\(\int \frac {(a+b x^2)^5 (A+B x^2)}{x^{19}} \, dx\) [51]

Optimal result

Integrand size = 20, antiderivative size = 117 \[ \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^{19}} \, dx=-\frac {a^5 A}{18 x^{18}}-\frac {a^4 (5 A b+a B)}{16 x^{16}}-\frac {5 a^3 b (2 A b+a B)}{14 x^{14}}-\frac {5 a^2 b^2 (A b+a B)}{6 x^{12}}-\frac {a b^3 (A b+2 a B)}{2 x^{10}}-\frac {b^4 (A b+5 a B)}{8 x^8}-\frac {b^5 B}{6 x^6} \]

[Out]

-1/18*a^5*A/x^18-1/16*a^4*(5*A*b+B*a)/x^16-5/14*a^3*b*(2*A*b+B*a)/x^14-5/6*a^2*b^2*(A*b+B*a)/x^12-1/2*a*b^3*(A
*b+2*B*a)/x^10-1/8*b^4*(A*b+5*B*a)/x^8-1/6*b^5*B/x^6

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {457, 77} \[ \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^{19}} \, dx=-\frac {a^5 A}{18 x^{18}}-\frac {a^4 (a B+5 A b)}{16 x^{16}}-\frac {5 a^3 b (a B+2 A b)}{14 x^{14}}-\frac {5 a^2 b^2 (a B+A b)}{6 x^{12}}-\frac {b^4 (5 a B+A b)}{8 x^8}-\frac {a b^3 (2 a B+A b)}{2 x^{10}}-\frac {b^5 B}{6 x^6} \]

[In]

Int[((a + b*x^2)^5*(A + B*x^2))/x^19,x]

[Out]

-1/18*(a^5*A)/x^18 - (a^4*(5*A*b + a*B))/(16*x^16) - (5*a^3*b*(2*A*b + a*B))/(14*x^14) - (5*a^2*b^2*(A*b + a*B
))/(6*x^12) - (a*b^3*(A*b + 2*a*B))/(2*x^10) - (b^4*(A*b + 5*a*B))/(8*x^8) - (b^5*B)/(6*x^6)

Rule 77

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && (NeQ[n, -1] || EqQ[p, 1]) && N
eQ[b*e + a*f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && Rational
Q[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 1])

Rule 457

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {(a+b x)^5 (A+B x)}{x^{10}} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (\frac {a^5 A}{x^{10}}+\frac {a^4 (5 A b+a B)}{x^9}+\frac {5 a^3 b (2 A b+a B)}{x^8}+\frac {10 a^2 b^2 (A b+a B)}{x^7}+\frac {5 a b^3 (A b+2 a B)}{x^6}+\frac {b^4 (A b+5 a B)}{x^5}+\frac {b^5 B}{x^4}\right ) \, dx,x,x^2\right ) \\ & = -\frac {a^5 A}{18 x^{18}}-\frac {a^4 (5 A b+a B)}{16 x^{16}}-\frac {5 a^3 b (2 A b+a B)}{14 x^{14}}-\frac {5 a^2 b^2 (A b+a B)}{6 x^{12}}-\frac {a b^3 (A b+2 a B)}{2 x^{10}}-\frac {b^4 (A b+5 a B)}{8 x^8}-\frac {b^5 B}{6 x^6} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.03 \[ \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^{19}} \, dx=-\frac {42 b^5 x^{10} \left (3 A+4 B x^2\right )+126 a b^4 x^8 \left (4 A+5 B x^2\right )+168 a^2 b^3 x^6 \left (5 A+6 B x^2\right )+120 a^3 b^2 x^4 \left (6 A+7 B x^2\right )+45 a^4 b x^2 \left (7 A+8 B x^2\right )+7 a^5 \left (8 A+9 B x^2\right )}{1008 x^{18}} \]

[In]

Integrate[((a + b*x^2)^5*(A + B*x^2))/x^19,x]

[Out]

-1/1008*(42*b^5*x^10*(3*A + 4*B*x^2) + 126*a*b^4*x^8*(4*A + 5*B*x^2) + 168*a^2*b^3*x^6*(5*A + 6*B*x^2) + 120*a
^3*b^2*x^4*(6*A + 7*B*x^2) + 45*a^4*b*x^2*(7*A + 8*B*x^2) + 7*a^5*(8*A + 9*B*x^2))/x^18

Maple [A] (verified)

Time = 2.57 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.89

method	result	size
default	\(-\frac {a^{5} A}{18 x^{18}}-\frac {a^{4} \left (5 A b +B a \right )}{16 x^{16}}-\frac {5 a^{3} b \left (2 A b +B a \right )}{14 x^{14}}-\frac {5 a^{2} b^{2} \left (A b +B a \right )}{6 x^{12}}-\frac {a \,b^{3} \left (A b +2 B a \right )}{2 x^{10}}-\frac {b^{4} \left (A b +5 B a \right )}{8 x^{8}}-\frac {b^{5} B}{6 x^{6}}\)	\(104\)
norman	\(\frac {-\frac {a^{5} A}{18}+\left (-\frac {5}{16} a^{4} b A -\frac {1}{16} a^{5} B \right ) x^{2}+\left (-\frac {5}{7} a^{3} b^{2} A -\frac {5}{14} a^{4} b B \right ) x^{4}+\left (-\frac {5}{6} a^{2} b^{3} A -\frac {5}{6} a^{3} b^{2} B \right ) x^{6}+\left (-\frac {1}{2} a \,b^{4} A -a^{2} b^{3} B \right ) x^{8}+\left (-\frac {1}{8} b^{5} A -\frac {5}{8} a \,b^{4} B \right ) x^{10}-\frac {b^{5} B \,x^{12}}{6}}{x^{18}}\)	\(122\)
risch	\(\frac {-\frac {a^{5} A}{18}+\left (-\frac {5}{16} a^{4} b A -\frac {1}{16} a^{5} B \right ) x^{2}+\left (-\frac {5}{7} a^{3} b^{2} A -\frac {5}{14} a^{4} b B \right ) x^{4}+\left (-\frac {5}{6} a^{2} b^{3} A -\frac {5}{6} a^{3} b^{2} B \right ) x^{6}+\left (-\frac {1}{2} a \,b^{4} A -a^{2} b^{3} B \right ) x^{8}+\left (-\frac {1}{8} b^{5} A -\frac {5}{8} a \,b^{4} B \right ) x^{10}-\frac {b^{5} B \,x^{12}}{6}}{x^{18}}\)	\(122\)
gosper	\(-\frac {168 b^{5} B \,x^{12}+126 A \,b^{5} x^{10}+630 B a \,b^{4} x^{10}+504 a A \,b^{4} x^{8}+1008 B \,a^{2} b^{3} x^{8}+840 a^{2} A \,b^{3} x^{6}+840 B \,a^{3} b^{2} x^{6}+720 a^{3} A \,b^{2} x^{4}+360 B \,a^{4} b \,x^{4}+315 a^{4} A b \,x^{2}+63 a^{5} B \,x^{2}+56 a^{5} A}{1008 x^{18}}\)	\(128\)
parallelrisch	\(-\frac {168 b^{5} B \,x^{12}+126 A \,b^{5} x^{10}+630 B a \,b^{4} x^{10}+504 a A \,b^{4} x^{8}+1008 B \,a^{2} b^{3} x^{8}+840 a^{2} A \,b^{3} x^{6}+840 B \,a^{3} b^{2} x^{6}+720 a^{3} A \,b^{2} x^{4}+360 B \,a^{4} b \,x^{4}+315 a^{4} A b \,x^{2}+63 a^{5} B \,x^{2}+56 a^{5} A}{1008 x^{18}}\)	\(128\)

[In]

int((b*x^2+a)^5*(B*x^2+A)/x^19,x,method=_RETURNVERBOSE)

[Out]

-1/18*a^5*A/x^18-1/16*a^4*(5*A*b+B*a)/x^16-5/14*a^3*b*(2*A*b+B*a)/x^14-5/6*a^2*b^2*(A*b+B*a)/x^12-1/2*a*b^3*(A
*b+2*B*a)/x^10-1/8*b^4*(A*b+5*B*a)/x^8-1/6*b^5*B/x^6

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.03 \[ \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^{19}} \, dx=-\frac {168 \, B b^{5} x^{12} + 126 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} + 504 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} + 840 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} + 56 \, A a^{5} + 360 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} + 63 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{1008 \, x^{18}} \]

[In]

integrate((b*x^2+a)^5*(B*x^2+A)/x^19,x, algorithm="fricas")

[Out]

-1/1008*(168*B*b^5*x^12 + 126*(5*B*a*b^4 + A*b^5)*x^10 + 504*(2*B*a^2*b^3 + A*a*b^4)*x^8 + 840*(B*a^3*b^2 + A*
a^2*b^3)*x^6 + 56*A*a^5 + 360*(B*a^4*b + 2*A*a^3*b^2)*x^4 + 63*(B*a^5 + 5*A*a^4*b)*x^2)/x^18

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^{19}} \, dx=\text {Timed out} \]

[In]

integrate((b*x**2+a)**5*(B*x**2+A)/x**19,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.03 \[ \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^{19}} \, dx=-\frac {168 \, B b^{5} x^{12} + 126 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} + 504 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} + 840 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} + 56 \, A a^{5} + 360 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} + 63 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{1008 \, x^{18}} \]

[In]

integrate((b*x^2+a)^5*(B*x^2+A)/x^19,x, algorithm="maxima")

[Out]

-1/1008*(168*B*b^5*x^12 + 126*(5*B*a*b^4 + A*b^5)*x^10 + 504*(2*B*a^2*b^3 + A*a*b^4)*x^8 + 840*(B*a^3*b^2 + A*
a^2*b^3)*x^6 + 56*A*a^5 + 360*(B*a^4*b + 2*A*a^3*b^2)*x^4 + 63*(B*a^5 + 5*A*a^4*b)*x^2)/x^18

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.09 \[ \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^{19}} \, dx=-\frac {168 \, B b^{5} x^{12} + 630 \, B a b^{4} x^{10} + 126 \, A b^{5} x^{10} + 1008 \, B a^{2} b^{3} x^{8} + 504 \, A a b^{4} x^{8} + 840 \, B a^{3} b^{2} x^{6} + 840 \, A a^{2} b^{3} x^{6} + 360 \, B a^{4} b x^{4} + 720 \, A a^{3} b^{2} x^{4} + 63 \, B a^{5} x^{2} + 315 \, A a^{4} b x^{2} + 56 \, A a^{5}}{1008 \, x^{18}} \]

[In]

integrate((b*x^2+a)^5*(B*x^2+A)/x^19,x, algorithm="giac")

[Out]

-1/1008*(168*B*b^5*x^12 + 630*B*a*b^4*x^10 + 126*A*b^5*x^10 + 1008*B*a^2*b^3*x^8 + 504*A*a*b^4*x^8 + 840*B*a^3
*b^2*x^6 + 840*A*a^2*b^3*x^6 + 360*B*a^4*b*x^4 + 720*A*a^3*b^2*x^4 + 63*B*a^5*x^2 + 315*A*a^4*b*x^2 + 56*A*a^5
)/x^18

Mupad [B] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.03 \[ \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^{19}} \, dx=-\frac {\frac {A\,a^5}{18}+x^8\,\left (B\,a^2\,b^3+\frac {A\,a\,b^4}{2}\right )+x^4\,\left (\frac {5\,B\,a^4\,b}{14}+\frac {5\,A\,a^3\,b^2}{7}\right )+x^2\,\left (\frac {B\,a^5}{16}+\frac {5\,A\,b\,a^4}{16}\right )+x^{10}\,\left (\frac {A\,b^5}{8}+\frac {5\,B\,a\,b^4}{8}\right )+x^6\,\left (\frac {5\,B\,a^3\,b^2}{6}+\frac {5\,A\,a^2\,b^3}{6}\right )+\frac {B\,b^5\,x^{12}}{6}}{x^{18}} \]

[In]

int(((A + B*x^2)*(a + b*x^2)^5)/x^19,x)

[Out]

-((A*a^5)/18 + x^8*(B*a^2*b^3 + (A*a*b^4)/2) + x^4*((5*A*a^3*b^2)/7 + (5*B*a^4*b)/14) + x^2*((B*a^5)/16 + (5*A
*a^4*b)/16) + x^10*((A*b^5)/8 + (5*B*a*b^4)/8) + x^6*((5*A*a^2*b^3)/6 + (5*B*a^3*b^2)/6) + (B*b^5*x^12)/6)/x^1
8