3.1.3 \(\int (a+b x^2) (A+B x^2) \, dx\)

Optimal. Leaf size=28 \[ \frac {1}{3} x^3 (a B+A b)+a A x+\frac {1}{5} b B x^5 \]

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Rubi [A]  time = 0.01, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {373} \begin {gather*} \frac {1}{3} x^3 (a B+A b)+a A x+\frac {1}{5} b B x^5 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)*(A + B*x^2),x]

[Out]

a*A*x + ((A*b + a*B)*x^3)/3 + (b*B*x^5)/5

Rule 373

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n
)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[p, 0] && IGtQ[q, 0]

Rubi steps

\begin {align*} \int \left (a+b x^2\right ) \left (A+B x^2\right ) \, dx &=\int \left (a A+(A b+a B) x^2+b B x^4\right ) \, dx\\ &=a A x+\frac {1}{3} (A b+a B) x^3+\frac {1}{5} b B x^5\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 28, normalized size = 1.00 \begin {gather*} \frac {1}{3} x^3 (a B+A b)+a A x+\frac {1}{5} b B x^5 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)*(A + B*x^2),x]

[Out]

a*A*x + ((A*b + a*B)*x^3)/3 + (b*B*x^5)/5

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a+b x^2\right ) \left (A+B x^2\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.40, size = 26, normalized size = 0.93 \begin {gather*} \frac {1}{5} x^{5} b B + \frac {1}{3} x^{3} a B + \frac {1}{3} x^{3} b A + x a A \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*x^5*b*B + 1/3*x^3*a*B + 1/3*x^3*b*A + x*a*A

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giac [A]  time = 0.41, size = 26, normalized size = 0.93 \begin {gather*} \frac {1}{5} \, B b x^{5} + \frac {1}{3} \, B a x^{3} + \frac {1}{3} \, A b x^{3} + A a x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*B*b*x^5 + 1/3*B*a*x^3 + 1/3*A*b*x^3 + A*a*x

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maple [A]  time = 0.00, size = 25, normalized size = 0.89 \begin {gather*} \frac {B b \,x^{5}}{5}+A a x +\frac {\left (A b +B a \right ) x^{3}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*A*x+1/3*(A*b+B*a)*x^3+1/5*b*B*x^5

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maxima [A]  time = 1.34, size = 24, normalized size = 0.86 \begin {gather*} \frac {1}{5} \, B b x^{5} + \frac {1}{3} \, {\left (B a + A b\right )} x^{3} + A a x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*B*b*x^5 + 1/3*(B*a + A*b)*x^3 + A*a*x

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mupad [B]  time = 0.04, size = 25, normalized size = 0.89 \begin {gather*} \frac {B\,b\,x^5}{5}+\left (\frac {A\,b}{3}+\frac {B\,a}{3}\right )\,x^3+A\,a\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x^2)*(a + b*x^2),x)

[Out]

x^3*((A*b)/3 + (B*a)/3) + A*a*x + (B*b*x^5)/5

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sympy [A]  time = 0.06, size = 26, normalized size = 0.93 \begin {gather*} A a x + \frac {B b x^{5}}{5} + x^{3} \left (\frac {A b}{3} + \frac {B a}{3}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

A*a*x + B*b*x**5/5 + x**3*(A*b/3 + B*a/3)

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