Integrand size = 38, antiderivative size = 202 \[ \int \frac {\sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+e x+d x^2}}{x^2} \, dx=-\frac {(a-b x) \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+e x+d x^2}}{x (a+b x)}+\frac {(2 a d+b e) \sqrt {a^2+2 a b x+b^2 x^2} \text {arctanh}\left (\frac {e+2 d x}{2 \sqrt {d} \sqrt {c+e x+d x^2}}\right )}{2 \sqrt {d} (a+b x)}-\frac {(2 b c+a e) \sqrt {a^2+2 a b x+b^2 x^2} \text {arctanh}\left (\frac {2 c+e x}{2 \sqrt {c} \sqrt {c+e x+d x^2}}\right )}{2 \sqrt {c} (a+b x)} \]
-1/2*(a*e+2*b*c)*arctanh(1/2*(e*x+2*c)/c^(1/2)/(d*x^2+e*x+c)^(1/2))*((b*x+ a)^2)^(1/2)/(b*x+a)/c^(1/2)+1/2*(2*a*d+b*e)*arctanh(1/2*(2*d*x+e)/d^(1/2)/ (d*x^2+e*x+c)^(1/2))*((b*x+a)^2)^(1/2)/(b*x+a)/d^(1/2)-(-b*x+a)*((b*x+a)^2 )^(1/2)*(d*x^2+e*x+c)^(1/2)/x/(b*x+a)
Time = 0.45 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.75 \[ \int \frac {\sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+e x+d x^2}}{x^2} \, dx=-\frac {\sqrt {(a+b x)^2} \left (2 \sqrt {d} (2 b c+a e) x \text {arctanh}\left (\frac {-\sqrt {d} x+\sqrt {c+x (e+d x)}}{\sqrt {c}}\right )+\sqrt {c} \left (2 \sqrt {d} (a-b x) \sqrt {c+x (e+d x)}+(2 a d+b e) x \log \left (e+2 d x-2 \sqrt {d} \sqrt {c+x (e+d x)}\right )\right )\right )}{2 \sqrt {c} \sqrt {d} x (a+b x)} \]
-1/2*(Sqrt[(a + b*x)^2]*(2*Sqrt[d]*(2*b*c + a*e)*x*ArcTanh[(-(Sqrt[d]*x) + Sqrt[c + x*(e + d*x)])/Sqrt[c]] + Sqrt[c]*(2*Sqrt[d]*(a - b*x)*Sqrt[c + x *(e + d*x)] + (2*a*d + b*e)*x*Log[e + 2*d*x - 2*Sqrt[d]*Sqrt[c + x*(e + d* x)]])))/(Sqrt[c]*Sqrt[d]*x*(a + b*x))
Time = 0.36 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.74, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.237, Rules used = {1333, 27, 1230, 25, 1269, 1092, 219, 1154, 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.
| \(\displaystyle \int \frac {\sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+d x^2+e x}}{x^2} \, dx\) |
| \(\Big \downarrow \) 1333 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {2 b (a+b x) \sqrt {d x^2+e x+c}}{x^2}dx}{2 b (a+b x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {(a+b x) \sqrt {d x^2+e x+c}}{x^2}dx}{a+b x}\) |
| \(\Big \downarrow \) 1230 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (-\frac {1}{2} \int -\frac {2 b c+a e+(2 a d+b e) x}{x \sqrt {d x^2+e x+c}}dx-\frac {(a-b x) \sqrt {c+d x^2+e x}}{x}\right )}{a+b x}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (\frac {1}{2} \int \frac {2 b c+a e+(2 a d+b e) x}{x \sqrt {d x^2+e x+c}}dx-\frac {(a-b x) \sqrt {c+d x^2+e x}}{x}\right )}{a+b x}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (\frac {1}{2} \left ((2 a d+b e) \int \frac {1}{\sqrt {d x^2+e x+c}}dx+(a e+2 b c) \int \frac {1}{x \sqrt {d x^2+e x+c}}dx\right )-\frac {(a-b x) \sqrt {c+d x^2+e x}}{x}\right )}{a+b x}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (\frac {1}{2} \left ((a e+2 b c) \int \frac {1}{x \sqrt {d x^2+e x+c}}dx+2 (2 a d+b e) \int \frac {1}{4 d-\frac {(e+2 d x)^2}{d x^2+e x+c}}d\frac {e+2 d x}{\sqrt {d x^2+e x+c}}\right )-\frac {(a-b x) \sqrt {c+d x^2+e x}}{x}\right )}{a+b x}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (\frac {1}{2} \left ((a e+2 b c) \int \frac {1}{x \sqrt {d x^2+e x+c}}dx+\frac {(2 a d+b e) \text {arctanh}\left (\frac {2 d x+e}{2 \sqrt {d} \sqrt {c+d x^2+e x}}\right )}{\sqrt {d}}\right )-\frac {(a-b x) \sqrt {c+d x^2+e x}}{x}\right )}{a+b x}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (\frac {1}{2} \left (\frac {(2 a d+b e) \text {arctanh}\left (\frac {2 d x+e}{2 \sqrt {d} \sqrt {c+d x^2+e x}}\right )}{\sqrt {d}}-2 (a e+2 b c) \int \frac {1}{4 c-\frac {(2 c+e x)^2}{d x^2+e x+c}}d\frac {2 c+e x}{\sqrt {d x^2+e x+c}}\right )-\frac {(a-b x) \sqrt {c+d x^2+e x}}{x}\right )}{a+b x}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (\frac {1}{2} \left (\frac {(2 a d+b e) \text {arctanh}\left (\frac {2 d x+e}{2 \sqrt {d} \sqrt {c+d x^2+e x}}\right )}{\sqrt {d}}-\frac {(a e+2 b c) \text {arctanh}\left (\frac {2 c+e x}{2 \sqrt {c} \sqrt {c+d x^2+e x}}\right )}{\sqrt {c}}\right )-\frac {(a-b x) \sqrt {c+d x^2+e x}}{x}\right )}{a+b x}\) |
(Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(-(((a - b*x)*Sqrt[c + e*x + d*x^2])/x) + ( ((2*a*d + b*e)*ArcTanh[(e + 2*d*x)/(2*Sqrt[d]*Sqrt[c + e*x + d*x^2])])/Sqr t[d] - ((2*b*c + a*e)*ArcTanh[(2*c + e*x)/(2*Sqrt[c]*Sqrt[c + e*x + d*x^2] )])/Sqrt[c])/2))/(a + b*x)
3.1.50.3.1 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_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2 Subst[I nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a , b, c}, x]
Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym bol] :> Simp[-2 Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c , d, e}, x]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(e*f*(m + 2*p + 2) - d*g*(2*p + 1) + e*g*(m + 1)*x)*((a + b*x + c*x^2)^p/(e^2*(m + 1)*(m + 2*p + 2))), x] + Simp[p/(e^2*(m + 1)*(m + 2*p + 2)) Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p - 1)*Simp[g*(b*d + 2*a*e + 2*a*e*m + 2*b*d*p) - f*b*e*(m + 2*p + 2) + (g*(2*c*d + b*e + b*e*m + 4*c*d*p) - 2*c*e*f*(m + 2*p + 2))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && GtQ[p, 0] && (LtQ[m, - 1] || EqQ[p, 1] || (IntegerQ[p] && !RationalQ[m])) && NeQ[m, -1] && !ILtQ [m + 2*p + 1, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Simp[(e*f - d*g)/e Int[(d + e*x)^m*(a + b*x + c*x^2)^ p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && !IGtQ[m, 0]
Int[((g_.) + (h_.)*(x_))^(m_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_ ) + (e_.)*(x_) + (f_.)*(x_)^2)^(q_), x_Symbol] :> Simp[(a + b*x + c*x^2)^Fr acPart[p]/((4*c)^IntPart[p]*(b + 2*c*x)^(2*FracPart[p])) Int[(g + h*x)^m* (b + 2*c*x)^(2*p)*(d + e*x + f*x^2)^q, x], x] /; FreeQ[{a, b, c, d, e, f, g , h, m, p, q}, x] && EqQ[b^2 - 4*a*c, 0] && !IntegerQ[p]
Time = 0.59 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.00
| method | result | size |
| risch | \(-\frac {a \sqrt {d \,x^{2}+e x +c}\, \sqrt {\left (b x +a \right )^{2}}}{x \left (b x +a \right )}+\frac {\left (a \sqrt {d}\, \ln \left (\frac {\frac {e}{2}+d x}{\sqrt {d}}+\sqrt {d \,x^{2}+e x +c}\right )+\frac {b e \ln \left (\frac {\frac {e}{2}+d x}{\sqrt {d}}+\sqrt {d \,x^{2}+e x +c}\right )}{2 \sqrt {d}}+b \sqrt {d \,x^{2}+e x +c}-\frac {\ln \left (\frac {2 c +e x +2 \sqrt {c}\, \sqrt {d \,x^{2}+e x +c}}{x}\right ) a e}{2 \sqrt {c}}-\sqrt {c}\, \ln \left (\frac {2 c +e x +2 \sqrt {c}\, \sqrt {d \,x^{2}+e x +c}}{x}\right ) b \right ) \sqrt {\left (b x +a \right )^{2}}}{b x +a}\) | \(201\) |
| default | \(-\frac {\operatorname {csgn}\left (b x +a \right ) \left (-2 \sqrt {d \,x^{2}+e x +c}\, d^{\frac {5}{2}} a \,x^{2}+2 d^{\frac {3}{2}} c^{\frac {3}{2}} \ln \left (\frac {2 c +e x +2 \sqrt {c}\, \sqrt {d \,x^{2}+e x +c}}{x}\right ) b x +d^{\frac {3}{2}} \sqrt {c}\, \ln \left (\frac {2 c +e x +2 \sqrt {c}\, \sqrt {d \,x^{2}+e x +c}}{x}\right ) a e x +2 \left (d \,x^{2}+e x +c \right )^{\frac {3}{2}} d^{\frac {3}{2}} a -2 \sqrt {d \,x^{2}+e x +c}\, d^{\frac {3}{2}} a e x -2 \sqrt {d \,x^{2}+e x +c}\, d^{\frac {3}{2}} b c x -2 \ln \left (\frac {2 \sqrt {d \,x^{2}+e x +c}\, \sqrt {d}+2 d x +e}{2 \sqrt {d}}\right ) a c \,d^{2} x -\ln \left (\frac {2 \sqrt {d \,x^{2}+e x +c}\, \sqrt {d}+2 d x +e}{2 \sqrt {d}}\right ) d b c e x \right )}{2 c x \,d^{\frac {3}{2}}}\) | \(249\) |
-a*(d*x^2+e*x+c)^(1/2)/x*((b*x+a)^2)^(1/2)/(b*x+a)+(a*d^(1/2)*ln((1/2*e+d* x)/d^(1/2)+(d*x^2+e*x+c)^(1/2))+1/2*b*e*ln((1/2*e+d*x)/d^(1/2)+(d*x^2+e*x+ c)^(1/2))/d^(1/2)+b*(d*x^2+e*x+c)^(1/2)-1/2/c^(1/2)*ln((2*c+e*x+2*c^(1/2)* (d*x^2+e*x+c)^(1/2))/x)*a*e-c^(1/2)*ln((2*c+e*x+2*c^(1/2)*(d*x^2+e*x+c)^(1 /2))/x)*b)*((b*x+a)^2)^(1/2)/(b*x+a)
Time = 0.45 (sec) , antiderivative size = 647, normalized size of antiderivative = 3.20 \[ \int \frac {\sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+e x+d x^2}}{x^2} \, dx=\left [\frac {{\left (2 \, a c d + b c e\right )} \sqrt {d} x \log \left (8 \, d^{2} x^{2} + 8 \, d e x + 4 \, \sqrt {d x^{2} + e x + c} {\left (2 \, d x + e\right )} \sqrt {d} + 4 \, c d + e^{2}\right ) + {\left (2 \, b c d + a d e\right )} \sqrt {c} x \log \left (\frac {8 \, c e x + {\left (4 \, c d + e^{2}\right )} x^{2} - 4 \, \sqrt {d x^{2} + e x + c} {\left (e x + 2 \, c\right )} \sqrt {c} + 8 \, c^{2}}{x^{2}}\right ) + 4 \, {\left (b c d x - a c d\right )} \sqrt {d x^{2} + e x + c}}{4 \, c d x}, -\frac {2 \, {\left (2 \, a c d + b c e\right )} \sqrt {-d} x \arctan \left (\frac {\sqrt {d x^{2} + e x + c} {\left (2 \, d x + e\right )} \sqrt {-d}}{2 \, {\left (d^{2} x^{2} + d e x + c d\right )}}\right ) - {\left (2 \, b c d + a d e\right )} \sqrt {c} x \log \left (\frac {8 \, c e x + {\left (4 \, c d + e^{2}\right )} x^{2} - 4 \, \sqrt {d x^{2} + e x + c} {\left (e x + 2 \, c\right )} \sqrt {c} + 8 \, c^{2}}{x^{2}}\right ) - 4 \, {\left (b c d x - a c d\right )} \sqrt {d x^{2} + e x + c}}{4 \, c d x}, \frac {2 \, {\left (2 \, b c d + a d e\right )} \sqrt {-c} x \arctan \left (\frac {\sqrt {d x^{2} + e x + c} {\left (e x + 2 \, c\right )} \sqrt {-c}}{2 \, {\left (c d x^{2} + c e x + c^{2}\right )}}\right ) + {\left (2 \, a c d + b c e\right )} \sqrt {d} x \log \left (8 \, d^{2} x^{2} + 8 \, d e x + 4 \, \sqrt {d x^{2} + e x + c} {\left (2 \, d x + e\right )} \sqrt {d} + 4 \, c d + e^{2}\right ) + 4 \, {\left (b c d x - a c d\right )} \sqrt {d x^{2} + e x + c}}{4 \, c d x}, \frac {{\left (2 \, b c d + a d e\right )} \sqrt {-c} x \arctan \left (\frac {\sqrt {d x^{2} + e x + c} {\left (e x + 2 \, c\right )} \sqrt {-c}}{2 \, {\left (c d x^{2} + c e x + c^{2}\right )}}\right ) - {\left (2 \, a c d + b c e\right )} \sqrt {-d} x \arctan \left (\frac {\sqrt {d x^{2} + e x + c} {\left (2 \, d x + e\right )} \sqrt {-d}}{2 \, {\left (d^{2} x^{2} + d e x + c d\right )}}\right ) + 2 \, {\left (b c d x - a c d\right )} \sqrt {d x^{2} + e x + c}}{2 \, c d x}\right ] \]
[1/4*((2*a*c*d + b*c*e)*sqrt(d)*x*log(8*d^2*x^2 + 8*d*e*x + 4*sqrt(d*x^2 + e*x + c)*(2*d*x + e)*sqrt(d) + 4*c*d + e^2) + (2*b*c*d + a*d*e)*sqrt(c)*x *log((8*c*e*x + (4*c*d + e^2)*x^2 - 4*sqrt(d*x^2 + e*x + c)*(e*x + 2*c)*sq rt(c) + 8*c^2)/x^2) + 4*(b*c*d*x - a*c*d)*sqrt(d*x^2 + e*x + c))/(c*d*x), -1/4*(2*(2*a*c*d + b*c*e)*sqrt(-d)*x*arctan(1/2*sqrt(d*x^2 + e*x + c)*(2*d *x + e)*sqrt(-d)/(d^2*x^2 + d*e*x + c*d)) - (2*b*c*d + a*d*e)*sqrt(c)*x*lo g((8*c*e*x + (4*c*d + e^2)*x^2 - 4*sqrt(d*x^2 + e*x + c)*(e*x + 2*c)*sqrt( c) + 8*c^2)/x^2) - 4*(b*c*d*x - a*c*d)*sqrt(d*x^2 + e*x + c))/(c*d*x), 1/4 *(2*(2*b*c*d + a*d*e)*sqrt(-c)*x*arctan(1/2*sqrt(d*x^2 + e*x + c)*(e*x + 2 *c)*sqrt(-c)/(c*d*x^2 + c*e*x + c^2)) + (2*a*c*d + b*c*e)*sqrt(d)*x*log(8* d^2*x^2 + 8*d*e*x + 4*sqrt(d*x^2 + e*x + c)*(2*d*x + e)*sqrt(d) + 4*c*d + e^2) + 4*(b*c*d*x - a*c*d)*sqrt(d*x^2 + e*x + c))/(c*d*x), 1/2*((2*b*c*d + a*d*e)*sqrt(-c)*x*arctan(1/2*sqrt(d*x^2 + e*x + c)*(e*x + 2*c)*sqrt(-c)/( c*d*x^2 + c*e*x + c^2)) - (2*a*c*d + b*c*e)*sqrt(-d)*x*arctan(1/2*sqrt(d*x ^2 + e*x + c)*(2*d*x + e)*sqrt(-d)/(d^2*x^2 + d*e*x + c*d)) + 2*(b*c*d*x - a*c*d)*sqrt(d*x^2 + e*x + c))/(c*d*x)]
\[ \int \frac {\sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+e x+d x^2}}{x^2} \, dx=\int \frac {\sqrt {c + d x^{2} + e x} \sqrt {\left (a + b x\right )^{2}}}{x^{2}}\, dx \]
\[ \int \frac {\sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+e x+d x^2}}{x^2} \, dx=\int { \frac {\sqrt {d x^{2} + e x + c} \sqrt {{\left (b x + a\right )}^{2}}}{x^{2}} \,d x } \]
Time = 0.66 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+e x+d x^2}}{x^2} \, dx=\sqrt {d x^{2} + e x + c} b \mathrm {sgn}\left (b x + a\right ) + \frac {{\left (2 \, b c \mathrm {sgn}\left (b x + a\right ) + a e \mathrm {sgn}\left (b x + a\right )\right )} \arctan \left (-\frac {\sqrt {d} x - \sqrt {d x^{2} + e x + c}}{\sqrt {-c}}\right )}{\sqrt {-c}} - \frac {{\left (2 \, a d \mathrm {sgn}\left (b x + a\right ) + b e \mathrm {sgn}\left (b x + a\right )\right )} \log \left ({\left | -2 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + e x + c}\right )} \sqrt {d} - e \right |}\right )}{2 \, \sqrt {d}} + \frac {{\left (\sqrt {d} x - \sqrt {d x^{2} + e x + c}\right )} a e \mathrm {sgn}\left (b x + a\right ) + 2 \, a c \sqrt {d} \mathrm {sgn}\left (b x + a\right )}{{\left (\sqrt {d} x - \sqrt {d x^{2} + e x + c}\right )}^{2} - c} \]
sqrt(d*x^2 + e*x + c)*b*sgn(b*x + a) + (2*b*c*sgn(b*x + a) + a*e*sgn(b*x + a))*arctan(-(sqrt(d)*x - sqrt(d*x^2 + e*x + c))/sqrt(-c))/sqrt(-c) - 1/2* (2*a*d*sgn(b*x + a) + b*e*sgn(b*x + a))*log(abs(-2*(sqrt(d)*x - sqrt(d*x^2 + e*x + c))*sqrt(d) - e))/sqrt(d) + ((sqrt(d)*x - sqrt(d*x^2 + e*x + c))* a*e*sgn(b*x + a) + 2*a*c*sqrt(d)*sgn(b*x + a))/((sqrt(d)*x - sqrt(d*x^2 + e*x + c))^2 - c)
Timed out. \[ \int \frac {\sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+e x+d x^2}}{x^2} \, dx=\int \frac {\sqrt {{\left (a+b\,x\right )}^2}\,\sqrt {d\,x^2+e\,x+c}}{x^2} \,d x \]