Integrand size = 21, antiderivative size = 133 \[ \int \frac {\sqrt {b x+c x^2}}{(d+e x)^3} \, dx=-\frac {b \sqrt {b x+c x^2}}{4 d (c d-b e) (d+e x)}+\frac {\left (b x+c x^2\right )^{3/2}}{2 (c d-b e) x (d+e x)^2}-\frac {b^2 \text {arctanh}\left (\frac {\sqrt {c d-b e} x}{\sqrt {d} \sqrt {b x+c x^2}}\right )}{4 d^{3/2} (c d-b e)^{3/2}} \] Output:
-1/4*b*(c*x^2+b*x)^(1/2)/d/(-b*e+c*d)/(e*x+d)+1/2*(c*x^2+b*x)^(3/2)/(-b*e+ c*d)/x/(e*x+d)^2-1/4*b^2*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 = 10.17 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.92 \[ \int \frac {\sqrt {b x+c x^2}}{(d+e x)^3} \, dx=\frac {\sqrt {x (b+c x)} \left (\frac {\sqrt {d} (2 c d x+b (d-e x))}{(c d-b e) (d+e x)^2}-\frac {b^2 \text {arctanh}\left (\frac {\sqrt {c d-b e} \sqrt {x}}{\sqrt {d} \sqrt {b+c x}}\right )}{(c d-b e)^{3/2} \sqrt {x} \sqrt {b+c x}}\right )}{4 d^{3/2}} \] Input:
Integrate[Sqrt[b*x + c*x^2]/(d + e*x)^3,x]
Output:
(Sqrt[x*(b + c*x)]*((Sqrt[d]*(2*c*d*x + b*(d - e*x)))/((c*d - b*e)*(d + e* x)^2) - (b^2*ArcTanh[(Sqrt[c*d - b*e]*Sqrt[x])/(Sqrt[d]*Sqrt[b + c*x])])/( (c*d - b*e)^(3/2)*Sqrt[x]*Sqrt[b + c*x])))/(4*d^(3/2))
Time = 0.43 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.95, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {1152, 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.
\(\displaystyle \int \frac {\sqrt {b x+c x^2}}{(d+e x)^3} \, dx\) |
\(\Big \downarrow \) 1152 |
\(\displaystyle \frac {\sqrt {b x+c x^2} (x (2 c d-b e)+b d)}{4 d (d+e x)^2 (c d-b e)}-\frac {b^2 \int \frac {1}{(d+e x) \sqrt {c x^2+b x}}dx}{8 d (c d-b e)}\) |
\(\Big \downarrow \) 1154 |
\(\displaystyle \frac {b^2 \int \frac {1}{4 d (c d-b e)-\frac {(b d+(2 c d-b e) x)^2}{c x^2+b x}}d\left (-\frac {b d+(2 c d-b e) x}{\sqrt {c x^2+b x}}\right )}{4 d (c d-b e)}+\frac {\sqrt {b x+c x^2} (x (2 c d-b e)+b d)}{4 d (d+e x)^2 (c d-b e)}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {\sqrt {b x+c x^2} (x (2 c d-b e)+b d)}{4 d (d+e x)^2 (c d-b e)}-\frac {b^2 \text {arctanh}\left (\frac {x (2 c d-b e)+b d}{2 \sqrt {d} \sqrt {b x+c x^2} \sqrt {c d-b e}}\right )}{8 d^{3/2} (c d-b e)^{3/2}}\) |
Input:
Int[Sqrt[b*x + c*x^2]/(d + e*x)^3,x]
Output:
((b*d + (2*c*d - b*e)*x)*Sqrt[b*x + c*x^2])/(4*d*(c*d - b*e)*(d + e*x)^2) - (b^2*ArcTanh[(b*d + (2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x + c*x^2])])/(8*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[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(-(d + e*x)^(m + 1))*(d*b - 2*a*e + (2*c*d - b*e)*x)*((a + b *x + c*x^2)^p/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[p*((b^2 - 4*a *c)/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2))) Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[m + 2*p + 2, 0] && GtQ[p, 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]
Time = 0.66 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.71
method | result | size |
pseudoelliptic | \(-\frac {b^{2} \left (\frac {\sqrt {x \left (c x +b \right )}\, \left (-b e x +2 c d x +b d \right )}{b^{2} \left (e x +d \right )^{2}}+\frac {\arctan \left (\frac {\sqrt {x \left (c x +b \right )}\, d}{x \sqrt {d \left (b e -c d \right )}}\right )}{\sqrt {d \left (b e -c d \right )}}\right )}{4 \left (b e -c d \right ) d}\) | \(94\) |
default | \(\text {Expression too large to display}\) | \(986\) |
Input:
int((c*x^2+b*x)^(1/2)/(e*x+d)^3,x,method=_RETURNVERBOSE)
Output:
-1/4*b^2/(b*e-c*d)/d*((x*(c*x+b))^(1/2)*(-b*e*x+2*c*d*x+b*d)/b^2/(e*x+d)^2 +1/(d*(b*e-c*d))^(1/2)*arctan((x*(c*x+b))^(1/2)/x*d/(d*(b*e-c*d))^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 229 vs. \(2 (113) = 226\).
Time = 0.10 (sec) , antiderivative size = 474, normalized size of antiderivative = 3.56 \[ \int \frac {\sqrt {b x+c x^2}}{(d+e x)^3} \, dx=\left [-\frac {{\left (b^{2} e^{2} x^{2} + 2 \, b^{2} d e x + b^{2} d^{2}\right )} \sqrt {c d^{2} - b d e} \log \left (\frac {b d + {\left (2 \, c d - b e\right )} x + 2 \, \sqrt {c d^{2} - b d e} \sqrt {c x^{2} + b x}}{e x + d}\right ) - 2 \, {\left (b c d^{3} - b^{2} d^{2} e + {\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e + b^{2} d e^{2}\right )} x\right )} \sqrt {c x^{2} + b x}}{8 \, {\left (c^{2} d^{6} - 2 \, b c d^{5} e + b^{2} d^{4} e^{2} + {\left (c^{2} d^{4} e^{2} - 2 \, b c d^{3} e^{3} + b^{2} d^{2} e^{4}\right )} x^{2} + 2 \, {\left (c^{2} d^{5} e - 2 \, b c d^{4} e^{2} + b^{2} d^{3} e^{3}\right )} x\right )}}, \frac {{\left (b^{2} e^{2} x^{2} + 2 \, b^{2} d e x + b^{2} d^{2}\right )} \sqrt {-c d^{2} + b d e} \arctan \left (\frac {\sqrt {-c d^{2} + b d e} \sqrt {c x^{2} + b x}}{c d x + b d}\right ) + {\left (b c d^{3} - b^{2} d^{2} e + {\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e + b^{2} d e^{2}\right )} x\right )} \sqrt {c x^{2} + b x}}{4 \, {\left (c^{2} d^{6} - 2 \, b c d^{5} e + b^{2} d^{4} e^{2} + {\left (c^{2} d^{4} e^{2} - 2 \, b c d^{3} e^{3} + b^{2} d^{2} e^{4}\right )} x^{2} + 2 \, {\left (c^{2} d^{5} e - 2 \, b c d^{4} e^{2} + b^{2} d^{3} e^{3}\right )} x\right )}}\right ] \] Input:
integrate((c*x^2+b*x)^(1/2)/(e*x+d)^3,x, algorithm="fricas")
Output:
[-1/8*((b^2*e^2*x^2 + 2*b^2*d*e*x + b^2*d^2)*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*(b*c*d^3 - b^2*d^2*e + (2*c^2*d^3 - 3*b*c*d^2*e + b^2*d*e^2)*x)*sqrt(c*x ^2 + b*x))/(c^2*d^6 - 2*b*c*d^5*e + b^2*d^4*e^2 + (c^2*d^4*e^2 - 2*b*c*d^3 *e^3 + b^2*d^2*e^4)*x^2 + 2*(c^2*d^5*e - 2*b*c*d^4*e^2 + b^2*d^3*e^3)*x), 1/4*((b^2*e^2*x^2 + 2*b^2*d*e*x + b^2*d^2)*sqrt(-c*d^2 + b*d*e)*arctan(sqr t(-c*d^2 + b*d*e)*sqrt(c*x^2 + b*x)/(c*d*x + b*d)) + (b*c*d^3 - b^2*d^2*e + (2*c^2*d^3 - 3*b*c*d^2*e + b^2*d*e^2)*x)*sqrt(c*x^2 + b*x))/(c^2*d^6 - 2 *b*c*d^5*e + b^2*d^4*e^2 + (c^2*d^4*e^2 - 2*b*c*d^3*e^3 + b^2*d^2*e^4)*x^2 + 2*(c^2*d^5*e - 2*b*c*d^4*e^2 + b^2*d^3*e^3)*x)]
\[ \int \frac {\sqrt {b x+c x^2}}{(d+e x)^3} \, dx=\int \frac {\sqrt {x \left (b + c x\right )}}{\left (d + e x\right )^{3}}\, dx \] Input:
integrate((c*x**2+b*x)**(1/2)/(e*x+d)**3,x)
Output:
Integral(sqrt(x*(b + c*x))/(d + e*x)**3, x)
Exception generated. \[ \int \frac {\sqrt {b x+c x^2}}{(d+e x)^3} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((c*x^2+b*x)^(1/2)/(e*x+d)^3,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
Leaf count of result is larger than twice the leaf count of optimal. 407 vs. \(2 (113) = 226\).
Time = 0.16 (sec) , antiderivative size = 407, normalized size of antiderivative = 3.06 \[ \int \frac {\sqrt {b x+c x^2}}{(d+e x)^3} \, dx=-\frac {b^{2} \arctan \left (-\frac {{\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} e + \sqrt {c} d}{\sqrt {-c d^{2} + b d e}}\right )}{4 \, {\left (c d^{2} - b d e\right )} \sqrt {-c d^{2} + b d e}} + \frac {8 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} c^{2} d^{2} e - 8 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} b c d e^{2} + {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} b^{2} e^{3} + 8 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} c^{\frac {5}{2}} d^{3} - 5 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} b^{2} \sqrt {c} d e^{2} + 8 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} b c^{2} d^{3} - 4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} b^{2} c d^{2} e - {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} b^{3} d e^{2} + 2 \, b^{2} c^{\frac {3}{2}} d^{3} - b^{3} \sqrt {c} d^{2} e}{4 \, {\left (c d^{2} e^{2} - b d e^{3}\right )} {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} e + 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} d + b d\right )}^{2}} \] Input:
integrate((c*x^2+b*x)^(1/2)/(e*x+d)^3,x, algorithm="giac")
Output:
-1/4*b^2*arctan(-((sqrt(c)*x - sqrt(c*x^2 + b*x))*e + sqrt(c)*d)/sqrt(-c*d ^2 + b*d*e))/((c*d^2 - b*d*e)*sqrt(-c*d^2 + b*d*e)) + 1/4*(8*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*c^2*d^2*e - 8*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*b*c*d *e^2 + (sqrt(c)*x - sqrt(c*x^2 + b*x))^3*b^2*e^3 + 8*(sqrt(c)*x - sqrt(c*x ^2 + b*x))^2*c^(5/2)*d^3 - 5*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*b^2*sqrt(c) *d*e^2 + 8*(sqrt(c)*x - sqrt(c*x^2 + b*x))*b*c^2*d^3 - 4*(sqrt(c)*x - sqrt (c*x^2 + b*x))*b^2*c*d^2*e - (sqrt(c)*x - sqrt(c*x^2 + b*x))*b^3*d*e^2 + 2 *b^2*c^(3/2)*d^3 - b^3*sqrt(c)*d^2*e)/((c*d^2*e^2 - b*d*e^3)*((sqrt(c)*x - sqrt(c*x^2 + b*x))^2*e + 2*(sqrt(c)*x - sqrt(c*x^2 + b*x))*sqrt(c)*d + b* d)^2)
Timed out. \[ \int \frac {\sqrt {b x+c x^2}}{(d+e x)^3} \, dx=\int \frac {\sqrt {c\,x^2+b\,x}}{{\left (d+e\,x\right )}^3} \,d x \] Input:
int((b*x + c*x^2)^(1/2)/(d + e*x)^3,x)
Output:
int((b*x + c*x^2)^(1/2)/(d + e*x)^3, x)
Time = 0.56 (sec) , antiderivative size = 1056, normalized size of antiderivative = 7.94 \[ \int \frac {\sqrt {b x+c x^2}}{(d+e x)^3} \, dx =\text {Too large to display} \] Input:
int((c*x^2+b*x)^(1/2)/(e*x+d)^3,x)
Output:
( - 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)))*b**3*d**2*e - 4*sqrt(d)*sq rt(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**3*d*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)))*b**3*e**3*x**2 + 4*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**2*c*d**3 + 8*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**2*c*d**2 *e*x + 4*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**2*c*d*e**2*x**2 - 2* sqrt(d)*sqrt(b*e - c*d)*atan((sqrt(b*e - c*d) + sqrt(e)*sqrt(b + c*x) + sq rt(x)*sqrt(e)*sqrt(c))/(sqrt(d)*sqrt(c)))*b**3*d**2*e - 4*sqrt(d)*sqrt(b*e - c*d)*atan((sqrt(b*e - c*d) + sqrt(e)*sqrt(b + c*x) + sqrt(x)*sqrt(e)*sq rt(c))/(sqrt(d)*sqrt(c)))*b**3*d*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)))*b**3*e**3*x**2 + 4*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 **2*c*d**3 + 8*sqrt(d)*sqrt(b*e - c*d)*atan((sqrt(b*e - c*d) + sqrt(e)*sqr t(b + c*x) + sqrt(x)*sqrt(e)*sqrt(c))/(sqrt(d)*sqrt(c)))*b**2*c*d**2*e*...