3.34 \(\int \frac {(a+b x^2)^5 (A+B x^2)}{x^2} \, dx\)

Optimal. Leaf size=108 \[ -\frac {a^5 A}{x}+a^4 x (a B+5 A b)+\frac {5}{3} a^3 b x^3 (a B+2 A b)+2 a^2 b^2 x^5 (a B+A b)+\frac {1}{9} b^4 x^9 (5 a B+A b)+\frac {5}{7} a b^3 x^7 (2 a B+A b)+\frac {1}{11} b^5 B x^{11} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.06, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {448} \[ 2 a^2 b^2 x^5 (a B+A b)+\frac {5}{3} a^3 b x^3 (a B+2 A b)+a^4 x (a B+5 A b)-\frac {a^5 A}{x}+\frac {1}{9} b^4 x^9 (5 a B+A b)+\frac {5}{7} a b^3 x^7 (2 a B+A b)+\frac {1}{11} b^5 B x^{11} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 448

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*} \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^2} \, dx &=\int \left (a^4 (5 A b+a B)+\frac {a^5 A}{x^2}+5 a^3 b (2 A b+a B) x^2+10 a^2 b^2 (A b+a B) x^4+5 a b^3 (A b+2 a B) x^6+b^4 (A b+5 a B) x^8+b^5 B x^{10}\right ) \, dx\\ &=-\frac {a^5 A}{x}+a^4 (5 A b+a B) x+\frac {5}{3} a^3 b (2 A b+a B) x^3+2 a^2 b^2 (A b+a B) x^5+\frac {5}{7} a b^3 (A b+2 a B) x^7+\frac {1}{9} b^4 (A b+5 a B) x^9+\frac {1}{11} b^5 B x^{11}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.03, size = 108, normalized size = 1.00 \[ -\frac {a^5 A}{x}+a^4 x (a B+5 A b)+\frac {5}{3} a^3 b x^3 (a B+2 A b)+2 a^2 b^2 x^5 (a B+A b)+\frac {1}{9} b^4 x^9 (5 a B+A b)+\frac {5}{7} a b^3 x^7 (2 a B+A b)+\frac {1}{11} b^5 B x^{11} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.42, size = 121, normalized size = 1.12 \[ \frac {63 \, B b^{5} x^{12} + 77 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} + 495 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} + 1386 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} - 693 \, A a^{5} + 1155 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} + 693 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{693 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/693*(63*B*b^5*x^12 + 77*(5*B*a*b^4 + A*b^5)*x^10 + 495*(2*B*a^2*b^3 + A*a*b^4)*x^8 + 1386*(B*a^3*b^2 + A*a^2
*b^3)*x^6 - 693*A*a^5 + 1155*(B*a^4*b + 2*A*a^3*b^2)*x^4 + 693*(B*a^5 + 5*A*a^4*b)*x^2)/x

________________________________________________________________________________________

giac [A]  time = 0.31, size = 120, normalized size = 1.11 \[ \frac {1}{11} \, B b^{5} x^{11} + \frac {5}{9} \, B a b^{4} x^{9} + \frac {1}{9} \, A b^{5} x^{9} + \frac {10}{7} \, B a^{2} b^{3} x^{7} + \frac {5}{7} \, A a b^{4} x^{7} + 2 \, B a^{3} b^{2} x^{5} + 2 \, A a^{2} b^{3} x^{5} + \frac {5}{3} \, B a^{4} b x^{3} + \frac {10}{3} \, A a^{3} b^{2} x^{3} + B a^{5} x + 5 \, A a^{4} b x - \frac {A a^{5}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.00, size = 121, normalized size = 1.12 \[ \frac {B \,b^{5} x^{11}}{11}+\frac {A \,b^{5} x^{9}}{9}+\frac {5 B a \,b^{4} x^{9}}{9}+\frac {5 A a \,b^{4} x^{7}}{7}+\frac {10 B \,a^{2} b^{3} x^{7}}{7}+2 A \,a^{2} b^{3} x^{5}+2 B \,a^{3} b^{2} x^{5}+\frac {10 A \,a^{3} b^{2} x^{3}}{3}+\frac {5 B \,a^{4} b \,x^{3}}{3}+5 A \,a^{4} b x +B \,a^{5} x -\frac {A \,a^{5}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 1.13, size = 116, normalized size = 1.07 \[ \frac {1}{11} \, B b^{5} x^{11} + \frac {1}{9} \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{9} + \frac {5}{7} \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{7} + 2 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{5} - \frac {A a^{5}}{x} + \frac {5}{3} \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{3} + {\left (B a^{5} + 5 \, A a^{4} b\right )} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 0.04, size = 104, normalized size = 0.96 \[ x\,\left (B\,a^5+5\,A\,b\,a^4\right )+x^9\,\left (\frac {A\,b^5}{9}+\frac {5\,B\,a\,b^4}{9}\right )-\frac {A\,a^5}{x}+\frac {B\,b^5\,x^{11}}{11}+2\,a^2\,b^2\,x^5\,\left (A\,b+B\,a\right )+\frac {5\,a^3\,b\,x^3\,\left (2\,A\,b+B\,a\right )}{3}+\frac {5\,a\,b^3\,x^7\,\left (A\,b+2\,B\,a\right )}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 0.24, size = 126, normalized size = 1.17 \[ - \frac {A a^{5}}{x} + \frac {B b^{5} x^{11}}{11} + x^{9} \left (\frac {A b^{5}}{9} + \frac {5 B a b^{4}}{9}\right ) + x^{7} \left (\frac {5 A a b^{4}}{7} + \frac {10 B a^{2} b^{3}}{7}\right ) + x^{5} \left (2 A a^{2} b^{3} + 2 B a^{3} b^{2}\right ) + x^{3} \left (\frac {10 A a^{3} b^{2}}{3} + \frac {5 B a^{4} b}{3}\right ) + x \left (5 A a^{4} b + B a^{5}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________