Integrand size = 25, antiderivative size = 101 \[ \int \frac {\sqrt {c+d x} (e+f x)}{x (a+b x)} \, dx=\frac {2 f \sqrt {c+d x}}{b}-\frac {2 \sqrt {c} e \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a}+\frac {2 \sqrt {b c-a d} (b e-a f) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{a b^{3/2}} \] Output:
2*f*(d*x+c)^(1/2)/b-2*c^(1/2)*e*arctanh((d*x+c)^(1/2)/c^(1/2))/a+2*(-a*d+b *c)^(1/2)*(-a*f+b*e)*arctanh(b^(1/2)*(d*x+c)^(1/2)/(-a*d+b*c)^(1/2))/a/b^( 3/2)
Time = 0.19 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {c+d x} (e+f x)}{x (a+b x)} \, dx=\frac {2 f \sqrt {c+d x}}{b}+\frac {2 \sqrt {-b c+a d} (b e-a f) \arctan \left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {-b c+a d}}\right )}{a b^{3/2}}-\frac {2 \sqrt {c} e \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a} \] Input:
Integrate[(Sqrt[c + d*x]*(e + f*x))/(x*(a + b*x)),x]
Output:
(2*f*Sqrt[c + d*x])/b + (2*Sqrt[-(b*c) + a*d]*(b*e - a*f)*ArcTan[(Sqrt[b]* Sqrt[c + d*x])/Sqrt[-(b*c) + a*d]])/(a*b^(3/2)) - (2*Sqrt[c]*e*ArcTanh[Sqr t[c + d*x]/Sqrt[c]])/a
Time = 0.29 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.06, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {171, 27, 174, 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.
\(\displaystyle \int \frac {\sqrt {c+d x} (e+f x)}{x (a+b x)} \, dx\) |
\(\Big \downarrow \) 171 |
\(\displaystyle \frac {2 \int \frac {b c e+(b d e+b c f-a d f) x}{2 x (a+b x) \sqrt {c+d x}}dx}{b}+\frac {2 f \sqrt {c+d x}}{b}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {b c e+(b d e+b c f-a d f) x}{x (a+b x) \sqrt {c+d x}}dx}{b}+\frac {2 f \sqrt {c+d x}}{b}\) |
\(\Big \downarrow \) 174 |
\(\displaystyle \frac {\frac {b c e \int \frac {1}{x \sqrt {c+d x}}dx}{a}-\frac {(b c-a d) (b e-a f) \int \frac {1}{(a+b x) \sqrt {c+d x}}dx}{a}}{b}+\frac {2 f \sqrt {c+d x}}{b}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {\frac {2 b c e \int \frac {1}{\frac {c+d x}{d}-\frac {c}{d}}d\sqrt {c+d x}}{a d}-\frac {2 (b c-a d) (b e-a f) \int \frac {1}{a+\frac {b (c+d x)}{d}-\frac {b c}{d}}d\sqrt {c+d x}}{a d}}{b}+\frac {2 f \sqrt {c+d x}}{b}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {\frac {2 \sqrt {b c-a d} (b e-a f) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{a \sqrt {b}}-\frac {2 b \sqrt {c} e \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a}}{b}+\frac {2 f \sqrt {c+d x}}{b}\) |
Input:
Int[(Sqrt[c + d*x]*(e + f*x))/(x*(a + b*x)),x]
Output:
(2*f*Sqrt[c + d*x])/b + ((-2*b*Sqrt[c]*e*ArcTanh[Sqrt[c + d*x]/Sqrt[c]])/a + (2*Sqrt[b*c - a*d]*(b*e - a*f)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/(a*Sqrt[b]))/b
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, 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_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[h*(a + b*x)^m*(c + d*x)^(n + 1)*(( e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Simp[1/(d*f*(m + n + p + 2)) Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2 ) - h*(b*c*e*m + a*(d*e*(n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x], x], x] /; Fre eQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegersQ[2*m, 2*n, 2*p]
Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))* ((c_.) + (d_.)*(x_))), x_] :> Simp[(b*g - a*h)/(b*c - a*d) Int[(e + f*x)^ p/(a + b*x), x], x] - Simp[(d*g - c*h)/(b*c - a*d) Int[(e + f*x)^p/(c + d *x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]
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 = 0.38 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.02
method | result | size |
derivativedivides | \(\frac {2 f \sqrt {x d +c}}{b}+\frac {2 \left (-a^{2} d f +a b c f +a b d e -c e \,b^{2}\right ) \arctan \left (\frac {b \sqrt {x d +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{a b \sqrt {\left (a d -b c \right ) b}}-\frac {2 \sqrt {c}\, e \,\operatorname {arctanh}\left (\frac {\sqrt {x d +c}}{\sqrt {c}}\right )}{a}\) | \(103\) |
default | \(\frac {2 f \sqrt {x d +c}}{b}+\frac {2 \left (-a^{2} d f +a b c f +a b d e -c e \,b^{2}\right ) \arctan \left (\frac {b \sqrt {x d +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{a b \sqrt {\left (a d -b c \right ) b}}-\frac {2 \sqrt {c}\, e \,\operatorname {arctanh}\left (\frac {\sqrt {x d +c}}{\sqrt {c}}\right )}{a}\) | \(103\) |
pseudoelliptic | \(\frac {-2 \left (a f -b e \right ) \left (a d -b c \right ) \arctan \left (\frac {b \sqrt {x d +c}}{\sqrt {\left (a d -b c \right ) b}}\right )+2 \left (-\sqrt {c}\, \operatorname {arctanh}\left (\frac {\sqrt {x d +c}}{\sqrt {c}}\right ) b e +\sqrt {x d +c}\, a f \right ) \sqrt {\left (a d -b c \right ) b}}{a b \sqrt {\left (a d -b c \right ) b}}\) | \(105\) |
Input:
int((d*x+c)^(1/2)*(f*x+e)/x/(b*x+a),x,method=_RETURNVERBOSE)
Output:
2*f*(d*x+c)^(1/2)/b+2*(-a^2*d*f+a*b*c*f+a*b*d*e-b^2*c*e)/a/b/((a*d-b*c)*b) ^(1/2)*arctan(b*(d*x+c)^(1/2)/((a*d-b*c)*b)^(1/2))-2*c^(1/2)*e*arctanh((d* x+c)^(1/2)/c^(1/2))/a
Time = 0.16 (sec) , antiderivative size = 443, normalized size of antiderivative = 4.39 \[ \int \frac {\sqrt {c+d x} (e+f x)}{x (a+b x)} \, dx=\left [\frac {b \sqrt {c} e \log \left (\frac {d x - 2 \, \sqrt {d x + c} \sqrt {c} + 2 \, c}{x}\right ) + 2 \, \sqrt {d x + c} a f - {\left (b e - a f\right )} \sqrt {\frac {b c - a d}{b}} \log \left (\frac {b d x + 2 \, b c - a d - 2 \, \sqrt {d x + c} b \sqrt {\frac {b c - a d}{b}}}{b x + a}\right )}{a b}, \frac {b \sqrt {c} e \log \left (\frac {d x - 2 \, \sqrt {d x + c} \sqrt {c} + 2 \, c}{x}\right ) + 2 \, \sqrt {d x + c} a f + 2 \, {\left (b e - a f\right )} \sqrt {-\frac {b c - a d}{b}} \arctan \left (-\frac {\sqrt {d x + c} b \sqrt {-\frac {b c - a d}{b}}}{b c - a d}\right )}{a b}, \frac {2 \, b \sqrt {-c} e \arctan \left (\frac {\sqrt {-c}}{\sqrt {d x + c}}\right ) + 2 \, \sqrt {d x + c} a f - {\left (b e - a f\right )} \sqrt {\frac {b c - a d}{b}} \log \left (\frac {b d x + 2 \, b c - a d - 2 \, \sqrt {d x + c} b \sqrt {\frac {b c - a d}{b}}}{b x + a}\right )}{a b}, \frac {2 \, {\left (b \sqrt {-c} e \arctan \left (\frac {\sqrt {-c}}{\sqrt {d x + c}}\right ) + \sqrt {d x + c} a f + {\left (b e - a f\right )} \sqrt {-\frac {b c - a d}{b}} \arctan \left (-\frac {\sqrt {d x + c} b \sqrt {-\frac {b c - a d}{b}}}{b c - a d}\right )\right )}}{a b}\right ] \] Input:
integrate((d*x+c)^(1/2)*(f*x+e)/x/(b*x+a),x, algorithm="fricas")
Output:
[(b*sqrt(c)*e*log((d*x - 2*sqrt(d*x + c)*sqrt(c) + 2*c)/x) + 2*sqrt(d*x + c)*a*f - (b*e - a*f)*sqrt((b*c - a*d)/b)*log((b*d*x + 2*b*c - a*d - 2*sqrt (d*x + c)*b*sqrt((b*c - a*d)/b))/(b*x + a)))/(a*b), (b*sqrt(c)*e*log((d*x - 2*sqrt(d*x + c)*sqrt(c) + 2*c)/x) + 2*sqrt(d*x + c)*a*f + 2*(b*e - a*f)* sqrt(-(b*c - a*d)/b)*arctan(-sqrt(d*x + c)*b*sqrt(-(b*c - a*d)/b)/(b*c - a *d)))/(a*b), (2*b*sqrt(-c)*e*arctan(sqrt(-c)/sqrt(d*x + c)) + 2*sqrt(d*x + c)*a*f - (b*e - a*f)*sqrt((b*c - a*d)/b)*log((b*d*x + 2*b*c - a*d - 2*sqr t(d*x + c)*b*sqrt((b*c - a*d)/b))/(b*x + a)))/(a*b), 2*(b*sqrt(-c)*e*arcta n(sqrt(-c)/sqrt(d*x + c)) + sqrt(d*x + c)*a*f + (b*e - a*f)*sqrt(-(b*c - a *d)/b)*arctan(-sqrt(d*x + c)*b*sqrt(-(b*c - a*d)/b)/(b*c - a*d)))/(a*b)]
Leaf count of result is larger than twice the leaf count of optimal. 196 vs. \(2 (88) = 176\).
Time = 13.55 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.94 \[ \int \frac {\sqrt {c+d x} (e+f x)}{x (a+b x)} \, dx=\begin {cases} \frac {2 f \sqrt {c + d x}}{b} + \frac {2 c e \operatorname {atan}{\left (\frac {\sqrt {c + d x}}{\sqrt {- c}} \right )}}{a \sqrt {- c}} - \frac {2 \left (a d - b c\right ) \left (a f - b e\right ) \operatorname {atan}{\left (\frac {\sqrt {c + d x}}{\sqrt {\frac {a d - b c}{b}}} \right )}}{a b^{2} \sqrt {\frac {a d - b c}{b}}} & \text {for}\: d \neq 0 \\\sqrt {c} \left (\left (- f + \frac {b e}{2 a}\right ) \left (\frac {2 a \left (\begin {cases} - \frac {\frac {1}{x} + \frac {b}{2 a}}{b} & \text {for}\: a = 0 \\\frac {\log {\left (2 a \left (\frac {1}{x} + \frac {b}{2 a}\right ) - b \right )}}{2 a} & \text {otherwise} \end {cases}\right )}{b} - \frac {2 a \left (\begin {cases} \frac {\frac {1}{x} + \frac {b}{2 a}}{b} & \text {for}\: a = 0 \\\frac {\log {\left (2 a \left (\frac {1}{x} + \frac {b}{2 a}\right ) + b \right )}}{2 a} & \text {otherwise} \end {cases}\right )}{b}\right ) - \frac {e \log {\left (\frac {a}{x^{2}} + \frac {b}{x} \right )}}{2 a}\right ) & \text {otherwise} \end {cases} \] Input:
integrate((d*x+c)**(1/2)*(f*x+e)/x/(b*x+a),x)
Output:
Piecewise((2*f*sqrt(c + d*x)/b + 2*c*e*atan(sqrt(c + d*x)/sqrt(-c))/(a*sqr t(-c)) - 2*(a*d - b*c)*(a*f - b*e)*atan(sqrt(c + d*x)/sqrt((a*d - b*c)/b)) /(a*b**2*sqrt((a*d - b*c)/b)), Ne(d, 0)), (sqrt(c)*((-f + b*e/(2*a))*(2*a* Piecewise((-(1/x + b/(2*a))/b, Eq(a, 0)), (log(2*a*(1/x + b/(2*a)) - b)/(2 *a), True))/b - 2*a*Piecewise(((1/x + b/(2*a))/b, Eq(a, 0)), (log(2*a*(1/x + b/(2*a)) + b)/(2*a), True))/b) - e*log(a/x**2 + b/x)/(2*a)), True))
Exception generated. \[ \int \frac {\sqrt {c+d x} (e+f x)}{x (a+b x)} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((d*x+c)^(1/2)*(f*x+e)/x/(b*x+a),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*d-b*c>0)', see `assume?` for m ore detail
Time = 0.13 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.08 \[ \int \frac {\sqrt {c+d x} (e+f x)}{x (a+b x)} \, dx=\frac {2 \, c e \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-c}}\right )}{a \sqrt {-c}} + \frac {2 \, \sqrt {d x + c} f}{b} - \frac {2 \, {\left (b^{2} c e - a b d e - a b c f + a^{2} d f\right )} \arctan \left (\frac {\sqrt {d x + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{\sqrt {-b^{2} c + a b d} a b} \] Input:
integrate((d*x+c)^(1/2)*(f*x+e)/x/(b*x+a),x, algorithm="giac")
Output:
2*c*e*arctan(sqrt(d*x + c)/sqrt(-c))/(a*sqrt(-c)) + 2*sqrt(d*x + c)*f/b - 2*(b^2*c*e - a*b*d*e - a*b*c*f + a^2*d*f)*arctan(sqrt(d*x + c)*b/sqrt(-b^2 *c + a*b*d))/(sqrt(-b^2*c + a*b*d)*a*b)
Time = 2.19 (sec) , antiderivative size = 2368, normalized size of antiderivative = 23.45 \[ \int \frac {\sqrt {c+d x} (e+f x)}{x (a+b x)} \, dx=\text {Too large to display} \] Input:
int(((e + f*x)*(c + d*x)^(1/2))/(x*(a + b*x)),x)
Output:
(2*f*(c + d*x)^(1/2))/b - (c^(1/2)*e*atan(((c^(1/2)*e*((8*(c + d*x)^(1/2)* (a^4*d^4*f^2 + a^2*b^2*d^4*e^2 + 2*b^4*c^2*d^2*e^2 - 2*a^3*b*d^4*e*f + a^2 *b^2*c^2*d^2*f^2 - 2*a*b^3*c*d^3*e^2 - 2*a^3*b*c*d^3*f^2 - 2*a*b^3*c^2*d^2 *e*f + 4*a^2*b^2*c*d^3*e*f))/b + (c^(1/2)*e*((8*(a^3*b^2*c*d^3*f - a^2*b^3 *c^2*d^2*f))/b + (8*c^(1/2)*e*(a^3*b^3*d^3 - 2*a^2*b^4*c*d^2)*(c + d*x)^(1 /2))/(a*b)))/a)*1i)/a + (c^(1/2)*e*((8*(c + d*x)^(1/2)*(a^4*d^4*f^2 + a^2* b^2*d^4*e^2 + 2*b^4*c^2*d^2*e^2 - 2*a^3*b*d^4*e*f + a^2*b^2*c^2*d^2*f^2 - 2*a*b^3*c*d^3*e^2 - 2*a^3*b*c*d^3*f^2 - 2*a*b^3*c^2*d^2*e*f + 4*a^2*b^2*c* d^3*e*f))/b - (c^(1/2)*e*((8*(a^3*b^2*c*d^3*f - a^2*b^3*c^2*d^2*f))/b - (8 *c^(1/2)*e*(a^3*b^3*d^3 - 2*a^2*b^4*c*d^2)*(c + d*x)^(1/2))/(a*b)))/a)*1i) /a)/((16*(b^3*c^2*d^3*e^3 - a*b^2*c*d^4*e^3 - a^3*c*d^4*e*f^2 + b^3*c^3*d^ 2*e^2*f - 3*a*b^2*c^2*d^3*e^2*f - a*b^2*c^3*d^2*e*f^2 + 2*a^2*b*c^2*d^3*e* f^2 + 2*a^2*b*c*d^4*e^2*f))/b - (c^(1/2)*e*((8*(c + d*x)^(1/2)*(a^4*d^4*f^ 2 + a^2*b^2*d^4*e^2 + 2*b^4*c^2*d^2*e^2 - 2*a^3*b*d^4*e*f + a^2*b^2*c^2*d^ 2*f^2 - 2*a*b^3*c*d^3*e^2 - 2*a^3*b*c*d^3*f^2 - 2*a*b^3*c^2*d^2*e*f + 4*a^ 2*b^2*c*d^3*e*f))/b + (c^(1/2)*e*((8*(a^3*b^2*c*d^3*f - a^2*b^3*c^2*d^2*f) )/b + (8*c^(1/2)*e*(a^3*b^3*d^3 - 2*a^2*b^4*c*d^2)*(c + d*x)^(1/2))/(a*b)) )/a))/a + (c^(1/2)*e*((8*(c + d*x)^(1/2)*(a^4*d^4*f^2 + a^2*b^2*d^4*e^2 + 2*b^4*c^2*d^2*e^2 - 2*a^3*b*d^4*e*f + a^2*b^2*c^2*d^2*f^2 - 2*a*b^3*c*d^3* e^2 - 2*a^3*b*c*d^3*f^2 - 2*a*b^3*c^2*d^2*e*f + 4*a^2*b^2*c*d^3*e*f))/b...
Time = 0.16 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.33 \[ \int \frac {\sqrt {c+d x} (e+f x)}{x (a+b x)} \, dx=\frac {-2 \sqrt {b}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {d x +c}\, b}{\sqrt {b}\, \sqrt {a d -b c}}\right ) a f +2 \sqrt {b}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {d x +c}\, b}{\sqrt {b}\, \sqrt {a d -b c}}\right ) b e +2 \sqrt {d x +c}\, a b f +\sqrt {c}\, \mathrm {log}\left (\sqrt {d x +c}-\sqrt {c}\right ) b^{2} e -\sqrt {c}\, \mathrm {log}\left (\sqrt {d x +c}+\sqrt {c}\right ) b^{2} e}{a \,b^{2}} \] Input:
int((d*x+c)^(1/2)*(f*x+e)/x/(b*x+a),x)
Output:
( - 2*sqrt(b)*sqrt(a*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(a*d - b *c)))*a*f + 2*sqrt(b)*sqrt(a*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt (a*d - b*c)))*b*e + 2*sqrt(c + d*x)*a*b*f + sqrt(c)*log(sqrt(c + d*x) - sq rt(c))*b**2*e - sqrt(c)*log(sqrt(c + d*x) + sqrt(c))*b**2*e)/(a*b**2)