3.12.43 \(\int \frac {x^3}{(a+b x^4)^{5/4}} \, dx\) [1143]

Optimal. Leaf size=16 \[ -\frac {1}{b \sqrt [4]{a+b x^4}} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 16, 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 {1}{b \sqrt [4]{a+b x^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3/(a + b*x^4)^(5/4),x]

[Out]

-(1/(b*(a + b*x^4)^(1/4)))

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 \frac {x^3}{\left (a+b x^4\right )^{5/4}} \, dx &=-\frac {1}{b \sqrt [4]{a+b x^4}}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 16, normalized size = 1.00 \begin {gather*} -\frac {1}{b \sqrt [4]{a+b x^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^3/(a + b*x^4)^(5/4),x]

[Out]

-(1/(b*(a + b*x^4)^(1/4)))

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Maple [A]
time = 0.15, size = 15, normalized size = 0.94

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

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

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Maxima [A]
time = 0.29, size = 14, normalized size = 0.88 \begin {gather*} -\frac {1}{{\left (b x^{4} + a\right )}^{\frac {1}{4}} 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/((b*x^4 + a)^(1/4)*b)

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Fricas [A]
time = 0.38, size = 24, normalized size = 1.50 \begin {gather*} -\frac {{\left (b x^{4} + a\right )}^{\frac {3}{4}}}{b^{2} x^{4} + a 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]

-(b*x^4 + a)^(3/4)/(b^2*x^4 + a*b)

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Sympy [A]
time = 0.29, size = 24, normalized size = 1.50 \begin {gather*} \begin {cases} - \frac {1}{b \sqrt [4]{a + b x^{4}}} & \text {for}\: b \neq 0 \\\frac {x^{4}}{4 a^{\frac {5}{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((-1/(b*(a + b*x**4)**(1/4)), Ne(b, 0)), (x**4/(4*a**(5/4)), True))

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Giac [A]
time = 1.27, size = 14, normalized size = 0.88 \begin {gather*} -\frac {1}{{\left (b x^{4} + a\right )}^{\frac {1}{4}} 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/((b*x^4 + a)^(1/4)*b)

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Mupad [B]
time = 1.10, size = 14, normalized size = 0.88 \begin {gather*} -\frac {1}{b\,{\left (b\,x^4+a\right )}^{1/4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(a + b*x^4)^(5/4),x)

[Out]

-1/(b*(a + b*x^4)^(1/4))

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