3.11.51 \(\int x^3 (a+b x^4)^{5/4} \, dx\) [1051]

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

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 18, normalized size of antiderivative = 1.00, 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 = {267} \begin {gather*} \frac {\left (a+b x^4\right )^{9/4}}{9 b} \end {gather*}

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 267

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*}

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Mathematica [A]
time = 0.02, size = 18, normalized size = 1.00 \begin {gather*} \frac {\left (a+b x^4\right )^{9/4}}{9 b} \end {gather*}

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 = 0.16, size = 15, normalized size = 0.83

method	result	size
gosper	\(\frac {\left (b \,x^{4}+a \right )^{\frac {9}{4}}}{9 b}\)	\(15\)
derivativedivides	\(\frac {\left (b \,x^{4}+a \right )^{\frac {9}{4}}}{9 b}\)	\(15\)
default	\(\frac {\left (b \,x^{4}+a \right )^{\frac {9}{4}}}{9 b}\)	\(15\)
trager	\(\frac {\left (b^{2} x^{8}+2 a b \,x^{4}+a^{2}\right ) \left (b \,x^{4}+a \right )^{\frac {1}{4}}}{9 b}\)	\(33\)
risch	\(\frac {\left (b^{2} x^{8}+2 a b \,x^{4}+a^{2}\right ) \left (b \,x^{4}+a \right )^{\frac {1}{4}}}{9 b}\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.30, size = 14, normalized size = 0.78 \begin {gather*} \frac {{\left (b x^{4} + a\right )}^{\frac {9}{4}}}{9 \, b} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 32 vs. \(2 (14) = 28\).
time = 0.40, size = 32, normalized size = 1.78 \begin {gather*} \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 {gather*}

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 [B] Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (12) = 24\).
time = 0.39, size = 61, normalized size = 3.39 \begin {gather*} \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 {gather*}

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 = 1.09, size = 14, normalized size = 0.78 \begin {gather*} \frac {{\left (b x^{4} + a\right )}^{\frac {9}{4}}}{9 \, b} \end {gather*}

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

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Mupad [B]
time = 1.11, size = 14, normalized size = 0.78 \begin {gather*} \frac {{\left (b\,x^4+a\right )}^{9/4}}{9\,b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a + b*x^4)^(5/4),x)

[Out]

(a + b*x^4)^(9/4)/(9*b)

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