3.8.94 \(\int \frac {(a+c x^4)^{3/2}}{x^{15}} \, dx\) [794]

Optimal. Leaf size=44 \[ -\frac {\left (a+c x^4\right )^{5/2}}{14 a x^{14}}+\frac {c \left (a+c x^4\right )^{5/2}}{35 a^2 x^{10}} \]

[Out]

-1/14*(c*x^4+a)^(5/2)/a/x^14+1/35*c*(c*x^4+a)^(5/2)/a^2/x^10

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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 )^{5/2}}{35 a^2 x^{10}}-\frac {\left (a+c x^4\right )^{5/2}}{14 a x^{14}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + c*x^4)^(3/2)/x^15,x]

[Out]

-1/14*(a + c*x^4)^(5/2)/(a*x^14) + (c*(a + c*x^4)^(5/2))/(35*a^2*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]

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 {\left (a+c x^4\right )^{3/2}}{x^{15}} \, dx &=-\frac {\left (a+c x^4\right )^{5/2}}{14 a x^{14}}-\frac {(2 c) \int \frac {\left (a+c x^4\right )^{3/2}}{x^{11}} \, dx}{7 a}\\ &=-\frac {\left (a+c x^4\right )^{5/2}}{14 a x^{14}}+\frac {c \left (a+c x^4\right )^{5/2}}{35 a^2 x^{10}}\\ \end {align*}

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Mathematica [A]
time = 0.15, size = 31, normalized size = 0.70 \begin {gather*} \frac {\left (a+c x^4\right )^{5/2} \left (-5 a+2 c x^4\right )}{70 a^2 x^{14}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + c*x^4)^(3/2)/x^15,x]

[Out]

((a + c*x^4)^(5/2)*(-5*a + 2*c*x^4))/(70*a^2*x^14)

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Maple [A]
time = 0.15, size = 46, normalized size = 1.05

method	result	size
gosper	\(-\frac {\left (x^{4} c +a \right )^{\frac {5}{2}} \left (-2 x^{4} c +5 a \right )}{70 x^{14} a^{2}}\)	\(28\)
default	\(-\frac {\sqrt {x^{4} c +a}\, \left (-2 x^{4} c +5 a \right ) \left (c^{2} x^{8}+2 a c \,x^{4}+a^{2}\right )}{70 a^{2} x^{14}}\)	\(46\)
elliptic	\(-\frac {\sqrt {x^{4} c +a}\, \left (-2 x^{4} c +5 a \right ) \left (c^{2} x^{8}+2 a c \,x^{4}+a^{2}\right )}{70 a^{2} x^{14}}\)	\(46\)
trager	\(-\frac {\left (-2 c^{3} x^{12}+a \,c^{2} x^{8}+8 a^{2} c \,x^{4}+5 a^{3}\right ) \sqrt {x^{4} c +a}}{70 x^{14} a^{2}}\)	\(49\)
risch	\(-\frac {\left (-2 c^{3} x^{12}+a \,c^{2} x^{8}+8 a^{2} c \,x^{4}+5 a^{3}\right ) \sqrt {x^{4} c +a}}{70 x^{14} a^{2}}\)	\(49\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+a)^(3/2)/x^15,x,method=_RETURNVERBOSE)

[Out]

-1/70*(c*x^4+a)^(1/2)*(-2*c*x^4+5*a)*(c^2*x^8+2*a*c*x^4+a^2)/a^2/x^14

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Maxima [A]
time = 0.30, size = 35, normalized size = 0.80 \begin {gather*} \frac {\frac {7 \, {\left (c x^{4} + a\right )}^{\frac {5}{2}} c}{x^{10}} - \frac {5 \, {\left (c x^{4} + a\right )}^{\frac {7}{2}}}{x^{14}}}{70 \, a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+a)^(3/2)/x^15,x, algorithm="maxima")

[Out]

1/70*(7*(c*x^4 + a)^(5/2)*c/x^10 - 5*(c*x^4 + a)^(7/2)/x^14)/a^2

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Fricas [A]
time = 0.37, size = 49, normalized size = 1.11 \begin {gather*} \frac {{\left (2 \, c^{3} x^{12} - a c^{2} x^{8} - 8 \, a^{2} c x^{4} - 5 \, a^{3}\right )} \sqrt {c x^{4} + a}}{70 \, a^{2} x^{14}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+a)^(3/2)/x^15,x, algorithm="fricas")

[Out]

1/70*(2*c^3*x^12 - a*c^2*x^8 - 8*a^2*c*x^4 - 5*a^3)*sqrt(c*x^4 + a)/(a^2*x^14)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 92 vs. \(2 (36) = 72\).
time = 1.05, size = 92, normalized size = 2.09 \begin {gather*} - \frac {a \sqrt {c} \sqrt {\frac {a}{c x^{4}} + 1}}{14 x^{12}} - \frac {4 c^{\frac {3}{2}} \sqrt {\frac {a}{c x^{4}} + 1}}{35 x^{8}} - \frac {c^{\frac {5}{2}} \sqrt {\frac {a}{c x^{4}} + 1}}{70 a x^{4}} + \frac {c^{\frac {7}{2}} \sqrt {\frac {a}{c x^{4}} + 1}}{35 a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+a)**(3/2)/x**15,x)

[Out]

-a*sqrt(c)*sqrt(a/(c*x**4) + 1)/(14*x**12) - 4*c**(3/2)*sqrt(a/(c*x**4) + 1)/(35*x**8) - c**(5/2)*sqrt(a/(c*x*
*4) + 1)/(70*a*x**4) + c**(7/2)*sqrt(a/(c*x**4) + 1)/(35*a**2)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 178 vs. \(2 (36) = 72\).
time = 1.01, size = 178, normalized size = 4.05 \begin {gather*} \frac {2 \, {\left (35 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + a}\right )}^{10} c^{\frac {7}{2}} + 35 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + a}\right )}^{8} a c^{\frac {7}{2}} + 70 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + a}\right )}^{6} a^{2} c^{\frac {7}{2}} + 14 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + a}\right )}^{4} a^{3} c^{\frac {7}{2}} + 7 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + a}\right )}^{2} a^{4} c^{\frac {7}{2}} - a^{5} c^{\frac {7}{2}}\right )}}{35 \, {\left ({\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + a}\right )}^{2} - a\right )}^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+a)^(3/2)/x^15,x, algorithm="giac")

[Out]

2/35*(35*(sqrt(c)*x^2 - sqrt(c*x^4 + a))^10*c^(7/2) + 35*(sqrt(c)*x^2 - sqrt(c*x^4 + a))^8*a*c^(7/2) + 70*(sqr
t(c)*x^2 - sqrt(c*x^4 + a))^6*a^2*c^(7/2) + 14*(sqrt(c)*x^2 - sqrt(c*x^4 + a))^4*a^3*c^(7/2) + 7*(sqrt(c)*x^2
- sqrt(c*x^4 + a))^2*a^4*c^(7/2) - a^5*c^(7/2))/((sqrt(c)*x^2 - sqrt(c*x^4 + a))^2 - a)^7

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Mupad [B]
time = 1.76, size = 71, normalized size = 1.61 \begin {gather*} \frac {c^3\,\sqrt {c\,x^4+a}}{35\,a^2\,x^2}-\frac {4\,c\,\sqrt {c\,x^4+a}}{35\,x^{10}}-\frac {a\,\sqrt {c\,x^4+a}}{14\,x^{14}}-\frac {c^2\,\sqrt {c\,x^4+a}}{70\,a\,x^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + c*x^4)^(3/2)/x^15,x)

[Out]

(c^3*(a + c*x^4)^(1/2))/(35*a^2*x^2) - (4*c*(a + c*x^4)^(1/2))/(35*x^10) - (a*(a + c*x^4)^(1/2))/(14*x^14) - (
c^2*(a + c*x^4)^(1/2))/(70*a*x^6)

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