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

Optimal. Leaf size=74 \[ -\frac {c}{2 d \left (c+d x^2\right ) (b c-a d)}-\frac {a \log \left (a+b x^2\right )}{2 (b c-a d)^2}+\frac {a \log \left (c+d x^2\right )}{2 (b c-a d)^2} \]

[Out]

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

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Rubi [A]  time = 0.06, antiderivative size = 74, 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, 77} \[ -\frac {c}{2 d \left (c+d x^2\right ) (b c-a d)}-\frac {a \log \left (a+b x^2\right )}{2 (b c-a d)^2}+\frac {a \log \left (c+d x^2\right )}{2 (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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]))))

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

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Mathematica [A]  time = 0.04, size = 74, normalized size = 1.00 \[ \frac {c}{2 d \left (c+d x^2\right ) (a d-b c)}-\frac {a \log \left (a+b x^2\right )}{2 (b c-a d)^2}+\frac {a \log \left (c+d x^2\right )}{2 (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.48, size = 117, normalized size = 1.58 \[ -\frac {b c^{2} - a c d + {\left (a d^{2} x^{2} + a c d\right )} \log \left (b x^{2} + a\right ) - {\left (a d^{2} x^{2} + a c d\right )} \log \left (d x^{2} + c\right )}{2 \, {\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3} + {\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.37, size = 91, normalized size = 1.23 \[ -\frac {\frac {a d^{2} \log \left ({\left | b - \frac {b c}{d x^{2} + c} + \frac {a d}{d x^{2} + c} \right |}\right )}{b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}} + \frac {c d}{{\left (b c d - a d^{2}\right )} {\left (d x^{2} + c\right )}}}{2 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.01, size = 95, normalized size = 1.28 \[ \frac {a c}{2 \left (a d -b c \right )^{2} \left (d \,x^{2}+c \right )}-\frac {a \ln \left (b \,x^{2}+a \right )}{2 \left (a d -b c \right )^{2}}+\frac {a \ln \left (d \,x^{2}+c \right )}{2 \left (a d -b c \right )^{2}}-\frac {b \,c^{2}}{2 \left (a d -b c \right )^{2} \left (d \,x^{2}+c \right ) d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.11, size = 105, normalized size = 1.42 \[ -\frac {a \log \left (b x^{2} + a\right )}{2 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )}} + \frac {a \log \left (d x^{2} + c\right )}{2 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )}} - \frac {c}{2 \, {\left (b c^{2} d - a c d^{2} + {\left (b c d^{2} - a d^{3}\right )} x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.27, size = 173, normalized size = 2.34 \[ -\frac {b\,c^2-c\,\left (a\,d-a\,d\,\mathrm {atan}\left (\frac {a\,d\,x^2\,1{}\mathrm {i}-b\,c\,x^2\,1{}\mathrm {i}}{2\,a\,c+a\,d\,x^2+b\,c\,x^2}\right )\,2{}\mathrm {i}\right )+a\,d^2\,x^2\,\mathrm {atan}\left (\frac {a\,d\,x^2\,1{}\mathrm {i}-b\,c\,x^2\,1{}\mathrm {i}}{2\,a\,c+a\,d\,x^2+b\,c\,x^2}\right )\,2{}\mathrm {i}}{2\,a^2\,c\,d^3+2\,a^2\,d^4\,x^2-4\,a\,b\,c^2\,d^2-4\,a\,b\,c\,d^3\,x^2+2\,b^2\,c^3\,d+2\,b^2\,c^2\,d^2\,x^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [B]  time = 1.92, size = 253, normalized size = 3.42 \[ \frac {a \log {\left (x^{2} + \frac {- \frac {a^{4} d^{3}}{\left (a d - b c\right )^{2}} + \frac {3 a^{3} b c d^{2}}{\left (a d - b c\right )^{2}} - \frac {3 a^{2} b^{2} c^{2} d}{\left (a d - b c\right )^{2}} + a^{2} d + \frac {a b^{3} c^{3}}{\left (a d - b c\right )^{2}} + a b c}{2 a b d} \right )}}{2 \left (a d - b c\right )^{2}} - \frac {a \log {\left (x^{2} + \frac {\frac {a^{4} d^{3}}{\left (a d - b c\right )^{2}} - \frac {3 a^{3} b c d^{2}}{\left (a d - b c\right )^{2}} + \frac {3 a^{2} b^{2} c^{2} d}{\left (a d - b c\right )^{2}} + a^{2} d - \frac {a b^{3} c^{3}}{\left (a d - b c\right )^{2}} + a b c}{2 a b d} \right )}}{2 \left (a d - b c\right )^{2}} + \frac {c}{2 a c d^{2} - 2 b c^{2} d + x^{2} \left (2 a d^{3} - 2 b c d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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