3.1.98 \(\int \frac {x^3}{(b x^2+c x^4)^3} \, dx\)

Optimal. Leaf size=67 \[ \frac {3 c \log \left (b+c x^2\right )}{2 b^4}-\frac {3 c \log (x)}{b^4}-\frac {c}{b^3 \left (b+c x^2\right )}-\frac {1}{2 b^3 x^2}-\frac {c}{4 b^2 \left (b+c x^2\right )^2} \]

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Rubi [A]  time = 0.06, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {1584, 266, 44} \begin {gather*} -\frac {c}{b^3 \left (b+c x^2\right )}-\frac {c}{4 b^2 \left (b+c x^2\right )^2}+\frac {3 c \log \left (b+c x^2\right )}{2 b^4}-\frac {3 c \log (x)}{b^4}-\frac {1}{2 b^3 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3/(b*x^2 + c*x^4)^3,x]

[Out]

-1/(2*b^3*x^2) - c/(4*b^2*(b + c*x^2)^2) - c/(b^3*(b + c*x^2)) - (3*c*Log[x])/b^4 + (3*c*Log[b + c*x^2])/(2*b^
4)

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 1584

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(m + n*p)*(a + b*x^(q -
p))^n, x] /; FreeQ[{a, b, m, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rubi steps

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

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Mathematica [A]  time = 0.06, size = 59, normalized size = 0.88 \begin {gather*} -\frac {\frac {b \left (2 b^2+9 b c x^2+6 c^2 x^4\right )}{x^2 \left (b+c x^2\right )^2}-6 c \log \left (b+c x^2\right )+12 c \log (x)}{4 b^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^3/(b*x^2 + c*x^4)^3,x]

[Out]

-1/4*((b*(2*b^2 + 9*b*c*x^2 + 6*c^2*x^4))/(x^2*(b + c*x^2)^2) + 12*c*Log[x] - 6*c*Log[b + c*x^2])/b^4

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^3}{\left (b x^2+c x^4\right )^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[x^3/(b*x^2 + c*x^4)^3,x]

[Out]

IntegrateAlgebraic[x^3/(b*x^2 + c*x^4)^3, x]

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fricas [A]  time = 0.79, size = 119, normalized size = 1.78 \begin {gather*} -\frac {6 \, b c^{2} x^{4} + 9 \, b^{2} c x^{2} + 2 \, b^{3} - 6 \, {\left (c^{3} x^{6} + 2 \, b c^{2} x^{4} + b^{2} c x^{2}\right )} \log \left (c x^{2} + b\right ) + 12 \, {\left (c^{3} x^{6} + 2 \, b c^{2} x^{4} + b^{2} c x^{2}\right )} \log \relax (x)}{4 \, {\left (b^{4} c^{2} x^{6} + 2 \, b^{5} c x^{4} + b^{6} x^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(c*x^4+b*x^2)^3,x, algorithm="fricas")

[Out]

-1/4*(6*b*c^2*x^4 + 9*b^2*c*x^2 + 2*b^3 - 6*(c^3*x^6 + 2*b*c^2*x^4 + b^2*c*x^2)*log(c*x^2 + b) + 12*(c^3*x^6 +
2*b*c^2*x^4 + b^2*c*x^2)*log(x))/(b^4*c^2*x^6 + 2*b^5*c*x^4 + b^6*x^2)

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giac [A]  time = 0.17, size = 66, normalized size = 0.99 \begin {gather*} \frac {3 \, c \log \left ({\left | c x^{2} + b \right |}\right )}{2 \, b^{4}} - \frac {3 \, c \log \left ({\left | x \right |}\right )}{b^{4}} - \frac {6 \, b c^{2} x^{4} + 9 \, b^{2} c x^{2} + 2 \, b^{3}}{4 \, {\left (c x^{2} + b\right )}^{2} b^{4} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(c*x^4+b*x^2)^3,x, algorithm="giac")

[Out]

3/2*c*log(abs(c*x^2 + b))/b^4 - 3*c*log(abs(x))/b^4 - 1/4*(6*b*c^2*x^4 + 9*b^2*c*x^2 + 2*b^3)/((c*x^2 + b)^2*b
^4*x^2)

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maple [A]  time = 0.02, size = 62, normalized size = 0.93 \begin {gather*} -\frac {c}{4 \left (c \,x^{2}+b \right )^{2} b^{2}}-\frac {c}{\left (c \,x^{2}+b \right ) b^{3}}-\frac {3 c \ln \relax (x )}{b^{4}}+\frac {3 c \ln \left (c \,x^{2}+b \right )}{2 b^{4}}-\frac {1}{2 b^{3} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(c*x^4+b*x^2)^3,x)

[Out]

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

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maxima [A]  time = 1.38, size = 77, normalized size = 1.15 \begin {gather*} -\frac {6 \, c^{2} x^{4} + 9 \, b c x^{2} + 2 \, b^{2}}{4 \, {\left (b^{3} c^{2} x^{6} + 2 \, b^{4} c x^{4} + b^{5} x^{2}\right )}} + \frac {3 \, c \log \left (c x^{2} + b\right )}{2 \, b^{4}} - \frac {3 \, c \log \left (x^{2}\right )}{2 \, b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(c*x^4+b*x^2)^3,x, algorithm="maxima")

[Out]

-1/4*(6*c^2*x^4 + 9*b*c*x^2 + 2*b^2)/(b^3*c^2*x^6 + 2*b^4*c*x^4 + b^5*x^2) + 3/2*c*log(c*x^2 + b)/b^4 - 3/2*c*
log(x^2)/b^4

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mupad [B]  time = 0.06, size = 75, normalized size = 1.12 \begin {gather*} \frac {3\,c\,\ln \left (c\,x^2+b\right )}{2\,b^4}-\frac {\frac {1}{2\,b}+\frac {9\,c\,x^2}{4\,b^2}+\frac {3\,c^2\,x^4}{2\,b^3}}{b^2\,x^2+2\,b\,c\,x^4+c^2\,x^6}-\frac {3\,c\,\ln \relax (x)}{b^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(b*x^2 + c*x^4)^3,x)

[Out]

(3*c*log(b + c*x^2))/(2*b^4) - (1/(2*b) + (9*c*x^2)/(4*b^2) + (3*c^2*x^4)/(2*b^3))/(b^2*x^2 + c^2*x^6 + 2*b*c*
x^4) - (3*c*log(x))/b^4

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sympy [A]  time = 0.63, size = 80, normalized size = 1.19 \begin {gather*} \frac {- 2 b^{2} - 9 b c x^{2} - 6 c^{2} x^{4}}{4 b^{5} x^{2} + 8 b^{4} c x^{4} + 4 b^{3} c^{2} x^{6}} - \frac {3 c \log {\relax (x )}}{b^{4}} + \frac {3 c \log {\left (\frac {b}{c} + x^{2} \right )}}{2 b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3/(c*x**4+b*x**2)**3,x)

[Out]

(-2*b**2 - 9*b*c*x**2 - 6*c**2*x**4)/(4*b**5*x**2 + 8*b**4*c*x**4 + 4*b**3*c**2*x**6) - 3*c*log(x)/b**4 + 3*c*
log(b/c + x**2)/(2*b**4)

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