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

Optimal. Leaf size=53 \[ \frac {c \log \left (c+d x^2\right )}{2 d (b c-a d)}-\frac {a \log \left (a+b x^2\right )}{2 b (b c-a d)} \]

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Rubi [A]  time = 0.05, antiderivative size = 53, 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, 72} \begin {gather*} \frac {c \log \left (c+d x^2\right )}{2 d (b c-a d)}-\frac {a \log \left (a+b x^2\right )}{2 b (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3/((a + b*x^2)*(c + d*x^2)),x]

[Out]

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

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]

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

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Mathematica [A]  time = 0.03, size = 43, normalized size = 0.81 \begin {gather*} -\frac {a d \log \left (a+b x^2\right )-b c \log \left (c+d x^2\right )}{2 b^2 c d-2 a b d^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^3}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.78, size = 42, normalized size = 0.79 \begin {gather*} -\frac {a d \log \left (b x^{2} + a\right ) - b c \log \left (d x^{2} + c\right )}{2 \, {\left (b^{2} c d - a b d^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.30, size = 51, normalized size = 0.96 \begin {gather*} -\frac {a \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, {\left (b^{2} c - a b d\right )}} + \frac {c \log \left ({\left | d x^{2} + c \right |}\right )}{2 \, {\left (b c d - a d^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.01, size = 50, normalized size = 0.94 \begin {gather*} \frac {a \ln \left (b \,x^{2}+a \right )}{2 \left (a d -b c \right ) b}-\frac {c \ln \left (d \,x^{2}+c \right )}{2 \left (a d -b c \right ) d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.03, size = 49, normalized size = 0.92 \begin {gather*} -\frac {a \log \left (b x^{2} + a\right )}{2 \, {\left (b^{2} c - a b d\right )}} + \frac {c \log \left (d x^{2} + c\right )}{2 \, {\left (b c d - a d^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.27, size = 51, normalized size = 0.96 \begin {gather*} -\frac {a\,\ln \left (b\,x^2+a\right )}{2\,b^2\,c-2\,a\,b\,d}-\frac {c\,\ln \left (d\,x^2+c\right )}{2\,a\,d^2-2\,b\,c\,d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/((a + b*x^2)*(c + d*x^2)),x)

[Out]

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

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sympy [B]  time = 2.34, size = 144, normalized size = 2.72 \begin {gather*} \frac {a \log {\left (x^{2} + \frac {\frac {a^{3} d^{2}}{b \left (a d - b c\right )} - \frac {2 a^{2} c d}{a d - b c} + \frac {a b c^{2}}{a d - b c} + 2 a c}{a d + b c} \right )}}{2 b \left (a d - b c\right )} - \frac {c \log {\left (x^{2} + \frac {- \frac {a^{2} c d}{a d - b c} + \frac {2 a b c^{2}}{a d - b c} + 2 a c - \frac {b^{2} c^{3}}{d \left (a d - b c\right )}}{a d + b c} \right )}}{2 d \left (a d - b c\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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