Integrand size = 21, antiderivative size = 140 \[ \int \frac {\sqrt {b x+c x^2}}{(d+e x)^2} \, dx=-\frac {\sqrt {b x+c x^2}}{e (d+e x)}+\frac {2 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{e^2}-\frac {(2 c d-b e) \text {arctanh}\left (\frac {b d+(2 c d-b e) x}{2 \sqrt {d} \sqrt {c d-b e} \sqrt {b x+c x^2}}\right )}{2 \sqrt {d} e^2 \sqrt {c d-b e}} \]
2*arctanh(x*c^(1/2)/(c*x^2+b*x)^(1/2))*c^(1/2)/e^2-1/2*(-b*e+2*c*d)*arctan h(1/2*(b*d+(-b*e+2*c*d)*x)/d^(1/2)/(-b*e+c*d)^(1/2)/(c*x^2+b*x)^(1/2))/e^2 /d^(1/2)/(-b*e+c*d)^(1/2)-(c*x^2+b*x)^(1/2)/e/(e*x+d)
Time = 0.70 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.14 \[ \int \frac {\sqrt {b x+c x^2}}{(d+e x)^2} \, dx=\frac {\sqrt {x (b+c x)} \left (-\frac {e}{d+e x}+\frac {(2 c d-b e) \arctan \left (\frac {-e \sqrt {x} \sqrt {b+c x}+\sqrt {c} (d+e x)}{\sqrt {d} \sqrt {-c d+b e}}\right )}{\sqrt {d} \sqrt {-c d+b e} \sqrt {x} \sqrt {b+c x}}-\frac {2 \sqrt {c} \log \left (-\sqrt {c} \sqrt {x}+\sqrt {b+c x}\right )}{\sqrt {x} \sqrt {b+c x}}\right )}{e^2} \]
(Sqrt[x*(b + c*x)]*(-(e/(d + e*x)) + ((2*c*d - b*e)*ArcTan[(-(e*Sqrt[x]*Sq rt[b + c*x]) + Sqrt[c]*(d + e*x))/(Sqrt[d]*Sqrt[-(c*d) + b*e])])/(Sqrt[d]* Sqrt[-(c*d) + b*e]*Sqrt[x]*Sqrt[b + c*x]) - (2*Sqrt[c]*Log[-(Sqrt[c]*Sqrt[ x]) + Sqrt[b + c*x]])/(Sqrt[x]*Sqrt[b + c*x])))/e^2
Time = 0.31 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.04, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {1161, 1269, 1091, 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.
| \(\displaystyle \int \frac {\sqrt {b x+c x^2}}{(d+e x)^2} \, dx\) |
| \(\Big \downarrow \) 1161 |
| \(\displaystyle \frac {\int \frac {b+2 c x}{(d+e x) \sqrt {c x^2+b x}}dx}{2 e}-\frac {\sqrt {b x+c x^2}}{e (d+e x)}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {\frac {2 c \int \frac {1}{\sqrt {c x^2+b x}}dx}{e}-\frac {(2 c d-b e) \int \frac {1}{(d+e x) \sqrt {c x^2+b x}}dx}{e}}{2 e}-\frac {\sqrt {b x+c x^2}}{e (d+e x)}\) |
| \(\Big \downarrow \) 1091 |
| \(\displaystyle \frac {\frac {4 c \int \frac {1}{1-\frac {c x^2}{c x^2+b x}}d\frac {x}{\sqrt {c x^2+b x}}}{e}-\frac {(2 c d-b e) \int \frac {1}{(d+e x) \sqrt {c x^2+b x}}dx}{e}}{2 e}-\frac {\sqrt {b x+c x^2}}{e (d+e x)}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {4 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{e}-\frac {(2 c d-b e) \int \frac {1}{(d+e x) \sqrt {c x^2+b x}}dx}{e}}{2 e}-\frac {\sqrt {b x+c x^2}}{e (d+e x)}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {\frac {2 (2 c d-b e) \int \frac {1}{4 d (c d-b e)-\frac {(b d+(2 c d-b e) x)^2}{c x^2+b x}}d\left (-\frac {b d+(2 c d-b e) x}{\sqrt {c x^2+b x}}\right )}{e}+\frac {4 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{e}}{2 e}-\frac {\sqrt {b x+c x^2}}{e (d+e x)}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {4 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{e}-\frac {(2 c d-b e) \text {arctanh}\left (\frac {x (2 c d-b e)+b d}{2 \sqrt {d} \sqrt {b x+c x^2} \sqrt {c d-b e}}\right )}{\sqrt {d} e \sqrt {c d-b e}}}{2 e}-\frac {\sqrt {b x+c x^2}}{e (d+e x)}\) |
-(Sqrt[b*x + c*x^2]/(e*(d + e*x))) + ((4*Sqrt[c]*ArcTanh[(Sqrt[c]*x)/Sqrt[ b*x + c*x^2]])/e - ((2*c*d - b*e)*ArcTanh[(b*d + (2*c*d - b*e)*x)/(2*Sqrt[ d]*Sqrt[c*d - b*e]*Sqrt[b*x + c*x^2])])/(Sqrt[d]*e*Sqrt[c*d - b*e]))/(2*e)
3.3.90.3.1 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[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2 Subst[Int[1/(1 - c*x^2), x], x, x/Sqrt[b*x + c*x^2]], x] /; FreeQ[{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_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(d + e*x)^(m + 1)*((a + b*x + c*x^2)^p/(e*(m + 1))), x] - Si mp[p/(e*(m + 1)) Int[(d + e*x)^(m + 1)*(b + 2*c*x)*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && GtQ[p, 0] && (IntegerQ[p] || LtQ[m, -1]) && NeQ[m, -1] && !ILtQ[m + 2*p + 1, 0] && IntQuadraticQ[a, b, c, d, e, m, p, x]
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]
Time = 2.37 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.86
| method | result | size |
| pseudoelliptic | \(-\frac {\left (e x +d \right ) \left (b e -2 c d \right ) \arctan \left (\frac {\sqrt {x \left (c x +b \right )}\, d}{x \sqrt {d \left (b e -c d \right )}}\right )-2 \sqrt {d \left (b e -c d \right )}\, \left (\sqrt {c}\, \left (e x +d \right ) \operatorname {arctanh}\left (\frac {\sqrt {x \left (c x +b \right )}}{x \sqrt {c}}\right )-\frac {\sqrt {x \left (c x +b \right )}\, e}{2}\right )}{\sqrt {d \left (b e -c d \right )}\, e^{2} \left (e x +d \right )}\) | \(120\) |
| default | \(\frac {\frac {e^{2} \left (c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}\right )^{\frac {3}{2}}}{d \left (b e -c d \right ) \left (x +\frac {d}{e}\right )}-\frac {e \left (b e -2 c d \right ) \left (\sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}+\frac {\left (b e -2 c d \right ) \ln \left (\frac {\frac {b e -2 c d}{2 e}+c \left (x +\frac {d}{e}\right )}{\sqrt {c}}+\sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}\right )}{2 e \sqrt {c}}+\frac {d \left (b e -c d \right ) \ln \left (\frac {-\frac {2 d \left (b e -c d \right )}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{2} \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}}\right )}{2 d \left (b e -c d \right )}-\frac {2 c \,e^{2} \left (\frac {\left (2 c \left (x +\frac {d}{e}\right )+\frac {b e -2 c d}{e}\right ) \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{4 c}+\frac {\left (-\frac {4 c d \left (b e -c d \right )}{e^{2}}-\frac {\left (b e -2 c d \right )^{2}}{e^{2}}\right ) \ln \left (\frac {\frac {b e -2 c d}{2 e}+c \left (x +\frac {d}{e}\right )}{\sqrt {c}}+\sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}\right )}{8 c^{\frac {3}{2}}}\right )}{d \left (b e -c d \right )}}{e^{2}}\) | \(589\) |
-1/(d*(b*e-c*d))^(1/2)*((e*x+d)*(b*e-2*c*d)*arctan((x*(c*x+b))^(1/2)/x*d/( d*(b*e-c*d))^(1/2))-2*(d*(b*e-c*d))^(1/2)*(c^(1/2)*(e*x+d)*arctanh((x*(c*x +b))^(1/2)/x/c^(1/2))-1/2*(x*(c*x+b))^(1/2)*e))/e^2/(e*x+d)
Time = 0.32 (sec) , antiderivative size = 846, normalized size of antiderivative = 6.04 \[ \int \frac {\sqrt {b x+c x^2}}{(d+e x)^2} \, dx=\left [\frac {2 \, {\left (c d^{3} - b d^{2} e + {\left (c d^{2} e - b d e^{2}\right )} x\right )} \sqrt {c} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) - {\left (2 \, c d^{2} - b d e + {\left (2 \, c d e - b e^{2}\right )} x\right )} \sqrt {c d^{2} - b d e} \log \left (\frac {b d + {\left (2 \, c d - b e\right )} x + 2 \, \sqrt {c d^{2} - b d e} \sqrt {c x^{2} + b x}}{e x + d}\right ) - 2 \, {\left (c d^{2} e - b d e^{2}\right )} \sqrt {c x^{2} + b x}}{2 \, {\left (c d^{3} e^{2} - b d^{2} e^{3} + {\left (c d^{2} e^{3} - b d e^{4}\right )} x\right )}}, -\frac {{\left (2 \, c d^{2} - b d e + {\left (2 \, c d e - b e^{2}\right )} x\right )} \sqrt {-c d^{2} + b d e} \arctan \left (-\frac {\sqrt {-c d^{2} + b d e} \sqrt {c x^{2} + b x}}{{\left (c d - b e\right )} x}\right ) - {\left (c d^{3} - b d^{2} e + {\left (c d^{2} e - b d e^{2}\right )} x\right )} \sqrt {c} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) + {\left (c d^{2} e - b d e^{2}\right )} \sqrt {c x^{2} + b x}}{c d^{3} e^{2} - b d^{2} e^{3} + {\left (c d^{2} e^{3} - b d e^{4}\right )} x}, -\frac {4 \, {\left (c d^{3} - b d^{2} e + {\left (c d^{2} e - b d e^{2}\right )} x\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) + {\left (2 \, c d^{2} - b d e + {\left (2 \, c d e - b e^{2}\right )} x\right )} \sqrt {c d^{2} - b d e} \log \left (\frac {b d + {\left (2 \, c d - b e\right )} x + 2 \, \sqrt {c d^{2} - b d e} \sqrt {c x^{2} + b x}}{e x + d}\right ) + 2 \, {\left (c d^{2} e - b d e^{2}\right )} \sqrt {c x^{2} + b x}}{2 \, {\left (c d^{3} e^{2} - b d^{2} e^{3} + {\left (c d^{2} e^{3} - b d e^{4}\right )} x\right )}}, -\frac {{\left (2 \, c d^{2} - b d e + {\left (2 \, c d e - b e^{2}\right )} x\right )} \sqrt {-c d^{2} + b d e} \arctan \left (-\frac {\sqrt {-c d^{2} + b d e} \sqrt {c x^{2} + b x}}{{\left (c d - b e\right )} x}\right ) + 2 \, {\left (c d^{3} - b d^{2} e + {\left (c d^{2} e - b d e^{2}\right )} x\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) + {\left (c d^{2} e - b d e^{2}\right )} \sqrt {c x^{2} + b x}}{c d^{3} e^{2} - b d^{2} e^{3} + {\left (c d^{2} e^{3} - b d e^{4}\right )} x}\right ] \]
[1/2*(2*(c*d^3 - b*d^2*e + (c*d^2*e - b*d*e^2)*x)*sqrt(c)*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c)) - (2*c*d^2 - b*d*e + (2*c*d*e - b*e^2)*x)*sqr t(c*d^2 - b*d*e)*log((b*d + (2*c*d - b*e)*x + 2*sqrt(c*d^2 - b*d*e)*sqrt(c *x^2 + b*x))/(e*x + d)) - 2*(c*d^2*e - b*d*e^2)*sqrt(c*x^2 + b*x))/(c*d^3* e^2 - b*d^2*e^3 + (c*d^2*e^3 - b*d*e^4)*x), -((2*c*d^2 - b*d*e + (2*c*d*e - b*e^2)*x)*sqrt(-c*d^2 + b*d*e)*arctan(-sqrt(-c*d^2 + b*d*e)*sqrt(c*x^2 + b*x)/((c*d - b*e)*x)) - (c*d^3 - b*d^2*e + (c*d^2*e - b*d*e^2)*x)*sqrt(c) *log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c)) + (c*d^2*e - b*d*e^2)*sqrt(c *x^2 + b*x))/(c*d^3*e^2 - b*d^2*e^3 + (c*d^2*e^3 - b*d*e^4)*x), -1/2*(4*(c *d^3 - b*d^2*e + (c*d^2*e - b*d*e^2)*x)*sqrt(-c)*arctan(sqrt(c*x^2 + b*x)* sqrt(-c)/(c*x)) + (2*c*d^2 - b*d*e + (2*c*d*e - b*e^2)*x)*sqrt(c*d^2 - b*d *e)*log((b*d + (2*c*d - b*e)*x + 2*sqrt(c*d^2 - b*d*e)*sqrt(c*x^2 + b*x))/ (e*x + d)) + 2*(c*d^2*e - b*d*e^2)*sqrt(c*x^2 + b*x))/(c*d^3*e^2 - b*d^2*e ^3 + (c*d^2*e^3 - b*d*e^4)*x), -((2*c*d^2 - b*d*e + (2*c*d*e - b*e^2)*x)*s qrt(-c*d^2 + b*d*e)*arctan(-sqrt(-c*d^2 + b*d*e)*sqrt(c*x^2 + b*x)/((c*d - b*e)*x)) + 2*(c*d^3 - b*d^2*e + (c*d^2*e - b*d*e^2)*x)*sqrt(-c)*arctan(sq rt(c*x^2 + b*x)*sqrt(-c)/(c*x)) + (c*d^2*e - b*d*e^2)*sqrt(c*x^2 + b*x))/( c*d^3*e^2 - b*d^2*e^3 + (c*d^2*e^3 - b*d*e^4)*x)]
\[ \int \frac {\sqrt {b x+c x^2}}{(d+e x)^2} \, dx=\int \frac {\sqrt {x \left (b + c x\right )}}{\left (d + e x\right )^{2}}\, dx \]
Exception generated. \[ \int \frac {\sqrt {b x+c x^2}}{(d+e x)^2} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(b*e-c*d>0)', see `assume?` for m ore detail
Exception generated. \[ \int \frac {\sqrt {b x+c x^2}}{(d+e x)^2} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
Timed out. \[ \int \frac {\sqrt {b x+c x^2}}{(d+e x)^2} \, dx=\int \frac {\sqrt {c\,x^2+b\,x}}{{\left (d+e\,x\right )}^2} \,d x \]