3.17 \(\int \frac{(a+b x^3)^2 (A+B x^3)}{x^4} \, dx\)

Optimal. Leaf size=51 \[ -\frac{a^2 A}{3 x^3}+\frac{1}{3} b x^3 (2 a B+A b)+a \log (x) (a B+2 A b)+\frac{1}{6} b^2 B x^6 \]

[Out]

-(a^2*A)/(3*x^3) + (b*(A*b + 2*a*B)*x^3)/3 + (b^2*B*x^6)/6 + a*(2*A*b + a*B)*Log[x]

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Rubi [A]  time = 0.0500513, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {446, 76} \[ -\frac{a^2 A}{3 x^3}+\frac{1}{3} b x^3 (2 a B+A b)+a \log (x) (a B+2 A b)+\frac{1}{6} b^2 B x^6 \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x^3)^2*(A + B*x^3))/x^4,x]

[Out]

-(a^2*A)/(3*x^3) + (b*(A*b + 2*a*B)*x^3)/3 + (b^2*B*x^6)/6 + a*(2*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 )^2 \left (A+B x^3\right )}{x^4} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{(a+b x)^2 (A+B x)}{x^2} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (b (A b+2 a B)+\frac{a^2 A}{x^2}+\frac{a (2 A b+a B)}{x}+b^2 B x\right ) \, dx,x,x^3\right )\\ &=-\frac{a^2 A}{3 x^3}+\frac{1}{3} b (A b+2 a B) x^3+\frac{1}{6} b^2 B x^6+a (2 A b+a B) \log (x)\\ \end{align*}

Mathematica [A]  time = 0.023097, size = 49, normalized size = 0.96 \[ \frac{1}{6} \left (-\frac{2 a^2 A}{x^3}+2 b x^3 (2 a B+A b)+6 a \log (x) (a B+2 A b)+b^2 B x^6\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x^3)^2*(A + B*x^3))/x^4,x]

[Out]

((-2*a^2*A)/x^3 + 2*b*(A*b + 2*a*B)*x^3 + b^2*B*x^6 + 6*a*(2*A*b + a*B)*Log[x])/6

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Maple [A]  time = 0.007, size = 51, normalized size = 1. \begin{align*}{\frac{{b}^{2}B{x}^{6}}{6}}+{\frac{A{x}^{3}{b}^{2}}{3}}+{\frac{2\,B{x}^{3}ab}{3}}-{\frac{A{a}^{2}}{3\,{x}^{3}}}+2\,A\ln \left ( x \right ) ab+B\ln \left ( x \right ){a}^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^2*(B*x^3+A)/x^4,x)

[Out]

1/6*b^2*B*x^6+1/3*A*x^3*b^2+2/3*B*x^3*a*b-1/3*a^2*A/x^3+2*A*ln(x)*a*b+B*ln(x)*a^2

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Maxima [A]  time = 1.11259, size = 70, normalized size = 1.37 \begin{align*} \frac{1}{6} \, B b^{2} x^{6} + \frac{1}{3} \,{\left (2 \, B a b + A b^{2}\right )} x^{3} + \frac{1}{3} \,{\left (B a^{2} + 2 \, A a b\right )} \log \left (x^{3}\right ) - \frac{A a^{2}}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^2*(B*x^3+A)/x^4,x, algorithm="maxima")

[Out]

1/6*B*b^2*x^6 + 1/3*(2*B*a*b + A*b^2)*x^3 + 1/3*(B*a^2 + 2*A*a*b)*log(x^3) - 1/3*A*a^2/x^3

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Fricas [A]  time = 1.61107, size = 122, normalized size = 2.39 \begin{align*} \frac{B b^{2} x^{9} + 2 \,{\left (2 \, B a b + A b^{2}\right )} x^{6} + 6 \,{\left (B a^{2} + 2 \, A a b\right )} x^{3} \log \left (x\right ) - 2 \, A a^{2}}{6 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^2*(B*x^3+A)/x^4,x, algorithm="fricas")

[Out]

1/6*(B*b^2*x^9 + 2*(2*B*a*b + A*b^2)*x^6 + 6*(B*a^2 + 2*A*a*b)*x^3*log(x) - 2*A*a^2)/x^3

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Sympy [A]  time = 0.400952, size = 51, normalized size = 1. \begin{align*} - \frac{A a^{2}}{3 x^{3}} + \frac{B b^{2} x^{6}}{6} + a \left (2 A b + B a\right ) \log{\left (x \right )} + x^{3} \left (\frac{A b^{2}}{3} + \frac{2 B a b}{3}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**2*(B*x**3+A)/x**4,x)

[Out]

-A*a**2/(3*x**3) + B*b**2*x**6/6 + a*(2*A*b + B*a)*log(x) + x**3*(A*b**2/3 + 2*B*a*b/3)

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Giac [A]  time = 1.26942, size = 93, normalized size = 1.82 \begin{align*} \frac{1}{6} \, B b^{2} x^{6} + \frac{2}{3} \, B a b x^{3} + \frac{1}{3} \, A b^{2} x^{3} +{\left (B a^{2} + 2 \, A a b\right )} \log \left ({\left | x \right |}\right ) - \frac{B a^{2} x^{3} + 2 \, A a b x^{3} + A a^{2}}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^2*(B*x^3+A)/x^4,x, algorithm="giac")

[Out]

1/6*B*b^2*x^6 + 2/3*B*a*b*x^3 + 1/3*A*b^2*x^3 + (B*a^2 + 2*A*a*b)*log(abs(x)) - 1/3*(B*a^2*x^3 + 2*A*a*b*x^3 +
A*a^2)/x^3