3.1017 \(\int x^{11} \left (a+b x^4\right )^{3/4} \, dx\)

Optimal. Leaf size=59 \[ \frac{a^2 \left (a+b x^4\right )^{7/4}}{7 b^3}+\frac{\left (a+b x^4\right )^{15/4}}{15 b^3}-\frac{2 a \left (a+b x^4\right )^{11/4}}{11 b^3} \]

[Out]

(a^2*(a + b*x^4)^(7/4))/(7*b^3) - (2*a*(a + b*x^4)^(11/4))/(11*b^3) + (a + b*x^4
)^(15/4)/(15*b^3)

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Rubi [A]  time = 0.0823975, 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 \[ \frac{a^2 \left (a+b x^4\right )^{7/4}}{7 b^3}+\frac{\left (a+b x^4\right )^{15/4}}{15 b^3}-\frac{2 a \left (a+b x^4\right )^{11/4}}{11 b^3} \]

Antiderivative was successfully verified.

[In]  Int[x^11*(a + b*x^4)^(3/4),x]

[Out]

(a^2*(a + b*x^4)^(7/4))/(7*b^3) - (2*a*(a + b*x^4)^(11/4))/(11*b^3) + (a + b*x^4
)^(15/4)/(15*b^3)

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Rubi in Sympy [A]  time = 10.601, size = 51, normalized size = 0.86 \[ \frac{a^{2} \left (a + b x^{4}\right )^{\frac{7}{4}}}{7 b^{3}} - \frac{2 a \left (a + b x^{4}\right )^{\frac{11}{4}}}{11 b^{3}} + \frac{\left (a + b x^{4}\right )^{\frac{15}{4}}}{15 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**11*(b*x**4+a)**(3/4),x)

[Out]

a**2*(a + b*x**4)**(7/4)/(7*b**3) - 2*a*(a + b*x**4)**(11/4)/(11*b**3) + (a + b*
x**4)**(15/4)/(15*b**3)

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Mathematica [A]  time = 0.0274331, size = 50, normalized size = 0.85 \[ \frac{\left (a+b x^4\right )^{3/4} \left (32 a^3-24 a^2 b x^4+21 a b^2 x^8+77 b^3 x^{12}\right )}{1155 b^3} \]

Antiderivative was successfully verified.

[In]  Integrate[x^11*(a + b*x^4)^(3/4),x]

[Out]

((a + b*x^4)^(3/4)*(32*a^3 - 24*a^2*b*x^4 + 21*a*b^2*x^8 + 77*b^3*x^12))/(1155*b
^3)

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Maple [A]  time = 0.008, size = 36, normalized size = 0.6 \[{\frac{77\,{b}^{2}{x}^{8}-56\,ab{x}^{4}+32\,{a}^{2}}{1155\,{b}^{3}} \left ( b{x}^{4}+a \right ) ^{{\frac{7}{4}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^11*(b*x^4+a)^(3/4),x)

[Out]

1/1155*(b*x^4+a)^(7/4)*(77*b^2*x^8-56*a*b*x^4+32*a^2)/b^3

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Maxima [A]  time = 1.42403, size = 63, normalized size = 1.07 \[ \frac{{\left (b x^{4} + a\right )}^{\frac{15}{4}}}{15 \, b^{3}} - \frac{2 \,{\left (b x^{4} + a\right )}^{\frac{11}{4}} a}{11 \, b^{3}} + \frac{{\left (b x^{4} + a\right )}^{\frac{7}{4}} a^{2}}{7 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(3/4)*x^11,x, algorithm="maxima")

[Out]

1/15*(b*x^4 + a)^(15/4)/b^3 - 2/11*(b*x^4 + a)^(11/4)*a/b^3 + 1/7*(b*x^4 + a)^(7
/4)*a^2/b^3

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Fricas [A]  time = 0.316823, size = 62, normalized size = 1.05 \[ \frac{{\left (77 \, b^{3} x^{12} + 21 \, a b^{2} x^{8} - 24 \, a^{2} b x^{4} + 32 \, a^{3}\right )}{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{1155 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(3/4)*x^11,x, algorithm="fricas")

[Out]

1/1155*(77*b^3*x^12 + 21*a*b^2*x^8 - 24*a^2*b*x^4 + 32*a^3)*(b*x^4 + a)^(3/4)/b^
3

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Sympy [A]  time = 34.1866, size = 87, normalized size = 1.47 \[ \begin{cases} \frac{32 a^{3} \left (a + b x^{4}\right )^{\frac{3}{4}}}{1155 b^{3}} - \frac{8 a^{2} x^{4} \left (a + b x^{4}\right )^{\frac{3}{4}}}{385 b^{2}} + \frac{a x^{8} \left (a + b x^{4}\right )^{\frac{3}{4}}}{55 b} + \frac{x^{12} \left (a + b x^{4}\right )^{\frac{3}{4}}}{15} & \text{for}\: b \neq 0 \\\frac{a^{\frac{3}{4}} x^{12}}{12} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**11*(b*x**4+a)**(3/4),x)

[Out]

Piecewise((32*a**3*(a + b*x**4)**(3/4)/(1155*b**3) - 8*a**2*x**4*(a + b*x**4)**(
3/4)/(385*b**2) + a*x**8*(a + b*x**4)**(3/4)/(55*b) + x**12*(a + b*x**4)**(3/4)/
15, Ne(b, 0)), (a**(3/4)*x**12/12, True))

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GIAC/XCAS [A]  time = 0.217737, size = 58, normalized size = 0.98 \[ \frac{77 \,{\left (b x^{4} + a\right )}^{\frac{15}{4}} - 210 \,{\left (b x^{4} + a\right )}^{\frac{11}{4}} a + 165 \,{\left (b x^{4} + a\right )}^{\frac{7}{4}} a^{2}}{1155 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(3/4)*x^11,x, algorithm="giac")

[Out]

1/1155*(77*(b*x^4 + a)^(15/4) - 210*(b*x^4 + a)^(11/4)*a + 165*(b*x^4 + a)^(7/4)
*a^2)/b^3