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

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

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {266, 43} \[ \frac {x^2}{2 a}-\frac {b \log \left (a x^2+b\right )}{2 a^2} \]

Antiderivative was successfully verified.

[In]

Int[x^3/(b + a*x^2),x]

[Out]

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

Rubi steps

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

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

Antiderivative was successfully verified.

[In]

Integrate[x^3/(b + a*x^2),x]

[Out]

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

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fricas [A]  time = 0.40, size = 22, normalized size = 0.81 \[ \frac {a x^{2} - b \log \left (a x^{2} + b\right )}{2 \, a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(a*x^2 - b*log(a*x^2 + b))/a^2

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giac [A]  time = 1.14, size = 24, normalized size = 0.89 \[ \frac {x^{2}}{2 \, a} - \frac {b \log \left ({\left | a x^{2} + b \right |}\right )}{2 \, a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.00, size = 24, normalized size = 0.89 \[ \frac {x^{2}}{2 a}-\frac {b \ln \left (a \,x^{2}+b \right )}{2 a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.42, size = 23, normalized size = 0.85 \[ \frac {x^{2}}{2 \, a} - \frac {b \log \left (a x^{2} + b\right )}{2 \, a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2/a - 1/2*b*log(a*x^2 + b)/a^2

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mupad [B]  time = 0.15, size = 22, normalized size = 0.81 \[ -\frac {b\,\ln \left (a\,x^2+b\right )-a\,x^2}{2\,a^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(b + a*x^2),x)

[Out]

-(b*log(b + a*x^2) - a*x^2)/(2*a^2)

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sympy [A]  time = 0.14, size = 20, normalized size = 0.74 \[ \frac {x^{2}}{2 a} - \frac {b \log {\left (a x^{2} + b \right )}}{2 a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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