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]

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

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 29, normalized size of antiderivative = 1.00, 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 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])

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]]

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 = 0.01, size = 29, normalized size = 1.00 \[ \frac {1}{3} x^3 (a B+A b)+a A \log (x)+\frac {1}{6} b B x^6 \]

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]

________________________________________________________________________________________

fricas [A]  time = 0.91, size = 25, normalized size = 0.86 \[ \frac {1}{6} \, B b x^{6} + \frac {1}{3} \, {\left (B a + A b\right )} x^{3} + A a \log \relax (x) \]

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)

________________________________________________________________________________________

giac [A]  time = 0.15, size = 28, normalized size = 0.97 \[ \frac {1}{6} \, B b x^{6} + \frac {1}{3} \, B a x^{3} + \frac {1}{3} \, A b x^{3} + A a \log \left ({\left | x \right |}\right ) \]

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))

________________________________________________________________________________________

maple [A]  time = 0.04, size = 28, normalized size = 0.97 \[ \frac {B b \,x^{6}}{6}+\frac {A b \,x^{3}}{3}+\frac {B a \,x^{3}}{3}+A a \ln \relax (x ) \]

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*a*x^3+A*a*ln(x)

________________________________________________________________________________________

maxima [A]  time = 0.57, size = 28, normalized size = 0.97 \[ \frac {1}{6} \, B b x^{6} + \frac {1}{3} \, {\left (B a + A b\right )} x^{3} + \frac {1}{3} \, A a \log \left (x^{3}\right ) \]

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)

________________________________________________________________________________________

mupad [B]  time = 0.03, size = 26, normalized size = 0.90 \[ x^3\,\left (\frac {A\,b}{3}+\frac {B\,a}{3}\right )+\frac {B\,b\,x^6}{6}+A\,a\,\ln \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x^3)*(a + b*x^3))/x,x)

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 0.11, size = 27, normalized size = 0.93 \[ A a \log {\relax (x )} + \frac {B b x^{6}}{6} + x^{3} \left (\frac {A b}{3} + \frac {B a}{3}\right ) \]

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)

________________________________________________________________________________________