3.983 \(\int x^7 \sqrt [3]{1+x^4} \, dx\)

Optimal. Leaf size=27 \[ \frac{3}{28} \left (x^4+1\right )^{7/3}-\frac{3}{16} \left (x^4+1\right )^{4/3} \]

[Out]

(-3*(1 + x^4)^(4/3))/16 + (3*(1 + x^4)^(7/3))/28

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Rubi [A]  time = 0.0313519, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{3}{28} \left (x^4+1\right )^{7/3}-\frac{3}{16} \left (x^4+1\right )^{4/3} \]

Antiderivative was successfully verified.

[In]  Int[x^7*(1 + x^4)^(1/3),x]

[Out]

(-3*(1 + x^4)^(4/3))/16 + (3*(1 + x^4)^(7/3))/28

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Rubi in Sympy [A]  time = 3.30984, size = 22, normalized size = 0.81 \[ \frac{3 \left (x^{4} + 1\right )^{\frac{7}{3}}}{28} - \frac{3 \left (x^{4} + 1\right )^{\frac{4}{3}}}{16} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**7*(x**4+1)**(1/3),x)

[Out]

3*(x**4 + 1)**(7/3)/28 - 3*(x**4 + 1)**(4/3)/16

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Mathematica [A]  time = 0.0116579, size = 20, normalized size = 0.74 \[ \frac{3}{112} \left (x^4+1\right )^{4/3} \left (4 x^4-3\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^7*(1 + x^4)^(1/3),x]

[Out]

(3*(1 + x^4)^(4/3)*(-3 + 4*x^4))/112

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Maple [A]  time = 0.006, size = 17, normalized size = 0.6 \[{\frac{12\,{x}^{4}-9}{112} \left ({x}^{4}+1 \right ) ^{{\frac{4}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^7*(x^4+1)^(1/3),x)

[Out]

3/112*(x^4+1)^(4/3)*(4*x^4-3)

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Maxima [A]  time = 1.43776, size = 26, normalized size = 0.96 \[ \frac{3}{28} \,{\left (x^{4} + 1\right )}^{\frac{7}{3}} - \frac{3}{16} \,{\left (x^{4} + 1\right )}^{\frac{4}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^4 + 1)^(1/3)*x^7,x, algorithm="maxima")

[Out]

3/28*(x^4 + 1)^(7/3) - 3/16*(x^4 + 1)^(4/3)

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Fricas [A]  time = 0.251414, size = 26, normalized size = 0.96 \[ \frac{3}{112} \,{\left (4 \, x^{8} + x^{4} - 3\right )}{\left (x^{4} + 1\right )}^{\frac{1}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^4 + 1)^(1/3)*x^7,x, algorithm="fricas")

[Out]

3/112*(4*x^8 + x^4 - 3)*(x^4 + 1)^(1/3)

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Sympy [A]  time = 2.18884, size = 41, normalized size = 1.52 \[ \frac{3 x^{8} \sqrt [3]{x^{4} + 1}}{28} + \frac{3 x^{4} \sqrt [3]{x^{4} + 1}}{112} - \frac{9 \sqrt [3]{x^{4} + 1}}{112} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**7*(x**4+1)**(1/3),x)

[Out]

3*x**8*(x**4 + 1)**(1/3)/28 + 3*x**4*(x**4 + 1)**(1/3)/112 - 9*(x**4 + 1)**(1/3)
/112

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GIAC/XCAS [A]  time = 0.213136, size = 26, normalized size = 0.96 \[ \frac{3}{28} \,{\left (x^{4} + 1\right )}^{\frac{7}{3}} - \frac{3}{16} \,{\left (x^{4} + 1\right )}^{\frac{4}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^4 + 1)^(1/3)*x^7,x, algorithm="giac")

[Out]

3/28*(x^4 + 1)^(7/3) - 3/16*(x^4 + 1)^(4/3)