Integrand size = 21, antiderivative size = 125 \[ \int \frac {\sqrt {d+e x}}{\left (b x+c x^2\right )^2} \, dx=-\frac {(b+2 c x) \sqrt {d+e x}}{b^2 \left (b x+c x^2\right )}+\frac {(4 c d-b e) \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b^3 \sqrt {d}}-\frac {\sqrt {c} (4 c d-3 b e) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b^3 \sqrt {c d-b e}} \]
(-b*e+4*c*d)*arctanh((e*x+d)^(1/2)/d^(1/2))/b^3/d^(1/2)-(-3*b*e+4*c*d)*arc tanh(c^(1/2)*(e*x+d)^(1/2)/(-b*e+c*d)^(1/2))*c^(1/2)/b^3/(-b*e+c*d)^(1/2)- (2*c*x+b)*(e*x+d)^(1/2)/b^2/(c*x^2+b*x)
Time = 0.85 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.95 \[ \int \frac {\sqrt {d+e x}}{\left (b x+c x^2\right )^2} \, dx=\frac {-\frac {b (b+2 c x) \sqrt {d+e x}}{x (b+c x)}+\frac {\sqrt {c} (4 c d-3 b e) \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {-c d+b e}}\right )}{\sqrt {-c d+b e}}+\frac {(4 c d-b e) \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{\sqrt {d}}}{b^3} \]
(-((b*(b + 2*c*x)*Sqrt[d + e*x])/(x*(b + c*x))) + (Sqrt[c]*(4*c*d - 3*b*e) *ArcTan[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[-(c*d) + b*e]])/Sqrt[-(c*d) + b*e] + ((4*c*d - b*e)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/Sqrt[d])/b^3
Time = 0.36 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.10, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {1163, 27, 1197, 27, 1480, 221}
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.
| \(\displaystyle \int \frac {\sqrt {d+e x}}{\left (b x+c x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 1163 |
| \(\displaystyle \frac {\int -\frac {4 c d-b e+2 c e x}{2 \sqrt {d+e x} \left (c x^2+b x\right )}dx}{b^2}-\frac {(b+2 c x) \sqrt {d+e x}}{b^2 \left (b x+c x^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {4 c d-b e+2 c e x}{\sqrt {d+e x} \left (c x^2+b x\right )}dx}{2 b^2}-\frac {(b+2 c x) \sqrt {d+e x}}{b^2 \left (b x+c x^2\right )}\) |
| \(\Big \downarrow \) 1197 |
| \(\displaystyle -\frac {\int \frac {e (2 c d-b e+2 c (d+e x))}{c (d+e x)^2-(2 c d-b e) (d+e x)+d (c d-b e)}d\sqrt {d+e x}}{b^2}-\frac {(b+2 c x) \sqrt {d+e x}}{b^2 \left (b x+c x^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {e \int \frac {2 c d-b e+2 c (d+e x)}{c (d+e x)^2-(2 c d-b e) (d+e x)+d (c d-b e)}d\sqrt {d+e x}}{b^2}-\frac {(b+2 c x) \sqrt {d+e x}}{b^2 \left (b x+c x^2\right )}\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle -\frac {e \left (\frac {c (4 c d-b e) \int \frac {1}{c (d+e x)-c d}d\sqrt {d+e x}}{b e}-\frac {c (4 c d-3 b e) \int \frac {1}{-c d+b e+c (d+e x)}d\sqrt {d+e x}}{b e}\right )}{b^2}-\frac {(b+2 c x) \sqrt {d+e x}}{b^2 \left (b x+c x^2\right )}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {e \left (\frac {\sqrt {c} (4 c d-3 b e) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b e \sqrt {c d-b e}}-\frac {\text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) (4 c d-b e)}{b \sqrt {d} e}\right )}{b^2}-\frac {(b+2 c x) \sqrt {d+e x}}{b^2 \left (b x+c x^2\right )}\) |
-(((b + 2*c*x)*Sqrt[d + e*x])/(b^2*(b*x + c*x^2))) - (e*(-(((4*c*d - b*e)* ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(b*Sqrt[d]*e)) + (Sqrt[c]*(4*c*d - 3*b*e)* ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(b*e*Sqrt[c*d - b*e])))/ b^2
3.4.73.3.1 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[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(d + e*x)^m*(b + 2*c*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)* (b^2 - 4*a*c))), x] - Simp[1/((p + 1)*(b^2 - 4*a*c)) Int[(d + e*x)^(m - 1 )*(b*e*m + 2*c*d*(2*p + 3) + 2*c*e*(m + 2*p + 3)*x)*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && LtQ[p, -1] && GtQ[m, 0] && (LtQ[ m, 1] || (ILtQ[m + 2*p + 3, 0] && NeQ[m, 2])) && IntQuadraticQ[a, b, c, d, e, m, p, x]
Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)), x_Symbol] :> Simp[2 Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; Fr eeQ[{a, b, c, d, e, f, g}, x]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : > With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q)) Int[1/( b/2 - q/2 + c*x^2), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q)) Int[1/(b/2 + q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^2 - 4*a*c]
Time = 1.99 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.06
| method | result | size |
| pseudoelliptic | \(-\frac {-4 x \left (c d -\frac {3 b e}{4}\right ) \left (c x +b \right ) c \sqrt {d}\, \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )+\left (x \left (c x +b \right ) \left (b e -4 c d \right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )+\sqrt {d}\, \sqrt {e x +d}\, \left (2 c x +b \right ) b \right ) \sqrt {\left (b e -c d \right ) c}}{\sqrt {d}\, \sqrt {\left (b e -c d \right ) c}\, b^{3} x \left (c x +b \right )}\) | \(133\) |
| derivativedivides | \(2 e^{3} \left (-\frac {c \left (\frac {b e \sqrt {e x +d}}{2 c \left (e x +d \right )+2 b e -2 c d}+\frac {\left (3 b e -4 c d \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{2 \sqrt {\left (b e -c d \right ) c}}\right )}{b^{3} e^{3}}+\frac {-\frac {b \sqrt {e x +d}}{2 x}-\frac {\left (b e -4 c d \right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{2 \sqrt {d}}}{b^{3} e^{3}}\right )\) | \(136\) |
| default | \(2 e^{3} \left (-\frac {c \left (\frac {b e \sqrt {e x +d}}{2 c \left (e x +d \right )+2 b e -2 c d}+\frac {\left (3 b e -4 c d \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{2 \sqrt {\left (b e -c d \right ) c}}\right )}{b^{3} e^{3}}+\frac {-\frac {b \sqrt {e x +d}}{2 x}-\frac {\left (b e -4 c d \right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{2 \sqrt {d}}}{b^{3} e^{3}}\right )\) | \(136\) |
| risch | \(-\frac {\sqrt {e x +d}}{b^{2} x}-\frac {e \left (-\frac {\left (-b e +4 c d \right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b e \sqrt {d}}+\frac {2 c \left (\frac {b e \sqrt {e x +d}}{2 c \left (e x +d \right )+2 b e -2 c d}+\frac {\left (3 b e -4 c d \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{2 \sqrt {\left (b e -c d \right ) c}}\right )}{b e}\right )}{b^{2}}\) | \(139\) |
-(-4*x*(c*d-3/4*b*e)*(c*x+b)*c*d^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c )^(1/2))+(x*(c*x+b)*(b*e-4*c*d)*arctanh((e*x+d)^(1/2)/d^(1/2))+d^(1/2)*(e* x+d)^(1/2)*(2*c*x+b)*b)*((b*e-c*d)*c)^(1/2))/d^(1/2)/((b*e-c*d)*c)^(1/2)/b ^3/x/(c*x+b)
Time = 0.30 (sec) , antiderivative size = 798, normalized size of antiderivative = 6.38 \[ \int \frac {\sqrt {d+e x}}{\left (b x+c x^2\right )^2} \, dx=\left [-\frac {{\left ({\left (4 \, c^{2} d^{2} - 3 \, b c d e\right )} x^{2} + {\left (4 \, b c d^{2} - 3 \, b^{2} d e\right )} x\right )} \sqrt {\frac {c}{c d - b e}} \log \left (\frac {c e x + 2 \, c d - b e + 2 \, {\left (c d - b e\right )} \sqrt {e x + d} \sqrt {\frac {c}{c d - b e}}}{c x + b}\right ) + {\left ({\left (4 \, c^{2} d - b c e\right )} x^{2} + {\left (4 \, b c d - b^{2} e\right )} x\right )} \sqrt {d} \log \left (\frac {e x - 2 \, \sqrt {e x + d} \sqrt {d} + 2 \, d}{x}\right ) + 2 \, {\left (2 \, b c d x + b^{2} d\right )} \sqrt {e x + d}}{2 \, {\left (b^{3} c d x^{2} + b^{4} d x\right )}}, -\frac {2 \, {\left ({\left (4 \, c^{2} d^{2} - 3 \, b c d e\right )} x^{2} + {\left (4 \, b c d^{2} - 3 \, b^{2} d e\right )} x\right )} \sqrt {-\frac {c}{c d - b e}} \arctan \left (-\frac {{\left (c d - b e\right )} \sqrt {e x + d} \sqrt {-\frac {c}{c d - b e}}}{c e x + c d}\right ) + {\left ({\left (4 \, c^{2} d - b c e\right )} x^{2} + {\left (4 \, b c d - b^{2} e\right )} x\right )} \sqrt {d} \log \left (\frac {e x - 2 \, \sqrt {e x + d} \sqrt {d} + 2 \, d}{x}\right ) + 2 \, {\left (2 \, b c d x + b^{2} d\right )} \sqrt {e x + d}}{2 \, {\left (b^{3} c d x^{2} + b^{4} d x\right )}}, -\frac {2 \, {\left ({\left (4 \, c^{2} d - b c e\right )} x^{2} + {\left (4 \, b c d - b^{2} e\right )} x\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {e x + d} \sqrt {-d}}{d}\right ) + {\left ({\left (4 \, c^{2} d^{2} - 3 \, b c d e\right )} x^{2} + {\left (4 \, b c d^{2} - 3 \, b^{2} d e\right )} x\right )} \sqrt {\frac {c}{c d - b e}} \log \left (\frac {c e x + 2 \, c d - b e + 2 \, {\left (c d - b e\right )} \sqrt {e x + d} \sqrt {\frac {c}{c d - b e}}}{c x + b}\right ) + 2 \, {\left (2 \, b c d x + b^{2} d\right )} \sqrt {e x + d}}{2 \, {\left (b^{3} c d x^{2} + b^{4} d x\right )}}, -\frac {{\left ({\left (4 \, c^{2} d^{2} - 3 \, b c d e\right )} x^{2} + {\left (4 \, b c d^{2} - 3 \, b^{2} d e\right )} x\right )} \sqrt {-\frac {c}{c d - b e}} \arctan \left (-\frac {{\left (c d - b e\right )} \sqrt {e x + d} \sqrt {-\frac {c}{c d - b e}}}{c e x + c d}\right ) + {\left ({\left (4 \, c^{2} d - b c e\right )} x^{2} + {\left (4 \, b c d - b^{2} e\right )} x\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {e x + d} \sqrt {-d}}{d}\right ) + {\left (2 \, b c d x + b^{2} d\right )} \sqrt {e x + d}}{b^{3} c d x^{2} + b^{4} d x}\right ] \]
[-1/2*(((4*c^2*d^2 - 3*b*c*d*e)*x^2 + (4*b*c*d^2 - 3*b^2*d*e)*x)*sqrt(c/(c *d - b*e))*log((c*e*x + 2*c*d - b*e + 2*(c*d - b*e)*sqrt(e*x + d)*sqrt(c/( c*d - b*e)))/(c*x + b)) + ((4*c^2*d - b*c*e)*x^2 + (4*b*c*d - b^2*e)*x)*sq rt(d)*log((e*x - 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) + 2*(2*b*c*d*x + b^2*d) *sqrt(e*x + d))/(b^3*c*d*x^2 + b^4*d*x), -1/2*(2*((4*c^2*d^2 - 3*b*c*d*e)* x^2 + (4*b*c*d^2 - 3*b^2*d*e)*x)*sqrt(-c/(c*d - b*e))*arctan(-(c*d - b*e)* sqrt(e*x + d)*sqrt(-c/(c*d - b*e))/(c*e*x + c*d)) + ((4*c^2*d - b*c*e)*x^2 + (4*b*c*d - b^2*e)*x)*sqrt(d)*log((e*x - 2*sqrt(e*x + d)*sqrt(d) + 2*d)/ x) + 2*(2*b*c*d*x + b^2*d)*sqrt(e*x + d))/(b^3*c*d*x^2 + b^4*d*x), -1/2*(2 *((4*c^2*d - b*c*e)*x^2 + (4*b*c*d - b^2*e)*x)*sqrt(-d)*arctan(sqrt(e*x + d)*sqrt(-d)/d) + ((4*c^2*d^2 - 3*b*c*d*e)*x^2 + (4*b*c*d^2 - 3*b^2*d*e)*x) *sqrt(c/(c*d - b*e))*log((c*e*x + 2*c*d - b*e + 2*(c*d - b*e)*sqrt(e*x + d )*sqrt(c/(c*d - b*e)))/(c*x + b)) + 2*(2*b*c*d*x + b^2*d)*sqrt(e*x + d))/( b^3*c*d*x^2 + b^4*d*x), -(((4*c^2*d^2 - 3*b*c*d*e)*x^2 + (4*b*c*d^2 - 3*b^ 2*d*e)*x)*sqrt(-c/(c*d - b*e))*arctan(-(c*d - b*e)*sqrt(e*x + d)*sqrt(-c/( c*d - b*e))/(c*e*x + c*d)) + ((4*c^2*d - b*c*e)*x^2 + (4*b*c*d - b^2*e)*x) *sqrt(-d)*arctan(sqrt(e*x + d)*sqrt(-d)/d) + (2*b*c*d*x + b^2*d)*sqrt(e*x + d))/(b^3*c*d*x^2 + b^4*d*x)]
\[ \int \frac {\sqrt {d+e x}}{\left (b x+c x^2\right )^2} \, dx=\int \frac {\sqrt {d + e x}}{x^{2} \left (b + c x\right )^{2}}\, dx \]
Exception generated. \[ \int \frac {\sqrt {d+e x}}{\left (b x+c x^2\right )^2} \, dx=\text {Exception raised: ValueError} \]
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
Time = 0.30 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.33 \[ \int \frac {\sqrt {d+e x}}{\left (b x+c x^2\right )^2} \, dx=\frac {{\left (4 \, c^{2} d - 3 \, b c e\right )} \arctan \left (\frac {\sqrt {e x + d} c}{\sqrt {-c^{2} d + b c e}}\right )}{\sqrt {-c^{2} d + b c e} b^{3}} - \frac {{\left (4 \, c d - b e\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-d}}\right )}{b^{3} \sqrt {-d}} - \frac {2 \, {\left (e x + d\right )}^{\frac {3}{2}} c e - 2 \, \sqrt {e x + d} c d e + \sqrt {e x + d} b e^{2}}{{\left ({\left (e x + d\right )}^{2} c - 2 \, {\left (e x + d\right )} c d + c d^{2} + {\left (e x + d\right )} b e - b d e\right )} b^{2}} \]
(4*c^2*d - 3*b*c*e)*arctan(sqrt(e*x + d)*c/sqrt(-c^2*d + b*c*e))/(sqrt(-c^ 2*d + b*c*e)*b^3) - (4*c*d - b*e)*arctan(sqrt(e*x + d)/sqrt(-d))/(b^3*sqrt (-d)) - (2*(e*x + d)^(3/2)*c*e - 2*sqrt(e*x + d)*c*d*e + sqrt(e*x + d)*b*e ^2)/(((e*x + d)^2*c - 2*(e*x + d)*c*d + c*d^2 + (e*x + d)*b*e - b*d*e)*b^2 )
Time = 9.43 (sec) , antiderivative size = 1174, normalized size of antiderivative = 9.39 \[ \int \frac {\sqrt {d+e x}}{\left (b x+c x^2\right )^2} \, dx=\text {Too large to display} \]
(atanh((2*c^2*e^6*(d + e*x)^(1/2))/(d^(3/2)*((8*c^3*e^5)/b - (2*c^2*e^6)/d )) - (8*c^3*e^5*(d + e*x)^(1/2))/(d^(1/2)*(8*c^3*e^5 - (2*b*c^2*e^6)/d)))* (b*e - 4*c*d))/(b^3*d^(1/2)) - ((2*c*e*(d + e*x)^(3/2))/b^2 + (e*(b*e - 2* c*d)*(d + e*x)^(1/2))/b^2)/((b*e - 2*c*d)*(d + e*x) + c*(d + e*x)^2 + c*d^ 2 - b*d*e) + (atan(((((4*(d + e*x)^(1/2)*(5*b^2*c^3*e^4 + 16*c^5*d^2*e^2 - 16*b*c^4*d*e^3))/b^4 - ((-c*(b*e - c*d))^(1/2)*(3*b*e - 4*c*d)*((2*(2*b^7 *c^2*e^4 - 4*b^6*c^3*d*e^3))/b^6 - (2*(2*b^7*c^2*e^3 - 4*b^6*c^3*d*e^2)*(- c*(b*e - c*d))^(1/2)*(3*b*e - 4*c*d)*(d + e*x)^(1/2))/(b^4*(b^4*e - b^3*c* d))))/(2*(b^4*e - b^3*c*d)))*(-c*(b*e - c*d))^(1/2)*(3*b*e - 4*c*d)*1i)/(2 *(b^4*e - b^3*c*d)) + (((4*(d + e*x)^(1/2)*(5*b^2*c^3*e^4 + 16*c^5*d^2*e^2 - 16*b*c^4*d*e^3))/b^4 + ((-c*(b*e - c*d))^(1/2)*(3*b*e - 4*c*d)*((2*(2*b ^7*c^2*e^4 - 4*b^6*c^3*d*e^3))/b^6 + (2*(2*b^7*c^2*e^3 - 4*b^6*c^3*d*e^2)* (-c*(b*e - c*d))^(1/2)*(3*b*e - 4*c*d)*(d + e*x)^(1/2))/(b^4*(b^4*e - b^3* c*d))))/(2*(b^4*e - b^3*c*d)))*(-c*(b*e - c*d))^(1/2)*(3*b*e - 4*c*d)*1i)/ (2*(b^4*e - b^3*c*d)))/((4*(3*b^2*c^3*e^5 + 16*c^5*d^2*e^3 - 16*b*c^4*d*e^ 4))/b^6 - (((4*(d + e*x)^(1/2)*(5*b^2*c^3*e^4 + 16*c^5*d^2*e^2 - 16*b*c^4* d*e^3))/b^4 - ((-c*(b*e - c*d))^(1/2)*(3*b*e - 4*c*d)*((2*(2*b^7*c^2*e^4 - 4*b^6*c^3*d*e^3))/b^6 - (2*(2*b^7*c^2*e^3 - 4*b^6*c^3*d*e^2)*(-c*(b*e - c *d))^(1/2)*(3*b*e - 4*c*d)*(d + e*x)^(1/2))/(b^4*(b^4*e - b^3*c*d))))/(2*( b^4*e - b^3*c*d)))*(-c*(b*e - c*d))^(1/2)*(3*b*e - 4*c*d))/(2*(b^4*e - ...