3.794 \(\int \frac{(a+c x^4)^{3/2}}{x^{15}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0107889, antiderivative size = 44, normalized size of antiderivative = 1., 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 = {271, 264} \[ \frac{c \left (a+c x^4\right )^{5/2}}{35 a^2 x^{10}}-\frac{\left (a+c x^4\right )^{5/2}}{14 a x^{14}} \]

Antiderivative was successfully verified.

[In]

Int[(a + c*x^4)^(3/2)/x^15,x]

[Out]

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

Rule 271

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]

Rule 264

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^{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*}

Mathematica [A]  time = 0.0110097, size = 31, normalized size = 0.7 \[ \frac{\left (a+c x^4\right )^{5/2} \left (2 c x^4-5 a\right )}{70 a^2 x^{14}} \]

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 = 0.003, size = 28, normalized size = 0.6 \begin{align*} -{\frac{-2\,c{x}^{4}+5\,a}{70\,{x}^{14}{a}^{2}} \left ( c{x}^{4}+a \right ) ^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.958227, size = 47, normalized size = 1.07 \begin{align*} \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{align*}

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 = 1.54392, size = 108, normalized size = 2.45 \begin{align*} \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{align*}

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]  time = 4.87371, size = 92, normalized size = 2.09 \begin{align*} - \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{align*}

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]  time = 1.1297, size = 105, normalized size = 2.39 \begin{align*} -\frac{\frac{7 \,{\left (3 \,{\left (c + \frac{a}{x^{4}}\right )}^{\frac{5}{2}} - 5 \,{\left (c + \frac{a}{x^{4}}\right )}^{\frac{3}{2}} c\right )} c}{a} + \frac{15 \,{\left (c + \frac{a}{x^{4}}\right )}^{\frac{7}{2}} - 42 \,{\left (c + \frac{a}{x^{4}}\right )}^{\frac{5}{2}} c + 35 \,{\left (c + \frac{a}{x^{4}}\right )}^{\frac{3}{2}} c^{2}}{a}}{210 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/210*(7*(3*(c + a/x^4)^(5/2) - 5*(c + a/x^4)^(3/2)*c)*c/a + (15*(c + a/x^4)^(7/2) - 42*(c + a/x^4)^(5/2)*c +
35*(c + a/x^4)^(3/2)*c^2)/a)/a