3.43 \(\int \frac {(a+b x^2)^5 (A+B x^2)}{x^{11}} \, dx\)

Optimal. Leaf size=113 \[ -\frac {a^5 A}{10 x^{10}}-\frac {a^4 (a B+5 A b)}{8 x^8}-\frac {5 a^3 b (a B+2 A b)}{6 x^6}-\frac {5 a^2 b^2 (a B+A b)}{2 x^4}+b^4 \log (x) (5 a B+A b)-\frac {5 a b^3 (2 a B+A b)}{2 x^2}+\frac {1}{2} b^5 B x^2 \]

[Out]

-1/10*a^5*A/x^10-1/8*a^4*(5*A*b+B*a)/x^8-5/6*a^3*b*(2*A*b+B*a)/x^6-5/2*a^2*b^2*(A*b+B*a)/x^4-5/2*a*b^3*(A*b+2*
B*a)/x^2+1/2*b^5*B*x^2+b^4*(A*b+5*B*a)*ln(x)

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Rubi [A]  time = 0.09, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {446, 76} \[ -\frac {5 a^2 b^2 (a B+A b)}{2 x^4}-\frac {a^4 (a B+5 A b)}{8 x^8}-\frac {5 a^3 b (a B+2 A b)}{6 x^6}-\frac {a^5 A}{10 x^{10}}-\frac {5 a b^3 (2 a B+A b)}{2 x^2}+b^4 \log (x) (5 a B+A b)+\frac {1}{2} b^5 B x^2 \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x^2)^5*(A + B*x^2))/x^11,x]

[Out]

-(a^5*A)/(10*x^10) - (a^4*(5*A*b + a*B))/(8*x^8) - (5*a^3*b*(2*A*b + a*B))/(6*x^6) - (5*a^2*b^2*(A*b + a*B))/(
2*x^4) - (5*a*b^3*(A*b + 2*a*B))/(2*x^2) + (b^5*B*x^2)/2 + b^4*(A*b + 5*a*B)*Log[x]

Rule 76

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && (NeQ[n, -1] || EqQ[p, 1]) && N
eQ[b*e + a*f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && Rational
Q[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 1])

Rule 446

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

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

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Mathematica [A]  time = 0.05, size = 116, normalized size = 1.03 \[ b^4 \log (x) (5 a B+A b)-\frac {3 a^5 \left (4 A+5 B x^2\right )+25 a^4 b x^2 \left (3 A+4 B x^2\right )+100 a^3 b^2 x^4 \left (2 A+3 B x^2\right )+300 a^2 b^3 x^6 \left (A+2 B x^2\right )+300 a A b^4 x^8-60 b^5 B x^{12}}{120 x^{10}} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x^2)^5*(A + B*x^2))/x^11,x]

[Out]

-1/120*(300*a*A*b^4*x^8 - 60*b^5*B*x^12 + 300*a^2*b^3*x^6*(A + 2*B*x^2) + 100*a^3*b^2*x^4*(2*A + 3*B*x^2) + 25
*a^4*b*x^2*(3*A + 4*B*x^2) + 3*a^5*(4*A + 5*B*x^2))/x^10 + b^4*(A*b + 5*a*B)*Log[x]

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fricas [A]  time = 0.43, size = 123, normalized size = 1.09 \[ \frac {60 \, B b^{5} x^{12} + 120 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} \log \relax (x) - 300 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} - 300 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} - 12 \, A a^{5} - 100 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} - 15 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{120 \, x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5*(B*x^2+A)/x^11,x, algorithm="fricas")

[Out]

1/120*(60*B*b^5*x^12 + 120*(5*B*a*b^4 + A*b^5)*x^10*log(x) - 300*(2*B*a^2*b^3 + A*a*b^4)*x^8 - 300*(B*a^3*b^2
+ A*a^2*b^3)*x^6 - 12*A*a^5 - 100*(B*a^4*b + 2*A*a^3*b^2)*x^4 - 15*(B*a^5 + 5*A*a^4*b)*x^2)/x^10

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giac [A]  time = 0.43, size = 147, normalized size = 1.30 \[ \frac {1}{2} \, B b^{5} x^{2} + \frac {1}{2} \, {\left (5 \, B a b^{4} + A b^{5}\right )} \log \left (x^{2}\right ) - \frac {685 \, B a b^{4} x^{10} + 137 \, A b^{5} x^{10} + 600 \, B a^{2} b^{3} x^{8} + 300 \, A a b^{4} x^{8} + 300 \, B a^{3} b^{2} x^{6} + 300 \, A a^{2} b^{3} x^{6} + 100 \, B a^{4} b x^{4} + 200 \, A a^{3} b^{2} x^{4} + 15 \, B a^{5} x^{2} + 75 \, A a^{4} b x^{2} + 12 \, A a^{5}}{120 \, x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5*(B*x^2+A)/x^11,x, algorithm="giac")

[Out]

1/2*B*b^5*x^2 + 1/2*(5*B*a*b^4 + A*b^5)*log(x^2) - 1/120*(685*B*a*b^4*x^10 + 137*A*b^5*x^10 + 600*B*a^2*b^3*x^
8 + 300*A*a*b^4*x^8 + 300*B*a^3*b^2*x^6 + 300*A*a^2*b^3*x^6 + 100*B*a^4*b*x^4 + 200*A*a^3*b^2*x^4 + 15*B*a^5*x
^2 + 75*A*a^4*b*x^2 + 12*A*a^5)/x^10

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maple [A]  time = 0.01, size = 123, normalized size = 1.09 \[ \frac {B \,b^{5} x^{2}}{2}+A \,b^{5} \ln \relax (x )+5 B a \,b^{4} \ln \relax (x )-\frac {5 A a \,b^{4}}{2 x^{2}}-\frac {5 B \,a^{2} b^{3}}{x^{2}}-\frac {5 A \,a^{2} b^{3}}{2 x^{4}}-\frac {5 B \,a^{3} b^{2}}{2 x^{4}}-\frac {5 A \,a^{3} b^{2}}{3 x^{6}}-\frac {5 B \,a^{4} b}{6 x^{6}}-\frac {5 A \,a^{4} b}{8 x^{8}}-\frac {B \,a^{5}}{8 x^{8}}-\frac {A \,a^{5}}{10 x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^5*(B*x^2+A)/x^11,x)

[Out]

1/2*b^5*B*x^2-5/2*a^2*b^3/x^4*A-5/2*a^3*b^2/x^4*B-1/10*a^5*A/x^10-5/3*a^3*b^2/x^6*A-5/6*a^4*b/x^6*B-5/8*a^4/x^
8*A*b-1/8*a^5/x^8*B-5/2*a*b^4/x^2*A-5*a^2*b^3/x^2*B+A*ln(x)*b^5+5*B*ln(x)*a*b^4

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maxima [A]  time = 1.02, size = 123, normalized size = 1.09 \[ \frac {1}{2} \, B b^{5} x^{2} + \frac {1}{2} \, {\left (5 \, B a b^{4} + A b^{5}\right )} \log \left (x^{2}\right ) - \frac {300 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} + 300 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} + 12 \, A a^{5} + 100 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} + 15 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{120 \, x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5*(B*x^2+A)/x^11,x, algorithm="maxima")

[Out]

1/2*B*b^5*x^2 + 1/2*(5*B*a*b^4 + A*b^5)*log(x^2) - 1/120*(300*(2*B*a^2*b^3 + A*a*b^4)*x^8 + 300*(B*a^3*b^2 + A
*a^2*b^3)*x^6 + 12*A*a^5 + 100*(B*a^4*b + 2*A*a^3*b^2)*x^4 + 15*(B*a^5 + 5*A*a^4*b)*x^2)/x^10

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mupad [B]  time = 0.11, size = 121, normalized size = 1.07 \[ \ln \relax (x)\,\left (A\,b^5+5\,B\,a\,b^4\right )-\frac {\frac {A\,a^5}{10}+x^8\,\left (5\,B\,a^2\,b^3+\frac {5\,A\,a\,b^4}{2}\right )+x^4\,\left (\frac {5\,B\,a^4\,b}{6}+\frac {5\,A\,a^3\,b^2}{3}\right )+x^2\,\left (\frac {B\,a^5}{8}+\frac {5\,A\,b\,a^4}{8}\right )+x^6\,\left (\frac {5\,B\,a^3\,b^2}{2}+\frac {5\,A\,a^2\,b^3}{2}\right )}{x^{10}}+\frac {B\,b^5\,x^2}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x^2)*(a + b*x^2)^5)/x^11,x)

[Out]

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

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sympy [A]  time = 5.00, size = 129, normalized size = 1.14 \[ \frac {B b^{5} x^{2}}{2} + b^{4} \left (A b + 5 B a\right ) \log {\relax (x )} + \frac {- 12 A a^{5} + x^{8} \left (- 300 A a b^{4} - 600 B a^{2} b^{3}\right ) + x^{6} \left (- 300 A a^{2} b^{3} - 300 B a^{3} b^{2}\right ) + x^{4} \left (- 200 A a^{3} b^{2} - 100 B a^{4} b\right ) + x^{2} \left (- 75 A a^{4} b - 15 B a^{5}\right )}{120 x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**5*(B*x**2+A)/x**11,x)

[Out]

B*b**5*x**2/2 + b**4*(A*b + 5*B*a)*log(x) + (-12*A*a**5 + x**8*(-300*A*a*b**4 - 600*B*a**2*b**3) + x**6*(-300*
A*a**2*b**3 - 300*B*a**3*b**2) + x**4*(-200*A*a**3*b**2 - 100*B*a**4*b) + x**2*(-75*A*a**4*b - 15*B*a**5))/(12
0*x**10)

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