Integrand size = 18, antiderivative size = 208 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{x^7} \, dx=\frac {b^2 (5 A b-12 a B) \sqrt {a+b x}}{192 a x^3}+\frac {b^3 (5 A b-12 a B) \sqrt {a+b x}}{768 a^2 x^2}-\frac {b^4 (5 A b-12 a B) \sqrt {a+b x}}{512 a^3 x}+\frac {b (5 A b-12 a B) (a+b x)^{3/2}}{96 a x^4}+\frac {(5 A b-12 a B) (a+b x)^{5/2}}{60 a x^5}-\frac {A (a+b x)^{7/2}}{6 a x^6}+\frac {b^5 (5 A b-12 a B) \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{512 a^{7/2}} \]
1/96*b*(5*A*b-12*B*a)*(b*x+a)^(3/2)/a/x^4+1/60*(5*A*b-12*B*a)*(b*x+a)^(5/2 )/a/x^5-1/6*A*(b*x+a)^(7/2)/a/x^6+1/512*b^5*(5*A*b-12*B*a)*arctanh((b*x+a) ^(1/2)/a^(1/2))/a^(7/2)+1/192*b^2*(5*A*b-12*B*a)*(b*x+a)^(1/2)/a/x^3+1/768 *b^3*(5*A*b-12*B*a)*(b*x+a)^(1/2)/a^2/x^2-1/512*b^4*(5*A*b-12*B*a)*(b*x+a) ^(1/2)/a^3/x
Time = 0.31 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.71 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{x^7} \, dx=-\frac {\sqrt {a+b x} \left (75 A b^5 x^5+40 a^2 b^3 x^3 (A+3 B x)+256 a^5 (5 A+6 B x)-10 a b^4 x^4 (5 A+18 B x)+48 a^3 b^2 x^2 (45 A+62 B x)+64 a^4 b x (50 A+63 B x)\right )}{7680 a^3 x^6}+\frac {b^5 (5 A b-12 a B) \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{512 a^{7/2}} \]
-1/7680*(Sqrt[a + b*x]*(75*A*b^5*x^5 + 40*a^2*b^3*x^3*(A + 3*B*x) + 256*a^ 5*(5*A + 6*B*x) - 10*a*b^4*x^4*(5*A + 18*B*x) + 48*a^3*b^2*x^2*(45*A + 62* B*x) + 64*a^4*b*x*(50*A + 63*B*x)))/(a^3*x^6) + (b^5*(5*A*b - 12*a*B)*ArcT anh[Sqrt[a + b*x]/Sqrt[a]])/(512*a^(7/2))
Time = 0.24 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.83, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {87, 51, 51, 51, 52, 52, 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.
| \(\displaystyle \int \frac {(a+b x)^{5/2} (A+B x)}{x^7} \, dx\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle -\frac {(5 A b-12 a B) \int \frac {(a+b x)^{5/2}}{x^6}dx}{12 a}-\frac {A (a+b x)^{7/2}}{6 a x^6}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle -\frac {(5 A b-12 a B) \left (\frac {1}{2} b \int \frac {(a+b x)^{3/2}}{x^5}dx-\frac {(a+b x)^{5/2}}{5 x^5}\right )}{12 a}-\frac {A (a+b x)^{7/2}}{6 a x^6}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle -\frac {(5 A b-12 a B) \left (\frac {1}{2} b \left (\frac {3}{8} b \int \frac {\sqrt {a+b x}}{x^4}dx-\frac {(a+b x)^{3/2}}{4 x^4}\right )-\frac {(a+b x)^{5/2}}{5 x^5}\right )}{12 a}-\frac {A (a+b x)^{7/2}}{6 a x^6}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle -\frac {(5 A b-12 a B) \left (\frac {1}{2} b \left (\frac {3}{8} b \left (\frac {1}{6} b \int \frac {1}{x^3 \sqrt {a+b x}}dx-\frac {\sqrt {a+b x}}{3 x^3}\right )-\frac {(a+b x)^{3/2}}{4 x^4}\right )-\frac {(a+b x)^{5/2}}{5 x^5}\right )}{12 a}-\frac {A (a+b x)^{7/2}}{6 a x^6}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle -\frac {(5 A b-12 a B) \left (\frac {1}{2} b \left (\frac {3}{8} b \left (\frac {1}{6} b \left (-\frac {3 b \int \frac {1}{x^2 \sqrt {a+b x}}dx}{4 a}-\frac {\sqrt {a+b x}}{2 a x^2}\right )-\frac {\sqrt {a+b x}}{3 x^3}\right )-\frac {(a+b x)^{3/2}}{4 x^4}\right )-\frac {(a+b x)^{5/2}}{5 x^5}\right )}{12 a}-\frac {A (a+b x)^{7/2}}{6 a x^6}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle -\frac {(5 A b-12 a B) \left (\frac {1}{2} b \left (\frac {3}{8} b \left (\frac {1}{6} b \left (-\frac {3 b \left (-\frac {b \int \frac {1}{x \sqrt {a+b x}}dx}{2 a}-\frac {\sqrt {a+b x}}{a x}\right )}{4 a}-\frac {\sqrt {a+b x}}{2 a x^2}\right )-\frac {\sqrt {a+b x}}{3 x^3}\right )-\frac {(a+b x)^{3/2}}{4 x^4}\right )-\frac {(a+b x)^{5/2}}{5 x^5}\right )}{12 a}-\frac {A (a+b x)^{7/2}}{6 a x^6}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {(5 A b-12 a B) \left (\frac {1}{2} b \left (\frac {3}{8} b \left (\frac {1}{6} b \left (-\frac {3 b \left (-\frac {\int \frac {1}{\frac {a+b x}{b}-\frac {a}{b}}d\sqrt {a+b x}}{a}-\frac {\sqrt {a+b x}}{a x}\right )}{4 a}-\frac {\sqrt {a+b x}}{2 a x^2}\right )-\frac {\sqrt {a+b x}}{3 x^3}\right )-\frac {(a+b x)^{3/2}}{4 x^4}\right )-\frac {(a+b x)^{5/2}}{5 x^5}\right )}{12 a}-\frac {A (a+b x)^{7/2}}{6 a x^6}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {(5 A b-12 a B) \left (\frac {1}{2} b \left (\frac {3}{8} b \left (\frac {1}{6} b \left (-\frac {3 b \left (\frac {b \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {\sqrt {a+b x}}{a x}\right )}{4 a}-\frac {\sqrt {a+b x}}{2 a x^2}\right )-\frac {\sqrt {a+b x}}{3 x^3}\right )-\frac {(a+b x)^{3/2}}{4 x^4}\right )-\frac {(a+b x)^{5/2}}{5 x^5}\right )}{12 a}-\frac {A (a+b x)^{7/2}}{6 a x^6}\) |
-1/6*(A*(a + b*x)^(7/2))/(a*x^6) - ((5*A*b - 12*a*B)*(-1/5*(a + b*x)^(5/2) /x^5 + (b*(-1/4*(a + b*x)^(3/2)/x^4 + (3*b*(-1/3*Sqrt[a + b*x]/x^3 + (b*(- 1/2*Sqrt[a + b*x]/(a*x^2) - (3*b*(-(Sqrt[a + b*x]/(a*x)) + (b*ArcTanh[Sqrt [a + b*x]/Sqrt[a]])/a^(3/2)))/(4*a)))/6))/8))/2))/(12*a)
3.5.21.3.1 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 + 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)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Time = 0.56 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.64
| method | result | size |
| pseudoelliptic | \(-\frac {-\frac {15 x^{6} \left (A b -\frac {12 B a}{5}\right ) b^{5} \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{8}+\sqrt {b x +a}\, \left (\left (\frac {192 B x}{5}+32 A \right ) a^{\frac {11}{2}}+x \left (-\frac {5 x^{3} \left (\frac {18 B x}{5}+A \right ) b^{3} a^{\frac {3}{2}}}{4}+b^{2} x^{2} \left (3 B x +A \right ) a^{\frac {5}{2}}+54 x \left (\frac {62 B x}{45}+A \right ) b \,a^{\frac {7}{2}}+\left (\frac {504 B x}{5}+80 A \right ) a^{\frac {9}{2}}+\frac {15 A \sqrt {a}\, b^{4} x^{4}}{8}\right ) b \right )}{192 a^{\frac {7}{2}} x^{6}}\) | \(133\) |
| risch | \(-\frac {\sqrt {b x +a}\, \left (75 A \,b^{5} x^{5}-180 B a \,b^{4} x^{5}-50 a A \,b^{4} x^{4}+120 B \,a^{2} b^{3} x^{4}+40 a^{2} A \,b^{3} x^{3}+2976 B \,a^{3} b^{2} x^{3}+2160 a^{3} A \,b^{2} x^{2}+4032 B \,a^{4} b \,x^{2}+3200 a^{4} A b x +1536 a^{5} B x +1280 a^{5} A \right )}{7680 x^{6} a^{3}}+\frac {b^{5} \left (5 A b -12 B a \right ) \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{512 a^{\frac {7}{2}}}\) | \(155\) |
| derivativedivides | \(2 b^{5} \left (-\frac {\frac {\left (5 A b -12 B a \right ) \left (b x +a \right )^{\frac {11}{2}}}{1024 a^{3}}-\frac {17 \left (5 A b -12 B a \right ) \left (b x +a \right )^{\frac {9}{2}}}{3072 a^{2}}+\frac {\left (165 A b +116 B a \right ) \left (b x +a \right )^{\frac {7}{2}}}{2560 a}+\left (\frac {33 A b}{512}-\frac {99 B a}{640}\right ) \left (b x +a \right )^{\frac {5}{2}}-\frac {17 a \left (5 A b -12 B a \right ) \left (b x +a \right )^{\frac {3}{2}}}{3072}+\frac {a^{2} \left (5 A b -12 B a \right ) \sqrt {b x +a}}{1024}}{b^{6} x^{6}}+\frac {\left (5 A b -12 B a \right ) \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{1024 a^{\frac {7}{2}}}\right )\) | \(162\) |
| default | \(2 b^{5} \left (-\frac {\frac {\left (5 A b -12 B a \right ) \left (b x +a \right )^{\frac {11}{2}}}{1024 a^{3}}-\frac {17 \left (5 A b -12 B a \right ) \left (b x +a \right )^{\frac {9}{2}}}{3072 a^{2}}+\frac {\left (165 A b +116 B a \right ) \left (b x +a \right )^{\frac {7}{2}}}{2560 a}+\left (\frac {33 A b}{512}-\frac {99 B a}{640}\right ) \left (b x +a \right )^{\frac {5}{2}}-\frac {17 a \left (5 A b -12 B a \right ) \left (b x +a \right )^{\frac {3}{2}}}{3072}+\frac {a^{2} \left (5 A b -12 B a \right ) \sqrt {b x +a}}{1024}}{b^{6} x^{6}}+\frac {\left (5 A b -12 B a \right ) \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{1024 a^{\frac {7}{2}}}\right )\) | \(162\) |
-1/192*(-15/8*x^6*(A*b-12/5*B*a)*b^5*arctanh((b*x+a)^(1/2)/a^(1/2))+(b*x+a )^(1/2)*((192/5*B*x+32*A)*a^(11/2)+x*(-5/4*x^3*(18/5*B*x+A)*b^3*a^(3/2)+b^ 2*x^2*(3*B*x+A)*a^(5/2)+54*x*(62/45*B*x+A)*b*a^(7/2)+(504/5*B*x+80*A)*a^(9 /2)+15/8*A*a^(1/2)*b^4*x^4)*b))/a^(7/2)/x^6
Time = 0.23 (sec) , antiderivative size = 356, normalized size of antiderivative = 1.71 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{x^7} \, dx=\left [-\frac {15 \, {\left (12 \, B a b^{5} - 5 \, A b^{6}\right )} \sqrt {a} x^{6} \log \left (\frac {b x + 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) + 2 \, {\left (1280 \, A a^{6} - 15 \, {\left (12 \, B a^{2} b^{4} - 5 \, A a b^{5}\right )} x^{5} + 10 \, {\left (12 \, B a^{3} b^{3} - 5 \, A a^{2} b^{4}\right )} x^{4} + 8 \, {\left (372 \, B a^{4} b^{2} + 5 \, A a^{3} b^{3}\right )} x^{3} + 144 \, {\left (28 \, B a^{5} b + 15 \, A a^{4} b^{2}\right )} x^{2} + 128 \, {\left (12 \, B a^{6} + 25 \, A a^{5} b\right )} x\right )} \sqrt {b x + a}}{15360 \, a^{4} x^{6}}, \frac {15 \, {\left (12 \, B a b^{5} - 5 \, A b^{6}\right )} \sqrt {-a} x^{6} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) - {\left (1280 \, A a^{6} - 15 \, {\left (12 \, B a^{2} b^{4} - 5 \, A a b^{5}\right )} x^{5} + 10 \, {\left (12 \, B a^{3} b^{3} - 5 \, A a^{2} b^{4}\right )} x^{4} + 8 \, {\left (372 \, B a^{4} b^{2} + 5 \, A a^{3} b^{3}\right )} x^{3} + 144 \, {\left (28 \, B a^{5} b + 15 \, A a^{4} b^{2}\right )} x^{2} + 128 \, {\left (12 \, B a^{6} + 25 \, A a^{5} b\right )} x\right )} \sqrt {b x + a}}{7680 \, a^{4} x^{6}}\right ] \]
[-1/15360*(15*(12*B*a*b^5 - 5*A*b^6)*sqrt(a)*x^6*log((b*x + 2*sqrt(b*x + a )*sqrt(a) + 2*a)/x) + 2*(1280*A*a^6 - 15*(12*B*a^2*b^4 - 5*A*a*b^5)*x^5 + 10*(12*B*a^3*b^3 - 5*A*a^2*b^4)*x^4 + 8*(372*B*a^4*b^2 + 5*A*a^3*b^3)*x^3 + 144*(28*B*a^5*b + 15*A*a^4*b^2)*x^2 + 128*(12*B*a^6 + 25*A*a^5*b)*x)*sqr t(b*x + a))/(a^4*x^6), 1/7680*(15*(12*B*a*b^5 - 5*A*b^6)*sqrt(-a)*x^6*arct an(sqrt(b*x + a)*sqrt(-a)/a) - (1280*A*a^6 - 15*(12*B*a^2*b^4 - 5*A*a*b^5) *x^5 + 10*(12*B*a^3*b^3 - 5*A*a^2*b^4)*x^4 + 8*(372*B*a^4*b^2 + 5*A*a^3*b^ 3)*x^3 + 144*(28*B*a^5*b + 15*A*a^4*b^2)*x^2 + 128*(12*B*a^6 + 25*A*a^5*b) *x)*sqrt(b*x + a))/(a^4*x^6)]
Timed out. \[ \int \frac {(a+b x)^{5/2} (A+B x)}{x^7} \, dx=\text {Timed out} \]
Time = 0.30 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.29 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{x^7} \, dx=\frac {1}{15360} \, b^{6} {\left (\frac {2 \, {\left (15 \, {\left (12 \, B a - 5 \, A b\right )} {\left (b x + a\right )}^{\frac {11}{2}} - 85 \, {\left (12 \, B a^{2} - 5 \, A a b\right )} {\left (b x + a\right )}^{\frac {9}{2}} - 6 \, {\left (116 \, B a^{3} + 165 \, A a^{2} b\right )} {\left (b x + a\right )}^{\frac {7}{2}} + 198 \, {\left (12 \, B a^{4} - 5 \, A a^{3} b\right )} {\left (b x + a\right )}^{\frac {5}{2}} - 85 \, {\left (12 \, B a^{5} - 5 \, A a^{4} b\right )} {\left (b x + a\right )}^{\frac {3}{2}} + 15 \, {\left (12 \, B a^{6} - 5 \, A a^{5} b\right )} \sqrt {b x + a}\right )}}{{\left (b x + a\right )}^{6} a^{3} b - 6 \, {\left (b x + a\right )}^{5} a^{4} b + 15 \, {\left (b x + a\right )}^{4} a^{5} b - 20 \, {\left (b x + a\right )}^{3} a^{6} b + 15 \, {\left (b x + a\right )}^{2} a^{7} b - 6 \, {\left (b x + a\right )} a^{8} b + a^{9} b} + \frac {15 \, {\left (12 \, B a - 5 \, A b\right )} \log \left (\frac {\sqrt {b x + a} - \sqrt {a}}{\sqrt {b x + a} + \sqrt {a}}\right )}{a^{\frac {7}{2}} b}\right )} \]
1/15360*b^6*(2*(15*(12*B*a - 5*A*b)*(b*x + a)^(11/2) - 85*(12*B*a^2 - 5*A* a*b)*(b*x + a)^(9/2) - 6*(116*B*a^3 + 165*A*a^2*b)*(b*x + a)^(7/2) + 198*( 12*B*a^4 - 5*A*a^3*b)*(b*x + a)^(5/2) - 85*(12*B*a^5 - 5*A*a^4*b)*(b*x + a )^(3/2) + 15*(12*B*a^6 - 5*A*a^5*b)*sqrt(b*x + a))/((b*x + a)^6*a^3*b - 6* (b*x + a)^5*a^4*b + 15*(b*x + a)^4*a^5*b - 20*(b*x + a)^3*a^6*b + 15*(b*x + a)^2*a^7*b - 6*(b*x + a)*a^8*b + a^9*b) + 15*(12*B*a - 5*A*b)*log((sqrt( b*x + a) - sqrt(a))/(sqrt(b*x + a) + sqrt(a)))/(a^(7/2)*b))
Time = 0.29 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.15 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{x^7} \, dx=\frac {\frac {15 \, {\left (12 \, B a b^{6} - 5 \, A b^{7}\right )} \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{3}} + \frac {180 \, {\left (b x + a\right )}^{\frac {11}{2}} B a b^{6} - 1020 \, {\left (b x + a\right )}^{\frac {9}{2}} B a^{2} b^{6} - 696 \, {\left (b x + a\right )}^{\frac {7}{2}} B a^{3} b^{6} + 2376 \, {\left (b x + a\right )}^{\frac {5}{2}} B a^{4} b^{6} - 1020 \, {\left (b x + a\right )}^{\frac {3}{2}} B a^{5} b^{6} + 180 \, \sqrt {b x + a} B a^{6} b^{6} - 75 \, {\left (b x + a\right )}^{\frac {11}{2}} A b^{7} + 425 \, {\left (b x + a\right )}^{\frac {9}{2}} A a b^{7} - 990 \, {\left (b x + a\right )}^{\frac {7}{2}} A a^{2} b^{7} - 990 \, {\left (b x + a\right )}^{\frac {5}{2}} A a^{3} b^{7} + 425 \, {\left (b x + a\right )}^{\frac {3}{2}} A a^{4} b^{7} - 75 \, \sqrt {b x + a} A a^{5} b^{7}}{a^{3} b^{6} x^{6}}}{7680 \, b} \]
1/7680*(15*(12*B*a*b^6 - 5*A*b^7)*arctan(sqrt(b*x + a)/sqrt(-a))/(sqrt(-a) *a^3) + (180*(b*x + a)^(11/2)*B*a*b^6 - 1020*(b*x + a)^(9/2)*B*a^2*b^6 - 6 96*(b*x + a)^(7/2)*B*a^3*b^6 + 2376*(b*x + a)^(5/2)*B*a^4*b^6 - 1020*(b*x + a)^(3/2)*B*a^5*b^6 + 180*sqrt(b*x + a)*B*a^6*b^6 - 75*(b*x + a)^(11/2)*A *b^7 + 425*(b*x + a)^(9/2)*A*a*b^7 - 990*(b*x + a)^(7/2)*A*a^2*b^7 - 990*( b*x + a)^(5/2)*A*a^3*b^7 + 425*(b*x + a)^(3/2)*A*a^4*b^7 - 75*sqrt(b*x + a )*A*a^5*b^7)/(a^3*b^6*x^6))/b
Time = 0.48 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.22 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{x^7} \, dx=\frac {b^5\,\mathrm {atanh}\left (\frac {\sqrt {a+b\,x}}{\sqrt {a}}\right )\,\left (5\,A\,b-12\,B\,a\right )}{512\,a^{7/2}}-\frac {\left (\frac {33\,A\,b^6}{256}-\frac {99\,B\,a\,b^5}{320}\right )\,{\left (a+b\,x\right )}^{5/2}+\left (\frac {5\,A\,a^2\,b^6}{512}-\frac {3\,B\,a^3\,b^5}{128}\right )\,\sqrt {a+b\,x}+\left (\frac {17\,B\,a^2\,b^5}{128}-\frac {85\,A\,a\,b^6}{1536}\right )\,{\left (a+b\,x\right )}^{3/2}-\frac {17\,\left (5\,A\,b^6-12\,B\,a\,b^5\right )\,{\left (a+b\,x\right )}^{9/2}}{1536\,a^2}+\frac {\left (5\,A\,b^6-12\,B\,a\,b^5\right )\,{\left (a+b\,x\right )}^{11/2}}{512\,a^3}+\frac {\left (165\,A\,b^6+116\,B\,a\,b^5\right )\,{\left (a+b\,x\right )}^{7/2}}{1280\,a}}{{\left (a+b\,x\right )}^6-6\,a^5\,\left (a+b\,x\right )-6\,a\,{\left (a+b\,x\right )}^5+15\,a^2\,{\left (a+b\,x\right )}^4-20\,a^3\,{\left (a+b\,x\right )}^3+15\,a^4\,{\left (a+b\,x\right )}^2+a^6} \]
(b^5*atanh((a + b*x)^(1/2)/a^(1/2))*(5*A*b - 12*B*a))/(512*a^(7/2)) - (((3 3*A*b^6)/256 - (99*B*a*b^5)/320)*(a + b*x)^(5/2) + ((5*A*a^2*b^6)/512 - (3 *B*a^3*b^5)/128)*(a + b*x)^(1/2) + ((17*B*a^2*b^5)/128 - (85*A*a*b^6)/1536 )*(a + b*x)^(3/2) - (17*(5*A*b^6 - 12*B*a*b^5)*(a + b*x)^(9/2))/(1536*a^2) + ((5*A*b^6 - 12*B*a*b^5)*(a + b*x)^(11/2))/(512*a^3) + ((165*A*b^6 + 116 *B*a*b^5)*(a + b*x)^(7/2))/(1280*a))/((a + b*x)^6 - 6*a^5*(a + b*x) - 6*a* (a + b*x)^5 + 15*a^2*(a + b*x)^4 - 20*a^3*(a + b*x)^3 + 15*a^4*(a + b*x)^2 + a^6)