Integrand size = 29, antiderivative size = 267 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^7} \, dx=-\frac {a^5 B \sqrt {a^2+2 a b x+b^2 x^2}}{5 x^5 (a+b x)}-\frac {5 a^4 b B \sqrt {a^2+2 a b x+b^2 x^2}}{4 x^4 (a+b x)}-\frac {10 a^3 b^2 B \sqrt {a^2+2 a b x+b^2 x^2}}{3 x^3 (a+b x)}-\frac {5 a^2 b^3 B \sqrt {a^2+2 a b x+b^2 x^2}}{x^2 (a+b x)}-\frac {5 a b^4 B \sqrt {a^2+2 a b x+b^2 x^2}}{x (a+b x)}-\frac {A (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{6 a x^6}+\frac {b^5 B \sqrt {a^2+2 a b x+b^2 x^2} \log (x)}{a+b x} \]
-1/5*a^5*B*((b*x+a)^2)^(1/2)/x^5/(b*x+a)-5/4*a^4*b*B*((b*x+a)^2)^(1/2)/x^4 /(b*x+a)-10/3*a^3*b^2*B*((b*x+a)^2)^(1/2)/x^3/(b*x+a)-5*a^2*b^3*B*((b*x+a) ^2)^(1/2)/x^2/(b*x+a)-5*a*b^4*B*((b*x+a)^2)^(1/2)/x/(b*x+a)-1/6*A*(b*x+a)^ 5*((b*x+a)^2)^(1/2)/a/x^6+b^5*B*ln(x)*((b*x+a)^2)^(1/2)/(b*x+a)
Time = 0.93 (sec) , antiderivative size = 354, normalized size of antiderivative = 1.33 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^7} \, dx=\frac {\sqrt {a^2} \left (60 A b^5 x^5+150 a b^4 x^4 (A+2 B x)+100 a^2 b^3 x^3 (2 A+3 B x)+50 a^3 b^2 x^2 (3 A+4 B x)+15 a^4 b x (4 A+5 B x)+2 a^5 (5 A+6 B x)\right )-\sqrt {(a+b x)^2} \left (10 A b^5 x^5+2 a^5 (5 A+6 B x)+a^4 b x (50 A+63 B x)+a b^4 x^4 (50 A+137 B x)+a^3 b^2 x^2 (100 A+137 B x)+a^2 b^3 x^3 (100 A+163 B x)\right )-120 a b^5 B x^6 \text {arctanh}\left (\frac {b x}{\sqrt {a^2}-\sqrt {(a+b x)^2}}\right )-120 \sqrt {a^2} b^5 B x^6 \log (x)+60 \sqrt {a^2} b^5 B x^6 \log \left (a \left (\sqrt {a^2}-b x-\sqrt {(a+b x)^2}\right )\right )+60 \sqrt {a^2} b^5 B x^6 \log \left (a \left (\sqrt {a^2}+b x-\sqrt {(a+b x)^2}\right )\right )}{120 a x^6} \]
(Sqrt[a^2]*(60*A*b^5*x^5 + 150*a*b^4*x^4*(A + 2*B*x) + 100*a^2*b^3*x^3*(2* A + 3*B*x) + 50*a^3*b^2*x^2*(3*A + 4*B*x) + 15*a^4*b*x*(4*A + 5*B*x) + 2*a ^5*(5*A + 6*B*x)) - Sqrt[(a + b*x)^2]*(10*A*b^5*x^5 + 2*a^5*(5*A + 6*B*x) + a^4*b*x*(50*A + 63*B*x) + a*b^4*x^4*(50*A + 137*B*x) + a^3*b^2*x^2*(100* A + 137*B*x) + a^2*b^3*x^3*(100*A + 163*B*x)) - 120*a*b^5*B*x^6*ArcTanh[(b *x)/(Sqrt[a^2] - Sqrt[(a + b*x)^2])] - 120*Sqrt[a^2]*b^5*B*x^6*Log[x] + 60 *Sqrt[a^2]*b^5*B*x^6*Log[a*(Sqrt[a^2] - b*x - Sqrt[(a + b*x)^2])] + 60*Sqr t[a^2]*b^5*B*x^6*Log[a*(Sqrt[a^2] + b*x - Sqrt[(a + b*x)^2])])/(120*a*x^6)
Time = 0.26 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.41, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {1187, 27, 87, 49, 2009}
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.
| \(\displaystyle \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2} (A+B x)}{x^7} \, dx\) |
| \(\Big \downarrow \) 1187 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {b^5 (a+b x)^5 (A+B x)}{x^7}dx}{b^5 (a+b x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {(a+b x)^5 (A+B x)}{x^7}dx}{a+b x}\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (B \int \frac {(a+b x)^5}{x^6}dx-\frac {A (a+b x)^6}{6 a x^6}\right )}{a+b x}\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (B \int \left (\frac {a^5}{x^6}+\frac {5 b a^4}{x^5}+\frac {10 b^2 a^3}{x^4}+\frac {10 b^3 a^2}{x^3}+\frac {5 b^4 a}{x^2}+\frac {b^5}{x}\right )dx-\frac {A (a+b x)^6}{6 a x^6}\right )}{a+b x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (B \left (-\frac {a^5}{5 x^5}-\frac {5 a^4 b}{4 x^4}-\frac {10 a^3 b^2}{3 x^3}-\frac {5 a^2 b^3}{x^2}-\frac {5 a b^4}{x}+b^5 \log (x)\right )-\frac {A (a+b x)^6}{6 a x^6}\right )}{a+b x}\) |
(Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(-1/6*(A*(a + b*x)^6)/(a*x^6) + B*(-1/5*a^5 /x^5 - (5*a^4*b)/(4*x^4) - (10*a^3*b^2)/(3*x^3) - (5*a^2*b^3)/x^2 - (5*a*b ^4)/x + b^5*Log[x])))/(a + b*x)
3.7.96.3.1 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] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
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[((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 = 0.67 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.53
| method | result | size |
| default | \(-\frac {\left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} \left (-60 B \,b^{5} \ln \left (x \right ) x^{6}+60 A \,b^{5} x^{5}+300 B a \,b^{4} x^{5}+150 A a \,b^{4} x^{4}+300 B \,a^{2} b^{3} x^{4}+200 A \,a^{2} b^{3} x^{3}+200 B \,a^{3} b^{2} x^{3}+150 A \,a^{3} b^{2} x^{2}+75 B \,a^{4} b \,x^{2}+60 A \,a^{4} b x +12 a^{5} B x +10 A \,a^{5}\right )}{60 \left (b x +a \right )^{5} x^{6}}\) | \(142\) |
| risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (\left (-A \,b^{5}-5 B a \,b^{4}\right ) x^{5}+\left (-\frac {5}{2} A a \,b^{4}-5 B \,a^{2} b^{3}\right ) x^{4}+\left (-\frac {10}{3} A \,a^{2} b^{3}-\frac {10}{3} B \,a^{3} b^{2}\right ) x^{3}+\left (-\frac {5}{2} A \,a^{3} b^{2}-\frac {5}{4} B \,a^{4} b \right ) x^{2}+\left (-A \,a^{4} b -\frac {1}{5} a^{5} B \right ) x -\frac {A \,a^{5}}{6}\right )}{\left (b x +a \right ) x^{6}}+\frac {b^{5} B \ln \left (x \right ) \sqrt {\left (b x +a \right )^{2}}}{b x +a}\) | \(151\) |
-1/60*((b*x+a)^2)^(5/2)*(-60*B*b^5*ln(x)*x^6+60*A*b^5*x^5+300*B*a*b^4*x^5+ 150*A*a*b^4*x^4+300*B*a^2*b^3*x^4+200*A*a^2*b^3*x^3+200*B*a^3*b^2*x^3+150* A*a^3*b^2*x^2+75*B*a^4*b*x^2+60*A*a^4*b*x+12*a^5*B*x+10*A*a^5)/(b*x+a)^5/x ^6
Time = 0.26 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.45 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^7} \, dx=\frac {60 \, B b^{5} x^{6} \log \left (x\right ) - 10 \, A a^{5} - 60 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} - 150 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} - 200 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} - 75 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} - 12 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x}{60 \, x^{6}} \]
1/60*(60*B*b^5*x^6*log(x) - 10*A*a^5 - 60*(5*B*a*b^4 + A*b^5)*x^5 - 150*(2 *B*a^2*b^3 + A*a*b^4)*x^4 - 200*(B*a^3*b^2 + A*a^2*b^3)*x^3 - 75*(B*a^4*b + 2*A*a^3*b^2)*x^2 - 12*(B*a^5 + 5*A*a^4*b)*x)/x^6
\[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^7} \, dx=\int \frac {\left (A + B x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}}{x^{7}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 554 vs. \(2 (182) = 364\).
Time = 0.21 (sec) , antiderivative size = 554, normalized size of antiderivative = 2.07 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^7} \, dx=\left (-1\right )^{2 \, b^{2} x + 2 \, a b} B b^{5} \log \left (2 \, b^{2} x + 2 \, a b\right ) - \left (-1\right )^{2 \, a b x + 2 \, a^{2}} B b^{5} \log \left (\frac {2 \, a b x}{{\left | x \right |}} + \frac {2 \, a^{2}}{{\left | x \right |}}\right ) + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} B b^{6} x}{2 \, a^{2}} + \frac {3 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} B b^{5}}{2 \, a} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} B b^{6} x}{4 \, a^{4}} + \frac {7 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} B b^{5}}{12 \, a^{3}} - \frac {2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B b^{5}}{15 \, a^{5}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A b^{6}}{6 \, a^{6}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B b^{4}}{3 \, a^{4} x} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A b^{5}}{6 \, a^{5} x} + \frac {2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b^{3}}{15 \, a^{5} x^{2}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{4}}{6 \, a^{6} x^{2}} - \frac {11 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b^{2}}{60 \, a^{4} x^{3}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{3}}{6 \, a^{5} x^{3}} + \frac {3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b}{20 \, a^{3} x^{4}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{2}}{6 \, a^{4} x^{4}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B}{5 \, a^{2} x^{5}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b}{6 \, a^{3} x^{5}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A}{6 \, a^{2} x^{6}} \]
(-1)^(2*b^2*x + 2*a*b)*B*b^5*log(2*b^2*x + 2*a*b) - (-1)^(2*a*b*x + 2*a^2) *B*b^5*log(2*a*b*x/abs(x) + 2*a^2/abs(x)) + 1/2*sqrt(b^2*x^2 + 2*a*b*x + a ^2)*B*b^6*x/a^2 + 3/2*sqrt(b^2*x^2 + 2*a*b*x + a^2)*B*b^5/a + 1/4*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*B*b^6*x/a^4 + 7/12*(b^2*x^2 + 2*a*b*x + a^2)^(3/2) *B*b^5/a^3 - 2/15*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*B*b^5/a^5 + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*A*b^6/a^6 - 1/3*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*B* b^4/(a^4*x) + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*A*b^5/(a^5*x) + 2/15*(b^ 2*x^2 + 2*a*b*x + a^2)^(7/2)*B*b^3/(a^5*x^2) - 1/6*(b^2*x^2 + 2*a*b*x + a^ 2)^(7/2)*A*b^4/(a^6*x^2) - 11/60*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*B*b^2/(a^ 4*x^3) + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*A*b^3/(a^5*x^3) + 3/20*(b^2*x ^2 + 2*a*b*x + a^2)^(7/2)*B*b/(a^3*x^4) - 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(7 /2)*A*b^2/(a^4*x^4) - 1/5*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*B/(a^2*x^5) + 1/ 6*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*A*b/(a^3*x^5) - 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*A/(a^2*x^6)
Time = 0.27 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.72 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^7} \, dx=B b^{5} \log \left ({\left | x \right |}\right ) \mathrm {sgn}\left (b x + a\right ) - \frac {10 \, A a^{5} \mathrm {sgn}\left (b x + a\right ) + 60 \, {\left (5 \, B a b^{4} \mathrm {sgn}\left (b x + a\right ) + A b^{5} \mathrm {sgn}\left (b x + a\right )\right )} x^{5} + 150 \, {\left (2 \, B a^{2} b^{3} \mathrm {sgn}\left (b x + a\right ) + A a b^{4} \mathrm {sgn}\left (b x + a\right )\right )} x^{4} + 200 \, {\left (B a^{3} b^{2} \mathrm {sgn}\left (b x + a\right ) + A a^{2} b^{3} \mathrm {sgn}\left (b x + a\right )\right )} x^{3} + 75 \, {\left (B a^{4} b \mathrm {sgn}\left (b x + a\right ) + 2 \, A a^{3} b^{2} \mathrm {sgn}\left (b x + a\right )\right )} x^{2} + 12 \, {\left (B a^{5} \mathrm {sgn}\left (b x + a\right ) + 5 \, A a^{4} b \mathrm {sgn}\left (b x + a\right )\right )} x}{60 \, x^{6}} \]
B*b^5*log(abs(x))*sgn(b*x + a) - 1/60*(10*A*a^5*sgn(b*x + a) + 60*(5*B*a*b ^4*sgn(b*x + a) + A*b^5*sgn(b*x + a))*x^5 + 150*(2*B*a^2*b^3*sgn(b*x + a) + A*a*b^4*sgn(b*x + a))*x^4 + 200*(B*a^3*b^2*sgn(b*x + a) + A*a^2*b^3*sgn( b*x + a))*x^3 + 75*(B*a^4*b*sgn(b*x + a) + 2*A*a^3*b^2*sgn(b*x + a))*x^2 + 12*(B*a^5*sgn(b*x + a) + 5*A*a^4*b*sgn(b*x + a))*x)/x^6
Timed out. \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^7} \, dx=\int \frac {\left (A+B\,x\right )\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}}{x^7} \,d x \]