Integrand size = 21, antiderivative size = 97 \[ \int \frac {1}{(d+e x)^2 \sqrt {b x+c x^2}} \, dx=-\frac {e \sqrt {b x+c x^2}}{d (c d-b e) (d+e x)}+\frac {(2 c d-b e) \text {arctanh}\left (\frac {\sqrt {c d-b e} x}{\sqrt {d} \sqrt {b x+c x^2}}\right )}{d^{3/2} (c d-b e)^{3/2}} \] Output:
-e*(c*x^2+b*x)^(1/2)/d/(-b*e+c*d)/(e*x+d)+(-b*e+2*c*d)*arctanh((-b*e+c*d)^ (1/2)*x/d^(1/2)/(c*x^2+b*x)^(1/2))/d^(3/2)/(-b*e+c*d)^(3/2)
Time = 0.40 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.41 \[ \int \frac {1}{(d+e x)^2 \sqrt {b x+c x^2}} \, dx=\frac {\sqrt {x} \left (-\frac {\sqrt {d} e \sqrt {x} (b+c x)}{(c d-b e) (d+e x)}+\frac {(2 c d-b e) \sqrt {b+c x} \arctan \left (\frac {-e \sqrt {x} \sqrt {b+c x}+\sqrt {c} (d+e x)}{\sqrt {d} \sqrt {-c d+b e}}\right )}{(-c d+b e)^{3/2}}\right )}{d^{3/2} \sqrt {x (b+c x)}} \] Input:
Integrate[1/((d + e*x)^2*Sqrt[b*x + c*x^2]),x]
Output:
(Sqrt[x]*(-((Sqrt[d]*e*Sqrt[x]*(b + c*x))/((c*d - b*e)*(d + e*x))) + ((2*c *d - b*e)*Sqrt[b + c*x]*ArcTan[(-(e*Sqrt[x]*Sqrt[b + c*x]) + Sqrt[c]*(d + e*x))/(Sqrt[d]*Sqrt[-(c*d) + b*e])])/(-(c*d) + b*e)^(3/2)))/(d^(3/2)*Sqrt[ x*(b + c*x)])
Time = 0.42 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.21, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {1157, 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.
\(\displaystyle \int \frac {1}{\sqrt {b x+c x^2} (d+e x)^2} \, dx\) |
\(\Big \downarrow \) 1157 |
\(\displaystyle \frac {(2 c d-b e) \int \frac {1}{(d+e x) \sqrt {c x^2+b x}}dx}{2 d (c d-b e)}-\frac {e \sqrt {b x+c x^2}}{d (d+e x) (c d-b e)}\) |
\(\Big \downarrow \) 1154 |
\(\displaystyle -\frac {(2 c d-b e) \int \frac {1}{4 d (c d-b e)-\frac {(b d+(2 c d-b e) x)^2}{c x^2+b x}}d\left (-\frac {b d+(2 c d-b e) x}{\sqrt {c x^2+b x}}\right )}{d (c d-b e)}-\frac {e \sqrt {b x+c x^2}}{d (d+e x) (c d-b e)}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {(2 c d-b e) \text {arctanh}\left (\frac {x (2 c d-b e)+b d}{2 \sqrt {d} \sqrt {b x+c x^2} \sqrt {c d-b e}}\right )}{2 d^{3/2} (c d-b e)^{3/2}}-\frac {e \sqrt {b x+c x^2}}{d (d+e x) (c d-b e)}\) |
Input:
Int[1/((d + e*x)^2*Sqrt[b*x + c*x^2]),x]
Output:
-((e*Sqrt[b*x + c*x^2])/(d*(c*d - b*e)*(d + e*x))) + ((2*c*d - b*e)*ArcTan h[(b*d + (2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x + c*x^2])])/ (2*d^(3/2)*(c*d - b*e)^(3/2))
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/(((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[e*(d + e*x)^(m + 1)*((a + b*x + c*x^2)^(p + 1)/((m + 1)*(c*d ^2 - b*d*e + a*e^2))), x] + Simp[(2*c*d - b*e)/(2*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e , m, p}, x] && EqQ[m + 2*p + 3, 0]
Time = 0.77 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.86
method | result | size |
pseudoelliptic | \(\frac {\frac {e \sqrt {x \left (c x +b \right )}}{e x +d}-\frac {\left (b e -2 c d \right ) \arctan \left (\frac {\sqrt {x \left (c x +b \right )}\, d}{x \sqrt {d \left (b e -c d \right )}}\right )}{\sqrt {d \left (b e -c d \right )}}}{\left (b e -c d \right ) d}\) | \(83\) |
default | \(\frac {\frac {e^{2} \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{d \left (b e -c d \right ) \left (x +\frac {d}{e}\right )}-\frac {\left (b e -2 c d \right ) e \ln \left (\frac {-\frac {2 d \left (b e -c d \right )}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{x +\frac {d}{e}}\right )}{2 d \left (b e -c d \right ) \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}}}{e^{2}}\) | \(229\) |
Input:
int(1/(e*x+d)^2/(c*x^2+b*x)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/(b*e-c*d)/d*(e*(x*(c*x+b))^(1/2)/(e*x+d)-(b*e-2*c*d)/(d*(b*e-c*d))^(1/2) *arctan((x*(c*x+b))^(1/2)/x*d/(d*(b*e-c*d))^(1/2)))
Time = 0.09 (sec) , antiderivative size = 344, normalized size of antiderivative = 3.55 \[ \int \frac {1}{(d+e x)^2 \sqrt {b x+c x^2}} \, dx=\left [\frac {{\left (2 \, c d^{2} - b d e + {\left (2 \, c d e - b e^{2}\right )} x\right )} \sqrt {c d^{2} - b d e} \log \left (\frac {b d + {\left (2 \, c d - b e\right )} x + 2 \, \sqrt {c d^{2} - b d e} \sqrt {c x^{2} + b x}}{e x + d}\right ) - 2 \, {\left (c d^{2} e - b d e^{2}\right )} \sqrt {c x^{2} + b x}}{2 \, {\left (c^{2} d^{5} - 2 \, b c d^{4} e + b^{2} d^{3} e^{2} + {\left (c^{2} d^{4} e - 2 \, b c d^{3} e^{2} + b^{2} d^{2} e^{3}\right )} x\right )}}, -\frac {{\left (2 \, c d^{2} - b d e + {\left (2 \, c d e - b e^{2}\right )} x\right )} \sqrt {-c d^{2} + b d e} \arctan \left (\frac {\sqrt {-c d^{2} + b d e} \sqrt {c x^{2} + b x}}{c d x + b d}\right ) + {\left (c d^{2} e - b d e^{2}\right )} \sqrt {c x^{2} + b x}}{c^{2} d^{5} - 2 \, b c d^{4} e + b^{2} d^{3} e^{2} + {\left (c^{2} d^{4} e - 2 \, b c d^{3} e^{2} + b^{2} d^{2} e^{3}\right )} x}\right ] \] Input:
integrate(1/(e*x+d)^2/(c*x^2+b*x)^(1/2),x, algorithm="fricas")
Output:
[1/2*((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^2*d^5 - 2*b*c*d^4*e + b^2*d^3 *e^2 + (c^2*d^4*e - 2*b*c*d^3*e^2 + b^2*d^2*e^3)*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*x + b*d)) + (c*d^2*e - b*d*e^2)*sqrt(c*x^2 + b*x))/(c^2 *d^5 - 2*b*c*d^4*e + b^2*d^3*e^2 + (c^2*d^4*e - 2*b*c*d^3*e^2 + b^2*d^2*e^ 3)*x)]
\[ \int \frac {1}{(d+e x)^2 \sqrt {b x+c x^2}} \, dx=\int \frac {1}{\sqrt {x \left (b + c x\right )} \left (d + e x\right )^{2}}\, dx \] Input:
integrate(1/(e*x+d)**2/(c*x**2+b*x)**(1/2),x)
Output:
Integral(1/(sqrt(x*(b + c*x))*(d + e*x)**2), x)
Exception generated. \[ \int \frac {1}{(d+e x)^2 \sqrt {b x+c x^2}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(1/(e*x+d)^2/(c*x^2+b*x)^(1/2),x, algorithm="maxima")
Output:
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. \[ \int \frac {1}{(d+e x)^2 \sqrt {b x+c x^2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(1/(e*x+d)^2/(c*x^2+b*x)^(1/2),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
Timed out. \[ \int \frac {1}{(d+e x)^2 \sqrt {b x+c x^2}} \, dx=\int \frac {1}{\sqrt {c\,x^2+b\,x}\,{\left (d+e\,x\right )}^2} \,d x \] Input:
int(1/((b*x + c*x^2)^(1/2)*(d + e*x)^2),x)
Output:
int(1/((b*x + c*x^2)^(1/2)*(d + e*x)^2), x)
Time = 0.26 (sec) , antiderivative size = 518, normalized size of antiderivative = 5.34 \[ \int \frac {1}{(d+e x)^2 \sqrt {b x+c x^2}} \, dx=\frac {-\sqrt {d}\, \sqrt {b e -c d}\, \mathit {atan} \left (\frac {\sqrt {b e -c d}-\sqrt {e}\, \sqrt {c x +b}-\sqrt {x}\, \sqrt {e}\, \sqrt {c}}{\sqrt {d}\, \sqrt {c}}\right ) b d e -\sqrt {d}\, \sqrt {b e -c d}\, \mathit {atan} \left (\frac {\sqrt {b e -c d}-\sqrt {e}\, \sqrt {c x +b}-\sqrt {x}\, \sqrt {e}\, \sqrt {c}}{\sqrt {d}\, \sqrt {c}}\right ) b \,e^{2} x +2 \sqrt {d}\, \sqrt {b e -c d}\, \mathit {atan} \left (\frac {\sqrt {b e -c d}-\sqrt {e}\, \sqrt {c x +b}-\sqrt {x}\, \sqrt {e}\, \sqrt {c}}{\sqrt {d}\, \sqrt {c}}\right ) c \,d^{2}+2 \sqrt {d}\, \sqrt {b e -c d}\, \mathit {atan} \left (\frac {\sqrt {b e -c d}-\sqrt {e}\, \sqrt {c x +b}-\sqrt {x}\, \sqrt {e}\, \sqrt {c}}{\sqrt {d}\, \sqrt {c}}\right ) c d e x -\sqrt {d}\, \sqrt {b e -c d}\, \mathit {atan} \left (\frac {\sqrt {b e -c d}+\sqrt {e}\, \sqrt {c x +b}+\sqrt {x}\, \sqrt {e}\, \sqrt {c}}{\sqrt {d}\, \sqrt {c}}\right ) b d e -\sqrt {d}\, \sqrt {b e -c d}\, \mathit {atan} \left (\frac {\sqrt {b e -c d}+\sqrt {e}\, \sqrt {c x +b}+\sqrt {x}\, \sqrt {e}\, \sqrt {c}}{\sqrt {d}\, \sqrt {c}}\right ) b \,e^{2} x +2 \sqrt {d}\, \sqrt {b e -c d}\, \mathit {atan} \left (\frac {\sqrt {b e -c d}+\sqrt {e}\, \sqrt {c x +b}+\sqrt {x}\, \sqrt {e}\, \sqrt {c}}{\sqrt {d}\, \sqrt {c}}\right ) c \,d^{2}+2 \sqrt {d}\, \sqrt {b e -c d}\, \mathit {atan} \left (\frac {\sqrt {b e -c d}+\sqrt {e}\, \sqrt {c x +b}+\sqrt {x}\, \sqrt {e}\, \sqrt {c}}{\sqrt {d}\, \sqrt {c}}\right ) c d e x +\sqrt {x}\, \sqrt {c x +b}\, b d \,e^{2}-\sqrt {x}\, \sqrt {c x +b}\, c \,d^{2} e}{d^{2} \left (b^{2} e^{3} x -2 b c d \,e^{2} x +c^{2} d^{2} e x +b^{2} d \,e^{2}-2 b c \,d^{2} e +c^{2} d^{3}\right )} \] Input:
int(1/(e*x+d)^2/(c*x^2+b*x)^(1/2),x)
Output:
( - sqrt(d)*sqrt(b*e - c*d)*atan((sqrt(b*e - c*d) - sqrt(e)*sqrt(b + c*x) - sqrt(x)*sqrt(e)*sqrt(c))/(sqrt(d)*sqrt(c)))*b*d*e - sqrt(d)*sqrt(b*e - c *d)*atan((sqrt(b*e - c*d) - sqrt(e)*sqrt(b + c*x) - sqrt(x)*sqrt(e)*sqrt(c ))/(sqrt(d)*sqrt(c)))*b*e**2*x + 2*sqrt(d)*sqrt(b*e - c*d)*atan((sqrt(b*e - c*d) - sqrt(e)*sqrt(b + c*x) - sqrt(x)*sqrt(e)*sqrt(c))/(sqrt(d)*sqrt(c) ))*c*d**2 + 2*sqrt(d)*sqrt(b*e - c*d)*atan((sqrt(b*e - c*d) - sqrt(e)*sqrt (b + c*x) - sqrt(x)*sqrt(e)*sqrt(c))/(sqrt(d)*sqrt(c)))*c*d*e*x - sqrt(d)* sqrt(b*e - c*d)*atan((sqrt(b*e - c*d) + sqrt(e)*sqrt(b + c*x) + sqrt(x)*sq rt(e)*sqrt(c))/(sqrt(d)*sqrt(c)))*b*d*e - sqrt(d)*sqrt(b*e - c*d)*atan((sq rt(b*e - c*d) + sqrt(e)*sqrt(b + c*x) + sqrt(x)*sqrt(e)*sqrt(c))/(sqrt(d)* sqrt(c)))*b*e**2*x + 2*sqrt(d)*sqrt(b*e - c*d)*atan((sqrt(b*e - c*d) + sqr t(e)*sqrt(b + c*x) + sqrt(x)*sqrt(e)*sqrt(c))/(sqrt(d)*sqrt(c)))*c*d**2 + 2*sqrt(d)*sqrt(b*e - c*d)*atan((sqrt(b*e - c*d) + sqrt(e)*sqrt(b + c*x) + sqrt(x)*sqrt(e)*sqrt(c))/(sqrt(d)*sqrt(c)))*c*d*e*x + sqrt(x)*sqrt(b + c*x )*b*d*e**2 - sqrt(x)*sqrt(b + c*x)*c*d**2*e)/(d**2*(b**2*d*e**2 + b**2*e** 3*x - 2*b*c*d**2*e - 2*b*c*d*e**2*x + c**2*d**3 + c**2*d**2*e*x))