3.918 \(\int \frac {x^7}{\sqrt {1+x^4}} \, dx\)

Optimal. Leaf size=27 \[ \frac {1}{6} \left (x^4+1\right )^{3/2}-\frac {\sqrt {x^4+1}}{2} \]

[Out]

1/6*(x^4+1)^(3/2)-1/2*(x^4+1)^(1/2)

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Rubi [A]  time = 0.01, 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} \[ \frac {1}{6} \left (x^4+1\right )^{3/2}-\frac {\sqrt {x^4+1}}{2} \]

Antiderivative was successfully verified.

[In]

Int[x^7/Sqrt[1 + x^4],x]

[Out]

-Sqrt[1 + x^4]/2 + (1 + x^4)^(3/2)/6

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^7}{\sqrt {1+x^4}} \, dx &=\frac {1}{4} \operatorname {Subst}\left (\int \frac {x}{\sqrt {1+x}} \, dx,x,x^4\right )\\ &=\frac {1}{4} \operatorname {Subst}\left (\int \left (-\frac {1}{\sqrt {1+x}}+\sqrt {1+x}\right ) \, dx,x,x^4\right )\\ &=-\frac {1}{2} \sqrt {1+x^4}+\frac {1}{6} \left (1+x^4\right )^{3/2}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 18, normalized size = 0.67 \[ \frac {1}{6} \left (x^4-2\right ) \sqrt {x^4+1} \]

Antiderivative was successfully verified.

[In]

Integrate[x^7/Sqrt[1 + x^4],x]

[Out]

((-2 + x^4)*Sqrt[1 + x^4])/6

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fricas [A]  time = 0.80, size = 14, normalized size = 0.52 \[ \frac {1}{6} \, \sqrt {x^{4} + 1} {\left (x^{4} - 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(x^4+1)^(1/2),x, algorithm="fricas")

[Out]

1/6*sqrt(x^4 + 1)*(x^4 - 2)

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giac [A]  time = 0.15, size = 19, normalized size = 0.70 \[ \frac {1}{6} \, {\left (x^{4} + 1\right )}^{\frac {3}{2}} - \frac {1}{2} \, \sqrt {x^{4} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(x^4+1)^(1/2),x, algorithm="giac")

[Out]

1/6*(x^4 + 1)^(3/2) - 1/2*sqrt(x^4 + 1)

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maple [A]  time = 0.00, size = 15, normalized size = 0.56 \[ \frac {\sqrt {x^{4}+1}\, \left (x^{4}-2\right )}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7/(x^4+1)^(1/2),x)

[Out]

1/6*(x^4+1)^(1/2)*(x^4-2)

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maxima [A]  time = 1.35, size = 19, normalized size = 0.70 \[ \frac {1}{6} \, {\left (x^{4} + 1\right )}^{\frac {3}{2}} - \frac {1}{2} \, \sqrt {x^{4} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(x^4+1)^(1/2),x, algorithm="maxima")

[Out]

1/6*(x^4 + 1)^(3/2) - 1/2*sqrt(x^4 + 1)

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mupad [B]  time = 1.15, size = 14, normalized size = 0.52 \[ \frac {\sqrt {x^4+1}\,\left (x^4-2\right )}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7/(x^4 + 1)^(1/2),x)

[Out]

((x^4 + 1)^(1/2)*(x^4 - 2))/6

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sympy [A]  time = 0.83, size = 22, normalized size = 0.81 \[ \frac {x^{4} \sqrt {x^{4} + 1}}{6} - \frac {\sqrt {x^{4} + 1}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**7/(x**4+1)**(1/2),x)

[Out]

x**4*sqrt(x**4 + 1)/6 - sqrt(x**4 + 1)/3

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