Integrand size = 17, antiderivative size = 167 \[ \int \frac {1}{(a+b x)^3 (c+d x)^{5/2}} \, dx=\frac {35 d^2}{12 (b c-a d)^3 (c+d x)^{3/2}}-\frac {1}{2 (b c-a d) (a+b x)^2 (c+d x)^{3/2}}+\frac {7 d}{4 (b c-a d)^2 (a+b x) (c+d x)^{3/2}}+\frac {35 b d^2}{4 (b c-a d)^4 \sqrt {c+d x}}-\frac {35 b^{3/2} d^2 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{4 (b c-a d)^{9/2}} \]
35/12*d^2/(-a*d+b*c)^3/(d*x+c)^(3/2)-1/2/(-a*d+b*c)/(b*x+a)^2/(d*x+c)^(3/2 )+7/4*d/(-a*d+b*c)^2/(b*x+a)/(d*x+c)^(3/2)-35/4*b^(3/2)*d^2*arctanh(b^(1/2 )*(d*x+c)^(1/2)/(-a*d+b*c)^(1/2))/(-a*d+b*c)^(9/2)+35/4*b*d^2/(-a*d+b*c)^4 /(d*x+c)^(1/2)
Time = 0.58 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.01 \[ \int \frac {1}{(a+b x)^3 (c+d x)^{5/2}} \, dx=\frac {-8 a^3 d^3+8 a^2 b d^2 (10 c+7 d x)+a b^2 d \left (39 c^2+238 c d x+175 d^2 x^2\right )+b^3 \left (-6 c^3+21 c^2 d x+140 c d^2 x^2+105 d^3 x^3\right )}{12 (b c-a d)^4 (a+b x)^2 (c+d x)^{3/2}}+\frac {35 b^{3/2} d^2 \arctan \left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {-b c+a d}}\right )}{4 (-b c+a d)^{9/2}} \]
(-8*a^3*d^3 + 8*a^2*b*d^2*(10*c + 7*d*x) + a*b^2*d*(39*c^2 + 238*c*d*x + 1 75*d^2*x^2) + b^3*(-6*c^3 + 21*c^2*d*x + 140*c*d^2*x^2 + 105*d^3*x^3))/(12 *(b*c - a*d)^4*(a + b*x)^2*(c + d*x)^(3/2)) + (35*b^(3/2)*d^2*ArcTan[(Sqrt [b]*Sqrt[c + d*x])/Sqrt[-(b*c) + a*d]])/(4*(-(b*c) + a*d)^(9/2))
Time = 0.25 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.17, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {52, 52, 61, 61, 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.
| \(\displaystyle \int \frac {1}{(a+b x)^3 (c+d x)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle -\frac {7 d \int \frac {1}{(a+b x)^2 (c+d x)^{5/2}}dx}{4 (b c-a d)}-\frac {1}{2 (a+b x)^2 (c+d x)^{3/2} (b c-a d)}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle -\frac {7 d \left (-\frac {5 d \int \frac {1}{(a+b x) (c+d x)^{5/2}}dx}{2 (b c-a d)}-\frac {1}{(a+b x) (c+d x)^{3/2} (b c-a d)}\right )}{4 (b c-a d)}-\frac {1}{2 (a+b x)^2 (c+d x)^{3/2} (b c-a d)}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle -\frac {7 d \left (-\frac {5 d \left (\frac {b \int \frac {1}{(a+b x) (c+d x)^{3/2}}dx}{b c-a d}+\frac {2}{3 (c+d x)^{3/2} (b c-a d)}\right )}{2 (b c-a d)}-\frac {1}{(a+b x) (c+d x)^{3/2} (b c-a d)}\right )}{4 (b c-a d)}-\frac {1}{2 (a+b x)^2 (c+d x)^{3/2} (b c-a d)}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle -\frac {7 d \left (-\frac {5 d \left (\frac {b \left (\frac {b \int \frac {1}{(a+b x) \sqrt {c+d x}}dx}{b c-a d}+\frac {2}{\sqrt {c+d x} (b c-a d)}\right )}{b c-a d}+\frac {2}{3 (c+d x)^{3/2} (b c-a d)}\right )}{2 (b c-a d)}-\frac {1}{(a+b x) (c+d x)^{3/2} (b c-a d)}\right )}{4 (b c-a d)}-\frac {1}{2 (a+b x)^2 (c+d x)^{3/2} (b c-a d)}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {7 d \left (-\frac {5 d \left (\frac {b \left (\frac {2 b \int \frac {1}{a+\frac {b (c+d x)}{d}-\frac {b c}{d}}d\sqrt {c+d x}}{d (b c-a d)}+\frac {2}{\sqrt {c+d x} (b c-a d)}\right )}{b c-a d}+\frac {2}{3 (c+d x)^{3/2} (b c-a d)}\right )}{2 (b c-a d)}-\frac {1}{(a+b x) (c+d x)^{3/2} (b c-a d)}\right )}{4 (b c-a d)}-\frac {1}{2 (a+b x)^2 (c+d x)^{3/2} (b c-a d)}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {7 d \left (-\frac {5 d \left (\frac {b \left (\frac {2}{\sqrt {c+d x} (b c-a d)}-\frac {2 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{(b c-a d)^{3/2}}\right )}{b c-a d}+\frac {2}{3 (c+d x)^{3/2} (b c-a d)}\right )}{2 (b c-a d)}-\frac {1}{(a+b x) (c+d x)^{3/2} (b c-a d)}\right )}{4 (b c-a d)}-\frac {1}{2 (a+b x)^2 (c+d x)^{3/2} (b c-a d)}\) |
-1/2*1/((b*c - a*d)*(a + b*x)^2*(c + d*x)^(3/2)) - (7*d*(-(1/((b*c - a*d)* (a + b*x)*(c + d*x)^(3/2))) - (5*d*(2/(3*(b*c - a*d)*(c + d*x)^(3/2)) + (b *(2/((b*c - a*d)*Sqrt[c + d*x]) - (2*Sqrt[b]*ArcTanh[(Sqrt[b]*Sqrt[c + d*x ])/Sqrt[b*c - a*d]])/(b*c - a*d)^(3/2)))/(b*c - a*d)))/(2*(b*c - a*d))))/( 4*(b*c - a*d))
3.15.42.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[m, -1] && !(LtQ[n, -1] && (EqQ[a, 0 ] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d , m, n, 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_)^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 = 0.34 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.86
| method | result | size |
| derivativedivides | \(2 d^{2} \left (-\frac {1}{3 \left (a d -b c \right )^{3} \left (d x +c \right )^{\frac {3}{2}}}+\frac {3 b}{\left (a d -b c \right )^{4} \sqrt {d x +c}}+\frac {b^{2} \left (\frac {\frac {11 b \left (d x +c \right )^{\frac {3}{2}}}{8}+\left (\frac {13 a d}{8}-\frac {13 b c}{8}\right ) \sqrt {d x +c}}{\left (\left (d x +c \right ) b +a d -b c \right )^{2}}+\frac {35 \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{8 \sqrt {\left (a d -b c \right ) b}}\right )}{\left (a d -b c \right )^{4}}\right )\) | \(143\) |
| default | \(2 d^{2} \left (-\frac {1}{3 \left (a d -b c \right )^{3} \left (d x +c \right )^{\frac {3}{2}}}+\frac {3 b}{\left (a d -b c \right )^{4} \sqrt {d x +c}}+\frac {b^{2} \left (\frac {\frac {11 b \left (d x +c \right )^{\frac {3}{2}}}{8}+\left (\frac {13 a d}{8}-\frac {13 b c}{8}\right ) \sqrt {d x +c}}{\left (\left (d x +c \right ) b +a d -b c \right )^{2}}+\frac {35 \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{8 \sqrt {\left (a d -b c \right ) b}}\right )}{\left (a d -b c \right )^{4}}\right )\) | \(143\) |
| pseudoelliptic | \(-\frac {2 \left (-\frac {105 \left (d x +c \right )^{\frac {3}{2}} b^{2} d^{2} \left (b x +a \right )^{2} \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{8}+\sqrt {\left (a d -b c \right ) b}\, \left (\left (-\frac {105}{8} d^{3} x^{3}-\frac {35}{2} c \,d^{2} x^{2}-\frac {21}{8} c^{2} d x +\frac {3}{4} c^{3}\right ) b^{3}-\frac {39 \left (\frac {175}{39} d^{2} x^{2}+\frac {238}{39} c d x +c^{2}\right ) d a \,b^{2}}{8}-10 \left (\frac {7 d x}{10}+c \right ) d^{2} a^{2} b +a^{3} d^{3}\right )\right )}{3 \left (d x +c \right )^{\frac {3}{2}} \sqrt {\left (a d -b c \right ) b}\, \left (b x +a \right )^{2} \left (a d -b c \right )^{4}}\) | \(178\) |
2*d^2*(-1/3/(a*d-b*c)^3/(d*x+c)^(3/2)+3/(a*d-b*c)^4*b/(d*x+c)^(1/2)+1/(a*d -b*c)^4*b^2*((11/8*b*(d*x+c)^(3/2)+(13/8*a*d-13/8*b*c)*(d*x+c)^(1/2))/((d* x+c)*b+a*d-b*c)^2+35/8/((a*d-b*c)*b)^(1/2)*arctan(b*(d*x+c)^(1/2)/((a*d-b* c)*b)^(1/2))))
Leaf count of result is larger than twice the leaf count of optimal. 608 vs. \(2 (139) = 278\).
Time = 0.27 (sec) , antiderivative size = 1226, normalized size of antiderivative = 7.34 \[ \int \frac {1}{(a+b x)^3 (c+d x)^{5/2}} \, dx=\text {Too large to display} \]
[1/24*(105*(b^3*d^4*x^4 + a^2*b*c^2*d^2 + 2*(b^3*c*d^3 + a*b^2*d^4)*x^3 + (b^3*c^2*d^2 + 4*a*b^2*c*d^3 + a^2*b*d^4)*x^2 + 2*(a*b^2*c^2*d^2 + a^2*b*c *d^3)*x)*sqrt(b/(b*c - a*d))*log((b*d*x + 2*b*c - a*d - 2*(b*c - a*d)*sqrt (d*x + c)*sqrt(b/(b*c - a*d)))/(b*x + a)) + 2*(105*b^3*d^3*x^3 - 6*b^3*c^3 + 39*a*b^2*c^2*d + 80*a^2*b*c*d^2 - 8*a^3*d^3 + 35*(4*b^3*c*d^2 + 5*a*b^2 *d^3)*x^2 + 7*(3*b^3*c^2*d + 34*a*b^2*c*d^2 + 8*a^2*b*d^3)*x)*sqrt(d*x + c ))/(a^2*b^4*c^6 - 4*a^3*b^3*c^5*d + 6*a^4*b^2*c^4*d^2 - 4*a^5*b*c^3*d^3 + a^6*c^2*d^4 + (b^6*c^4*d^2 - 4*a*b^5*c^3*d^3 + 6*a^2*b^4*c^2*d^4 - 4*a^3*b ^3*c*d^5 + a^4*b^2*d^6)*x^4 + 2*(b^6*c^5*d - 3*a*b^5*c^4*d^2 + 2*a^2*b^4*c ^3*d^3 + 2*a^3*b^3*c^2*d^4 - 3*a^4*b^2*c*d^5 + a^5*b*d^6)*x^3 + (b^6*c^6 - 9*a^2*b^4*c^4*d^2 + 16*a^3*b^3*c^3*d^3 - 9*a^4*b^2*c^2*d^4 + a^6*d^6)*x^2 + 2*(a*b^5*c^6 - 3*a^2*b^4*c^5*d + 2*a^3*b^3*c^4*d^2 + 2*a^4*b^2*c^3*d^3 - 3*a^5*b*c^2*d^4 + a^6*c*d^5)*x), -1/12*(105*(b^3*d^4*x^4 + a^2*b*c^2*d^2 + 2*(b^3*c*d^3 + a*b^2*d^4)*x^3 + (b^3*c^2*d^2 + 4*a*b^2*c*d^3 + a^2*b*d^ 4)*x^2 + 2*(a*b^2*c^2*d^2 + a^2*b*c*d^3)*x)*sqrt(-b/(b*c - a*d))*arctan(-( b*c - a*d)*sqrt(d*x + c)*sqrt(-b/(b*c - a*d))/(b*d*x + b*c)) - (105*b^3*d^ 3*x^3 - 6*b^3*c^3 + 39*a*b^2*c^2*d + 80*a^2*b*c*d^2 - 8*a^3*d^3 + 35*(4*b^ 3*c*d^2 + 5*a*b^2*d^3)*x^2 + 7*(3*b^3*c^2*d + 34*a*b^2*c*d^2 + 8*a^2*b*d^3 )*x)*sqrt(d*x + c))/(a^2*b^4*c^6 - 4*a^3*b^3*c^5*d + 6*a^4*b^2*c^4*d^2 - 4 *a^5*b*c^3*d^3 + a^6*c^2*d^4 + (b^6*c^4*d^2 - 4*a*b^5*c^3*d^3 + 6*a^2*b...
Timed out. \[ \int \frac {1}{(a+b x)^3 (c+d x)^{5/2}} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {1}{(a+b x)^3 (c+d x)^{5/2}} \, dx=\text {Exception raised: ValueError} \]
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
Leaf count of result is larger than twice the leaf count of optimal. 298 vs. \(2 (139) = 278\).
Time = 0.34 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.78 \[ \int \frac {1}{(a+b x)^3 (c+d x)^{5/2}} \, dx=\frac {35 \, b^{2} d^{2} \arctan \left (\frac {\sqrt {d x + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{4 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \sqrt {-b^{2} c + a b d}} + \frac {2 \, {\left (9 \, {\left (d x + c\right )} b d^{2} + b c d^{2} - a d^{3}\right )}}{3 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} {\left (d x + c\right )}^{\frac {3}{2}}} + \frac {11 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{3} d^{2} - 13 \, \sqrt {d x + c} b^{3} c d^{2} + 13 \, \sqrt {d x + c} a b^{2} d^{3}}{4 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} {\left ({\left (d x + c\right )} b - b c + a d\right )}^{2}} \]
35/4*b^2*d^2*arctan(sqrt(d*x + c)*b/sqrt(-b^2*c + a*b*d))/((b^4*c^4 - 4*a* b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*sqrt(-b^2*c + a*b *d)) + 2/3*(9*(d*x + c)*b*d^2 + b*c*d^2 - a*d^3)/((b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*(d*x + c)^(3/2)) + 1/4*(11 *(d*x + c)^(3/2)*b^3*d^2 - 13*sqrt(d*x + c)*b^3*c*d^2 + 13*sqrt(d*x + c)*a *b^2*d^3)/((b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*((d*x + c)*b - b*c + a*d)^2)
Time = 0.25 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.46 \[ \int \frac {1}{(a+b x)^3 (c+d x)^{5/2}} \, dx=\frac {\frac {175\,b^2\,d^2\,{\left (c+d\,x\right )}^2}{12\,{\left (a\,d-b\,c\right )}^3}-\frac {2\,d^2}{3\,\left (a\,d-b\,c\right )}+\frac {35\,b^3\,d^2\,{\left (c+d\,x\right )}^3}{4\,{\left (a\,d-b\,c\right )}^4}+\frac {14\,b\,d^2\,\left (c+d\,x\right )}{3\,{\left (a\,d-b\,c\right )}^2}}{b^2\,{\left (c+d\,x\right )}^{7/2}-\left (2\,b^2\,c-2\,a\,b\,d\right )\,{\left (c+d\,x\right )}^{5/2}+{\left (c+d\,x\right )}^{3/2}\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {35\,b^{3/2}\,d^2\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {c+d\,x}\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}{{\left (a\,d-b\,c\right )}^{9/2}}\right )}{4\,{\left (a\,d-b\,c\right )}^{9/2}} \]
((175*b^2*d^2*(c + d*x)^2)/(12*(a*d - b*c)^3) - (2*d^2)/(3*(a*d - b*c)) + (35*b^3*d^2*(c + d*x)^3)/(4*(a*d - b*c)^4) + (14*b*d^2*(c + d*x))/(3*(a*d - b*c)^2))/(b^2*(c + d*x)^(7/2) - (2*b^2*c - 2*a*b*d)*(c + d*x)^(5/2) + (c + d*x)^(3/2)*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)) + (35*b^(3/2)*d^2*atan((b^( 1/2)*(c + d*x)^(1/2)*(a^4*d^4 + b^4*c^4 + 6*a^2*b^2*c^2*d^2 - 4*a*b^3*c^3* d - 4*a^3*b*c*d^3))/(a*d - b*c)^(9/2)))/(4*(a*d - b*c)^(9/2))
Time = 0.00 (sec) , antiderivative size = 804, normalized size of antiderivative = 4.81 \[ \int \frac {1}{(a+b x)^3 (c+d x)^{5/2}} \, dx=\frac {105 \sqrt {b}\, \sqrt {d x +c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {d x +c}\, b}{\sqrt {b}\, \sqrt {a d -b c}}\right ) a^{2} b c \,d^{2}+105 \sqrt {b}\, \sqrt {d x +c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {d x +c}\, b}{\sqrt {b}\, \sqrt {a d -b c}}\right ) a^{2} b \,d^{3} x +210 \sqrt {b}\, \sqrt {d x +c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {d x +c}\, b}{\sqrt {b}\, \sqrt {a d -b c}}\right ) a \,b^{2} c \,d^{2} x +210 \sqrt {b}\, \sqrt {d x +c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {d x +c}\, b}{\sqrt {b}\, \sqrt {a d -b c}}\right ) a \,b^{2} d^{3} x^{2}+105 \sqrt {b}\, \sqrt {d x +c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {d x +c}\, b}{\sqrt {b}\, \sqrt {a d -b c}}\right ) b^{3} c \,d^{2} x^{2}+105 \sqrt {b}\, \sqrt {d x +c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {d x +c}\, b}{\sqrt {b}\, \sqrt {a d -b c}}\right ) b^{3} d^{3} x^{3}-8 a^{4} d^{4}+88 a^{3} b c \,d^{3}+56 a^{3} b \,d^{4} x -41 a^{2} b^{2} c^{2} d^{2}+182 a^{2} b^{2} c \,d^{3} x +175 a^{2} b^{2} d^{4} x^{2}-45 a \,b^{3} c^{3} d -217 a \,b^{3} c^{2} d^{2} x -35 a \,b^{3} c \,d^{3} x^{2}+105 a \,b^{3} d^{4} x^{3}+6 b^{4} c^{4}-21 b^{4} c^{3} d x -140 b^{4} c^{2} d^{2} x^{2}-105 b^{4} c \,d^{3} x^{3}}{12 \sqrt {d x +c}\, \left (a^{5} b^{2} d^{6} x^{3}-5 a^{4} b^{3} c \,d^{5} x^{3}+10 a^{3} b^{4} c^{2} d^{4} x^{3}-10 a^{2} b^{5} c^{3} d^{3} x^{3}+5 a \,b^{6} c^{4} d^{2} x^{3}-b^{7} c^{5} d \,x^{3}+2 a^{6} b \,d^{6} x^{2}-9 a^{5} b^{2} c \,d^{5} x^{2}+15 a^{4} b^{3} c^{2} d^{4} x^{2}-10 a^{3} b^{4} c^{3} d^{3} x^{2}+3 a \,b^{6} c^{5} d \,x^{2}-b^{7} c^{6} x^{2}+a^{7} d^{6} x -3 a^{6} b c \,d^{5} x +10 a^{4} b^{3} c^{3} d^{3} x -15 a^{3} b^{4} c^{4} d^{2} x +9 a^{2} b^{5} c^{5} d x -2 a \,b^{6} c^{6} x +a^{7} c \,d^{5}-5 a^{6} b \,c^{2} d^{4}+10 a^{5} b^{2} c^{3} d^{3}-10 a^{4} b^{3} c^{4} d^{2}+5 a^{3} b^{4} c^{5} d -a^{2} b^{5} c^{6}\right )} \]
int(1/(sqrt(c + d*x)*(a**3*c**2 + 2*a**3*c*d*x + a**3*d**2*x**2 + 3*a**2*b *c**2*x + 6*a**2*b*c*d*x**2 + 3*a**2*b*d**2*x**3 + 3*a*b**2*c**2*x**2 + 6* a*b**2*c*d*x**3 + 3*a*b**2*d**2*x**4 + b**3*c**2*x**3 + 2*b**3*c*d*x**4 + b**3*d**2*x**5)),x)
(105*sqrt(b)*sqrt(c + d*x)*sqrt(a*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b) *sqrt(a*d - b*c)))*a**2*b*c*d**2 + 105*sqrt(b)*sqrt(c + d*x)*sqrt(a*d - b* c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(a*d - b*c)))*a**2*b*d**3*x + 210*s qrt(b)*sqrt(c + d*x)*sqrt(a*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt( a*d - b*c)))*a*b**2*c*d**2*x + 210*sqrt(b)*sqrt(c + d*x)*sqrt(a*d - b*c)*a tan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(a*d - b*c)))*a*b**2*d**3*x**2 + 105*sq rt(b)*sqrt(c + d*x)*sqrt(a*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(a *d - b*c)))*b**3*c*d**2*x**2 + 105*sqrt(b)*sqrt(c + d*x)*sqrt(a*d - b*c)*a tan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(a*d - b*c)))*b**3*d**3*x**3 - 8*a**4*d **4 + 88*a**3*b*c*d**3 + 56*a**3*b*d**4*x - 41*a**2*b**2*c**2*d**2 + 182*a **2*b**2*c*d**3*x + 175*a**2*b**2*d**4*x**2 - 45*a*b**3*c**3*d - 217*a*b** 3*c**2*d**2*x - 35*a*b**3*c*d**3*x**2 + 105*a*b**3*d**4*x**3 + 6*b**4*c**4 - 21*b**4*c**3*d*x - 140*b**4*c**2*d**2*x**2 - 105*b**4*c*d**3*x**3)/(12* sqrt(c + d*x)*(a**7*c*d**5 + a**7*d**6*x - 5*a**6*b*c**2*d**4 - 3*a**6*b*c *d**5*x + 2*a**6*b*d**6*x**2 + 10*a**5*b**2*c**3*d**3 - 9*a**5*b**2*c*d**5 *x**2 + a**5*b**2*d**6*x**3 - 10*a**4*b**3*c**4*d**2 + 10*a**4*b**3*c**3*d **3*x + 15*a**4*b**3*c**2*d**4*x**2 - 5*a**4*b**3*c*d**5*x**3 + 5*a**3*b** 4*c**5*d - 15*a**3*b**4*c**4*d**2*x - 10*a**3*b**4*c**3*d**3*x**2 + 10*a** 3*b**4*c**2*d**4*x**3 - a**2*b**5*c**6 + 9*a**2*b**5*c**5*d*x - 10*a**2*b* *5*c**3*d**3*x**3 - 2*a*b**6*c**6*x + 3*a*b**6*c**5*d*x**2 + 5*a*b**6*c...