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

Optimal. Leaf size=88 \[ \frac{10}{9} a^2 A b^3 x^9+\frac{5}{3} a^3 A b^2 x^6+\frac{5}{3} a^4 A b x^3+a^5 A \log (x)+\frac{5}{12} a A b^4 x^{12}+\frac{B \left (a+b x^3\right )^6}{18 b}+\frac{1}{15} A b^5 x^{15} \]

[Out]

(5*a^4*A*b*x^3)/3 + (5*a^3*A*b^2*x^6)/3 + (10*a^2*A*b^3*x^9)/9 + (5*a*A*b^4*x^12)/12 + (A*b^5*x^15)/15 + (B*(a
+ b*x^3)^6)/(18*b) + a^5*A*Log[x]

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Rubi [A]  time = 0.0672439, antiderivative size = 88, 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, 80, 43} \[ \frac{10}{9} a^2 A b^3 x^9+\frac{5}{3} a^3 A b^2 x^6+\frac{5}{3} a^4 A b x^3+a^5 A \log (x)+\frac{5}{12} a A b^4 x^{12}+\frac{B \left (a+b x^3\right )^6}{18 b}+\frac{1}{15} A b^5 x^{15} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x^3)^5*(A + B*x^3))/x,x]

[Out]

(5*a^4*A*b*x^3)/3 + (5*a^3*A*b^2*x^6)/3 + (10*a^2*A*b^3*x^9)/9 + (5*a*A*b^4*x^12)/12 + (A*b^5*x^15)/15 + (B*(a
+ b*x^3)^6)/(18*b) + a^5*A*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 80

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

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^3\right )^5 \left (A+B x^3\right )}{x} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{(a+b x)^5 (A+B x)}{x} \, dx,x,x^3\right )\\ &=\frac{B \left (a+b x^3\right )^6}{18 b}+\frac{1}{3} A \operatorname{Subst}\left (\int \frac{(a+b x)^5}{x} \, dx,x,x^3\right )\\ &=\frac{B \left (a+b x^3\right )^6}{18 b}+\frac{1}{3} A \operatorname{Subst}\left (\int \left (5 a^4 b+\frac{a^5}{x}+10 a^3 b^2 x+10 a^2 b^3 x^2+5 a b^4 x^3+b^5 x^4\right ) \, dx,x,x^3\right )\\ &=\frac{5}{3} a^4 A b x^3+\frac{5}{3} a^3 A b^2 x^6+\frac{10}{9} a^2 A b^3 x^9+\frac{5}{12} a A b^4 x^{12}+\frac{1}{15} A b^5 x^{15}+\frac{B \left (a+b x^3\right )^6}{18 b}+a^5 A \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0271966, size = 113, normalized size = 1.28 \[ \frac{10}{9} a^2 b^2 x^9 (a B+A b)+\frac{5}{6} a^3 b x^6 (a B+2 A b)+\frac{1}{3} a^4 x^3 (a B+5 A b)+a^5 A \log (x)+\frac{1}{15} b^4 x^{15} (5 a B+A b)+\frac{5}{12} a b^3 x^{12} (2 a B+A b)+\frac{1}{18} b^5 B x^{18} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^4*(5*A*b + a*B)*x^3)/3 + (5*a^3*b*(2*A*b + a*B)*x^6)/6 + (10*a^2*b^2*(A*b + a*B)*x^9)/9 + (5*a*b^3*(A*b + 2
*a*B)*x^12)/12 + (b^4*(A*b + 5*a*B)*x^15)/15 + (b^5*B*x^18)/18 + a^5*A*Log[x]

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Maple [A]  time = 0.002, size = 124, normalized size = 1.4 \begin{align*}{\frac{B{b}^{5}{x}^{18}}{18}}+{\frac{A{b}^{5}{x}^{15}}{15}}+{\frac{B{x}^{15}a{b}^{4}}{3}}+{\frac{5\,aA{b}^{4}{x}^{12}}{12}}+{\frac{5\,B{x}^{12}{a}^{2}{b}^{3}}{6}}+{\frac{10\,{a}^{2}A{b}^{3}{x}^{9}}{9}}+{\frac{10\,B{x}^{9}{a}^{3}{b}^{2}}{9}}+{\frac{5\,{a}^{3}A{b}^{2}{x}^{6}}{3}}+{\frac{5\,B{x}^{6}{a}^{4}b}{6}}+{\frac{5\,{a}^{4}Ab{x}^{3}}{3}}+{\frac{B{x}^{3}{a}^{5}}{3}}+{a}^{5}A\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/18*B*b^5*x^18+1/15*A*b^5*x^15+1/3*B*x^15*a*b^4+5/12*a*A*b^4*x^12+5/6*B*x^12*a^2*b^3+10/9*a^2*A*b^3*x^9+10/9*
B*x^9*a^3*b^2+5/3*a^3*A*b^2*x^6+5/6*B*x^6*a^4*b+5/3*a^4*A*b*x^3+1/3*B*x^3*a^5+a^5*A*ln(x)

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Maxima [A]  time = 1.71705, size = 162, normalized size = 1.84 \begin{align*} \frac{1}{18} \, B b^{5} x^{18} + \frac{1}{15} \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{15} + \frac{5}{12} \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{12} + \frac{10}{9} \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{9} + \frac{5}{6} \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{6} + \frac{1}{3} \, A a^{5} \log \left (x^{3}\right ) + \frac{1}{3} \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/18*B*b^5*x^18 + 1/15*(5*B*a*b^4 + A*b^5)*x^15 + 5/12*(2*B*a^2*b^3 + A*a*b^4)*x^12 + 10/9*(B*a^3*b^2 + A*a^2*
b^3)*x^9 + 5/6*(B*a^4*b + 2*A*a^3*b^2)*x^6 + 1/3*A*a^5*log(x^3) + 1/3*(B*a^5 + 5*A*a^4*b)*x^3

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Fricas [A]  time = 1.43029, size = 269, normalized size = 3.06 \begin{align*} \frac{1}{18} \, B b^{5} x^{18} + \frac{1}{15} \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{15} + \frac{5}{12} \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{12} + \frac{10}{9} \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{9} + \frac{5}{6} \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{6} + A a^{5} \log \left (x\right ) + \frac{1}{3} \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/18*B*b^5*x^18 + 1/15*(5*B*a*b^4 + A*b^5)*x^15 + 5/12*(2*B*a^2*b^3 + A*a*b^4)*x^12 + 10/9*(B*a^3*b^2 + A*a^2*
b^3)*x^9 + 5/6*(B*a^4*b + 2*A*a^3*b^2)*x^6 + A*a^5*log(x) + 1/3*(B*a^5 + 5*A*a^4*b)*x^3

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Sympy [A]  time = 0.376198, size = 134, normalized size = 1.52 \begin{align*} A a^{5} \log{\left (x \right )} + \frac{B b^{5} x^{18}}{18} + x^{15} \left (\frac{A b^{5}}{15} + \frac{B a b^{4}}{3}\right ) + x^{12} \left (\frac{5 A a b^{4}}{12} + \frac{5 B a^{2} b^{3}}{6}\right ) + x^{9} \left (\frac{10 A a^{2} b^{3}}{9} + \frac{10 B a^{3} b^{2}}{9}\right ) + x^{6} \left (\frac{5 A a^{3} b^{2}}{3} + \frac{5 B a^{4} b}{6}\right ) + x^{3} \left (\frac{5 A a^{4} b}{3} + \frac{B a^{5}}{3}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

A*a**5*log(x) + B*b**5*x**18/18 + x**15*(A*b**5/15 + B*a*b**4/3) + x**12*(5*A*a*b**4/12 + 5*B*a**2*b**3/6) + x
**9*(10*A*a**2*b**3/9 + 10*B*a**3*b**2/9) + x**6*(5*A*a**3*b**2/3 + 5*B*a**4*b/6) + x**3*(5*A*a**4*b/3 + B*a**
5/3)

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Giac [A]  time = 1.13411, size = 167, normalized size = 1.9 \begin{align*} \frac{1}{18} \, B b^{5} x^{18} + \frac{1}{3} \, B a b^{4} x^{15} + \frac{1}{15} \, A b^{5} x^{15} + \frac{5}{6} \, B a^{2} b^{3} x^{12} + \frac{5}{12} \, A a b^{4} x^{12} + \frac{10}{9} \, B a^{3} b^{2} x^{9} + \frac{10}{9} \, A a^{2} b^{3} x^{9} + \frac{5}{6} \, B a^{4} b x^{6} + \frac{5}{3} \, A a^{3} b^{2} x^{6} + \frac{1}{3} \, B a^{5} x^{3} + \frac{5}{3} \, A a^{4} b x^{3} + A a^{5} \log \left ({\left | x \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/18*B*b^5*x^18 + 1/3*B*a*b^4*x^15 + 1/15*A*b^5*x^15 + 5/6*B*a^2*b^3*x^12 + 5/12*A*a*b^4*x^12 + 10/9*B*a^3*b^2
*x^9 + 10/9*A*a^2*b^3*x^9 + 5/6*B*a^4*b*x^6 + 5/3*A*a^3*b^2*x^6 + 1/3*B*a^5*x^3 + 5/3*A*a^4*b*x^3 + A*a^5*log(
abs(x))