\(\int \frac {(a+b x^2)^5 (A+B x^2)}{x^{10}} \, dx\) [42]

Optimal result

Integrand size = 20, antiderivative size = 108 \[ \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^{10}} \, dx=-\frac {a^5 A}{9 x^9}-\frac {a^4 (5 A b+a B)}{7 x^7}-\frac {a^3 b (2 A b+a B)}{x^5}-\frac {10 a^2 b^2 (A b+a B)}{3 x^3}-\frac {5 a b^3 (A b+2 a B)}{x}+b^4 (A b+5 a B) x+\frac {1}{3} b^5 B x^3 \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 108, 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} \[ \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^{10}} \, dx=-\frac {a^5 A}{9 x^9}-\frac {a^4 (a B+5 A b)}{7 x^7}-\frac {a^3 b (a B+2 A b)}{x^5}-\frac {10 a^2 b^2 (a B+A b)}{3 x^3}+b^4 x (5 a B+A b)-\frac {5 a b^3 (2 a B+A b)}{x}+\frac {1}{3} b^5 B x^3 \]

[In]

Int[((a + b*x^2)^5*(A + B*x^2))/x^10,x]

[Out]

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

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 (b^4 (A b+5 a B)+\frac {a^5 A}{x^{10}}+\frac {a^4 (5 A b+a B)}{x^8}+\frac {5 a^3 b (2 A b+a B)}{x^6}+\frac {10 a^2 b^2 (A b+a B)}{x^4}+\frac {5 a b^3 (A b+2 a B)}{x^2}+b^5 B x^2\right ) \, dx \\ & = -\frac {a^5 A}{9 x^9}-\frac {a^4 (5 A b+a B)}{7 x^7}-\frac {a^3 b (2 A b+a B)}{x^5}-\frac {10 a^2 b^2 (A b+a B)}{3 x^3}-\frac {5 a b^3 (A b+2 a B)}{x}+b^4 (A b+5 a B) x+\frac {1}{3} b^5 B x^3 \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.06 \[ \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^{10}} \, dx=-\frac {315 a b^4 x^8 \left (A-B x^2\right )-21 b^5 x^{10} \left (3 A+B x^2\right )+210 a^2 b^3 x^6 \left (A+3 B x^2\right )+42 a^3 b^2 x^4 \left (3 A+5 B x^2\right )+9 a^4 b x^2 \left (5 A+7 B x^2\right )+a^5 \left (7 A+9 B x^2\right )}{63 x^9} \]

[In]

Integrate[((a + b*x^2)^5*(A + B*x^2))/x^10,x]

[Out]

-1/63*(315*a*b^4*x^8*(A - B*x^2) - 21*b^5*x^10*(3*A + B*x^2) + 210*a^2*b^3*x^6*(A + 3*B*x^2) + 42*a^3*b^2*x^4*
(3*A + 5*B*x^2) + 9*a^4*b*x^2*(5*A + 7*B*x^2) + a^5*(7*A + 9*B*x^2))/x^9

Maple [A] (verified)

Time = 2.45 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.94

method	result	size
default	\(\frac {b^{5} B \,x^{3}}{3}+A \,b^{5} x +5 B a \,b^{4} x -\frac {a^{4} \left (5 A b +B a \right )}{7 x^{7}}-\frac {10 a^{2} b^{2} \left (A b +B a \right )}{3 x^{3}}-\frac {5 a \,b^{3} \left (A b +2 B a \right )}{x}-\frac {a^{3} b \left (2 A b +B a \right )}{x^{5}}-\frac {a^{5} A}{9 x^{9}}\)	\(102\)
risch	\(\frac {b^{5} B \,x^{3}}{3}+A \,b^{5} x +5 B a \,b^{4} x +\frac {\left (-5 a \,b^{4} A -10 a^{2} b^{3} B \right ) x^{8}+\left (-\frac {10}{3} a^{2} b^{3} A -\frac {10}{3} a^{3} b^{2} B \right ) x^{6}+\left (-2 a^{3} b^{2} A -a^{4} b B \right ) x^{4}+\left (-\frac {5}{7} a^{4} b A -\frac {1}{7} a^{5} B \right ) x^{2}-\frac {a^{5} A}{9}}{x^{9}}\)	\(119\)
norman	\(\frac {\frac {b^{5} B \,x^{12}}{3}+\left (b^{5} A +5 a \,b^{4} B \right ) x^{10}+\left (-5 a \,b^{4} A -10 a^{2} b^{3} B \right ) x^{8}+\left (-\frac {10}{3} a^{2} b^{3} A -\frac {10}{3} a^{3} b^{2} B \right ) x^{6}+\left (-2 a^{3} b^{2} A -a^{4} b B \right ) x^{4}+\left (-\frac {5}{7} a^{4} b A -\frac {1}{7} a^{5} B \right ) x^{2}-\frac {a^{5} A}{9}}{x^{9}}\)	\(121\)
gosper	\(-\frac {-21 b^{5} B \,x^{12}-63 A \,b^{5} x^{10}-315 B a \,b^{4} x^{10}+315 a A \,b^{4} x^{8}+630 B \,a^{2} b^{3} x^{8}+210 a^{2} A \,b^{3} x^{6}+210 B \,a^{3} b^{2} x^{6}+126 a^{3} A \,b^{2} x^{4}+63 B \,a^{4} b \,x^{4}+45 a^{4} A b \,x^{2}+9 a^{5} B \,x^{2}+7 a^{5} A}{63 x^{9}}\)	\(128\)
parallelrisch	\(\frac {21 b^{5} B \,x^{12}+63 A \,b^{5} x^{10}+315 B a \,b^{4} x^{10}-315 a A \,b^{4} x^{8}-630 B \,a^{2} b^{3} x^{8}-210 a^{2} A \,b^{3} x^{6}-210 B \,a^{3} b^{2} x^{6}-126 a^{3} A \,b^{2} x^{4}-63 B \,a^{4} b \,x^{4}-45 a^{4} A b \,x^{2}-9 a^{5} B \,x^{2}-7 a^{5} A}{63 x^{9}}\)	\(128\)

[In]

int((b*x^2+a)^5*(B*x^2+A)/x^10,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.12 \[ \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^{10}} \, dx=\frac {21 \, B b^{5} x^{12} + 63 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} - 315 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} - 210 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} - 7 \, A a^{5} - 63 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} - 9 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{63 \, x^{9}} \]

[In]

integrate((b*x^2+a)^5*(B*x^2+A)/x^10,x, algorithm="fricas")

[Out]

1/63*(21*B*b^5*x^12 + 63*(5*B*a*b^4 + A*b^5)*x^10 - 315*(2*B*a^2*b^3 + A*a*b^4)*x^8 - 210*(B*a^3*b^2 + A*a^2*b
^3)*x^6 - 7*A*a^5 - 63*(B*a^4*b + 2*A*a^3*b^2)*x^4 - 9*(B*a^5 + 5*A*a^4*b)*x^2)/x^9

Sympy [A] (verification not implemented)

Time = 3.12 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.19 \[ \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^{10}} \, dx=\frac {B b^{5} x^{3}}{3} + x \left (A b^{5} + 5 B a b^{4}\right ) + \frac {- 7 A a^{5} + x^{8} \left (- 315 A a b^{4} - 630 B a^{2} b^{3}\right ) + x^{6} \left (- 210 A a^{2} b^{3} - 210 B a^{3} b^{2}\right ) + x^{4} \left (- 126 A a^{3} b^{2} - 63 B a^{4} b\right ) + x^{2} \left (- 45 A a^{4} b - 9 B a^{5}\right )}{63 x^{9}} \]

[In]

integrate((b*x**2+a)**5*(B*x**2+A)/x**10,x)

[Out]

B*b**5*x**3/3 + x*(A*b**5 + 5*B*a*b**4) + (-7*A*a**5 + x**8*(-315*A*a*b**4 - 630*B*a**2*b**3) + x**6*(-210*A*a
**2*b**3 - 210*B*a**3*b**2) + x**4*(-126*A*a**3*b**2 - 63*B*a**4*b) + x**2*(-45*A*a**4*b - 9*B*a**5))/(63*x**9
)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.10 \[ \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^{10}} \, dx=\frac {1}{3} \, B b^{5} x^{3} + {\left (5 \, B a b^{4} + A b^{5}\right )} x - \frac {315 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} + 210 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} + 7 \, A a^{5} + 63 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} + 9 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{63 \, x^{9}} \]

[In]

integrate((b*x^2+a)^5*(B*x^2+A)/x^10,x, algorithm="maxima")

[Out]

1/3*B*b^5*x^3 + (5*B*a*b^4 + A*b^5)*x - 1/63*(315*(2*B*a^2*b^3 + A*a*b^4)*x^8 + 210*(B*a^3*b^2 + A*a^2*b^3)*x^
6 + 7*A*a^5 + 63*(B*a^4*b + 2*A*a^3*b^2)*x^4 + 9*(B*a^5 + 5*A*a^4*b)*x^2)/x^9

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.14 \[ \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^{10}} \, dx=\frac {1}{3} \, B b^{5} x^{3} + 5 \, B a b^{4} x + A b^{5} x - \frac {630 \, B a^{2} b^{3} x^{8} + 315 \, A a b^{4} x^{8} + 210 \, B a^{3} b^{2} x^{6} + 210 \, A a^{2} b^{3} x^{6} + 63 \, B a^{4} b x^{4} + 126 \, A a^{3} b^{2} x^{4} + 9 \, B a^{5} x^{2} + 45 \, A a^{4} b x^{2} + 7 \, A a^{5}}{63 \, x^{9}} \]

[In]

integrate((b*x^2+a)^5*(B*x^2+A)/x^10,x, algorithm="giac")

[Out]

1/3*B*b^5*x^3 + 5*B*a*b^4*x + A*b^5*x - 1/63*(630*B*a^2*b^3*x^8 + 315*A*a*b^4*x^8 + 210*B*a^3*b^2*x^6 + 210*A*
a^2*b^3*x^6 + 63*B*a^4*b*x^4 + 126*A*a^3*b^2*x^4 + 9*B*a^5*x^2 + 45*A*a^4*b*x^2 + 7*A*a^5)/x^9

Mupad [B] (verification not implemented)

Time = 4.92 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.10 \[ \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^{10}} \, dx=x\,\left (A\,b^5+5\,B\,a\,b^4\right )-\frac {\frac {A\,a^5}{9}+x^4\,\left (B\,a^4\,b+2\,A\,a^3\,b^2\right )+x^8\,\left (10\,B\,a^2\,b^3+5\,A\,a\,b^4\right )+x^2\,\left (\frac {B\,a^5}{7}+\frac {5\,A\,b\,a^4}{7}\right )+x^6\,\left (\frac {10\,B\,a^3\,b^2}{3}+\frac {10\,A\,a^2\,b^3}{3}\right )}{x^9}+\frac {B\,b^5\,x^3}{3} \]

[In]

int(((A + B*x^2)*(a + b*x^2)^5)/x^10,x)

[Out]

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