\(\int \frac {\sqrt {c+d x} (e+f x)}{x} \, dx\) [11]

Optimal result

Integrand size = 18, antiderivative size = 54 \[ \int \frac {\sqrt {c+d x} (e+f x)}{x} \, dx=2 e \sqrt {c+d x}+\frac {2 f (c+d x)^{3/2}}{3 d}-2 \sqrt {c} e \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right ) \]

[Out]

2/3*f*(d*x+c)^(3/2)/d-2*e*arctanh((d*x+c)^(1/2)/c^(1/2))*c^(1/2)+2*e*(d*x+c)^(1/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {81, 52, 65, 214} \[ \int \frac {\sqrt {c+d x} (e+f x)}{x} \, dx=-2 \sqrt {c} e \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )+2 e \sqrt {c+d x}+\frac {2 f (c+d x)^{3/2}}{3 d} \]

[In]

Int[(Sqrt[c + d*x]*(e + f*x))/x,x]

[Out]

2*e*Sqrt[c + d*x] + (2*f*(c + d*x)^(3/2))/(3*d) - 2*Sqrt[c]*e*ArcTanh[Sqrt[c + d*x]/Sqrt[c]]

Rule 52

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] + Dist[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] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[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] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 81

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(c + d*x)^
(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Dist[(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]

Rule 214

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]

Rubi steps \begin{align*} \text {integral}& = \frac {2 f (c+d x)^{3/2}}{3 d}+e \int \frac {\sqrt {c+d x}}{x} \, dx \\ & = 2 e \sqrt {c+d x}+\frac {2 f (c+d x)^{3/2}}{3 d}+(c e) \int \frac {1}{x \sqrt {c+d x}} \, dx \\ & = 2 e \sqrt {c+d x}+\frac {2 f (c+d x)^{3/2}}{3 d}+\frac {(2 c e) \text {Subst}\left (\int \frac {1}{-\frac {c}{d}+\frac {x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{d} \\ & = 2 e \sqrt {c+d x}+\frac {2 f (c+d x)^{3/2}}{3 d}-2 \sqrt {c} e \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.98 \[ \int \frac {\sqrt {c+d x} (e+f x)}{x} \, dx=\frac {2 \sqrt {c+d x} (3 d e+c f+d f x)}{3 d}-2 \sqrt {c} e \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right ) \]

[In]

Integrate[(Sqrt[c + d*x]*(e + f*x))/x,x]

[Out]

(2*Sqrt[c + d*x]*(3*d*e + c*f + d*f*x))/(3*d) - 2*Sqrt[c]*e*ArcTanh[Sqrt[c + d*x]/Sqrt[c]]

Maple [A] (verified)

Time = 5.25 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.85

method	result	size
derivativedivides	\(\frac {\frac {2 f \left (d x +c \right )^{\frac {3}{2}}}{3}+2 d e \sqrt {d x +c}-2 \sqrt {c}\, d e \,\operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{d}\)	\(46\)
default	\(\frac {\frac {2 f \left (d x +c \right )^{\frac {3}{2}}}{3}+2 d e \sqrt {d x +c}-2 \sqrt {c}\, d e \,\operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{d}\)	\(46\)
pseudoelliptic	\(\frac {-6 \sqrt {c}\, d e \,\operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )+2 \left (\left (f x +3 e \right ) d +c f \right ) \sqrt {d x +c}}{3 d}\)	\(48\)

[In]

int((f*x+e)*(d*x+c)^(1/2)/x,x,method=_RETURNVERBOSE)

[Out]

2/d*(1/3*f*(d*x+c)^(3/2)+d*e*(d*x+c)^(1/2)-c^(1/2)*d*e*arctanh((d*x+c)^(1/2)/c^(1/2)))

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 111, normalized size of antiderivative = 2.06 \[ \int \frac {\sqrt {c+d x} (e+f x)}{x} \, dx=\left [\frac {3 \, \sqrt {c} d e \log \left (\frac {d x - 2 \, \sqrt {d x + c} \sqrt {c} + 2 \, c}{x}\right ) + 2 \, {\left (d f x + 3 \, d e + c f\right )} \sqrt {d x + c}}{3 \, d}, \frac {2 \, {\left (3 \, \sqrt {-c} d e \arctan \left (\frac {\sqrt {d x + c} \sqrt {-c}}{c}\right ) + {\left (d f x + 3 \, d e + c f\right )} \sqrt {d x + c}\right )}}{3 \, d}\right ] \]

[In]

integrate((f*x+e)*(d*x+c)^(1/2)/x,x, algorithm="fricas")

[Out]

[1/3*(3*sqrt(c)*d*e*log((d*x - 2*sqrt(d*x + c)*sqrt(c) + 2*c)/x) + 2*(d*f*x + 3*d*e + c*f)*sqrt(d*x + c))/d, 2
/3*(3*sqrt(-c)*d*e*arctan(sqrt(d*x + c)*sqrt(-c)/c) + (d*f*x + 3*d*e + c*f)*sqrt(d*x + c))/d]

Sympy [A] (verification not implemented)

Time = 1.67 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.30 \[ \int \frac {\sqrt {c+d x} (e+f x)}{x} \, dx=\begin {cases} \frac {2 c e \operatorname {atan}{\left (\frac {\sqrt {c + d x}}{\sqrt {- c}} \right )}}{\sqrt {- c}} + 2 e \sqrt {c + d x} + \frac {2 f \left (c + d x\right )^{\frac {3}{2}}}{3 d} & \text {for}\: d \neq 0 \\\sqrt {c} \left (e \log {\left (f x \right )} + f x\right ) & \text {otherwise} \end {cases} \]

[In]

integrate((f*x+e)*(d*x+c)**(1/2)/x,x)

[Out]

Piecewise((2*c*e*atan(sqrt(c + d*x)/sqrt(-c))/sqrt(-c) + 2*e*sqrt(c + d*x) + 2*f*(c + d*x)**(3/2)/(3*d), Ne(d,
0)), (sqrt(c)*(e*log(f*x) + f*x), True))

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.11 \[ \int \frac {\sqrt {c+d x} (e+f x)}{x} \, dx=\sqrt {c} e \log \left (\frac {\sqrt {d x + c} - \sqrt {c}}{\sqrt {d x + c} + \sqrt {c}}\right ) + \frac {2 \, {\left (3 \, \sqrt {d x + c} d e + {\left (d x + c\right )}^{\frac {3}{2}} f\right )}}{3 \, d} \]

[In]

integrate((f*x+e)*(d*x+c)^(1/2)/x,x, algorithm="maxima")

[Out]

sqrt(c)*e*log((sqrt(d*x + c) - sqrt(c))/(sqrt(d*x + c) + sqrt(c))) + 2/3*(3*sqrt(d*x + c)*d*e + (d*x + c)^(3/2
)*f)/d

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.02 \[ \int \frac {\sqrt {c+d x} (e+f x)}{x} \, dx=\frac {2 \, c e \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-c}}\right )}{\sqrt {-c}} + \frac {2 \, {\left (3 \, \sqrt {d x + c} d^{3} e + {\left (d x + c\right )}^{\frac {3}{2}} d^{2} f\right )}}{3 \, d^{3}} \]

[In]

integrate((f*x+e)*(d*x+c)^(1/2)/x,x, algorithm="giac")

[Out]

2*c*e*arctan(sqrt(d*x + c)/sqrt(-c))/sqrt(-c) + 2/3*(3*sqrt(d*x + c)*d^3*e + (d*x + c)^(3/2)*d^2*f)/d^3

Mupad [B] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.83 \[ \int \frac {\sqrt {c+d x} (e+f x)}{x} \, dx=2\,e\,\sqrt {c+d\,x}+\frac {2\,f\,{\left (c+d\,x\right )}^{3/2}}{3\,d}+\sqrt {c}\,e\,\mathrm {atan}\left (\frac {\sqrt {c+d\,x}\,1{}\mathrm {i}}{\sqrt {c}}\right )\,2{}\mathrm {i} \]

[In]

int(((e + f*x)*(c + d*x)^(1/2))/x,x)

[Out]

2*e*(c + d*x)^(1/2) + c^(1/2)*e*atan(((c + d*x)^(1/2)*1i)/c^(1/2))*2i + (2*f*(c + d*x)^(3/2))/(3*d)