3.938 \(\int \frac{x^7}{(1+x^4)^{3/2}} \, dx\)

Optimal. Leaf size=27 \[ \frac{\sqrt{x^4+1}}{2}+\frac{1}{2 \sqrt{x^4+1}} \]

[Out]

1/(2*Sqrt[1 + x^4]) + Sqrt[1 + x^4]/2

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Rubi [A]  time = 0.0096739, antiderivative size = 27, normalized size of antiderivative = 1., 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} \[ \frac{\sqrt{x^4+1}}{2}+\frac{1}{2 \sqrt{x^4+1}} \]

Antiderivative was successfully verified.

[In]

Int[x^7/(1 + x^4)^(3/2),x]

[Out]

1/(2*Sqrt[1 + x^4]) + Sqrt[1 + x^4]/2

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 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])

Rubi steps

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

Mathematica [A]  time = 0.0052901, size = 18, normalized size = 0.67 \[ \frac{x^4+2}{2 \sqrt{x^4+1}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^7/(1 + x^4)^(3/2),x]

[Out]

(2 + x^4)/(2*Sqrt[1 + x^4])

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Maple [A]  time = 0.008, size = 15, normalized size = 0.6 \begin{align*}{\frac{{x}^{4}+2}{2}{\frac{1}{\sqrt{{x}^{4}+1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7/(x^4+1)^(3/2),x)

[Out]

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

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Maxima [A]  time = 0.982288, size = 26, normalized size = 0.96 \begin{align*} \frac{1}{2} \, \sqrt{x^{4} + 1} + \frac{1}{2 \, \sqrt{x^{4} + 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(x^4+1)^(3/2),x, algorithm="maxima")

[Out]

1/2*sqrt(x^4 + 1) + 1/2/sqrt(x^4 + 1)

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Fricas [A]  time = 1.45773, size = 39, normalized size = 1.44 \begin{align*} \frac{x^{4} + 2}{2 \, \sqrt{x^{4} + 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(x^4+1)^(3/2),x, algorithm="fricas")

[Out]

1/2*(x^4 + 2)/sqrt(x^4 + 1)

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Sympy [A]  time = 0.896962, size = 22, normalized size = 0.81 \begin{align*} \frac{x^{4}}{2 \sqrt{x^{4} + 1}} + \frac{1}{\sqrt{x^{4} + 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**7/(x**4+1)**(3/2),x)

[Out]

x**4/(2*sqrt(x**4 + 1)) + 1/sqrt(x**4 + 1)

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Giac [A]  time = 1.16527, size = 26, normalized size = 0.96 \begin{align*} \frac{1}{2} \, \sqrt{x^{4} + 1} + \frac{1}{2 \, \sqrt{x^{4} + 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(x^4+1)^(3/2),x, algorithm="giac")

[Out]

1/2*sqrt(x^4 + 1) + 1/2/sqrt(x^4 + 1)