3.36 \(\int \frac {(a+b x^2)^5 (A+B x^2)}{x^4} \, dx\)

Optimal. Leaf size=108 \[ -\frac {a^5 A}{3 x^3}-\frac {a^4 (a B+5 A b)}{x}+5 a^3 b x (a B+2 A b)+\frac {10}{3} a^2 b^2 x^3 (a B+A b)+\frac {1}{7} b^4 x^7 (5 a B+A b)+a b^3 x^5 (2 a B+A b)+\frac {1}{9} b^5 B x^9 \]

[Out]

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

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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} \[ \frac {10}{3} a^2 b^2 x^3 (a B+A b)+5 a^3 b x (a B+2 A b)-\frac {a^4 (a B+5 A b)}{x}-\frac {a^5 A}{3 x^3}+\frac {1}{7} b^4 x^7 (5 a B+A b)+a b^3 x^5 (2 a B+A b)+\frac {1}{9} b^5 B x^9 \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x^2)^5*(A + B*x^2))/x^4,x]

[Out]

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

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

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Mathematica [A]  time = 0.04, size = 110, normalized size = 1.02 \[ -\frac {a^5 A}{3 x^3}+5 a^3 b x (a B+2 A b)+\frac {10}{3} a^2 b^2 x^3 (a B+A b)+\frac {a^5 (-B)-5 a^4 A b}{x}+\frac {1}{7} b^4 x^7 (5 a B+A b)+a b^3 x^5 (2 a B+A b)+\frac {1}{9} b^5 B x^9 \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x^2)^5*(A + B*x^2))/x^4,x]

[Out]

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

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fricas [A]  time = 0.43, size = 121, normalized size = 1.12 \[ \frac {7 \, B b^{5} x^{12} + 9 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} + 63 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} + 210 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} - 21 \, A a^{5} + 315 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} - 63 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{63 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5*(B*x^2+A)/x^4,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.37, size = 122, normalized size = 1.13 \[ \frac {1}{9} \, B b^{5} x^{9} + \frac {5}{7} \, B a b^{4} x^{7} + \frac {1}{7} \, A b^{5} x^{7} + 2 \, B a^{2} b^{3} x^{5} + A a b^{4} x^{5} + \frac {10}{3} \, B a^{3} b^{2} x^{3} + \frac {10}{3} \, A a^{2} b^{3} x^{3} + 5 \, B a^{4} b x + 10 \, A a^{3} b^{2} x - \frac {3 \, B a^{5} x^{2} + 15 \, A a^{4} b x^{2} + A a^{5}}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5*(B*x^2+A)/x^4,x, algorithm="giac")

[Out]

1/9*B*b^5*x^9 + 5/7*B*a*b^4*x^7 + 1/7*A*b^5*x^7 + 2*B*a^2*b^3*x^5 + A*a*b^4*x^5 + 10/3*B*a^3*b^2*x^3 + 10/3*A*
a^2*b^3*x^3 + 5*B*a^4*b*x + 10*A*a^3*b^2*x - 1/3*(3*B*a^5*x^2 + 15*A*a^4*b*x^2 + A*a^5)/x^3

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maple [A]  time = 0.00, size = 118, normalized size = 1.09 \[ \frac {B \,b^{5} x^{9}}{9}+\frac {A \,b^{5} x^{7}}{7}+\frac {5 B a \,b^{4} x^{7}}{7}+A a \,b^{4} x^{5}+2 B \,a^{2} b^{3} x^{5}+\frac {10 A \,a^{2} b^{3} x^{3}}{3}+\frac {10 B \,a^{3} b^{2} x^{3}}{3}+10 A \,a^{3} b^{2} x +5 B \,a^{4} b x -\frac {A \,a^{5}}{3 x^{3}}-\frac {\left (5 A b +B a \right ) a^{4}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^5*(B*x^2+A)/x^4,x)

[Out]

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

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maxima [A]  time = 1.03, size = 118, normalized size = 1.09 \[ \frac {1}{9} \, B b^{5} x^{9} + \frac {1}{7} \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{7} + {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{5} + \frac {10}{3} \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} + 5 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x - \frac {A a^{5} + 3 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5*(B*x^2+A)/x^4,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.04, size = 106, normalized size = 0.98 \[ x^7\,\left (\frac {A\,b^5}{7}+\frac {5\,B\,a\,b^4}{7}\right )-\frac {\frac {A\,a^5}{3}+x^2\,\left (B\,a^5+5\,A\,b\,a^4\right )}{x^3}+\frac {B\,b^5\,x^9}{9}+\frac {10\,a^2\,b^2\,x^3\,\left (A\,b+B\,a\right )}{3}+5\,a^3\,b\,x\,\left (2\,A\,b+B\,a\right )+a\,b^3\,x^5\,\left (A\,b+2\,B\,a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x^2)*(a + b*x^2)^5)/x^4,x)

[Out]

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

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sympy [A]  time = 0.36, size = 128, normalized size = 1.19 \[ \frac {B b^{5} x^{9}}{9} + x^{7} \left (\frac {A b^{5}}{7} + \frac {5 B a b^{4}}{7}\right ) + x^{5} \left (A a b^{4} + 2 B a^{2} b^{3}\right ) + x^{3} \left (\frac {10 A a^{2} b^{3}}{3} + \frac {10 B a^{3} b^{2}}{3}\right ) + x \left (10 A a^{3} b^{2} + 5 B a^{4} b\right ) + \frac {- A a^{5} + x^{2} \left (- 15 A a^{4} b - 3 B a^{5}\right )}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**5*(B*x**2+A)/x**4,x)

[Out]

B*b**5*x**9/9 + x**7*(A*b**5/7 + 5*B*a*b**4/7) + x**5*(A*a*b**4 + 2*B*a**2*b**3) + x**3*(10*A*a**2*b**3/3 + 10
*B*a**3*b**2/3) + x*(10*A*a**3*b**2 + 5*B*a**4*b) + (-A*a**5 + x**2*(-15*A*a**4*b - 3*B*a**5))/(3*x**3)

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