Integrand size = 24, antiderivative size = 109 \[ \int \frac {\left (a x+b x^2\right )^{3/2}}{x^3 (c+d x)} \, dx=-\frac {2 a \sqrt {a x+b x^2}}{c x}+\frac {2 b^{3/2} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right )}{d}-\frac {2 (b c-a d)^{3/2} \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a x+b x^2}}\right )}{c^{3/2} d} \] Output:
-2*a*(b*x^2+a*x)^(1/2)/c/x+2*b^(3/2)*arctanh(b^(1/2)*x/(b*x^2+a*x)^(1/2))/ d-2*(-a*d+b*c)^(3/2)*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 = 1.77 (sec) , antiderivative size = 418, normalized size of antiderivative = 3.83 \[ \int \frac {\left (a x+b x^2\right )^{3/2}}{x^3 (c+d x)} \, dx=\frac {2 (x (a+b x))^{3/2} \left (-a b c^{3/2} d \sqrt {a+b x}+(b c-a d) \left (b c-a d-i \sqrt {a} \sqrt {d} \sqrt {b c-a d}\right ) \sqrt {-b c+2 a d-2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \sqrt {x} \arctan \left (\frac {\sqrt {-b c+2 a d-2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \sqrt {x}}{\sqrt {c} \left (-\sqrt {a}+\sqrt {a+b x}\right )}\right )+(b c-a d) \left (b c-a d+i \sqrt {a} \sqrt {d} \sqrt {b c-a d}\right ) \sqrt {-b c+2 a d+2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \sqrt {x} \arctan \left (\frac {\sqrt {-b c+2 a d+2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \sqrt {x}}{\sqrt {c} \left (-\sqrt {a}+\sqrt {a+b x}\right )}\right )+2 b^{5/2} c^{5/2} \sqrt {x} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{-\sqrt {a}+\sqrt {a+b x}}\right )\right )}{b c^{5/2} d x^2 (a+b x)^{3/2}} \] Input:
Integrate[(a*x + b*x^2)^(3/2)/(x^3*(c + d*x)),x]
Output:
(2*(x*(a + b*x))^(3/2)*(-(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)*Sqr t[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]*(-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[(Sqr t[-(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]))] + 2*b^(5/2)*c^(5/2)*Sqrt[x]*ArcTanh[(Sqrt[b ]*Sqrt[x])/(-Sqrt[a] + Sqrt[a + b*x])]))/(b*c^(5/2)*d*x^2*(a + b*x)^(3/2))
Leaf count is larger than twice the leaf count of optimal. \(427\) vs. \(2(109)=218\).
Time = 1.23 (sec) , antiderivative size = 427, normalized size of antiderivative = 3.92, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {1260, 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.
| \(\displaystyle \int \frac {\left (a x+b x^2\right )^{3/2}}{x^3 (c+d x)} \, dx\) |
| \(\Big \downarrow \) 1260 |
| \(\displaystyle \int \left (-\frac {d^3 \left (a x+b x^2\right )^{3/2}}{c^3 (c+d x)}+\frac {d^2 \left (a x+b x^2\right )^{3/2}}{c^3 x}-\frac {d \left (a x+b x^2\right )^{3/2}}{c^2 x^2}+\frac {\left (a x+b x^2\right )^{3/2}}{c x^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a^3 d^2 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right )}{8 b^{3/2} c^3}+\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right ) (2 b c-a d) \left (-a^2 d^2-8 a b c d+8 b^2 c^2\right )}{8 b^{3/2} c^3 d}-\frac {3 a^2 d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right )}{4 \sqrt {b} c^2}-\frac {\sqrt {a x+b x^2} \left (a^2 d^2-2 b d x (2 b c-a d)-10 a b c d+8 b^2 c^2\right )}{8 b c^3}-\frac {(b c-a d)^{3/2} \text {arctanh}\left (\frac {x (2 b c-a d)+a c}{2 \sqrt {c} \sqrt {a x+b x^2} \sqrt {b c-a d}}\right )}{c^{3/2} d}+\frac {3 a \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right )}{c}+\frac {a d^2 (a+2 b x) \sqrt {a x+b x^2}}{8 b c^3}-\frac {3 a d \sqrt {a x+b x^2}}{4 c^2}-\frac {d \left (a x+b x^2\right )^{3/2}}{2 c^2 x}-\frac {2 a \sqrt {a x+b x^2}}{c x}+\frac {b \sqrt {a x+b x^2}}{c}\) |
Input:
Int[(a*x + b*x^2)^(3/2)/(x^3*(c + d*x)),x]
Output:
(b*Sqrt[a*x + b*x^2])/c - (3*a*d*Sqrt[a*x + b*x^2])/(4*c^2) - (2*a*Sqrt[a* x + b*x^2])/(c*x) + (a*d^2*(a + 2*b*x)*Sqrt[a*x + b*x^2])/(8*b*c^3) - ((8* b^2*c^2 - 10*a*b*c*d + a^2*d^2 - 2*b*d*(2*b*c - a*d)*x)*Sqrt[a*x + b*x^2]) /(8*b*c^3) - (d*(a*x + b*x^2)^(3/2))/(2*c^2*x) + (3*a*Sqrt[b]*ArcTanh[(Sqr t[b]*x)/Sqrt[a*x + b*x^2]])/c - (3*a^2*d*ArcTanh[(Sqrt[b]*x)/Sqrt[a*x + b* x^2]])/(4*Sqrt[b]*c^2) - (a^3*d^2*ArcTanh[(Sqrt[b]*x)/Sqrt[a*x + b*x^2]])/ (8*b^(3/2)*c^3) + ((2*b*c - a*d)*(8*b^2*c^2 - 8*a*b*c*d - a^2*d^2)*ArcTanh [(Sqrt[b]*x)/Sqrt[a*x + b*x^2]])/(8*b^(3/2)*c^3*d) - ((b*c - a*d)^(3/2)*Ar cTanh[(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[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x + c*x^2)^p , (d + e*x)^m*(f + g*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ [c*d^2 - b*d*e + a*e^2, 0] && EqQ[m + n + 2*p + 1, 0] && ILtQ[m, 0] && ILtQ [n, 0]
Time = 0.62 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.06
| method | result | size |
| pseudoelliptic | \(-\frac {2 \left (-x \left (a d -b c \right )^{2} \arctan \left (\frac {\sqrt {x \left (b x +a \right )}\, c}{x \sqrt {c \left (a d -b c \right )}}\right )+\left (-b^{\frac {3}{2}} \operatorname {arctanh}\left (\frac {\sqrt {x \left (b x +a \right )}}{x \sqrt {b}}\right ) c x +\sqrt {x \left (b x +a \right )}\, a d \right ) \sqrt {c \left (a d -b c \right )}\right )}{\sqrt {c \left (a d -b c \right )}\, c d x}\) | \(115\) |
| risch | \(-\frac {2 a \left (b x +a \right )}{c \sqrt {x \left (b x +a \right )}}+\frac {\frac {c \,b^{\frac {3}{2}} \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{d}+\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \ln \left (\frac {-\frac {2 c \left (a d -b c \right )}{d^{2}}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {-\frac {c \left (a d -b c \right )}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}-\frac {c \left (a d -b c \right )}{d^{2}}}}{x +\frac {c}{d}}\right )}{d^{2} \sqrt {-\frac {c \left (a d -b c \right )}{d^{2}}}}}{c}\) | \(210\) |
| default | \(\frac {-\frac {2 \left (b \,x^{2}+a x \right )^{\frac {5}{2}}}{a \,x^{3}}+\frac {4 b \left (\frac {2 \left (b \,x^{2}+a x \right )^{\frac {5}{2}}}{a \,x^{2}}-\frac {6 b \left (\frac {\left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{3}+\frac {a \left (\frac {\left (2 b x +a \right ) \sqrt {b \,x^{2}+a x}}{4 b}-\frac {a^{2} \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{8 b^{\frac {3}{2}}}\right )}{2}\right )}{a}\right )}{a}}{c}+\frac {d^{2} \left (\frac {\left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{3}+\frac {a \left (\frac {\left (2 b x +a \right ) \sqrt {b \,x^{2}+a x}}{4 b}-\frac {a^{2} \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{8 b^{\frac {3}{2}}}\right )}{2}\right )}{c^{3}}-\frac {d \left (\frac {2 \left (b \,x^{2}+a x \right )^{\frac {5}{2}}}{a \,x^{2}}-\frac {6 b \left (\frac {\left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{3}+\frac {a \left (\frac {\left (2 b x +a \right ) \sqrt {b \,x^{2}+a x}}{4 b}-\frac {a^{2} \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{8 b^{\frac {3}{2}}}\right )}{2}\right )}{a}\right )}{c^{2}}-\frac {d^{2} \left (\frac {\left (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 )^{\frac {3}{2}}}{3}+\frac {\left (a d -2 b c \right ) \left (\frac {\left (2 b \left (x +\frac {c}{d}\right )+\frac {a d -2 b c}{d}\right ) \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}}}}{4 b}+\frac {\left (-\frac {4 b c \left (a d -b c \right )}{d^{2}}-\frac {\left (a d -2 b c \right )^{2}}{d^{2}}\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 )}{8 b^{\frac {3}{2}}}\right )}{2 d}-\frac {c \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 )}{d^{2}}\right )}{c^{3}}\) | \(865\) |
Input:
int((b*x^2+a*x)^(3/2)/x^3/(d*x+c),x,method=_RETURNVERBOSE)
Output:
-2/(c*(a*d-b*c))^(1/2)*(-x*(a*d-b*c)^2*arctan((x*(b*x+a))^(1/2)/x*c/(c*(a* d-b*c))^(1/2))+(-b^(3/2)*arctanh((x*(b*x+a))^(1/2)/x/b^(1/2))*c*x+(x*(b*x+ a))^(1/2)*a*d)*(c*(a*d-b*c))^(1/2))/c/d/x
Time = 0.14 (sec) , antiderivative size = 523, normalized size of antiderivative = 4.80 \[ \int \frac {\left (a x+b x^2\right )^{3/2}}{x^3 (c+d x)} \, dx=\left [\frac {b^{\frac {3}{2}} c x \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right ) - {\left (b c - a d\right )} x \sqrt {\frac {b c - a d}{c}} \log \left (\frac {a c + {\left (2 \, b c - a d\right )} x + 2 \, \sqrt {b x^{2} + a x} c \sqrt {\frac {b c - a d}{c}}}{d x + c}\right ) - 2 \, \sqrt {b x^{2} + a x} a d}{c d x}, \frac {b^{\frac {3}{2}} c x \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right ) - 2 \, {\left (b c - a d\right )} x \sqrt {-\frac {b c - a d}{c}} \arctan \left (-\frac {\sqrt {b x^{2} + a x} c \sqrt {-\frac {b c - a d}{c}}}{{\left (b c - a d\right )} x}\right ) - 2 \, \sqrt {b x^{2} + a x} a d}{c d x}, -\frac {2 \, \sqrt {-b} b c x \arctan \left (\frac {\sqrt {b x^{2} + a x} \sqrt {-b}}{b x + a}\right ) + {\left (b c - a d\right )} x \sqrt {\frac {b c - a d}{c}} \log \left (\frac {a c + {\left (2 \, b c - a d\right )} x + 2 \, \sqrt {b x^{2} + a x} c \sqrt {\frac {b c - a d}{c}}}{d x + c}\right ) + 2 \, \sqrt {b x^{2} + a x} a d}{c d x}, -\frac {2 \, {\left (\sqrt {-b} b c x \arctan \left (\frac {\sqrt {b x^{2} + a x} \sqrt {-b}}{b x + a}\right ) + {\left (b c - a d\right )} x \sqrt {-\frac {b c - a d}{c}} \arctan \left (-\frac {\sqrt {b x^{2} + a x} c \sqrt {-\frac {b c - a d}{c}}}{{\left (b c - a d\right )} x}\right ) + \sqrt {b x^{2} + a x} a d\right )}}{c d x}\right ] \] Input:
integrate((b*x^2+a*x)^(3/2)/x^3/(d*x+c),x, algorithm="fricas")
Output:
[(b^(3/2)*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*sq rt((b*c - a*d)/c))/(d*x + c)) - 2*sqrt(b*x^2 + a*x)*a*d)/(c*d*x), (b^(3/2) *c*x*log(2*b*x + a + 2*sqrt(b*x^2 + a*x)*sqrt(b)) - 2*(b*c - a*d)*x*sqrt(- (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*sqrt(-b)*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)*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), -2*(sqrt(-b)*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)*arctan( -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)]
\[ \int \frac {\left (a x+b x^2\right )^{3/2}}{x^3 (c+d x)} \, dx=\int \frac {\left (x \left (a + b x\right )\right )^{\frac {3}{2}}}{x^{3} \left (c + d x\right )}\, dx \] Input:
integrate((b*x**2+a*x)**(3/2)/x**3/(d*x+c),x)
Output:
Integral((x*(a + b*x))**(3/2)/(x**3*(c + d*x)), x)
\[ \int \frac {\left (a x+b x^2\right )^{3/2}}{x^3 (c+d x)} \, dx=\int { \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}}}{{\left (d x + c\right )} x^{3}} \,d x } \] Input:
integrate((b*x^2+a*x)^(3/2)/x^3/(d*x+c),x, algorithm="maxima")
Output:
integrate((b*x^2 + a*x)^(3/2)/((d*x + c)*x^3), x)
Exception generated. \[ \int \frac {\left (a x+b x^2\right )^{3/2}}{x^3 (c+d x)} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((b*x^2+a*x)^(3/2)/x^3/(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. \[ \int \frac {\left (a x+b x^2\right )^{3/2}}{x^3 (c+d x)} \, dx=\int \frac {{\left (b\,x^2+a\,x\right )}^{3/2}}{x^3\,\left (c+d\,x\right )} \,d x \] Input:
int((a*x + b*x^2)^(3/2)/(x^3*(c + d*x)),x)
Output:
int((a*x + b*x^2)^(3/2)/(x^3*(c + d*x)), x)
Time = 0.24 (sec) , antiderivative size = 269, normalized size of antiderivative = 2.47 \[ \int \frac {\left (a x+b x^2\right )^{3/2}}{x^3 (c+d x)} \, dx=\frac {2 \sqrt {c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {a d -b c}-\sqrt {d}\, \sqrt {b x +a}-\sqrt {x}\, \sqrt {d}\, \sqrt {b}}{\sqrt {c}\, \sqrt {b}}\right ) a d x -2 \sqrt {c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {a d -b c}-\sqrt {d}\, \sqrt {b x +a}-\sqrt {x}\, \sqrt {d}\, \sqrt {b}}{\sqrt {c}\, \sqrt {b}}\right ) b c x +2 \sqrt {c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {a d -b c}+\sqrt {d}\, \sqrt {b x +a}+\sqrt {x}\, \sqrt {d}\, \sqrt {b}}{\sqrt {c}\, \sqrt {b}}\right ) a d x -2 \sqrt {c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {a d -b c}+\sqrt {d}\, \sqrt {b x +a}+\sqrt {x}\, \sqrt {d}\, \sqrt {b}}{\sqrt {c}\, \sqrt {b}}\right ) b c x -2 \sqrt {x}\, \sqrt {b x +a}\, a c d +2 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b x +a}+\sqrt {x}\, \sqrt {b}}{\sqrt {a}}\right ) b \,c^{2} x -2 \sqrt {b}\, a c d x}{c^{2} d x} \] Input:
int((b*x^2+a*x)^(3/2)/x^3/(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)