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

Optimal. Leaf size=100 \[ -\frac{a}{2 \left (c+d x^2\right ) (b c-a d)^2}-\frac{c}{4 d \left (c+d x^2\right )^2 (b c-a d)}-\frac{a b \log \left (a+b x^2\right )}{2 (b c-a d)^3}+\frac{a b \log \left (c+d x^2\right )}{2 (b c-a d)^3} \]

[Out]

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

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Rubi [A]  time = 0.0924123, antiderivative size = 100, 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, 77} \[ -\frac{a}{2 \left (c+d x^2\right ) (b c-a d)^2}-\frac{c}{4 d \left (c+d x^2\right )^2 (b c-a d)}-\frac{a b \log \left (a+b x^2\right )}{2 (b c-a d)^3}+\frac{a b \log \left (c+d x^2\right )}{2 (b c-a d)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-c/(4*d*(b*c - a*d)*(c + d*x^2)^2) - a/(2*(b*c - a*d)^2*(c + d*x^2)) - (a*b*Log[a + b*x^2])/(2*(b*c - a*d)^3)
+ (a*b*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 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

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

Mathematica [A]  time = 0.114326, size = 77, normalized size = 0.77 \[ \frac{\frac{(a d-b c) \left (a d \left (c+2 d x^2\right )+b c^2\right )}{d \left (c+d x^2\right )^2}+2 a b \log \left (c+d x^2\right )-2 a b \log \left (a+b x^2\right )}{4 (b c-a d)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.01, size = 177, normalized size = 1.8 \begin{align*} -{\frac{ab\ln \left ( d{x}^{2}+c \right ) }{2\, \left ( ad-bc \right ) ^{3}}}+{\frac{{a}^{2}cd}{4\, \left ( ad-bc \right ) ^{3} \left ( d{x}^{2}+c \right ) ^{2}}}-{\frac{ab{c}^{2}}{2\, \left ( ad-bc \right ) ^{3} \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{{b}^{2}{c}^{3}}{4\, \left ( ad-bc \right ) ^{3}d \left ( d{x}^{2}+c \right ) ^{2}}}-{\frac{{a}^{2}d}{2\, \left ( ad-bc \right ) ^{3} \left ( d{x}^{2}+c \right ) }}+{\frac{abc}{2\, \left ( ad-bc \right ) ^{3} \left ( d{x}^{2}+c \right ) }}+{\frac{ab\ln \left ( b{x}^{2}+a \right ) }{2\, \left ( ad-bc \right ) ^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.09098, size = 293, normalized size = 2.93 \begin{align*} -\frac{a b \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{a b \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{2 \, a d^{2} x^{2} + b c^{2} + a c d}{4 \,{\left (b^{2} c^{4} d - 2 \, a b c^{3} d^{2} + a^{2} c^{2} d^{3} +{\left (b^{2} c^{2} d^{3} - 2 \, a b c d^{4} + a^{2} d^{5}\right )} x^{4} + 2 \,{\left (b^{2} c^{3} d^{2} - 2 \, a b c^{2} d^{3} + a^{2} c d^{4}\right )} x^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.65148, size = 513, normalized size = 5.13 \begin{align*} -\frac{b^{2} c^{3} - a^{2} c d^{2} + 2 \,{\left (a b c d^{2} - a^{2} d^{3}\right )} x^{2} + 2 \,{\left (a b d^{3} x^{4} + 2 \, a b c d^{2} x^{2} + a b c^{2} d\right )} \log \left (b x^{2} + a\right ) - 2 \,{\left (a b d^{3} x^{4} + 2 \, a b c d^{2} x^{2} + a b c^{2} d\right )} \log \left (d x^{2} + c\right )}{4 \,{\left (b^{3} c^{5} d - 3 \, a b^{2} c^{4} d^{2} + 3 \, a^{2} b c^{3} d^{3} - a^{3} c^{2} d^{4} +{\left (b^{3} c^{3} d^{3} - 3 \, a b^{2} c^{2} d^{4} + 3 \, a^{2} b c d^{5} - a^{3} d^{6}\right )} x^{4} + 2 \,{\left (b^{3} c^{4} d^{2} - 3 \, a b^{2} c^{3} d^{3} + 3 \, a^{2} b c^{2} d^{4} - a^{3} c d^{5}\right )} x^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 4.31335, size = 410, normalized size = 4.1 \begin{align*} - \frac{a b \log{\left (x^{2} + \frac{- \frac{a^{5} b d^{4}}{\left (a d - b c\right )^{3}} + \frac{4 a^{4} b^{2} c d^{3}}{\left (a d - b c\right )^{3}} - \frac{6 a^{3} b^{3} c^{2} d^{2}}{\left (a d - b c\right )^{3}} + \frac{4 a^{2} b^{4} c^{3} d}{\left (a d - b c\right )^{3}} + a^{2} b d - \frac{a b^{5} c^{4}}{\left (a d - b c\right )^{3}} + a b^{2} c}{2 a b^{2} d} \right )}}{2 \left (a d - b c\right )^{3}} + \frac{a b \log{\left (x^{2} + \frac{\frac{a^{5} b d^{4}}{\left (a d - b c\right )^{3}} - \frac{4 a^{4} b^{2} c d^{3}}{\left (a d - b c\right )^{3}} + \frac{6 a^{3} b^{3} c^{2} d^{2}}{\left (a d - b c\right )^{3}} - \frac{4 a^{2} b^{4} c^{3} d}{\left (a d - b c\right )^{3}} + a^{2} b d + \frac{a b^{5} c^{4}}{\left (a d - b c\right )^{3}} + a b^{2} c}{2 a b^{2} d} \right )}}{2 \left (a d - b c\right )^{3}} - \frac{a c d + 2 a d^{2} x^{2} + b c^{2}}{4 a^{2} c^{2} d^{3} - 8 a b c^{3} d^{2} + 4 b^{2} c^{4} d + x^{4} \left (4 a^{2} d^{5} - 8 a b c d^{4} + 4 b^{2} c^{2} d^{3}\right ) + x^{2} \left (8 a^{2} c d^{4} - 16 a b c^{2} d^{3} + 8 b^{2} c^{3} d^{2}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.17873, size = 235, normalized size = 2.35 \begin{align*} -\frac{a b^{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{a b 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{b^{2} c^{3} - a^{2} c d^{2} + 2 \,{\left (a b c d^{2} - a^{2} d^{3}\right )} x^{2}}{4 \,{\left (d x^{2} + c\right )}^{2}{\left (b c - a d\right )}^{3} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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