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

Optimal. Leaf size=29 \[ \frac{1}{3} x^3 (a B+A b)+a A \log (x)+\frac{1}{6} b B x^6 \]

[Out]

((A*b + a*B)*x^3)/3 + (b*B*x^6)/6 + a*A*Log[x]

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Rubi [A]  time = 0.0213559, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {446, 76} \[ \frac{1}{3} x^3 (a B+A b)+a A \log (x)+\frac{1}{6} b B x^6 \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x^3)*(A + B*x^3))/x,x]

[Out]

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

Mathematica [A]  time = 0.0084466, size = 29, normalized size = 1. \[ \frac{1}{3} x^3 (a B+A b)+a A \log (x)+\frac{1}{6} b B x^6 \]

Antiderivative was successfully verified.

[In]

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

[Out]

((A*b + a*B)*x^3)/3 + (b*B*x^6)/6 + a*A*Log[x]

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Maple [A]  time = 0.02, size = 28, normalized size = 1. \begin{align*}{\frac{bB{x}^{6}}{6}}+{\frac{A{x}^{3}b}{3}}+{\frac{B{x}^{3}a}{3}}+aA\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*b*B*x^6+1/3*A*x^3*b+1/3*B*x^3*a+a*A*ln(x)

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Maxima [A]  time = 1.36668, size = 38, normalized size = 1.31 \begin{align*} \frac{1}{6} \, B b x^{6} + \frac{1}{3} \,{\left (B a + A b\right )} x^{3} + \frac{1}{3} \, A a \log \left (x^{3}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*B*b*x^6 + 1/3*(B*a + A*b)*x^3 + 1/3*A*a*log(x^3)

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Fricas [A]  time = 1.42804, size = 65, normalized size = 2.24 \begin{align*} \frac{1}{6} \, B b x^{6} + \frac{1}{3} \,{\left (B a + A b\right )} x^{3} + A a \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*B*b*x^6 + 1/3*(B*a + A*b)*x^3 + A*a*log(x)

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Sympy [A]  time = 0.253742, size = 27, normalized size = 0.93 \begin{align*} A a \log{\left (x \right )} + \frac{B b x^{6}}{6} + x^{3} \left (\frac{A b}{3} + \frac{B a}{3}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

A*a*log(x) + B*b*x**6/6 + x**3*(A*b/3 + B*a/3)

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Giac [A]  time = 1.19681, size = 38, normalized size = 1.31 \begin{align*} \frac{1}{6} \, B b x^{6} + \frac{1}{3} \, B a x^{3} + \frac{1}{3} \, A b x^{3} + A a \log \left ({\left | x \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*B*b*x^6 + 1/3*B*a*x^3 + 1/3*A*b*x^3 + A*a*log(abs(x))