3.2.77 \(\int \frac {(a+b x^2)^2}{x^5 (c+d x^2)} \, dx\)

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

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Rubi [A]  time = 0.07, antiderivative size = 75, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\frac {a^2}{4 c x^4}-\frac {a (2 b c-a d)}{2 c^2 x^2}-\frac {(b c-a d)^2 \log \left (c+d x^2\right )}{2 c^3}+\frac {\log (x) (b c-a d)^2}{c^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^2/(x^5*(c + d*x^2)),x]

[Out]

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

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]))

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]]

Rubi steps

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

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Mathematica [A]  time = 0.04, size = 72, normalized size = 0.96 \begin {gather*} -\frac {-4 x^4 \log (x) (b c-a d)^2+a c \left (a c-2 a d x^2+4 b c x^2\right )+2 x^4 (b c-a d)^2 \log \left (c+d x^2\right )}{4 c^3 x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^2/(x^5*(c + d*x^2)),x]

[Out]

-1/4*(a*c*(a*c + 4*b*c*x^2 - 2*a*d*x^2) - 4*(b*c - a*d)^2*x^4*Log[x] + 2*(b*c - a*d)^2*x^4*Log[c + d*x^2])/(c^
3*x^4)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a+b x^2\right )^2}{x^5 \left (c+d x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + b*x^2)^2/(x^5*(c + d*x^2)),x]

[Out]

IntegrateAlgebraic[(a + b*x^2)^2/(x^5*(c + d*x^2)), x]

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fricas [A]  time = 0.89, size = 98, normalized size = 1.31 \begin {gather*} -\frac {2 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x^{4} \log \left (d x^{2} + c\right ) - 4 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x^{4} \log \relax (x) + a^{2} c^{2} + 2 \, {\left (2 \, a b c^{2} - a^{2} c d\right )} x^{2}}{4 \, c^{3} x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2/x^5/(d*x^2+c),x, algorithm="fricas")

[Out]

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

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giac [B]  time = 0.33, size = 139, normalized size = 1.85 \begin {gather*} \frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (x^{2}\right )}{2 \, c^{3}} - \frac {{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} \log \left ({\left | d x^{2} + c \right |}\right )}{2 \, c^{3} d} - \frac {3 \, b^{2} c^{2} x^{4} - 6 \, a b c d x^{4} + 3 \, a^{2} d^{2} x^{4} + 4 \, a b c^{2} x^{2} - 2 \, a^{2} c d x^{2} + a^{2} c^{2}}{4 \, c^{3} x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2/x^5/(d*x^2+c),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.01, size = 116, normalized size = 1.55 \begin {gather*} \frac {a^{2} d^{2} \ln \relax (x )}{c^{3}}-\frac {a^{2} d^{2} \ln \left (d \,x^{2}+c \right )}{2 c^{3}}-\frac {2 a b d \ln \relax (x )}{c^{2}}+\frac {a b d \ln \left (d \,x^{2}+c \right )}{c^{2}}+\frac {b^{2} \ln \relax (x )}{c}-\frac {b^{2} \ln \left (d \,x^{2}+c \right )}{2 c}+\frac {a^{2} d}{2 c^{2} x^{2}}-\frac {a b}{c \,x^{2}}-\frac {a^{2}}{4 c \,x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^2/x^5/(d*x^2+c),x)

[Out]

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

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maxima [A]  time = 0.98, size = 96, normalized size = 1.28 \begin {gather*} -\frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (d x^{2} + c\right )}{2 \, c^{3}} + \frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (x^{2}\right )}{2 \, c^{3}} - \frac {a^{2} c + 2 \, {\left (2 \, a b c - a^{2} d\right )} x^{2}}{4 \, c^{2} x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2/x^5/(d*x^2+c),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.16, size = 93, normalized size = 1.24 \begin {gather*} \frac {\ln \relax (x)\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}{c^3}-\frac {\frac {a^2}{4\,c}-\frac {a\,x^2\,\left (a\,d-2\,b\,c\right )}{2\,c^2}}{x^4}-\frac {\ln \left (d\,x^2+c\right )\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}{2\,c^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)^2/(x^5*(c + d*x^2)),x)

[Out]

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

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sympy [A]  time = 1.32, size = 66, normalized size = 0.88 \begin {gather*} \frac {- a^{2} c + x^{2} \left (2 a^{2} d - 4 a b c\right )}{4 c^{2} x^{4}} + \frac {\left (a d - b c\right )^{2} \log {\relax (x )}}{c^{3}} - \frac {\left (a d - b c\right )^{2} \log {\left (\frac {c}{d} + x^{2} \right )}}{2 c^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**2/x**5/(d*x**2+c),x)

[Out]

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

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