3.812 \(\int \frac{x^7}{\sqrt{a+b x^4}} \, dx\)

Optimal. Leaf size=38 \[ \frac{\left (a+b x^4\right )^{3/2}}{6 b^2}-\frac{a \sqrt{a+b x^4}}{2 b^2} \]

[Out]

-(a*Sqrt[a + b*x^4])/(2*b^2) + (a + b*x^4)^(3/2)/(6*b^2)

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Rubi [A]  time = 0.0221903, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {266, 43} \[ \frac{\left (a+b x^4\right )^{3/2}}{6 b^2}-\frac{a \sqrt{a+b x^4}}{2 b^2} \]

Antiderivative was successfully verified.

[In]

Int[x^7/Sqrt[a + b*x^4],x]

[Out]

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

Mathematica [A]  time = 0.0123844, size = 27, normalized size = 0.71 \[ \frac{\left (b x^4-2 a\right ) \sqrt{a+b x^4}}{6 b^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x^7/Sqrt[a + b*x^4],x]

[Out]

((-2*a + b*x^4)*Sqrt[a + b*x^4])/(6*b^2)

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Maple [A]  time = 0.005, size = 25, normalized size = 0.7 \begin{align*} -{\frac{-b{x}^{4}+2\,a}{6\,{b}^{2}}\sqrt{b{x}^{4}+a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.966302, size = 41, normalized size = 1.08 \begin{align*} \frac{{\left (b x^{4} + a\right )}^{\frac{3}{2}}}{6 \, b^{2}} - \frac{\sqrt{b x^{4} + a} a}{2 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.41924, size = 53, normalized size = 1.39 \begin{align*} \frac{\sqrt{b x^{4} + a}{\left (b x^{4} - 2 \, a\right )}}{6 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*sqrt(b*x^4 + a)*(b*x^4 - 2*a)/b^2

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Sympy [A]  time = 1.5115, size = 42, normalized size = 1.11 \begin{align*} \begin{cases} - \frac{a \sqrt{a + b x^{4}}}{3 b^{2}} + \frac{x^{4} \sqrt{a + b x^{4}}}{6 b} & \text{for}\: b \neq 0 \\\frac{x^{8}}{8 \sqrt{a}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-a*sqrt(a + b*x**4)/(3*b**2) + x**4*sqrt(a + b*x**4)/(6*b), Ne(b, 0)), (x**8/(8*sqrt(a)), True))

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Giac [A]  time = 1.10248, size = 36, normalized size = 0.95 \begin{align*} \frac{{\left (b x^{4} + a\right )}^{\frac{3}{2}} - 3 \, \sqrt{b x^{4} + a} a}{6 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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