3.3.69 \(\int \frac {c+d x^2}{x^3 (a+b x^2)^2} \, dx\)

Optimal. Leaf size=76 \[ \frac {(2 b c-a d) \log \left (a+b x^2\right )}{2 a^3}-\frac {\log (x) (2 b c-a d)}{a^3}-\frac {b c-a d}{2 a^2 \left (a+b x^2\right )}-\frac {c}{2 a^2 x^2} \]

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Rubi [A]  time = 0.07, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {446, 77} \begin {gather*} -\frac {b c-a d}{2 a^2 \left (a+b x^2\right )}+\frac {(2 b c-a d) \log \left (a+b x^2\right )}{2 a^3}-\frac {\log (x) (2 b c-a d)}{a^3}-\frac {c}{2 a^2 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x^2)/(x^3*(a + b*x^2)^2),x]

[Out]

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

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]))))

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

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Mathematica [A]  time = 0.05, size = 64, normalized size = 0.84 \begin {gather*} \frac {\frac {a (a d-b c)}{a+b x^2}+(2 b c-a d) \log \left (a+b x^2\right )+2 \log (x) (a d-2 b c)-\frac {a c}{x^2}}{2 a^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x^2)/(x^3*(a + b*x^2)^2),x]

[Out]

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

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {c+d x^2}{x^3 \left (a+b x^2\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(c + d*x^2)/(x^3*(a + b*x^2)^2),x]

[Out]

IntegrateAlgebraic[(c + d*x^2)/(x^3*(a + b*x^2)^2), x]

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fricas [A]  time = 0.85, size = 122, normalized size = 1.61 \begin {gather*} -\frac {a^{2} c + {\left (2 \, a b c - a^{2} d\right )} x^{2} - {\left ({\left (2 \, b^{2} c - a b d\right )} x^{4} + {\left (2 \, a b c - a^{2} d\right )} x^{2}\right )} \log \left (b x^{2} + a\right ) + 2 \, {\left ({\left (2 \, b^{2} c - a b d\right )} x^{4} + {\left (2 \, a b c - a^{2} d\right )} x^{2}\right )} \log \relax (x)}{2 \, {\left (a^{3} b x^{4} + a^{4} x^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^2+c)/x^3/(b*x^2+a)^2,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.29, size = 84, normalized size = 1.11 \begin {gather*} -\frac {{\left (2 \, b c - a d\right )} \log \left (x^{2}\right )}{2 \, a^{3}} - \frac {2 \, b c x^{2} - a d x^{2} + a c}{2 \, {\left (b x^{4} + a x^{2}\right )} a^{2}} + \frac {{\left (2 \, b^{2} c - a b d\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{3} b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^2+c)/x^3/(b*x^2+a)^2,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.02, size = 86, normalized size = 1.13 \begin {gather*} \frac {d}{2 \left (b \,x^{2}+a \right ) a}-\frac {b c}{2 \left (b \,x^{2}+a \right ) a^{2}}+\frac {d \ln \relax (x )}{a^{2}}-\frac {d \ln \left (b \,x^{2}+a \right )}{2 a^{2}}-\frac {2 b c \ln \relax (x )}{a^{3}}+\frac {b c \ln \left (b \,x^{2}+a \right )}{a^{3}}-\frac {c}{2 a^{2} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x^2+c)/x^3/(b*x^2+a)^2,x)

[Out]

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

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maxima [A]  time = 1.00, size = 78, normalized size = 1.03 \begin {gather*} -\frac {{\left (2 \, b c - a d\right )} x^{2} + a c}{2 \, {\left (a^{2} b x^{4} + a^{3} x^{2}\right )}} + \frac {{\left (2 \, b c - a d\right )} \log \left (b x^{2} + a\right )}{2 \, a^{3}} - \frac {{\left (2 \, b c - a d\right )} \log \left (x^{2}\right )}{2 \, a^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^2+c)/x^3/(b*x^2+a)^2,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.11, size = 74, normalized size = 0.97 \begin {gather*} \frac {\ln \relax (x)\,\left (a\,d-2\,b\,c\right )}{a^3}-\frac {\ln \left (b\,x^2+a\right )\,\left (a\,d-2\,b\,c\right )}{2\,a^3}-\frac {\frac {c}{2\,a}-\frac {x^2\,\left (a\,d-2\,b\,c\right )}{2\,a^2}}{b\,x^4+a\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x^2)/(x^3*(a + b*x^2)^2),x)

[Out]

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

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sympy [A]  time = 0.87, size = 70, normalized size = 0.92 \begin {gather*} \frac {- a c + x^{2} \left (a d - 2 b c\right )}{2 a^{3} x^{2} + 2 a^{2} b x^{4}} + \frac {\left (a d - 2 b c\right ) \log {\relax (x )}}{a^{3}} - \frac {\left (a d - 2 b c\right ) \log {\left (\frac {a}{b} + x^{2} \right )}}{2 a^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x**2+c)/x**3/(b*x**2+a)**2,x)

[Out]

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

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