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

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

[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

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Rubi [A]  time = 0.0266763, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {448} \[ -\frac{a^2 A}{2 x^2}+\frac{1}{4} b x^4 (2 a B+A b)+a x (a B+2 A b)+\frac{1}{7} b^2 B x^7 \]

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*}

Mathematica [A]  time = 0.0142723, size = 50, normalized size = 1. \[ -\frac{a^2 A}{2 x^2}+\frac{1}{4} b x^4 (2 a B+A b)+a x (a B+2 A b)+\frac{1}{7} b^2 B x^7 \]

Antiderivative was successfully verified.

[In]

Integrate[((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

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Maple [A]  time = 0.003, size = 49, normalized size = 1. \begin{align*}{\frac{{b}^{2}B{x}^{7}}{7}}+{\frac{A{x}^{4}{b}^{2}}{4}}+{\frac{B{x}^{4}ab}{2}}+2\,abAx+{a}^{2}Bx-{\frac{A{a}^{2}}{2\,{x}^{2}}} \end{align*}

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

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Maxima [A]  time = 1.15342, size = 65, normalized size = 1.3 \begin{align*} \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{align*}

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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Fricas [A]  time = 1.38677, size = 119, normalized size = 2.38 \begin{align*} \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{align*}

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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Sympy [A]  time = 0.284578, size = 49, normalized size = 0.98 \begin{align*} - \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{align*}

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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Giac [A]  time = 1.2148, size = 65, normalized size = 1.3 \begin{align*} \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{align*}

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