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

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 = 0.0614255, 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 \[ -\frac{4 c^2 \left (a+c x^4\right )^{3/2}}{105 a^3 x^6}+\frac{2 c \left (a+c x^4\right )^{3/2}}{35 a^2 x^{10}}-\frac{\left (a+c x^4\right )^{3/2}}{14 a x^{14}} \]

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)

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Rubi in Sympy [A]  time = 6.74217, size = 61, normalized size = 0.9 \[ - \frac{\left (a + c x^{4}\right )^{\frac{3}{2}}}{14 a x^{14}} + \frac{2 c \left (a + c x^{4}\right )^{\frac{3}{2}}}{35 a^{2} x^{10}} - \frac{4 c^{2} \left (a + c x^{4}\right )^{\frac{3}{2}}}{105 a^{3} x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**4+a)**(1/2)/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)

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Mathematica [A]  time = 0.029864, size = 53, normalized size = 0.78 \[ -\frac{\sqrt{a+c x^4} \left (15 a^3+3 a^2 c x^4-4 a c^2 x^8+8 c^3 x^{12}\right )}{210 a^3 x^{14}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a + c*x^4]/x^15,x]

[Out]

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

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Maple [A]  time = 0.009, size = 39, normalized size = 0.6 \[ -{\frac{8\,{c}^{2}{x}^{8}-12\,c{x}^{4}a+15\,{a}^{2}}{210\,{x}^{14}{a}^{3}} \left ( c{x}^{4}+a \right ) ^{{\frac{3}{2}}}} \]

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 = 1.45509, size = 70, normalized size = 1.03 \[ -\frac{\frac{35 \,{\left (c x^{4} + a\right )}^{\frac{3}{2}} c^{2}}{x^{6}} - \frac{42 \,{\left (c x^{4} + a\right )}^{\frac{5}{2}} c}{x^{10}} + \frac{15 \,{\left (c x^{4} + a\right )}^{\frac{7}{2}}}{x^{14}}}{210 \, a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^4 + a)/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 = 0.296953, size = 66, normalized size = 0.97 \[ -\frac{{\left (8 \, c^{3} x^{12} - 4 \, a c^{2} x^{8} + 3 \, a^{2} c x^{4} + 15 \, a^{3}\right )} \sqrt{c x^{4} + a}}{210 \, a^{3} x^{14}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^4 + a)/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 [A]  time = 13.4143, 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**1
6 + 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) - 17*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 + 42
0*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**(1
9/2)*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/XCAS [A]  time = 0.218101, size = 58, normalized size = 0.85 \[ -\frac{15 \,{\left (c + \frac{a}{x^{4}}\right )}^{\frac{7}{2}} - 42 \,{\left (c + \frac{a}{x^{4}}\right )}^{\frac{5}{2}} c + 35 \,{\left (c + \frac{a}{x^{4}}\right )}^{\frac{3}{2}} c^{2}}{210 \, a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^4 + a)/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