Integrand size = 29, antiderivative size = 118 \[ \int \frac {(A+B x) \sqrt {a x+b x^2}}{x^2 (c+d x)} \, dx=-\frac {2 A \sqrt {a x+b x^2}}{c x}+\frac {2 \sqrt {b} B \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right )}{d}-\frac {2 \sqrt {b c-a d} (B c-A d) \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a x+b x^2}}\right )}{c^{3/2} d} \] Output:
-2*A*(b*x^2+a*x)^(1/2)/c/x+2*b^(1/2)*B*arctanh(b^(1/2)*x/(b*x^2+a*x)^(1/2) )/d-2*(-a*d+b*c)^(1/2)*(-A*d+B*c)*arctanh((-a*d+b*c)^(1/2)*x/c^(1/2)/(b*x^ 2+a*x)^(1/2))/c^(3/2)/d
Result contains complex when optimal does not.
Time = 2.15 (sec) , antiderivative size = 416, normalized size of antiderivative = 3.53 \[ \int \frac {(A+B x) \sqrt {a x+b x^2}}{x^2 (c+d x)} \, dx=\frac {2 \sqrt {a+b x} \left (-A b c^{3/2} d \sqrt {a+b x}+(B c-A d) \left (b c-a d-i \sqrt {a} \sqrt {d} \sqrt {b c-a d}\right ) \sqrt {-b c+2 a d-2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \sqrt {x} \arctan \left (\frac {\sqrt {-b c+2 a d-2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \sqrt {x}}{\sqrt {c} \left (-\sqrt {a}+\sqrt {a+b x}\right )}\right )+(B c-A d) \left (b c-a d+i \sqrt {a} \sqrt {d} \sqrt {b c-a d}\right ) \sqrt {-b c+2 a d+2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \sqrt {x} \arctan \left (\frac {\sqrt {-b c+2 a d+2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \sqrt {x}}{\sqrt {c} \left (-\sqrt {a}+\sqrt {a+b x}\right )}\right )+2 b^{3/2} B c^{5/2} \sqrt {x} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{-\sqrt {a}+\sqrt {a+b x}}\right )\right )}{b c^{5/2} d \sqrt {x (a+b x)}} \] Input:
Integrate[((A + B*x)*Sqrt[a*x + b*x^2])/(x^2*(c + d*x)),x]
Output:
(2*Sqrt[a + b*x]*(-(A*b*c^(3/2)*d*Sqrt[a + b*x]) + (B*c - A*d)*(b*c - a*d - I*Sqrt[a]*Sqrt[d]*Sqrt[b*c - a*d])*Sqrt[-(b*c) + 2*a*d - (2*I)*Sqrt[a]*S qrt[d]*Sqrt[b*c - a*d]]*Sqrt[x]*ArcTan[(Sqrt[-(b*c) + 2*a*d - (2*I)*Sqrt[a ]*Sqrt[d]*Sqrt[b*c - a*d]]*Sqrt[x])/(Sqrt[c]*(-Sqrt[a] + Sqrt[a + b*x]))] + (B*c - A*d)*(b*c - a*d + I*Sqrt[a]*Sqrt[d]*Sqrt[b*c - a*d])*Sqrt[-(b*c) + 2*a*d + (2*I)*Sqrt[a]*Sqrt[d]*Sqrt[b*c - a*d]]*Sqrt[x]*ArcTan[(Sqrt[-(b* c) + 2*a*d + (2*I)*Sqrt[a]*Sqrt[d]*Sqrt[b*c - a*d]]*Sqrt[x])/(Sqrt[c]*(-Sq rt[a] + Sqrt[a + b*x]))] + 2*b^(3/2)*B*c^(5/2)*Sqrt[x]*ArcTanh[(Sqrt[b]*Sq rt[x])/(-Sqrt[a] + Sqrt[a + b*x])]))/(b*c^(5/2)*d*Sqrt[x*(a + b*x)])
Time = 0.99 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.90, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {2153, 2009}
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.
| \(\displaystyle \int \frac {\sqrt {a x+b x^2} (A+B x)}{x^2 (c+d x)} \, dx\) |
| \(\Big \downarrow \) 2153 |
| \(\displaystyle \int \left (\frac {\sqrt {a x+b x^2} (B c-A d)}{c^2 x}-\frac {d \sqrt {a x+b x^2} (B c-A d)}{c^2 (c+d x)}+\frac {A \sqrt {a x+b x^2}}{c x^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\sqrt {b c-a d} (B c-A d) \text {arctanh}\left (\frac {x (2 b c-a d)+a c}{2 \sqrt {c} \sqrt {a x+b x^2} \sqrt {b c-a d}}\right )}{c^{3/2} d}+\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right ) (2 b c-a d) (B c-A d)}{\sqrt {b} c^2 d}+\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right ) (B c-A d)}{\sqrt {b} c^2}+\frac {2 A \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right )}{c}-\frac {2 A \sqrt {a x+b x^2}}{c x}\) |
Input:
Int[((A + B*x)*Sqrt[a*x + b*x^2])/(x^2*(c + d*x)),x]
Output:
(-2*A*Sqrt[a*x + b*x^2])/(c*x) + (2*A*Sqrt[b]*ArcTanh[(Sqrt[b]*x)/Sqrt[a*x + b*x^2]])/c + (a*(B*c - A*d)*ArcTanh[(Sqrt[b]*x)/Sqrt[a*x + b*x^2]])/(Sq rt[b]*c^2) + ((2*b*c - a*d)*(B*c - A*d)*ArcTanh[(Sqrt[b]*x)/Sqrt[a*x + b*x ^2]])/(Sqrt[b]*c^2*d) - (Sqrt[b*c - a*d]*(B*c - A*d)*ArcTanh[(a*c + (2*b*c - a*d)*x)/(2*Sqrt[c]*Sqrt[b*c - a*d]*Sqrt[a*x + b*x^2])])/(c^(3/2)*d)
Int[(Px_)*((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b _.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Px*(d + e* x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && PolyQ[Px, x] && (IntegerQ[p] || (IntegerQ[2*p] && IntegerQ [m] && ILtQ[n, 0])) && !(IGtQ[m, 0] && IGtQ[n, 0])
Time = 0.67 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.03
| method | result | size |
| pseudoelliptic | \(-\frac {2 \left (-x \left (a d -b c \right ) \left (A d -B c \right ) \arctan \left (\frac {\sqrt {x \left (b x +a \right )}\, c}{x \sqrt {c \left (a d -b c \right )}}\right )+\left (-B \sqrt {b}\, \operatorname {arctanh}\left (\frac {\sqrt {x \left (b x +a \right )}}{x \sqrt {b}}\right ) c x +A \sqrt {x \left (b x +a \right )}\, d \right ) \sqrt {c \left (a d -b c \right )}\right )}{\sqrt {c \left (a d -b c \right )}\, c x d}\) | \(122\) |
| risch | \(-\frac {2 A \left (b x +a \right )}{c \sqrt {x \left (b x +a \right )}}-\frac {-\frac {\left (A a \,d^{2}-A b c d -B a c d +B b \,c^{2}\right ) \ln \left (\frac {-\frac {2 c \left (a d -b c \right )}{d^{2}}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {-\frac {c \left (a d -b c \right )}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}-\frac {c \left (a d -b c \right )}{d^{2}}}}{x +\frac {c}{d}}\right )}{d^{2} \sqrt {-\frac {c \left (a d -b c \right )}{d^{2}}}}-\frac {B \sqrt {b}\, c \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{d}}{c}\) | \(218\) |
| default | \(\frac {A \left (-\frac {2 \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{a \,x^{2}}+\frac {2 b \left (\sqrt {b \,x^{2}+a x}+\frac {a \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{2 \sqrt {b}}\right )}{a}\right )}{c}-\frac {\left (A d -B c \right ) \left (\sqrt {b \,x^{2}+a x}+\frac {a \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{2 \sqrt {b}}\right )}{c^{2}}+\frac {\left (A d -B c \right ) \left (\sqrt {b \left (x +\frac {c}{d}\right )^{2}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}-\frac {c \left (a d -b c \right )}{d^{2}}}+\frac {\left (a d -2 b c \right ) \ln \left (\frac {\frac {a d -2 b c}{2 d}+b \left (x +\frac {c}{d}\right )}{\sqrt {b}}+\sqrt {b \left (x +\frac {c}{d}\right )^{2}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}-\frac {c \left (a d -b c \right )}{d^{2}}}\right )}{2 d \sqrt {b}}+\frac {c \left (a d -b c \right ) \ln \left (\frac {-\frac {2 c \left (a d -b c \right )}{d^{2}}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {-\frac {c \left (a d -b c \right )}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}-\frac {c \left (a d -b c \right )}{d^{2}}}}{x +\frac {c}{d}}\right )}{d^{2} \sqrt {-\frac {c \left (a d -b c \right )}{d^{2}}}}\right )}{c^{2}}\) | \(421\) |
Input:
int((B*x+A)*(b*x^2+a*x)^(1/2)/x^2/(d*x+c),x,method=_RETURNVERBOSE)
Output:
-2*(-x*(a*d-b*c)*(A*d-B*c)*arctan((x*(b*x+a))^(1/2)/x*c/(c*(a*d-b*c))^(1/2 ))+(-B*b^(1/2)*arctanh((x*(b*x+a))^(1/2)/x/b^(1/2))*c*x+A*(x*(b*x+a))^(1/2 )*d)*(c*(a*d-b*c))^(1/2))/(c*(a*d-b*c))^(1/2)/c/x/d
Time = 0.12 (sec) , antiderivative size = 525, normalized size of antiderivative = 4.45 \[ \int \frac {(A+B x) \sqrt {a x+b x^2}}{x^2 (c+d x)} \, dx=\left [\frac {B \sqrt {b} c x \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right ) - {\left (B c - A d\right )} x \sqrt {\frac {b c - a d}{c}} \log \left (\frac {a c + {\left (2 \, b c - a d\right )} x + 2 \, \sqrt {b x^{2} + a x} c \sqrt {\frac {b c - a d}{c}}}{d x + c}\right ) - 2 \, \sqrt {b x^{2} + a x} A d}{c d x}, \frac {B \sqrt {b} c x \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right ) - 2 \, {\left (B c - A d\right )} x \sqrt {-\frac {b c - a d}{c}} \arctan \left (-\frac {\sqrt {b x^{2} + a x} c \sqrt {-\frac {b c - a d}{c}}}{{\left (b c - a d\right )} x}\right ) - 2 \, \sqrt {b x^{2} + a x} A d}{c d x}, -\frac {2 \, B \sqrt {-b} c x \arctan \left (\frac {\sqrt {b x^{2} + a x} \sqrt {-b}}{b x + a}\right ) + {\left (B c - A d\right )} x \sqrt {\frac {b c - a d}{c}} \log \left (\frac {a c + {\left (2 \, b c - a d\right )} x + 2 \, \sqrt {b x^{2} + a x} c \sqrt {\frac {b c - a d}{c}}}{d x + c}\right ) + 2 \, \sqrt {b x^{2} + a x} A d}{c d x}, -\frac {2 \, {\left (B \sqrt {-b} c x \arctan \left (\frac {\sqrt {b x^{2} + a x} \sqrt {-b}}{b x + a}\right ) + {\left (B c - A d\right )} x \sqrt {-\frac {b c - a d}{c}} \arctan \left (-\frac {\sqrt {b x^{2} + a x} c \sqrt {-\frac {b c - a d}{c}}}{{\left (b c - a d\right )} x}\right ) + \sqrt {b x^{2} + a x} A d\right )}}{c d x}\right ] \] Input:
integrate((B*x+A)*(b*x^2+a*x)^(1/2)/x^2/(d*x+c),x, algorithm="fricas")
Output:
[(B*sqrt(b)*c*x*log(2*b*x + a + 2*sqrt(b*x^2 + a*x)*sqrt(b)) - (B*c - A*d) *x*sqrt((b*c - a*d)/c)*log((a*c + (2*b*c - a*d)*x + 2*sqrt(b*x^2 + a*x)*c* sqrt((b*c - a*d)/c))/(d*x + c)) - 2*sqrt(b*x^2 + a*x)*A*d)/(c*d*x), (B*sqr t(b)*c*x*log(2*b*x + a + 2*sqrt(b*x^2 + a*x)*sqrt(b)) - 2*(B*c - A*d)*x*sq rt(-(b*c - a*d)/c)*arctan(-sqrt(b*x^2 + a*x)*c*sqrt(-(b*c - a*d)/c)/((b*c - a*d)*x)) - 2*sqrt(b*x^2 + a*x)*A*d)/(c*d*x), -(2*B*sqrt(-b)*c*x*arctan(s qrt(b*x^2 + a*x)*sqrt(-b)/(b*x + a)) + (B*c - A*d)*x*sqrt((b*c - a*d)/c)*l og((a*c + (2*b*c - a*d)*x + 2*sqrt(b*x^2 + a*x)*c*sqrt((b*c - a*d)/c))/(d* x + c)) + 2*sqrt(b*x^2 + a*x)*A*d)/(c*d*x), -2*(B*sqrt(-b)*c*x*arctan(sqrt (b*x^2 + a*x)*sqrt(-b)/(b*x + a)) + (B*c - A*d)*x*sqrt(-(b*c - a*d)/c)*arc tan(-sqrt(b*x^2 + a*x)*c*sqrt(-(b*c - a*d)/c)/((b*c - a*d)*x)) + sqrt(b*x^ 2 + a*x)*A*d)/(c*d*x)]
\[ \int \frac {(A+B x) \sqrt {a x+b x^2}}{x^2 (c+d x)} \, dx=\int \frac {\sqrt {x \left (a + b x\right )} \left (A + B x\right )}{x^{2} \left (c + d x\right )}\, dx \] Input:
integrate((B*x+A)*(b*x**2+a*x)**(1/2)/x**2/(d*x+c),x)
Output:
Integral(sqrt(x*(a + b*x))*(A + B*x)/(x**2*(c + d*x)), x)
\[ \int \frac {(A+B x) \sqrt {a x+b x^2}}{x^2 (c+d x)} \, dx=\int { \frac {\sqrt {b x^{2} + a x} {\left (B x + A\right )}}{{\left (d x + c\right )} x^{2}} \,d x } \] Input:
integrate((B*x+A)*(b*x^2+a*x)^(1/2)/x^2/(d*x+c),x, algorithm="maxima")
Output:
integrate(sqrt(b*x^2 + a*x)*(B*x + A)/((d*x + c)*x^2), x)
Exception generated. \[ \int \frac {(A+B x) \sqrt {a x+b x^2}}{x^2 (c+d x)} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((B*x+A)*(b*x^2+a*x)^(1/2)/x^2/(d*x+c),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
Timed out. \[ \int \frac {(A+B x) \sqrt {a x+b x^2}}{x^2 (c+d x)} \, dx=\int \frac {\sqrt {b\,x^2+a\,x}\,\left (A+B\,x\right )}{x^2\,\left (c+d\,x\right )} \,d x \] Input:
int(((a*x + b*x^2)^(1/2)*(A + B*x))/(x^2*(c + d*x)),x)
Output:
int(((a*x + b*x^2)^(1/2)*(A + B*x))/(x^2*(c + d*x)), x)
Time = 0.23 (sec) , antiderivative size = 269, normalized size of antiderivative = 2.28 \[ \int \frac {(A+B x) \sqrt {a x+b x^2}}{x^2 (c+d x)} \, dx=\frac {2 \sqrt {c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {a d -b c}-\sqrt {d}\, \sqrt {b x +a}-\sqrt {x}\, \sqrt {d}\, \sqrt {b}}{\sqrt {c}\, \sqrt {b}}\right ) a d x -2 \sqrt {c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {a d -b c}-\sqrt {d}\, \sqrt {b x +a}-\sqrt {x}\, \sqrt {d}\, \sqrt {b}}{\sqrt {c}\, \sqrt {b}}\right ) b c x +2 \sqrt {c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {a d -b c}+\sqrt {d}\, \sqrt {b x +a}+\sqrt {x}\, \sqrt {d}\, \sqrt {b}}{\sqrt {c}\, \sqrt {b}}\right ) a d x -2 \sqrt {c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {a d -b c}+\sqrt {d}\, \sqrt {b x +a}+\sqrt {x}\, \sqrt {d}\, \sqrt {b}}{\sqrt {c}\, \sqrt {b}}\right ) b c x -2 \sqrt {x}\, \sqrt {b x +a}\, a c d +2 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b x +a}+\sqrt {x}\, \sqrt {b}}{\sqrt {a}}\right ) b \,c^{2} x -2 \sqrt {b}\, a c d x}{c^{2} d x} \] Input:
int((B*x+A)*(b*x^2+a*x)^(1/2)/x^2/(d*x+c),x)
Output:
(2*(sqrt(c)*sqrt(a*d - b*c)*atan((sqrt(a*d - b*c) - sqrt(d)*sqrt(a + b*x) - sqrt(x)*sqrt(d)*sqrt(b))/(sqrt(c)*sqrt(b)))*a*d*x - sqrt(c)*sqrt(a*d - b *c)*atan((sqrt(a*d - b*c) - sqrt(d)*sqrt(a + b*x) - sqrt(x)*sqrt(d)*sqrt(b ))/(sqrt(c)*sqrt(b)))*b*c*x + sqrt(c)*sqrt(a*d - b*c)*atan((sqrt(a*d - b*c ) + sqrt(d)*sqrt(a + b*x) + sqrt(x)*sqrt(d)*sqrt(b))/(sqrt(c)*sqrt(b)))*a* d*x - sqrt(c)*sqrt(a*d - b*c)*atan((sqrt(a*d - b*c) + sqrt(d)*sqrt(a + b*x ) + sqrt(x)*sqrt(d)*sqrt(b))/(sqrt(c)*sqrt(b)))*b*c*x - sqrt(x)*sqrt(a + b *x)*a*c*d + sqrt(b)*log((sqrt(a + b*x) + sqrt(x)*sqrt(b))/sqrt(a))*b*c**2* x - sqrt(b)*a*c*d*x))/(c**2*d*x)