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

Optimal. Leaf size=29 \[ \frac{1}{2} x^2 (a B+A b)+a A \log (x)+\frac{1}{4} b B x^4 \]

[Out]

((A*b + a*B)*x^2)/2 + (b*B*x^4)/4 + a*A*Log[x]

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Rubi [A]  time = 0.0208088, 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} \[ \frac{1}{2} x^2 (a B+A b)+a A \log (x)+\frac{1}{4} b B x^4 \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x^2)*(A + B*x^2))/x,x]

[Out]

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

Mathematica [A]  time = 0.0086718, size = 29, normalized size = 1. \[ \frac{1}{2} x^2 (a B+A b)+a A \log (x)+\frac{1}{4} b B x^4 \]

Antiderivative was successfully verified.

[In]

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

[Out]

((A*b + a*B)*x^2)/2 + (b*B*x^4)/4 + a*A*Log[x]

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Maple [A]  time = 0.002, size = 28, normalized size = 1. \begin{align*}{\frac{bB{x}^{4}}{4}}+{\frac{A{x}^{2}b}{2}}+{\frac{B{x}^{2}a}{2}}+aA\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*b*B*x^4+1/2*A*x^2*b+1/2*B*x^2*a+a*A*ln(x)

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Maxima [A]  time = 1.02901, size = 38, normalized size = 1.31 \begin{align*} \frac{1}{4} \, B b x^{4} + \frac{1}{2} \,{\left (B a + A b\right )} x^{2} + \frac{1}{2} \, A a \log \left (x^{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*B*b*x^4 + 1/2*(B*a + A*b)*x^2 + 1/2*A*a*log(x^2)

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Fricas [A]  time = 1.39864, size = 65, normalized size = 2.24 \begin{align*} \frac{1}{4} \, B b x^{4} + \frac{1}{2} \,{\left (B a + A b\right )} x^{2} + A a \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*B*b*x^4 + 1/2*(B*a + A*b)*x^2 + A*a*log(x)

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Sympy [A]  time = 0.250676, size = 27, normalized size = 0.93 \begin{align*} A a \log{\left (x \right )} + \frac{B b x^{4}}{4} + x^{2} \left (\frac{A b}{2} + \frac{B a}{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

A*a*log(x) + B*b*x**4/4 + x**2*(A*b/2 + B*a/2)

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Giac [A]  time = 1.17047, size = 41, normalized size = 1.41 \begin{align*} \frac{1}{4} \, B b x^{4} + \frac{1}{2} \, B a x^{2} + \frac{1}{2} \, A b x^{2} + \frac{1}{2} \, A a \log \left (x^{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*B*b*x^4 + 1/2*B*a*x^2 + 1/2*A*b*x^2 + 1/2*A*a*log(x^2)