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

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

[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]

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Rubi [A]  time = 0.0965514, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {446, 76} \[ -\frac{5 a^2 b^2 (a B+A b)}{x^2}-\frac{a^4 (a B+5 A b)}{6 x^6}-\frac{5 a^3 b (a B+2 A b)}{4 x^4}-\frac{a^5 A}{8 x^8}+\frac{1}{2} b^4 x^2 (5 a B+A b)+5 a b^3 \log (x) (2 a B+A b)+\frac{1}{4} b^5 B x^4 \]

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 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]]

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])

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*}

Mathematica [A]  time = 0.0505843, size = 116, normalized size = 1.04 \[ 5 a b^3 \log (x) (2 a B+A b)-\frac{60 a^3 b^2 x^4 \left (A+2 B x^2\right )+120 a^2 A b^3 x^6+10 a^4 b x^2 \left (2 A+3 B x^2\right )+a^5 \left (3 A+4 B x^2\right )-60 a b^4 B x^{10}-6 b^5 x^{10} \left (2 A+B x^2\right )}{24 x^8} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(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))/(24*x^8) + 5*a*b^3*(A*b + 2*a*B)*Log[x]

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Maple [A]  time = 0.006, size = 124, normalized size = 1.1 \begin{align*}{\frac{{b}^{5}B{x}^{4}}{4}}+{\frac{A{x}^{2}{b}^{5}}{2}}+{\frac{5\,B{x}^{2}a{b}^{4}}{2}}+5\,A\ln \left ( x \right ) a{b}^{4}+10\,B\ln \left ( x \right ){a}^{2}{b}^{3}-{\frac{5\,{a}^{3}{b}^{2}A}{2\,{x}^{4}}}-{\frac{5\,{a}^{4}bB}{4\,{x}^{4}}}-{\frac{A{a}^{5}}{8\,{x}^{8}}}-5\,{\frac{{a}^{2}{b}^{3}A}{{x}^{2}}}-5\,{\frac{{a}^{3}{b}^{2}B}{{x}^{2}}}-{\frac{5\,{a}^{4}bA}{6\,{x}^{6}}}-{\frac{{a}^{5}B}{6\,{x}^{6}}} \end{align*}

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

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Maxima [A]  time = 1.00189, size = 166, normalized size = 1.48 \begin{align*} \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{align*}

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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Fricas [A]  time = 1.4237, size = 271, normalized size = 2.42 \begin{align*} \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 \left (x\right ) - 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{align*}

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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Sympy [A]  time = 3.03705, size = 124, normalized size = 1.11 \begin{align*} \frac{B b^{5} x^{4}}{4} + 5 a b^{3} \left (A b + 2 B a\right ) \log{\left (x \right )} + 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{align*}

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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Giac [A]  time = 1.20431, size = 203, normalized size = 1.81 \begin{align*} \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{align*}

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