\(\int \frac {(a+b x^2)^5 (A+B x^2)}{x^{14}} \, dx\) [46]

Optimal result

Integrand size = 20, antiderivative size = 113 \[ \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^{14}} \, dx=-\frac {a^5 A}{13 x^{13}}-\frac {a^4 (5 A b+a B)}{11 x^{11}}-\frac {5 a^3 b (2 A b+a B)}{9 x^9}-\frac {10 a^2 b^2 (A b+a B)}{7 x^7}-\frac {a b^3 (A b+2 a B)}{x^5}-\frac {b^4 (A b+5 a B)}{3 x^3}-\frac {b^5 B}{x} \]

[Out]

-1/13*a^5*A/x^13-1/11*a^4*(5*A*b+B*a)/x^11-5/9*a^3*b*(2*A*b+B*a)/x^9-10/7*a^2*b^2*(A*b+B*a)/x^7-a*b^3*(A*b+2*B
*a)/x^5-1/3*b^4*(A*b+5*B*a)/x^3-b^5*B/x

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 113, 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} \[ \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^{14}} \, dx=-\frac {a^5 A}{13 x^{13}}-\frac {a^4 (a B+5 A b)}{11 x^{11}}-\frac {5 a^3 b (a B+2 A b)}{9 x^9}-\frac {10 a^2 b^2 (a B+A b)}{7 x^7}-\frac {b^4 (5 a B+A b)}{3 x^3}-\frac {a b^3 (2 a B+A b)}{x^5}-\frac {b^5 B}{x} \]

[In]

Int[((a + b*x^2)^5*(A + B*x^2))/x^14,x]

[Out]

-1/13*(a^5*A)/x^13 - (a^4*(5*A*b + a*B))/(11*x^11) - (5*a^3*b*(2*A*b + a*B))/(9*x^9) - (10*a^2*b^2*(A*b + a*B)
)/(7*x^7) - (a*b^3*(A*b + 2*a*B))/x^5 - (b^4*(A*b + 5*a*B))/(3*x^3) - (b^5*B)/x

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^{14}}+\frac {a^4 (5 A b+a B)}{x^{12}}+\frac {5 a^3 b (2 A b+a B)}{x^{10}}+\frac {10 a^2 b^2 (A b+a B)}{x^8}+\frac {5 a b^3 (A b+2 a B)}{x^6}+\frac {b^4 (A b+5 a B)}{x^4}+\frac {b^5 B}{x^2}\right ) \, dx \\ & = -\frac {a^5 A}{13 x^{13}}-\frac {a^4 (5 A b+a B)}{11 x^{11}}-\frac {5 a^3 b (2 A b+a B)}{9 x^9}-\frac {10 a^2 b^2 (A b+a B)}{7 x^7}-\frac {a b^3 (A b+2 a B)}{x^5}-\frac {b^4 (A b+5 a B)}{3 x^3}-\frac {b^5 B}{x} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.05 \[ \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^{14}} \, dx=-\frac {3003 b^5 x^{10} \left (A+3 B x^2\right )+3003 a b^4 x^8 \left (3 A+5 B x^2\right )+2574 a^2 b^3 x^6 \left (5 A+7 B x^2\right )+1430 a^3 b^2 x^4 \left (7 A+9 B x^2\right )+455 a^4 b x^2 \left (9 A+11 B x^2\right )+63 a^5 \left (11 A+13 B x^2\right )}{9009 x^{13}} \]

[In]

Integrate[((a + b*x^2)^5*(A + B*x^2))/x^14,x]

[Out]

-1/9009*(3003*b^5*x^10*(A + 3*B*x^2) + 3003*a*b^4*x^8*(3*A + 5*B*x^2) + 2574*a^2*b^3*x^6*(5*A + 7*B*x^2) + 143
0*a^3*b^2*x^4*(7*A + 9*B*x^2) + 455*a^4*b*x^2*(9*A + 11*B*x^2) + 63*a^5*(11*A + 13*B*x^2))/x^13

Maple [A] (verified)

Time = 2.55 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.92

method	result	size
default	\(-\frac {a^{5} A}{13 x^{13}}-\frac {a^{4} \left (5 A b +B a \right )}{11 x^{11}}-\frac {5 a^{3} b \left (2 A b +B a \right )}{9 x^{9}}-\frac {10 a^{2} b^{2} \left (A b +B a \right )}{7 x^{7}}-\frac {a \,b^{3} \left (A b +2 B a \right )}{x^{5}}-\frac {b^{4} \left (A b +5 B a \right )}{3 x^{3}}-\frac {b^{5} B}{x}\)	\(104\)
norman	\(\frac {-b^{5} B \,x^{12}+\left (-\frac {1}{3} b^{5} A -\frac {5}{3} a \,b^{4} B \right ) x^{10}+\left (-a \,b^{4} A -2 a^{2} b^{3} B \right ) x^{8}+\left (-\frac {10}{7} a^{2} b^{3} A -\frac {10}{7} a^{3} b^{2} B \right ) x^{6}+\left (-\frac {10}{9} a^{3} b^{2} A -\frac {5}{9} a^{4} b B \right ) x^{4}+\left (-\frac {5}{11} a^{4} b A -\frac {1}{11} a^{5} B \right ) x^{2}-\frac {a^{5} A}{13}}{x^{13}}\)	\(122\)
risch	\(\frac {-b^{5} B \,x^{12}+\left (-\frac {1}{3} b^{5} A -\frac {5}{3} a \,b^{4} B \right ) x^{10}+\left (-a \,b^{4} A -2 a^{2} b^{3} B \right ) x^{8}+\left (-\frac {10}{7} a^{2} b^{3} A -\frac {10}{7} a^{3} b^{2} B \right ) x^{6}+\left (-\frac {10}{9} a^{3} b^{2} A -\frac {5}{9} a^{4} b B \right ) x^{4}+\left (-\frac {5}{11} a^{4} b A -\frac {1}{11} a^{5} B \right ) x^{2}-\frac {a^{5} A}{13}}{x^{13}}\)	\(122\)
gosper	\(-\frac {9009 b^{5} B \,x^{12}+3003 A \,b^{5} x^{10}+15015 B a \,b^{4} x^{10}+9009 a A \,b^{4} x^{8}+18018 B \,a^{2} b^{3} x^{8}+12870 a^{2} A \,b^{3} x^{6}+12870 B \,a^{3} b^{2} x^{6}+10010 a^{3} A \,b^{2} x^{4}+5005 B \,a^{4} b \,x^{4}+4095 a^{4} A b \,x^{2}+819 a^{5} B \,x^{2}+693 a^{5} A}{9009 x^{13}}\)	\(128\)
parallelrisch	\(-\frac {9009 b^{5} B \,x^{12}+3003 A \,b^{5} x^{10}+15015 B a \,b^{4} x^{10}+9009 a A \,b^{4} x^{8}+18018 B \,a^{2} b^{3} x^{8}+12870 a^{2} A \,b^{3} x^{6}+12870 B \,a^{3} b^{2} x^{6}+10010 a^{3} A \,b^{2} x^{4}+5005 B \,a^{4} b \,x^{4}+4095 a^{4} A b \,x^{2}+819 a^{5} B \,x^{2}+693 a^{5} A}{9009 x^{13}}\)	\(128\)

[In]

int((b*x^2+a)^5*(B*x^2+A)/x^14,x,method=_RETURNVERBOSE)

[Out]

-1/13*a^5*A/x^13-1/11*a^4*(5*A*b+B*a)/x^11-5/9*a^3*b*(2*A*b+B*a)/x^9-10/7*a^2*b^2*(A*b+B*a)/x^7-a*b^3*(A*b+2*B
*a)/x^5-1/3*b^4*(A*b+5*B*a)/x^3-b^5*B/x

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.07 \[ \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^{14}} \, dx=-\frac {9009 \, B b^{5} x^{12} + 3003 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} + 9009 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} + 12870 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} + 693 \, A a^{5} + 5005 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} + 819 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{9009 \, x^{13}} \]

[In]

integrate((b*x^2+a)^5*(B*x^2+A)/x^14,x, algorithm="fricas")

[Out]

-1/9009*(9009*B*b^5*x^12 + 3003*(5*B*a*b^4 + A*b^5)*x^10 + 9009*(2*B*a^2*b^3 + A*a*b^4)*x^8 + 12870*(B*a^3*b^2
+ A*a^2*b^3)*x^6 + 693*A*a^5 + 5005*(B*a^4*b + 2*A*a^3*b^2)*x^4 + 819*(B*a^5 + 5*A*a^4*b)*x^2)/x^13

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^{14}} \, dx=\text {Timed out} \]

[In]

integrate((b*x**2+a)**5*(B*x**2+A)/x**14,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.07 \[ \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^{14}} \, dx=-\frac {9009 \, B b^{5} x^{12} + 3003 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} + 9009 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} + 12870 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} + 693 \, A a^{5} + 5005 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} + 819 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{9009 \, x^{13}} \]

[In]

integrate((b*x^2+a)^5*(B*x^2+A)/x^14,x, algorithm="maxima")

[Out]

-1/9009*(9009*B*b^5*x^12 + 3003*(5*B*a*b^4 + A*b^5)*x^10 + 9009*(2*B*a^2*b^3 + A*a*b^4)*x^8 + 12870*(B*a^3*b^2
+ A*a^2*b^3)*x^6 + 693*A*a^5 + 5005*(B*a^4*b + 2*A*a^3*b^2)*x^4 + 819*(B*a^5 + 5*A*a^4*b)*x^2)/x^13

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.12 \[ \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^{14}} \, dx=-\frac {9009 \, B b^{5} x^{12} + 15015 \, B a b^{4} x^{10} + 3003 \, A b^{5} x^{10} + 18018 \, B a^{2} b^{3} x^{8} + 9009 \, A a b^{4} x^{8} + 12870 \, B a^{3} b^{2} x^{6} + 12870 \, A a^{2} b^{3} x^{6} + 5005 \, B a^{4} b x^{4} + 10010 \, A a^{3} b^{2} x^{4} + 819 \, B a^{5} x^{2} + 4095 \, A a^{4} b x^{2} + 693 \, A a^{5}}{9009 \, x^{13}} \]

[In]

integrate((b*x^2+a)^5*(B*x^2+A)/x^14,x, algorithm="giac")

[Out]

-1/9009*(9009*B*b^5*x^12 + 15015*B*a*b^4*x^10 + 3003*A*b^5*x^10 + 18018*B*a^2*b^3*x^8 + 9009*A*a*b^4*x^8 + 128
70*B*a^3*b^2*x^6 + 12870*A*a^2*b^3*x^6 + 5005*B*a^4*b*x^4 + 10010*A*a^3*b^2*x^4 + 819*B*a^5*x^2 + 4095*A*a^4*b
*x^2 + 693*A*a^5)/x^13

Mupad [B] (verification not implemented)

Time = 4.98 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.06 \[ \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^{14}} \, dx=-\frac {\frac {A\,a^5}{13}+x^8\,\left (2\,B\,a^2\,b^3+A\,a\,b^4\right )+x^4\,\left (\frac {5\,B\,a^4\,b}{9}+\frac {10\,A\,a^3\,b^2}{9}\right )+x^2\,\left (\frac {B\,a^5}{11}+\frac {5\,A\,b\,a^4}{11}\right )+x^{10}\,\left (\frac {A\,b^5}{3}+\frac {5\,B\,a\,b^4}{3}\right )+x^6\,\left (\frac {10\,B\,a^3\,b^2}{7}+\frac {10\,A\,a^2\,b^3}{7}\right )+B\,b^5\,x^{12}}{x^{13}} \]

[In]

int(((A + B*x^2)*(a + b*x^2)^5)/x^14,x)

[Out]

-((A*a^5)/13 + x^8*(2*B*a^2*b^3 + A*a*b^4) + x^4*((10*A*a^3*b^2)/9 + (5*B*a^4*b)/9) + x^2*((B*a^5)/11 + (5*A*a
^4*b)/11) + x^10*((A*b^5)/3 + (5*B*a*b^4)/3) + x^6*((10*A*a^2*b^3)/7 + (10*B*a^3*b^2)/7) + B*b^5*x^12)/x^13