3.251 \(\int \frac{x^5}{(a+b x^2)^3 (c+d x^2)} \, dx\)

Optimal. Leaf size=116 \[ -\frac{a^2}{4 b^2 \left (a+b x^2\right )^2 (b c-a d)}+\frac{a (2 b c-a d)}{2 b^2 \left (a+b x^2\right ) (b c-a d)^2}+\frac{c^2 \log \left (a+b x^2\right )}{2 (b c-a d)^3}-\frac{c^2 \log \left (c+d x^2\right )}{2 (b c-a d)^3} \]

[Out]

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

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Rubi [A]  time = 0.113149, antiderivative size = 116, 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} \[ -\frac{a^2}{4 b^2 \left (a+b x^2\right )^2 (b c-a d)}+\frac{a (2 b c-a d)}{2 b^2 \left (a+b x^2\right ) (b c-a d)^2}+\frac{c^2 \log \left (a+b x^2\right )}{2 (b c-a d)^3}-\frac{c^2 \log \left (c+d x^2\right )}{2 (b c-a d)^3} \]

Antiderivative was successfully verified.

[In]

Int[x^5/((a + b*x^2)^3*(c + d*x^2)),x]

[Out]

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

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

Mathematica [A]  time = 0.0999876, size = 99, normalized size = 0.85 \[ \frac{-\frac{a^2 (b c-a d)^2}{b^2 \left (a+b x^2\right )^2}+\frac{2 a (a d-2 b c) (a d-b c)}{b^2 \left (a+b x^2\right )}+2 c^2 \log \left (a+b x^2\right )-2 c^2 \log \left (c+d x^2\right )}{4 (b c-a d)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5/((a + b*x^2)^3*(c + d*x^2)),x]

[Out]

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

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Maple [B]  time = 0.011, size = 218, normalized size = 1.9 \begin{align*}{\frac{{c}^{2}\ln \left ( d{x}^{2}+c \right ) }{2\, \left ( ad-bc \right ) ^{3}}}+{\frac{{a}^{4}{d}^{2}}{4\, \left ( ad-bc \right ) ^{3}{b}^{2} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{{a}^{3}cd}{2\, \left ( ad-bc \right ) ^{3}b \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{{a}^{2}{c}^{2}}{4\, \left ( ad-bc \right ) ^{3} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{{c}^{2}\ln \left ( b{x}^{2}+a \right ) }{2\, \left ( ad-bc \right ) ^{3}}}-{\frac{{a}^{3}{d}^{2}}{2\, \left ( ad-bc \right ) ^{3}{b}^{2} \left ( b{x}^{2}+a \right ) }}+{\frac{3\,{a}^{2}cd}{2\, \left ( ad-bc \right ) ^{3}b \left ( b{x}^{2}+a \right ) }}-{\frac{a{c}^{2}}{ \left ( ad-bc \right ) ^{3} \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)^3/(d*x^2+c),x)

[Out]

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

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Maxima [B]  time = 1.244, size = 319, normalized size = 2.75 \begin{align*} \frac{c^{2} \log \left (b x^{2} + a\right )}{2 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )}} - \frac{c^{2} \log \left (d x^{2} + c\right )}{2 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )}} + \frac{3 \, a^{2} b c - a^{3} d + 2 \,{\left (2 \, a b^{2} c - a^{2} b d\right )} x^{2}}{4 \,{\left (a^{2} b^{4} c^{2} - 2 \, a^{3} b^{3} c d + a^{4} b^{2} d^{2} +{\left (b^{6} c^{2} - 2 \, a b^{5} c d + a^{2} b^{4} d^{2}\right )} x^{4} + 2 \,{\left (a b^{5} c^{2} - 2 \, a^{2} b^{4} c d + a^{3} b^{3} d^{2}\right )} x^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(b*x^2+a)^3/(d*x^2+c),x, algorithm="maxima")

[Out]

1/2*c^2*log(b*x^2 + a)/(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3) - 1/2*c^2*log(d*x^2 + c)/(b^3*c^3 -
3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3) + 1/4*(3*a^2*b*c - a^3*d + 2*(2*a*b^2*c - a^2*b*d)*x^2)/(a^2*b^4*c^2
- 2*a^3*b^3*c*d + a^4*b^2*d^2 + (b^6*c^2 - 2*a*b^5*c*d + a^2*b^4*d^2)*x^4 + 2*(a*b^5*c^2 - 2*a^2*b^4*c*d + a^
3*b^3*d^2)*x^2)

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Fricas [B]  time = 1.60294, size = 571, normalized size = 4.92 \begin{align*} \frac{3 \, a^{2} b^{2} c^{2} - 4 \, a^{3} b c d + a^{4} d^{2} + 2 \,{\left (2 \, a b^{3} c^{2} - 3 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x^{2} + 2 \,{\left (b^{4} c^{2} x^{4} + 2 \, a b^{3} c^{2} x^{2} + a^{2} b^{2} c^{2}\right )} \log \left (b x^{2} + a\right ) - 2 \,{\left (b^{4} c^{2} x^{4} + 2 \, a b^{3} c^{2} x^{2} + a^{2} b^{2} c^{2}\right )} \log \left (d x^{2} + c\right )}{4 \,{\left (a^{2} b^{5} c^{3} - 3 \, a^{3} b^{4} c^{2} d + 3 \, a^{4} b^{3} c d^{2} - a^{5} b^{2} d^{3} +{\left (b^{7} c^{3} - 3 \, a b^{6} c^{2} d + 3 \, a^{2} b^{5} c d^{2} - a^{3} b^{4} d^{3}\right )} x^{4} + 2 \,{\left (a b^{6} c^{3} - 3 \, a^{2} b^{5} c^{2} d + 3 \, a^{3} b^{4} c d^{2} - a^{4} 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)^3/(d*x^2+c),x, algorithm="fricas")

[Out]

1/4*(3*a^2*b^2*c^2 - 4*a^3*b*c*d + a^4*d^2 + 2*(2*a*b^3*c^2 - 3*a^2*b^2*c*d + a^3*b*d^2)*x^2 + 2*(b^4*c^2*x^4
+ 2*a*b^3*c^2*x^2 + a^2*b^2*c^2)*log(b*x^2 + a) - 2*(b^4*c^2*x^4 + 2*a*b^3*c^2*x^2 + a^2*b^2*c^2)*log(d*x^2 +
c))/(a^2*b^5*c^3 - 3*a^3*b^4*c^2*d + 3*a^4*b^3*c*d^2 - a^5*b^2*d^3 + (b^7*c^3 - 3*a*b^6*c^2*d + 3*a^2*b^5*c*d^
2 - a^3*b^4*d^3)*x^4 + 2*(a*b^6*c^3 - 3*a^2*b^5*c^2*d + 3*a^3*b^4*c*d^2 - a^4*b^3*d^3)*x^2)

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Sympy [B]  time = 4.60615, size = 418, normalized size = 3.6 \begin{align*} \frac{c^{2} \log{\left (x^{2} + \frac{- \frac{a^{4} c^{2} d^{4}}{\left (a d - b c\right )^{3}} + \frac{4 a^{3} b c^{3} d^{3}}{\left (a d - b c\right )^{3}} - \frac{6 a^{2} b^{2} c^{4} d^{2}}{\left (a d - b c\right )^{3}} + \frac{4 a b^{3} c^{5} d}{\left (a d - b c\right )^{3}} + a c^{2} d - \frac{b^{4} c^{6}}{\left (a d - b c\right )^{3}} + b c^{3}}{2 b c^{2} d} \right )}}{2 \left (a d - b c\right )^{3}} - \frac{c^{2} \log{\left (x^{2} + \frac{\frac{a^{4} c^{2} d^{4}}{\left (a d - b c\right )^{3}} - \frac{4 a^{3} b c^{3} d^{3}}{\left (a d - b c\right )^{3}} + \frac{6 a^{2} b^{2} c^{4} d^{2}}{\left (a d - b c\right )^{3}} - \frac{4 a b^{3} c^{5} d}{\left (a d - b c\right )^{3}} + a c^{2} d + \frac{b^{4} c^{6}}{\left (a d - b c\right )^{3}} + b c^{3}}{2 b c^{2} d} \right )}}{2 \left (a d - b c\right )^{3}} - \frac{a^{3} d - 3 a^{2} b c + x^{2} \left (2 a^{2} b d - 4 a b^{2} c\right )}{4 a^{4} b^{2} d^{2} - 8 a^{3} b^{3} c d + 4 a^{2} b^{4} c^{2} + x^{4} \left (4 a^{2} b^{4} d^{2} - 8 a b^{5} c d + 4 b^{6} c^{2}\right ) + x^{2} \left (8 a^{3} b^{3} d^{2} - 16 a^{2} b^{4} c d + 8 a b^{5} c^{2}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5/(b*x**2+a)**3/(d*x**2+c),x)

[Out]

c**2*log(x**2 + (-a**4*c**2*d**4/(a*d - b*c)**3 + 4*a**3*b*c**3*d**3/(a*d - b*c)**3 - 6*a**2*b**2*c**4*d**2/(a
*d - b*c)**3 + 4*a*b**3*c**5*d/(a*d - b*c)**3 + a*c**2*d - b**4*c**6/(a*d - b*c)**3 + b*c**3)/(2*b*c**2*d))/(2
*(a*d - b*c)**3) - c**2*log(x**2 + (a**4*c**2*d**4/(a*d - b*c)**3 - 4*a**3*b*c**3*d**3/(a*d - b*c)**3 + 6*a**2
*b**2*c**4*d**2/(a*d - b*c)**3 - 4*a*b**3*c**5*d/(a*d - b*c)**3 + a*c**2*d + b**4*c**6/(a*d - b*c)**3 + b*c**3
)/(2*b*c**2*d))/(2*(a*d - b*c)**3) - (a**3*d - 3*a**2*b*c + x**2*(2*a**2*b*d - 4*a*b**2*c))/(4*a**4*b**2*d**2
- 8*a**3*b**3*c*d + 4*a**2*b**4*c**2 + x**4*(4*a**2*b**4*d**2 - 8*a*b**5*c*d + 4*b**6*c**2) + x**2*(8*a**3*b**
3*d**2 - 16*a**2*b**4*c*d + 8*a*b**5*c**2))

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Giac [B]  time = 1.19138, size = 313, normalized size = 2.7 \begin{align*} \frac{b c^{2} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \,{\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )}} - \frac{c^{2} d \log \left ({\left | d x^{2} + c \right |}\right )}{2 \,{\left (b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3} - a^{3} d^{4}\right )}} - \frac{3 \, b^{4} c^{2} x^{4} + 2 \, a b^{3} c^{2} x^{2} + 6 \, a^{2} b^{2} c d x^{2} - 2 \, a^{3} b d^{2} x^{2} + 4 \, a^{3} b c d - a^{4} d^{2}}{4 \,{\left (b^{5} c^{3} - 3 \, a b^{4} c^{2} d + 3 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )}{\left (b x^{2} + a\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(b*x^2+a)^3/(d*x^2+c),x, algorithm="giac")

[Out]

1/2*b*c^2*log(abs(b*x^2 + a))/(b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2*c*d^2 - a^3*b*d^3) - 1/2*c^2*d*log(abs(d*x^
2 + c))/(b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 - a^3*d^4) - 1/4*(3*b^4*c^2*x^4 + 2*a*b^3*c^2*x^2 + 6*a^2
*b^2*c*d*x^2 - 2*a^3*b*d^2*x^2 + 4*a^3*b*c*d - a^4*d^2)/((b^5*c^3 - 3*a*b^4*c^2*d + 3*a^2*b^3*c*d^2 - a^3*b^2*
d^3)*(b*x^2 + a)^2)