3.1.41 \(\int \frac {(a+b x^2)^5 (A+B x^2)}{x^9} \, dx\)

Optimal. Leaf size=112 \[ -\frac {a^5 A}{8 x^8}-\frac {a^4 (a B+5 A b)}{6 x^6}-\frac {5 a^3 b (a B+2 A b)}{4 x^4}-\frac {5 a^2 b^2 (a B+A b)}{x^2}+\frac {1}{2} b^4 x^2 (5 a B+A b)+5 a b^3 \log (x) (2 a B+A b)+\frac {1}{4} b^5 B x^4 \]

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Rubi [A]  time = 0.10, antiderivative size = 112, 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} \begin {gather*} -\frac {5 a^2 b^2 (a B+A b)}{x^2}-\frac {a^4 (a B+5 A b)}{6 x^6}-\frac {5 a^3 b (a B+2 A b)}{4 x^4}-\frac {a^5 A}{8 x^8}+\frac {1}{2} b^4 x^2 (5 a B+A b)+5 a b^3 \log (x) (2 a B+A b)+\frac {1}{4} b^5 B x^4 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x^2)^5*(A + B*x^2))/x^9,x]

[Out]

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

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Mathematica [A]  time = 0.05, size = 116, normalized size = 1.04 \begin {gather*} 5 a b^3 \log (x) (2 a B+A b)-\frac {a^5 \left (3 A+4 B x^2\right )+10 a^4 b x^2 \left (2 A+3 B x^2\right )+60 a^3 b^2 x^4 \left (A+2 B x^2\right )+120 a^2 A b^3 x^6-60 a b^4 B x^{10}-6 b^5 x^{10} \left (2 A+B x^2\right )}{24 x^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x^2)^5*(A + B*x^2))/x^9,x]

[Out]

-1/24*(120*a^2*A*b^3*x^6 - 60*a*b^4*B*x^10 - 6*b^5*x^10*(2*A + B*x^2) + 60*a^3*b^2*x^4*(A + 2*B*x^2) + 10*a^4*
b*x^2*(2*A + 3*B*x^2) + a^5*(3*A + 4*B*x^2))/x^8 + 5*a*b^3*(A*b + 2*a*B)*Log[x]

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^9} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((a + b*x^2)^5*(A + B*x^2))/x^9,x]

[Out]

IntegrateAlgebraic[((a + b*x^2)^5*(A + B*x^2))/x^9, x]

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fricas [A]  time = 0.43, size = 123, normalized size = 1.10 \begin {gather*} \frac {6 \, B b^{5} x^{12} + 12 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} + 120 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} \log \relax (x) - 120 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} - 3 \, A a^{5} - 30 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} - 4 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{24 \, x^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5*(B*x^2+A)/x^9,x, algorithm="fricas")

[Out]

1/24*(6*B*b^5*x^12 + 12*(5*B*a*b^4 + A*b^5)*x^10 + 120*(2*B*a^2*b^3 + A*a*b^4)*x^8*log(x) - 120*(B*a^3*b^2 + A
*a^2*b^3)*x^6 - 3*A*a^5 - 30*(B*a^4*b + 2*A*a^3*b^2)*x^4 - 4*(B*a^5 + 5*A*a^4*b)*x^2)/x^8

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giac [A]  time = 0.37, size = 150, normalized size = 1.34 \begin {gather*} \frac {1}{4} \, B b^{5} x^{4} + \frac {5}{2} \, B a b^{4} x^{2} + \frac {1}{2} \, A b^{5} x^{2} + \frac {5}{2} \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} \log \left (x^{2}\right ) - \frac {250 \, B a^{2} b^{3} x^{8} + 125 \, A a b^{4} x^{8} + 120 \, B a^{3} b^{2} x^{6} + 120 \, A a^{2} b^{3} x^{6} + 30 \, B a^{4} b x^{4} + 60 \, A a^{3} b^{2} x^{4} + 4 \, B a^{5} x^{2} + 20 \, A a^{4} b x^{2} + 3 \, A a^{5}}{24 \, x^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5*(B*x^2+A)/x^9,x, algorithm="giac")

[Out]

1/4*B*b^5*x^4 + 5/2*B*a*b^4*x^2 + 1/2*A*b^5*x^2 + 5/2*(2*B*a^2*b^3 + A*a*b^4)*log(x^2) - 1/24*(250*B*a^2*b^3*x
^8 + 125*A*a*b^4*x^8 + 120*B*a^3*b^2*x^6 + 120*A*a^2*b^3*x^6 + 30*B*a^4*b*x^4 + 60*A*a^3*b^2*x^4 + 4*B*a^5*x^2
+ 20*A*a^4*b*x^2 + 3*A*a^5)/x^8

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maple [A]  time = 0.01, size = 124, normalized size = 1.11 \begin {gather*} \frac {B \,b^{5} x^{4}}{4}+\frac {A \,b^{5} x^{2}}{2}+\frac {5 B a \,b^{4} x^{2}}{2}+5 A a \,b^{4} \ln \relax (x )+10 B \,a^{2} b^{3} \ln \relax (x )-\frac {5 A \,a^{2} b^{3}}{x^{2}}-\frac {5 B \,a^{3} b^{2}}{x^{2}}-\frac {5 A \,a^{3} b^{2}}{2 x^{4}}-\frac {5 B \,a^{4} b}{4 x^{4}}-\frac {5 A \,a^{4} b}{6 x^{6}}-\frac {B \,a^{5}}{6 x^{6}}-\frac {A \,a^{5}}{8 x^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^5*(B*x^2+A)/x^9,x)

[Out]

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

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maxima [A]  time = 0.96, size = 123, normalized size = 1.10 \begin {gather*} \frac {1}{4} \, B b^{5} x^{4} + \frac {1}{2} \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{2} + \frac {5}{2} \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} \log \left (x^{2}\right ) - \frac {120 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} + 3 \, A a^{5} + 30 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} + 4 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{24 \, x^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5*(B*x^2+A)/x^9,x, algorithm="maxima")

[Out]

1/4*B*b^5*x^4 + 1/2*(5*B*a*b^4 + A*b^5)*x^2 + 5/2*(2*B*a^2*b^3 + A*a*b^4)*log(x^2) - 1/24*(120*(B*a^3*b^2 + A*
a^2*b^3)*x^6 + 3*A*a^5 + 30*(B*a^4*b + 2*A*a^3*b^2)*x^4 + 4*(B*a^5 + 5*A*a^4*b)*x^2)/x^8

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mupad [B]  time = 0.06, size = 122, normalized size = 1.09 \begin {gather*} \ln \relax (x)\,\left (10\,B\,a^2\,b^3+5\,A\,a\,b^4\right )-\frac {\frac {A\,a^5}{8}+x^4\,\left (\frac {5\,B\,a^4\,b}{4}+\frac {5\,A\,a^3\,b^2}{2}\right )+x^2\,\left (\frac {B\,a^5}{6}+\frac {5\,A\,b\,a^4}{6}\right )+x^6\,\left (5\,B\,a^3\,b^2+5\,A\,a^2\,b^3\right )}{x^8}+x^2\,\left (\frac {A\,b^5}{2}+\frac {5\,B\,a\,b^4}{2}\right )+\frac {B\,b^5\,x^4}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x^2)*(a + b*x^2)^5)/x^9,x)

[Out]

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

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sympy [A]  time = 2.73, size = 129, normalized size = 1.15 \begin {gather*} \frac {B b^{5} x^{4}}{4} + 5 a b^{3} \left (A b + 2 B a\right ) \log {\relax (x )} + x^{2} \left (\frac {A b^{5}}{2} + \frac {5 B a b^{4}}{2}\right ) + \frac {- 3 A a^{5} + x^{6} \left (- 120 A a^{2} b^{3} - 120 B a^{3} b^{2}\right ) + x^{4} \left (- 60 A a^{3} b^{2} - 30 B a^{4} b\right ) + x^{2} \left (- 20 A a^{4} b - 4 B a^{5}\right )}{24 x^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**5*(B*x**2+A)/x**9,x)

[Out]

B*b**5*x**4/4 + 5*a*b**3*(A*b + 2*B*a)*log(x) + x**2*(A*b**5/2 + 5*B*a*b**4/2) + (-3*A*a**5 + x**6*(-120*A*a**
2*b**3 - 120*B*a**3*b**2) + x**4*(-60*A*a**3*b**2 - 30*B*a**4*b) + x**2*(-20*A*a**4*b - 4*B*a**5))/(24*x**8)

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