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

Optimal. Leaf size=50 \[ -\frac {a^2 A}{2 x^2}+\frac {1}{4} b x^4 (2 a B+A b)+a x (a B+2 A b)+\frac {1}{7} b^2 B x^7 \]

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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}{2 x^2}+\frac {1}{4} b x^4 (2 a B+A b)+a x (a B+2 A b)+\frac {1}{7} b^2 B x^7 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(a^2*A)/(2*x^2) + a*(2*A*b + a*B)*x + (b*(A*b + 2*a*B)*x^4)/4 + (b^2*B*x^7)/7

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^3} \, dx &=\int \left (a (2 A b+a B)+\frac {a^2 A}{x^3}+b (A b+2 a B) x^3+b^2 B x^6\right ) \, dx\\ &=-\frac {a^2 A}{2 x^2}+a (2 A b+a B) x+\frac {1}{4} b (A b+2 a B) x^4+\frac {1}{7} b^2 B x^7\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 50, normalized size = 1.00 \begin {gather*} -\frac {a^2 A}{2 x^2}+\frac {1}{4} b x^4 (2 a B+A b)+a x (a B+2 A b)+\frac {1}{7} b^2 B x^7 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*(a^2*A)/x^2 + a*(2*A*b + a*B)*x + (b*(A*b + 2*a*B)*x^4)/4 + (b^2*B*x^7)/7

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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^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.83, size = 53, normalized size = 1.06 \begin {gather*} \frac {4 \, B b^{2} x^{9} + 7 \, {\left (2 \, B a b + A b^{2}\right )} x^{6} + 28 \, {\left (B a^{2} + 2 \, A a b\right )} x^{3} - 14 \, A a^{2}}{28 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/28*(4*B*b^2*x^9 + 7*(2*B*a*b + A*b^2)*x^6 + 28*(B*a^2 + 2*A*a*b)*x^3 - 14*A*a^2)/x^2

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giac [A]  time = 0.17, size = 48, normalized size = 0.96 \begin {gather*} \frac {1}{7} \, B b^{2} x^{7} + \frac {1}{2} \, B a b x^{4} + \frac {1}{4} \, A b^{2} x^{4} + B a^{2} x + 2 \, A a b x - \frac {A a^{2}}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*B*b^2*x^7 + 1/2*B*a*b*x^4 + 1/4*A*b^2*x^4 + B*a^2*x + 2*A*a*b*x - 1/2*A*a^2/x^2

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maple [A]  time = 0.04, size = 49, normalized size = 0.98 \begin {gather*} \frac {B \,b^{2} x^{7}}{7}+\frac {A \,b^{2} x^{4}}{4}+\frac {B a b \,x^{4}}{2}+2 A a b x +B \,a^{2} x -\frac {A \,a^{2}}{2 x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*b^2*B*x^7+1/4*A*b^2*x^4+1/2*B*x^4*a*b+2*a*b*A*x+B*a^2*x-1/2*A*a^2/x^2

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maxima [A]  time = 0.49, size = 48, normalized size = 0.96 \begin {gather*} \frac {1}{7} \, B b^{2} x^{7} + \frac {1}{4} \, {\left (2 \, B a b + A b^{2}\right )} x^{4} + {\left (B a^{2} + 2 \, A a b\right )} x - \frac {A a^{2}}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*B*b^2*x^7 + 1/4*(2*B*a*b + A*b^2)*x^4 + (B*a^2 + 2*A*a*b)*x - 1/2*A*a^2/x^2

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mupad [B]  time = 0.05, size = 48, normalized size = 0.96 \begin {gather*} x^4\,\left (\frac {A\,b^2}{4}+\frac {B\,a\,b}{2}\right )+x\,\left (B\,a^2+2\,A\,b\,a\right )-\frac {A\,a^2}{2\,x^2}+\frac {B\,b^2\,x^7}{7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^4*((A*b^2)/4 + (B*a*b)/2) + x*(B*a^2 + 2*A*a*b) - (A*a^2)/(2*x^2) + (B*b^2*x^7)/7

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sympy [A]  time = 0.15, size = 49, normalized size = 0.98 \begin {gather*} - \frac {A a^{2}}{2 x^{2}} + \frac {B b^{2} x^{7}}{7} + x^{4} \left (\frac {A b^{2}}{4} + \frac {B a b}{2}\right ) + x \left (2 A a b + B a^{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-A*a**2/(2*x**2) + B*b**2*x**7/7 + x**4*(A*b**2/4 + B*a*b/2) + x*(2*A*a*b + B*a**2)

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