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

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

[Out]

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

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Rubi [A]  time = 0.0215009, 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} \[ \log (x) (a B+A b)-\frac{a A}{3 x^3}+\frac{1}{3} b B x^3 \]

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 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^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*}

Mathematica [A]  time = 0.0105077, size = 29, normalized size = 1. \[ \log (x) (a B+A b)-\frac{a A}{3 x^3}+\frac{1}{3} b B x^3 \]

Antiderivative was successfully verified.

[In]

Integrate[((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]

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Maple [A]  time = 0.007, size = 26, normalized size = 0.9 \begin{align*}{\frac{bB{x}^{3}}{3}}-{\frac{Aa}{3\,{x}^{3}}}+A\ln \left ( x \right ) b+B\ln \left ( x \right ) a \end{align*}

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*ln(x)*b+B*ln(x)*a

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Maxima [A]  time = 1.04531, size = 38, normalized size = 1.31 \begin{align*} \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{align*}

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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Fricas [A]  time = 1.44147, size = 70, normalized size = 2.41 \begin{align*} \frac{B b x^{6} + 3 \,{\left (B a + A b\right )} x^{3} \log \left (x\right ) - A a}{3 \, x^{3}} \end{align*}

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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Sympy [A]  time = 0.336231, size = 26, normalized size = 0.9 \begin{align*} - \frac{A a}{3 x^{3}} + \frac{B b x^{3}}{3} + \left (A b + B a\right ) \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**4,x)

[Out]

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

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Giac [A]  time = 1.17242, size = 54, normalized size = 1.86 \begin{align*} \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{align*}

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