Integrand size = 18, antiderivative size = 154 \[ \int \frac {x^{7/2} (A+B x)}{(a+b x)^2} \, dx=\frac {a^2 (7 A b-9 a B) \sqrt {x}}{b^5}-\frac {a (7 A b-9 a B) x^{3/2}}{3 b^4}+\frac {(7 A b-9 a B) x^{5/2}}{5 b^3}-\frac {(7 A b-9 a B) x^{7/2}}{7 a b^2}+\frac {(A b-a B) x^{9/2}}{a b (a+b x)}-\frac {a^{5/2} (7 A b-9 a B) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{11/2}} \]
-1/3*a*(7*A*b-9*B*a)*x^(3/2)/b^4+1/5*(7*A*b-9*B*a)*x^(5/2)/b^3-1/7*(7*A*b- 9*B*a)*x^(7/2)/a/b^2+(A*b-B*a)*x^(9/2)/a/b/(b*x+a)-a^(5/2)*(7*A*b-9*B*a)*a rctan(b^(1/2)*x^(1/2)/a^(1/2))/b^(11/2)+a^2*(7*A*b-9*B*a)*x^(1/2)/b^5
Time = 0.17 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.83 \[ \int \frac {x^{7/2} (A+B x)}{(a+b x)^2} \, dx=\frac {\sqrt {x} \left (-945 a^4 B+105 a^3 b (7 A-6 B x)+6 b^4 x^3 (7 A+5 B x)+14 a^2 b^2 x (35 A+9 B x)-2 a b^3 x^2 (49 A+27 B x)\right )}{105 b^5 (a+b x)}+\frac {a^{5/2} (-7 A b+9 a B) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{11/2}} \]
(Sqrt[x]*(-945*a^4*B + 105*a^3*b*(7*A - 6*B*x) + 6*b^4*x^3*(7*A + 5*B*x) + 14*a^2*b^2*x*(35*A + 9*B*x) - 2*a*b^3*x^2*(49*A + 27*B*x)))/(105*b^5*(a + b*x)) + (a^(5/2)*(-7*A*b + 9*a*B)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/b^(1 1/2)
Time = 0.22 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.94, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {87, 60, 60, 60, 60, 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.
| \(\displaystyle \int \frac {x^{7/2} (A+B x)}{(a+b x)^2} \, dx\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {x^{9/2} (A b-a B)}{a b (a+b x)}-\frac {(7 A b-9 a B) \int \frac {x^{7/2}}{a+b x}dx}{2 a b}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {x^{9/2} (A b-a B)}{a b (a+b x)}-\frac {(7 A b-9 a B) \left (\frac {2 x^{7/2}}{7 b}-\frac {a \int \frac {x^{5/2}}{a+b x}dx}{b}\right )}{2 a b}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {x^{9/2} (A b-a B)}{a b (a+b x)}-\frac {(7 A b-9 a B) \left (\frac {2 x^{7/2}}{7 b}-\frac {a \left (\frac {2 x^{5/2}}{5 b}-\frac {a \int \frac {x^{3/2}}{a+b x}dx}{b}\right )}{b}\right )}{2 a b}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {x^{9/2} (A b-a B)}{a b (a+b x)}-\frac {(7 A b-9 a B) \left (\frac {2 x^{7/2}}{7 b}-\frac {a \left (\frac {2 x^{5/2}}{5 b}-\frac {a \left (\frac {2 x^{3/2}}{3 b}-\frac {a \int \frac {\sqrt {x}}{a+b x}dx}{b}\right )}{b}\right )}{b}\right )}{2 a b}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {x^{9/2} (A b-a B)}{a b (a+b x)}-\frac {(7 A b-9 a B) \left (\frac {2 x^{7/2}}{7 b}-\frac {a \left (\frac {2 x^{5/2}}{5 b}-\frac {a \left (\frac {2 x^{3/2}}{3 b}-\frac {a \left (\frac {2 \sqrt {x}}{b}-\frac {a \int \frac {1}{\sqrt {x} (a+b x)}dx}{b}\right )}{b}\right )}{b}\right )}{b}\right )}{2 a b}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {x^{9/2} (A b-a B)}{a b (a+b x)}-\frac {(7 A b-9 a B) \left (\frac {2 x^{7/2}}{7 b}-\frac {a \left (\frac {2 x^{5/2}}{5 b}-\frac {a \left (\frac {2 x^{3/2}}{3 b}-\frac {a \left (\frac {2 \sqrt {x}}{b}-\frac {2 a \int \frac {1}{a+b x}d\sqrt {x}}{b}\right )}{b}\right )}{b}\right )}{b}\right )}{2 a b}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {x^{9/2} (A b-a B)}{a b (a+b x)}-\frac {(7 A b-9 a B) \left (\frac {2 x^{7/2}}{7 b}-\frac {a \left (\frac {2 x^{5/2}}{5 b}-\frac {a \left (\frac {2 x^{3/2}}{3 b}-\frac {a \left (\frac {2 \sqrt {x}}{b}-\frac {2 \sqrt {a} \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{3/2}}\right )}{b}\right )}{b}\right )}{b}\right )}{2 a b}\) |
((A*b - a*B)*x^(9/2))/(a*b*(a + b*x)) - ((7*A*b - 9*a*B)*((2*x^(7/2))/(7*b ) - (a*((2*x^(5/2))/(5*b) - (a*((2*x^(3/2))/(3*b) - (a*((2*Sqrt[x])/b - (2 *Sqrt[a]*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/b^(3/2)))/b))/b))/b))/(2*a*b)
3.4.54.3.1 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/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[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]
Time = 1.32 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.81
| method | result | size |
| risch | \(\frac {2 \left (15 b^{3} B \,x^{3}+21 A \,b^{3} x^{2}-42 B a \,b^{2} x^{2}-70 a \,b^{2} A x +105 a^{2} b B x +315 a^{2} b A -420 a^{3} B \right ) \sqrt {x}}{105 b^{5}}-\frac {a^{3} \left (\frac {2 \left (-\frac {A b}{2}+\frac {B a}{2}\right ) \sqrt {x}}{b x +a}+\frac {\left (7 A b -9 B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b}}\right )}{b^{5}}\) | \(124\) |
| derivativedivides | \(\frac {\frac {2 b^{3} B \,x^{\frac {7}{2}}}{7}+\frac {2 A \,b^{3} x^{\frac {5}{2}}}{5}-\frac {4 B a \,b^{2} x^{\frac {5}{2}}}{5}-\frac {4 A a \,b^{2} x^{\frac {3}{2}}}{3}+2 B \,a^{2} b \,x^{\frac {3}{2}}+6 a^{2} b A \sqrt {x}-8 a^{3} B \sqrt {x}}{b^{5}}-\frac {2 a^{3} \left (\frac {\left (-\frac {A b}{2}+\frac {B a}{2}\right ) \sqrt {x}}{b x +a}+\frac {\left (7 A b -9 B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{b^{5}}\) | \(130\) |
| default | \(\frac {\frac {2 b^{3} B \,x^{\frac {7}{2}}}{7}+\frac {2 A \,b^{3} x^{\frac {5}{2}}}{5}-\frac {4 B a \,b^{2} x^{\frac {5}{2}}}{5}-\frac {4 A a \,b^{2} x^{\frac {3}{2}}}{3}+2 B \,a^{2} b \,x^{\frac {3}{2}}+6 a^{2} b A \sqrt {x}-8 a^{3} B \sqrt {x}}{b^{5}}-\frac {2 a^{3} \left (\frac {\left (-\frac {A b}{2}+\frac {B a}{2}\right ) \sqrt {x}}{b x +a}+\frac {\left (7 A b -9 B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{b^{5}}\) | \(130\) |
2/105*(15*B*b^3*x^3+21*A*b^3*x^2-42*B*a*b^2*x^2-70*A*a*b^2*x+105*B*a^2*b*x +315*A*a^2*b-420*B*a^3)*x^(1/2)/b^5-a^3/b^5*(2*(-1/2*A*b+1/2*B*a)*x^(1/2)/ (b*x+a)+(7*A*b-9*B*a)/(a*b)^(1/2)*arctan(b*x^(1/2)/(a*b)^(1/2)))
Time = 0.25 (sec) , antiderivative size = 341, normalized size of antiderivative = 2.21 \[ \int \frac {x^{7/2} (A+B x)}{(a+b x)^2} \, dx=\left [-\frac {105 \, {\left (9 \, B a^{4} - 7 \, A a^{3} b + {\left (9 \, B a^{3} b - 7 \, A a^{2} b^{2}\right )} x\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x - 2 \, b \sqrt {x} \sqrt {-\frac {a}{b}} - a}{b x + a}\right ) - 2 \, {\left (30 \, B b^{4} x^{4} - 945 \, B a^{4} + 735 \, A a^{3} b - 6 \, {\left (9 \, B a b^{3} - 7 \, A b^{4}\right )} x^{3} + 14 \, {\left (9 \, B a^{2} b^{2} - 7 \, A a b^{3}\right )} x^{2} - 70 \, {\left (9 \, B a^{3} b - 7 \, A a^{2} b^{2}\right )} x\right )} \sqrt {x}}{210 \, {\left (b^{6} x + a b^{5}\right )}}, \frac {105 \, {\left (9 \, B a^{4} - 7 \, A a^{3} b + {\left (9 \, B a^{3} b - 7 \, A a^{2} b^{2}\right )} x\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b \sqrt {x} \sqrt {\frac {a}{b}}}{a}\right ) + {\left (30 \, B b^{4} x^{4} - 945 \, B a^{4} + 735 \, A a^{3} b - 6 \, {\left (9 \, B a b^{3} - 7 \, A b^{4}\right )} x^{3} + 14 \, {\left (9 \, B a^{2} b^{2} - 7 \, A a b^{3}\right )} x^{2} - 70 \, {\left (9 \, B a^{3} b - 7 \, A a^{2} b^{2}\right )} x\right )} \sqrt {x}}{105 \, {\left (b^{6} x + a b^{5}\right )}}\right ] \]
[-1/210*(105*(9*B*a^4 - 7*A*a^3*b + (9*B*a^3*b - 7*A*a^2*b^2)*x)*sqrt(-a/b )*log((b*x - 2*b*sqrt(x)*sqrt(-a/b) - a)/(b*x + a)) - 2*(30*B*b^4*x^4 - 94 5*B*a^4 + 735*A*a^3*b - 6*(9*B*a*b^3 - 7*A*b^4)*x^3 + 14*(9*B*a^2*b^2 - 7* A*a*b^3)*x^2 - 70*(9*B*a^3*b - 7*A*a^2*b^2)*x)*sqrt(x))/(b^6*x + a*b^5), 1 /105*(105*(9*B*a^4 - 7*A*a^3*b + (9*B*a^3*b - 7*A*a^2*b^2)*x)*sqrt(a/b)*ar ctan(b*sqrt(x)*sqrt(a/b)/a) + (30*B*b^4*x^4 - 945*B*a^4 + 735*A*a^3*b - 6* (9*B*a*b^3 - 7*A*b^4)*x^3 + 14*(9*B*a^2*b^2 - 7*A*a*b^3)*x^2 - 70*(9*B*a^3 *b - 7*A*a^2*b^2)*x)*sqrt(x))/(b^6*x + a*b^5)]
Leaf count of result is larger than twice the leaf count of optimal. 986 vs. \(2 (143) = 286\).
Time = 37.95 (sec) , antiderivative size = 986, normalized size of antiderivative = 6.40 \[ \int \frac {x^{7/2} (A+B x)}{(a+b x)^2} \, dx=\begin {cases} \tilde {\infty } \left (\frac {2 A x^{\frac {5}{2}}}{5} + \frac {2 B x^{\frac {7}{2}}}{7}\right ) & \text {for}\: a = 0 \wedge b = 0 \\\frac {\frac {2 A x^{\frac {9}{2}}}{9} + \frac {2 B x^{\frac {11}{2}}}{11}}{a^{2}} & \text {for}\: b = 0 \\\frac {\frac {2 A x^{\frac {5}{2}}}{5} + \frac {2 B x^{\frac {7}{2}}}{7}}{b^{2}} & \text {for}\: a = 0 \\- \frac {735 A a^{4} b \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{210 a b^{6} \sqrt {- \frac {a}{b}} + 210 b^{7} x \sqrt {- \frac {a}{b}}} + \frac {735 A a^{4} b \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{210 a b^{6} \sqrt {- \frac {a}{b}} + 210 b^{7} x \sqrt {- \frac {a}{b}}} + \frac {1470 A a^{3} b^{2} \sqrt {x} \sqrt {- \frac {a}{b}}}{210 a b^{6} \sqrt {- \frac {a}{b}} + 210 b^{7} x \sqrt {- \frac {a}{b}}} - \frac {735 A a^{3} b^{2} x \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{210 a b^{6} \sqrt {- \frac {a}{b}} + 210 b^{7} x \sqrt {- \frac {a}{b}}} + \frac {735 A a^{3} b^{2} x \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{210 a b^{6} \sqrt {- \frac {a}{b}} + 210 b^{7} x \sqrt {- \frac {a}{b}}} + \frac {980 A a^{2} b^{3} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}}}{210 a b^{6} \sqrt {- \frac {a}{b}} + 210 b^{7} x \sqrt {- \frac {a}{b}}} - \frac {196 A a b^{4} x^{\frac {5}{2}} \sqrt {- \frac {a}{b}}}{210 a b^{6} \sqrt {- \frac {a}{b}} + 210 b^{7} x \sqrt {- \frac {a}{b}}} + \frac {84 A b^{5} x^{\frac {7}{2}} \sqrt {- \frac {a}{b}}}{210 a b^{6} \sqrt {- \frac {a}{b}} + 210 b^{7} x \sqrt {- \frac {a}{b}}} + \frac {945 B a^{5} \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{210 a b^{6} \sqrt {- \frac {a}{b}} + 210 b^{7} x \sqrt {- \frac {a}{b}}} - \frac {945 B a^{5} \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{210 a b^{6} \sqrt {- \frac {a}{b}} + 210 b^{7} x \sqrt {- \frac {a}{b}}} - \frac {1890 B a^{4} b \sqrt {x} \sqrt {- \frac {a}{b}}}{210 a b^{6} \sqrt {- \frac {a}{b}} + 210 b^{7} x \sqrt {- \frac {a}{b}}} + \frac {945 B a^{4} b x \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{210 a b^{6} \sqrt {- \frac {a}{b}} + 210 b^{7} x \sqrt {- \frac {a}{b}}} - \frac {945 B a^{4} b x \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{210 a b^{6} \sqrt {- \frac {a}{b}} + 210 b^{7} x \sqrt {- \frac {a}{b}}} - \frac {1260 B a^{3} b^{2} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}}}{210 a b^{6} \sqrt {- \frac {a}{b}} + 210 b^{7} x \sqrt {- \frac {a}{b}}} + \frac {252 B a^{2} b^{3} x^{\frac {5}{2}} \sqrt {- \frac {a}{b}}}{210 a b^{6} \sqrt {- \frac {a}{b}} + 210 b^{7} x \sqrt {- \frac {a}{b}}} - \frac {108 B a b^{4} x^{\frac {7}{2}} \sqrt {- \frac {a}{b}}}{210 a b^{6} \sqrt {- \frac {a}{b}} + 210 b^{7} x \sqrt {- \frac {a}{b}}} + \frac {60 B b^{5} x^{\frac {9}{2}} \sqrt {- \frac {a}{b}}}{210 a b^{6} \sqrt {- \frac {a}{b}} + 210 b^{7} x \sqrt {- \frac {a}{b}}} & \text {otherwise} \end {cases} \]
Piecewise((zoo*(2*A*x**(5/2)/5 + 2*B*x**(7/2)/7), Eq(a, 0) & Eq(b, 0)), (( 2*A*x**(9/2)/9 + 2*B*x**(11/2)/11)/a**2, Eq(b, 0)), ((2*A*x**(5/2)/5 + 2*B *x**(7/2)/7)/b**2, Eq(a, 0)), (-735*A*a**4*b*log(sqrt(x) - sqrt(-a/b))/(21 0*a*b**6*sqrt(-a/b) + 210*b**7*x*sqrt(-a/b)) + 735*A*a**4*b*log(sqrt(x) + sqrt(-a/b))/(210*a*b**6*sqrt(-a/b) + 210*b**7*x*sqrt(-a/b)) + 1470*A*a**3* b**2*sqrt(x)*sqrt(-a/b)/(210*a*b**6*sqrt(-a/b) + 210*b**7*x*sqrt(-a/b)) - 735*A*a**3*b**2*x*log(sqrt(x) - sqrt(-a/b))/(210*a*b**6*sqrt(-a/b) + 210*b **7*x*sqrt(-a/b)) + 735*A*a**3*b**2*x*log(sqrt(x) + sqrt(-a/b))/(210*a*b** 6*sqrt(-a/b) + 210*b**7*x*sqrt(-a/b)) + 980*A*a**2*b**3*x**(3/2)*sqrt(-a/b )/(210*a*b**6*sqrt(-a/b) + 210*b**7*x*sqrt(-a/b)) - 196*A*a*b**4*x**(5/2)* sqrt(-a/b)/(210*a*b**6*sqrt(-a/b) + 210*b**7*x*sqrt(-a/b)) + 84*A*b**5*x** (7/2)*sqrt(-a/b)/(210*a*b**6*sqrt(-a/b) + 210*b**7*x*sqrt(-a/b)) + 945*B*a **5*log(sqrt(x) - sqrt(-a/b))/(210*a*b**6*sqrt(-a/b) + 210*b**7*x*sqrt(-a/ b)) - 945*B*a**5*log(sqrt(x) + sqrt(-a/b))/(210*a*b**6*sqrt(-a/b) + 210*b* *7*x*sqrt(-a/b)) - 1890*B*a**4*b*sqrt(x)*sqrt(-a/b)/(210*a*b**6*sqrt(-a/b) + 210*b**7*x*sqrt(-a/b)) + 945*B*a**4*b*x*log(sqrt(x) - sqrt(-a/b))/(210* a*b**6*sqrt(-a/b) + 210*b**7*x*sqrt(-a/b)) - 945*B*a**4*b*x*log(sqrt(x) + sqrt(-a/b))/(210*a*b**6*sqrt(-a/b) + 210*b**7*x*sqrt(-a/b)) - 1260*B*a**3* b**2*x**(3/2)*sqrt(-a/b)/(210*a*b**6*sqrt(-a/b) + 210*b**7*x*sqrt(-a/b)) + 252*B*a**2*b**3*x**(5/2)*sqrt(-a/b)/(210*a*b**6*sqrt(-a/b) + 210*b**7*...
Time = 0.29 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.90 \[ \int \frac {x^{7/2} (A+B x)}{(a+b x)^2} \, dx=-\frac {{\left (B a^{4} - A a^{3} b\right )} \sqrt {x}}{b^{6} x + a b^{5}} + \frac {{\left (9 \, B a^{4} - 7 \, A a^{3} b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} b^{5}} + \frac {2 \, {\left (15 \, B b^{3} x^{\frac {7}{2}} - 21 \, {\left (2 \, B a b^{2} - A b^{3}\right )} x^{\frac {5}{2}} + 35 \, {\left (3 \, B a^{2} b - 2 \, A a b^{2}\right )} x^{\frac {3}{2}} - 105 \, {\left (4 \, B a^{3} - 3 \, A a^{2} b\right )} \sqrt {x}\right )}}{105 \, b^{5}} \]
-(B*a^4 - A*a^3*b)*sqrt(x)/(b^6*x + a*b^5) + (9*B*a^4 - 7*A*a^3*b)*arctan( b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*b^5) + 2/105*(15*B*b^3*x^(7/2) - 21*(2*B*a *b^2 - A*b^3)*x^(5/2) + 35*(3*B*a^2*b - 2*A*a*b^2)*x^(3/2) - 105*(4*B*a^3 - 3*A*a^2*b)*sqrt(x))/b^5
Time = 0.29 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.95 \[ \int \frac {x^{7/2} (A+B x)}{(a+b x)^2} \, dx=\frac {{\left (9 \, B a^{4} - 7 \, A a^{3} b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} b^{5}} - \frac {B a^{4} \sqrt {x} - A a^{3} b \sqrt {x}}{{\left (b x + a\right )} b^{5}} + \frac {2 \, {\left (15 \, B b^{12} x^{\frac {7}{2}} - 42 \, B a b^{11} x^{\frac {5}{2}} + 21 \, A b^{12} x^{\frac {5}{2}} + 105 \, B a^{2} b^{10} x^{\frac {3}{2}} - 70 \, A a b^{11} x^{\frac {3}{2}} - 420 \, B a^{3} b^{9} \sqrt {x} + 315 \, A a^{2} b^{10} \sqrt {x}\right )}}{105 \, b^{14}} \]
(9*B*a^4 - 7*A*a^3*b)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*b^5) - (B*a^4 *sqrt(x) - A*a^3*b*sqrt(x))/((b*x + a)*b^5) + 2/105*(15*B*b^12*x^(7/2) - 4 2*B*a*b^11*x^(5/2) + 21*A*b^12*x^(5/2) + 105*B*a^2*b^10*x^(3/2) - 70*A*a*b ^11*x^(3/2) - 420*B*a^3*b^9*sqrt(x) + 315*A*a^2*b^10*sqrt(x))/b^14
Time = 0.40 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.36 \[ \int \frac {x^{7/2} (A+B x)}{(a+b x)^2} \, dx=\sqrt {x}\,\left (\frac {2\,a\,\left (\frac {2\,a\,\left (\frac {2\,A}{b^2}-\frac {4\,B\,a}{b^3}\right )}{b}+\frac {2\,B\,a^2}{b^4}\right )}{b}-\frac {a^2\,\left (\frac {2\,A}{b^2}-\frac {4\,B\,a}{b^3}\right )}{b^2}\right )+x^{5/2}\,\left (\frac {2\,A}{5\,b^2}-\frac {4\,B\,a}{5\,b^3}\right )-x^{3/2}\,\left (\frac {2\,a\,\left (\frac {2\,A}{b^2}-\frac {4\,B\,a}{b^3}\right )}{3\,b}+\frac {2\,B\,a^2}{3\,b^4}\right )+\frac {2\,B\,x^{7/2}}{7\,b^2}-\frac {\sqrt {x}\,\left (B\,a^4-A\,a^3\,b\right )}{x\,b^6+a\,b^5}+\frac {a^{5/2}\,\mathrm {atan}\left (\frac {a^{5/2}\,\sqrt {b}\,\sqrt {x}\,\left (7\,A\,b-9\,B\,a\right )}{9\,B\,a^4-7\,A\,a^3\,b}\right )\,\left (7\,A\,b-9\,B\,a\right )}{b^{11/2}} \]
x^(1/2)*((2*a*((2*a*((2*A)/b^2 - (4*B*a)/b^3))/b + (2*B*a^2)/b^4))/b - (a^ 2*((2*A)/b^2 - (4*B*a)/b^3))/b^2) + x^(5/2)*((2*A)/(5*b^2) - (4*B*a)/(5*b^ 3)) - x^(3/2)*((2*a*((2*A)/b^2 - (4*B*a)/b^3))/(3*b) + (2*B*a^2)/(3*b^4)) + (2*B*x^(7/2))/(7*b^2) - (x^(1/2)*(B*a^4 - A*a^3*b))/(a*b^5 + b^6*x) + (a ^(5/2)*atan((a^(5/2)*b^(1/2)*x^(1/2)*(7*A*b - 9*B*a))/(9*B*a^4 - 7*A*a^3*b ))*(7*A*b - 9*B*a))/b^(11/2)