3.784 \(\int x^7 (a+c x^4)^{3/2} \, dx\)

Optimal. Leaf size=38 \[ \frac{\left (a+c x^4\right )^{7/2}}{14 c^2}-\frac{a \left (a+c x^4\right )^{5/2}}{10 c^2} \]

[Out]

-(a*(a + c*x^4)^(5/2))/(10*c^2) + (a + c*x^4)^(7/2)/(14*c^2)

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Rubi [A]  time = 0.023506, 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} \[ \frac{\left (a+c x^4\right )^{7/2}}{14 c^2}-\frac{a \left (a+c x^4\right )^{5/2}}{10 c^2} \]

Antiderivative was successfully verified.

[In]

Int[x^7*(a + c*x^4)^(3/2),x]

[Out]

-(a*(a + c*x^4)^(5/2))/(10*c^2) + (a + c*x^4)^(7/2)/(14*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 \left (a+c x^4\right )^{3/2} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int x (a+c x)^{3/2} \, dx,x,x^4\right )\\ &=\frac{1}{4} \operatorname{Subst}\left (\int \left (-\frac{a (a+c x)^{3/2}}{c}+\frac{(a+c x)^{5/2}}{c}\right ) \, dx,x,x^4\right )\\ &=-\frac{a \left (a+c x^4\right )^{5/2}}{10 c^2}+\frac{\left (a+c x^4\right )^{7/2}}{14 c^2}\\ \end{align*}

Mathematica [A]  time = 0.0152567, size = 28, normalized size = 0.74 \[ \frac{\left (a+c x^4\right )^{5/2} \left (5 c x^4-2 a\right )}{70 c^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x^7*(a + c*x^4)^(3/2),x]

[Out]

((a + c*x^4)^(5/2)*(-2*a + 5*c*x^4))/(70*c^2)

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Maple [A]  time = 0.003, size = 25, normalized size = 0.7 \begin{align*} -{\frac{-5\,c{x}^{4}+2\,a}{70\,{c}^{2}} \left ( c{x}^{4}+a \right ) ^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7*(c*x^4+a)^(3/2),x)

[Out]

-1/70*(c*x^4+a)^(5/2)*(-5*c*x^4+2*a)/c^2

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Maxima [A]  time = 0.965312, size = 41, normalized size = 1.08 \begin{align*} \frac{{\left (c x^{4} + a\right )}^{\frac{7}{2}}}{14 \, c^{2}} - \frac{{\left (c x^{4} + a\right )}^{\frac{5}{2}} a}{10 \, c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/14*(c*x^4 + a)^(7/2)/c^2 - 1/10*(c*x^4 + a)^(5/2)*a/c^2

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Fricas [A]  time = 1.41216, size = 99, normalized size = 2.61 \begin{align*} \frac{{\left (5 \, c^{3} x^{12} + 8 \, a c^{2} x^{8} + a^{2} c x^{4} - 2 \, a^{3}\right )} \sqrt{c x^{4} + a}}{70 \, c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/70*(5*c^3*x^12 + 8*a*c^2*x^8 + a^2*c*x^4 - 2*a^3)*sqrt(c*x^4 + a)/c^2

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Sympy [A]  time = 3.86053, size = 83, normalized size = 2.18 \begin{align*} \begin{cases} - \frac{a^{3} \sqrt{a + c x^{4}}}{35 c^{2}} + \frac{a^{2} x^{4} \sqrt{a + c x^{4}}}{70 c} + \frac{4 a x^{8} \sqrt{a + c x^{4}}}{35} + \frac{c x^{12} \sqrt{a + c x^{4}}}{14} & \text{for}\: c \neq 0 \\\frac{a^{\frac{3}{2}} 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)**(3/2),x)

[Out]

Piecewise((-a**3*sqrt(a + c*x**4)/(35*c**2) + a**2*x**4*sqrt(a + c*x**4)/(70*c) + 4*a*x**8*sqrt(a + c*x**4)/35
+ c*x**12*sqrt(a + c*x**4)/14, Ne(c, 0)), (a**(3/2)*x**8/8, True))

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Giac [B]  time = 1.13815, size = 105, normalized size = 2.76 \begin{align*} \frac{\frac{7 \,{\left (3 \,{\left (c x^{4} + a\right )}^{\frac{5}{2}} - 5 \,{\left (c x^{4} + a\right )}^{\frac{3}{2}} a\right )} a}{c} + \frac{15 \,{\left (c x^{4} + a\right )}^{\frac{7}{2}} - 42 \,{\left (c x^{4} + a\right )}^{\frac{5}{2}} a + 35 \,{\left (c x^{4} + a\right )}^{\frac{3}{2}} a^{2}}{c}}{210 \, c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/210*(7*(3*(c*x^4 + a)^(5/2) - 5*(c*x^4 + a)^(3/2)*a)*a/c + (15*(c*x^4 + a)^(7/2) - 42*(c*x^4 + a)^(5/2)*a +
35*(c*x^4 + a)^(3/2)*a^2)/c)/c