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]

-1/14*(c*x^4+a)^(3/2)/a/x^14+2/35*c*(c*x^4+a)^(3/2)/a^2/x^10-4/105*c^2*(c*x^4+a)^(3/2)/a^3/x^6

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 68, normalized size of antiderivative = 1.00, 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 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]

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]

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 = 0.01, 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]

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

________________________________________________________________________________________

fricas [A]  time = 0.63, size = 49, normalized size = 0.72 \[ -\frac {{\left (8 \, c^{3} x^{12} - 4 \, a c^{2} x^{8} + 3 \, a^{2} c x^{4} + 15 \, a^{3}\right )} \sqrt {c x^{4} + a}}{210 \, a^{3} x^{14}} \]

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)

________________________________________________________________________________________

giac [B]  time = 0.20, size = 148, normalized size = 2.18 \[ \frac {8 \, {\left (70 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + a}\right )}^{8} c^{\frac {7}{2}} + 35 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + a}\right )}^{6} a c^{\frac {7}{2}} + 21 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + a}\right )}^{4} a^{2} c^{\frac {7}{2}} - 7 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + a}\right )}^{2} a^{3} c^{\frac {7}{2}} + a^{4} c^{\frac {7}{2}}\right )}}{105 \, {\left ({\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + a}\right )}^{2} - a\right )}^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

8/105*(70*(sqrt(c)*x^2 - sqrt(c*x^4 + a))^8*c^(7/2) + 35*(sqrt(c)*x^2 - sqrt(c*x^4 + a))^6*a*c^(7/2) + 21*(sqr
t(c)*x^2 - sqrt(c*x^4 + a))^4*a^2*c^(7/2) - 7*(sqrt(c)*x^2 - sqrt(c*x^4 + a))^2*a^3*c^(7/2) + a^4*c^(7/2))/((s
qrt(c)*x^2 - sqrt(c*x^4 + a))^2 - a)^7

________________________________________________________________________________________

maple [A]  time = 0.00, size = 39, normalized size = 0.57 \[ -\frac {\left (c \,x^{4}+a \right )^{\frac {3}{2}} \left (8 c^{2} x^{8}-12 a c \,x^{4}+15 a^{2}\right )}{210 a^{3} x^{14}} \]

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

________________________________________________________________________________________

maxima [A]  time = 1.40, size = 52, normalized size = 0.76 \[ -\frac {\frac {35 \, {\left (c x^{4} + a\right )}^{\frac {3}{2}} c^{2}}{x^{6}} - \frac {42 \, {\left (c x^{4} + a\right )}^{\frac {5}{2}} c}{x^{10}} + \frac {15 \, {\left (c x^{4} + a\right )}^{\frac {7}{2}}}{x^{14}}}{210 \, a^{3}} \]

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

________________________________________________________________________________________

mupad [B]  time = 1.51, size = 73, normalized size = 1.07 \[ \frac {2\,c^2\,\sqrt {c\,x^4+a}}{105\,a^2\,x^6}-\frac {c\,\sqrt {c\,x^4+a}}{70\,a\,x^{10}}-\frac {4\,c^3\,\sqrt {c\,x^4+a}}{105\,a^3\,x^2}-\frac {\sqrt {c\,x^4+a}}{14\,x^{14}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + c*x^4)^(1/2)/x^15,x)

[Out]

(2*c^2*(a + c*x^4)^(1/2))/(105*a^2*x^6) - (c*(a + c*x^4)^(1/2))/(70*a*x^10) - (4*c^3*(a + c*x^4)^(1/2))/(105*a
^3*x^2) - (a + c*x^4)^(1/2)/(14*x^14)

________________________________________________________________________________________

sympy [B]  time = 3.43, size = 359, normalized size = 5.28 \[ - \frac {15 a^{5} c^{\frac {9}{2}} \sqrt {\frac {a}{c x^{4}} + 1}}{210 a^{5} c^{4} x^{12} + 420 a^{4} c^{5} x^{16} + 210 a^{3} c^{6} x^{20}} - \frac {33 a^{4} c^{\frac {11}{2}} x^{4} \sqrt {\frac {a}{c x^{4}} + 1}}{210 a^{5} c^{4} x^{12} + 420 a^{4} c^{5} x^{16} + 210 a^{3} c^{6} x^{20}} - \frac {17 a^{3} c^{\frac {13}{2}} x^{8} \sqrt {\frac {a}{c x^{4}} + 1}}{210 a^{5} c^{4} x^{12} + 420 a^{4} c^{5} x^{16} + 210 a^{3} c^{6} x^{20}} - \frac {3 a^{2} c^{\frac {15}{2}} x^{12} \sqrt {\frac {a}{c x^{4}} + 1}}{210 a^{5} c^{4} x^{12} + 420 a^{4} c^{5} x^{16} + 210 a^{3} c^{6} x^{20}} - \frac {12 a c^{\frac {17}{2}} x^{16} \sqrt {\frac {a}{c x^{4}} + 1}}{210 a^{5} c^{4} x^{12} + 420 a^{4} c^{5} x^{16} + 210 a^{3} c^{6} x^{20}} - \frac {8 c^{\frac {19}{2}} x^{20} \sqrt {\frac {a}{c x^{4}} + 1}}{210 a^{5} c^{4} x^{12} + 420 a^{4} c^{5} x^{16} + 210 a^{3} c^{6} x^{20}} \]

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)

________________________________________________________________________________________