Integrand size = 22, antiderivative size = 98 \[ \int \frac {\sqrt {e+f x} (g+h x)}{c+d x} \, dx=\frac {2 (d g-c h) \sqrt {e+f x}}{d^2}+\frac {2 h (e+f x)^{3/2}}{3 d f}-\frac {2 \sqrt {d e-c f} (d g-c h) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{d^{5/2}} \] Output:
2*(-c*h+d*g)*(f*x+e)^(1/2)/d^2+2/3*h*(f*x+e)^(3/2)/d/f-2*(-c*f+d*e)^(1/2)* (-c*h+d*g)*arctanh(d^(1/2)*(f*x+e)^(1/2)/(-c*f+d*e)^(1/2))/d^(5/2)
Time = 0.17 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.97 \[ \int \frac {\sqrt {e+f x} (g+h x)}{c+d x} \, dx=\frac {2 \sqrt {e+f x} (-3 c f h+d (3 f g+e h+f h x))}{3 d^2 f}-\frac {2 \sqrt {-d e+c f} (d g-c h) \arctan \left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {-d e+c f}}\right )}{d^{5/2}} \] Input:
Integrate[(Sqrt[e + f*x]*(g + h*x))/(c + d*x),x]
Output:
(2*Sqrt[e + f*x]*(-3*c*f*h + d*(3*f*g + e*h + f*h*x)))/(3*d^2*f) - (2*Sqrt [-(d*e) + c*f]*(d*g - c*h)*ArcTan[(Sqrt[d]*Sqrt[e + f*x])/Sqrt[-(d*e) + c* f]])/d^(5/2)
Time = 0.23 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.97, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {90, 60, 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.
\(\displaystyle \int \frac {\sqrt {e+f x} (g+h x)}{c+d x} \, dx\) |
\(\Big \downarrow \) 90 |
\(\displaystyle \frac {(d g-c h) \int \frac {\sqrt {e+f x}}{c+d x}dx}{d}+\frac {2 h (e+f x)^{3/2}}{3 d f}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {(d g-c h) \left (\frac {(d e-c f) \int \frac {1}{(c+d x) \sqrt {e+f x}}dx}{d}+\frac {2 \sqrt {e+f x}}{d}\right )}{d}+\frac {2 h (e+f x)^{3/2}}{3 d f}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {(d g-c h) \left (\frac {2 (d e-c f) \int \frac {1}{c+\frac {d (e+f x)}{f}-\frac {d e}{f}}d\sqrt {e+f x}}{d f}+\frac {2 \sqrt {e+f x}}{d}\right )}{d}+\frac {2 h (e+f x)^{3/2}}{3 d f}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {(d g-c h) \left (\frac {2 \sqrt {e+f x}}{d}-\frac {2 \sqrt {d e-c f} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{d^{3/2}}\right )}{d}+\frac {2 h (e+f x)^{3/2}}{3 d f}\) |
Input:
Int[(Sqrt[e + f*x]*(g + h*x))/(c + d*x),x]
Output:
(2*h*(e + f*x)^(3/2))/(3*d*f) + ((d*g - c*h)*((2*Sqrt[e + f*x])/d - (2*Sqr t[d*e - c*f]*ArcTanh[(Sqrt[d]*Sqrt[e + f*x])/Sqrt[d*e - c*f]])/d^(3/2)))/d
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
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_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p + 2)) Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 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 = 0.39 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.05
method | result | size |
risch | \(-\frac {2 \left (-f d h x +3 c f h -d e h -3 d f g \right ) \sqrt {f x +e}}{3 f \,d^{2}}+\frac {2 \left (c^{2} f h -c d e h -c d f g +d^{2} e g \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{d^{2} \sqrt {\left (c f -d e \right ) d}}\) | \(103\) |
pseudoelliptic | \(-\frac {2 \left (-f \left (c h -d g \right ) \left (c f -d e \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )+\left (\left (\left (-\frac {h x}{3}-g \right ) f -\frac {e h}{3}\right ) d +c f h \right ) \sqrt {f x +e}\, \sqrt {\left (c f -d e \right ) d}\right )}{\sqrt {\left (c f -d e \right ) d}\, f \,d^{2}}\) | \(105\) |
derivativedivides | \(\frac {-\frac {2 \left (-\frac {h \left (f x +e \right )^{\frac {3}{2}} d}{3}+c f h \sqrt {f x +e}-d f g \sqrt {f x +e}\right )}{d^{2}}+\frac {2 f \left (c^{2} f h -c d e h -c d f g +d^{2} e g \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{d^{2} \sqrt {\left (c f -d e \right ) d}}}{f}\) | \(111\) |
default | \(\frac {-\frac {2 \left (-\frac {h \left (f x +e \right )^{\frac {3}{2}} d}{3}+c f h \sqrt {f x +e}-d f g \sqrt {f x +e}\right )}{d^{2}}+\frac {2 f \left (c^{2} f h -c d e h -c d f g +d^{2} e g \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{d^{2} \sqrt {\left (c f -d e \right ) d}}}{f}\) | \(111\) |
Input:
int((f*x+e)^(1/2)*(h*x+g)/(d*x+c),x,method=_RETURNVERBOSE)
Output:
-2/3*(-d*f*h*x+3*c*f*h-d*e*h-3*d*f*g)*(f*x+e)^(1/2)/f/d^2+2*(c^2*f*h-c*d*e *h-c*d*f*g+d^2*e*g)/d^2/((c*f-d*e)*d)^(1/2)*arctan(d*(f*x+e)^(1/2)/((c*f-d *e)*d)^(1/2))
Time = 0.09 (sec) , antiderivative size = 214, normalized size of antiderivative = 2.18 \[ \int \frac {\sqrt {e+f x} (g+h x)}{c+d x} \, dx=\left [-\frac {3 \, {\left (d f g - c f h\right )} \sqrt {\frac {d e - c f}{d}} \log \left (\frac {d f x + 2 \, d e - c f + 2 \, \sqrt {f x + e} d \sqrt {\frac {d e - c f}{d}}}{d x + c}\right ) - 2 \, {\left (d f h x + 3 \, d f g + {\left (d e - 3 \, c f\right )} h\right )} \sqrt {f x + e}}{3 \, d^{2} f}, -\frac {2 \, {\left (3 \, {\left (d f g - c f h\right )} \sqrt {-\frac {d e - c f}{d}} \arctan \left (-\frac {\sqrt {f x + e} d \sqrt {-\frac {d e - c f}{d}}}{d e - c f}\right ) - {\left (d f h x + 3 \, d f g + {\left (d e - 3 \, c f\right )} h\right )} \sqrt {f x + e}\right )}}{3 \, d^{2} f}\right ] \] Input:
integrate((f*x+e)^(1/2)*(h*x+g)/(d*x+c),x, algorithm="fricas")
Output:
[-1/3*(3*(d*f*g - c*f*h)*sqrt((d*e - c*f)/d)*log((d*f*x + 2*d*e - c*f + 2* sqrt(f*x + e)*d*sqrt((d*e - c*f)/d))/(d*x + c)) - 2*(d*f*h*x + 3*d*f*g + ( d*e - 3*c*f)*h)*sqrt(f*x + e))/(d^2*f), -2/3*(3*(d*f*g - c*f*h)*sqrt(-(d*e - c*f)/d)*arctan(-sqrt(f*x + e)*d*sqrt(-(d*e - c*f)/d)/(d*e - c*f)) - (d* f*h*x + 3*d*f*g + (d*e - 3*c*f)*h)*sqrt(f*x + e))/(d^2*f)]
Time = 3.67 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.27 \[ \int \frac {\sqrt {e+f x} (g+h x)}{c+d x} \, dx=\begin {cases} \frac {2 \left (\frac {h \left (e + f x\right )^{\frac {3}{2}}}{3 d} + \frac {\sqrt {e + f x} \left (- c f h + d f g\right )}{d^{2}} + \frac {f \left (c f - d e\right ) \left (c h - d g\right ) \operatorname {atan}{\left (\frac {\sqrt {e + f x}}{\sqrt {\frac {c f - d e}{d}}} \right )}}{d^{3} \sqrt {\frac {c f - d e}{d}}}\right )}{f} & \text {for}\: f \neq 0 \\\sqrt {e} \left (\frac {h x}{d} - \frac {\left (c h - d g\right ) \left (\begin {cases} \frac {x}{c} & \text {for}\: d = 0 \\\frac {\log {\left (c + d x \right )}}{d} & \text {otherwise} \end {cases}\right )}{d}\right ) & \text {otherwise} \end {cases} \] Input:
integrate((f*x+e)**(1/2)*(h*x+g)/(d*x+c),x)
Output:
Piecewise((2*(h*(e + f*x)**(3/2)/(3*d) + sqrt(e + f*x)*(-c*f*h + d*f*g)/d* *2 + f*(c*f - d*e)*(c*h - d*g)*atan(sqrt(e + f*x)/sqrt((c*f - d*e)/d))/(d* *3*sqrt((c*f - d*e)/d)))/f, Ne(f, 0)), (sqrt(e)*(h*x/d - (c*h - d*g)*Piece wise((x/c, Eq(d, 0)), (log(c + d*x)/d, True))/d), True))
Exception generated. \[ \int \frac {\sqrt {e+f x} (g+h x)}{c+d x} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((f*x+e)^(1/2)*(h*x+g)/(d*x+c),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(c*f-d*e>0)', see `assume?` for m ore detail
Time = 0.11 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.24 \[ \int \frac {\sqrt {e+f x} (g+h x)}{c+d x} \, dx=\frac {2 \, {\left (d^{2} e g - c d f g - c d e h + c^{2} f h\right )} \arctan \left (\frac {\sqrt {f x + e} d}{\sqrt {-d^{2} e + c d f}}\right )}{\sqrt {-d^{2} e + c d f} d^{2}} + \frac {2 \, {\left (3 \, \sqrt {f x + e} d^{2} f^{3} g + {\left (f x + e\right )}^{\frac {3}{2}} d^{2} f^{2} h - 3 \, \sqrt {f x + e} c d f^{3} h\right )}}{3 \, d^{3} f^{3}} \] Input:
integrate((f*x+e)^(1/2)*(h*x+g)/(d*x+c),x, algorithm="giac")
Output:
2*(d^2*e*g - c*d*f*g - c*d*e*h + c^2*f*h)*arctan(sqrt(f*x + e)*d/sqrt(-d^2 *e + c*d*f))/(sqrt(-d^2*e + c*d*f)*d^2) + 2/3*(3*sqrt(f*x + e)*d^2*f^3*g + (f*x + e)^(3/2)*d^2*f^2*h - 3*sqrt(f*x + e)*c*d*f^3*h)/(d^3*f^3)
Time = 1.99 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.10 \[ \int \frac {\sqrt {e+f x} (g+h x)}{c+d x} \, dx=\frac {2\,h\,{\left (e+f\,x\right )}^{3/2}}{3\,d\,f}-\sqrt {e+f\,x}\,\left (\frac {2\,e\,h-2\,f\,g}{d\,f}+\frac {2\,h\,\left (c\,f^2-d\,e\,f\right )}{d^2\,f^2}\right )+\frac {2\,\mathrm {atanh}\left (\frac {\sqrt {d}\,\sqrt {e+f\,x}}{\sqrt {d\,e-c\,f}}\right )\,\sqrt {d\,e-c\,f}\,\left (c\,h-d\,g\right )}{d^{5/2}} \] Input:
int(((e + f*x)^(1/2)*(g + h*x))/(c + d*x),x)
Output:
(2*h*(e + f*x)^(3/2))/(3*d*f) - (e + f*x)^(1/2)*((2*e*h - 2*f*g)/(d*f) + ( 2*h*(c*f^2 - d*e*f))/(d^2*f^2)) + (2*atanh((d^(1/2)*(e + f*x)^(1/2))/(d*e - c*f)^(1/2))*(d*e - c*f)^(1/2)*(c*h - d*g))/d^(5/2)
Time = 0.16 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.42 \[ \int \frac {\sqrt {e+f x} (g+h x)}{c+d x} \, dx=\frac {2 \sqrt {d}\, \sqrt {c f -d e}\, \mathit {atan} \left (\frac {\sqrt {f x +e}\, d}{\sqrt {d}\, \sqrt {c f -d e}}\right ) c f h -2 \sqrt {d}\, \sqrt {c f -d e}\, \mathit {atan} \left (\frac {\sqrt {f x +e}\, d}{\sqrt {d}\, \sqrt {c f -d e}}\right ) d f g -2 \sqrt {f x +e}\, c d f h +\frac {2 \sqrt {f x +e}\, d^{2} e h}{3}+2 \sqrt {f x +e}\, d^{2} f g +\frac {2 \sqrt {f x +e}\, d^{2} f h x}{3}}{d^{3} f} \] Input:
int((f*x+e)^(1/2)*(h*x+g)/(d*x+c),x)
Output:
(2*(3*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d *e)))*c*f*h - 3*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sq rt(c*f - d*e)))*d*f*g - 3*sqrt(e + f*x)*c*d*f*h + sqrt(e + f*x)*d**2*e*h + 3*sqrt(e + f*x)*d**2*f*g + sqrt(e + f*x)*d**2*f*h*x))/(3*d**3*f)