3.45 \(\int \frac{(a+b x^2)^5 (A+B x^2)}{x^{13}} \, dx\)

Optimal. Leaf size=91 \[ -\frac{5 a^2 b^3 B}{2 x^4}-\frac{5 a^3 b^2 B}{3 x^6}-\frac{5 a^4 b B}{8 x^8}-\frac{a^5 B}{10 x^{10}}-\frac{A \left (a+b x^2\right )^6}{12 a x^{12}}-\frac{5 a b^4 B}{2 x^2}+b^5 B \log (x) \]

[Out]

-(a^5*B)/(10*x^10) - (5*a^4*b*B)/(8*x^8) - (5*a^3*b^2*B)/(3*x^6) - (5*a^2*b^3*B)/(2*x^4) - (5*a*b^4*B)/(2*x^2)
- (A*(a + b*x^2)^6)/(12*a*x^12) + b^5*B*Log[x]

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Rubi [A]  time = 0.055555, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {446, 78, 43} \[ -\frac{5 a^2 b^3 B}{2 x^4}-\frac{5 a^3 b^2 B}{3 x^6}-\frac{5 a^4 b B}{8 x^8}-\frac{a^5 B}{10 x^{10}}-\frac{A \left (a+b x^2\right )^6}{12 a x^{12}}-\frac{5 a b^4 B}{2 x^2}+b^5 B \log (x) \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x^2)^5*(A + B*x^2))/x^13,x]

[Out]

-(a^5*B)/(10*x^10) - (5*a^4*b*B)/(8*x^8) - (5*a^3*b^2*B)/(3*x^6) - (5*a^2*b^3*B)/(2*x^4) - (5*a*b^4*B)/(2*x^2)
- (A*(a + b*x^2)^6)/(12*a*x^12) + b^5*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 78

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ
[p, n]))))

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^{13}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(a+b x)^5 (A+B x)}{x^7} \, dx,x,x^2\right )\\ &=-\frac{A \left (a+b x^2\right )^6}{12 a x^{12}}+\frac{1}{2} B \operatorname{Subst}\left (\int \frac{(a+b x)^5}{x^6} \, dx,x,x^2\right )\\ &=-\frac{A \left (a+b x^2\right )^6}{12 a x^{12}}+\frac{1}{2} B \operatorname{Subst}\left (\int \left (\frac{a^5}{x^6}+\frac{5 a^4 b}{x^5}+\frac{10 a^3 b^2}{x^4}+\frac{10 a^2 b^3}{x^3}+\frac{5 a b^4}{x^2}+\frac{b^5}{x}\right ) \, dx,x,x^2\right )\\ &=-\frac{a^5 B}{10 x^{10}}-\frac{5 a^4 b B}{8 x^8}-\frac{5 a^3 b^2 B}{3 x^6}-\frac{5 a^2 b^3 B}{2 x^4}-\frac{5 a b^4 B}{2 x^2}-\frac{A \left (a+b x^2\right )^6}{12 a x^{12}}+b^5 B \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0570483, size = 118, normalized size = 1.3 \[ b^5 B \log (x)-\frac{100 a^2 b^3 x^6 \left (2 A+3 B x^2\right )+50 a^3 b^2 x^4 \left (3 A+4 B x^2\right )+15 a^4 b x^2 \left (4 A+5 B x^2\right )+2 a^5 \left (5 A+6 B x^2\right )+150 a b^4 x^8 \left (A+2 B x^2\right )+60 A b^5 x^{10}}{120 x^{12}} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x^2)^5*(A + B*x^2))/x^13,x]

[Out]

-(60*A*b^5*x^10 + 150*a*b^4*x^8*(A + 2*B*x^2) + 100*a^2*b^3*x^6*(2*A + 3*B*x^2) + 50*a^3*b^2*x^4*(3*A + 4*B*x^
2) + 15*a^4*b*x^2*(4*A + 5*B*x^2) + 2*a^5*(5*A + 6*B*x^2))/(120*x^12) + b^5*B*Log[x]

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Maple [A]  time = 0.006, size = 124, normalized size = 1.4 \begin{align*}{b}^{5}B\ln \left ( x \right ) -{\frac{A{a}^{5}}{12\,{x}^{12}}}-{\frac{5\,a{b}^{4}A}{4\,{x}^{4}}}-{\frac{5\,{a}^{2}{b}^{3}B}{2\,{x}^{4}}}-{\frac{{b}^{5}A}{2\,{x}^{2}}}-{\frac{5\,a{b}^{4}B}{2\,{x}^{2}}}-{\frac{5\,{a}^{2}{b}^{3}A}{3\,{x}^{6}}}-{\frac{5\,{a}^{3}{b}^{2}B}{3\,{x}^{6}}}-{\frac{5\,{a}^{3}{b}^{2}A}{4\,{x}^{8}}}-{\frac{5\,{a}^{4}bB}{8\,{x}^{8}}}-{\frac{{a}^{4}bA}{2\,{x}^{10}}}-{\frac{{a}^{5}B}{10\,{x}^{10}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^5*(B*x^2+A)/x^13,x)

[Out]

b^5*B*ln(x)-1/12*A*a^5/x^12-5/4*a*b^4/x^4*A-5/2*a^2*b^3*B/x^4-1/2*b^5/x^2*A-5/2*a*b^4*B/x^2-5/3*b^3*a^2/x^6*A-
5/3*a^3*b^2*B/x^6-5/4*a^3*b^2/x^8*A-5/8*a^4*b*B/x^8-1/2*a^4/x^10*A*b-1/10*a^5*B/x^10

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Maxima [A]  time = 1.01765, size = 166, normalized size = 1.82 \begin{align*} \frac{1}{2} \, B b^{5} \log \left (x^{2}\right ) - \frac{60 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} + 150 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} + 200 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} + 10 \, A a^{5} + 75 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} + 12 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{120 \, x^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5*(B*x^2+A)/x^13,x, algorithm="maxima")

[Out]

1/2*B*b^5*log(x^2) - 1/120*(60*(5*B*a*b^4 + A*b^5)*x^10 + 150*(2*B*a^2*b^3 + A*a*b^4)*x^8 + 200*(B*a^3*b^2 + A
*a^2*b^3)*x^6 + 10*A*a^5 + 75*(B*a^4*b + 2*A*a^3*b^2)*x^4 + 12*(B*a^5 + 5*A*a^4*b)*x^2)/x^12

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Fricas [A]  time = 1.46626, size = 279, normalized size = 3.07 \begin{align*} \frac{120 \, B b^{5} x^{12} \log \left (x\right ) - 60 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} - 150 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} - 200 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} - 10 \, A a^{5} - 75 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} - 12 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{120 \, x^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5*(B*x^2+A)/x^13,x, algorithm="fricas")

[Out]

1/120*(120*B*b^5*x^12*log(x) - 60*(5*B*a*b^4 + A*b^5)*x^10 - 150*(2*B*a^2*b^3 + A*a*b^4)*x^8 - 200*(B*a^3*b^2
+ A*a^2*b^3)*x^6 - 10*A*a^5 - 75*(B*a^4*b + 2*A*a^3*b^2)*x^4 - 12*(B*a^5 + 5*A*a^4*b)*x^2)/x^12

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Sympy [A]  time = 14.2837, size = 124, normalized size = 1.36 \begin{align*} B b^{5} \log{\left (x \right )} - \frac{10 A a^{5} + x^{10} \left (60 A b^{5} + 300 B a b^{4}\right ) + x^{8} \left (150 A a b^{4} + 300 B a^{2} b^{3}\right ) + x^{6} \left (200 A a^{2} b^{3} + 200 B a^{3} b^{2}\right ) + x^{4} \left (150 A a^{3} b^{2} + 75 B a^{4} b\right ) + x^{2} \left (60 A a^{4} b + 12 B a^{5}\right )}{120 x^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**5*(B*x**2+A)/x**13,x)

[Out]

B*b**5*log(x) - (10*A*a**5 + x**10*(60*A*b**5 + 300*B*a*b**4) + x**8*(150*A*a*b**4 + 300*B*a**2*b**3) + x**6*(
200*A*a**2*b**3 + 200*B*a**3*b**2) + x**4*(150*A*a**3*b**2 + 75*B*a**4*b) + x**2*(60*A*a**4*b + 12*B*a**5))/(1
20*x**12)

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Giac [A]  time = 1.13902, size = 186, normalized size = 2.04 \begin{align*} \frac{1}{2} \, B b^{5} \log \left (x^{2}\right ) - \frac{147 \, B b^{5} x^{12} + 300 \, B a b^{4} x^{10} + 60 \, A b^{5} x^{10} + 300 \, B a^{2} b^{3} x^{8} + 150 \, A a b^{4} x^{8} + 200 \, B a^{3} b^{2} x^{6} + 200 \, A a^{2} b^{3} x^{6} + 75 \, B a^{4} b x^{4} + 150 \, A a^{3} b^{2} x^{4} + 12 \, B a^{5} x^{2} + 60 \, A a^{4} b x^{2} + 10 \, A a^{5}}{120 \, x^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5*(B*x^2+A)/x^13,x, algorithm="giac")

[Out]

1/2*B*b^5*log(x^2) - 1/120*(147*B*b^5*x^12 + 300*B*a*b^4*x^10 + 60*A*b^5*x^10 + 300*B*a^2*b^3*x^8 + 150*A*a*b^
4*x^8 + 200*B*a^3*b^2*x^6 + 200*A*a^2*b^3*x^6 + 75*B*a^4*b*x^4 + 150*A*a^3*b^2*x^4 + 12*B*a^5*x^2 + 60*A*a^4*b
*x^2 + 10*A*a^5)/x^12