3.44 \(\int \frac{x^7 (A+B x^2)}{b x^2+c x^4} \, dx\)

Optimal. Leaf size=75 \[ -\frac{b^2 (b B-A c) \log \left (b+c x^2\right )}{2 c^4}-\frac{x^4 (b B-A c)}{4 c^2}+\frac{b x^2 (b B-A c)}{2 c^3}+\frac{B x^6}{6 c} \]

[Out]

(b*(b*B - A*c)*x^2)/(2*c^3) - ((b*B - A*c)*x^4)/(4*c^2) + (B*x^6)/(6*c) - (b^2*(b*B - A*c)*Log[b + c*x^2])/(2*
c^4)

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Rubi [A]  time = 0.0938306, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {1584, 446, 77} \[ -\frac{b^2 (b B-A c) \log \left (b+c x^2\right )}{2 c^4}-\frac{x^4 (b B-A c)}{4 c^2}+\frac{b x^2 (b B-A c)}{2 c^3}+\frac{B x^6}{6 c} \]

Antiderivative was successfully verified.

[In]

Int[(x^7*(A + B*x^2))/(b*x^2 + c*x^4),x]

[Out]

(b*(b*B - A*c)*x^2)/(2*c^3) - ((b*B - A*c)*x^4)/(4*c^2) + (B*x^6)/(6*c) - (b^2*(b*B - A*c)*Log[b + c*x^2])/(2*
c^4)

Rule 1584

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(m + n*p)*(a + b*x^(q -
p))^n, x] /; FreeQ[{a, b, m, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 446

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin{align*} \int \frac{x^7 \left (A+B x^2\right )}{b x^2+c x^4} \, dx &=\int \frac{x^5 \left (A+B x^2\right )}{b+c x^2} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^2 (A+B x)}{b+c x} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{b (b B-A c)}{c^3}+\frac{(-b B+A c) x}{c^2}+\frac{B x^2}{c}-\frac{b^2 (b B-A c)}{c^3 (b+c x)}\right ) \, dx,x,x^2\right )\\ &=\frac{b (b B-A c) x^2}{2 c^3}-\frac{(b B-A c) x^4}{4 c^2}+\frac{B x^6}{6 c}-\frac{b^2 (b B-A c) \log \left (b+c x^2\right )}{2 c^4}\\ \end{align*}

Mathematica [A]  time = 0.0289057, size = 71, normalized size = 0.95 \[ \frac{c x^2 \left (-3 b c \left (2 A+B x^2\right )+c^2 x^2 \left (3 A+2 B x^2\right )+6 b^2 B\right )+6 b^2 (A c-b B) \log \left (b+c x^2\right )}{12 c^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^7*(A + B*x^2))/(b*x^2 + c*x^4),x]

[Out]

(c*x^2*(6*b^2*B - 3*b*c*(2*A + B*x^2) + c^2*x^2*(3*A + 2*B*x^2)) + 6*b^2*(-(b*B) + A*c)*Log[b + c*x^2])/(12*c^
4)

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Maple [A]  time = 0.004, size = 86, normalized size = 1.2 \begin{align*}{\frac{B{x}^{6}}{6\,c}}+{\frac{A{x}^{4}}{4\,c}}-{\frac{B{x}^{4}b}{4\,{c}^{2}}}-{\frac{Ab{x}^{2}}{2\,{c}^{2}}}+{\frac{B{x}^{2}{b}^{2}}{2\,{c}^{3}}}+{\frac{{b}^{2}\ln \left ( c{x}^{2}+b \right ) A}{2\,{c}^{3}}}-{\frac{{b}^{3}\ln \left ( c{x}^{2}+b \right ) B}{2\,{c}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7*(B*x^2+A)/(c*x^4+b*x^2),x)

[Out]

1/6*B*x^6/c+1/4/c*A*x^4-1/4/c^2*B*x^4*b-1/2/c^2*A*x^2*b+1/2/c^3*B*x^2*b^2+1/2*b^2/c^3*ln(c*x^2+b)*A-1/2*b^3/c^
4*ln(c*x^2+b)*B

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Maxima [A]  time = 1.50688, size = 100, normalized size = 1.33 \begin{align*} \frac{2 \, B c^{2} x^{6} - 3 \,{\left (B b c - A c^{2}\right )} x^{4} + 6 \,{\left (B b^{2} - A b c\right )} x^{2}}{12 \, c^{3}} - \frac{{\left (B b^{3} - A b^{2} c\right )} \log \left (c x^{2} + b\right )}{2 \, c^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(B*x^2+A)/(c*x^4+b*x^2),x, algorithm="maxima")

[Out]

1/12*(2*B*c^2*x^6 - 3*(B*b*c - A*c^2)*x^4 + 6*(B*b^2 - A*b*c)*x^2)/c^3 - 1/2*(B*b^3 - A*b^2*c)*log(c*x^2 + b)/
c^4

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Fricas [A]  time = 0.489839, size = 155, normalized size = 2.07 \begin{align*} \frac{2 \, B c^{3} x^{6} - 3 \,{\left (B b c^{2} - A c^{3}\right )} x^{4} + 6 \,{\left (B b^{2} c - A b c^{2}\right )} x^{2} - 6 \,{\left (B b^{3} - A b^{2} c\right )} \log \left (c x^{2} + b\right )}{12 \, c^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(B*x^2+A)/(c*x^4+b*x^2),x, algorithm="fricas")

[Out]

1/12*(2*B*c^3*x^6 - 3*(B*b*c^2 - A*c^3)*x^4 + 6*(B*b^2*c - A*b*c^2)*x^2 - 6*(B*b^3 - A*b^2*c)*log(c*x^2 + b))/
c^4

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Sympy [A]  time = 0.445902, size = 65, normalized size = 0.87 \begin{align*} \frac{B x^{6}}{6 c} - \frac{b^{2} \left (- A c + B b\right ) \log{\left (b + c x^{2} \right )}}{2 c^{4}} - \frac{x^{4} \left (- A c + B b\right )}{4 c^{2}} + \frac{x^{2} \left (- A b c + B b^{2}\right )}{2 c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**7*(B*x**2+A)/(c*x**4+b*x**2),x)

[Out]

B*x**6/(6*c) - b**2*(-A*c + B*b)*log(b + c*x**2)/(2*c**4) - x**4*(-A*c + B*b)/(4*c**2) + x**2*(-A*b*c + B*b**2
)/(2*c**3)

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Giac [A]  time = 1.24536, size = 104, normalized size = 1.39 \begin{align*} \frac{2 \, B c^{2} x^{6} - 3 \, B b c x^{4} + 3 \, A c^{2} x^{4} + 6 \, B b^{2} x^{2} - 6 \, A b c x^{2}}{12 \, c^{3}} - \frac{{\left (B b^{3} - A b^{2} c\right )} \log \left ({\left | c x^{2} + b \right |}\right )}{2 \, c^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(B*x^2+A)/(c*x^4+b*x^2),x, algorithm="giac")

[Out]

1/12*(2*B*c^2*x^6 - 3*B*b*c*x^4 + 3*A*c^2*x^4 + 6*B*b^2*x^2 - 6*A*b*c*x^2)/c^3 - 1/2*(B*b^3 - A*b^2*c)*log(abs
(c*x^2 + b))/c^4