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

Optimal. Leaf size=31 \[ \frac {1}{2} x^2 (a B+A b)-\frac {a A}{x}+\frac {1}{5} b B x^5 \]

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Rubi [A]  time = 0.02, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {448} \begin {gather*} \frac {1}{2} x^2 (a B+A b)-\frac {a A}{x}+\frac {1}{5} b B x^5 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x^3)*(A + B*x^3))/x^2,x]

[Out]

-((a*A)/x) + ((A*b + a*B)*x^2)/2 + (b*B*x^5)/5

Rule 448

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

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 31, normalized size = 1.00 \begin {gather*} \frac {1}{2} x^2 (a B+A b)-\frac {a A}{x}+\frac {1}{5} b B x^5 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((a*A)/x) + ((A*b + a*B)*x^2)/2 + (b*B*x^5)/5

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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^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.73, size = 29, normalized size = 0.94 \begin {gather*} \frac {2 \, B b x^{6} + 5 \, {\left (B a + A b\right )} x^{3} - 10 \, A a}{10 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*(2*B*b*x^6 + 5*(B*a + A*b)*x^3 - 10*A*a)/x

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giac [A]  time = 0.15, size = 29, normalized size = 0.94 \begin {gather*} \frac {1}{5} \, B b x^{5} + \frac {1}{2} \, B a x^{2} + \frac {1}{2} \, A b x^{2} - \frac {A a}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*B*b*x^5 + 1/2*B*a*x^2 + 1/2*A*b*x^2 - A*a/x

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maple [A]  time = 0.05, size = 30, normalized size = 0.97 \begin {gather*} \frac {B b \,x^{5}}{5}+\frac {A b \,x^{2}}{2}+\frac {B a \,x^{2}}{2}-\frac {A a}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*b*B*x^5+1/2*A*b*x^2+1/2*B*a*x^2-A*a/x

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maxima [A]  time = 0.50, size = 27, normalized size = 0.87 \begin {gather*} \frac {1}{5} \, B b x^{5} + \frac {1}{2} \, {\left (B a + A b\right )} x^{2} - \frac {A a}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*B*b*x^5 + 1/2*(B*a + A*b)*x^2 - A*a/x

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mupad [B]  time = 0.04, size = 28, normalized size = 0.90 \begin {gather*} x^2\,\left (\frac {A\,b}{2}+\frac {B\,a}{2}\right )-\frac {A\,a}{x}+\frac {B\,b\,x^5}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x^3)*(a + b*x^3))/x^2,x)

[Out]

x^2*((A*b)/2 + (B*a)/2) - (A*a)/x + (B*b*x^5)/5

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sympy [A]  time = 0.11, size = 26, normalized size = 0.84 \begin {gather*} - \frac {A a}{x} + \frac {B b x^{5}}{5} + x^{2} \left (\frac {A b}{2} + \frac {B a}{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-A*a/x + B*b*x**5/5 + x**2*(A*b/2 + B*a/2)

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