Integrand size = 27, antiderivative size = 154 \[ \int \frac {(c+d x) (g+h x)}{(a+b x)^2 \sqrt {e+f x}} \, dx=\frac {2 d h \sqrt {e+f x}}{b^2 f}-\frac {(b c-a d) (b g-a h) \sqrt {e+f x}}{b^2 (b e-a f) (a+b x)}+\frac {\left (b^2 c f g-a^2 d f h-(2 b e-a f) (b d g+b c h-2 a d h)\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {b e-a f}}\right )}{b^{5/2} (b e-a f)^{3/2}} \] Output:
2*d*h*(f*x+e)^(1/2)/b^2/f-(-a*d+b*c)*(-a*h+b*g)*(f*x+e)^(1/2)/b^2/(-a*f+b* e)/(b*x+a)+(b^2*c*f*g-a^2*d*f*h-(-a*f+2*b*e)*(-2*a*d*h+b*c*h+b*d*g))*arcta nh(b^(1/2)*(f*x+e)^(1/2)/(-a*f+b*e)^(1/2))/b^(5/2)/(-a*f+b*e)^(3/2)
Time = 0.46 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.14 \[ \int \frac {(c+d x) (g+h x)}{(a+b x)^2 \sqrt {e+f x}} \, dx=\frac {\sqrt {e+f x} \left (-3 a^2 d f h+b^2 (-c f g+2 d e h x)+a b (2 d e h+c f h+d f (g-2 h x))\right )}{b^2 f (b e-a f) (a+b x)}-\frac {\left (3 a^2 d f h+b^2 (2 d e g-c f g+2 c e h)-a b (d f g+4 d e h+c f h)\right ) \arctan \left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {-b e+a f}}\right )}{b^{5/2} (-b e+a f)^{3/2}} \] Input:
Integrate[((c + d*x)*(g + h*x))/((a + b*x)^2*Sqrt[e + f*x]),x]
Output:
(Sqrt[e + f*x]*(-3*a^2*d*f*h + b^2*(-(c*f*g) + 2*d*e*h*x) + a*b*(2*d*e*h + c*f*h + d*f*(g - 2*h*x))))/(b^2*f*(b*e - a*f)*(a + b*x)) - ((3*a^2*d*f*h + b^2*(2*d*e*g - c*f*g + 2*c*e*h) - a*b*(d*f*g + 4*d*e*h + c*f*h))*ArcTan[ (Sqrt[b]*Sqrt[e + f*x])/Sqrt[-(b*e) + a*f]])/(b^(5/2)*(-(b*e) + a*f)^(3/2) )
Time = 0.31 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.15, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {163, 73, 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.
\(\displaystyle \int \frac {(c+d x) (g+h x)}{(a+b x)^2 \sqrt {e+f x}} \, dx\) |
\(\Big \downarrow \) 163 |
\(\displaystyle \frac {\left (3 a^2 d f h-a b (c f h+4 d e h+d f g)+b^2 (2 c e h-c f g+2 d e g)\right ) \int \frac {1}{(a+b x) \sqrt {e+f x}}dx}{2 b^2 (b e-a f)}-\frac {\sqrt {e+f x} \left (3 a^2 d f h-a b (c f h+2 d e h+d f g)-2 b d h x (b e-a f)+b^2 c f g\right )}{b^2 f (a+b x) (b e-a f)}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {\left (3 a^2 d f h-a b (c f h+4 d e h+d f g)+b^2 (2 c e h-c f g+2 d e g)\right ) \int \frac {1}{a+\frac {b (e+f x)}{f}-\frac {b e}{f}}d\sqrt {e+f x}}{b^2 f (b e-a f)}-\frac {\sqrt {e+f x} \left (3 a^2 d f h-a b (c f h+2 d e h+d f g)-2 b d h x (b e-a f)+b^2 c f g\right )}{b^2 f (a+b x) (b e-a f)}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle -\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {b e-a f}}\right ) \left (3 a^2 d f h-a b (c f h+4 d e h+d f g)+b^2 (2 c e h-c f g+2 d e g)\right )}{b^{5/2} (b e-a f)^{3/2}}-\frac {\sqrt {e+f x} \left (3 a^2 d f h-a b (c f h+2 d e h+d f g)-2 b d h x (b e-a f)+b^2 c f g\right )}{b^2 f (a+b x) (b e-a f)}\) |
Input:
Int[((c + d*x)*(g + h*x))/((a + b*x)^2*Sqrt[e + f*x]),x]
Output:
-((Sqrt[e + f*x]*(b^2*c*f*g + 3*a^2*d*f*h - a*b*(d*f*g + 2*d*e*h + c*f*h) - 2*b*d*(b*e - a*f)*h*x))/(b^2*f*(b*e - a*f)*(a + b*x))) - ((3*a^2*d*f*h + b^2*(2*d*e*g - c*f*g + 2*c*e*h) - a*b*(d*f*g + 4*d*e*h + c*f*h))*ArcTanh[ (Sqrt[b]*Sqrt[e + f*x])/Sqrt[b*e - a*f]])/(b^(5/2)*(b*e - a*f)^(3/2))
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_) )*((g_.) + (h_.)*(x_)), x_] :> Simp[((a^2*d*f*h*(n + 2) + b^2*d*e*g*(m + n + 3) + a*b*(c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b*f*h*(b*c - a*d)* (m + 1)*x)/(b^2*d*(b*c - a*d)*(m + 1)*(m + n + 3)))*(a + b*x)^(m + 1)*(c + d*x)^(n + 1), x] - Simp[(a^2*d^2*f*h*(n + 1)*(n + 2) + a*b*d*(n + 1)*(2*c*f *h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c* d*(f*g + e*h)*(m + 1)*(m + n + 3) + d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2* d*(b*c - a*d)*(m + 1)*(m + n + 3)) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && ((GeQ[m, -2] && LtQ[m, - 1]) || SumSimplerQ[m, 1]) && NeQ[m, -1] && NeQ[m + n + 3, 0]
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]
Time = 0.44 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.20
method | result | size |
pseudoelliptic | \(\frac {-3 \left (\frac {\left (-c f g +2 e \left (c h +d g \right )\right ) b^{2}}{3}-\frac {a \left (f \left (c h +d g \right )+4 d e h \right ) b}{3}+a^{2} d f h \right ) \left (b x +a \right ) f \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )+3 \left (\frac {\left (-2 d e h x +c f g \right ) b^{2}}{3}-\frac {a \left (\left (\left (-2 h x +g \right ) d +c h \right ) f +2 d e h \right ) b}{3}+a^{2} d f h \right ) \sqrt {\left (a f -b e \right ) b}\, \sqrt {f x +e}}{b^{2} \left (a f -b e \right ) f \left (b x +a \right ) \sqrt {\left (a f -b e \right ) b}}\) | \(185\) |
risch | \(\frac {2 d h \sqrt {f x +e}}{b^{2} f}-\frac {-\frac {f \left (a^{2} d h -a b c h -a b d g +b^{2} c g \right ) \sqrt {f x +e}}{\left (a f -b e \right ) \left (\left (f x +e \right ) b +a f -b e \right )}+\frac {\left (3 a^{2} d f h -a b c f h -4 a b d e h -a b d f g +2 b^{2} c e h -b^{2} c f g +2 b^{2} d e g \right ) \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{\left (a f -b e \right ) \sqrt {\left (a f -b e \right ) b}}}{b^{2}}\) | \(186\) |
derivativedivides | \(\frac {\frac {2 d h \sqrt {f x +e}}{b^{2}}-\frac {2 f \left (-\frac {f \left (a^{2} d h -a b c h -a b d g +b^{2} c g \right ) \sqrt {f x +e}}{2 \left (a f -b e \right ) \left (\left (f x +e \right ) b +a f -b e \right )}+\frac {\left (3 a^{2} d f h -a b c f h -4 a b d e h -a b d f g +2 b^{2} c e h -b^{2} c f g +2 b^{2} d e g \right ) \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{2 \left (a f -b e \right ) \sqrt {\left (a f -b e \right ) b}}\right )}{b^{2}}}{f}\) | \(189\) |
default | \(\frac {\frac {2 d h \sqrt {f x +e}}{b^{2}}-\frac {2 f \left (-\frac {f \left (a^{2} d h -a b c h -a b d g +b^{2} c g \right ) \sqrt {f x +e}}{2 \left (a f -b e \right ) \left (\left (f x +e \right ) b +a f -b e \right )}+\frac {\left (3 a^{2} d f h -a b c f h -4 a b d e h -a b d f g +2 b^{2} c e h -b^{2} c f g +2 b^{2} d e g \right ) \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{2 \left (a f -b e \right ) \sqrt {\left (a f -b e \right ) b}}\right )}{b^{2}}}{f}\) | \(189\) |
Input:
int((d*x+c)*(h*x+g)/(b*x+a)^2/(f*x+e)^(1/2),x,method=_RETURNVERBOSE)
Output:
3*(-(1/3*(-c*f*g+2*e*(c*h+d*g))*b^2-1/3*a*(f*(c*h+d*g)+4*d*e*h)*b+a^2*d*f* h)*(b*x+a)*f*arctan(b*(f*x+e)^(1/2)/((a*f-b*e)*b)^(1/2))+(1/3*(-2*d*e*h*x+ c*f*g)*b^2-1/3*a*(((-2*h*x+g)*d+c*h)*f+2*d*e*h)*b+a^2*d*f*h)*((a*f-b*e)*b) ^(1/2)*(f*x+e)^(1/2))/((a*f-b*e)*b)^(1/2)/f/b^2/(a*f-b*e)/(b*x+a)
Leaf count of result is larger than twice the leaf count of optimal. 389 vs. \(2 (140) = 280\).
Time = 0.12 (sec) , antiderivative size = 793, normalized size of antiderivative = 5.15 \[ \int \frac {(c+d x) (g+h x)}{(a+b x)^2 \sqrt {e+f x}} \, dx =\text {Too large to display} \] Input:
integrate((d*x+c)*(h*x+g)/(b*x+a)^2/(f*x+e)^(1/2),x, algorithm="fricas")
Output:
[-1/2*(sqrt(b^2*e - a*b*f)*((2*a*b^2*d*e*f - (a*b^2*c + a^2*b*d)*f^2)*g + (2*(a*b^2*c - 2*a^2*b*d)*e*f - (a^2*b*c - 3*a^3*d)*f^2)*h + ((2*b^3*d*e*f - (b^3*c + a*b^2*d)*f^2)*g + (2*(b^3*c - 2*a*b^2*d)*e*f - (a*b^2*c - 3*a^2 *b*d)*f^2)*h)*x)*log((b*f*x + 2*b*e - a*f + 2*sqrt(b^2*e - a*b*f)*sqrt(f*x + e))/(b*x + a)) - 2*(2*(b^4*d*e^2 - 2*a*b^3*d*e*f + a^2*b^2*d*f^2)*h*x - ((b^4*c - a*b^3*d)*e*f - (a*b^3*c - a^2*b^2*d)*f^2)*g + (2*a*b^3*d*e^2 + (a*b^3*c - 5*a^2*b^2*d)*e*f - (a^2*b^2*c - 3*a^3*b*d)*f^2)*h)*sqrt(f*x + e ))/(a*b^5*e^2*f - 2*a^2*b^4*e*f^2 + a^3*b^3*f^3 + (b^6*e^2*f - 2*a*b^5*e*f ^2 + a^2*b^4*f^3)*x), (sqrt(-b^2*e + a*b*f)*((2*a*b^2*d*e*f - (a*b^2*c + a ^2*b*d)*f^2)*g + (2*(a*b^2*c - 2*a^2*b*d)*e*f - (a^2*b*c - 3*a^3*d)*f^2)*h + ((2*b^3*d*e*f - (b^3*c + a*b^2*d)*f^2)*g + (2*(b^3*c - 2*a*b^2*d)*e*f - (a*b^2*c - 3*a^2*b*d)*f^2)*h)*x)*arctan(sqrt(-b^2*e + a*b*f)*sqrt(f*x + e )/(b*f*x + b*e)) + (2*(b^4*d*e^2 - 2*a*b^3*d*e*f + a^2*b^2*d*f^2)*h*x - (( b^4*c - a*b^3*d)*e*f - (a*b^3*c - a^2*b^2*d)*f^2)*g + (2*a*b^3*d*e^2 + (a* b^3*c - 5*a^2*b^2*d)*e*f - (a^2*b^2*c - 3*a^3*b*d)*f^2)*h)*sqrt(f*x + e))/ (a*b^5*e^2*f - 2*a^2*b^4*e*f^2 + a^3*b^3*f^3 + (b^6*e^2*f - 2*a*b^5*e*f^2 + a^2*b^4*f^3)*x)]
Timed out. \[ \int \frac {(c+d x) (g+h x)}{(a+b x)^2 \sqrt {e+f x}} \, dx=\text {Timed out} \] Input:
integrate((d*x+c)*(h*x+g)/(b*x+a)**2/(f*x+e)**(1/2),x)
Output:
Timed out
Exception generated. \[ \int \frac {(c+d x) (g+h x)}{(a+b x)^2 \sqrt {e+f x}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((d*x+c)*(h*x+g)/(b*x+a)^2/(f*x+e)^(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(a*f-b*e>0)', see `assume?` for m ore detail
Time = 0.13 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.40 \[ \int \frac {(c+d x) (g+h x)}{(a+b x)^2 \sqrt {e+f x}} \, dx=\frac {{\left (2 \, b^{2} d e g - b^{2} c f g - a b d f g + 2 \, b^{2} c e h - 4 \, a b d e h - a b c f h + 3 \, a^{2} d f h\right )} \arctan \left (\frac {\sqrt {f x + e} b}{\sqrt {-b^{2} e + a b f}}\right )}{{\left (b^{3} e - a b^{2} f\right )} \sqrt {-b^{2} e + a b f}} + \frac {2 \, \sqrt {f x + e} d h}{b^{2} f} - \frac {\sqrt {f x + e} b^{2} c f g - \sqrt {f x + e} a b d f g - \sqrt {f x + e} a b c f h + \sqrt {f x + e} a^{2} d f h}{{\left (b^{3} e - a b^{2} f\right )} {\left ({\left (f x + e\right )} b - b e + a f\right )}} \] Input:
integrate((d*x+c)*(h*x+g)/(b*x+a)^2/(f*x+e)^(1/2),x, algorithm="giac")
Output:
(2*b^2*d*e*g - b^2*c*f*g - a*b*d*f*g + 2*b^2*c*e*h - 4*a*b*d*e*h - a*b*c*f *h + 3*a^2*d*f*h)*arctan(sqrt(f*x + e)*b/sqrt(-b^2*e + a*b*f))/((b^3*e - a *b^2*f)*sqrt(-b^2*e + a*b*f)) + 2*sqrt(f*x + e)*d*h/(b^2*f) - (sqrt(f*x + e)*b^2*c*f*g - sqrt(f*x + e)*a*b*d*f*g - sqrt(f*x + e)*a*b*c*f*h + sqrt(f* x + e)*a^2*d*f*h)/((b^3*e - a*b^2*f)*((f*x + e)*b - b*e + a*f))
Time = 0.21 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.14 \[ \int \frac {(c+d x) (g+h x)}{(a+b x)^2 \sqrt {e+f x}} \, dx=\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {e+f\,x}}{\sqrt {a\,f-b\,e}}\right )\,\left (b^2\,c\,f\,g-2\,b^2\,c\,e\,h-2\,b^2\,d\,e\,g-3\,a^2\,d\,f\,h+a\,b\,c\,f\,h+4\,a\,b\,d\,e\,h+a\,b\,d\,f\,g\right )}{b^{5/2}\,{\left (a\,f-b\,e\right )}^{3/2}}+\frac {\sqrt {e+f\,x}\,\left (b^2\,c\,f\,g+a^2\,d\,f\,h-a\,b\,c\,f\,h-a\,b\,d\,f\,g\right )}{\left (a\,f-b\,e\right )\,\left (b^3\,\left (e+f\,x\right )-b^3\,e+a\,b^2\,f\right )}+\frac {2\,d\,h\,\sqrt {e+f\,x}}{b^2\,f} \] Input:
int(((g + h*x)*(c + d*x))/((e + f*x)^(1/2)*(a + b*x)^2),x)
Output:
(atan((b^(1/2)*(e + f*x)^(1/2))/(a*f - b*e)^(1/2))*(b^2*c*f*g - 2*b^2*c*e* h - 2*b^2*d*e*g - 3*a^2*d*f*h + a*b*c*f*h + 4*a*b*d*e*h + a*b*d*f*g))/(b^( 5/2)*(a*f - b*e)^(3/2)) + ((e + f*x)^(1/2)*(b^2*c*f*g + a^2*d*f*h - a*b*c* f*h - a*b*d*f*g))/((a*f - b*e)*(b^3*(e + f*x) - b^3*e + a*b^2*f)) + (2*d*h *(e + f*x)^(1/2))/(b^2*f)
Time = 0.18 (sec) , antiderivative size = 900, normalized size of antiderivative = 5.84 \[ \int \frac {(c+d x) (g+h x)}{(a+b x)^2 \sqrt {e+f x}} \, dx =\text {Too large to display} \] Input:
int((d*x+c)*(h*x+g)/(b*x+a)^2/(f*x+e)^(1/2),x)
Output:
( - 3*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b *e)))*a**3*d*f**2*h + sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt (b)*sqrt(a*f - b*e)))*a**2*b*c*f**2*h + 4*sqrt(b)*sqrt(a*f - b*e)*atan((sq rt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**2*b*d*e*f*h + sqrt(b)*sqrt(a* f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**2*b*d*f**2*g - 3*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b* e)))*a**2*b*d*f**2*h*x - 2*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/ (sqrt(b)*sqrt(a*f - b*e)))*a*b**2*c*e*f*h + sqrt(b)*sqrt(a*f - b*e)*atan(( sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a*b**2*c*f**2*g + sqrt(b)*sqrt (a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a*b**2*c*f** 2*h*x - 2*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a*b**2*d*e*f*g + 4*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b )/(sqrt(b)*sqrt(a*f - b*e)))*a*b**2*d*e*f*h*x + sqrt(b)*sqrt(a*f - b*e)*at an((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a*b**2*d*f**2*g*x - 2*sqrt (b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*b**3 *c*e*f*h*x + sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt( a*f - b*e)))*b**3*c*f**2*g*x - 2*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f* x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*b**3*d*e*f*g*x + 3*sqrt(e + f*x)*a**3*b*d *f**2*h - sqrt(e + f*x)*a**2*b**2*c*f**2*h - 5*sqrt(e + f*x)*a**2*b**2*d*e *f*h - sqrt(e + f*x)*a**2*b**2*d*f**2*g + 2*sqrt(e + f*x)*a**2*b**2*d*f...