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

Optimal. Leaf size=80 \[ \frac {c (b c-a d) \log \left (a+b x^2\right )}{a^3}-\frac {2 c \log (x) (b c-a d)}{a^3}-\frac {(b c-a d)^2}{2 a^2 b \left (a+b x^2\right )}-\frac {c^2}{2 a^2 x^2} \]

[Out]

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

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Rubi [A]  time = 0.08, antiderivative size = 80, 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} \[ -\frac {(b c-a d)^2}{2 a^2 b \left (a+b x^2\right )}+\frac {c (b c-a d) \log \left (a+b x^2\right )}{a^3}-\frac {2 c \log (x) (b c-a d)}{a^3}-\frac {c^2}{2 a^2 x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]  time = 0.10, size = 72, normalized size = 0.90 \[ -\frac {\frac {a (b c-a d)^2}{b \left (a+b x^2\right )}-2 c (b c-a d) \log \left (a+b x^2\right )+4 c \log (x) (b c-a d)+\frac {a c^2}{x^2}}{2 a^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.49, size = 159, normalized size = 1.99 \[ -\frac {a^{2} b c^{2} + {\left (2 \, a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )} x^{2} - 2 \, {\left ({\left (b^{3} c^{2} - a b^{2} c d\right )} x^{4} + {\left (a b^{2} c^{2} - a^{2} b c d\right )} x^{2}\right )} \log \left (b x^{2} + a\right ) + 4 \, {\left ({\left (b^{3} c^{2} - a b^{2} c d\right )} x^{4} + {\left (a b^{2} c^{2} - a^{2} b c d\right )} x^{2}\right )} \log \relax (x)}{2 \, {\left (a^{3} b^{2} x^{4} + a^{4} b x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.34, size = 109, normalized size = 1.36 \[ -\frac {{\left (b c^{2} - a c d\right )} \log \left (x^{2}\right )}{a^{3}} + \frac {{\left (b^{2} c^{2} - a b c d\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{a^{3} b} - \frac {2 \, b^{2} c^{2} x^{2} - 2 \, a b c d x^{2} + a^{2} d^{2} x^{2} + a b c^{2}}{2 \, {\left (b x^{4} + a x^{2}\right )} a^{2} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.01, size = 114, normalized size = 1.42 \[ \frac {c d}{\left (b \,x^{2}+a \right ) a}-\frac {b \,c^{2}}{2 \left (b \,x^{2}+a \right ) a^{2}}+\frac {2 c d \ln \relax (x )}{a^{2}}-\frac {c d \ln \left (b \,x^{2}+a \right )}{a^{2}}-\frac {2 b \,c^{2} \ln \relax (x )}{a^{3}}+\frac {b \,c^{2} \ln \left (b \,x^{2}+a \right )}{a^{3}}-\frac {d^{2}}{2 \left (b \,x^{2}+a \right ) b}-\frac {c^{2}}{2 a^{2} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.00, size = 100, normalized size = 1.25 \[ -\frac {a b c^{2} + {\left (2 \, b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x^{2}}{2 \, {\left (a^{2} b^{2} x^{4} + a^{3} b x^{2}\right )}} + \frac {{\left (b c^{2} - a c d\right )} \log \left (b x^{2} + a\right )}{a^{3}} - \frac {{\left (b c^{2} - a c d\right )} \log \left (x^{2}\right )}{a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.20, size = 100, normalized size = 1.25 \[ \frac {\ln \left (b\,x^2+a\right )\,\left (b\,c^2-a\,c\,d\right )}{a^3}-\frac {\frac {c^2}{2\,a}+\frac {x^2\,\left (a^2\,d^2-2\,a\,b\,c\,d+2\,b^2\,c^2\right )}{2\,a^2\,b}}{b\,x^4+a\,x^2}-\frac {\ln \relax (x)\,\left (2\,b\,c^2-2\,a\,c\,d\right )}{a^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 1.36, size = 92, normalized size = 1.15 \[ \frac {- a b c^{2} + x^{2} \left (- a^{2} d^{2} + 2 a b c d - 2 b^{2} c^{2}\right )}{2 a^{3} b x^{2} + 2 a^{2} b^{2} x^{4}} + \frac {2 c \left (a d - b c\right ) \log {\relax (x )}}{a^{3}} - \frac {c \left (a d - b c\right ) \log {\left (\frac {a}{b} + x^{2} \right )}}{a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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