3.1.12 \(\int \frac {x^4}{a x+b x^3} \, dx\)

Optimal. Leaf size=27 \[ \frac {x^2}{2 b}-\frac {a \log \left (a+b x^2\right )}{2 b^2} \]

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Rubi [A]  time = 0.02, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1584, 266, 43} \begin {gather*} \frac {x^2}{2 b}-\frac {a \log \left (a+b x^2\right )}{2 b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^4/(a*x + b*x^3),x]

[Out]

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

Rule 43

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[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^4}{a x+b x^3} \, dx &=\int \frac {x^3}{a+b x^2} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x}{a+b x} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {1}{b}-\frac {a}{b (a+b x)}\right ) \, dx,x,x^2\right )\\ &=\frac {x^2}{2 b}-\frac {a \log \left (a+b x^2\right )}{2 b^2}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 27, normalized size = 1.00 \begin {gather*} \frac {x^2}{2 b}-\frac {a \log \left (a+b x^2\right )}{2 b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^4/(a*x + b*x^3),x]

[Out]

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

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^4}{a x+b x^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.39, size = 22, normalized size = 0.81 \begin {gather*} \frac {b x^{2} - a \log \left (b x^{2} + a\right )}{2 \, b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(b*x^2 - a*log(b*x^2 + a))/b^2

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giac [A]  time = 0.15, size = 24, normalized size = 0.89 \begin {gather*} \frac {x^{2}}{2 \, b} - \frac {a \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.04, size = 24, normalized size = 0.89 \begin {gather*} \frac {x^{2}}{2 b}-\frac {a \ln \left (b \,x^{2}+a \right )}{2 b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.32, size = 23, normalized size = 0.85 \begin {gather*} \frac {x^{2}}{2 \, b} - \frac {a \log \left (b x^{2} + a\right )}{2 \, b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2/b - 1/2*a*log(b*x^2 + a)/b^2

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mupad [B]  time = 0.04, size = 22, normalized size = 0.81 \begin {gather*} -\frac {a\,\ln \left (b\,x^2+a\right )-b\,x^2}{2\,b^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/(a*x + b*x^3),x)

[Out]

-(a*log(a + b*x^2) - b*x^2)/(2*b^2)

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sympy [A]  time = 0.15, size = 20, normalized size = 0.74 \begin {gather*} - \frac {a \log {\left (a + b x^{2} \right )}}{2 b^{2}} + \frac {x^{2}}{2 b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4/(b*x**3+a*x),x)

[Out]

-a*log(a + b*x**2)/(2*b**2) + x**2/(2*b)

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