3.15.54 \(\int \frac {x^3}{a+b x^8} \, dx\) [1454]

Optimal. Leaf size=29 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {b} x^4}{\sqrt {a}}\right )}{4 \sqrt {a} \sqrt {b}} \]

[Out]

1/4*arctan(x^4*b^(1/2)/a^(1/2))/a^(1/2)/b^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {281, 211} \begin {gather*} \frac {\text {ArcTan}\left (\frac {\sqrt {b} x^4}{\sqrt {a}}\right )}{4 \sqrt {a} \sqrt {b}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3/(a + b*x^8),x]

[Out]

ArcTan[(Sqrt[b]*x^4)/Sqrt[a]]/(4*Sqrt[a]*Sqrt[b])

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 281

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

Rubi steps

\begin {align*} \int \frac {x^3}{a+b x^8} \, dx &=\frac {1}{4} \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,x^4\right )\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt {b} x^4}{\sqrt {a}}\right )}{4 \sqrt {a} \sqrt {b}}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 29, normalized size = 1.00 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt {b} x^4}{\sqrt {a}}\right )}{4 \sqrt {a} \sqrt {b}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^3/(a + b*x^8),x]

[Out]

ArcTan[(Sqrt[b]*x^4)/Sqrt[a]]/(4*Sqrt[a]*Sqrt[b])

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Maple [A]
time = 0.18, size = 19, normalized size = 0.66

method	result	size
default	\(\frac {\arctan \left (\frac {b \,x^{4}}{\sqrt {a b}}\right )}{4 \sqrt {a b}}\)	\(19\)
risch	\(-\frac {\ln \left (x^{4} \sqrt {-a b}-a \right )}{8 \sqrt {-a b}}+\frac {\ln \left (x^{4} \sqrt {-a b}+a \right )}{8 \sqrt {-a b}}\)	\(46\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(b*x^8+a),x,method=_RETURNVERBOSE)

[Out]

1/4/(a*b)^(1/2)*arctan(b*x^4/(a*b)^(1/2))

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Maxima [A]
time = 0.49, size = 18, normalized size = 0.62 \begin {gather*} \frac {\arctan \left (\frac {b x^{4}}{\sqrt {a b}}\right )}{4 \, \sqrt {a b}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(b*x^8+a),x, algorithm="maxima")

[Out]

1/4*arctan(b*x^4/sqrt(a*b))/sqrt(a*b)

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Fricas [A]
time = 0.36, size = 72, normalized size = 2.48 \begin {gather*} \left [-\frac {\sqrt {-a b} \log \left (\frac {b x^{8} - 2 \, \sqrt {-a b} x^{4} - a}{b x^{8} + a}\right )}{8 \, a b}, -\frac {\sqrt {a b} \arctan \left (\frac {\sqrt {a b}}{b x^{4}}\right )}{4 \, a b}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(b*x^8+a),x, algorithm="fricas")

[Out]

[-1/8*sqrt(-a*b)*log((b*x^8 - 2*sqrt(-a*b)*x^4 - a)/(b*x^8 + a))/(a*b), -1/4*sqrt(a*b)*arctan(sqrt(a*b)/(b*x^4
))/(a*b)]

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (26) = 52\).
time = 0.10, size = 56, normalized size = 1.93 \begin {gather*} - \frac {\sqrt {- \frac {1}{a b}} \log {\left (- a \sqrt {- \frac {1}{a b}} + x^{4} \right )}}{8} + \frac {\sqrt {- \frac {1}{a b}} \log {\left (a \sqrt {- \frac {1}{a b}} + x^{4} \right )}}{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3/(b*x**8+a),x)

[Out]

-sqrt(-1/(a*b))*log(-a*sqrt(-1/(a*b)) + x**4)/8 + sqrt(-1/(a*b))*log(a*sqrt(-1/(a*b)) + x**4)/8

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Giac [A]
time = 1.09, size = 18, normalized size = 0.62 \begin {gather*} \frac {\arctan \left (\frac {b x^{4}}{\sqrt {a b}}\right )}{4 \, \sqrt {a b}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(b*x^8+a),x, algorithm="giac")

[Out]

1/4*arctan(b*x^4/sqrt(a*b))/sqrt(a*b)

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Mupad [B]
time = 0.06, size = 19, normalized size = 0.66 \begin {gather*} \frac {\mathrm {atan}\left (\frac {\sqrt {b}\,x^4}{\sqrt {a}}\right )}{4\,\sqrt {a}\,\sqrt {b}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(a + b*x^8),x)

[Out]

atan((b^(1/2)*x^4)/a^(1/2))/(4*a^(1/2)*b^(1/2))

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