Integrand size = 22, antiderivative size = 98 \[ \int \frac {(a+b x) \sqrt {e+f x}}{c+d x} \, dx=-\frac {2 (b c-a d) \sqrt {e+f x}}{d^2}+\frac {2 b (e+f x)^{3/2}}{3 d f}+\frac {2 (b c-a d) \sqrt {d e-c f} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{d^{5/2}} \] Output:
-2*(-a*d+b*c)*(f*x+e)^(1/2)/d^2+2/3*b*(f*x+e)^(3/2)/d/f+2*(-a*d+b*c)*(-c*f +d*e)^(1/2)*arctanh(d^(1/2)*(f*x+e)^(1/2)/(-c*f+d*e)^(1/2))/d^(5/2)
Time = 0.19 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.96 \[ \int \frac {(a+b x) \sqrt {e+f x}}{c+d x} \, dx=\frac {2 \sqrt {e+f x} (-3 b c f+3 a d f+b d (e+f x))}{3 d^2 f}+\frac {2 (b c-a d) \sqrt {-d e+c f} \arctan \left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {-d e+c f}}\right )}{d^{5/2}} \] Input:
Integrate[((a + b*x)*Sqrt[e + f*x])/(c + d*x),x]
Output:
(2*Sqrt[e + f*x]*(-3*b*c*f + 3*a*d*f + b*d*(e + f*x)))/(3*d^2*f) + (2*(b*c - a*d)*Sqrt[-(d*e) + c*f]*ArcTan[(Sqrt[d]*Sqrt[e + f*x])/Sqrt[-(d*e) + c* f]])/d^(5/2)
Time = 0.22 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.98, 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.
\(\displaystyle \int \frac {(a+b x) \sqrt {e+f x}}{c+d x} \, dx\) |
\(\Big \downarrow \) 90 |
\(\displaystyle \frac {2 b (e+f x)^{3/2}}{3 d f}-\frac {(b c-a d) \int \frac {\sqrt {e+f x}}{c+d x}dx}{d}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {2 b (e+f x)^{3/2}}{3 d f}-\frac {(b c-a d) \left (\frac {(d e-c f) \int \frac {1}{(c+d x) \sqrt {e+f x}}dx}{d}+\frac {2 \sqrt {e+f x}}{d}\right )}{d}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {2 b (e+f x)^{3/2}}{3 d f}-\frac {(b c-a d) \left (\frac {2 (d e-c f) \int \frac {1}{c+\frac {d (e+f x)}{f}-\frac {d e}{f}}d\sqrt {e+f x}}{d f}+\frac {2 \sqrt {e+f x}}{d}\right )}{d}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {2 b (e+f x)^{3/2}}{3 d f}-\frac {(b c-a d) \left (\frac {2 \sqrt {e+f x}}{d}-\frac {2 \sqrt {d e-c f} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{d^{3/2}}\right )}{d}\) |
Input:
Int[((a + b*x)*Sqrt[e + f*x])/(c + d*x),x]
Output:
(2*b*(e + f*x)^(3/2))/(3*d*f) - ((b*c - a*d)*((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 = 0.40 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.03
method | result | size |
risch | \(\frac {2 \left (f b d x +3 a d f -3 b c f +b d e \right ) \sqrt {f x +e}}{3 f \,d^{2}}-\frac {2 \left (a c d f -a e \,d^{2}-c^{2} b f +b c d e \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{d^{2} \sqrt {\left (c f -d e \right ) d}}\) | \(101\) |
pseudoelliptic | \(\frac {-2 f \left (c f -d e \right ) \left (a d -b c \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )+2 \sqrt {f x +e}\, \sqrt {\left (c f -d e \right ) d}\, \left (\left (\left (\frac {b x}{3}+a \right ) f +\frac {b e}{3}\right ) d -b c f \right )}{f \,d^{2} \sqrt {\left (c f -d e \right ) d}}\) | \(104\) |
derivativedivides | \(\frac {\frac {2 \left (\frac {b \left (f x +e \right )^{\frac {3}{2}} d}{3}+a d f \sqrt {f x +e}-b c f \sqrt {f x +e}\right )}{d^{2}}-\frac {2 f \left (a c d f -a e \,d^{2}-c^{2} b f +b c d e \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{d^{2} \sqrt {\left (c f -d e \right ) d}}}{f}\) | \(111\) |
default | \(\frac {\frac {2 \left (\frac {b \left (f x +e \right )^{\frac {3}{2}} d}{3}+a d f \sqrt {f x +e}-b c f \sqrt {f x +e}\right )}{d^{2}}-\frac {2 f \left (a c d f -a e \,d^{2}-c^{2} b f +b c d e \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{d^{2} \sqrt {\left (c f -d e \right ) d}}}{f}\) | \(111\) |
Input:
int((b*x+a)*(f*x+e)^(1/2)/(d*x+c),x,method=_RETURNVERBOSE)
Output:
2/3*(b*d*f*x+3*a*d*f-3*b*c*f+b*d*e)*(f*x+e)^(1/2)/f/d^2-2*(a*c*d*f-a*d^2*e -b*c^2*f+b*c*d*e)/d^2/((c*f-d*e)*d)^(1/2)*arctan(d*(f*x+e)^(1/2)/((c*f-d*e )*d)^(1/2))
Time = 0.11 (sec) , antiderivative size = 211, normalized size of antiderivative = 2.15 \[ \int \frac {(a+b x) \sqrt {e+f x}}{c+d x} \, dx=\left [-\frac {3 \, {\left (b c - a d\right )} f \sqrt {\frac {d e - c f}{d}} \log \left (\frac {d f x + 2 \, d e - c f - 2 \, \sqrt {f x + e} d \sqrt {\frac {d e - c f}{d}}}{d x + c}\right ) - 2 \, {\left (b d f x + b d e - 3 \, {\left (b c - a d\right )} f\right )} \sqrt {f x + e}}{3 \, d^{2} f}, \frac {2 \, {\left (3 \, {\left (b c - a d\right )} f \sqrt {-\frac {d e - c f}{d}} \arctan \left (-\frac {\sqrt {f x + e} d \sqrt {-\frac {d e - c f}{d}}}{d e - c f}\right ) + {\left (b d f x + b d e - 3 \, {\left (b c - a d\right )} f\right )} \sqrt {f x + e}\right )}}{3 \, d^{2} f}\right ] \] Input:
integrate((b*x+a)*(f*x+e)^(1/2)/(d*x+c),x, algorithm="fricas")
Output:
[-1/3*(3*(b*c - a*d)*f*sqrt((d*e - c*f)/d)*log((d*f*x + 2*d*e - c*f - 2*sq rt(f*x + e)*d*sqrt((d*e - c*f)/d))/(d*x + c)) - 2*(b*d*f*x + b*d*e - 3*(b* c - a*d)*f)*sqrt(f*x + e))/(d^2*f), 2/3*(3*(b*c - a*d)*f*sqrt(-(d*e - c*f) /d)*arctan(-sqrt(f*x + e)*d*sqrt(-(d*e - c*f)/d)/(d*e - c*f)) + (b*d*f*x + b*d*e - 3*(b*c - a*d)*f)*sqrt(f*x + e))/(d^2*f)]
Time = 3.67 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.27 \[ \int \frac {(a+b x) \sqrt {e+f x}}{c+d x} \, dx=\begin {cases} \frac {2 \left (\frac {b \left (e + f x\right )^{\frac {3}{2}}}{3 d} + \frac {\sqrt {e + f x} \left (a d f - b c f\right )}{d^{2}} - \frac {f \left (a d - b c\right ) \left (c f - d e\right ) \operatorname {atan}{\left (\frac {\sqrt {e + f x}}{\sqrt {\frac {c f - d e}{d}}} \right )}}{d^{3} \sqrt {\frac {c f - d e}{d}}}\right )}{f} & \text {for}\: f \neq 0 \\\sqrt {e} \left (\frac {b x}{d} + \frac {\left (a d - b c\right ) \left (\begin {cases} \frac {x}{c} & \text {for}\: d = 0 \\\frac {\log {\left (c + d x \right )}}{d} & \text {otherwise} \end {cases}\right )}{d}\right ) & \text {otherwise} \end {cases} \] Input:
integrate((b*x+a)*(f*x+e)**(1/2)/(d*x+c),x)
Output:
Piecewise((2*(b*(e + f*x)**(3/2)/(3*d) + sqrt(e + f*x)*(a*d*f - b*c*f)/d** 2 - f*(a*d - b*c)*(c*f - d*e)*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)*(b*x/d + (a*d - b*c)*Piecew ise((x/c, Eq(d, 0)), (log(c + d*x)/d, True))/d), True))
Exception generated. \[ \int \frac {(a+b x) \sqrt {e+f x}}{c+d x} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((b*x+a)*(f*x+e)^(1/2)/(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 = 0.13 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.24 \[ \int \frac {(a+b x) \sqrt {e+f x}}{c+d x} \, dx=-\frac {2 \, {\left (b c d e - a d^{2} e - b c^{2} f + a c d f\right )} \arctan \left (\frac {\sqrt {f x + e} d}{\sqrt {-d^{2} e + c d f}}\right )}{\sqrt {-d^{2} e + c d f} d^{2}} + \frac {2 \, {\left ({\left (f x + e\right )}^{\frac {3}{2}} b d^{2} f^{2} - 3 \, \sqrt {f x + e} b c d f^{3} + 3 \, \sqrt {f x + e} a d^{2} f^{3}\right )}}{3 \, d^{3} f^{3}} \] Input:
integrate((b*x+a)*(f*x+e)^(1/2)/(d*x+c),x, algorithm="giac")
Output:
-2*(b*c*d*e - a*d^2*e - b*c^2*f + a*c*d*f)*arctan(sqrt(f*x + e)*d/sqrt(-d^ 2*e + c*d*f))/(sqrt(-d^2*e + c*d*f)*d^2) + 2/3*((f*x + e)^(3/2)*b*d^2*f^2 - 3*sqrt(f*x + e)*b*c*d*f^3 + 3*sqrt(f*x + e)*a*d^2*f^3)/(d^3*f^3)
Time = 0.08 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.09 \[ \int \frac {(a+b x) \sqrt {e+f x}}{c+d x} \, dx=\sqrt {e+f\,x}\,\left (\frac {2\,a\,f-2\,b\,e}{d\,f}-\frac {2\,b\,\left (c\,f^2-d\,e\,f\right )}{d^2\,f^2}\right )+\frac {2\,b\,{\left (e+f\,x\right )}^{3/2}}{3\,d\,f}-\frac {2\,\mathrm {atanh}\left (\frac {\sqrt {d}\,\sqrt {e+f\,x}}{\sqrt {d\,e-c\,f}}\right )\,\left (a\,d-b\,c\right )\,\sqrt {d\,e-c\,f}}{d^{5/2}} \] Input:
int(((e + f*x)^(1/2)*(a + b*x))/(c + d*x),x)
Output:
(e + f*x)^(1/2)*((2*a*f - 2*b*e)/(d*f) - (2*b*(c*f^2 - d*e*f))/(d^2*f^2)) + (2*b*(e + f*x)^(3/2))/(3*d*f) - (2*atanh((d^(1/2)*(e + f*x)^(1/2))/(d*e - c*f)^(1/2))*(a*d - b*c)*(d*e - c*f)^(1/2))/d^(5/2)
Time = 0.16 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.42 \[ \int \frac {(a+b x) \sqrt {e+f x}}{c+d x} \, dx=\frac {-2 \sqrt {d}\, \sqrt {c f -d e}\, \mathit {atan} \left (\frac {\sqrt {f x +e}\, d}{\sqrt {d}\, \sqrt {c f -d e}}\right ) a d f +2 \sqrt {d}\, \sqrt {c f -d e}\, \mathit {atan} \left (\frac {\sqrt {f x +e}\, d}{\sqrt {d}\, \sqrt {c f -d e}}\right ) b c f +2 \sqrt {f x +e}\, a \,d^{2} f -2 \sqrt {f x +e}\, b c d f +\frac {2 \sqrt {f x +e}\, b \,d^{2} e}{3}+\frac {2 \sqrt {f x +e}\, b \,d^{2} f x}{3}}{d^{3} f} \] Input:
int((b*x+a)*(f*x+e)^(1/2)/(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)))*a*d*f + 3*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d) *sqrt(c*f - d*e)))*b*c*f + 3*sqrt(e + f*x)*a*d**2*f - 3*sqrt(e + f*x)*b*c* d*f + sqrt(e + f*x)*b*d**2*e + sqrt(e + f*x)*b*d**2*f*x))/(3*d**3*f)