3.21 \(\int \frac{(a+b x^2)^2 (A+B x^2)}{x^8} \, dx\)

Optimal. Leaf size=53 \[ -\frac{a^2 A}{7 x^7}-\frac{a (a B+2 A b)}{5 x^5}-\frac{b (2 a B+A b)}{3 x^3}-\frac{b^2 B}{x} \]

[Out]

-(a^2*A)/(7*x^7) - (a*(2*A*b + a*B))/(5*x^5) - (b*(A*b + 2*a*B))/(3*x^3) - (b^2*B)/x

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Rubi [A]  time = 0.0274398, antiderivative size = 53, 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} \[ -\frac{a^2 A}{7 x^7}-\frac{a (a B+2 A b)}{5 x^5}-\frac{b (2 a B+A b)}{3 x^3}-\frac{b^2 B}{x} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x^2)^2*(A + B*x^2))/x^8,x]

[Out]

-(a^2*A)/(7*x^7) - (a*(2*A*b + a*B))/(5*x^5) - (b*(A*b + 2*a*B))/(3*x^3) - (b^2*B)/x

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^2\right )^2 \left (A+B x^2\right )}{x^8} \, dx &=\int \left (\frac{a^2 A}{x^8}+\frac{a (2 A b+a B)}{x^6}+\frac{b (A b+2 a B)}{x^4}+\frac{b^2 B}{x^2}\right ) \, dx\\ &=-\frac{a^2 A}{7 x^7}-\frac{a (2 A b+a B)}{5 x^5}-\frac{b (A b+2 a B)}{3 x^3}-\frac{b^2 B}{x}\\ \end{align*}

Mathematica [A]  time = 0.0161532, size = 56, normalized size = 1.06 \[ -\frac{3 a^2 \left (5 A+7 B x^2\right )+14 a b x^2 \left (3 A+5 B x^2\right )+35 b^2 x^4 \left (A+3 B x^2\right )}{105 x^7} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x^2)^2*(A + B*x^2))/x^8,x]

[Out]

-(35*b^2*x^4*(A + 3*B*x^2) + 14*a*b*x^2*(3*A + 5*B*x^2) + 3*a^2*(5*A + 7*B*x^2))/(105*x^7)

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Maple [A]  time = 0.004, size = 48, normalized size = 0.9 \begin{align*} -{\frac{A{a}^{2}}{7\,{x}^{7}}}-{\frac{a \left ( 2\,Ab+Ba \right ) }{5\,{x}^{5}}}-{\frac{b \left ( Ab+2\,Ba \right ) }{3\,{x}^{3}}}-{\frac{B{b}^{2}}{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^2*(B*x^2+A)/x^8,x)

[Out]

-1/7*a^2*A/x^7-1/5*a*(2*A*b+B*a)/x^5-1/3*b*(A*b+2*B*a)/x^3-b^2*B/x

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Maxima [A]  time = 1.00985, size = 72, normalized size = 1.36 \begin{align*} -\frac{105 \, B b^{2} x^{6} + 35 \,{\left (2 \, B a b + A b^{2}\right )} x^{4} + 15 \, A a^{2} + 21 \,{\left (B a^{2} + 2 \, A a b\right )} x^{2}}{105 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2*(B*x^2+A)/x^8,x, algorithm="maxima")

[Out]

-1/105*(105*B*b^2*x^6 + 35*(2*B*a*b + A*b^2)*x^4 + 15*A*a^2 + 21*(B*a^2 + 2*A*a*b)*x^2)/x^7

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Fricas [A]  time = 1.38145, size = 126, normalized size = 2.38 \begin{align*} -\frac{105 \, B b^{2} x^{6} + 35 \,{\left (2 \, B a b + A b^{2}\right )} x^{4} + 15 \, A a^{2} + 21 \,{\left (B a^{2} + 2 \, A a b\right )} x^{2}}{105 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2*(B*x^2+A)/x^8,x, algorithm="fricas")

[Out]

-1/105*(105*B*b^2*x^6 + 35*(2*B*a*b + A*b^2)*x^4 + 15*A*a^2 + 21*(B*a^2 + 2*A*a*b)*x^2)/x^7

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Sympy [A]  time = 1.13365, size = 56, normalized size = 1.06 \begin{align*} - \frac{15 A a^{2} + 105 B b^{2} x^{6} + x^{4} \left (35 A b^{2} + 70 B a b\right ) + x^{2} \left (42 A a b + 21 B a^{2}\right )}{105 x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**2*(B*x**2+A)/x**8,x)

[Out]

-(15*A*a**2 + 105*B*b**2*x**6 + x**4*(35*A*b**2 + 70*B*a*b) + x**2*(42*A*a*b + 21*B*a**2))/(105*x**7)

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Giac [A]  time = 1.16392, size = 74, normalized size = 1.4 \begin{align*} -\frac{105 \, B b^{2} x^{6} + 70 \, B a b x^{4} + 35 \, A b^{2} x^{4} + 21 \, B a^{2} x^{2} + 42 \, A a b x^{2} + 15 \, A a^{2}}{105 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2*(B*x^2+A)/x^8,x, algorithm="giac")

[Out]

-1/105*(105*B*b^2*x^6 + 70*B*a*b*x^4 + 35*A*b^2*x^4 + 21*B*a^2*x^2 + 42*A*a*b*x^2 + 15*A*a^2)/x^7