3.850 \(\int \frac{x^7}{\left (a+b x^4\right )^{3/2}} \, dx\)

Optimal. Leaf size=38 \[ \frac{a}{2 b^2 \sqrt{a+b x^4}}+\frac{\sqrt{a+b x^4}}{2 b^2} \]

[Out]

a/(2*b^2*Sqrt[a + b*x^4]) + Sqrt[a + b*x^4]/(2*b^2)

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Rubi [A]  time = 0.0582628, antiderivative size = 38, 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 \[ \frac{a}{2 b^2 \sqrt{a+b x^4}}+\frac{\sqrt{a+b x^4}}{2 b^2} \]

Antiderivative was successfully verified.

[In]  Int[x^7/(a + b*x^4)^(3/2),x]

[Out]

a/(2*b^2*Sqrt[a + b*x^4]) + Sqrt[a + b*x^4]/(2*b^2)

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Rubi in Sympy [A]  time = 7.10198, size = 31, normalized size = 0.82 \[ \frac{a}{2 b^{2} \sqrt{a + b x^{4}}} + \frac{\sqrt{a + b x^{4}}}{2 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**7/(b*x**4+a)**(3/2),x)

[Out]

a/(2*b**2*sqrt(a + b*x**4)) + sqrt(a + b*x**4)/(2*b**2)

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Mathematica [A]  time = 0.0253046, size = 27, normalized size = 0.71 \[ \frac{2 a+b x^4}{2 b^2 \sqrt{a+b x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^7/(a + b*x^4)^(3/2),x]

[Out]

(2*a + b*x^4)/(2*b^2*Sqrt[a + b*x^4])

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Maple [A]  time = 0.007, size = 24, normalized size = 0.6 \[{\frac{b{x}^{4}+2\,a}{2\,{b}^{2}}{\frac{1}{\sqrt{b{x}^{4}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^7/(b*x^4+a)^(3/2),x)

[Out]

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

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Maxima [A]  time = 1.44289, size = 41, normalized size = 1.08 \[ \frac{\sqrt{b x^{4} + a}}{2 \, b^{2}} + \frac{a}{2 \, \sqrt{b x^{4} + a} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^7/(b*x^4 + a)^(3/2),x, algorithm="maxima")

[Out]

1/2*sqrt(b*x^4 + a)/b^2 + 1/2*a/(sqrt(b*x^4 + a)*b^2)

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Fricas [A]  time = 0.253775, size = 31, normalized size = 0.82 \[ \frac{b x^{4} + 2 \, a}{2 \, \sqrt{b x^{4} + a} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^7/(b*x^4 + a)^(3/2),x, algorithm="fricas")

[Out]

1/2*(b*x^4 + 2*a)/(sqrt(b*x^4 + a)*b^2)

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Sympy [A]  time = 4.00967, size = 41, normalized size = 1.08 \[ \begin{cases} \frac{a}{b^{2} \sqrt{a + b x^{4}}} + \frac{x^{4}}{2 b \sqrt{a + b x^{4}}} & \text{for}\: b \neq 0 \\\frac{x^{8}}{8 a^{\frac{3}{2}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**7/(b*x**4+a)**(3/2),x)

[Out]

Piecewise((a/(b**2*sqrt(a + b*x**4)) + x**4/(2*b*sqrt(a + b*x**4)), Ne(b, 0)), (
x**8/(8*a**(3/2)), True))

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GIAC/XCAS [A]  time = 0.216919, size = 35, normalized size = 0.92 \[ \frac{\sqrt{b x^{4} + a} + \frac{a}{\sqrt{b x^{4} + a}}}{2 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^7/(b*x^4 + a)^(3/2),x, algorithm="giac")

[Out]

1/2*(sqrt(b*x^4 + a) + a/sqrt(b*x^4 + a))/b^2