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

Optimal. Leaf size=93 \[ -\frac{a^2}{2 b^2 \left (a+b x^2\right ) (b c-a d)}-\frac{a (2 b c-a d) \log \left (a+b x^2\right )}{2 b^2 (b c-a d)^2}+\frac{c^2 \log \left (c+d x^2\right )}{2 d (b c-a d)^2} \]

[Out]

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

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Rubi [A]  time = 0.0865865, antiderivative size = 93, normalized size of antiderivative = 1., 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} \[ -\frac{a^2}{2 b^2 \left (a+b x^2\right ) (b c-a d)}-\frac{a (2 b c-a d) \log \left (a+b x^2\right )}{2 b^2 (b c-a d)^2}+\frac{c^2 \log \left (c+d x^2\right )}{2 d (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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]]

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]))

Rubi steps

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

Mathematica [A]  time = 0.0460986, size = 91, normalized size = 0.98 \[ \frac{a^2 d (a d-b c)+b^2 c^2 \left (a+b x^2\right ) \log \left (c+d x^2\right )+a d \left (a+b x^2\right ) (a d-2 b c) \log \left (a+b x^2\right )}{2 b^2 d \left (a+b x^2\right ) (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.012, size = 136, normalized size = 1.5 \begin{align*}{\frac{{c}^{2}\ln \left ( d{x}^{2}+c \right ) }{2\, \left ( ad-bc \right ) ^{2}d}}+{\frac{{a}^{2}\ln \left ( b{x}^{2}+a \right ) d}{2\, \left ( ad-bc \right ) ^{2}{b}^{2}}}-{\frac{a\ln \left ( b{x}^{2}+a \right ) c}{ \left ( ad-bc \right ) ^{2}b}}+{\frac{{a}^{3}d}{2\, \left ( ad-bc \right ) ^{2}{b}^{2} \left ( b{x}^{2}+a \right ) }}-{\frac{{a}^{2}c}{2\, \left ( ad-bc \right ) ^{2}b \left ( b{x}^{2}+a \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.08163, size = 176, normalized size = 1.89 \begin{align*} \frac{c^{2} \log \left (d x^{2} + c\right )}{2 \,{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )}} - \frac{a^{2}}{2 \,{\left (a b^{3} c - a^{2} b^{2} d +{\left (b^{4} c - a b^{3} d\right )} x^{2}\right )}} - \frac{{\left (2 \, a b c - a^{2} d\right )} \log \left (b x^{2} + a\right )}{2 \,{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.7337, size = 321, normalized size = 3.45 \begin{align*} -\frac{a^{2} b c d - a^{3} d^{2} +{\left (2 \, a^{2} b c d - a^{3} d^{2} +{\left (2 \, a b^{2} c d - a^{2} b d^{2}\right )} x^{2}\right )} \log \left (b x^{2} + a\right ) -{\left (b^{3} c^{2} x^{2} + a b^{2} c^{2}\right )} \log \left (d x^{2} + c\right )}{2 \,{\left (a b^{4} c^{2} d - 2 \, a^{2} b^{3} c d^{2} + a^{3} b^{2} d^{3} +{\left (b^{5} c^{2} d - 2 \, a b^{4} c d^{2} + a^{2} b^{3} d^{3}\right )} x^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 5.72102, size = 348, normalized size = 3.74 \begin{align*} \frac{a^{2}}{2 a^{2} b^{2} d - 2 a b^{3} c + x^{2} \left (2 a b^{3} d - 2 b^{4} c\right )} + \frac{a \left (a d - 2 b c\right ) \log{\left (x^{2} + \frac{\frac{a^{4} d^{3} \left (a d - 2 b c\right )}{b \left (a d - b c\right )^{2}} - \frac{3 a^{3} c d^{2} \left (a d - 2 b c\right )}{\left (a d - b c\right )^{2}} + \frac{3 a^{2} b c^{2} d \left (a d - 2 b c\right )}{\left (a d - b c\right )^{2}} + a^{2} c d - \frac{a b^{2} c^{3} \left (a d - 2 b c\right )}{\left (a d - b c\right )^{2}} - 3 a b c^{2}}{a^{2} d^{2} - 2 a b c d - b^{2} c^{2}} \right )}}{2 b^{2} \left (a d - b c\right )^{2}} + \frac{c^{2} \log{\left (x^{2} + \frac{\frac{a^{3} b c^{2} d^{2}}{\left (a d - b c\right )^{2}} - \frac{3 a^{2} b^{2} c^{3} d}{\left (a d - b c\right )^{2}} + a^{2} c d + \frac{3 a b^{3} c^{4}}{\left (a d - b c\right )^{2}} - 3 a b c^{2} - \frac{b^{4} c^{5}}{d \left (a d - b c\right )^{2}}}{a^{2} d^{2} - 2 a b c d - b^{2} c^{2}} \right )}}{2 d \left (a d - b c\right )^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.16598, size = 205, normalized size = 2.2 \begin{align*} \frac{c^{2} \log \left ({\left | d x^{2} + c \right |}\right )}{2 \,{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )}} - \frac{{\left (2 \, a b c - a^{2} d\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \,{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )}} + \frac{2 \, a b c x^{2} - a^{2} d x^{2} + a^{2} c}{2 \,{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )}{\left (b x^{2} + a\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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