3.1139 \(\int \frac{x^{11}}{\left (a+b x^4\right )^{5/4}} \, dx\)

Optimal. Leaf size=57 \[ -\frac{a^2}{b^3 \sqrt [4]{a+b x^4}}-\frac{2 a \left (a+b x^4\right )^{3/4}}{3 b^3}+\frac{\left (a+b x^4\right )^{7/4}}{7 b^3} \]

[Out]

-(a^2/(b^3*(a + b*x^4)^(1/4))) - (2*a*(a + b*x^4)^(3/4))/(3*b^3) + (a + b*x^4)^(
7/4)/(7*b^3)

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Rubi [A]  time = 0.0839911, antiderivative size = 57, 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 \[ -\frac{a^2}{b^3 \sqrt [4]{a+b x^4}}-\frac{2 a \left (a+b x^4\right )^{3/4}}{3 b^3}+\frac{\left (a+b x^4\right )^{7/4}}{7 b^3} \]

Antiderivative was successfully verified.

[In]  Int[x^11/(a + b*x^4)^(5/4),x]

[Out]

-(a^2/(b^3*(a + b*x^4)^(1/4))) - (2*a*(a + b*x^4)^(3/4))/(3*b^3) + (a + b*x^4)^(
7/4)/(7*b^3)

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Rubi in Sympy [A]  time = 10.5703, size = 49, normalized size = 0.86 \[ - \frac{a^{2}}{b^{3} \sqrt [4]{a + b x^{4}}} - \frac{2 a \left (a + b x^{4}\right )^{\frac{3}{4}}}{3 b^{3}} + \frac{\left (a + b x^{4}\right )^{\frac{7}{4}}}{7 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**11/(b*x**4+a)**(5/4),x)

[Out]

-a**2/(b**3*(a + b*x**4)**(1/4)) - 2*a*(a + b*x**4)**(3/4)/(3*b**3) + (a + b*x**
4)**(7/4)/(7*b**3)

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Mathematica [A]  time = 0.0331995, size = 39, normalized size = 0.68 \[ \frac{-32 a^2-8 a b x^4+3 b^2 x^8}{21 b^3 \sqrt [4]{a+b x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^11/(a + b*x^4)^(5/4),x]

[Out]

(-32*a^2 - 8*a*b*x^4 + 3*b^2*x^8)/(21*b^3*(a + b*x^4)^(1/4))

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Maple [A]  time = 0.008, size = 36, normalized size = 0.6 \[ -{\frac{-3\,{b}^{2}{x}^{8}+8\,ab{x}^{4}+32\,{a}^{2}}{21\,{b}^{3}}{\frac{1}{\sqrt [4]{b{x}^{4}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^11/(b*x^4+a)^(5/4),x)

[Out]

-1/21*(-3*b^2*x^8+8*a*b*x^4+32*a^2)/(b*x^4+a)^(1/4)/b^3

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Maxima [A]  time = 1.41638, size = 63, normalized size = 1.11 \[ \frac{{\left (b x^{4} + a\right )}^{\frac{7}{4}}}{7 \, b^{3}} - \frac{2 \,{\left (b x^{4} + a\right )}^{\frac{3}{4}} a}{3 \, b^{3}} - \frac{a^{2}}{{\left (b x^{4} + a\right )}^{\frac{1}{4}} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^11/(b*x^4 + a)^(5/4),x, algorithm="maxima")

[Out]

1/7*(b*x^4 + a)^(7/4)/b^3 - 2/3*(b*x^4 + a)^(3/4)*a/b^3 - a^2/((b*x^4 + a)^(1/4)
*b^3)

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Fricas [A]  time = 0.231002, size = 47, normalized size = 0.82 \[ \frac{3 \, b^{2} x^{8} - 8 \, a b x^{4} - 32 \, a^{2}}{21 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^11/(b*x^4 + a)^(5/4),x, algorithm="fricas")

[Out]

1/21*(3*b^2*x^8 - 8*a*b*x^4 - 32*a^2)/((b*x^4 + a)^(1/4)*b^3)

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Sympy [A]  time = 10.2773, size = 68, normalized size = 1.19 \[ \begin{cases} - \frac{32 a^{2}}{21 b^{3} \sqrt [4]{a + b x^{4}}} - \frac{8 a x^{4}}{21 b^{2} \sqrt [4]{a + b x^{4}}} + \frac{x^{8}}{7 b \sqrt [4]{a + b x^{4}}} & \text{for}\: b \neq 0 \\\frac{x^{12}}{12 a^{\frac{5}{4}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**11/(b*x**4+a)**(5/4),x)

[Out]

Piecewise((-32*a**2/(21*b**3*(a + b*x**4)**(1/4)) - 8*a*x**4/(21*b**2*(a + b*x**
4)**(1/4)) + x**8/(7*b*(a + b*x**4)**(1/4)), Ne(b, 0)), (x**12/(12*a**(5/4)), Tr
ue))

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GIAC/XCAS [A]  time = 0.214441, size = 58, normalized size = 1.02 \[ \frac{3 \,{\left (b x^{4} + a\right )}^{\frac{7}{4}} - 14 \,{\left (b x^{4} + a\right )}^{\frac{3}{4}} a - \frac{21 \, a^{2}}{{\left (b x^{4} + a\right )}^{\frac{1}{4}}}}{21 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^11/(b*x^4 + a)^(5/4),x, algorithm="giac")

[Out]

1/21*(3*(b*x^4 + a)^(7/4) - 14*(b*x^4 + a)^(3/4)*a - 21*a^2/(b*x^4 + a)^(1/4))/b
^3