Integrand size = 29, antiderivative size = 195 \[ \int \frac {\sqrt {x} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {(A b-a B) x^{3/2}}{5 a b (a+b x)^5}-\frac {(7 A b+3 a B) \sqrt {x}}{40 a b^2 (a+b x)^4}+\frac {(7 A b+3 a B) \sqrt {x}}{240 a^2 b^2 (a+b x)^3}+\frac {(7 A b+3 a B) \sqrt {x}}{192 a^3 b^2 (a+b x)^2}+\frac {(7 A b+3 a B) \sqrt {x}}{128 a^4 b^2 (a+b x)}+\frac {(7 A b+3 a B) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{128 a^{9/2} b^{5/2}} \]
1/5*(A*b-B*a)*x^(3/2)/a/b/(b*x+a)^5+1/128*(7*A*b+3*B*a)*arctan(b^(1/2)*x^( 1/2)/a^(1/2))/a^(9/2)/b^(5/2)-1/40*(7*A*b+3*B*a)*x^(1/2)/a/b^2/(b*x+a)^4+1 /240*(7*A*b+3*B*a)*x^(1/2)/a^2/b^2/(b*x+a)^3+1/192*(7*A*b+3*B*a)*x^(1/2)/a ^3/b^2/(b*x+a)^2+1/128*(7*A*b+3*B*a)*x^(1/2)/a^4/b^2/(b*x+a)
Time = 0.24 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.74 \[ \int \frac {\sqrt {x} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {\sqrt {x} \left (-45 a^5 B+105 A b^5 x^4-105 a^4 b (A+2 B x)+5 a b^4 x^3 (98 A+9 B x)+14 a^2 b^3 x^2 (64 A+15 B x)+2 a^3 b^2 x (395 A+192 B x)\right )}{1920 a^4 b^2 (a+b x)^5}+\frac {(7 A b+3 a B) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{128 a^{9/2} b^{5/2}} \]
(Sqrt[x]*(-45*a^5*B + 105*A*b^5*x^4 - 105*a^4*b*(A + 2*B*x) + 5*a*b^4*x^3* (98*A + 9*B*x) + 14*a^2*b^3*x^2*(64*A + 15*B*x) + 2*a^3*b^2*x*(395*A + 192 *B*x)))/(1920*a^4*b^2*(a + b*x)^5) + ((7*A*b + 3*a*B)*ArcTan[(Sqrt[b]*Sqrt [x])/Sqrt[a]])/(128*a^(9/2)*b^(5/2))
Time = 0.28 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.90, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.310, Rules used = {1184, 27, 87, 51, 52, 52, 52, 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.
| \(\displaystyle \int \frac {\sqrt {x} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 1184 |
| \(\displaystyle b^6 \int \frac {\sqrt {x} (A+B x)}{b^6 (a+b x)^6}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {\sqrt {x} (A+B x)}{(a+b x)^6}dx\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {(3 a B+7 A b) \int \frac {\sqrt {x}}{(a+b x)^5}dx}{10 a b}+\frac {x^{3/2} (A b-a B)}{5 a b (a+b x)^5}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {(3 a B+7 A b) \left (\frac {\int \frac {1}{\sqrt {x} (a+b x)^4}dx}{8 b}-\frac {\sqrt {x}}{4 b (a+b x)^4}\right )}{10 a b}+\frac {x^{3/2} (A b-a B)}{5 a b (a+b x)^5}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {(3 a B+7 A b) \left (\frac {\frac {5 \int \frac {1}{\sqrt {x} (a+b x)^3}dx}{6 a}+\frac {\sqrt {x}}{3 a (a+b x)^3}}{8 b}-\frac {\sqrt {x}}{4 b (a+b x)^4}\right )}{10 a b}+\frac {x^{3/2} (A b-a B)}{5 a b (a+b x)^5}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {(3 a B+7 A b) \left (\frac {\frac {5 \left (\frac {3 \int \frac {1}{\sqrt {x} (a+b x)^2}dx}{4 a}+\frac {\sqrt {x}}{2 a (a+b x)^2}\right )}{6 a}+\frac {\sqrt {x}}{3 a (a+b x)^3}}{8 b}-\frac {\sqrt {x}}{4 b (a+b x)^4}\right )}{10 a b}+\frac {x^{3/2} (A b-a B)}{5 a b (a+b x)^5}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {(3 a B+7 A b) \left (\frac {\frac {5 \left (\frac {3 \left (\frac {\int \frac {1}{\sqrt {x} (a+b x)}dx}{2 a}+\frac {\sqrt {x}}{a (a+b x)}\right )}{4 a}+\frac {\sqrt {x}}{2 a (a+b x)^2}\right )}{6 a}+\frac {\sqrt {x}}{3 a (a+b x)^3}}{8 b}-\frac {\sqrt {x}}{4 b (a+b x)^4}\right )}{10 a b}+\frac {x^{3/2} (A b-a B)}{5 a b (a+b x)^5}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {(3 a B+7 A b) \left (\frac {\frac {5 \left (\frac {3 \left (\frac {\int \frac {1}{a+b x}d\sqrt {x}}{a}+\frac {\sqrt {x}}{a (a+b x)}\right )}{4 a}+\frac {\sqrt {x}}{2 a (a+b x)^2}\right )}{6 a}+\frac {\sqrt {x}}{3 a (a+b x)^3}}{8 b}-\frac {\sqrt {x}}{4 b (a+b x)^4}\right )}{10 a b}+\frac {x^{3/2} (A b-a B)}{5 a b (a+b x)^5}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {(3 a B+7 A b) \left (\frac {\frac {5 \left (\frac {3 \left (\frac {\arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{3/2} \sqrt {b}}+\frac {\sqrt {x}}{a (a+b x)}\right )}{4 a}+\frac {\sqrt {x}}{2 a (a+b x)^2}\right )}{6 a}+\frac {\sqrt {x}}{3 a (a+b x)^3}}{8 b}-\frac {\sqrt {x}}{4 b (a+b x)^4}\right )}{10 a b}+\frac {x^{3/2} (A b-a B)}{5 a b (a+b x)^5}\) |
((A*b - a*B)*x^(3/2))/(5*a*b*(a + b*x)^5) + ((7*A*b + 3*a*B)*(-1/4*Sqrt[x] /(b*(a + b*x)^4) + (Sqrt[x]/(3*a*(a + b*x)^3) + (5*(Sqrt[x]/(2*a*(a + b*x) ^2) + (3*(Sqrt[x]/(a*(a + b*x)) + ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]]/(a^(3/ 2)*Sqrt[b])))/(4*a)))/(6*a))/(8*b)))/(10*a*b)
3.8.78.3.1 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/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n}, x ] && ILtQ[m, -1] && FractionQ[n] && GtQ[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] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
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[1/c^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}, x] && E qQ[b^2 - 4*a*c, 0] && IntegerQ[p]
Time = 0.13 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.70
| method | result | size |
| derivativedivides | \(\frac {\frac {\left (7 A b +3 B a \right ) b^{2} x^{\frac {9}{2}}}{128 a^{4}}+\frac {7 b \left (7 A b +3 B a \right ) x^{\frac {7}{2}}}{192 a^{3}}+\frac {\left (7 A b +3 B a \right ) x^{\frac {5}{2}}}{15 a^{2}}+\frac {\left (79 A b -21 B a \right ) x^{\frac {3}{2}}}{192 b a}-\frac {\left (7 A b +3 B a \right ) \sqrt {x}}{128 b^{2}}}{\left (b x +a \right )^{5}}+\frac {\left (7 A b +3 B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right )}{128 a^{4} b^{2} \sqrt {b a}}\) | \(137\) |
| default | \(\frac {\frac {\left (7 A b +3 B a \right ) b^{2} x^{\frac {9}{2}}}{128 a^{4}}+\frac {7 b \left (7 A b +3 B a \right ) x^{\frac {7}{2}}}{192 a^{3}}+\frac {\left (7 A b +3 B a \right ) x^{\frac {5}{2}}}{15 a^{2}}+\frac {\left (79 A b -21 B a \right ) x^{\frac {3}{2}}}{192 b a}-\frac {\left (7 A b +3 B a \right ) \sqrt {x}}{128 b^{2}}}{\left (b x +a \right )^{5}}+\frac {\left (7 A b +3 B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right )}{128 a^{4} b^{2} \sqrt {b a}}\) | \(137\) |
2*(1/256*(7*A*b+3*B*a)/a^4*b^2*x^(9/2)+7/384/a^3*b*(7*A*b+3*B*a)*x^(7/2)+1 /30/a^2*(7*A*b+3*B*a)*x^(5/2)+1/384*(79*A*b-21*B*a)/b/a*x^(3/2)-1/256*(7*A *b+3*B*a)/b^2*x^(1/2))/(b*x+a)^5+1/128*(7*A*b+3*B*a)/a^4/b^2/(b*a)^(1/2)*a rctan(b*x^(1/2)/(b*a)^(1/2))
Time = 0.41 (sec) , antiderivative size = 657, normalized size of antiderivative = 3.37 \[ \int \frac {\sqrt {x} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\left [-\frac {15 \, {\left (3 \, B a^{6} + 7 \, A a^{5} b + {\left (3 \, B a b^{5} + 7 \, A b^{6}\right )} x^{5} + 5 \, {\left (3 \, B a^{2} b^{4} + 7 \, A a b^{5}\right )} x^{4} + 10 \, {\left (3 \, B a^{3} b^{3} + 7 \, A a^{2} b^{4}\right )} x^{3} + 10 \, {\left (3 \, B a^{4} b^{2} + 7 \, A a^{3} b^{3}\right )} x^{2} + 5 \, {\left (3 \, B a^{5} b + 7 \, A a^{4} b^{2}\right )} x\right )} \sqrt {-a b} \log \left (\frac {b x - a - 2 \, \sqrt {-a b} \sqrt {x}}{b x + a}\right ) + 2 \, {\left (45 \, B a^{6} b + 105 \, A a^{5} b^{2} - 15 \, {\left (3 \, B a^{2} b^{5} + 7 \, A a b^{6}\right )} x^{4} - 70 \, {\left (3 \, B a^{3} b^{4} + 7 \, A a^{2} b^{5}\right )} x^{3} - 128 \, {\left (3 \, B a^{4} b^{3} + 7 \, A a^{3} b^{4}\right )} x^{2} + 10 \, {\left (21 \, B a^{5} b^{2} - 79 \, A a^{4} b^{3}\right )} x\right )} \sqrt {x}}{3840 \, {\left (a^{5} b^{8} x^{5} + 5 \, a^{6} b^{7} x^{4} + 10 \, a^{7} b^{6} x^{3} + 10 \, a^{8} b^{5} x^{2} + 5 \, a^{9} b^{4} x + a^{10} b^{3}\right )}}, -\frac {15 \, {\left (3 \, B a^{6} + 7 \, A a^{5} b + {\left (3 \, B a b^{5} + 7 \, A b^{6}\right )} x^{5} + 5 \, {\left (3 \, B a^{2} b^{4} + 7 \, A a b^{5}\right )} x^{4} + 10 \, {\left (3 \, B a^{3} b^{3} + 7 \, A a^{2} b^{4}\right )} x^{3} + 10 \, {\left (3 \, B a^{4} b^{2} + 7 \, A a^{3} b^{3}\right )} x^{2} + 5 \, {\left (3 \, B a^{5} b + 7 \, A a^{4} b^{2}\right )} x\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b}}{b \sqrt {x}}\right ) + {\left (45 \, B a^{6} b + 105 \, A a^{5} b^{2} - 15 \, {\left (3 \, B a^{2} b^{5} + 7 \, A a b^{6}\right )} x^{4} - 70 \, {\left (3 \, B a^{3} b^{4} + 7 \, A a^{2} b^{5}\right )} x^{3} - 128 \, {\left (3 \, B a^{4} b^{3} + 7 \, A a^{3} b^{4}\right )} x^{2} + 10 \, {\left (21 \, B a^{5} b^{2} - 79 \, A a^{4} b^{3}\right )} x\right )} \sqrt {x}}{1920 \, {\left (a^{5} b^{8} x^{5} + 5 \, a^{6} b^{7} x^{4} + 10 \, a^{7} b^{6} x^{3} + 10 \, a^{8} b^{5} x^{2} + 5 \, a^{9} b^{4} x + a^{10} b^{3}\right )}}\right ] \]
[-1/3840*(15*(3*B*a^6 + 7*A*a^5*b + (3*B*a*b^5 + 7*A*b^6)*x^5 + 5*(3*B*a^2 *b^4 + 7*A*a*b^5)*x^4 + 10*(3*B*a^3*b^3 + 7*A*a^2*b^4)*x^3 + 10*(3*B*a^4*b ^2 + 7*A*a^3*b^3)*x^2 + 5*(3*B*a^5*b + 7*A*a^4*b^2)*x)*sqrt(-a*b)*log((b*x - a - 2*sqrt(-a*b)*sqrt(x))/(b*x + a)) + 2*(45*B*a^6*b + 105*A*a^5*b^2 - 15*(3*B*a^2*b^5 + 7*A*a*b^6)*x^4 - 70*(3*B*a^3*b^4 + 7*A*a^2*b^5)*x^3 - 12 8*(3*B*a^4*b^3 + 7*A*a^3*b^4)*x^2 + 10*(21*B*a^5*b^2 - 79*A*a^4*b^3)*x)*sq rt(x))/(a^5*b^8*x^5 + 5*a^6*b^7*x^4 + 10*a^7*b^6*x^3 + 10*a^8*b^5*x^2 + 5* a^9*b^4*x + a^10*b^3), -1/1920*(15*(3*B*a^6 + 7*A*a^5*b + (3*B*a*b^5 + 7*A *b^6)*x^5 + 5*(3*B*a^2*b^4 + 7*A*a*b^5)*x^4 + 10*(3*B*a^3*b^3 + 7*A*a^2*b^ 4)*x^3 + 10*(3*B*a^4*b^2 + 7*A*a^3*b^3)*x^2 + 5*(3*B*a^5*b + 7*A*a^4*b^2)* x)*sqrt(a*b)*arctan(sqrt(a*b)/(b*sqrt(x))) + (45*B*a^6*b + 105*A*a^5*b^2 - 15*(3*B*a^2*b^5 + 7*A*a*b^6)*x^4 - 70*(3*B*a^3*b^4 + 7*A*a^2*b^5)*x^3 - 1 28*(3*B*a^4*b^3 + 7*A*a^3*b^4)*x^2 + 10*(21*B*a^5*b^2 - 79*A*a^4*b^3)*x)*s qrt(x))/(a^5*b^8*x^5 + 5*a^6*b^7*x^4 + 10*a^7*b^6*x^3 + 10*a^8*b^5*x^2 + 5 *a^9*b^4*x + a^10*b^3)]
Leaf count of result is larger than twice the leaf count of optimal. 4933 vs. \(2 (187) = 374\).
Time = 140.64 (sec) , antiderivative size = 4933, normalized size of antiderivative = 25.30 \[ \int \frac {\sqrt {x} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {Too large to display} \]
Piecewise((zoo*(-2*A/(9*x**(9/2)) - 2*B/(7*x**(7/2))), Eq(a, 0) & Eq(b, 0) ), ((2*A*x**(3/2)/3 + 2*B*x**(5/2)/5)/a**6, Eq(b, 0)), ((-2*A/(9*x**(9/2)) - 2*B/(7*x**(7/2)))/b**6, Eq(a, 0)), (105*A*a**5*b*log(sqrt(x) - sqrt(-a/ b))/(3840*a**9*b**3*sqrt(-a/b) + 19200*a**8*b**4*x*sqrt(-a/b) + 38400*a**7 *b**5*x**2*sqrt(-a/b) + 38400*a**6*b**6*x**3*sqrt(-a/b) + 19200*a**5*b**7* x**4*sqrt(-a/b) + 3840*a**4*b**8*x**5*sqrt(-a/b)) - 105*A*a**5*b*log(sqrt( x) + sqrt(-a/b))/(3840*a**9*b**3*sqrt(-a/b) + 19200*a**8*b**4*x*sqrt(-a/b) + 38400*a**7*b**5*x**2*sqrt(-a/b) + 38400*a**6*b**6*x**3*sqrt(-a/b) + 192 00*a**5*b**7*x**4*sqrt(-a/b) + 3840*a**4*b**8*x**5*sqrt(-a/b)) - 210*A*a** 4*b**2*sqrt(x)*sqrt(-a/b)/(3840*a**9*b**3*sqrt(-a/b) + 19200*a**8*b**4*x*s qrt(-a/b) + 38400*a**7*b**5*x**2*sqrt(-a/b) + 38400*a**6*b**6*x**3*sqrt(-a /b) + 19200*a**5*b**7*x**4*sqrt(-a/b) + 3840*a**4*b**8*x**5*sqrt(-a/b)) + 525*A*a**4*b**2*x*log(sqrt(x) - sqrt(-a/b))/(3840*a**9*b**3*sqrt(-a/b) + 1 9200*a**8*b**4*x*sqrt(-a/b) + 38400*a**7*b**5*x**2*sqrt(-a/b) + 38400*a**6 *b**6*x**3*sqrt(-a/b) + 19200*a**5*b**7*x**4*sqrt(-a/b) + 3840*a**4*b**8*x **5*sqrt(-a/b)) - 525*A*a**4*b**2*x*log(sqrt(x) + sqrt(-a/b))/(3840*a**9*b **3*sqrt(-a/b) + 19200*a**8*b**4*x*sqrt(-a/b) + 38400*a**7*b**5*x**2*sqrt( -a/b) + 38400*a**6*b**6*x**3*sqrt(-a/b) + 19200*a**5*b**7*x**4*sqrt(-a/b) + 3840*a**4*b**8*x**5*sqrt(-a/b)) + 1580*A*a**3*b**3*x**(3/2)*sqrt(-a/b)/( 3840*a**9*b**3*sqrt(-a/b) + 19200*a**8*b**4*x*sqrt(-a/b) + 38400*a**7*b...
Time = 0.29 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.05 \[ \int \frac {\sqrt {x} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {15 \, {\left (3 \, B a b^{4} + 7 \, A b^{5}\right )} x^{\frac {9}{2}} + 70 \, {\left (3 \, B a^{2} b^{3} + 7 \, A a b^{4}\right )} x^{\frac {7}{2}} + 128 \, {\left (3 \, B a^{3} b^{2} + 7 \, A a^{2} b^{3}\right )} x^{\frac {5}{2}} - 10 \, {\left (21 \, B a^{4} b - 79 \, A a^{3} b^{2}\right )} x^{\frac {3}{2}} - 15 \, {\left (3 \, B a^{5} + 7 \, A a^{4} b\right )} \sqrt {x}}{1920 \, {\left (a^{4} b^{7} x^{5} + 5 \, a^{5} b^{6} x^{4} + 10 \, a^{6} b^{5} x^{3} + 10 \, a^{7} b^{4} x^{2} + 5 \, a^{8} b^{3} x + a^{9} b^{2}\right )}} + \frac {{\left (3 \, B a + 7 \, A b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{128 \, \sqrt {a b} a^{4} b^{2}} \]
1/1920*(15*(3*B*a*b^4 + 7*A*b^5)*x^(9/2) + 70*(3*B*a^2*b^3 + 7*A*a*b^4)*x^ (7/2) + 128*(3*B*a^3*b^2 + 7*A*a^2*b^3)*x^(5/2) - 10*(21*B*a^4*b - 79*A*a^ 3*b^2)*x^(3/2) - 15*(3*B*a^5 + 7*A*a^4*b)*sqrt(x))/(a^4*b^7*x^5 + 5*a^5*b^ 6*x^4 + 10*a^6*b^5*x^3 + 10*a^7*b^4*x^2 + 5*a^8*b^3*x + a^9*b^2) + 1/128*( 3*B*a + 7*A*b)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a^4*b^2)
Time = 0.28 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.80 \[ \int \frac {\sqrt {x} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {{\left (3 \, B a + 7 \, A b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{128 \, \sqrt {a b} a^{4} b^{2}} + \frac {45 \, B a b^{4} x^{\frac {9}{2}} + 105 \, A b^{5} x^{\frac {9}{2}} + 210 \, B a^{2} b^{3} x^{\frac {7}{2}} + 490 \, A a b^{4} x^{\frac {7}{2}} + 384 \, B a^{3} b^{2} x^{\frac {5}{2}} + 896 \, A a^{2} b^{3} x^{\frac {5}{2}} - 210 \, B a^{4} b x^{\frac {3}{2}} + 790 \, A a^{3} b^{2} x^{\frac {3}{2}} - 45 \, B a^{5} \sqrt {x} - 105 \, A a^{4} b \sqrt {x}}{1920 \, {\left (b x + a\right )}^{5} a^{4} b^{2}} \]
1/128*(3*B*a + 7*A*b)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a^4*b^2) + 1/ 1920*(45*B*a*b^4*x^(9/2) + 105*A*b^5*x^(9/2) + 210*B*a^2*b^3*x^(7/2) + 490 *A*a*b^4*x^(7/2) + 384*B*a^3*b^2*x^(5/2) + 896*A*a^2*b^3*x^(5/2) - 210*B*a ^4*b*x^(3/2) + 790*A*a^3*b^2*x^(3/2) - 45*B*a^5*sqrt(x) - 105*A*a^4*b*sqrt (x))/((b*x + a)^5*a^4*b^2)
Time = 10.16 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.89 \[ \int \frac {\sqrt {x} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {\frac {x^{5/2}\,\left (7\,A\,b+3\,B\,a\right )}{15\,a^2}-\frac {\sqrt {x}\,\left (7\,A\,b+3\,B\,a\right )}{128\,b^2}+\frac {b^2\,x^{9/2}\,\left (7\,A\,b+3\,B\,a\right )}{128\,a^4}+\frac {x^{3/2}\,\left (79\,A\,b-21\,B\,a\right )}{192\,a\,b}+\frac {7\,b\,x^{7/2}\,\left (7\,A\,b+3\,B\,a\right )}{192\,a^3}}{a^5+5\,a^4\,b\,x+10\,a^3\,b^2\,x^2+10\,a^2\,b^3\,x^3+5\,a\,b^4\,x^4+b^5\,x^5}+\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )\,\left (7\,A\,b+3\,B\,a\right )}{128\,a^{9/2}\,b^{5/2}} \]
((x^(5/2)*(7*A*b + 3*B*a))/(15*a^2) - (x^(1/2)*(7*A*b + 3*B*a))/(128*b^2) + (b^2*x^(9/2)*(7*A*b + 3*B*a))/(128*a^4) + (x^(3/2)*(79*A*b - 21*B*a))/(1 92*a*b) + (7*b*x^(7/2)*(7*A*b + 3*B*a))/(192*a^3))/(a^5 + b^5*x^5 + 5*a*b^ 4*x^4 + 10*a^3*b^2*x^2 + 10*a^2*b^3*x^3 + 5*a^4*b*x) + (atan((b^(1/2)*x^(1 /2))/a^(1/2))*(7*A*b + 3*B*a))/(128*a^(9/2)*b^(5/2))