3.2.87 \(\int \frac {(a+b x^2)^2}{x^3 (c+d x^2)^2} \, dx\) [187]

Optimal. Leaf size=81 \[ -\frac {a^2}{2 c^2 x^2}-\frac {(b c-a d)^2}{2 c^2 d \left (c+d x^2\right )}+\frac {2 a (b c-a d) \log (x)}{c^3}-\frac {a (b c-a d) \log \left (c+d x^2\right )}{c^3} \]

[Out]

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

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Rubi [A]
time = 0.06, antiderivative size = 81, 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 = {457, 90} \begin {gather*} -\frac {a^2}{2 c^2 x^2}-\frac {a (b c-a d) \log \left (c+d x^2\right )}{c^3}+\frac {2 a \log (x) (b c-a d)}{c^3}-\frac {(b c-a d)^2}{2 c^2 d \left (c+d x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 90

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 457

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

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Mathematica [A]
time = 0.07, size = 72, normalized size = 0.89 \begin {gather*} -\frac {\frac {a^2 c}{x^2}+\frac {c (b c-a d)^2}{d \left (c+d x^2\right )}+4 a (-b c+a d) \log (x)-2 a (-b c+a d) \log \left (c+d x^2\right )}{2 c^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.09, size = 77, normalized size = 0.95

method	result	size
default	\(\frac {\left (a d -b c \right ) \left (-\frac {c \left (a d -b c \right )}{d \left (d \,x^{2}+c \right )}+2 a \ln \left (d \,x^{2}+c \right )\right )}{2 c^{3}}-\frac {a^{2}}{2 c^{2} x^{2}}-\frac {2 a \left (a d -b c \right ) \ln \left (x \right )}{c^{3}}\)	\(77\)
norman	\(\frac {-\frac {a^{2}}{2 c}+\frac {\left (2 a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) x^{4}}{2 c^{3}}}{x^{2} \left (d \,x^{2}+c \right )}+\frac {a \left (a d -b c \right ) \ln \left (d \,x^{2}+c \right )}{c^{3}}-\frac {2 a \left (a d -b c \right ) \ln \left (x \right )}{c^{3}}\)	\(91\)
risch	\(\frac {-\frac {\left (2 a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) x^{2}}{2 c^{2} d}-\frac {a^{2}}{2 c}}{x^{2} \left (d \,x^{2}+c \right )}-\frac {2 a^{2} \ln \left (x \right ) d}{c^{3}}+\frac {2 a \ln \left (x \right ) b}{c^{2}}+\frac {a^{2} \ln \left (-d \,x^{2}-c \right ) d}{c^{3}}-\frac {a \ln \left (-d \,x^{2}-c \right ) b}{c^{2}}\)	\(114\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.27, size = 100, normalized size = 1.23 \begin {gather*} -\frac {a^{2} c d + {\left (b^{2} c^{2} - 2 \, a b c d + 2 \, a^{2} d^{2}\right )} x^{2}}{2 \, {\left (c^{2} d^{2} x^{4} + c^{3} d x^{2}\right )}} - \frac {{\left (a b c - a^{2} d\right )} \log \left (d x^{2} + c\right )}{c^{3}} + \frac {{\left (a b c - a^{2} d\right )} \log \left (x^{2}\right )}{c^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 159 vs. \(2 (77) = 154\).
time = 1.25, size = 159, normalized size = 1.96 \begin {gather*} -\frac {a^{2} c^{2} d + {\left (b^{2} c^{3} - 2 \, a b c^{2} d + 2 \, a^{2} c d^{2}\right )} x^{2} + 2 \, {\left ({\left (a b c d^{2} - a^{2} d^{3}\right )} x^{4} + {\left (a b c^{2} d - a^{2} c d^{2}\right )} x^{2}\right )} \log \left (d x^{2} + c\right ) - 4 \, {\left ({\left (a b c d^{2} - a^{2} d^{3}\right )} x^{4} + {\left (a b c^{2} d - a^{2} c d^{2}\right )} x^{2}\right )} \log \left (x\right )}{2 \, {\left (c^{3} d^{2} x^{4} + c^{4} d x^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 0.82, size = 92, normalized size = 1.14 \begin {gather*} - \frac {2 a \left (a d - b c\right ) \log {\left (x \right )}}{c^{3}} + \frac {a \left (a d - b c\right ) \log {\left (\frac {c}{d} + x^{2} \right )}}{c^{3}} + \frac {- a^{2} c d + x^{2} \left (- 2 a^{2} d^{2} + 2 a b c d - b^{2} c^{2}\right )}{2 c^{3} d x^{2} + 2 c^{2} d^{2} x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 1.65, size = 109, normalized size = 1.35 \begin {gather*} \frac {{\left (a b c - a^{2} d\right )} \log \left (x^{2}\right )}{c^{3}} - \frac {{\left (a b c d - a^{2} d^{2}\right )} \log \left ({\left | d x^{2} + c \right |}\right )}{c^{3} d} - \frac {b^{2} c^{2} x^{2} - 2 \, a b c d x^{2} + 2 \, a^{2} d^{2} x^{2} + a^{2} c d}{2 \, {\left (d x^{4} + c x^{2}\right )} c^{2} d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.05, size = 100, normalized size = 1.23 \begin {gather*} \frac {\ln \left (d\,x^2+c\right )\,\left (a^2\,d-a\,b\,c\right )}{c^3}-\frac {\frac {a^2}{2\,c}+\frac {x^2\,\left (2\,a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}{2\,c^2\,d}}{d\,x^4+c\,x^2}-\frac {\ln \left (x\right )\,\left (2\,a^2\,d-2\,a\,b\,c\right )}{c^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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