3.765 \(\int x^7 \sqrt{a+c x^4} \, dx\)

Optimal. Leaf size=38 \[ \frac{\left (a+c x^4\right )^{5/2}}{10 c^2}-\frac{a \left (a+c x^4\right )^{3/2}}{6 c^2} \]

[Out]

-(a*(a + c*x^4)^(3/2))/(6*c^2) + (a + c*x^4)^(5/2)/(10*c^2)

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Rubi [A]  time = 0.0227148, 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} \[ \frac{\left (a+c x^4\right )^{5/2}}{10 c^2}-\frac{a \left (a+c x^4\right )^{3/2}}{6 c^2} \]

Antiderivative was successfully verified.

[In]

Int[x^7*Sqrt[a + c*x^4],x]

[Out]

-(a*(a + c*x^4)^(3/2))/(6*c^2) + (a + c*x^4)^(5/2)/(10*c^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 x^7 \sqrt{a+c x^4} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int x \sqrt{a+c x} \, dx,x,x^4\right )\\ &=\frac{1}{4} \operatorname{Subst}\left (\int \left (-\frac{a \sqrt{a+c x}}{c}+\frac{(a+c x)^{3/2}}{c}\right ) \, dx,x,x^4\right )\\ &=-\frac{a \left (a+c x^4\right )^{3/2}}{6 c^2}+\frac{\left (a+c x^4\right )^{5/2}}{10 c^2}\\ \end{align*}

Mathematica [A]  time = 0.0130241, size = 28, normalized size = 0.74 \[ \frac{\left (a+c x^4\right )^{3/2} \left (3 c x^4-2 a\right )}{30 c^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x^7*Sqrt[a + c*x^4],x]

[Out]

((a + c*x^4)^(3/2)*(-2*a + 3*c*x^4))/(30*c^2)

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Maple [A]  time = 0.004, size = 25, normalized size = 0.7 \begin{align*} -{\frac{-3\,c{x}^{4}+2\,a}{30\,{c}^{2}} \left ( c{x}^{4}+a \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/30*(c*x^4+a)^(3/2)*(-3*c*x^4+2*a)/c^2

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Maxima [A]  time = 0.982079, size = 41, normalized size = 1.08 \begin{align*} \frac{{\left (c x^{4} + a\right )}^{\frac{5}{2}}}{10 \, c^{2}} - \frac{{\left (c x^{4} + a\right )}^{\frac{3}{2}} a}{6 \, c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*(c*x^4 + a)^(5/2)/c^2 - 1/6*(c*x^4 + a)^(3/2)*a/c^2

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Fricas [A]  time = 1.42882, size = 76, normalized size = 2. \begin{align*} \frac{{\left (3 \, c^{2} x^{8} + a c x^{4} - 2 \, a^{2}\right )} \sqrt{c x^{4} + a}}{30 \, c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/30*(3*c^2*x^8 + a*c*x^4 - 2*a^2)*sqrt(c*x^4 + a)/c^2

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Sympy [A]  time = 1.38985, size = 61, normalized size = 1.61 \begin{align*} \begin{cases} - \frac{a^{2} \sqrt{a + c x^{4}}}{15 c^{2}} + \frac{a x^{4} \sqrt{a + c x^{4}}}{30 c} + \frac{x^{8} \sqrt{a + c x^{4}}}{10} & \text{for}\: c \neq 0 \\\frac{\sqrt{a} x^{8}}{8} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-a**2*sqrt(a + c*x**4)/(15*c**2) + a*x**4*sqrt(a + c*x**4)/(30*c) + x**8*sqrt(a + c*x**4)/10, Ne(c,
0)), (sqrt(a)*x**8/8, True))

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Giac [A]  time = 1.10484, size = 39, normalized size = 1.03 \begin{align*} \frac{3 \,{\left (c x^{4} + a\right )}^{\frac{5}{2}} - 5 \,{\left (c x^{4} + a\right )}^{\frac{3}{2}} a}{30 \, c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/30*(3*(c*x^4 + a)^(5/2) - 5*(c*x^4 + a)^(3/2)*a)/c^2