3.179 \(\int \frac{(a+b x^2)^2}{x^7 (c+d x^2)} \, dx\)

Optimal. Leaf size=98 \[ -\frac{a^2}{6 c x^6}-\frac{a (2 b c-a d)}{4 c^2 x^4}-\frac{(b c-a d)^2}{2 c^3 x^2}+\frac{d (b c-a d)^2 \log \left (c+d x^2\right )}{2 c^4}-\frac{d \log (x) (b c-a d)^2}{c^4} \]

[Out]

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

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Rubi [A]  time = 0.0833537, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {446, 88} \[ -\frac{a^2}{6 c x^6}-\frac{a (2 b c-a d)}{4 c^2 x^4}-\frac{(b c-a d)^2}{2 c^3 x^2}+\frac{d (b c-a d)^2 \log \left (c+d x^2\right )}{2 c^4}-\frac{d \log (x) (b c-a d)^2}{c^4} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^2/(x^7*(c + d*x^2)),x]

[Out]

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

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 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps

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

Mathematica [A]  time = 0.0607518, size = 108, normalized size = 1.1 \[ -\frac{c \left (a^2 \left (2 c^2-3 c d x^2+6 d^2 x^4\right )+6 a b c x^2 \left (c-2 d x^2\right )+6 b^2 c^2 x^4\right )+12 d x^6 \log (x) (b c-a d)^2-6 d x^6 (b c-a d)^2 \log \left (c+d x^2\right )}{12 c^4 x^6} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^2/(x^7*(c + d*x^2)),x]

[Out]

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

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Maple [A]  time = 0.006, size = 160, normalized size = 1.6 \begin{align*}{\frac{{d}^{3}\ln \left ( d{x}^{2}+c \right ){a}^{2}}{2\,{c}^{4}}}-{\frac{{d}^{2}\ln \left ( d{x}^{2}+c \right ) ab}{{c}^{3}}}+{\frac{d\ln \left ( d{x}^{2}+c \right ){b}^{2}}{2\,{c}^{2}}}-{\frac{{a}^{2}}{6\,c{x}^{6}}}-{\frac{{a}^{2}{d}^{2}}{2\,{c}^{3}{x}^{2}}}+{\frac{abd}{{c}^{2}{x}^{2}}}-{\frac{{b}^{2}}{2\,c{x}^{2}}}+{\frac{{a}^{2}d}{4\,{c}^{2}{x}^{4}}}-{\frac{ab}{2\,c{x}^{4}}}-{\frac{{d}^{3}\ln \left ( x \right ){a}^{2}}{{c}^{4}}}+2\,{\frac{{d}^{2}\ln \left ( x \right ) ab}{{c}^{3}}}-{\frac{d\ln \left ( x \right ){b}^{2}}{{c}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^2/x^7/(d*x^2+c),x)

[Out]

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

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Maxima [A]  time = 0.983237, size = 181, normalized size = 1.85 \begin{align*} \frac{{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} \log \left (d x^{2} + c\right )}{2 \, c^{4}} - \frac{{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} \log \left (x^{2}\right )}{2 \, c^{4}} - \frac{6 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x^{4} + 2 \, a^{2} c^{2} + 3 \,{\left (2 \, a b c^{2} - a^{2} c d\right )} x^{2}}{12 \, c^{3} x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2/x^7/(d*x^2+c),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.28096, size = 290, normalized size = 2.96 \begin{align*} \frac{6 \,{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} x^{6} \log \left (d x^{2} + c\right ) - 12 \,{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} x^{6} \log \left (x\right ) - 2 \, a^{2} c^{3} - 6 \,{\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} x^{4} - 3 \,{\left (2 \, a b c^{3} - a^{2} c^{2} d\right )} x^{2}}{12 \, c^{4} x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2/x^7/(d*x^2+c),x, algorithm="fricas")

[Out]

1/12*(6*(b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*x^6*log(d*x^2 + c) - 12*(b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*x^6*lo
g(x) - 2*a^2*c^3 - 6*(b^2*c^3 - 2*a*b*c^2*d + a^2*c*d^2)*x^4 - 3*(2*a*b*c^3 - a^2*c^2*d)*x^2)/(c^4*x^6)

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Sympy [A]  time = 1.6226, size = 105, normalized size = 1.07 \begin{align*} - \frac{2 a^{2} c^{2} + x^{4} \left (6 a^{2} d^{2} - 12 a b c d + 6 b^{2} c^{2}\right ) + x^{2} \left (- 3 a^{2} c d + 6 a b c^{2}\right )}{12 c^{3} x^{6}} - \frac{d \left (a d - b c\right )^{2} \log{\left (x \right )}}{c^{4}} + \frac{d \left (a d - b c\right )^{2} \log{\left (\frac{c}{d} + x^{2} \right )}}{2 c^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**2/x**7/(d*x**2+c),x)

[Out]

-(2*a**2*c**2 + x**4*(6*a**2*d**2 - 12*a*b*c*d + 6*b**2*c**2) + x**2*(-3*a**2*c*d + 6*a*b*c**2))/(12*c**3*x**6
) - d*(a*d - b*c)**2*log(x)/c**4 + d*(a*d - b*c)**2*log(c/d + x**2)/(2*c**4)

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Giac [B]  time = 1.15046, size = 248, normalized size = 2.53 \begin{align*} -\frac{{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} \log \left (x^{2}\right )}{2 \, c^{4}} + \frac{{\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} \log \left ({\left | d x^{2} + c \right |}\right )}{2 \, c^{4} d} + \frac{11 \, b^{2} c^{2} d x^{6} - 22 \, a b c d^{2} x^{6} + 11 \, a^{2} d^{3} x^{6} - 6 \, b^{2} c^{3} x^{4} + 12 \, a b c^{2} d x^{4} - 6 \, a^{2} c d^{2} x^{4} - 6 \, a b c^{3} x^{2} + 3 \, a^{2} c^{2} d x^{2} - 2 \, a^{2} c^{3}}{12 \, c^{4} x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2/x^7/(d*x^2+c),x, algorithm="giac")

[Out]

-1/2*(b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*log(x^2)/c^4 + 1/2*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*log(abs(d*x^
2 + c))/(c^4*d) + 1/12*(11*b^2*c^2*d*x^6 - 22*a*b*c*d^2*x^6 + 11*a^2*d^3*x^6 - 6*b^2*c^3*x^4 + 12*a*b*c^2*d*x^
4 - 6*a^2*c*d^2*x^4 - 6*a*b*c^3*x^2 + 3*a^2*c^2*d*x^2 - 2*a^2*c^3)/(c^4*x^6)