3.8.93 \(\int \frac {(a+c x^4)^{3/2}}{x^{11}} \, dx\) [793]

Optimal. Leaf size=21 \[ -\frac {\left (a+c x^4\right )^{5/2}}{10 a x^{10}} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {270} \begin {gather*} -\frac {\left (a+c x^4\right )^{5/2}}{10 a x^{10}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + c*x^4)^(3/2)/x^11,x]

[Out]

-1/10*(a + c*x^4)^(5/2)/(a*x^10)

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]

Rubi steps

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

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Mathematica [A]
time = 0.12, size = 21, normalized size = 1.00 \begin {gather*} -\frac {\left (a+c x^4\right )^{5/2}}{10 a x^{10}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + c*x^4)^(3/2)/x^11,x]

[Out]

-1/10*(a + c*x^4)^(5/2)/(a*x^10)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(35\) vs. \(2(17)=34\).
time = 0.15, size = 36, normalized size = 1.71

method	result	size
gosper	\(-\frac {\left (x^{4} c +a \right )^{\frac {5}{2}}}{10 a \,x^{10}}\)	\(18\)
default	\(-\frac {\sqrt {x^{4} c +a}\, \left (c^{2} x^{8}+2 a c \,x^{4}+a^{2}\right )}{10 x^{10} a}\)	\(36\)
trager	\(-\frac {\sqrt {x^{4} c +a}\, \left (c^{2} x^{8}+2 a c \,x^{4}+a^{2}\right )}{10 x^{10} a}\)	\(36\)
risch	\(-\frac {\sqrt {x^{4} c +a}\, \left (c^{2} x^{8}+2 a c \,x^{4}+a^{2}\right )}{10 x^{10} a}\)	\(36\)
elliptic	\(-\frac {\sqrt {x^{4} c +a}\, \left (c^{2} x^{8}+2 a c \,x^{4}+a^{2}\right )}{10 x^{10} a}\)	\(36\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+a)^(3/2)/x^11,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.31, size = 17, normalized size = 0.81 \begin {gather*} -\frac {{\left (c x^{4} + a\right )}^{\frac {5}{2}}}{10 \, a x^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+a)^(3/2)/x^11,x, algorithm="maxima")

[Out]

-1/10*(c*x^4 + a)^(5/2)/(a*x^10)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 35 vs. \(2 (17) = 34\).
time = 0.37, size = 35, normalized size = 1.67 \begin {gather*} -\frac {{\left (c^{2} x^{8} + 2 \, a c x^{4} + a^{2}\right )} \sqrt {c x^{4} + a}}{10 \, a x^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+a)^(3/2)/x^11,x, algorithm="fricas")

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (17) = 34\).
time = 0.67, size = 66, normalized size = 3.14 \begin {gather*} - \frac {a \sqrt {c} \sqrt {\frac {a}{c x^{4}} + 1}}{10 x^{8}} - \frac {c^{\frac {3}{2}} \sqrt {\frac {a}{c x^{4}} + 1}}{5 x^{4}} - \frac {c^{\frac {5}{2}} \sqrt {\frac {a}{c x^{4}} + 1}}{10 a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+a)**(3/2)/x**11,x)

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 92 vs. \(2 (17) = 34\).
time = 0.71, size = 92, normalized size = 4.38 \begin {gather*} \frac {5 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + a}\right )}^{8} c^{\frac {5}{2}} + 10 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + a}\right )}^{4} a^{2} c^{\frac {5}{2}} + a^{4} c^{\frac {5}{2}}}{5 \, {\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)^(3/2)/x^11,x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 1.52, size = 17, normalized size = 0.81 \begin {gather*} -\frac {{\left (c\,x^4+a\right )}^{5/2}}{10\,a\,x^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + c*x^4)^(3/2)/x^11,x)

[Out]

-(a + c*x^4)^(5/2)/(10*a*x^10)

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