Integrand size = 35, antiderivative size = 276 \[ \int \frac {a+b x}{(d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=-\frac {35 e^2}{24 (b d-a e)^3 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {1}{3 (b d-a e) (a+b x)^2 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 e}{12 (b d-a e)^2 (a+b x) \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {35 e^3 (a+b x)}{8 (b d-a e)^4 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {35 \sqrt {b} e^3 (a+b x) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{8 (b d-a e)^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}} \]
35/8*e^3*(b*x+a)*arctanh(b^(1/2)*(e*x+d)^(1/2)/(-a*e+b*d)^(1/2))*b^(1/2)/( -a*e+b*d)^(9/2)/((b*x+a)^2)^(1/2)-35/24*e^2/(-a*e+b*d)^3/(e*x+d)^(1/2)/((b *x+a)^2)^(1/2)-1/3/(-a*e+b*d)/(b*x+a)^2/(e*x+d)^(1/2)/((b*x+a)^2)^(1/2)+7/ 12*e/(-a*e+b*d)^2/(b*x+a)/(e*x+d)^(1/2)/((b*x+a)^2)^(1/2)-35/8*e^3*(b*x+a) /(-a*e+b*d)^4/(e*x+d)^(1/2)/((b*x+a)^2)^(1/2)
Time = 0.12 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.69 \[ \int \frac {a+b x}{(d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {-\sqrt {-b d+a e} \left (48 a^3 e^3+3 a^2 b e^2 (29 d+77 e x)+2 a b^2 e \left (-19 d^2+49 d e x+140 e^2 x^2\right )+b^3 \left (8 d^3-14 d^2 e x+35 d e^2 x^2+105 e^3 x^3\right )\right )-105 \sqrt {b} e^3 (a+b x)^3 \sqrt {d+e x} \arctan \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{24 (-b d+a e)^{9/2} \left ((a+b x)^2\right )^{3/2} \sqrt {d+e x}} \]
(-(Sqrt[-(b*d) + a*e]*(48*a^3*e^3 + 3*a^2*b*e^2*(29*d + 77*e*x) + 2*a*b^2* e*(-19*d^2 + 49*d*e*x + 140*e^2*x^2) + b^3*(8*d^3 - 14*d^2*e*x + 35*d*e^2* x^2 + 105*e^3*x^3))) - 105*Sqrt[b]*e^3*(a + b*x)^3*Sqrt[d + e*x]*ArcTan[(S qrt[b]*Sqrt[d + e*x])/Sqrt[-(b*d) + a*e]])/(24*(-(b*d) + a*e)^(9/2)*((a + b*x)^2)^(3/2)*Sqrt[d + e*x])
Time = 0.34 (sec) , antiderivative size = 231, normalized size of antiderivative = 0.84, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.229, Rules used = {1187, 27, 52, 52, 52, 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.
| \(\displaystyle \int \frac {a+b x}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2} (d+e x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1187 |
| \(\displaystyle \frac {b^5 (a+b x) \int \frac {1}{b^5 (a+b x)^4 (d+e x)^{3/2}}dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(a+b x) \int \frac {1}{(a+b x)^4 (d+e x)^{3/2}}dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {(a+b x) \left (-\frac {7 e \int \frac {1}{(a+b x)^3 (d+e x)^{3/2}}dx}{6 (b d-a e)}-\frac {1}{3 (a+b x)^3 \sqrt {d+e x} (b d-a e)}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {(a+b x) \left (-\frac {7 e \left (-\frac {5 e \int \frac {1}{(a+b x)^2 (d+e x)^{3/2}}dx}{4 (b d-a e)}-\frac {1}{2 (a+b x)^2 \sqrt {d+e x} (b d-a e)}\right )}{6 (b d-a e)}-\frac {1}{3 (a+b x)^3 \sqrt {d+e x} (b d-a e)}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {(a+b x) \left (-\frac {7 e \left (-\frac {5 e \left (-\frac {3 e \int \frac {1}{(a+b x) (d+e x)^{3/2}}dx}{2 (b d-a e)}-\frac {1}{(a+b x) \sqrt {d+e x} (b d-a e)}\right )}{4 (b d-a e)}-\frac {1}{2 (a+b x)^2 \sqrt {d+e x} (b d-a e)}\right )}{6 (b d-a e)}-\frac {1}{3 (a+b x)^3 \sqrt {d+e x} (b d-a e)}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle \frac {(a+b x) \left (-\frac {7 e \left (-\frac {5 e \left (-\frac {3 e \left (\frac {b \int \frac {1}{(a+b x) \sqrt {d+e x}}dx}{b d-a e}+\frac {2}{\sqrt {d+e x} (b d-a e)}\right )}{2 (b d-a e)}-\frac {1}{(a+b x) \sqrt {d+e x} (b d-a e)}\right )}{4 (b d-a e)}-\frac {1}{2 (a+b x)^2 \sqrt {d+e x} (b d-a e)}\right )}{6 (b d-a e)}-\frac {1}{3 (a+b x)^3 \sqrt {d+e x} (b d-a e)}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {(a+b x) \left (-\frac {7 e \left (-\frac {5 e \left (-\frac {3 e \left (\frac {2 b \int \frac {1}{a+\frac {b (d+e x)}{e}-\frac {b d}{e}}d\sqrt {d+e x}}{e (b d-a e)}+\frac {2}{\sqrt {d+e x} (b d-a e)}\right )}{2 (b d-a e)}-\frac {1}{(a+b x) \sqrt {d+e x} (b d-a e)}\right )}{4 (b d-a e)}-\frac {1}{2 (a+b x)^2 \sqrt {d+e x} (b d-a e)}\right )}{6 (b d-a e)}-\frac {1}{3 (a+b x)^3 \sqrt {d+e x} (b d-a e)}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {(a+b x) \left (-\frac {7 e \left (-\frac {5 e \left (-\frac {3 e \left (\frac {2}{\sqrt {d+e x} (b d-a e)}-\frac {2 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{(b d-a e)^{3/2}}\right )}{2 (b d-a e)}-\frac {1}{(a+b x) \sqrt {d+e x} (b d-a e)}\right )}{4 (b d-a e)}-\frac {1}{2 (a+b x)^2 \sqrt {d+e x} (b d-a e)}\right )}{6 (b d-a e)}-\frac {1}{3 (a+b x)^3 \sqrt {d+e x} (b d-a e)}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
((a + b*x)*(-1/3*1/((b*d - a*e)*(a + b*x)^3*Sqrt[d + e*x]) - (7*e*(-1/2*1/ ((b*d - a*e)*(a + b*x)^2*Sqrt[d + e*x]) - (5*e*(-(1/((b*d - a*e)*(a + b*x) *Sqrt[d + e*x])) - (3*e*(2/((b*d - a*e)*Sqrt[d + e*x]) - (2*Sqrt[b]*ArcTan h[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(b*d - a*e)^(3/2)))/(2*(b*d - a*e))))/(4*(b*d - a*e))))/(6*(b*d - a*e))))/Sqrt[a^2 + 2*a*b*x + b^2*x^2]
3.22.43.3.1 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] :> 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]
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (b_.)*(x_ ) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a + b*x + c*x^2)^FracPart[p]/(c^ IntPart[p]*(b/2 + c*x)^(2*FracPart[p])) Int[(d + e*x)^m*(f + g*x)^n*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && EqQ[b^2 - 4*a*c, 0] && !IntegerQ[p]
Leaf count of result is larger than twice the leaf count of optimal. \(430\) vs. \(2(193)=386\).
Time = 0.29 (sec) , antiderivative size = 431, normalized size of antiderivative = 1.56
| method | result | size |
| default | \(-\frac {\left (105 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) b^{4} e^{3} x^{3} \sqrt {e x +d}+315 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) \sqrt {e x +d}\, a \,b^{3} e^{3} x^{2}+105 \sqrt {\left (a e -b d \right ) b}\, b^{3} e^{3} x^{3}+315 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) \sqrt {e x +d}\, a^{2} b^{2} e^{3} x +280 \sqrt {\left (a e -b d \right ) b}\, a \,b^{2} e^{3} x^{2}+35 \sqrt {\left (a e -b d \right ) b}\, b^{3} d \,e^{2} x^{2}+105 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) \sqrt {e x +d}\, a^{3} b \,e^{3}+231 \sqrt {\left (a e -b d \right ) b}\, a^{2} b \,e^{3} x +98 \sqrt {\left (a e -b d \right ) b}\, a \,b^{2} d \,e^{2} x -14 \sqrt {\left (a e -b d \right ) b}\, b^{3} d^{2} e x +48 \sqrt {\left (a e -b d \right ) b}\, a^{3} e^{3}+87 \sqrt {\left (a e -b d \right ) b}\, a^{2} b d \,e^{2}-38 \sqrt {\left (a e -b d \right ) b}\, a \,b^{2} d^{2} e +8 \sqrt {\left (a e -b d \right ) b}\, b^{3} d^{3}\right ) \left (b x +a \right )^{2}}{24 \sqrt {e x +d}\, \sqrt {\left (a e -b d \right ) b}\, \left (a e -b d \right )^{4} \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}\) | \(431\) |
-1/24*(105*arctan(b*(e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2))*b^4*e^3*x^3*(e*x+d) ^(1/2)+315*arctan(b*(e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2))*(e*x+d)^(1/2)*a*b^3 *e^3*x^2+105*((a*e-b*d)*b)^(1/2)*b^3*e^3*x^3+315*arctan(b*(e*x+d)^(1/2)/(( a*e-b*d)*b)^(1/2))*(e*x+d)^(1/2)*a^2*b^2*e^3*x+280*((a*e-b*d)*b)^(1/2)*a*b ^2*e^3*x^2+35*((a*e-b*d)*b)^(1/2)*b^3*d*e^2*x^2+105*arctan(b*(e*x+d)^(1/2) /((a*e-b*d)*b)^(1/2))*(e*x+d)^(1/2)*a^3*b*e^3+231*((a*e-b*d)*b)^(1/2)*a^2* b*e^3*x+98*((a*e-b*d)*b)^(1/2)*a*b^2*d*e^2*x-14*((a*e-b*d)*b)^(1/2)*b^3*d^ 2*e*x+48*((a*e-b*d)*b)^(1/2)*a^3*e^3+87*((a*e-b*d)*b)^(1/2)*a^2*b*d*e^2-38 *((a*e-b*d)*b)^(1/2)*a*b^2*d^2*e+8*((a*e-b*d)*b)^(1/2)*b^3*d^3)*(b*x+a)^2/ (e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)/(a*e-b*d)^4/((b*x+a)^2)^(5/2)
Leaf count of result is larger than twice the leaf count of optimal. 597 vs. \(2 (193) = 386\).
Time = 0.35 (sec) , antiderivative size = 1204, normalized size of antiderivative = 4.36 \[ \int \frac {a+b x}{(d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\left [\frac {105 \, {\left (b^{3} e^{4} x^{4} + a^{3} d e^{3} + {\left (b^{3} d e^{3} + 3 \, a b^{2} e^{4}\right )} x^{3} + 3 \, {\left (a b^{2} d e^{3} + a^{2} b e^{4}\right )} x^{2} + {\left (3 \, a^{2} b d e^{3} + a^{3} e^{4}\right )} x\right )} \sqrt {\frac {b}{b d - a e}} \log \left (\frac {b e x + 2 \, b d - a e + 2 \, {\left (b d - a e\right )} \sqrt {e x + d} \sqrt {\frac {b}{b d - a e}}}{b x + a}\right ) - 2 \, {\left (105 \, b^{3} e^{3} x^{3} + 8 \, b^{3} d^{3} - 38 \, a b^{2} d^{2} e + 87 \, a^{2} b d e^{2} + 48 \, a^{3} e^{3} + 35 \, {\left (b^{3} d e^{2} + 8 \, a b^{2} e^{3}\right )} x^{2} - 7 \, {\left (2 \, b^{3} d^{2} e - 14 \, a b^{2} d e^{2} - 33 \, a^{2} b e^{3}\right )} x\right )} \sqrt {e x + d}}{48 \, {\left (a^{3} b^{4} d^{5} - 4 \, a^{4} b^{3} d^{4} e + 6 \, a^{5} b^{2} d^{3} e^{2} - 4 \, a^{6} b d^{2} e^{3} + a^{7} d e^{4} + {\left (b^{7} d^{4} e - 4 \, a b^{6} d^{3} e^{2} + 6 \, a^{2} b^{5} d^{2} e^{3} - 4 \, a^{3} b^{4} d e^{4} + a^{4} b^{3} e^{5}\right )} x^{4} + {\left (b^{7} d^{5} - a b^{6} d^{4} e - 6 \, a^{2} b^{5} d^{3} e^{2} + 14 \, a^{3} b^{4} d^{2} e^{3} - 11 \, a^{4} b^{3} d e^{4} + 3 \, a^{5} b^{2} e^{5}\right )} x^{3} + 3 \, {\left (a b^{6} d^{5} - 3 \, a^{2} b^{5} d^{4} e + 2 \, a^{3} b^{4} d^{3} e^{2} + 2 \, a^{4} b^{3} d^{2} e^{3} - 3 \, a^{5} b^{2} d e^{4} + a^{6} b e^{5}\right )} x^{2} + {\left (3 \, a^{2} b^{5} d^{5} - 11 \, a^{3} b^{4} d^{4} e + 14 \, a^{4} b^{3} d^{3} e^{2} - 6 \, a^{5} b^{2} d^{2} e^{3} - a^{6} b d e^{4} + a^{7} e^{5}\right )} x\right )}}, \frac {105 \, {\left (b^{3} e^{4} x^{4} + a^{3} d e^{3} + {\left (b^{3} d e^{3} + 3 \, a b^{2} e^{4}\right )} x^{3} + 3 \, {\left (a b^{2} d e^{3} + a^{2} b e^{4}\right )} x^{2} + {\left (3 \, a^{2} b d e^{3} + a^{3} e^{4}\right )} x\right )} \sqrt {-\frac {b}{b d - a e}} \arctan \left (-\frac {{\left (b d - a e\right )} \sqrt {e x + d} \sqrt {-\frac {b}{b d - a e}}}{b e x + b d}\right ) - {\left (105 \, b^{3} e^{3} x^{3} + 8 \, b^{3} d^{3} - 38 \, a b^{2} d^{2} e + 87 \, a^{2} b d e^{2} + 48 \, a^{3} e^{3} + 35 \, {\left (b^{3} d e^{2} + 8 \, a b^{2} e^{3}\right )} x^{2} - 7 \, {\left (2 \, b^{3} d^{2} e - 14 \, a b^{2} d e^{2} - 33 \, a^{2} b e^{3}\right )} x\right )} \sqrt {e x + d}}{24 \, {\left (a^{3} b^{4} d^{5} - 4 \, a^{4} b^{3} d^{4} e + 6 \, a^{5} b^{2} d^{3} e^{2} - 4 \, a^{6} b d^{2} e^{3} + a^{7} d e^{4} + {\left (b^{7} d^{4} e - 4 \, a b^{6} d^{3} e^{2} + 6 \, a^{2} b^{5} d^{2} e^{3} - 4 \, a^{3} b^{4} d e^{4} + a^{4} b^{3} e^{5}\right )} x^{4} + {\left (b^{7} d^{5} - a b^{6} d^{4} e - 6 \, a^{2} b^{5} d^{3} e^{2} + 14 \, a^{3} b^{4} d^{2} e^{3} - 11 \, a^{4} b^{3} d e^{4} + 3 \, a^{5} b^{2} e^{5}\right )} x^{3} + 3 \, {\left (a b^{6} d^{5} - 3 \, a^{2} b^{5} d^{4} e + 2 \, a^{3} b^{4} d^{3} e^{2} + 2 \, a^{4} b^{3} d^{2} e^{3} - 3 \, a^{5} b^{2} d e^{4} + a^{6} b e^{5}\right )} x^{2} + {\left (3 \, a^{2} b^{5} d^{5} - 11 \, a^{3} b^{4} d^{4} e + 14 \, a^{4} b^{3} d^{3} e^{2} - 6 \, a^{5} b^{2} d^{2} e^{3} - a^{6} b d e^{4} + a^{7} e^{5}\right )} x\right )}}\right ] \]
[1/48*(105*(b^3*e^4*x^4 + a^3*d*e^3 + (b^3*d*e^3 + 3*a*b^2*e^4)*x^3 + 3*(a *b^2*d*e^3 + a^2*b*e^4)*x^2 + (3*a^2*b*d*e^3 + a^3*e^4)*x)*sqrt(b/(b*d - a *e))*log((b*e*x + 2*b*d - a*e + 2*(b*d - a*e)*sqrt(e*x + d)*sqrt(b/(b*d - a*e)))/(b*x + a)) - 2*(105*b^3*e^3*x^3 + 8*b^3*d^3 - 38*a*b^2*d^2*e + 87*a ^2*b*d*e^2 + 48*a^3*e^3 + 35*(b^3*d*e^2 + 8*a*b^2*e^3)*x^2 - 7*(2*b^3*d^2* e - 14*a*b^2*d*e^2 - 33*a^2*b*e^3)*x)*sqrt(e*x + d))/(a^3*b^4*d^5 - 4*a^4* b^3*d^4*e + 6*a^5*b^2*d^3*e^2 - 4*a^6*b*d^2*e^3 + a^7*d*e^4 + (b^7*d^4*e - 4*a*b^6*d^3*e^2 + 6*a^2*b^5*d^2*e^3 - 4*a^3*b^4*d*e^4 + a^4*b^3*e^5)*x^4 + (b^7*d^5 - a*b^6*d^4*e - 6*a^2*b^5*d^3*e^2 + 14*a^3*b^4*d^2*e^3 - 11*a^4 *b^3*d*e^4 + 3*a^5*b^2*e^5)*x^3 + 3*(a*b^6*d^5 - 3*a^2*b^5*d^4*e + 2*a^3*b ^4*d^3*e^2 + 2*a^4*b^3*d^2*e^3 - 3*a^5*b^2*d*e^4 + a^6*b*e^5)*x^2 + (3*a^2 *b^5*d^5 - 11*a^3*b^4*d^4*e + 14*a^4*b^3*d^3*e^2 - 6*a^5*b^2*d^2*e^3 - a^6 *b*d*e^4 + a^7*e^5)*x), 1/24*(105*(b^3*e^4*x^4 + a^3*d*e^3 + (b^3*d*e^3 + 3*a*b^2*e^4)*x^3 + 3*(a*b^2*d*e^3 + a^2*b*e^4)*x^2 + (3*a^2*b*d*e^3 + a^3* e^4)*x)*sqrt(-b/(b*d - a*e))*arctan(-(b*d - a*e)*sqrt(e*x + d)*sqrt(-b/(b* d - a*e))/(b*e*x + b*d)) - (105*b^3*e^3*x^3 + 8*b^3*d^3 - 38*a*b^2*d^2*e + 87*a^2*b*d*e^2 + 48*a^3*e^3 + 35*(b^3*d*e^2 + 8*a*b^2*e^3)*x^2 - 7*(2*b^3 *d^2*e - 14*a*b^2*d*e^2 - 33*a^2*b*e^3)*x)*sqrt(e*x + d))/(a^3*b^4*d^5 - 4 *a^4*b^3*d^4*e + 6*a^5*b^2*d^3*e^2 - 4*a^6*b*d^2*e^3 + a^7*d*e^4 + (b^7*d^ 4*e - 4*a*b^6*d^3*e^2 + 6*a^2*b^5*d^2*e^3 - 4*a^3*b^4*d*e^4 + a^4*b^3*e...
Timed out. \[ \int \frac {a+b x}{(d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\text {Timed out} \]
\[ \int \frac {a+b x}{(d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\int { \frac {b x + a}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} {\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 416 vs. \(2 (193) = 386\).
Time = 0.28 (sec) , antiderivative size = 416, normalized size of antiderivative = 1.51 \[ \int \frac {a+b x}{(d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=-\frac {35 \, b e^{3} \arctan \left (\frac {\sqrt {e x + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{8 \, {\left (b^{4} d^{4} \mathrm {sgn}\left (b x + a\right ) - 4 \, a b^{3} d^{3} e \mathrm {sgn}\left (b x + a\right ) + 6 \, a^{2} b^{2} d^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) - 4 \, a^{3} b d e^{3} \mathrm {sgn}\left (b x + a\right ) + a^{4} e^{4} \mathrm {sgn}\left (b x + a\right )\right )} \sqrt {-b^{2} d + a b e}} - \frac {2 \, e^{3}}{{\left (b^{4} d^{4} \mathrm {sgn}\left (b x + a\right ) - 4 \, a b^{3} d^{3} e \mathrm {sgn}\left (b x + a\right ) + 6 \, a^{2} b^{2} d^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) - 4 \, a^{3} b d e^{3} \mathrm {sgn}\left (b x + a\right ) + a^{4} e^{4} \mathrm {sgn}\left (b x + a\right )\right )} \sqrt {e x + d}} - \frac {57 \, {\left (e x + d\right )}^{\frac {5}{2}} b^{3} e^{3} - 136 \, {\left (e x + d\right )}^{\frac {3}{2}} b^{3} d e^{3} + 87 \, \sqrt {e x + d} b^{3} d^{2} e^{3} + 136 \, {\left (e x + d\right )}^{\frac {3}{2}} a b^{2} e^{4} - 174 \, \sqrt {e x + d} a b^{2} d e^{4} + 87 \, \sqrt {e x + d} a^{2} b e^{5}}{24 \, {\left (b^{4} d^{4} \mathrm {sgn}\left (b x + a\right ) - 4 \, a b^{3} d^{3} e \mathrm {sgn}\left (b x + a\right ) + 6 \, a^{2} b^{2} d^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) - 4 \, a^{3} b d e^{3} \mathrm {sgn}\left (b x + a\right ) + a^{4} e^{4} \mathrm {sgn}\left (b x + a\right )\right )} {\left ({\left (e x + d\right )} b - b d + a e\right )}^{3}} \]
-35/8*b*e^3*arctan(sqrt(e*x + d)*b/sqrt(-b^2*d + a*b*e))/((b^4*d^4*sgn(b*x + a) - 4*a*b^3*d^3*e*sgn(b*x + a) + 6*a^2*b^2*d^2*e^2*sgn(b*x + a) - 4*a^ 3*b*d*e^3*sgn(b*x + a) + a^4*e^4*sgn(b*x + a))*sqrt(-b^2*d + a*b*e)) - 2*e ^3/((b^4*d^4*sgn(b*x + a) - 4*a*b^3*d^3*e*sgn(b*x + a) + 6*a^2*b^2*d^2*e^2 *sgn(b*x + a) - 4*a^3*b*d*e^3*sgn(b*x + a) + a^4*e^4*sgn(b*x + a))*sqrt(e* x + d)) - 1/24*(57*(e*x + d)^(5/2)*b^3*e^3 - 136*(e*x + d)^(3/2)*b^3*d*e^3 + 87*sqrt(e*x + d)*b^3*d^2*e^3 + 136*(e*x + d)^(3/2)*a*b^2*e^4 - 174*sqrt (e*x + d)*a*b^2*d*e^4 + 87*sqrt(e*x + d)*a^2*b*e^5)/((b^4*d^4*sgn(b*x + a) - 4*a*b^3*d^3*e*sgn(b*x + a) + 6*a^2*b^2*d^2*e^2*sgn(b*x + a) - 4*a^3*b*d *e^3*sgn(b*x + a) + a^4*e^4*sgn(b*x + a))*((e*x + d)*b - b*d + a*e)^3)
Timed out. \[ \int \frac {a+b x}{(d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\int \frac {a+b\,x}{{\left (d+e\,x\right )}^{3/2}\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}} \,d x \]