3.8.84 \(\int x^7 (a+c x^4)^{3/2} \, dx\) [784]

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

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 38, normalized size of antiderivative = 1.00, 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 = {272, 45} \begin {gather*} \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} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^7*(a + c*x^4)^(3/2),x]

[Out]

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

Rule 45

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 272

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 x^7 \left (a+c x^4\right )^{3/2} \, dx &=\frac {1}{4} \text {Subst}\left (\int x (a+c x)^{3/2} \, dx,x,x^4\right )\\ &=\frac {1}{4} \text {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*}

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Mathematica [A]
time = 0.03, size = 28, normalized size = 0.74 \begin {gather*} \frac {\left (a+c x^4\right )^{5/2} \left (-2 a+5 c x^4\right )}{70 c^2} \end {gather*}

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 = 0.14, size = 43, normalized size = 1.13

method	result	size
gosper	\(-\frac {\left (x^{4} c +a \right )^{\frac {5}{2}} \left (-5 x^{4} c +2 a \right )}{70 c^{2}}\)	\(25\)
default	\(-\frac {\sqrt {x^{4} c +a}\, \left (-5 x^{4} c +2 a \right ) \left (c^{2} x^{8}+2 a c \,x^{4}+a^{2}\right )}{70 c^{2}}\)	\(43\)
elliptic	\(-\frac {\sqrt {x^{4} c +a}\, \left (-5 x^{4} c +2 a \right ) \left (c^{2} x^{8}+2 a c \,x^{4}+a^{2}\right )}{70 c^{2}}\)	\(43\)
trager	\(-\frac {\left (-5 c^{3} x^{12}-8 a \,c^{2} x^{8}-a^{2} c \,x^{4}+2 a^{3}\right ) \sqrt {x^{4} c +a}}{70 c^{2}}\)	\(47\)
risch	\(-\frac {\left (-5 c^{3} x^{12}-8 a \,c^{2} x^{8}-a^{2} c \,x^{4}+2 a^{3}\right ) \sqrt {x^{4} c +a}}{70 c^{2}}\)	\(47\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.30, size = 30, normalized size = 0.79 \begin {gather*} \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 {gather*}

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 = 0.35, size = 45, normalized size = 1.18 \begin {gather*} \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 {gather*}

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 [B] Leaf count of result is larger than twice the leaf count of optimal. 83 vs. \(2 (31) = 62\).
time = 0.34, size = 83, normalized size = 2.18 \begin {gather*} \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 {gather*}

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 [A]
time = 0.52, size = 29, normalized size = 0.76 \begin {gather*} \frac {5 \, {\left (c x^{4} + a\right )}^{\frac {7}{2}} - 7 \, {\left (c x^{4} + a\right )}^{\frac {5}{2}} a}{70 \, c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/70*(5*(c*x^4 + a)^(7/2) - 7*(c*x^4 + a)^(5/2)*a)/c^2

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Mupad [B]
time = 1.14, size = 42, normalized size = 1.11 \begin {gather*} \sqrt {c\,x^4+a}\,\left (\frac {4\,a\,x^8}{35}+\frac {c\,x^{12}}{14}-\frac {a^3}{35\,c^2}+\frac {a^2\,x^4}{70\,c}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7*(a + c*x^4)^(3/2),x)

[Out]

(a + c*x^4)^(1/2)*((4*a*x^8)/35 + (c*x^12)/14 - a^3/(35*c^2) + (a^2*x^4)/(70*c))

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