3.48 \(\int \frac{(a+b x^3)^5 (A+B x^3)}{x^{16}} \, dx\)

Optimal. Leaf size=113 \[ -\frac{5 a^2 b^2 (a B+A b)}{3 x^6}-\frac{a^4 (a B+5 A b)}{12 x^{12}}-\frac{5 a^3 b (a B+2 A b)}{9 x^9}-\frac{a^5 A}{15 x^{15}}-\frac{5 a b^3 (2 a B+A b)}{3 x^3}+b^4 \log (x) (5 a B+A b)+\frac{1}{3} b^5 B x^3 \]

[Out]

-(a^5*A)/(15*x^15) - (a^4*(5*A*b + a*B))/(12*x^12) - (5*a^3*b*(2*A*b + a*B))/(9*x^9) - (5*a^2*b^2*(A*b + a*B))
/(3*x^6) - (5*a*b^3*(A*b + 2*a*B))/(3*x^3) + (b^5*B*x^3)/3 + b^4*(A*b + 5*a*B)*Log[x]

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Rubi [A]  time = 0.0911369, antiderivative size = 113, 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} \[ -\frac{5 a^2 b^2 (a B+A b)}{3 x^6}-\frac{a^4 (a B+5 A b)}{12 x^{12}}-\frac{5 a^3 b (a B+2 A b)}{9 x^9}-\frac{a^5 A}{15 x^{15}}-\frac{5 a b^3 (2 a B+A b)}{3 x^3}+b^4 \log (x) (5 a B+A b)+\frac{1}{3} b^5 B x^3 \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x^3)^5*(A + B*x^3))/x^16,x]

[Out]

-(a^5*A)/(15*x^15) - (a^4*(5*A*b + a*B))/(12*x^12) - (5*a^3*b*(2*A*b + a*B))/(9*x^9) - (5*a^2*b^2*(A*b + a*B))
/(3*x^6) - (5*a*b^3*(A*b + 2*a*B))/(3*x^3) + (b^5*B*x^3)/3 + b^4*(A*b + 5*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^3\right )^5 \left (A+B x^3\right )}{x^{16}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{(a+b x)^5 (A+B x)}{x^6} \, dx,x,x^3\right )\\ &=\frac{1}{3} \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^3\right )\\ &=-\frac{a^5 A}{15 x^{15}}-\frac{a^4 (5 A b+a B)}{12 x^{12}}-\frac{5 a^3 b (2 A b+a B)}{9 x^9}-\frac{5 a^2 b^2 (A b+a B)}{3 x^6}-\frac{5 a b^3 (A b+2 a B)}{3 x^3}+\frac{1}{3} b^5 B x^3+b^4 (A b+5 a B) \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0549728, size = 116, normalized size = 1.03 \[ b^4 \log (x) (5 a B+A b)-\frac{300 a^2 b^3 x^9 \left (A+2 B x^3\right )+100 a^3 b^2 x^6 \left (2 A+3 B x^3\right )+25 a^4 b x^3 \left (3 A+4 B x^3\right )+3 a^5 \left (4 A+5 B x^3\right )+300 a A b^4 x^{12}-60 b^5 B x^{18}}{180 x^{15}} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x^3)^5*(A + B*x^3))/x^16,x]

[Out]

-(300*a*A*b^4*x^12 - 60*b^5*B*x^18 + 300*a^2*b^3*x^9*(A + 2*B*x^3) + 100*a^3*b^2*x^6*(2*A + 3*B*x^3) + 25*a^4*
b*x^3*(3*A + 4*B*x^3) + 3*a^5*(4*A + 5*B*x^3))/(180*x^15) + b^4*(A*b + 5*a*B)*Log[x]

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Maple [A]  time = 0.009, size = 123, normalized size = 1.1 \begin{align*}{\frac{{b}^{5}B{x}^{3}}{3}}-{\frac{5\,{a}^{2}{b}^{3}A}{3\,{x}^{6}}}-{\frac{5\,{a}^{3}{b}^{2}B}{3\,{x}^{6}}}-{\frac{5\,a{b}^{4}A}{3\,{x}^{3}}}-{\frac{10\,{a}^{2}{b}^{3}B}{3\,{x}^{3}}}-{\frac{A{a}^{5}}{15\,{x}^{15}}}-{\frac{10\,{a}^{3}{b}^{2}A}{9\,{x}^{9}}}-{\frac{5\,{a}^{4}bB}{9\,{x}^{9}}}-{\frac{5\,{a}^{4}bA}{12\,{x}^{12}}}-{\frac{{a}^{5}B}{12\,{x}^{12}}}+A\ln \left ( x \right ){b}^{5}+5\,B\ln \left ( x \right ) a{b}^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^5*(B*x^3+A)/x^16,x)

[Out]

1/3*b^5*B*x^3-5/3*a^2*b^3/x^6*A-5/3*a^3*b^2/x^6*B-5/3*a*b^4/x^3*A-10/3*a^2*b^3/x^3*B-1/15*a^5*A/x^15-10/9*a^3*
b^2/x^9*A-5/9*a^4*b/x^9*B-5/12*a^4/x^12*A*b-1/12*a^5/x^12*B+A*ln(x)*b^5+5*B*ln(x)*a*b^4

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Maxima [A]  time = 1.10972, size = 166, normalized size = 1.47 \begin{align*} \frac{1}{3} \, B b^{5} x^{3} + \frac{1}{3} \,{\left (5 \, B a b^{4} + A b^{5}\right )} \log \left (x^{3}\right ) - \frac{300 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{12} + 300 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{9} + 100 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{6} + 12 \, A a^{5} + 15 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3}}{180 \, x^{15}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5*(B*x^3+A)/x^16,x, algorithm="maxima")

[Out]

1/3*B*b^5*x^3 + 1/3*(5*B*a*b^4 + A*b^5)*log(x^3) - 1/180*(300*(2*B*a^2*b^3 + A*a*b^4)*x^12 + 300*(B*a^3*b^2 +
A*a^2*b^3)*x^9 + 100*(B*a^4*b + 2*A*a^3*b^2)*x^6 + 12*A*a^5 + 15*(B*a^5 + 5*A*a^4*b)*x^3)/x^15

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Fricas [A]  time = 1.44939, size = 282, normalized size = 2.5 \begin{align*} \frac{60 \, B b^{5} x^{18} + 180 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{15} \log \left (x\right ) - 300 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{12} - 300 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{9} - 100 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{6} - 12 \, A a^{5} - 15 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3}}{180 \, x^{15}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5*(B*x^3+A)/x^16,x, algorithm="fricas")

[Out]

1/180*(60*B*b^5*x^18 + 180*(5*B*a*b^4 + A*b^5)*x^15*log(x) - 300*(2*B*a^2*b^3 + A*a*b^4)*x^12 - 300*(B*a^3*b^2
+ A*a^2*b^3)*x^9 - 100*(B*a^4*b + 2*A*a^3*b^2)*x^6 - 12*A*a^5 - 15*(B*a^5 + 5*A*a^4*b)*x^3)/x^15

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Sympy [A]  time = 36.7863, size = 122, normalized size = 1.08 \begin{align*} \frac{B b^{5} x^{3}}{3} + b^{4} \left (A b + 5 B a\right ) \log{\left (x \right )} - \frac{12 A a^{5} + x^{12} \left (300 A a b^{4} + 600 B a^{2} b^{3}\right ) + x^{9} \left (300 A a^{2} b^{3} + 300 B a^{3} b^{2}\right ) + x^{6} \left (200 A a^{3} b^{2} + 100 B a^{4} b\right ) + x^{3} \left (75 A a^{4} b + 15 B a^{5}\right )}{180 x^{15}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**5*(B*x**3+A)/x**16,x)

[Out]

B*b**5*x**3/3 + b**4*(A*b + 5*B*a)*log(x) - (12*A*a**5 + x**12*(300*A*a*b**4 + 600*B*a**2*b**3) + x**9*(300*A*
a**2*b**3 + 300*B*a**3*b**2) + x**6*(200*A*a**3*b**2 + 100*B*a**4*b) + x**3*(75*A*a**4*b + 15*B*a**5))/(180*x*
*15)

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Giac [A]  time = 1.19474, size = 196, normalized size = 1.73 \begin{align*} \frac{1}{3} \, B b^{5} x^{3} +{\left (5 \, B a b^{4} + A b^{5}\right )} \log \left ({\left | x \right |}\right ) - \frac{685 \, B a b^{4} x^{15} + 137 \, A b^{5} x^{15} + 600 \, B a^{2} b^{3} x^{12} + 300 \, A a b^{4} x^{12} + 300 \, B a^{3} b^{2} x^{9} + 300 \, A a^{2} b^{3} x^{9} + 100 \, B a^{4} b x^{6} + 200 \, A a^{3} b^{2} x^{6} + 15 \, B a^{5} x^{3} + 75 \, A a^{4} b x^{3} + 12 \, A a^{5}}{180 \, x^{15}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5*(B*x^3+A)/x^16,x, algorithm="giac")

[Out]

1/3*B*b^5*x^3 + (5*B*a*b^4 + A*b^5)*log(abs(x)) - 1/180*(685*B*a*b^4*x^15 + 137*A*b^5*x^15 + 600*B*a^2*b^3*x^1
2 + 300*A*a*b^4*x^12 + 300*B*a^3*b^2*x^9 + 300*A*a^2*b^3*x^9 + 100*B*a^4*b*x^6 + 200*A*a^3*b^2*x^6 + 15*B*a^5*
x^3 + 75*A*a^4*b*x^3 + 12*A*a^5)/x^15