3.4.37 \(\int \frac {1}{(b x^{1-2 m}+a x^m)^3} \, dx\) [337]

Optimal. Leaf size=27 \[ -\frac {1}{2 b (1-3 m) \left (a+b x^{1-3 m}\right )^2} \]

[Out]

-1/2/b/(1-3*m)/(a+b*x^(1-3*m))^2

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Rubi [A]
time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {1607, 267} \begin {gather*} -\frac {1}{2 b (1-3 m) \left (a+b x^{1-3 m}\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(b*x^(1 - 2*m) + a*x^m)^(-3),x]

[Out]

-1/2*1/(b*(1 - 3*m)*(a + b*x^(1 - 3*m))^2)

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]

Rule 1607

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rubi steps

\begin {align*} \int \frac {1}{\left (b x^{1-2 m}+a x^m\right )^3} \, dx &=\int \frac {x^{-3 m}}{\left (a+b x^{1-3 m}\right )^3} \, dx\\ &=-\frac {1}{2 b (1-3 m) \left (a+b x^{1-3 m}\right )^2}\\ \end {align*}

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Mathematica [A]
time = 0.05, size = 27, normalized size = 1.00 \begin {gather*} -\frac {1}{2 b (1-3 m) \left (a+b x^{1-3 m}\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(b*x^(1 - 2*m) + a*x^m)^(-3),x]

[Out]

-1/2*1/(b*(1 - 3*m)*(a + b*x^(1 - 3*m))^2)

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Maple [A]
time = 0.44, size = 39, normalized size = 1.44

method	result	size
risch	\(-\frac {x \left (2 a \,x^{3 m}+b x \right )}{2 \left (3 m -1\right ) a^{2} \left (a \,x^{3 m}+b x \right )^{2}}\)	\(39\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x^(1-2*m)+a*x^m)^3,x,method=_RETURNVERBOSE)

[Out]

-1/2*x*(2*a*(x^m)^3+b*x)/(3*m-1)/a^2/(a*(x^m)^3+b*x)^2

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (25) = 50\).
time = 0.30, size = 66, normalized size = 2.44 \begin {gather*} -\frac {2 \, a x x^{3 \, m} + b x^{2}}{2 \, {\left (2 \, a^{3} b {\left (3 \, m - 1\right )} x x^{3 \, m} + a^{2} b^{2} {\left (3 \, m - 1\right )} x^{2} + a^{4} {\left (3 \, m - 1\right )} x^{6 \, m}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^(1-2*m)+a*x^m)^3,x, algorithm="maxima")

[Out]

-1/2*(2*a*x*x^(3*m) + b*x^2)/(2*a^3*b*(3*m - 1)*x*x^(3*m) + a^2*b^2*(3*m - 1)*x^2 + a^4*(3*m - 1)*x^(6*m))

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 82 vs. \(2 (25) = 50\).
time = 1.88, size = 82, normalized size = 3.04 \begin {gather*} -\frac {2 \, a x x^{3 \, m} + b x^{2}}{2 \, {\left (2 \, {\left (3 \, a^{3} b m - a^{3} b\right )} x x^{3 \, m} + {\left (3 \, a^{2} b^{2} m - a^{2} b^{2}\right )} x^{2} + {\left (3 \, a^{4} m - a^{4}\right )} x^{6 \, m}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^(1-2*m)+a*x^m)^3,x, algorithm="fricas")

[Out]

-1/2*(2*a*x*x^(3*m) + b*x^2)/(2*(3*a^3*b*m - a^3*b)*x*x^(3*m) + (3*a^2*b^2*m - a^2*b^2)*x^2 + (3*a^4*m - a^4)*
x^(6*m))

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: HeuristicGCDFailed} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x**(1-2*m)+a*x**m)**3,x)

[Out]

Exception raised: HeuristicGCDFailed >> no luck

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^(1-2*m)+a*x^m)^3,x, algorithm="giac")

[Out]

integrate((a*x^m + b*x^(-2*m + 1))^(-3), x)

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Mupad [B]
time = 5.21, size = 38, normalized size = 1.41 \begin {gather*} -\frac {x\,\left (b\,x+2\,a\,x^{3\,m}\right )}{2\,a^2\,\left (3\,m-1\right )\,{\left (b\,x+a\,x^{3\,m}\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*x^m + b*x^(1 - 2*m))^3,x)

[Out]

-(x*(b*x + 2*a*x^(3*m)))/(2*a^2*(3*m - 1)*(b*x + a*x^(3*m))^2)

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