3.8.64 \(\int x^{11} \sqrt {a+c x^4} \, dx\) [764]

Optimal. Leaf size=59 \[ \frac {a^2 \left (a+c x^4\right )^{3/2}}{6 c^3}-\frac {a \left (a+c x^4\right )^{5/2}}{5 c^3}+\frac {\left (a+c x^4\right )^{7/2}}{14 c^3} \]

[Out]

1/6*a^2*(c*x^4+a)^(3/2)/c^3-1/5*a*(c*x^4+a)^(5/2)/c^3+1/14*(c*x^4+a)^(7/2)/c^3

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Rubi [A]
time = 0.03, antiderivative size = 59, 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 {a^2 \left (a+c x^4\right )^{3/2}}{6 c^3}+\frac {\left (a+c x^4\right )^{7/2}}{14 c^3}-\frac {a \left (a+c x^4\right )^{5/2}}{5 c^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^11*Sqrt[a + c*x^4],x]

[Out]

(a^2*(a + c*x^4)^(3/2))/(6*c^3) - (a*(a + c*x^4)^(5/2))/(5*c^3) + (a + c*x^4)^(7/2)/(14*c^3)

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^{11} \sqrt {a+c x^4} \, dx &=\frac {1}{4} \text {Subst}\left (\int x^2 \sqrt {a+c x} \, dx,x,x^4\right )\\ &=\frac {1}{4} \text {Subst}\left (\int \left (\frac {a^2 \sqrt {a+c x}}{c^2}-\frac {2 a (a+c x)^{3/2}}{c^2}+\frac {(a+c x)^{5/2}}{c^2}\right ) \, dx,x,x^4\right )\\ &=\frac {a^2 \left (a+c x^4\right )^{3/2}}{6 c^3}-\frac {a \left (a+c x^4\right )^{5/2}}{5 c^3}+\frac {\left (a+c x^4\right )^{7/2}}{14 c^3}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 50, normalized size = 0.85 \begin {gather*} \frac {\sqrt {a+c x^4} \left (8 a^3-4 a^2 c x^4+3 a c^2 x^8+15 c^3 x^{12}\right )}{210 c^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^11*Sqrt[a + c*x^4],x]

[Out]

(Sqrt[a + c*x^4]*(8*a^3 - 4*a^2*c*x^4 + 3*a*c^2*x^8 + 15*c^3*x^12))/(210*c^3)

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Maple [A]
time = 0.14, size = 36, normalized size = 0.61

method	result	size
gosper	\(\frac {\left (x^{4} c +a \right )^{\frac {3}{2}} \left (15 c^{2} x^{8}-12 a c \,x^{4}+8 a^{2}\right )}{210 c^{3}}\)	\(36\)
default	\(\frac {\left (x^{4} c +a \right )^{\frac {3}{2}} \left (15 c^{2} x^{8}-12 a c \,x^{4}+8 a^{2}\right )}{210 c^{3}}\)	\(36\)
elliptic	\(\frac {\left (x^{4} c +a \right )^{\frac {3}{2}} \left (15 c^{2} x^{8}-12 a c \,x^{4}+8 a^{2}\right )}{210 c^{3}}\)	\(36\)
trager	\(\frac {\left (15 c^{3} x^{12}+3 a \,c^{2} x^{8}-4 a^{2} c \,x^{4}+8 a^{3}\right ) \sqrt {x^{4} c +a}}{210 c^{3}}\)	\(47\)
risch	\(\frac {\left (15 c^{3} x^{12}+3 a \,c^{2} x^{8}-4 a^{2} c \,x^{4}+8 a^{3}\right ) \sqrt {x^{4} c +a}}{210 c^{3}}\)	\(47\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^11*(c*x^4+a)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/210*(c*x^4+a)^(3/2)*(15*c^2*x^8-12*a*c*x^4+8*a^2)/c^3

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Maxima [A]
time = 0.29, size = 47, normalized size = 0.80 \begin {gather*} \frac {{\left (c x^{4} + a\right )}^{\frac {7}{2}}}{14 \, c^{3}} - \frac {{\left (c x^{4} + a\right )}^{\frac {5}{2}} a}{5 \, c^{3}} + \frac {{\left (c x^{4} + a\right )}^{\frac {3}{2}} a^{2}}{6 \, c^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11*(c*x^4+a)^(1/2),x, algorithm="maxima")

[Out]

1/14*(c*x^4 + a)^(7/2)/c^3 - 1/5*(c*x^4 + a)^(5/2)*a/c^3 + 1/6*(c*x^4 + a)^(3/2)*a^2/c^3

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Fricas [A]
time = 0.36, size = 46, normalized size = 0.78 \begin {gather*} \frac {{\left (15 \, c^{3} x^{12} + 3 \, a c^{2} x^{8} - 4 \, a^{2} c x^{4} + 8 \, a^{3}\right )} \sqrt {c x^{4} + a}}{210 \, c^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11*(c*x^4+a)^(1/2),x, algorithm="fricas")

[Out]

1/210*(15*c^3*x^12 + 3*a*c^2*x^8 - 4*a^2*c*x^4 + 8*a^3)*sqrt(c*x^4 + a)/c^3

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Sympy [A]
time = 0.35, size = 87, normalized size = 1.47 \begin {gather*} \begin {cases} \frac {4 a^{3} \sqrt {a + c x^{4}}}{105 c^{3}} - \frac {2 a^{2} x^{4} \sqrt {a + c x^{4}}}{105 c^{2}} + \frac {a x^{8} \sqrt {a + c x^{4}}}{70 c} + \frac {x^{12} \sqrt {a + c x^{4}}}{14} & \text {for}\: c \neq 0 \\\frac {\sqrt {a} x^{12}}{12} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**11*(c*x**4+a)**(1/2),x)

[Out]

Piecewise((4*a**3*sqrt(a + c*x**4)/(105*c**3) - 2*a**2*x**4*sqrt(a + c*x**4)/(105*c**2) + a*x**8*sqrt(a + c*x*
*4)/(70*c) + x**12*sqrt(a + c*x**4)/14, Ne(c, 0)), (sqrt(a)*x**12/12, True))

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Giac [A]
time = 0.57, size = 43, normalized size = 0.73 \begin {gather*} \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}}{210 \, c^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11*(c*x^4+a)^(1/2),x, algorithm="giac")

[Out]

1/210*(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^3

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Mupad [B]
time = 1.17, size = 44, normalized size = 0.75 \begin {gather*} \sqrt {c\,x^4+a}\,\left (\frac {x^{12}}{14}+\frac {4\,a^3}{105\,c^3}+\frac {a\,x^8}{70\,c}-\frac {2\,a^2\,x^4}{105\,c^2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^11*(a + c*x^4)^(1/2),x)

[Out]

(a + c*x^4)^(1/2)*(x^12/14 + (4*a^3)/(105*c^3) + (a*x^8)/(70*c) - (2*a^2*x^4)/(105*c^2))

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