3.988 \(\int x^{11} \sqrt [4]{a+b x^4} \, dx\)

Optimal. Leaf size=59 \[ \frac{a^2 \left (a+b x^4\right )^{5/4}}{5 b^3}+\frac{\left (a+b x^4\right )^{13/4}}{13 b^3}-\frac{2 a \left (a+b x^4\right )^{9/4}}{9 b^3} \]

[Out]

(a^2*(a + b*x^4)^(5/4))/(5*b^3) - (2*a*(a + b*x^4)^(9/4))/(9*b^3) + (a + b*x^4)^
(13/4)/(13*b^3)

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Rubi [A]  time = 0.0831764, antiderivative size = 59, 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 \[ \frac{a^2 \left (a+b x^4\right )^{5/4}}{5 b^3}+\frac{\left (a+b x^4\right )^{13/4}}{13 b^3}-\frac{2 a \left (a+b x^4\right )^{9/4}}{9 b^3} \]

Antiderivative was successfully verified.

[In]  Int[x^11*(a + b*x^4)^(1/4),x]

[Out]

(a^2*(a + b*x^4)^(5/4))/(5*b^3) - (2*a*(a + b*x^4)^(9/4))/(9*b^3) + (a + b*x^4)^
(13/4)/(13*b^3)

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Rubi in Sympy [A]  time = 10.5631, size = 51, normalized size = 0.86 \[ \frac{a^{2} \left (a + b x^{4}\right )^{\frac{5}{4}}}{5 b^{3}} - \frac{2 a \left (a + b x^{4}\right )^{\frac{9}{4}}}{9 b^{3}} + \frac{\left (a + b x^{4}\right )^{\frac{13}{4}}}{13 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**11*(b*x**4+a)**(1/4),x)

[Out]

a**2*(a + b*x**4)**(5/4)/(5*b**3) - 2*a*(a + b*x**4)**(9/4)/(9*b**3) + (a + b*x*
*4)**(13/4)/(13*b**3)

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Mathematica [A]  time = 0.0249289, size = 50, normalized size = 0.85 \[ \frac{\sqrt [4]{a+b x^4} \left (32 a^3-8 a^2 b x^4+5 a b^2 x^8+45 b^3 x^{12}\right )}{585 b^3} \]

Antiderivative was successfully verified.

[In]  Integrate[x^11*(a + b*x^4)^(1/4),x]

[Out]

((a + b*x^4)^(1/4)*(32*a^3 - 8*a^2*b*x^4 + 5*a*b^2*x^8 + 45*b^3*x^12))/(585*b^3)

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Maple [A]  time = 0.008, size = 36, normalized size = 0.6 \[{\frac{45\,{b}^{2}{x}^{8}-40\,ab{x}^{4}+32\,{a}^{2}}{585\,{b}^{3}} \left ( b{x}^{4}+a \right ) ^{{\frac{5}{4}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^11*(b*x^4+a)^(1/4),x)

[Out]

1/585*(b*x^4+a)^(5/4)*(45*b^2*x^8-40*a*b*x^4+32*a^2)/b^3

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Maxima [A]  time = 1.44974, size = 63, normalized size = 1.07 \[ \frac{{\left (b x^{4} + a\right )}^{\frac{13}{4}}}{13 \, b^{3}} - \frac{2 \,{\left (b x^{4} + a\right )}^{\frac{9}{4}} a}{9 \, b^{3}} + \frac{{\left (b x^{4} + a\right )}^{\frac{5}{4}} a^{2}}{5 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(1/4)*x^11,x, algorithm="maxima")

[Out]

1/13*(b*x^4 + a)^(13/4)/b^3 - 2/9*(b*x^4 + a)^(9/4)*a/b^3 + 1/5*(b*x^4 + a)^(5/4
)*a^2/b^3

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Fricas [A]  time = 0.265969, size = 62, normalized size = 1.05 \[ \frac{{\left (45 \, b^{3} x^{12} + 5 \, a b^{2} x^{8} - 8 \, a^{2} b x^{4} + 32 \, a^{3}\right )}{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{585 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(1/4)*x^11,x, algorithm="fricas")

[Out]

1/585*(45*b^3*x^12 + 5*a*b^2*x^8 - 8*a^2*b*x^4 + 32*a^3)*(b*x^4 + a)^(1/4)/b^3

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Sympy [A]  time = 14.904, size = 87, normalized size = 1.47 \[ \begin{cases} \frac{32 a^{3} \sqrt [4]{a + b x^{4}}}{585 b^{3}} - \frac{8 a^{2} x^{4} \sqrt [4]{a + b x^{4}}}{585 b^{2}} + \frac{a x^{8} \sqrt [4]{a + b x^{4}}}{117 b} + \frac{x^{12} \sqrt [4]{a + b x^{4}}}{13} & \text{for}\: b \neq 0 \\\frac{\sqrt [4]{a} x^{12}}{12} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**11*(b*x**4+a)**(1/4),x)

[Out]

Piecewise((32*a**3*(a + b*x**4)**(1/4)/(585*b**3) - 8*a**2*x**4*(a + b*x**4)**(1
/4)/(585*b**2) + a*x**8*(a + b*x**4)**(1/4)/(117*b) + x**12*(a + b*x**4)**(1/4)/
13, Ne(b, 0)), (a**(1/4)*x**12/12, True))

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GIAC/XCAS [A]  time = 0.215709, size = 58, normalized size = 0.98 \[ \frac{45 \,{\left (b x^{4} + a\right )}^{\frac{13}{4}} - 130 \,{\left (b x^{4} + a\right )}^{\frac{9}{4}} a + 117 \,{\left (b x^{4} + a\right )}^{\frac{5}{4}} a^{2}}{585 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(1/4)*x^11,x, algorithm="giac")

[Out]

1/585*(45*(b*x^4 + a)^(13/4) - 130*(b*x^4 + a)^(9/4)*a + 117*(b*x^4 + a)^(5/4)*a
^2)/b^3