Integrand size = 24, antiderivative size = 165 \[ \int \frac {\left (a x+b x^2\right )^{3/2}}{x^3 (c+d x)^3} \, dx=\frac {(b c-5 a d) \sqrt {a x+b x^2}}{2 c^2 (c+d x)^2}-\frac {2 a \sqrt {a x+b x^2}}{c x (c+d x)^2}+\frac {(2 b c-15 a d) \sqrt {a x+b x^2}}{4 c^3 (c+d x)}+\frac {3 a (4 b c-5 a d) \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a x+b x^2}}\right )}{4 c^{7/2} \sqrt {b c-a d}} \] Output:
1/2*(-5*a*d+b*c)*(b*x^2+a*x)^(1/2)/c^2/(d*x+c)^2-2*a*(b*x^2+a*x)^(1/2)/c/x /(d*x+c)^2+1/4*(-15*a*d+2*b*c)*(b*x^2+a*x)^(1/2)/c^3/(d*x+c)+3/4*a*(-5*a*d +4*b*c)*arctanh((-a*d+b*c)^(1/2)*x/c^(1/2)/(b*x^2+a*x)^(1/2))/c^(7/2)/(-a* d+b*c)^(1/2)
Time = 10.21 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.05 \[ \int \frac {\left (a x+b x^2\right )^{3/2}}{x^3 (c+d x)^3} \, dx=\frac {(x (a+b x))^{3/2} \left (-8 (a+b x)+\frac {(4 b c-5 a d) \sqrt {x} \left (\sqrt {c} \sqrt {b c-a d} \sqrt {x} \sqrt {a+b x} (5 a c+2 b c x+3 a d x)+3 a^2 (c+d x)^2 \text {arctanh}\left (\frac {\sqrt {b c-a d} \sqrt {x}}{\sqrt {c} \sqrt {a+b x}}\right )\right )}{c^{5/2} \sqrt {b c-a d} (a+b x)^{3/2}}\right )}{4 a c x^2 (c+d x)^2} \] Input:
Integrate[(a*x + b*x^2)^(3/2)/(x^3*(c + d*x)^3),x]
Output:
((x*(a + b*x))^(3/2)*(-8*(a + b*x) + ((4*b*c - 5*a*d)*Sqrt[x]*(Sqrt[c]*Sqr t[b*c - a*d]*Sqrt[x]*Sqrt[a + b*x]*(5*a*c + 2*b*c*x + 3*a*d*x) + 3*a^2*(c + d*x)^2*ArcTanh[(Sqrt[b*c - a*d]*Sqrt[x])/(Sqrt[c]*Sqrt[a + b*x])]))/(c^( 5/2)*Sqrt[b*c - a*d]*(a + b*x)^(3/2))))/(4*a*c*x^2*(c + d*x)^2)
Time = 0.50 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.15, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1261, 107, 105, 105, 104, 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.
| \(\displaystyle \int \frac {\left (a x+b x^2\right )^{3/2}}{x^3 (c+d x)^3} \, dx\) |
| \(\Big \downarrow \) 1261 |
| \(\displaystyle \frac {\left (a x+b x^2\right )^{3/2} \int \frac {(a+b x)^{3/2}}{x^{3/2} (c+d x)^3}dx}{x^{3/2} (a+b x)^{3/2}}\) |
| \(\Big \downarrow \) 107 |
| \(\displaystyle \frac {\left (a x+b x^2\right )^{3/2} \left (\frac {(4 b c-5 a d) \int \frac {(a+b x)^{3/2}}{\sqrt {x} (c+d x)^3}dx}{a c}-\frac {2 (a+b x)^{5/2}}{a c \sqrt {x} (c+d x)^2}\right )}{x^{3/2} (a+b x)^{3/2}}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle \frac {\left (a x+b x^2\right )^{3/2} \left (\frac {(4 b c-5 a d) \left (\frac {3 a \int \frac {\sqrt {a+b x}}{\sqrt {x} (c+d x)^2}dx}{4 c}+\frac {\sqrt {x} (a+b x)^{3/2}}{2 c (c+d x)^2}\right )}{a c}-\frac {2 (a+b x)^{5/2}}{a c \sqrt {x} (c+d x)^2}\right )}{x^{3/2} (a+b x)^{3/2}}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle \frac {\left (a x+b x^2\right )^{3/2} \left (\frac {(4 b c-5 a d) \left (\frac {3 a \left (\frac {a \int \frac {1}{\sqrt {x} \sqrt {a+b x} (c+d x)}dx}{2 c}+\frac {\sqrt {x} \sqrt {a+b x}}{c (c+d x)}\right )}{4 c}+\frac {\sqrt {x} (a+b x)^{3/2}}{2 c (c+d x)^2}\right )}{a c}-\frac {2 (a+b x)^{5/2}}{a c \sqrt {x} (c+d x)^2}\right )}{x^{3/2} (a+b x)^{3/2}}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {\left (a x+b x^2\right )^{3/2} \left (\frac {(4 b c-5 a d) \left (\frac {3 a \left (\frac {a \int \frac {1}{c-\frac {(b c-a d) x}{a+b x}}d\frac {\sqrt {x}}{\sqrt {a+b x}}}{c}+\frac {\sqrt {x} \sqrt {a+b x}}{c (c+d x)}\right )}{4 c}+\frac {\sqrt {x} (a+b x)^{3/2}}{2 c (c+d x)^2}\right )}{a c}-\frac {2 (a+b x)^{5/2}}{a c \sqrt {x} (c+d x)^2}\right )}{x^{3/2} (a+b x)^{3/2}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\left (a x+b x^2\right )^{3/2} \left (\frac {(4 b c-5 a d) \left (\frac {3 a \left (\frac {a \text {arctanh}\left (\frac {\sqrt {x} \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x}}\right )}{c^{3/2} \sqrt {b c-a d}}+\frac {\sqrt {x} \sqrt {a+b x}}{c (c+d x)}\right )}{4 c}+\frac {\sqrt {x} (a+b x)^{3/2}}{2 c (c+d x)^2}\right )}{a c}-\frac {2 (a+b x)^{5/2}}{a c \sqrt {x} (c+d x)^2}\right )}{x^{3/2} (a+b x)^{3/2}}\) |
Input:
Int[(a*x + b*x^2)^(3/2)/(x^3*(c + d*x)^3),x]
Output:
((a*x + b*x^2)^(3/2)*((-2*(a + b*x)^(5/2))/(a*c*Sqrt[x]*(c + d*x)^2) + ((4 *b*c - 5*a*d)*((Sqrt[x]*(a + b*x)^(3/2))/(2*c*(c + d*x)^2) + (3*a*((Sqrt[x ]*Sqrt[a + b*x])/(c*(c + d*x)) + (a*ArcTanh[(Sqrt[b*c - a*d]*Sqrt[x])/(Sqr t[c]*Sqrt[a + b*x])])/(c^(3/2)*Sqrt[b*c - a*d])))/(4*c)))/(a*c)))/(x^(3/2) *(a + b*x)^(3/2))
Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x _)), x_] :> With[{q = Denominator[m]}, Simp[q Subst[Int[x^(q*(m + 1) - 1) /(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] ] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 1)*(b*e - a*f))), x] - Simp[n*((d*e - c*f)/((m + 1)*(b*e - a*f))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[m + n + p + 2, 0] && GtQ[n, 0] && (SumSimplerQ[m, 1] || !SumSimplerQ[p, 1]) && NeQ[m, -1]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 )/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[(a*d*f*(m + 1) + b*c*f*(n + 1) + b*d*e*(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x ] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || SumSimplerQ[m, 1])
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]
Int[((e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((b_.)*(x_) + (c_.)*(x_)^2) ^(p_), x_Symbol] :> Simp[(e*x)^m*((b*x + c*x^2)^p/(x^(m + p)*(b + c*x)^p)) Int[x^(m + p)*(f + g*x)^n*(b + c*x)^p, x], x] /; FreeQ[{b, c, e, f, g, m, n}, x] && !IGtQ[n, 0]
Time = 0.74 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.01
| method | result | size |
| pseudoelliptic | \(-\frac {2 \left (-\frac {15 \left (d x +c \right )^{2} \left (a d -\frac {4 b c}{5}\right ) x \,a^{2} \arctan \left (\frac {\sqrt {x \left (b x +a \right )}\, c}{x \sqrt {c \left (a d -b c \right )}}\right )}{8}+\sqrt {c \left (a d -b c \right )}\, \left (\left (\frac {9}{8} a c d -\frac {1}{2} b \,c^{2}\right ) \left (x \left (b x +a \right )\right )^{\frac {3}{2}}+\sqrt {x \left (b x +a \right )}\, \left (\left (\frac {b^{2} x^{2}}{2}+a^{2}\right ) c^{2}+2 d x a \left (-\frac {11 b x}{16}+a \right ) c +\frac {15 a^{2} d^{2} x^{2}}{8}\right )\right )\right )}{\sqrt {c \left (a d -b c \right )}\, a \,c^{3} x \left (d x +c \right )^{2}}\) | \(166\) |
| risch | \(-\frac {2 a \left (b x +a \right )}{c^{3} \sqrt {x \left (b x +a \right )}}-\frac {-\frac {a^{2} \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 )}{\sqrt {-\frac {c \left (a d -b c \right )}{d^{2}}}}+\frac {c \left (a^{2} d^{2}-b^{2} c^{2}\right ) \left (\frac {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}}}}{c \left (a d -b c \right ) \left (x +\frac {c}{d}\right )}-\frac {\left (a d -2 b c \right ) d \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 )}{2 c \left (a d -b c \right ) \sqrt {-\frac {c \left (a d -b c \right )}{d^{2}}}}\right )}{d^{3}}+\frac {c^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (\frac {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}}}}{2 c \left (a d -b c \right ) \left (x +\frac {c}{d}\right )^{2}}+\frac {3 \left (a d -2 b c \right ) d \left (\frac {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}}}}{c \left (a d -b c \right ) \left (x +\frac {c}{d}\right )}-\frac {\left (a d -2 b c \right ) d \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 )}{2 c \left (a d -b c \right ) \sqrt {-\frac {c \left (a d -b c \right )}{d^{2}}}}\right )}{4 c \left (a d -b c \right )}-\frac {b \,d^{2} \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 )}{2 c \left (a d -b c \right ) \sqrt {-\frac {c \left (a d -b c \right )}{d^{2}}}}\right )}{d^{4}}}{c^{3}}\) | \(900\) |
| default | \(\text {Expression too large to display}\) | \(3461\) |
Input:
int((b*x^2+a*x)^(3/2)/x^3/(d*x+c)^3,x,method=_RETURNVERBOSE)
Output:
-2/(c*(a*d-b*c))^(1/2)*(-15/8*(d*x+c)^2*(a*d-4/5*b*c)*x*a^2*arctan((x*(b*x +a))^(1/2)/x*c/(c*(a*d-b*c))^(1/2))+(c*(a*d-b*c))^(1/2)*((9/8*a*c*d-1/2*b* c^2)*(x*(b*x+a))^(3/2)+(x*(b*x+a))^(1/2)*((1/2*b^2*x^2+a^2)*c^2+2*d*x*a*(- 11/16*b*x+a)*c+15/8*a^2*d^2*x^2)))/a/c^3/x/(d*x+c)^2
Time = 0.10 (sec) , antiderivative size = 562, normalized size of antiderivative = 3.41 \[ \int \frac {\left (a x+b x^2\right )^{3/2}}{x^3 (c+d x)^3} \, dx=\left [-\frac {3 \, {\left ({\left (4 \, a b c d^{2} - 5 \, a^{2} d^{3}\right )} x^{3} + 2 \, {\left (4 \, a b c^{2} d - 5 \, a^{2} c d^{2}\right )} x^{2} + {\left (4 \, a b c^{3} - 5 \, a^{2} c^{2} d\right )} x\right )} \sqrt {b c^{2} - a c d} \log \left (\frac {a c + {\left (2 \, b c - a d\right )} x - 2 \, \sqrt {b c^{2} - a c d} \sqrt {b x^{2} + a x}}{d x + c}\right ) + 2 \, {\left (8 \, a b c^{4} - 8 \, a^{2} c^{3} d - {\left (2 \, b^{2} c^{3} d - 17 \, a b c^{2} d^{2} + 15 \, a^{2} c d^{3}\right )} x^{2} - {\left (4 \, b^{2} c^{4} - 29 \, a b c^{3} d + 25 \, a^{2} c^{2} d^{2}\right )} x\right )} \sqrt {b x^{2} + a x}}{8 \, {\left ({\left (b c^{5} d^{2} - a c^{4} d^{3}\right )} x^{3} + 2 \, {\left (b c^{6} d - a c^{5} d^{2}\right )} x^{2} + {\left (b c^{7} - a c^{6} d\right )} x\right )}}, -\frac {3 \, {\left ({\left (4 \, a b c d^{2} - 5 \, a^{2} d^{3}\right )} x^{3} + 2 \, {\left (4 \, a b c^{2} d - 5 \, a^{2} c d^{2}\right )} x^{2} + {\left (4 \, a b c^{3} - 5 \, a^{2} c^{2} d\right )} x\right )} \sqrt {-b c^{2} + a c d} \arctan \left (\frac {\sqrt {-b c^{2} + a c d} \sqrt {b x^{2} + a x}}{b c x + a c}\right ) + {\left (8 \, a b c^{4} - 8 \, a^{2} c^{3} d - {\left (2 \, b^{2} c^{3} d - 17 \, a b c^{2} d^{2} + 15 \, a^{2} c d^{3}\right )} x^{2} - {\left (4 \, b^{2} c^{4} - 29 \, a b c^{3} d + 25 \, a^{2} c^{2} d^{2}\right )} x\right )} \sqrt {b x^{2} + a x}}{4 \, {\left ({\left (b c^{5} d^{2} - a c^{4} d^{3}\right )} x^{3} + 2 \, {\left (b c^{6} d - a c^{5} d^{2}\right )} x^{2} + {\left (b c^{7} - a c^{6} d\right )} x\right )}}\right ] \] Input:
integrate((b*x^2+a*x)^(3/2)/x^3/(d*x+c)^3,x, algorithm="fricas")
Output:
[-1/8*(3*((4*a*b*c*d^2 - 5*a^2*d^3)*x^3 + 2*(4*a*b*c^2*d - 5*a^2*c*d^2)*x^ 2 + (4*a*b*c^3 - 5*a^2*c^2*d)*x)*sqrt(b*c^2 - a*c*d)*log((a*c + (2*b*c - a *d)*x - 2*sqrt(b*c^2 - a*c*d)*sqrt(b*x^2 + a*x))/(d*x + c)) + 2*(8*a*b*c^4 - 8*a^2*c^3*d - (2*b^2*c^3*d - 17*a*b*c^2*d^2 + 15*a^2*c*d^3)*x^2 - (4*b^ 2*c^4 - 29*a*b*c^3*d + 25*a^2*c^2*d^2)*x)*sqrt(b*x^2 + a*x))/((b*c^5*d^2 - a*c^4*d^3)*x^3 + 2*(b*c^6*d - a*c^5*d^2)*x^2 + (b*c^7 - a*c^6*d)*x), -1/4 *(3*((4*a*b*c*d^2 - 5*a^2*d^3)*x^3 + 2*(4*a*b*c^2*d - 5*a^2*c*d^2)*x^2 + ( 4*a*b*c^3 - 5*a^2*c^2*d)*x)*sqrt(-b*c^2 + a*c*d)*arctan(sqrt(-b*c^2 + a*c* d)*sqrt(b*x^2 + a*x)/(b*c*x + a*c)) + (8*a*b*c^4 - 8*a^2*c^3*d - (2*b^2*c^ 3*d - 17*a*b*c^2*d^2 + 15*a^2*c*d^3)*x^2 - (4*b^2*c^4 - 29*a*b*c^3*d + 25* a^2*c^2*d^2)*x)*sqrt(b*x^2 + a*x))/((b*c^5*d^2 - a*c^4*d^3)*x^3 + 2*(b*c^6 *d - a*c^5*d^2)*x^2 + (b*c^7 - a*c^6*d)*x)]
\[ \int \frac {\left (a x+b x^2\right )^{3/2}}{x^3 (c+d x)^3} \, dx=\int \frac {\left (x \left (a + b x\right )\right )^{\frac {3}{2}}}{x^{3} \left (c + d x\right )^{3}}\, dx \] Input:
integrate((b*x**2+a*x)**(3/2)/x**3/(d*x+c)**3,x)
Output:
Integral((x*(a + b*x))**(3/2)/(x**3*(c + d*x)**3), x)
\[ \int \frac {\left (a x+b x^2\right )^{3/2}}{x^3 (c+d x)^3} \, dx=\int { \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}}}{{\left (d x + c\right )}^{3} x^{3}} \,d x } \] Input:
integrate((b*x^2+a*x)^(3/2)/x^3/(d*x+c)^3,x, algorithm="maxima")
Output:
integrate((b*x^2 + a*x)^(3/2)/((d*x + c)^3*x^3), x)
Exception generated. \[ \int \frac {\left (a x+b x^2\right )^{3/2}}{x^3 (c+d x)^3} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((b*x^2+a*x)^(3/2)/x^3/(d*x+c)^3,x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {\left (a x+b x^2\right )^{3/2}}{x^3 (c+d x)^3} \, dx=\int \frac {{\left (b\,x^2+a\,x\right )}^{3/2}}{x^3\,{\left (c+d\,x\right )}^3} \,d x \] Input:
int((a*x + b*x^2)^(3/2)/(x^3*(c + d*x)^3),x)
Output:
int((a*x + b*x^2)^(3/2)/(x^3*(c + d*x)^3), x)
Time = 99.58 (sec) , antiderivative size = 1613, normalized size of antiderivative = 9.78 \[ \int \frac {\left (a x+b x^2\right )^{3/2}}{x^3 (c+d x)^3} \, dx =\text {Too large to display} \] Input:
int((b*x^2+a*x)^(3/2)/x^3/(d*x+c)^3,x)
Output:
(75*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**3*c**2*d**3*x + 150*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**3*c*d**4*x**2 + 75*sqrt(c)*sqrt(a* d - b*c)*atan((sqrt(a*d - b*c) - sqrt(d)*sqrt(a + b*x) - sqrt(x)*sqrt(d)*s qrt(b))/(sqrt(c)*sqrt(b)))*a**3*d**5*x**3 - 180*sqrt(c)*sqrt(a*d - b*c)*at an((sqrt(a*d - b*c) - sqrt(d)*sqrt(a + b*x) - sqrt(x)*sqrt(d)*sqrt(b))/(sq rt(c)*sqrt(b)))*a**2*b*c**3*d**2*x - 360*sqrt(c)*sqrt(a*d - b*c)*atan((sqr t(a*d - b*c) - sqrt(d)*sqrt(a + b*x) - sqrt(x)*sqrt(d)*sqrt(b))/(sqrt(c)*s qrt(b)))*a**2*b*c**2*d**3*x**2 - 180*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**2*b*c*d**4*x**3 + 96*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*b **2*c**4*d*x + 192*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*b**2*c**3*d **2*x**2 + 96*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*b**2*c**2*d**3*x **3 + 75*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**3*c**2*d**3*x + 150* sqrt(c)*sqrt(a*d - b*c)*atan((sqrt(a*d - b*c) + sqrt(d)*sqrt(a + b*x) +...