3.4.25 \(\int \frac {(c+d x^n)^{-1/n}}{a+b x^n} \, dx\) [325]

Optimal. Leaf size=53 \[ \frac {x \left (c+d x^n\right )^{-1/n} \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {(b c-a d) x^n}{a \left (c+d x^n\right )}\right )}{a} \]

[Out]

x*hypergeom([1, 1/n],[1+1/n],-(-a*d+b*c)*x^n/a/(c+d*x^n))/a/((c+d*x^n)^(1/n))

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Rubi [A]
time = 0.01, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {387} \begin {gather*} \frac {x \left (c+d x^n\right )^{-1/n} \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {(b c-a d) x^n}{a \left (d x^n+c\right )}\right )}{a} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((a + b*x^n)*(c + d*x^n)^n^(-1)),x]

[Out]

(x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), -(((b*c - a*d)*x^n)/(a*(c + d*x^n)))])/(a*(c + d*x^n)^n^(-1))

Rule 387

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^p*(x/(c^(p + 1)*(c + d*x^
n)^(1/n)))*Hypergeometric2F1[1/n, -p, 1 + 1/n, (-(b*c - a*d))*(x^n/(a*(c + d*x^n)))], x] /; FreeQ[{a, b, c, d,
n, q}, x] && NeQ[b*c - a*d, 0] && EqQ[n*(p + q + 1) + 1, 0] && ILtQ[p, 0]

Rubi steps

\begin {align*} \int \frac {\left (c+d x^n\right )^{-1/n}}{a+b x^n} \, dx &=\frac {x \left (c+d x^n\right )^{-1/n} \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {(b c-a d) x^n}{a \left (c+d x^n\right )}\right )}{a}\\ \end {align*}

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Mathematica [A]
time = 0.04, size = 52, normalized size = 0.98 \begin {gather*} \frac {x \left (c+d x^n\right )^{-1/n} \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};\frac {(-b c+a d) x^n}{a \left (c+d x^n\right )}\right )}{a} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b*x^n)*(c + d*x^n)^n^(-1)),x]

[Out]

(x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), ((-(b*c) + a*d)*x^n)/(a*(c + d*x^n))])/(a*(c + d*x^n)^n^(-1))

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Maple [F]
time = 0.13, size = 0, normalized size = 0.00 \[\int \frac {\left (c +d \,x^{n}\right )^{-\frac {1}{n}}}{a +b \,x^{n}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b*x^n)/((c+d*x^n)^(1/n)),x)

[Out]

int(1/(a+b*x^n)/((c+d*x^n)^(1/n)),x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*x^n)/((c+d*x^n)^(1/n)),x, algorithm="maxima")

[Out]

integrate(1/((b*x^n + a)*(d*x^n + c)^(1/n)), x)

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Fricas [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/(a+b*x^n)/((c+d*x^n)^(1/n)),x, algorithm="fricas")

[Out]

integral(1/((b*x^n + a)*(d*x^n + c)^(1/n)), x)

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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/(a+b*x**n)/((c+d*x**n)**(1/n)),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/(a+b*x^n)/((c+d*x^n)^(1/n)),x, algorithm="giac")

[Out]

integrate(1/((b*x^n + a)*(d*x^n + c)^(1/n)), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {1}{\left (a+b\,x^n\right )\,{\left (c+d\,x^n\right )}^{1/n}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a + b*x^n)*(c + d*x^n)^(1/n)),x)

[Out]

int(1/((a + b*x^n)*(c + d*x^n)^(1/n)), x)

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