Integrand size = 18, antiderivative size = 177 \[ \int \frac {A+B x}{x^6 \sqrt {a+b x}} \, dx=-\frac {A \sqrt {a+b x}}{5 a x^5}+\frac {(9 A b-10 a B) \sqrt {a+b x}}{40 a^2 x^4}-\frac {7 b (9 A b-10 a B) \sqrt {a+b x}}{240 a^3 x^3}+\frac {7 b^2 (9 A b-10 a B) \sqrt {a+b x}}{192 a^4 x^2}-\frac {7 b^3 (9 A b-10 a B) \sqrt {a+b x}}{128 a^5 x}+\frac {7 b^4 (9 A b-10 a B) \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{128 a^{11/2}} \]
7/128*b^4*(9*A*b-10*B*a)*arctanh((b*x+a)^(1/2)/a^(1/2))/a^(11/2)-1/5*A*(b* x+a)^(1/2)/a/x^5+1/40*(9*A*b-10*B*a)*(b*x+a)^(1/2)/a^2/x^4-7/240*b*(9*A*b- 10*B*a)*(b*x+a)^(1/2)/a^3/x^3+7/192*b^2*(9*A*b-10*B*a)*(b*x+a)^(1/2)/a^4/x ^2-7/128*b^3*(9*A*b-10*B*a)*(b*x+a)^(1/2)/a^5/x
Time = 0.26 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.74 \[ \int \frac {A+B x}{x^6 \sqrt {a+b x}} \, dx=\frac {\sqrt {a+b x} \left (-945 A b^4 x^4+210 a b^3 x^3 (3 A+5 B x)-96 a^4 (4 A+5 B x)-28 a^2 b^2 x^2 (18 A+25 B x)+16 a^3 b x (27 A+35 B x)\right )}{1920 a^5 x^5}+\frac {7 b^4 (9 A b-10 a B) \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{128 a^{11/2}} \]
(Sqrt[a + b*x]*(-945*A*b^4*x^4 + 210*a*b^3*x^3*(3*A + 5*B*x) - 96*a^4*(4*A + 5*B*x) - 28*a^2*b^2*x^2*(18*A + 25*B*x) + 16*a^3*b*x*(27*A + 35*B*x)))/ (1920*a^5*x^5) + (7*b^4*(9*A*b - 10*a*B)*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/( 128*a^(11/2))
Time = 0.23 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.92, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {87, 52, 52, 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.
| \(\displaystyle \int \frac {A+B x}{x^6 \sqrt {a+b x}} \, dx\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle -\frac {(9 A b-10 a B) \int \frac {1}{x^5 \sqrt {a+b x}}dx}{10 a}-\frac {A \sqrt {a+b x}}{5 a x^5}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle -\frac {(9 A b-10 a B) \left (-\frac {7 b \int \frac {1}{x^4 \sqrt {a+b x}}dx}{8 a}-\frac {\sqrt {a+b x}}{4 a x^4}\right )}{10 a}-\frac {A \sqrt {a+b x}}{5 a x^5}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle -\frac {(9 A b-10 a B) \left (-\frac {7 b \left (-\frac {5 b \int \frac {1}{x^3 \sqrt {a+b x}}dx}{6 a}-\frac {\sqrt {a+b x}}{3 a x^3}\right )}{8 a}-\frac {\sqrt {a+b x}}{4 a x^4}\right )}{10 a}-\frac {A \sqrt {a+b x}}{5 a x^5}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle -\frac {(9 A b-10 a B) \left (-\frac {7 b \left (-\frac {5 b \left (-\frac {3 b \int \frac {1}{x^2 \sqrt {a+b x}}dx}{4 a}-\frac {\sqrt {a+b x}}{2 a x^2}\right )}{6 a}-\frac {\sqrt {a+b x}}{3 a x^3}\right )}{8 a}-\frac {\sqrt {a+b x}}{4 a x^4}\right )}{10 a}-\frac {A \sqrt {a+b x}}{5 a x^5}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle -\frac {(9 A b-10 a B) \left (-\frac {7 b \left (-\frac {5 b \left (-\frac {3 b \left (-\frac {b \int \frac {1}{x \sqrt {a+b x}}dx}{2 a}-\frac {\sqrt {a+b x}}{a x}\right )}{4 a}-\frac {\sqrt {a+b x}}{2 a x^2}\right )}{6 a}-\frac {\sqrt {a+b x}}{3 a x^3}\right )}{8 a}-\frac {\sqrt {a+b x}}{4 a x^4}\right )}{10 a}-\frac {A \sqrt {a+b x}}{5 a x^5}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {(9 A b-10 a B) \left (-\frac {7 b \left (-\frac {5 b \left (-\frac {3 b \left (-\frac {\int \frac {1}{\frac {a+b x}{b}-\frac {a}{b}}d\sqrt {a+b x}}{a}-\frac {\sqrt {a+b x}}{a x}\right )}{4 a}-\frac {\sqrt {a+b x}}{2 a x^2}\right )}{6 a}-\frac {\sqrt {a+b x}}{3 a x^3}\right )}{8 a}-\frac {\sqrt {a+b x}}{4 a x^4}\right )}{10 a}-\frac {A \sqrt {a+b x}}{5 a x^5}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {(9 A b-10 a B) \left (-\frac {7 b \left (-\frac {5 b \left (-\frac {3 b \left (\frac {b \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {\sqrt {a+b x}}{a x}\right )}{4 a}-\frac {\sqrt {a+b x}}{2 a x^2}\right )}{6 a}-\frac {\sqrt {a+b x}}{3 a x^3}\right )}{8 a}-\frac {\sqrt {a+b x}}{4 a x^4}\right )}{10 a}-\frac {A \sqrt {a+b x}}{5 a x^5}\) |
-1/5*(A*Sqrt[a + b*x])/(a*x^5) - ((9*A*b - 10*a*B)*(-1/4*Sqrt[a + b*x]/(a* x^4) - (7*b*(-1/3*Sqrt[a + b*x]/(a*x^3) - (5*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*a)))/(8*a)))/(10*a)
3.5.33.3.1 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 + 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 = 0.54 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.67
| method | result | size |
| pseudoelliptic | \(\frac {\frac {63 x^{5} \left (A b -\frac {10 B a}{9}\right ) b^{4} \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{128}+\frac {9 \sqrt {b x +a}\, \left (\frac {35 x^{3} \left (\frac {5 B x}{3}+A \right ) b^{3} a^{\frac {3}{2}}}{24}-\frac {7 x^{2} \left (\frac {25 B x}{18}+A \right ) b^{2} a^{\frac {5}{2}}}{6}+b x \left (\frac {35 B x}{27}+A \right ) a^{\frac {7}{2}}+\frac {2 \left (-5 B x -4 A \right ) a^{\frac {9}{2}}}{9}-\frac {35 A \sqrt {a}\, b^{4} x^{4}}{16}\right )}{40}}{a^{\frac {11}{2}} x^{5}}\) | \(118\) |
| risch | \(-\frac {\sqrt {b x +a}\, \left (945 A \,b^{4} x^{4}-1050 B a \,b^{3} x^{4}-630 A a \,b^{3} x^{3}+700 B \,a^{2} b^{2} x^{3}+504 A \,a^{2} b^{2} x^{2}-560 B \,a^{3} b \,x^{2}-432 A \,a^{3} b x +480 B \,a^{4} x +384 A \,a^{4}\right )}{1920 a^{5} x^{5}}+\frac {7 b^{4} \left (9 A b -10 B a \right ) \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{128 a^{\frac {11}{2}}}\) | \(131\) |
| derivativedivides | \(2 b^{4} \left (-\frac {\frac {7 \left (9 A b -10 B a \right ) \left (b x +a \right )^{\frac {9}{2}}}{256 a^{5}}-\frac {49 \left (9 A b -10 B a \right ) \left (b x +a \right )^{\frac {7}{2}}}{384 a^{4}}+\frac {7 \left (9 A b -10 B a \right ) \left (b x +a \right )^{\frac {5}{2}}}{30 a^{3}}-\frac {79 \left (9 A b -10 B a \right ) \left (b x +a \right )^{\frac {3}{2}}}{384 a^{2}}+\frac {\left (193 A b -186 B a \right ) \sqrt {b x +a}}{256 a}}{b^{5} x^{5}}+\frac {7 \left (9 A b -10 B a \right ) \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{256 a^{\frac {11}{2}}}\right )\) | \(147\) |
| default | \(2 b^{4} \left (-\frac {\frac {7 \left (9 A b -10 B a \right ) \left (b x +a \right )^{\frac {9}{2}}}{256 a^{5}}-\frac {49 \left (9 A b -10 B a \right ) \left (b x +a \right )^{\frac {7}{2}}}{384 a^{4}}+\frac {7 \left (9 A b -10 B a \right ) \left (b x +a \right )^{\frac {5}{2}}}{30 a^{3}}-\frac {79 \left (9 A b -10 B a \right ) \left (b x +a \right )^{\frac {3}{2}}}{384 a^{2}}+\frac {\left (193 A b -186 B a \right ) \sqrt {b x +a}}{256 a}}{b^{5} x^{5}}+\frac {7 \left (9 A b -10 B a \right ) \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{256 a^{\frac {11}{2}}}\right )\) | \(147\) |
9/40*(35/16*x^5*(A*b-10/9*B*a)*b^4*arctanh((b*x+a)^(1/2)/a^(1/2))+(b*x+a)^ (1/2)*(35/24*x^3*(5/3*B*x+A)*b^3*a^(3/2)-7/6*x^2*(25/18*B*x+A)*b^2*a^(5/2) +b*x*(35/27*B*x+A)*a^(7/2)+2/9*(-5*B*x-4*A)*a^(9/2)-35/16*A*a^(1/2)*b^4*x^ 4))/a^(11/2)/x^5
Time = 0.25 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.74 \[ \int \frac {A+B x}{x^6 \sqrt {a+b x}} \, dx=\left [-\frac {105 \, {\left (10 \, B a b^{4} - 9 \, A b^{5}\right )} \sqrt {a} x^{5} \log \left (\frac {b x + 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) + 2 \, {\left (384 \, A a^{5} - 105 \, {\left (10 \, B a^{2} b^{3} - 9 \, A a b^{4}\right )} x^{4} + 70 \, {\left (10 \, B a^{3} b^{2} - 9 \, A a^{2} b^{3}\right )} x^{3} - 56 \, {\left (10 \, B a^{4} b - 9 \, A a^{3} b^{2}\right )} x^{2} + 48 \, {\left (10 \, B a^{5} - 9 \, A a^{4} b\right )} x\right )} \sqrt {b x + a}}{3840 \, a^{6} x^{5}}, \frac {105 \, {\left (10 \, B a b^{4} - 9 \, A b^{5}\right )} \sqrt {-a} x^{5} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) - {\left (384 \, A a^{5} - 105 \, {\left (10 \, B a^{2} b^{3} - 9 \, A a b^{4}\right )} x^{4} + 70 \, {\left (10 \, B a^{3} b^{2} - 9 \, A a^{2} b^{3}\right )} x^{3} - 56 \, {\left (10 \, B a^{4} b - 9 \, A a^{3} b^{2}\right )} x^{2} + 48 \, {\left (10 \, B a^{5} - 9 \, A a^{4} b\right )} x\right )} \sqrt {b x + a}}{1920 \, a^{6} x^{5}}\right ] \]
[-1/3840*(105*(10*B*a*b^4 - 9*A*b^5)*sqrt(a)*x^5*log((b*x + 2*sqrt(b*x + a )*sqrt(a) + 2*a)/x) + 2*(384*A*a^5 - 105*(10*B*a^2*b^3 - 9*A*a*b^4)*x^4 + 70*(10*B*a^3*b^2 - 9*A*a^2*b^3)*x^3 - 56*(10*B*a^4*b - 9*A*a^3*b^2)*x^2 + 48*(10*B*a^5 - 9*A*a^4*b)*x)*sqrt(b*x + a))/(a^6*x^5), 1/1920*(105*(10*B*a *b^4 - 9*A*b^5)*sqrt(-a)*x^5*arctan(sqrt(b*x + a)*sqrt(-a)/a) - (384*A*a^5 - 105*(10*B*a^2*b^3 - 9*A*a*b^4)*x^4 + 70*(10*B*a^3*b^2 - 9*A*a^2*b^3)*x^ 3 - 56*(10*B*a^4*b - 9*A*a^3*b^2)*x^2 + 48*(10*B*a^5 - 9*A*a^4*b)*x)*sqrt( b*x + a))/(a^6*x^5)]
Timed out. \[ \int \frac {A+B x}{x^6 \sqrt {a+b x}} \, dx=\text {Timed out} \]
Time = 0.28 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.32 \[ \int \frac {A+B x}{x^6 \sqrt {a+b x}} \, dx=\frac {1}{3840} \, b^{5} {\left (\frac {2 \, {\left (105 \, {\left (10 \, B a - 9 \, A b\right )} {\left (b x + a\right )}^{\frac {9}{2}} - 490 \, {\left (10 \, B a^{2} - 9 \, A a b\right )} {\left (b x + a\right )}^{\frac {7}{2}} + 896 \, {\left (10 \, B a^{3} - 9 \, A a^{2} b\right )} {\left (b x + a\right )}^{\frac {5}{2}} - 790 \, {\left (10 \, B a^{4} - 9 \, A a^{3} b\right )} {\left (b x + a\right )}^{\frac {3}{2}} + 15 \, {\left (186 \, B a^{5} - 193 \, A a^{4} b\right )} \sqrt {b x + a}\right )}}{{\left (b x + a\right )}^{5} a^{5} b - 5 \, {\left (b x + a\right )}^{4} a^{6} b + 10 \, {\left (b x + a\right )}^{3} a^{7} b - 10 \, {\left (b x + a\right )}^{2} a^{8} b + 5 \, {\left (b x + a\right )} a^{9} b - a^{10} b} + \frac {105 \, {\left (10 \, B a - 9 \, A b\right )} \log \left (\frac {\sqrt {b x + a} - \sqrt {a}}{\sqrt {b x + a} + \sqrt {a}}\right )}{a^{\frac {11}{2}} b}\right )} \]
1/3840*b^5*(2*(105*(10*B*a - 9*A*b)*(b*x + a)^(9/2) - 490*(10*B*a^2 - 9*A* a*b)*(b*x + a)^(7/2) + 896*(10*B*a^3 - 9*A*a^2*b)*(b*x + a)^(5/2) - 790*(1 0*B*a^4 - 9*A*a^3*b)*(b*x + a)^(3/2) + 15*(186*B*a^5 - 193*A*a^4*b)*sqrt(b *x + a))/((b*x + a)^5*a^5*b - 5*(b*x + a)^4*a^6*b + 10*(b*x + a)^3*a^7*b - 10*(b*x + a)^2*a^8*b + 5*(b*x + a)*a^9*b - a^10*b) + 105*(10*B*a - 9*A*b) *log((sqrt(b*x + a) - sqrt(a))/(sqrt(b*x + a) + sqrt(a)))/(a^(11/2)*b))
Time = 0.27 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.18 \[ \int \frac {A+B x}{x^6 \sqrt {a+b x}} \, dx=\frac {\frac {105 \, {\left (10 \, B a b^{5} - 9 \, A b^{6}\right )} \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{5}} + \frac {1050 \, {\left (b x + a\right )}^{\frac {9}{2}} B a b^{5} - 4900 \, {\left (b x + a\right )}^{\frac {7}{2}} B a^{2} b^{5} + 8960 \, {\left (b x + a\right )}^{\frac {5}{2}} B a^{3} b^{5} - 7900 \, {\left (b x + a\right )}^{\frac {3}{2}} B a^{4} b^{5} + 2790 \, \sqrt {b x + a} B a^{5} b^{5} - 945 \, {\left (b x + a\right )}^{\frac {9}{2}} A b^{6} + 4410 \, {\left (b x + a\right )}^{\frac {7}{2}} A a b^{6} - 8064 \, {\left (b x + a\right )}^{\frac {5}{2}} A a^{2} b^{6} + 7110 \, {\left (b x + a\right )}^{\frac {3}{2}} A a^{3} b^{6} - 2895 \, \sqrt {b x + a} A a^{4} b^{6}}{a^{5} b^{5} x^{5}}}{1920 \, b} \]
1/1920*(105*(10*B*a*b^5 - 9*A*b^6)*arctan(sqrt(b*x + a)/sqrt(-a))/(sqrt(-a )*a^5) + (1050*(b*x + a)^(9/2)*B*a*b^5 - 4900*(b*x + a)^(7/2)*B*a^2*b^5 + 8960*(b*x + a)^(5/2)*B*a^3*b^5 - 7900*(b*x + a)^(3/2)*B*a^4*b^5 + 2790*sqr t(b*x + a)*B*a^5*b^5 - 945*(b*x + a)^(9/2)*A*b^6 + 4410*(b*x + a)^(7/2)*A* a*b^6 - 8064*(b*x + a)^(5/2)*A*a^2*b^6 + 7110*(b*x + a)^(3/2)*A*a^3*b^6 - 2895*sqrt(b*x + a)*A*a^4*b^6)/(a^5*b^5*x^5))/b
Time = 0.12 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.25 \[ \int \frac {A+B x}{x^6 \sqrt {a+b x}} \, dx=\frac {\frac {7\,\left (9\,A\,b^5-10\,B\,a\,b^4\right )\,{\left (a+b\,x\right )}^{5/2}}{15\,a^3}-\frac {79\,\left (9\,A\,b^5-10\,B\,a\,b^4\right )\,{\left (a+b\,x\right )}^{3/2}}{192\,a^2}-\frac {49\,\left (9\,A\,b^5-10\,B\,a\,b^4\right )\,{\left (a+b\,x\right )}^{7/2}}{192\,a^4}+\frac {7\,\left (9\,A\,b^5-10\,B\,a\,b^4\right )\,{\left (a+b\,x\right )}^{9/2}}{128\,a^5}+\frac {\left (193\,A\,b^5-186\,B\,a\,b^4\right )\,\sqrt {a+b\,x}}{128\,a}}{5\,a\,{\left (a+b\,x\right )}^4-5\,a^4\,\left (a+b\,x\right )-{\left (a+b\,x\right )}^5-10\,a^2\,{\left (a+b\,x\right )}^3+10\,a^3\,{\left (a+b\,x\right )}^2+a^5}+\frac {7\,b^4\,\mathrm {atanh}\left (\frac {\sqrt {a+b\,x}}{\sqrt {a}}\right )\,\left (9\,A\,b-10\,B\,a\right )}{128\,a^{11/2}} \]
((7*(9*A*b^5 - 10*B*a*b^4)*(a + b*x)^(5/2))/(15*a^3) - (79*(9*A*b^5 - 10*B *a*b^4)*(a + b*x)^(3/2))/(192*a^2) - (49*(9*A*b^5 - 10*B*a*b^4)*(a + b*x)^ (7/2))/(192*a^4) + (7*(9*A*b^5 - 10*B*a*b^4)*(a + b*x)^(9/2))/(128*a^5) + ((193*A*b^5 - 186*B*a*b^4)*(a + b*x)^(1/2))/(128*a))/(5*a*(a + b*x)^4 - 5* a^4*(a + b*x) - (a + b*x)^5 - 10*a^2*(a + b*x)^3 + 10*a^3*(a + b*x)^2 + a^ 5) + (7*b^4*atanh((a + b*x)^(1/2)/a^(1/2))*(9*A*b - 10*B*a))/(128*a^(11/2) )