3.1.19 \(\int \frac {(a+b x^3)^2 (A+B x^3)}{x^6} \, dx\)

Optimal. Leaf size=50 \[ -\frac {a^2 A}{5 x^5}-\frac {a (a B+2 A b)}{2 x^2}+b x (2 a B+A b)+\frac {1}{4} b^2 B x^4 \]

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Rubi [A]  time = 0.03, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {448} \begin {gather*} -\frac {a^2 A}{5 x^5}-\frac {a (a B+2 A b)}{2 x^2}+b x (2 a B+A b)+\frac {1}{4} b^2 B x^4 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(a^2*A)/(5*x^5) - (a*(2*A*b + a*B))/(2*x^2) + b*(A*b + 2*a*B)*x + (b^2*B*x^4)/4

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 )^2 \left (A+B x^3\right )}{x^6} \, dx &=\int \left (b (A b+2 a B)+\frac {a^2 A}{x^6}+\frac {a (2 A b+a B)}{x^3}+b^2 B x^3\right ) \, dx\\ &=-\frac {a^2 A}{5 x^5}-\frac {a (2 A b+a B)}{2 x^2}+b (A b+2 a B) x+\frac {1}{4} b^2 B x^4\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 50, normalized size = 1.00 \begin {gather*} -\frac {a^2 A}{5 x^5}-\frac {a (a B+2 A b)}{2 x^2}+b x (2 a B+A b)+\frac {1}{4} b^2 B x^4 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/5*(a^2*A)/x^5 - (a*(2*A*b + a*B))/(2*x^2) + b*(A*b + 2*a*B)*x + (b^2*B*x^4)/4

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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 )^2 \left (A+B x^3\right )}{x^6} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.37, size = 53, normalized size = 1.06 \begin {gather*} \frac {5 \, B b^{2} x^{9} + 20 \, {\left (2 \, B a b + A b^{2}\right )} x^{6} - 10 \, {\left (B a^{2} + 2 \, A a b\right )} x^{3} - 4 \, A a^{2}}{20 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^2*(B*x^3+A)/x^6,x, algorithm="fricas")

[Out]

1/20*(5*B*b^2*x^9 + 20*(2*B*a*b + A*b^2)*x^6 - 10*(B*a^2 + 2*A*a*b)*x^3 - 4*A*a^2)/x^5

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giac [A]  time = 0.15, size = 51, normalized size = 1.02 \begin {gather*} \frac {1}{4} \, B b^{2} x^{4} + 2 \, B a b x + A b^{2} x - \frac {5 \, B a^{2} x^{3} + 10 \, A a b x^{3} + 2 \, A a^{2}}{10 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^2*(B*x^3+A)/x^6,x, algorithm="giac")

[Out]

1/4*B*b^2*x^4 + 2*B*a*b*x + A*b^2*x - 1/10*(5*B*a^2*x^3 + 10*A*a*b*x^3 + 2*A*a^2)/x^5

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maple [A]  time = 0.06, size = 46, normalized size = 0.92 \begin {gather*} \frac {B \,b^{2} x^{4}}{4}+A \,b^{2} x +2 B a b x -\frac {\left (2 A b +B a \right ) a}{2 x^{2}}-\frac {A \,a^{2}}{5 x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^2*(B*x^3+A)/x^6,x)

[Out]

1/4*B*b^2*x^4+b^2*A*x+2*a*b*B*x-1/5*a^2*A/x^5-1/2*a*(2*A*b+B*a)/x^2

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maxima [A]  time = 0.57, size = 51, normalized size = 1.02 \begin {gather*} \frac {1}{4} \, B b^{2} x^{4} + {\left (2 \, B a b + A b^{2}\right )} x - \frac {5 \, {\left (B a^{2} + 2 \, A a b\right )} x^{3} + 2 \, A a^{2}}{10 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^2*(B*x^3+A)/x^6,x, algorithm="maxima")

[Out]

1/4*B*b^2*x^4 + (2*B*a*b + A*b^2)*x - 1/10*(5*(B*a^2 + 2*A*a*b)*x^3 + 2*A*a^2)/x^5

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mupad [B]  time = 2.37, size = 50, normalized size = 1.00 \begin {gather*} x\,\left (A\,b^2+2\,B\,a\,b\right )-\frac {x^3\,\left (\frac {B\,a^2}{2}+A\,b\,a\right )+\frac {A\,a^2}{5}}{x^5}+\frac {B\,b^2\,x^4}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x^3)*(a + b*x^3)^2)/x^6,x)

[Out]

x*(A*b^2 + 2*B*a*b) - (x^3*((B*a^2)/2 + A*a*b) + (A*a^2)/5)/x^5 + (B*b^2*x^4)/4

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sympy [A]  time = 0.35, size = 53, normalized size = 1.06 \begin {gather*} \frac {B b^{2} x^{4}}{4} + x \left (A b^{2} + 2 B a b\right ) + \frac {- 2 A a^{2} + x^{3} \left (- 10 A a b - 5 B a^{2}\right )}{10 x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**2*(B*x**3+A)/x**6,x)

[Out]

B*b**2*x**4/4 + x*(A*b**2 + 2*B*a*b) + (-2*A*a**2 + x**3*(-10*A*a*b - 5*B*a**2))/(10*x**5)

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