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

Optimal. Leaf size=70 \[ \frac{a^2 \log \left (a+b x^2\right )}{2 b^2 (b c-a d)}-\frac{c^2 \log \left (c+d x^2\right )}{2 d^2 (b c-a d)}+\frac{x^2}{2 b d} \]

[Out]

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

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Rubi [A]  time = 0.0668481, antiderivative size = 70, 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, 72} \[ \frac{a^2 \log \left (a+b x^2\right )}{2 b^2 (b c-a d)}-\frac{c^2 \log \left (c+d x^2\right )}{2 d^2 (b c-a d)}+\frac{x^2}{2 b d} \]

Antiderivative was successfully verified.

[In]

Int[x^5/((a + b*x^2)*(c + d*x^2)),x]

[Out]

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

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 72

Int[((e_.) + (f_.)*(x_))^(p_.)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Int[ExpandIntegrand[(
e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[p]

Rubi steps

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

Mathematica [A]  time = 0.0302386, size = 66, normalized size = 0.94 \[ \frac{a^2 d^2 \log \left (a+b x^2\right )-b \left (d x^2 (a d-b c)+b c^2 \log \left (c+d x^2\right )\right )}{2 b^2 d^2 (b c-a d)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.007, size = 65, normalized size = 0.9 \begin{align*}{\frac{{x}^{2}}{2\,bd}}+{\frac{{c}^{2}\ln \left ( d{x}^{2}+c \right ) }{ \left ( 2\,ad-2\,bc \right ){d}^{2}}}-{\frac{{a}^{2}\ln \left ( b{x}^{2}+a \right ) }{ \left ( 2\,ad-2\,bc \right ){b}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.985118, size = 92, normalized size = 1.31 \begin{align*} \frac{a^{2} \log \left (b x^{2} + a\right )}{2 \,{\left (b^{3} c - a b^{2} d\right )}} - \frac{c^{2} \log \left (d x^{2} + c\right )}{2 \,{\left (b c d^{2} - a d^{3}\right )}} + \frac{x^{2}}{2 \, b d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.57231, size = 142, normalized size = 2.03 \begin{align*} \frac{a^{2} d^{2} \log \left (b x^{2} + a\right ) - b^{2} c^{2} \log \left (d x^{2} + c\right ) +{\left (b^{2} c d - a b d^{2}\right )} x^{2}}{2 \,{\left (b^{3} c d^{2} - a b^{2} d^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 3.12081, size = 201, normalized size = 2.87 \begin{align*} - \frac{a^{2} \log{\left (x^{2} + \frac{\frac{a^{4} d^{3}}{b \left (a d - b c\right )} - \frac{2 a^{3} c d^{2}}{a d - b c} + \frac{a^{2} b c^{2} d}{a d - b c} + a^{2} c d + a b c^{2}}{a^{2} d^{2} + b^{2} c^{2}} \right )}}{2 b^{2} \left (a d - b c\right )} + \frac{c^{2} \log{\left (x^{2} + \frac{- \frac{a^{2} b c^{2} d}{a d - b c} + a^{2} c d + \frac{2 a b^{2} c^{3}}{a d - b c} + a b c^{2} - \frac{b^{3} c^{4}}{d \left (a d - b c\right )}}{a^{2} d^{2} + b^{2} c^{2}} \right )}}{2 d^{2} \left (a d - b c\right )} + \frac{x^{2}}{2 b d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError