3.775 \(\int \frac{\sqrt{a+c x^4}}{x^{15}} \, dx\)

Optimal. Leaf size=68 \[ -\frac{4 c^2 \left (a+c x^4\right )^{3/2}}{105 a^3 x^6}+\frac{2 c \left (a+c x^4\right )^{3/2}}{35 a^2 x^{10}}-\frac{\left (a+c x^4\right )^{3/2}}{14 a x^{14}} \]

[Out]

-(a + c*x^4)^(3/2)/(14*a*x^14) + (2*c*(a + c*x^4)^(3/2))/(35*a^2*x^10) - (4*c^2*(a + c*x^4)^(3/2))/(105*a^3*x^
6)

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Rubi [A]  time = 0.0185057, antiderivative size = 68, normalized size of antiderivative = 1., 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 = {271, 264} \[ -\frac{4 c^2 \left (a+c x^4\right )^{3/2}}{105 a^3 x^6}+\frac{2 c \left (a+c x^4\right )^{3/2}}{35 a^2 x^{10}}-\frac{\left (a+c x^4\right )^{3/2}}{14 a x^{14}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + c*x^4]/x^15,x]

[Out]

-(a + c*x^4)^(3/2)/(14*a*x^14) + (2*c*(a + c*x^4)^(3/2))/(35*a^2*x^10) - (4*c^2*(a + c*x^4)^(3/2))/(105*a^3*x^
6)

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{\sqrt{a+c x^4}}{x^{15}} \, dx &=-\frac{\left (a+c x^4\right )^{3/2}}{14 a x^{14}}-\frac{(4 c) \int \frac{\sqrt{a+c x^4}}{x^{11}} \, dx}{7 a}\\ &=-\frac{\left (a+c x^4\right )^{3/2}}{14 a x^{14}}+\frac{2 c \left (a+c x^4\right )^{3/2}}{35 a^2 x^{10}}+\frac{\left (8 c^2\right ) \int \frac{\sqrt{a+c x^4}}{x^7} \, dx}{35 a^2}\\ &=-\frac{\left (a+c x^4\right )^{3/2}}{14 a x^{14}}+\frac{2 c \left (a+c x^4\right )^{3/2}}{35 a^2 x^{10}}-\frac{4 c^2 \left (a+c x^4\right )^{3/2}}{105 a^3 x^6}\\ \end{align*}

Mathematica [A]  time = 0.0106359, size = 42, normalized size = 0.62 \[ -\frac{\left (a+c x^4\right )^{3/2} \left (15 a^2-12 a c x^4+8 c^2 x^8\right )}{210 a^3 x^{14}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + c*x^4]/x^15,x]

[Out]

-((a + c*x^4)^(3/2)*(15*a^2 - 12*a*c*x^4 + 8*c^2*x^8))/(210*a^3*x^14)

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Maple [A]  time = 0.004, size = 39, normalized size = 0.6 \begin{align*} -{\frac{8\,{c}^{2}{x}^{8}-12\,c{x}^{4}a+15\,{a}^{2}}{210\,{x}^{14}{a}^{3}} \left ( c{x}^{4}+a \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+a)^(1/2)/x^15,x)

[Out]

-1/210*(c*x^4+a)^(3/2)*(8*c^2*x^8-12*a*c*x^4+15*a^2)/x^14/a^3

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Maxima [A]  time = 0.954035, size = 70, normalized size = 1.03 \begin{align*} -\frac{\frac{35 \,{\left (c x^{4} + a\right )}^{\frac{3}{2}} c^{2}}{x^{6}} - \frac{42 \,{\left (c x^{4} + a\right )}^{\frac{5}{2}} c}{x^{10}} + \frac{15 \,{\left (c x^{4} + a\right )}^{\frac{7}{2}}}{x^{14}}}{210 \, a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+a)^(1/2)/x^15,x, algorithm="maxima")

[Out]

-1/210*(35*(c*x^4 + a)^(3/2)*c^2/x^6 - 42*(c*x^4 + a)^(5/2)*c/x^10 + 15*(c*x^4 + a)^(7/2)/x^14)/a^3

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Fricas [A]  time = 1.89919, size = 115, normalized size = 1.69 \begin{align*} -\frac{{\left (8 \, c^{3} x^{12} - 4 \, a c^{2} x^{8} + 3 \, a^{2} c x^{4} + 15 \, a^{3}\right )} \sqrt{c x^{4} + a}}{210 \, a^{3} x^{14}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+a)^(1/2)/x^15,x, algorithm="fricas")

[Out]

-1/210*(8*c^3*x^12 - 4*a*c^2*x^8 + 3*a^2*c*x^4 + 15*a^3)*sqrt(c*x^4 + a)/(a^3*x^14)

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Sympy [B]  time = 3.98208, size = 359, normalized size = 5.28 \begin{align*} - \frac{15 a^{5} c^{\frac{9}{2}} \sqrt{\frac{a}{c x^{4}} + 1}}{210 a^{5} c^{4} x^{12} + 420 a^{4} c^{5} x^{16} + 210 a^{3} c^{6} x^{20}} - \frac{33 a^{4} c^{\frac{11}{2}} x^{4} \sqrt{\frac{a}{c x^{4}} + 1}}{210 a^{5} c^{4} x^{12} + 420 a^{4} c^{5} x^{16} + 210 a^{3} c^{6} x^{20}} - \frac{17 a^{3} c^{\frac{13}{2}} x^{8} \sqrt{\frac{a}{c x^{4}} + 1}}{210 a^{5} c^{4} x^{12} + 420 a^{4} c^{5} x^{16} + 210 a^{3} c^{6} x^{20}} - \frac{3 a^{2} c^{\frac{15}{2}} x^{12} \sqrt{\frac{a}{c x^{4}} + 1}}{210 a^{5} c^{4} x^{12} + 420 a^{4} c^{5} x^{16} + 210 a^{3} c^{6} x^{20}} - \frac{12 a c^{\frac{17}{2}} x^{16} \sqrt{\frac{a}{c x^{4}} + 1}}{210 a^{5} c^{4} x^{12} + 420 a^{4} c^{5} x^{16} + 210 a^{3} c^{6} x^{20}} - \frac{8 c^{\frac{19}{2}} x^{20} \sqrt{\frac{a}{c x^{4}} + 1}}{210 a^{5} c^{4} x^{12} + 420 a^{4} c^{5} x^{16} + 210 a^{3} c^{6} x^{20}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+a)**(1/2)/x**15,x)

[Out]

-15*a**5*c**(9/2)*sqrt(a/(c*x**4) + 1)/(210*a**5*c**4*x**12 + 420*a**4*c**5*x**16 + 210*a**3*c**6*x**20) - 33*
a**4*c**(11/2)*x**4*sqrt(a/(c*x**4) + 1)/(210*a**5*c**4*x**12 + 420*a**4*c**5*x**16 + 210*a**3*c**6*x**20) - 1
7*a**3*c**(13/2)*x**8*sqrt(a/(c*x**4) + 1)/(210*a**5*c**4*x**12 + 420*a**4*c**5*x**16 + 210*a**3*c**6*x**20) -
3*a**2*c**(15/2)*x**12*sqrt(a/(c*x**4) + 1)/(210*a**5*c**4*x**12 + 420*a**4*c**5*x**16 + 210*a**3*c**6*x**20)
- 12*a*c**(17/2)*x**16*sqrt(a/(c*x**4) + 1)/(210*a**5*c**4*x**12 + 420*a**4*c**5*x**16 + 210*a**3*c**6*x**20)
- 8*c**(19/2)*x**20*sqrt(a/(c*x**4) + 1)/(210*a**5*c**4*x**12 + 420*a**4*c**5*x**16 + 210*a**3*c**6*x**20)

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Giac [A]  time = 1.107, size = 58, normalized size = 0.85 \begin{align*} -\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}}{210 \, a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+a)^(1/2)/x^15,x, algorithm="giac")

[Out]

-1/210*(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^3