3.1051 \(\int x^3 (a+b x^4)^{5/4} \, dx\)

Optimal. Leaf size=18 \[ \frac{\left (a+b x^4\right )^{9/4}}{9 b} \]

[Out]

(a + b*x^4)^(9/4)/(9*b)

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Rubi [A]  time = 0.0041643, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {261} \[ \frac{\left (a+b x^4\right )^{9/4}}{9 b} \]

Antiderivative was successfully verified.

[In]

Int[x^3*(a + b*x^4)^(5/4),x]

[Out]

(a + b*x^4)^(9/4)/(9*b)

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rubi steps

\begin{align*} \int x^3 \left (a+b x^4\right )^{5/4} \, dx &=\frac{\left (a+b x^4\right )^{9/4}}{9 b}\\ \end{align*}

Mathematica [A]  time = 0.0044347, size = 18, normalized size = 1. \[ \frac{\left (a+b x^4\right )^{9/4}}{9 b} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*(a + b*x^4)^(5/4),x]

[Out]

(a + b*x^4)^(9/4)/(9*b)

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Maple [A]  time = 0.004, size = 15, normalized size = 0.8 \begin{align*}{\frac{1}{9\,b} \left ( b{x}^{4}+a \right ) ^{{\frac{9}{4}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(b*x^4+a)^(5/4),x)

[Out]

1/9*(b*x^4+a)^(9/4)/b

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Maxima [A]  time = 0.958315, size = 19, normalized size = 1.06 \begin{align*} \frac{{\left (b x^{4} + a\right )}^{\frac{9}{4}}}{9 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x^4+a)^(5/4),x, algorithm="maxima")

[Out]

1/9*(b*x^4 + a)^(9/4)/b

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Fricas [B]  time = 1.64556, size = 72, normalized size = 4. \begin{align*} \frac{{\left (b^{2} x^{8} + 2 \, a b x^{4} + a^{2}\right )}{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{9 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x^4+a)^(5/4),x, algorithm="fricas")

[Out]

1/9*(b^2*x^8 + 2*a*b*x^4 + a^2)*(b*x^4 + a)^(1/4)/b

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Sympy [A]  time = 4.66337, size = 61, normalized size = 3.39 \begin{align*} \begin{cases} \frac{a^{2} \sqrt [4]{a + b x^{4}}}{9 b} + \frac{2 a x^{4} \sqrt [4]{a + b x^{4}}}{9} + \frac{b x^{8} \sqrt [4]{a + b x^{4}}}{9} & \text{for}\: b \neq 0 \\\frac{a^{\frac{5}{4}} x^{4}}{4} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(b*x**4+a)**(5/4),x)

[Out]

Piecewise((a**2*(a + b*x**4)**(1/4)/(9*b) + 2*a*x**4*(a + b*x**4)**(1/4)/9 + b*x**8*(a + b*x**4)**(1/4)/9, Ne(
b, 0)), (a**(5/4)*x**4/4, True))

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Giac [A]  time = 1.10771, size = 19, normalized size = 1.06 \begin{align*} \frac{{\left (b x^{4} + a\right )}^{\frac{9}{4}}}{9 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x^4+a)^(5/4),x, algorithm="giac")

[Out]

1/9*(b*x^4 + a)^(9/4)/b