3.8.74 \(\int \frac {\sqrt {a+c x^4}}{x^{11}} \, dx\) [774]

Optimal. Leaf size=44 \[ -\frac {\left (a+c x^4\right )^{3/2}}{10 a x^{10}}+\frac {c \left (a+c x^4\right )^{3/2}}{15 a^2 x^6} \]

[Out]

-1/10*(c*x^4+a)^(3/2)/a/x^10+1/15*c*(c*x^4+a)^(3/2)/a^2/x^6

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Rubi [A]
time = 0.01, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {277, 270} \begin {gather*} \frac {c \left (a+c x^4\right )^{3/2}}{15 a^2 x^6}-\frac {\left (a+c x^4\right )^{3/2}}{10 a x^{10}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + c*x^4]/x^11,x]

[Out]

-1/10*(a + c*x^4)^(3/2)/(a*x^10) + (c*(a + c*x^4)^(3/2))/(15*a^2*x^6)

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*
c*(m + 1))), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rule 277

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x^(m + 1)*((a + b*x^n)^(p + 1)/(a*(m + 1))), x]
- Dist[b*((m + n*(p + 1) + 1)/(a*(m + 1))), Int[x^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, m, n, p}, x]
&& ILtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]

Rubi steps

\begin {align*} \int \frac {\sqrt {a+c x^4}}{x^{11}} \, dx &=-\frac {\left (a+c x^4\right )^{3/2}}{10 a x^{10}}-\frac {(2 c) \int \frac {\sqrt {a+c x^4}}{x^7} \, dx}{5 a}\\ &=-\frac {\left (a+c x^4\right )^{3/2}}{10 a x^{10}}+\frac {c \left (a+c x^4\right )^{3/2}}{15 a^2 x^6}\\ \end {align*}

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Mathematica [A]
time = 0.15, size = 42, normalized size = 0.95 \begin {gather*} \frac {\sqrt {a+c x^4} \left (-3 a^2-a c x^4+2 c^2 x^8\right )}{30 a^2 x^{10}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + c*x^4]/x^11,x]

[Out]

(Sqrt[a + c*x^4]*(-3*a^2 - a*c*x^4 + 2*c^2*x^8))/(30*a^2*x^10)

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Maple [A]
time = 0.16, size = 28, normalized size = 0.64

method	result	size
gosper	\(-\frac {\left (x^{4} c +a \right )^{\frac {3}{2}} \left (-2 x^{4} c +3 a \right )}{30 x^{10} a^{2}}\)	\(28\)
default	\(-\frac {\left (x^{4} c +a \right )^{\frac {3}{2}} \left (-2 x^{4} c +3 a \right )}{30 x^{10} a^{2}}\)	\(28\)
elliptic	\(-\frac {\left (x^{4} c +a \right )^{\frac {3}{2}} \left (-2 x^{4} c +3 a \right )}{30 x^{10} a^{2}}\)	\(28\)
trager	\(-\frac {\left (-2 c^{2} x^{8}+a c \,x^{4}+3 a^{2}\right ) \sqrt {x^{4} c +a}}{30 x^{10} a^{2}}\)	\(38\)
risch	\(-\frac {\left (-2 c^{2} x^{8}+a c \,x^{4}+3 a^{2}\right ) \sqrt {x^{4} c +a}}{30 x^{10} a^{2}}\)	\(38\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/30*(c*x^4+a)^(3/2)*(-2*c*x^4+3*a)/x^10/a^2

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Maxima [A]
time = 0.29, size = 35, normalized size = 0.80 \begin {gather*} \frac {\frac {5 \, {\left (c x^{4} + a\right )}^{\frac {3}{2}} c}{x^{6}} - \frac {3 \, {\left (c x^{4} + a\right )}^{\frac {5}{2}}}{x^{10}}}{30 \, a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/30*(5*(c*x^4 + a)^(3/2)*c/x^6 - 3*(c*x^4 + a)^(5/2)/x^10)/a^2

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Fricas [A]
time = 0.35, size = 38, normalized size = 0.86 \begin {gather*} \frac {{\left (2 \, c^{2} x^{8} - a c x^{4} - 3 \, a^{2}\right )} \sqrt {c x^{4} + a}}{30 \, a^{2} x^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/30*(2*c^2*x^8 - a*c*x^4 - 3*a^2)*sqrt(c*x^4 + a)/(a^2*x^10)

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Sympy [A]
time = 0.61, size = 66, normalized size = 1.50 \begin {gather*} - \frac {\sqrt {c} \sqrt {\frac {a}{c x^{4}} + 1}}{10 x^{8}} - \frac {c^{\frac {3}{2}} \sqrt {\frac {a}{c x^{4}} + 1}}{30 a x^{4}} + \frac {c^{\frac {5}{2}} \sqrt {\frac {a}{c x^{4}} + 1}}{15 a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(c)*sqrt(a/(c*x**4) + 1)/(10*x**8) - c**(3/2)*sqrt(a/(c*x**4) + 1)/(30*a*x**4) + c**(5/2)*sqrt(a/(c*x**4)
+ 1)/(15*a**2)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 120 vs. \(2 (36) = 72\).
time = 0.55, size = 120, normalized size = 2.73 \begin {gather*} \frac {2 \, {\left (15 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + a}\right )}^{6} c^{\frac {5}{2}} + 5 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + a}\right )}^{4} a c^{\frac {5}{2}} + 5 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + a}\right )}^{2} a^{2} c^{\frac {5}{2}} - a^{3} c^{\frac {5}{2}}\right )}}{15 \, {\left ({\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + a}\right )}^{2} - a\right )}^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*(15*(sqrt(c)*x^2 - sqrt(c*x^4 + a))^6*c^(5/2) + 5*(sqrt(c)*x^2 - sqrt(c*x^4 + a))^4*a*c^(5/2) + 5*(sqrt(c
)*x^2 - sqrt(c*x^4 + a))^2*a^2*c^(5/2) - a^3*c^(5/2))/((sqrt(c)*x^2 - sqrt(c*x^4 + a))^2 - a)^5

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Mupad [B]
time = 1.32, size = 37, normalized size = 0.84 \begin {gather*} -\frac {\sqrt {c\,x^4+a}\,\left (3\,a^2+a\,c\,x^4-2\,c^2\,x^8\right )}{30\,a^2\,x^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + c*x^4)^(1/2)/x^11,x)

[Out]

-((a + c*x^4)^(1/2)*(3*a^2 - 2*c^2*x^8 + a*c*x^4))/(30*a^2*x^10)

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