3.1.7 \(\int \frac {(a+b x^3) (A+B x^3)}{x^4} \, dx\)

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

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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} \begin {gather*} \log (x) (a B+A b)-\frac {a A}{3 x^3}+\frac {1}{3} b B x^3 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x^3)*(A + B*x^3))/x^4,x]

[Out]

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

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Mathematica [A]  time = 0.01, size = 29, normalized size = 1.00 \begin {gather*} \log (x) (a B+A b)-\frac {a A}{3 x^3}+\frac {1}{3} b B x^3 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x^3)*(A + B*x^3))/x^4,x]

[Out]

-1/3*(a*A)/x^3 + (b*B*x^3)/3 + (A*b + a*B)*Log[x]

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a+b x^3\right ) \left (A+B x^3\right )}{x^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((a + b*x^3)*(A + B*x^3))/x^4,x]

[Out]

IntegrateAlgebraic[((a + b*x^3)*(A + B*x^3))/x^4, x]

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fricas [A]  time = 0.81, size = 30, normalized size = 1.03 \begin {gather*} \frac {B b x^{6} + 3 \, {\left (B a + A b\right )} x^{3} \log \relax (x) - A a}{3 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)*(B*x^3+A)/x^4,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.16, size = 40, normalized size = 1.38 \begin {gather*} \frac {1}{3} \, B b x^{3} + {\left (B a + A b\right )} \log \left ({\left | x \right |}\right ) - \frac {B a x^{3} + A b x^{3} + A a}{3 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)*(B*x^3+A)/x^4,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.05, size = 26, normalized size = 0.90 \begin {gather*} \frac {B b \,x^{3}}{3}+A b \ln \relax (x )+B a \ln \relax (x )-\frac {A a}{3 x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)*(B*x^3+A)/x^4,x)

[Out]

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

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maxima [A]  time = 0.48, size = 28, normalized size = 0.97 \begin {gather*} \frac {1}{3} \, B b x^{3} + \frac {1}{3} \, {\left (B a + A b\right )} \log \left (x^{3}\right ) - \frac {A a}{3 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)*(B*x^3+A)/x^4,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.04, size = 25, normalized size = 0.86 \begin {gather*} \ln \relax (x)\,\left (A\,b+B\,a\right )-\frac {A\,a}{3\,x^3}+\frac {B\,b\,x^3}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x^3)*(a + b*x^3))/x^4,x)

[Out]

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

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sympy [A]  time = 0.21, size = 26, normalized size = 0.90 \begin {gather*} - \frac {A a}{3 x^{3}} + \frac {B b x^{3}}{3} + \left (A b + B a\right ) \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)*(B*x**3+A)/x**4,x)

[Out]

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

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