3.2.30 \(\int \frac {x^5}{a+b x^2} \, dx\) [130]

Optimal. Leaf size=40 \[ -\frac {a x^2}{2 b^2}+\frac {x^4}{4 b}+\frac {a^2 \log \left (a+b x^2\right )}{2 b^3} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 40, 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 = {272, 45} \begin {gather*} \frac {a^2 \log \left (a+b x^2\right )}{2 b^3}-\frac {a x^2}{2 b^2}+\frac {x^4}{4 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^5/(a + b*x^2),x]

[Out]

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

Rule 45

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 272

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^5}{a+b x^2} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x^2}{a+b x} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (-\frac {a}{b^2}+\frac {x}{b}+\frac {a^2}{b^2 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {a x^2}{2 b^2}+\frac {x^4}{4 b}+\frac {a^2 \log \left (a+b x^2\right )}{2 b^3}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 40, normalized size = 1.00 \begin {gather*} -\frac {a x^2}{2 b^2}+\frac {x^4}{4 b}+\frac {a^2 \log \left (a+b x^2\right )}{2 b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^5/(a + b*x^2),x]

[Out]

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

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Maple [A]
time = 0.04, size = 35, normalized size = 0.88

method	result	size
default	\(-\frac {-\frac {1}{2} b \,x^{4}+a \,x^{2}}{2 b^{2}}+\frac {a^{2} \ln \left (b \,x^{2}+a \right )}{2 b^{3}}\)	\(35\)
norman	\(-\frac {a \,x^{2}}{2 b^{2}}+\frac {x^{4}}{4 b}+\frac {a^{2} \ln \left (b \,x^{2}+a \right )}{2 b^{3}}\)	\(35\)
risch	\(\frac {x^{4}}{4 b}-\frac {a \,x^{2}}{2 b^{2}}+\frac {a^{2}}{4 b^{3}}+\frac {a^{2} \ln \left (b \,x^{2}+a \right )}{2 b^{3}}\)	\(43\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(b*x^2+a),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.27, size = 34, normalized size = 0.85 \begin {gather*} \frac {a^{2} \log \left (b x^{2} + a\right )}{2 \, b^{3}} + \frac {b x^{4} - 2 \, a x^{2}}{4 \, b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(b*x^2+a),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.98, size = 33, normalized size = 0.82 \begin {gather*} \frac {b^{2} x^{4} - 2 \, a b x^{2} + 2 \, a^{2} \log \left (b x^{2} + a\right )}{4 \, b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(b*x^2+a),x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 0.05, size = 32, normalized size = 0.80 \begin {gather*} \frac {a^{2} \log {\left (a + b x^{2} \right )}}{2 b^{3}} - \frac {a x^{2}}{2 b^{2}} + \frac {x^{4}}{4 b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5/(b*x**2+a),x)

[Out]

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

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Giac [A]
time = 0.75, size = 35, normalized size = 0.88 \begin {gather*} \frac {a^{2} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{3}} + \frac {b x^{4} - 2 \, a x^{2}}{4 \, b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(b*x^2+a),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 4.65, size = 33, normalized size = 0.82 \begin {gather*} \frac {2\,a^2\,\ln \left (b\,x^2+a\right )+b^2\,x^4-2\,a\,b\,x^2}{4\,b^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(a + b*x^2),x)

[Out]

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

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