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

Optimal. Leaf size=86 \[ \frac {\frac {a^2}{c^2}-\frac {b^2}{d^2}}{2 \left (c+d x^2\right )}-\frac {a^2 \log \left (c+d x^2\right )}{2 c^3}+\frac {a^2 \log (x)}{c^3}+\frac {(b c-a d)^2}{4 c d^2 \left (c+d x^2\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.08, antiderivative size = 86, 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} \[ \frac {\frac {a^2}{c^2}-\frac {b^2}{d^2}}{2 \left (c+d x^2\right )}-\frac {a^2 \log \left (c+d x^2\right )}{2 c^3}+\frac {a^2 \log (x)}{c^3}+\frac {(b c-a d)^2}{4 c d^2 \left (c+d x^2\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]  time = 0.05, size = 103, normalized size = 1.20 \[ \frac {a^2 d^2-b^2 c^2}{2 c^2 d^2 \left (c+d x^2\right )}+\frac {a^2 d^2-2 a b c d+b^2 c^2}{4 c d^2 \left (c+d x^2\right )^2}-\frac {a^2 \log \left (c+d x^2\right )}{2 c^3}+\frac {a^2 \log (x)}{c^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.45, size = 163, normalized size = 1.90 \[ -\frac {b^{2} c^{4} + 2 \, a b c^{3} d - 3 \, a^{2} c^{2} d^{2} + 2 \, {\left (b^{2} c^{3} d - a^{2} c d^{3}\right )} x^{2} + 2 \, {\left (a^{2} d^{4} x^{4} + 2 \, a^{2} c d^{3} x^{2} + a^{2} c^{2} d^{2}\right )} \log \left (d x^{2} + c\right ) - 4 \, {\left (a^{2} d^{4} x^{4} + 2 \, a^{2} c d^{3} x^{2} + a^{2} c^{2} d^{2}\right )} \log \relax (x)}{4 \, {\left (c^{3} d^{4} x^{4} + 2 \, c^{4} d^{3} x^{2} + c^{5} d^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(b^2*c^4 + 2*a*b*c^3*d - 3*a^2*c^2*d^2 + 2*(b^2*c^3*d - a^2*c*d^3)*x^2 + 2*(a^2*d^4*x^4 + 2*a^2*c*d^3*x^2
+ a^2*c^2*d^2)*log(d*x^2 + c) - 4*(a^2*d^4*x^4 + 2*a^2*c*d^3*x^2 + a^2*c^2*d^2)*log(x))/(c^3*d^4*x^4 + 2*c^4*
d^3*x^2 + c^5*d^2)

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giac [A]  time = 0.41, size = 110, normalized size = 1.28 \[ \frac {a^{2} \log \left (x^{2}\right )}{2 \, c^{3}} - \frac {a^{2} \log \left ({\left | d x^{2} + c \right |}\right )}{2 \, c^{3}} + \frac {3 \, a^{2} d^{4} x^{4} - 2 \, b^{2} c^{3} d x^{2} + 8 \, a^{2} c d^{3} x^{2} - b^{2} c^{4} - 2 \, a b c^{3} d + 6 \, a^{2} c^{2} d^{2}}{4 \, {\left (d x^{2} + c\right )}^{2} c^{3} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*a^2*log(x^2)/c^3 - 1/2*a^2*log(abs(d*x^2 + c))/c^3 + 1/4*(3*a^2*d^4*x^4 - 2*b^2*c^3*d*x^2 + 8*a^2*c*d^3*x^
2 - b^2*c^4 - 2*a*b*c^3*d + 6*a^2*c^2*d^2)/((d*x^2 + c)^2*c^3*d^2)

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maple [A]  time = 0.02, size = 112, normalized size = 1.30 \[ \frac {a^{2}}{4 \left (d \,x^{2}+c \right )^{2} c}-\frac {a b}{2 \left (d \,x^{2}+c \right )^{2} d}+\frac {b^{2} c}{4 \left (d \,x^{2}+c \right )^{2} d^{2}}+\frac {a^{2}}{2 \left (d \,x^{2}+c \right ) c^{2}}+\frac {a^{2} \ln \relax (x )}{c^{3}}-\frac {a^{2} \ln \left (d \,x^{2}+c \right )}{2 c^{3}}-\frac {b^{2}}{2 \left (d \,x^{2}+c \right ) d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.01, size = 109, normalized size = 1.27 \[ -\frac {b^{2} c^{3} + 2 \, a b c^{2} d - 3 \, a^{2} c d^{2} + 2 \, {\left (b^{2} c^{2} d - a^{2} d^{3}\right )} x^{2}}{4 \, {\left (c^{2} d^{4} x^{4} + 2 \, c^{3} d^{3} x^{2} + c^{4} d^{2}\right )}} - \frac {a^{2} \log \left (d x^{2} + c\right )}{2 \, c^{3}} + \frac {a^{2} \log \left (x^{2}\right )}{2 \, c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.13, size = 106, normalized size = 1.23 \[ \frac {a^2\,\ln \relax (x)}{c^3}-\frac {a^2\,\ln \left (d\,x^2+c\right )}{2\,c^3}-\frac {\frac {-3\,a^2\,d^2+2\,a\,b\,c\,d+b^2\,c^2}{4\,c\,d^2}-\frac {x^2\,\left (a^2\,d^2-b^2\,c^2\right )}{2\,c^2\,d}}{c^2+2\,c\,d\,x^2+d^2\,x^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 1.18, size = 107, normalized size = 1.24 \[ \frac {a^{2} \log {\relax (x )}}{c^{3}} - \frac {a^{2} \log {\left (\frac {c}{d} + x^{2} \right )}}{2 c^{3}} + \frac {3 a^{2} c d^{2} - 2 a b c^{2} d - b^{2} c^{3} + x^{2} \left (2 a^{2} d^{3} - 2 b^{2} c^{2} d\right )}{4 c^{4} d^{2} + 8 c^{3} d^{3} x^{2} + 4 c^{2} d^{4} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**2*log(x)/c**3 - a**2*log(c/d + x**2)/(2*c**3) + (3*a**2*c*d**2 - 2*a*b*c**2*d - b**2*c**3 + x**2*(2*a**2*d*
*3 - 2*b**2*c**2*d))/(4*c**4*d**2 + 8*c**3*d**3*x**2 + 4*c**2*d**4*x**4)

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