(m + 2*p + 2) - g*(c^2*d^2*(2*p + 1) + a*c*e^2*(m + 2*p + 1)))*x, x], x ], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && GtQ[p, 0] && (IntegerQ[p] || ! RationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) && !ILtQ[m + 2*p, 0] && (Intege rQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
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]
Time = 0.56 (sec) , antiderivative size = 515, normalized size of antiderivative = 1.67
| method | result | size |
| risch | \(\frac {\left (f x h +2 e h -4 f g \right ) \sqrt {c \,x^{2}+a}}{2 h^{3}}+\frac {\frac {\left (a f \,h^{2}+2 c d \,h^{2}-4 c e g h +6 c f \,g^{2}\right ) \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )}{h \sqrt {c}}-\frac {\left (2 a e \,h^{3}-4 a f g \,h^{2}-4 c d g \,h^{2}+6 c e \,g^{2} h -8 c f \,g^{3}\right ) \ln \left (\frac {\frac {2 a \,h^{2}+2 c \,g^{2}}{h^{2}}-\frac {2 c g \left (x +\frac {g}{h}\right )}{h}+2 \sqrt {\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}\, \sqrt {\left (x +\frac {g}{h}\right )^{2} c -\frac {2 c g \left (x +\frac {g}{h}\right )}{h}+\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}}{x +\frac {g}{h}}\right )}{h^{2} \sqrt {\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}}+\frac {\left (2 a d \,h^{4}-2 a e g \,h^{3}+2 a f \,g^{2} h^{2}+2 c d \,g^{2} h^{2}-2 g^{3} c e h +2 g^{4} c f \right ) \left (-\frac {h^{2} \sqrt {\left (x +\frac {g}{h}\right )^{2} c -\frac {2 c g \left (x +\frac {g}{h}\right )}{h}+\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}}{\left (a \,h^{2}+c \,g^{2}\right ) \left (x +\frac {g}{h}\right )}-\frac {c g h \ln \left (\frac {\frac {2 a \,h^{2}+2 c \,g^{2}}{h^{2}}-\frac {2 c g \left (x +\frac {g}{h}\right )}{h}+2 \sqrt {\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}\, \sqrt {\left (x +\frac {g}{h}\right )^{2} c -\frac {2 c g \left (x +\frac {g}{h}\right )}{h}+\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}}{x +\frac {g}{h}}\right )}{\left (a \,h^{2}+c \,g^{2}\right ) \sqrt {\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}}\right )}{h^{3}}}{2 h^{3}}\) | \(515\) |
| default | \(\frac {f \left (\frac {x \sqrt {c \,x^{2}+a}}{2}+\frac {a \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )}{2 \sqrt {c}}\right )}{h^{2}}+\frac {\left (e h -2 f g \right ) \left (\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 {\sqrt {c}\, g \ln \left (\frac {-\frac {c g}{h}+c \left (x +\frac {g}{h}\right )}{\sqrt {c}}+\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}}}\right )}{h}-\frac {\left (a \,h^{2}+c \,g^{2}\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^{2} \sqrt {\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}}\right )}{h^{3}}+\frac {\left (d \,h^{2}-e g h +f \,g^{2}\right ) \left (-\frac {h^{2} \left (\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}}\right )^{\frac {3}{2}}}{\left (a \,h^{2}+c \,g^{2}\right ) \left (x +\frac {g}{h}\right )}-\frac {c g h \left (\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 {\sqrt {c}\, g \ln \left (\frac {-\frac {c g}{h}+c \left (x +\frac {g}{h}\right )}{\sqrt {c}}+\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}}}\right )}{h}-\frac {\left (a \,h^{2}+c \,g^{2}\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^{2} \sqrt {\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}}\right )}{a \,h^{2}+c \,g^{2}}+\frac {2 c \,h^{2} \left (\frac {\left (2 c \left (x +\frac {g}{h}\right )-\frac {2 c g}{h}\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}}}}{4 c}+\frac {\left (\frac {4 c \left (a \,h^{2}+c \,g^{2}\right )}{h^{2}}-\frac {4 c^{2} g^{2}}{h^{2}}\right ) \ln \left (\frac {-\frac {c g}{h}+c \left (x +\frac {g}{h}\right )}{\sqrt {c}}+\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}}}\right )}{8 c^{\frac {3}{2}}}\right )}{a \,h^{2}+c \,g^{2}}\right )}{h^{4}}\) | \(861\) |
1/2*(f*h*x+2*e*h-4*f*g)*(c*x^2+a)^(1/2)/h^3+1/2/h^3*((a*f*h^2+2*c*d*h^2-4* c*e*g*h+6*c*f*g^2)/h*ln(x*c^(1/2)+(c*x^2+a)^(1/2))/c^(1/2)-(2*a*e*h^3-4*a* f*g*h^2-4*c*d*g*h^2+6*c*e*g^2*h-8*c*f*g^3)/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))+1/h^3*(2* a*d*h^4-2*a*e*g*h^3+2*a*f*g^2*h^2+2*c*d*g^2*h^2-2*c*e*g^3*h+2*c*f*g^4)*(-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. \[ \int \frac {\sqrt {a+c x^2} \left (d+e x+f x^2\right )}{(g+h x)^2} \, dx=\text {Timed out} \]
\[ \int \frac {\sqrt {a+c x^2} \left (d+e x+f x^2\right )}{(g+h x)^2} \, dx=\int \frac {\sqrt {a + c x^{2}} \left (d + e x + f x^{2}\right )}{\left (g + h x\right )^{2}}\, dx \]
Time = 0.23 (sec) , antiderivative size = 478, normalized size of antiderivative = 1.55 \[ \int \frac {\sqrt {a+c x^2} \left (d+e x+f x^2\right )}{(g+h x)^2} \, dx=-\frac {\sqrt {c x^{2} + a} f g^{2}}{h^{4} x + g h^{3}} + \frac {\sqrt {c x^{2} + a} e g}{h^{3} x + g h^{2}} - \frac {\sqrt {c x^{2} + a} d}{h^{2} x + g h} + \frac {\sqrt {c x^{2} + a} f x}{2 \, h^{2}} + \frac {3 \, \sqrt {c} f g^{2} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{h^{4}} - \frac {2 \, \sqrt {c} e g \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{h^{3}} + \frac {\sqrt {c} d \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{h^{2}} + \frac {a f \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{2 \, \sqrt {c} h^{2}} - \frac {c f g^{3} \operatorname {arsinh}\left (\frac {c g x}{\sqrt {a c} {\left | h x + g \right |}} - \frac {a h}{\sqrt {a c} {\left | h x + g \right |}}\right )}{\sqrt {a + \frac {c g^{2}}{h^{2}}} h^{5}} + \frac {c e g^{2} \operatorname {arsinh}\left (\frac {c g x}{\sqrt {a c} {\left | h x + g \right |}} - \frac {a h}{\sqrt {a c} {\left | h x + g \right |}}\right )}{\sqrt {a + \frac {c g^{2}}{h^{2}}} h^{4}} - \frac {c d g \operatorname {arsinh}\left (\frac {c g x}{\sqrt {a c} {\left | h x + g \right |}} - \frac {a h}{\sqrt {a c} {\left | h x + g \right |}}\right )}{\sqrt {a + \frac {c g^{2}}{h^{2}}} h^{3}} - \frac {2 \, \sqrt {a + \frac {c g^{2}}{h^{2}}} f g \operatorname {arsinh}\left (\frac {c g x}{\sqrt {a c} {\left | h x + g \right |}} - \frac {a h}{\sqrt {a c} {\left | h x + g \right |}}\right )}{h^{3}} + \frac {\sqrt {a + \frac {c g^{2}}{h^{2}}} e \operatorname {arsinh}\left (\frac {c g x}{\sqrt {a c} {\left | h x + g \right |}} - \frac {a h}{\sqrt {a c} {\left | h x + g \right |}}\right )}{h^{2}} - \frac {2 \, \sqrt {c x^{2} + a} f g}{h^{3}} + \frac {\sqrt {c x^{2} + a} e}{h^{2}} \]
-sqrt(c*x^2 + a)*f*g^2/(h^4*x + g*h^3) + sqrt(c*x^2 + a)*e*g/(h^3*x + g*h^ 2) - sqrt(c*x^2 + a)*d/(h^2*x + g*h) + 1/2*sqrt(c*x^2 + a)*f*x/h^2 + 3*sqr t(c)*f*g^2*arcsinh(c*x/sqrt(a*c))/h^4 - 2*sqrt(c)*e*g*arcsinh(c*x/sqrt(a*c ))/h^3 + sqrt(c)*d*arcsinh(c*x/sqrt(a*c))/h^2 + 1/2*a*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)*abs(h*x + g)))/(sqrt(a + c*g^2/h^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)))/(sqrt(a + c*g^2 /h^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)))/(sqrt(a + c*g^2/h^2)*h^3) - 2*sqrt(a + c*g^2/h^2)*f*g*arcs inh(c*g*x/(sqrt(a*c)*abs(h*x + g)) - a*h/(sqrt(a*c)*abs(h*x + g)))/h^3 + s qrt(a + c*g^2/h^2)*e*arcsinh(c*g*x/(sqrt(a*c)*abs(h*x + g)) - a*h/(sqrt(a* c)*abs(h*x + g)))/h^2 - 2*sqrt(c*x^2 + a)*f*g/h^3 + sqrt(c*x^2 + a)*e/h^2
\[ \int \frac {\sqrt {a+c x^2} \left (d+e x+f x^2\right )}{(g+h x)^2} \, dx=\int { \frac {\sqrt {c x^{2} + a} {\left (f x^{2} + e x + d\right )}}{{\left (h x + g\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {\sqrt {a+c x^2} \left (d+e x+f x^2\right )}{(g+h x)^2} \, dx=\int \frac {\sqrt {c\,x^2+a}\,\left (f\,x^2+e\,x+d\right )}{{\left (g+h\,x\right )}^2} \,d x \]