Integrand size = 25, antiderivative size = 128 \[ \int \frac {\sqrt {a+b x} (e+f x)}{x (c+d x)^2} \, dx=\frac {(d e-c f) \sqrt {a+b x}}{c d (c+d x)}-\frac {\left (2 a d^2 e-b c (d e+c f)\right ) \arctan \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )}{c^2 d^{3/2} \sqrt {b c-a d}}-\frac {2 \sqrt {a} e \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{c^2} \]
-2*e*arctanh((b*x+a)^(1/2)/a^(1/2))*a^(1/2)/c^2-(2*a*d^2*e-b*c*(c*f+d*e))* arctan(d^(1/2)*(b*x+a)^(1/2)/(-a*d+b*c)^(1/2))/c^2/d^(3/2)/(-a*d+b*c)^(1/2 )+(-c*f+d*e)*(b*x+a)^(1/2)/c/d/(d*x+c)
Time = 0.46 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.95 \[ \int \frac {\sqrt {a+b x} (e+f x)}{x (c+d x)^2} \, dx=\frac {\frac {c (d e-c f) \sqrt {a+b x}}{d (c+d x)}+\frac {\left (-2 a d^2 e+b c (d e+c f)\right ) \arctan \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )}{d^{3/2} \sqrt {b c-a d}}-2 \sqrt {a} e \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{c^2} \]
((c*(d*e - c*f)*Sqrt[a + b*x])/(d*(c + d*x)) + ((-2*a*d^2*e + b*c*(d*e + c *f))*ArcTan[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[b*c - a*d]])/(d^(3/2)*Sqrt[b*c - a*d]) - 2*Sqrt[a]*e*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/c^2
Time = 0.26 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.09, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {166, 27, 174, 73, 218, 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.
| \(\displaystyle \int \frac {\sqrt {a+b x} (e+f x)}{x (c+d x)^2} \, dx\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle \frac {\sqrt {a+b x} (d e-c f)}{c d (c+d x)}-\frac {\int -\frac {2 a d e+b (d e+c f) x}{2 x \sqrt {a+b x} (c+d x)}dx}{c d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {2 a d e+b (d e+c f) x}{x \sqrt {a+b x} (c+d x)}dx}{2 c d}+\frac {\sqrt {a+b x} (d e-c f)}{c d (c+d x)}\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle \frac {\frac {2 a d e \int \frac {1}{x \sqrt {a+b x}}dx}{c}-\frac {\left (2 a d^2 e-b c (c f+d e)\right ) \int \frac {1}{\sqrt {a+b x} (c+d x)}dx}{c}}{2 c d}+\frac {\sqrt {a+b x} (d e-c f)}{c d (c+d x)}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {\frac {4 a d e \int \frac {1}{\frac {a+b x}{b}-\frac {a}{b}}d\sqrt {a+b x}}{b c}-\frac {2 \left (2 a d^2 e-b c (c f+d e)\right ) \int \frac {1}{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}d\sqrt {a+b x}}{b c}}{2 c d}+\frac {\sqrt {a+b x} (d e-c f)}{c d (c+d x)}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\frac {4 a d e \int \frac {1}{\frac {a+b x}{b}-\frac {a}{b}}d\sqrt {a+b x}}{b c}-\frac {2 \arctan \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right ) \left (2 a d^2 e-b c (c f+d e)\right )}{c \sqrt {d} \sqrt {b c-a d}}}{2 c d}+\frac {\sqrt {a+b x} (d e-c f)}{c d (c+d x)}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {-\frac {2 \arctan \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right ) \left (2 a d^2 e-b c (c f+d e)\right )}{c \sqrt {d} \sqrt {b c-a d}}-\frac {4 \sqrt {a} d e \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{c}}{2 c d}+\frac {\sqrt {a+b x} (d e-c f)}{c d (c+d x)}\) |
((d*e - c*f)*Sqrt[a + b*x])/(c*d*(c + d*x)) + ((-2*(2*a*d^2*e - b*c*(d*e + c*f))*ArcTan[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[b*c - a*d]])/(c*Sqrt[d]*Sqrt[b* c - a*d]) - (4*Sqrt[a]*d*e*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/c)/(2*c*d)
3.1.20.3.1 Defintions of rubi rules used
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[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] - Simp[1/(b*(b*e - a*f)*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[b* c*(f*g - e*h)*(m + 1) + (b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h )*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, p}, x] && ILtQ[m, -1] && GtQ[n, 0]
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)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
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 = 1.62 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.86
| method | result | size |
| pseudoelliptic | \(\frac {-2 e \,\operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right ) \sqrt {a}+\frac {-\frac {c \left (c f -d e \right ) \sqrt {b x +a}}{d x +c}+\frac {\left (2 a e \,d^{2}-c^{2} b f -b c d e \right ) \operatorname {arctanh}\left (\frac {d \sqrt {b x +a}}{\sqrt {\left (a d -b c \right ) d}}\right )}{\sqrt {\left (a d -b c \right ) d}}}{d}}{c^{2}}\) | \(110\) |
| derivativedivides | \(2 b \left (-\frac {e \sqrt {a}\, \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{b \,c^{2}}+\frac {\frac {b c \left (c f -d e \right ) \sqrt {b x +a}}{2 d \left (-d \left (b x +a \right )+a d -b c \right )}+\frac {\left (2 a e \,d^{2}-c^{2} b f -b c d e \right ) \operatorname {arctanh}\left (\frac {d \sqrt {b x +a}}{\sqrt {\left (a d -b c \right ) d}}\right )}{2 d \sqrt {\left (a d -b c \right ) d}}}{c^{2} b}\right )\) | \(137\) |
| default | \(2 b \left (-\frac {e \sqrt {a}\, \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{b \,c^{2}}+\frac {\frac {b c \left (c f -d e \right ) \sqrt {b x +a}}{2 d \left (-d \left (b x +a \right )+a d -b c \right )}+\frac {\left (2 a e \,d^{2}-c^{2} b f -b c d e \right ) \operatorname {arctanh}\left (\frac {d \sqrt {b x +a}}{\sqrt {\left (a d -b c \right ) d}}\right )}{2 d \sqrt {\left (a d -b c \right ) d}}}{c^{2} b}\right )\) | \(137\) |
1/c^2*(-2*e*arctanh((b*x+a)^(1/2)/a^(1/2))*a^(1/2)+1/d*(-c*(c*f-d*e)*(b*x+ a)^(1/2)/(d*x+c)+(2*a*d^2*e-b*c^2*f-b*c*d*e)/((a*d-b*c)*d)^(1/2)*arctanh(d *(b*x+a)^(1/2)/((a*d-b*c)*d)^(1/2))))
Leaf count of result is larger than twice the leaf count of optimal. 243 vs. \(2 (110) = 220\).
Time = 0.32 (sec) , antiderivative size = 1008, normalized size of antiderivative = 7.88 \[ \int \frac {\sqrt {a+b x} (e+f x)}{x (c+d x)^2} \, dx=\left [-\frac {{\left (b c^{3} f + {\left (b c^{2} d - 2 \, a c d^{2}\right )} e + {\left (b c^{2} d f + {\left (b c d^{2} - 2 \, a d^{3}\right )} e\right )} x\right )} \sqrt {-b c d + a d^{2}} \log \left (\frac {b d x - b c + 2 \, a d - 2 \, \sqrt {-b c d + a d^{2}} \sqrt {b x + a}}{d x + c}\right ) - 2 \, {\left ({\left (b c d^{3} - a d^{4}\right )} e x + {\left (b c^{2} d^{2} - a c d^{3}\right )} e\right )} \sqrt {a} \log \left (\frac {b x - 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) - 2 \, {\left ({\left (b c^{2} d^{2} - a c d^{3}\right )} e - {\left (b c^{3} d - a c^{2} d^{2}\right )} f\right )} \sqrt {b x + a}}{2 \, {\left (b c^{4} d^{2} - a c^{3} d^{3} + {\left (b c^{3} d^{3} - a c^{2} d^{4}\right )} x\right )}}, \frac {4 \, {\left ({\left (b c d^{3} - a d^{4}\right )} e x + {\left (b c^{2} d^{2} - a c d^{3}\right )} e\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) - {\left (b c^{3} f + {\left (b c^{2} d - 2 \, a c d^{2}\right )} e + {\left (b c^{2} d f + {\left (b c d^{2} - 2 \, a d^{3}\right )} e\right )} x\right )} \sqrt {-b c d + a d^{2}} \log \left (\frac {b d x - b c + 2 \, a d - 2 \, \sqrt {-b c d + a d^{2}} \sqrt {b x + a}}{d x + c}\right ) + 2 \, {\left ({\left (b c^{2} d^{2} - a c d^{3}\right )} e - {\left (b c^{3} d - a c^{2} d^{2}\right )} f\right )} \sqrt {b x + a}}{2 \, {\left (b c^{4} d^{2} - a c^{3} d^{3} + {\left (b c^{3} d^{3} - a c^{2} d^{4}\right )} x\right )}}, -\frac {{\left (b c^{3} f + {\left (b c^{2} d - 2 \, a c d^{2}\right )} e + {\left (b c^{2} d f + {\left (b c d^{2} - 2 \, a d^{3}\right )} e\right )} x\right )} \sqrt {b c d - a d^{2}} \arctan \left (\frac {\sqrt {b c d - a d^{2}} \sqrt {b x + a}}{b d x + a d}\right ) - {\left ({\left (b c d^{3} - a d^{4}\right )} e x + {\left (b c^{2} d^{2} - a c d^{3}\right )} e\right )} \sqrt {a} \log \left (\frac {b x - 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) - {\left ({\left (b c^{2} d^{2} - a c d^{3}\right )} e - {\left (b c^{3} d - a c^{2} d^{2}\right )} f\right )} \sqrt {b x + a}}{b c^{4} d^{2} - a c^{3} d^{3} + {\left (b c^{3} d^{3} - a c^{2} d^{4}\right )} x}, -\frac {{\left (b c^{3} f + {\left (b c^{2} d - 2 \, a c d^{2}\right )} e + {\left (b c^{2} d f + {\left (b c d^{2} - 2 \, a d^{3}\right )} e\right )} x\right )} \sqrt {b c d - a d^{2}} \arctan \left (\frac {\sqrt {b c d - a d^{2}} \sqrt {b x + a}}{b d x + a d}\right ) - 2 \, {\left ({\left (b c d^{3} - a d^{4}\right )} e x + {\left (b c^{2} d^{2} - a c d^{3}\right )} e\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) - {\left ({\left (b c^{2} d^{2} - a c d^{3}\right )} e - {\left (b c^{3} d - a c^{2} d^{2}\right )} f\right )} \sqrt {b x + a}}{b c^{4} d^{2} - a c^{3} d^{3} + {\left (b c^{3} d^{3} - a c^{2} d^{4}\right )} x}\right ] \]
[-1/2*((b*c^3*f + (b*c^2*d - 2*a*c*d^2)*e + (b*c^2*d*f + (b*c*d^2 - 2*a*d^ 3)*e)*x)*sqrt(-b*c*d + a*d^2)*log((b*d*x - b*c + 2*a*d - 2*sqrt(-b*c*d + a *d^2)*sqrt(b*x + a))/(d*x + c)) - 2*((b*c*d^3 - a*d^4)*e*x + (b*c^2*d^2 - a*c*d^3)*e)*sqrt(a)*log((b*x - 2*sqrt(b*x + a)*sqrt(a) + 2*a)/x) - 2*((b*c ^2*d^2 - a*c*d^3)*e - (b*c^3*d - a*c^2*d^2)*f)*sqrt(b*x + a))/(b*c^4*d^2 - a*c^3*d^3 + (b*c^3*d^3 - a*c^2*d^4)*x), 1/2*(4*((b*c*d^3 - a*d^4)*e*x + ( b*c^2*d^2 - a*c*d^3)*e)*sqrt(-a)*arctan(sqrt(b*x + a)*sqrt(-a)/a) - (b*c^3 *f + (b*c^2*d - 2*a*c*d^2)*e + (b*c^2*d*f + (b*c*d^2 - 2*a*d^3)*e)*x)*sqrt (-b*c*d + a*d^2)*log((b*d*x - b*c + 2*a*d - 2*sqrt(-b*c*d + a*d^2)*sqrt(b* x + a))/(d*x + c)) + 2*((b*c^2*d^2 - a*c*d^3)*e - (b*c^3*d - a*c^2*d^2)*f) *sqrt(b*x + a))/(b*c^4*d^2 - a*c^3*d^3 + (b*c^3*d^3 - a*c^2*d^4)*x), -((b* c^3*f + (b*c^2*d - 2*a*c*d^2)*e + (b*c^2*d*f + (b*c*d^2 - 2*a*d^3)*e)*x)*s qrt(b*c*d - a*d^2)*arctan(sqrt(b*c*d - a*d^2)*sqrt(b*x + a)/(b*d*x + a*d)) - ((b*c*d^3 - a*d^4)*e*x + (b*c^2*d^2 - a*c*d^3)*e)*sqrt(a)*log((b*x - 2* sqrt(b*x + a)*sqrt(a) + 2*a)/x) - ((b*c^2*d^2 - a*c*d^3)*e - (b*c^3*d - a* c^2*d^2)*f)*sqrt(b*x + a))/(b*c^4*d^2 - a*c^3*d^3 + (b*c^3*d^3 - a*c^2*d^4 )*x), -((b*c^3*f + (b*c^2*d - 2*a*c*d^2)*e + (b*c^2*d*f + (b*c*d^2 - 2*a*d ^3)*e)*x)*sqrt(b*c*d - a*d^2)*arctan(sqrt(b*c*d - a*d^2)*sqrt(b*x + a)/(b* d*x + a*d)) - 2*((b*c*d^3 - a*d^4)*e*x + (b*c^2*d^2 - a*c*d^3)*e)*sqrt(-a) *arctan(sqrt(b*x + a)*sqrt(-a)/a) - ((b*c^2*d^2 - a*c*d^3)*e - (b*c^3*d...
Timed out. \[ \int \frac {\sqrt {a+b x} (e+f x)}{x (c+d x)^2} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {\sqrt {a+b x} (e+f x)}{x (c+d x)^2} \, dx=\text {Exception raised: ValueError} \]
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 = 0.28 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.07 \[ \int \frac {\sqrt {a+b x} (e+f x)}{x (c+d x)^2} \, dx=\frac {2 \, a e \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a} c^{2}} + \frac {{\left (b c d e - 2 \, a d^{2} e + b c^{2} f\right )} \arctan \left (\frac {\sqrt {b x + a} d}{\sqrt {b c d - a d^{2}}}\right )}{\sqrt {b c d - a d^{2}} c^{2} d} + \frac {\sqrt {b x + a} b d e - \sqrt {b x + a} b c f}{{\left (b c + {\left (b x + a\right )} d - a d\right )} c d} \]
2*a*e*arctan(sqrt(b*x + a)/sqrt(-a))/(sqrt(-a)*c^2) + (b*c*d*e - 2*a*d^2*e + b*c^2*f)*arctan(sqrt(b*x + a)*d/sqrt(b*c*d - a*d^2))/(sqrt(b*c*d - a*d^ 2)*c^2*d) + (sqrt(b*x + a)*b*d*e - sqrt(b*x + a)*b*c*f)/((b*c + (b*x + a)* d - a*d)*c*d)
Time = 3.39 (sec) , antiderivative size = 1814, normalized size of antiderivative = 14.17 \[ \int \frac {\sqrt {a+b x} (e+f x)}{x (c+d x)^2} \, dx=\text {Too large to display} \]
(atan(((((((2*(2*a*b^3*c^4*d^3*e - 2*a*b^3*c^5*d^2*f))/(c^3*d) + ((4*b^3*c ^5*d^3 - 8*a*b^2*c^4*d^4)*(d^3*(a*d - b*c))^(1/2)*(a + b*x)^(1/2)*(b*c^2*f - 2*a*d^2*e + b*c*d*e))/(c^2*d*(a*c^2*d^4 - b*c^3*d^3)))*(d^3*(a*d - b*c) )^(1/2)*(b*c^2*f - 2*a*d^2*e + b*c*d*e))/(2*(a*c^2*d^4 - b*c^3*d^3)) + (2* (a + b*x)^(1/2)*(b^4*c^4*f^2 + 8*a^2*b^2*d^4*e^2 + b^4*c^2*d^2*e^2 + 2*b^4 *c^3*d*e*f - 4*a*b^3*c*d^3*e^2 - 4*a*b^3*c^2*d^2*e*f))/(c^2*d))*(d^3*(a*d - b*c))^(1/2)*(b*c^2*f - 2*a*d^2*e + b*c*d*e)*1i)/(2*(a*c^2*d^4 - b*c^3*d^ 3)) - (((((2*(2*a*b^3*c^4*d^3*e - 2*a*b^3*c^5*d^2*f))/(c^3*d) - ((4*b^3*c^ 5*d^3 - 8*a*b^2*c^4*d^4)*(d^3*(a*d - b*c))^(1/2)*(a + b*x)^(1/2)*(b*c^2*f - 2*a*d^2*e + b*c*d*e))/(c^2*d*(a*c^2*d^4 - b*c^3*d^3)))*(d^3*(a*d - b*c)) ^(1/2)*(b*c^2*f - 2*a*d^2*e + b*c*d*e))/(2*(a*c^2*d^4 - b*c^3*d^3)) - (2*( a + b*x)^(1/2)*(b^4*c^4*f^2 + 8*a^2*b^2*d^4*e^2 + b^4*c^2*d^2*e^2 + 2*b^4* c^3*d*e*f - 4*a*b^3*c*d^3*e^2 - 4*a*b^3*c^2*d^2*e*f))/(c^2*d))*(d^3*(a*d - b*c))^(1/2)*(b*c^2*f - 2*a*d^2*e + b*c*d*e)*1i)/(2*(a*c^2*d^4 - b*c^3*d^3 )))/((4*(a*b^4*c*d^2*e^3 - 2*a^2*b^3*d^3*e^3 + a*b^4*c^3*e*f^2 - 2*a^2*b^3 *c*d^2*e^2*f + 2*a*b^4*c^2*d*e^2*f))/(c^3*d) + (((((2*(2*a*b^3*c^4*d^3*e - 2*a*b^3*c^5*d^2*f))/(c^3*d) + ((4*b^3*c^5*d^3 - 8*a*b^2*c^4*d^4)*(d^3*(a* d - b*c))^(1/2)*(a + b*x)^(1/2)*(b*c^2*f - 2*a*d^2*e + b*c*d*e))/(c^2*d*(a *c^2*d^4 - b*c^3*d^3)))*(d^3*(a*d - b*c))^(1/2)*(b*c^2*f - 2*a*d^2*e + b*c *d*e))/(2*(a*c^2*d^4 - b*c^3*d^3)) + (2*(a + b*x)^(1/2)*(b^4*c^4*f^2 + ...