Integrand size = 31, antiderivative size = 311 \[ \int \frac {A+B x}{x^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {35 (9 A b-a B)}{192 a^4 b \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {A b-a B}{4 a b \sqrt {x} (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {9 A b-a B}{24 a^2 b \sqrt {x} (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 (9 A b-a B)}{96 a^3 b \sqrt {x} (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {35 (9 A b-a B) (a+b x)}{64 a^5 b \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {35 (9 A b-a B) (a+b x) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{64 a^{11/2} \sqrt {b} \sqrt {a^2+2 a b x+b^2 x^2}} \]
-35/64*(9*A*b-B*a)*(b*x+a)*arctan(b^(1/2)*x^(1/2)/a^(1/2))/a^(11/2)/b^(1/2 )/((b*x+a)^2)^(1/2)+35/192*(9*A*b-B*a)/a^4/b/x^(1/2)/((b*x+a)^2)^(1/2)+1/4 *(A*b-B*a)/a/b/(b*x+a)^3/x^(1/2)/((b*x+a)^2)^(1/2)+1/24*(9*A*b-B*a)/a^2/b/ (b*x+a)^2/x^(1/2)/((b*x+a)^2)^(1/2)+7/96*(9*A*b-B*a)/a^3/b/(b*x+a)/x^(1/2) /((b*x+a)^2)^(1/2)-35/64*(9*A*b-B*a)*(b*x+a)/a^5/b/x^(1/2)/((b*x+a)^2)^(1/ 2)
Time = 0.25 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.48 \[ \int \frac {A+B x}{x^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {\frac {\sqrt {a} \left (-945 A b^4 x^4+105 a b^3 x^3 (-33 A+B x)+7 a^2 b^2 x^2 (-657 A+55 B x)+a^4 (-384 A+279 B x)+a^3 b x (-2511 A+511 B x)\right )}{\sqrt {x}}+\frac {105 (-9 A b+a B) (a+b x)^4 \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{\sqrt {b}}}{192 a^{11/2} (a+b x)^3 \sqrt {(a+b x)^2}} \]
((Sqrt[a]*(-945*A*b^4*x^4 + 105*a*b^3*x^3*(-33*A + B*x) + 7*a^2*b^2*x^2*(- 657*A + 55*B*x) + a^4*(-384*A + 279*B*x) + a^3*b*x*(-2511*A + 511*B*x)))/S qrt[x] + (105*(-9*A*b + a*B)*(a + b*x)^4*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]] )/Sqrt[b])/(192*a^(11/2)*(a + b*x)^3*Sqrt[(a + b*x)^2])
Time = 0.29 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.62, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.290, Rules used = {1187, 27, 87, 52, 52, 52, 61, 73, 218}
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.
| \(\displaystyle \int \frac {A+B x}{x^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 1187 |
| \(\displaystyle \frac {b^5 (a+b x) \int \frac {A+B x}{b^5 x^{3/2} (a+b x)^5}dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(a+b x) \int \frac {A+B x}{x^{3/2} (a+b x)^5}dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {(a+b x) \left (\frac {(9 A b-a B) \int \frac {1}{x^{3/2} (a+b x)^4}dx}{8 a b}+\frac {A b-a B}{4 a b \sqrt {x} (a+b x)^4}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {(a+b x) \left (\frac {(9 A b-a B) \left (\frac {7 \int \frac {1}{x^{3/2} (a+b x)^3}dx}{6 a}+\frac {1}{3 a \sqrt {x} (a+b x)^3}\right )}{8 a b}+\frac {A b-a B}{4 a b \sqrt {x} (a+b x)^4}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {(a+b x) \left (\frac {(9 A b-a B) \left (\frac {7 \left (\frac {5 \int \frac {1}{x^{3/2} (a+b x)^2}dx}{4 a}+\frac {1}{2 a \sqrt {x} (a+b x)^2}\right )}{6 a}+\frac {1}{3 a \sqrt {x} (a+b x)^3}\right )}{8 a b}+\frac {A b-a B}{4 a b \sqrt {x} (a+b x)^4}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {(a+b x) \left (\frac {(9 A b-a B) \left (\frac {7 \left (\frac {5 \left (\frac {3 \int \frac {1}{x^{3/2} (a+b x)}dx}{2 a}+\frac {1}{a \sqrt {x} (a+b x)}\right )}{4 a}+\frac {1}{2 a \sqrt {x} (a+b x)^2}\right )}{6 a}+\frac {1}{3 a \sqrt {x} (a+b x)^3}\right )}{8 a b}+\frac {A b-a B}{4 a b \sqrt {x} (a+b x)^4}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle \frac {(a+b x) \left (\frac {(9 A b-a B) \left (\frac {7 \left (\frac {5 \left (\frac {3 \left (-\frac {b \int \frac {1}{\sqrt {x} (a+b x)}dx}{a}-\frac {2}{a \sqrt {x}}\right )}{2 a}+\frac {1}{a \sqrt {x} (a+b x)}\right )}{4 a}+\frac {1}{2 a \sqrt {x} (a+b x)^2}\right )}{6 a}+\frac {1}{3 a \sqrt {x} (a+b x)^3}\right )}{8 a b}+\frac {A b-a B}{4 a b \sqrt {x} (a+b x)^4}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {(a+b x) \left (\frac {(9 A b-a B) \left (\frac {7 \left (\frac {5 \left (\frac {3 \left (-\frac {2 b \int \frac {1}{a+b x}d\sqrt {x}}{a}-\frac {2}{a \sqrt {x}}\right )}{2 a}+\frac {1}{a \sqrt {x} (a+b x)}\right )}{4 a}+\frac {1}{2 a \sqrt {x} (a+b x)^2}\right )}{6 a}+\frac {1}{3 a \sqrt {x} (a+b x)^3}\right )}{8 a b}+\frac {A b-a B}{4 a b \sqrt {x} (a+b x)^4}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {(a+b x) \left (\frac {(9 A b-a B) \left (\frac {7 \left (\frac {5 \left (\frac {3 \left (-\frac {2 \sqrt {b} \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {2}{a \sqrt {x}}\right )}{2 a}+\frac {1}{a \sqrt {x} (a+b x)}\right )}{4 a}+\frac {1}{2 a \sqrt {x} (a+b x)^2}\right )}{6 a}+\frac {1}{3 a \sqrt {x} (a+b x)^3}\right )}{8 a b}+\frac {A b-a B}{4 a b \sqrt {x} (a+b x)^4}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
((a + b*x)*((A*b - a*B)/(4*a*b*Sqrt[x]*(a + b*x)^4) + ((9*A*b - a*B)*(1/(3 *a*Sqrt[x]*(a + b*x)^3) + (7*(1/(2*a*Sqrt[x]*(a + b*x)^2) + (5*(1/(a*Sqrt[ x]*(a + b*x)) + (3*(-2/(a*Sqrt[x]) - (2*Sqrt[b]*ArcTan[(Sqrt[b]*Sqrt[x])/S qrt[a]])/a^(3/2)))/(2*a)))/(4*a)))/(6*a)))/(8*a*b)))/Sqrt[a^2 + 2*a*b*x + b^2*x^2]
3.9.33.3.1 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_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[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]
Time = 0.19 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.52
| method | result | size |
| risch | \(-\frac {2 A \sqrt {\left (b x +a \right )^{2}}}{a^{5} \sqrt {x}\, \left (b x +a \right )}-\frac {\left (\frac {2 \left (\frac {187}{128} A \,b^{4}-\frac {35}{128} B \,b^{3} a \right ) x^{\frac {7}{2}}+\frac {b^{2} a \left (1929 A b -385 B a \right ) x^{\frac {5}{2}}}{192}+2 \left (\frac {765}{128} A \,a^{2} b^{2}-\frac {511}{384} B \,a^{3} b \right ) x^{\frac {3}{2}}+2 \left (\frac {325}{128} A \,a^{3} b -\frac {93}{128} B \,a^{4}\right ) \sqrt {x}}{\left (b x +a \right )^{4}}+\frac {35 \left (9 A b -B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right )}{64 \sqrt {b a}}\right ) \sqrt {\left (b x +a \right )^{2}}}{a^{5} \left (b x +a \right )}\) | \(161\) |
| default | \(-\frac {\left (945 A \,x^{4} \sqrt {b a}\, b^{4}-105 B \,x^{4} \sqrt {b a}\, a \,b^{3}+3465 A \,x^{3} \sqrt {b a}\, a \,b^{3}+945 A \,x^{\frac {9}{2}} \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right ) b^{5}-385 B \,x^{3} \sqrt {b a}\, a^{2} b^{2}-105 B \,x^{\frac {9}{2}} \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right ) a \,b^{4}+3780 A \,x^{\frac {7}{2}} \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right ) a \,b^{4}-420 B \,x^{\frac {7}{2}} \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right ) a^{2} b^{3}+4599 A \,x^{2} \sqrt {b a}\, a^{2} b^{2}+5670 A \,x^{\frac {5}{2}} \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right ) a^{2} b^{3}-511 B \,x^{2} \sqrt {b a}\, a^{3} b -630 B \,x^{\frac {5}{2}} \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right ) a^{3} b^{2}+3780 A \,x^{\frac {3}{2}} \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right ) a^{3} b^{2}-420 B \,x^{\frac {3}{2}} \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right ) a^{4} b +2511 A x \sqrt {b a}\, a^{3} b +945 A \sqrt {x}\, \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right ) a^{4} b -279 B x \sqrt {b a}\, a^{4}-105 B \sqrt {x}\, \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right ) a^{5}+384 A \sqrt {b a}\, a^{4}\right ) \left (b x +a \right )}{192 \sqrt {x}\, \sqrt {b a}\, a^{5} \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}\) | \(374\) |
-2*A/a^5/x^(1/2)*((b*x+a)^2)^(1/2)/(b*x+a)-1/a^5*(2*((187/128*A*b^4-35/128 *B*b^3*a)*x^(7/2)+1/384*b^2*a*(1929*A*b-385*B*a)*x^(5/2)+(765/128*A*a^2*b^ 2-511/384*B*a^3*b)*x^(3/2)+(325/128*A*a^3*b-93/128*B*a^4)*x^(1/2))/(b*x+a) ^4+35/64*(9*A*b-B*a)/(b*a)^(1/2)*arctan(b*x^(1/2)/(b*a)^(1/2)))*((b*x+a)^2 )^(1/2)/(b*x+a)
Time = 0.34 (sec) , antiderivative size = 559, normalized size of antiderivative = 1.80 \[ \int \frac {A+B x}{x^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\left [\frac {105 \, {\left ({\left (B a b^{4} - 9 \, A b^{5}\right )} x^{5} + 4 \, {\left (B a^{2} b^{3} - 9 \, A a b^{4}\right )} x^{4} + 6 \, {\left (B a^{3} b^{2} - 9 \, A a^{2} b^{3}\right )} x^{3} + 4 \, {\left (B a^{4} b - 9 \, A a^{3} b^{2}\right )} x^{2} + {\left (B a^{5} - 9 \, A a^{4} b\right )} x\right )} \sqrt {-a b} \log \left (\frac {b x - a + 2 \, \sqrt {-a b} \sqrt {x}}{b x + a}\right ) - 2 \, {\left (384 \, A a^{5} b - 105 \, {\left (B a^{2} b^{4} - 9 \, A a b^{5}\right )} x^{4} - 385 \, {\left (B a^{3} b^{3} - 9 \, A a^{2} b^{4}\right )} x^{3} - 511 \, {\left (B a^{4} b^{2} - 9 \, A a^{3} b^{3}\right )} x^{2} - 279 \, {\left (B a^{5} b - 9 \, A a^{4} b^{2}\right )} x\right )} \sqrt {x}}{384 \, {\left (a^{6} b^{5} x^{5} + 4 \, a^{7} b^{4} x^{4} + 6 \, a^{8} b^{3} x^{3} + 4 \, a^{9} b^{2} x^{2} + a^{10} b x\right )}}, -\frac {105 \, {\left ({\left (B a b^{4} - 9 \, A b^{5}\right )} x^{5} + 4 \, {\left (B a^{2} b^{3} - 9 \, A a b^{4}\right )} x^{4} + 6 \, {\left (B a^{3} b^{2} - 9 \, A a^{2} b^{3}\right )} x^{3} + 4 \, {\left (B a^{4} b - 9 \, A a^{3} b^{2}\right )} x^{2} + {\left (B a^{5} - 9 \, A a^{4} b\right )} x\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b}}{b \sqrt {x}}\right ) + {\left (384 \, A a^{5} b - 105 \, {\left (B a^{2} b^{4} - 9 \, A a b^{5}\right )} x^{4} - 385 \, {\left (B a^{3} b^{3} - 9 \, A a^{2} b^{4}\right )} x^{3} - 511 \, {\left (B a^{4} b^{2} - 9 \, A a^{3} b^{3}\right )} x^{2} - 279 \, {\left (B a^{5} b - 9 \, A a^{4} b^{2}\right )} x\right )} \sqrt {x}}{192 \, {\left (a^{6} b^{5} x^{5} + 4 \, a^{7} b^{4} x^{4} + 6 \, a^{8} b^{3} x^{3} + 4 \, a^{9} b^{2} x^{2} + a^{10} b x\right )}}\right ] \]
[1/384*(105*((B*a*b^4 - 9*A*b^5)*x^5 + 4*(B*a^2*b^3 - 9*A*a*b^4)*x^4 + 6*( B*a^3*b^2 - 9*A*a^2*b^3)*x^3 + 4*(B*a^4*b - 9*A*a^3*b^2)*x^2 + (B*a^5 - 9* A*a^4*b)*x)*sqrt(-a*b)*log((b*x - a + 2*sqrt(-a*b)*sqrt(x))/(b*x + a)) - 2 *(384*A*a^5*b - 105*(B*a^2*b^4 - 9*A*a*b^5)*x^4 - 385*(B*a^3*b^3 - 9*A*a^2 *b^4)*x^3 - 511*(B*a^4*b^2 - 9*A*a^3*b^3)*x^2 - 279*(B*a^5*b - 9*A*a^4*b^2 )*x)*sqrt(x))/(a^6*b^5*x^5 + 4*a^7*b^4*x^4 + 6*a^8*b^3*x^3 + 4*a^9*b^2*x^2 + a^10*b*x), -1/192*(105*((B*a*b^4 - 9*A*b^5)*x^5 + 4*(B*a^2*b^3 - 9*A*a* b^4)*x^4 + 6*(B*a^3*b^2 - 9*A*a^2*b^3)*x^3 + 4*(B*a^4*b - 9*A*a^3*b^2)*x^2 + (B*a^5 - 9*A*a^4*b)*x)*sqrt(a*b)*arctan(sqrt(a*b)/(b*sqrt(x))) + (384*A *a^5*b - 105*(B*a^2*b^4 - 9*A*a*b^5)*x^4 - 385*(B*a^3*b^3 - 9*A*a^2*b^4)*x ^3 - 511*(B*a^4*b^2 - 9*A*a^3*b^3)*x^2 - 279*(B*a^5*b - 9*A*a^4*b^2)*x)*sq rt(x))/(a^6*b^5*x^5 + 4*a^7*b^4*x^4 + 6*a^8*b^3*x^3 + 4*a^9*b^2*x^2 + a^10 *b*x)]
Timed out. \[ \int \frac {A+B x}{x^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 427 vs. \(2 (208) = 416\).
Time = 0.37 (sec) , antiderivative size = 427, normalized size of antiderivative = 1.37 \[ \int \frac {A+B x}{x^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=-\frac {35 \, {\left ({\left (B a b^{6} + 9 \, A b^{7}\right )} x^{2} - 27 \, {\left (B a^{2} b^{5} - 11 \, A a b^{6}\right )} x\right )} x^{\frac {9}{2}} + 70 \, {\left ({\left (B a^{2} b^{5} + 9 \, A a b^{6}\right )} x^{2} - 81 \, {\left (B a^{3} b^{4} - 11 \, A a^{2} b^{5}\right )} x\right )} x^{\frac {7}{2}} - 140 \, {\left (2 \, {\left (B a^{3} b^{4} + 9 \, A a^{2} b^{5}\right )} x^{2} + 99 \, {\left (B a^{4} b^{3} - 11 \, A a^{3} b^{4}\right )} x\right )} x^{\frac {5}{2}} - 14 \, {\left (85 \, {\left (B a^{4} b^{3} + 9 \, A a^{3} b^{4}\right )} x^{2} + 1251 \, {\left (B a^{5} b^{2} - 11 \, A a^{4} b^{3}\right )} x\right )} x^{\frac {3}{2}} - {\left (1771 \, {\left (B a^{5} b^{2} + 9 \, A a^{4} b^{3}\right )} x^{2} + 11835 \, {\left (B a^{6} b - 11 \, A a^{5} b^{2}\right )} x\right )} \sqrt {x} - \frac {1280 \, {\left ({\left (B a^{6} b + 9 \, A a^{5} b^{2}\right )} x^{2} + 3 \, {\left (B a^{7} - 11 \, A a^{6} b\right )} x\right )}}{\sqrt {x}} - \frac {3840 \, {\left (A a^{6} b x^{2} - A a^{7} x\right )}}{x^{\frac {3}{2}}}}{1920 \, {\left (a^{7} b^{5} x^{5} + 5 \, a^{8} b^{4} x^{4} + 10 \, a^{9} b^{3} x^{3} + 10 \, a^{10} b^{2} x^{2} + 5 \, a^{11} b x + a^{12}\right )}} + \frac {35 \, {\left (B a - 9 \, A b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{64 \, \sqrt {a b} a^{5}} + \frac {7 \, {\left ({\left (B a b + 9 \, A b^{2}\right )} x^{\frac {3}{2}} - 30 \, {\left (B a^{2} - 9 \, A a b\right )} \sqrt {x}\right )}}{384 \, a^{7}} \]
-1/1920*(35*((B*a*b^6 + 9*A*b^7)*x^2 - 27*(B*a^2*b^5 - 11*A*a*b^6)*x)*x^(9 /2) + 70*((B*a^2*b^5 + 9*A*a*b^6)*x^2 - 81*(B*a^3*b^4 - 11*A*a^2*b^5)*x)*x ^(7/2) - 140*(2*(B*a^3*b^4 + 9*A*a^2*b^5)*x^2 + 99*(B*a^4*b^3 - 11*A*a^3*b ^4)*x)*x^(5/2) - 14*(85*(B*a^4*b^3 + 9*A*a^3*b^4)*x^2 + 1251*(B*a^5*b^2 - 11*A*a^4*b^3)*x)*x^(3/2) - (1771*(B*a^5*b^2 + 9*A*a^4*b^3)*x^2 + 11835*(B* a^6*b - 11*A*a^5*b^2)*x)*sqrt(x) - 1280*((B*a^6*b + 9*A*a^5*b^2)*x^2 + 3*( B*a^7 - 11*A*a^6*b)*x)/sqrt(x) - 3840*(A*a^6*b*x^2 - A*a^7*x)/x^(3/2))/(a^ 7*b^5*x^5 + 5*a^8*b^4*x^4 + 10*a^9*b^3*x^3 + 10*a^10*b^2*x^2 + 5*a^11*b*x + a^12) + 35/64*(B*a - 9*A*b)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a^5) + 7/384*((B*a*b + 9*A*b^2)*x^(3/2) - 30*(B*a^2 - 9*A*a*b)*sqrt(x))/a^7
Time = 0.28 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.51 \[ \int \frac {A+B x}{x^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {35 \, {\left (B a - 9 \, A b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{64 \, \sqrt {a b} a^{5} \mathrm {sgn}\left (b x + a\right )} - \frac {2 \, A}{a^{5} \sqrt {x} \mathrm {sgn}\left (b x + a\right )} + \frac {105 \, B a b^{3} x^{\frac {7}{2}} - 561 \, A b^{4} x^{\frac {7}{2}} + 385 \, B a^{2} b^{2} x^{\frac {5}{2}} - 1929 \, A a b^{3} x^{\frac {5}{2}} + 511 \, B a^{3} b x^{\frac {3}{2}} - 2295 \, A a^{2} b^{2} x^{\frac {3}{2}} + 279 \, B a^{4} \sqrt {x} - 975 \, A a^{3} b \sqrt {x}}{192 \, {\left (b x + a\right )}^{4} a^{5} \mathrm {sgn}\left (b x + a\right )} \]
35/64*(B*a - 9*A*b)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a^5*sgn(b*x + a )) - 2*A/(a^5*sqrt(x)*sgn(b*x + a)) + 1/192*(105*B*a*b^3*x^(7/2) - 561*A*b ^4*x^(7/2) + 385*B*a^2*b^2*x^(5/2) - 1929*A*a*b^3*x^(5/2) + 511*B*a^3*b*x^ (3/2) - 2295*A*a^2*b^2*x^(3/2) + 279*B*a^4*sqrt(x) - 975*A*a^3*b*sqrt(x))/ ((b*x + a)^4*a^5*sgn(b*x + a))
Timed out. \[ \int \frac {A+B x}{x^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\int \frac {A+B\,x}{x^{3/2}\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}} \,d x \]