\(\int \frac {(a+b x^2)^5 (A+B x^2)}{x^{22}} \, dx\) [54]

Optimal result

Integrand size = 20, antiderivative size = 117 \[ \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^{22}} \, dx=-\frac {a^5 A}{21 x^{21}}-\frac {a^4 (5 A b+a B)}{19 x^{19}}-\frac {5 a^3 b (2 A b+a B)}{17 x^{17}}-\frac {2 a^2 b^2 (A b+a B)}{3 x^{15}}-\frac {5 a b^3 (A b+2 a B)}{13 x^{13}}-\frac {b^4 (A b+5 a B)}{11 x^{11}}-\frac {b^5 B}{9 x^9} \]

[Out]

-1/21*a^5*A/x^21-1/19*a^4*(5*A*b+B*a)/x^19-5/17*a^3*b*(2*A*b+B*a)/x^17-2/3*a^2*b^2*(A*b+B*a)/x^15-5/13*a*b^3*(
A*b+2*B*a)/x^13-1/11*b^4*(A*b+5*B*a)/x^11-1/9*b^5*B/x^9

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {459} \[ \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^{22}} \, dx=-\frac {a^5 A}{21 x^{21}}-\frac {a^4 (a B+5 A b)}{19 x^{19}}-\frac {5 a^3 b (a B+2 A b)}{17 x^{17}}-\frac {2 a^2 b^2 (a B+A b)}{3 x^{15}}-\frac {b^4 (5 a B+A b)}{11 x^{11}}-\frac {5 a b^3 (2 a B+A b)}{13 x^{13}}-\frac {b^5 B}{9 x^9} \]

[In]

Int[((a + b*x^2)^5*(A + B*x^2))/x^22,x]

[Out]

-1/21*(a^5*A)/x^21 - (a^4*(5*A*b + a*B))/(19*x^19) - (5*a^3*b*(2*A*b + a*B))/(17*x^17) - (2*a^2*b^2*(A*b + a*B
))/(3*x^15) - (5*a*b^3*(A*b + 2*a*B))/(13*x^13) - (b^4*(A*b + 5*a*B))/(11*x^11) - (b^5*B)/(9*x^9)

Rule 459

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

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a^5 A}{x^{22}}+\frac {a^4 (5 A b+a B)}{x^{20}}+\frac {5 a^3 b (2 A b+a B)}{x^{18}}+\frac {10 a^2 b^2 (A b+a B)}{x^{16}}+\frac {5 a b^3 (A b+2 a B)}{x^{14}}+\frac {b^4 (A b+5 a B)}{x^{12}}+\frac {b^5 B}{x^{10}}\right ) \, dx \\ & = -\frac {a^5 A}{21 x^{21}}-\frac {a^4 (5 A b+a B)}{19 x^{19}}-\frac {5 a^3 b (2 A b+a B)}{17 x^{17}}-\frac {2 a^2 b^2 (A b+a B)}{3 x^{15}}-\frac {5 a b^3 (A b+2 a B)}{13 x^{13}}-\frac {b^4 (A b+5 a B)}{11 x^{11}}-\frac {b^5 B}{9 x^9} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^{22}} \, dx=-\frac {a^5 A}{21 x^{21}}-\frac {a^4 (5 A b+a B)}{19 x^{19}}-\frac {5 a^3 b (2 A b+a B)}{17 x^{17}}-\frac {2 a^2 b^2 (A b+a B)}{3 x^{15}}-\frac {5 a b^3 (A b+2 a B)}{13 x^{13}}-\frac {b^4 (A b+5 a B)}{11 x^{11}}-\frac {b^5 B}{9 x^9} \]

[In]

Integrate[((a + b*x^2)^5*(A + B*x^2))/x^22,x]

[Out]

-1/21*(a^5*A)/x^21 - (a^4*(5*A*b + a*B))/(19*x^19) - (5*a^3*b*(2*A*b + a*B))/(17*x^17) - (2*a^2*b^2*(A*b + a*B
))/(3*x^15) - (5*a*b^3*(A*b + 2*a*B))/(13*x^13) - (b^4*(A*b + 5*a*B))/(11*x^11) - (b^5*B)/(9*x^9)

Maple [A] (verified)

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

method	result	size
default	\(-\frac {a^{5} A}{21 x^{21}}-\frac {a^{4} \left (5 A b +B a \right )}{19 x^{19}}-\frac {5 a^{3} b \left (2 A b +B a \right )}{17 x^{17}}-\frac {2 a^{2} b^{2} \left (A b +B a \right )}{3 x^{15}}-\frac {5 a \,b^{3} \left (A b +2 B a \right )}{13 x^{13}}-\frac {b^{4} \left (A b +5 B a \right )}{11 x^{11}}-\frac {b^{5} B}{9 x^{9}}\)	\(104\)
norman	\(\frac {-\frac {a^{5} A}{21}+\left (-\frac {5}{19} a^{4} b A -\frac {1}{19} a^{5} B \right ) x^{2}+\left (-\frac {10}{17} a^{3} b^{2} A -\frac {5}{17} a^{4} b B \right ) x^{4}+\left (-\frac {2}{3} a^{2} b^{3} A -\frac {2}{3} a^{3} b^{2} B \right ) x^{6}+\left (-\frac {5}{13} a \,b^{4} A -\frac {10}{13} a^{2} b^{3} B \right ) x^{8}+\left (-\frac {1}{11} b^{5} A -\frac {5}{11} a \,b^{4} B \right ) x^{10}-\frac {b^{5} B \,x^{12}}{9}}{x^{21}}\)	\(122\)
risch	\(\frac {-\frac {a^{5} A}{21}+\left (-\frac {5}{19} a^{4} b A -\frac {1}{19} a^{5} B \right ) x^{2}+\left (-\frac {10}{17} a^{3} b^{2} A -\frac {5}{17} a^{4} b B \right ) x^{4}+\left (-\frac {2}{3} a^{2} b^{3} A -\frac {2}{3} a^{3} b^{2} B \right ) x^{6}+\left (-\frac {5}{13} a \,b^{4} A -\frac {10}{13} a^{2} b^{3} B \right ) x^{8}+\left (-\frac {1}{11} b^{5} A -\frac {5}{11} a \,b^{4} B \right ) x^{10}-\frac {b^{5} B \,x^{12}}{9}}{x^{21}}\)	\(122\)
gosper	\(-\frac {323323 b^{5} B \,x^{12}+264537 A \,b^{5} x^{10}+1322685 B a \,b^{4} x^{10}+1119195 a A \,b^{4} x^{8}+2238390 B \,a^{2} b^{3} x^{8}+1939938 a^{2} A \,b^{3} x^{6}+1939938 B \,a^{3} b^{2} x^{6}+1711710 a^{3} A \,b^{2} x^{4}+855855 B \,a^{4} b \,x^{4}+765765 a^{4} A b \,x^{2}+153153 a^{5} B \,x^{2}+138567 a^{5} A}{2909907 x^{21}}\)	\(128\)
parallelrisch	\(-\frac {323323 b^{5} B \,x^{12}+264537 A \,b^{5} x^{10}+1322685 B a \,b^{4} x^{10}+1119195 a A \,b^{4} x^{8}+2238390 B \,a^{2} b^{3} x^{8}+1939938 a^{2} A \,b^{3} x^{6}+1939938 B \,a^{3} b^{2} x^{6}+1711710 a^{3} A \,b^{2} x^{4}+855855 B \,a^{4} b \,x^{4}+765765 a^{4} A b \,x^{2}+153153 a^{5} B \,x^{2}+138567 a^{5} A}{2909907 x^{21}}\)	\(128\)

[In]

int((b*x^2+a)^5*(B*x^2+A)/x^22,x,method=_RETURNVERBOSE)

[Out]

-1/21*a^5*A/x^21-1/19*a^4*(5*A*b+B*a)/x^19-5/17*a^3*b*(2*A*b+B*a)/x^17-2/3*a^2*b^2*(A*b+B*a)/x^15-5/13*a*b^3*(
A*b+2*B*a)/x^13-1/11*b^4*(A*b+5*B*a)/x^11-1/9*b^5*B/x^9

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.03 \[ \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^{22}} \, dx=-\frac {323323 \, B b^{5} x^{12} + 264537 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} + 1119195 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} + 1939938 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} + 138567 \, A a^{5} + 855855 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} + 153153 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{2909907 \, x^{21}} \]

[In]

integrate((b*x^2+a)^5*(B*x^2+A)/x^22,x, algorithm="fricas")

[Out]

-1/2909907*(323323*B*b^5*x^12 + 264537*(5*B*a*b^4 + A*b^5)*x^10 + 1119195*(2*B*a^2*b^3 + A*a*b^4)*x^8 + 193993
8*(B*a^3*b^2 + A*a^2*b^3)*x^6 + 138567*A*a^5 + 855855*(B*a^4*b + 2*A*a^3*b^2)*x^4 + 153153*(B*a^5 + 5*A*a^4*b)
*x^2)/x^21

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^{22}} \, dx=\text {Timed out} \]

[In]

integrate((b*x**2+a)**5*(B*x**2+A)/x**22,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.03 \[ \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^{22}} \, dx=-\frac {323323 \, B b^{5} x^{12} + 264537 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} + 1119195 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} + 1939938 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} + 138567 \, A a^{5} + 855855 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} + 153153 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{2909907 \, x^{21}} \]

[In]

integrate((b*x^2+a)^5*(B*x^2+A)/x^22,x, algorithm="maxima")

[Out]

-1/2909907*(323323*B*b^5*x^12 + 264537*(5*B*a*b^4 + A*b^5)*x^10 + 1119195*(2*B*a^2*b^3 + A*a*b^4)*x^8 + 193993
8*(B*a^3*b^2 + A*a^2*b^3)*x^6 + 138567*A*a^5 + 855855*(B*a^4*b + 2*A*a^3*b^2)*x^4 + 153153*(B*a^5 + 5*A*a^4*b)
*x^2)/x^21

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.09 \[ \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^{22}} \, dx=-\frac {323323 \, B b^{5} x^{12} + 1322685 \, B a b^{4} x^{10} + 264537 \, A b^{5} x^{10} + 2238390 \, B a^{2} b^{3} x^{8} + 1119195 \, A a b^{4} x^{8} + 1939938 \, B a^{3} b^{2} x^{6} + 1939938 \, A a^{2} b^{3} x^{6} + 855855 \, B a^{4} b x^{4} + 1711710 \, A a^{3} b^{2} x^{4} + 153153 \, B a^{5} x^{2} + 765765 \, A a^{4} b x^{2} + 138567 \, A a^{5}}{2909907 \, x^{21}} \]

[In]

integrate((b*x^2+a)^5*(B*x^2+A)/x^22,x, algorithm="giac")

[Out]

-1/2909907*(323323*B*b^5*x^12 + 1322685*B*a*b^4*x^10 + 264537*A*b^5*x^10 + 2238390*B*a^2*b^3*x^8 + 1119195*A*a
*b^4*x^8 + 1939938*B*a^3*b^2*x^6 + 1939938*A*a^2*b^3*x^6 + 855855*B*a^4*b*x^4 + 1711710*A*a^3*b^2*x^4 + 153153
*B*a^5*x^2 + 765765*A*a^4*b*x^2 + 138567*A*a^5)/x^21

Mupad [B] (verification not implemented)

Time = 5.03 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.04 \[ \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^{22}} \, dx=-\frac {\frac {A\,a^5}{21}+x^8\,\left (\frac {10\,B\,a^2\,b^3}{13}+\frac {5\,A\,a\,b^4}{13}\right )+x^4\,\left (\frac {5\,B\,a^4\,b}{17}+\frac {10\,A\,a^3\,b^2}{17}\right )+x^2\,\left (\frac {B\,a^5}{19}+\frac {5\,A\,b\,a^4}{19}\right )+x^{10}\,\left (\frac {A\,b^5}{11}+\frac {5\,B\,a\,b^4}{11}\right )+x^6\,\left (\frac {2\,B\,a^3\,b^2}{3}+\frac {2\,A\,a^2\,b^3}{3}\right )+\frac {B\,b^5\,x^{12}}{9}}{x^{21}} \]

[In]

int(((A + B*x^2)*(a + b*x^2)^5)/x^22,x)

[Out]

-((A*a^5)/21 + x^8*((10*B*a^2*b^3)/13 + (5*A*a*b^4)/13) + x^4*((10*A*a^3*b^2)/17 + (5*B*a^4*b)/17) + x^2*((B*a
^5)/19 + (5*A*a^4*b)/19) + x^10*((A*b^5)/11 + (5*B*a*b^4)/11) + x^6*((2*A*a^2*b^3)/3 + (2*B*a^3*b^2)/3) + (B*b
^5*x^12)/9)/x^21