3.803 \(\int x^7 \sqrt{5+3 x^4} \, dx\)

Optimal. Leaf size=31 \[ \frac{1}{90} \left (3 x^4+5\right )^{5/2}-\frac{5}{54} \left (3 x^4+5\right )^{3/2} \]

[Out]

(-5*(5 + 3*x^4)^(3/2))/54 + (5 + 3*x^4)^(5/2)/90

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Rubi [A]  time = 0.0394817, antiderivative size = 31, 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 \[ \frac{1}{90} \left (3 x^4+5\right )^{5/2}-\frac{5}{54} \left (3 x^4+5\right )^{3/2} \]

Antiderivative was successfully verified.

[In]  Int[x^7*Sqrt[5 + 3*x^4],x]

[Out]

(-5*(5 + 3*x^4)^(3/2))/54 + (5 + 3*x^4)^(5/2)/90

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Rubi in Sympy [A]  time = 4.84401, size = 24, normalized size = 0.77 \[ \frac{\left (3 x^{4} + 5\right )^{\frac{5}{2}}}{90} - \frac{5 \left (3 x^{4} + 5\right )^{\frac{3}{2}}}{54} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**7*(3*x**4+5)**(1/2),x)

[Out]

(3*x**4 + 5)**(5/2)/90 - 5*(3*x**4 + 5)**(3/2)/54

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Mathematica [A]  time = 0.0133727, size = 22, normalized size = 0.71 \[ \frac{1}{270} \left (3 x^4+5\right )^{3/2} \left (9 x^4-10\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^7*Sqrt[5 + 3*x^4],x]

[Out]

((5 + 3*x^4)^(3/2)*(-10 + 9*x^4))/270

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Maple [A]  time = 0.006, size = 19, normalized size = 0.6 \[{\frac{9\,{x}^{4}-10}{270} \left ( 3\,{x}^{4}+5 \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^7*(3*x^4+5)^(1/2),x)

[Out]

1/270*(3*x^4+5)^(3/2)*(9*x^4-10)

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Maxima [A]  time = 1.4372, size = 31, normalized size = 1. \[ \frac{1}{90} \,{\left (3 \, x^{4} + 5\right )}^{\frac{5}{2}} - \frac{5}{54} \,{\left (3 \, x^{4} + 5\right )}^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(3*x^4 + 5)*x^7,x, algorithm="maxima")

[Out]

1/90*(3*x^4 + 5)^(5/2) - 5/54*(3*x^4 + 5)^(3/2)

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Fricas [A]  time = 0.23149, size = 31, normalized size = 1. \[ \frac{1}{270} \,{\left (27 \, x^{8} + 15 \, x^{4} - 50\right )} \sqrt{3 \, x^{4} + 5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(3*x^4 + 5)*x^7,x, algorithm="fricas")

[Out]

1/270*(27*x^8 + 15*x^4 - 50)*sqrt(3*x^4 + 5)

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Sympy [A]  time = 1.66159, size = 42, normalized size = 1.35 \[ \frac{x^{8} \sqrt{3 x^{4} + 5}}{10} + \frac{x^{4} \sqrt{3 x^{4} + 5}}{18} - \frac{5 \sqrt{3 x^{4} + 5}}{27} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**7*(3*x**4+5)**(1/2),x)

[Out]

x**8*sqrt(3*x**4 + 5)/10 + x**4*sqrt(3*x**4 + 5)/18 - 5*sqrt(3*x**4 + 5)/27

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GIAC/XCAS [A]  time = 0.215625, size = 31, normalized size = 1. \[ \frac{1}{90} \,{\left (3 \, x^{4} + 5\right )}^{\frac{5}{2}} - \frac{5}{54} \,{\left (3 \, x^{4} + 5\right )}^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(3*x^4 + 5)*x^7,x, algorithm="giac")

[Out]

1/90*(3*x^4 + 5)^(5/2) - 5/54*(3*x^4 + 5)^(3/2)