3.326 \(\int \frac {(c+d x^n)^{1-\frac {1}{n}}}{(a+b x^n)^2} \, dx\)

Optimal. Leaf size=54 \[ \frac {c x \left (c+d x^n\right )^{-1/n} \, _2F_1\left (2,\frac {1}{n};1+\frac {1}{n};-\frac {(b c-a d) x^n}{a \left (d x^n+c\right )}\right )}{a^2} \]

[Out]

c*x*hypergeom([2, 1/n],[1+1/n],-(-a*d+b*c)*x^n/a/(c+d*x^n))/a^2/((c+d*x^n)^(1/n))

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Rubi [A]  time = 0.01, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {379} \[ \frac {c x \left (c+d x^n\right )^{-1/n} \, _2F_1\left (2,\frac {1}{n};1+\frac {1}{n};-\frac {(b c-a d) x^n}{a \left (d x^n+c\right )}\right )}{a^2} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x^n)^(1 - n^(-1))/(a + b*x^n)^2,x]

[Out]

(c*x*Hypergeometric2F1[2, n^(-1), 1 + n^(-1), -(((b*c - a*d)*x^n)/(a*(c + d*x^n)))])/(a^2*(c + d*x^n)^n^(-1))

Rule 379

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(a^p*x*Hypergeometric2F1[1/
n, -p, 1 + 1/n, -(((b*c - a*d)*x^n)/(a*(c + d*x^n)))])/(c^(p + 1)*(c + d*x^n)^(1/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-\frac {1}{n}}}{\left (a+b x^n\right )^2} \, dx &=\frac {c x \left (c+d x^n\right )^{-1/n} \, _2F_1\left (2,\frac {1}{n};1+\frac {1}{n};-\frac {(b c-a d) x^n}{a \left (c+d x^n\right )}\right )}{a^2}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 53, normalized size = 0.98 \[ \frac {c x \left (c+d x^n\right )^{-1/n} \, _2F_1\left (2,\frac {1}{n};1+\frac {1}{n};\frac {(a d-b c) x^n}{a \left (d x^n+c\right )}\right )}{a^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x^n)^(1 - n^(-1))/(a + b*x^n)^2,x]

[Out]

(c*x*Hypergeometric2F1[2, n^(-1), 1 + n^(-1), ((-(b*c) + a*d)*x^n)/(a*(c + d*x^n))])/(a^2*(c + d*x^n)^n^(-1))

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fricas [F]  time = 1.16, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (d x^{n} + c\right )}^{\frac {n - 1}{n}}}{b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c+d*x^n)^(1-1/n)/(a+b*x^n)^2,x, algorithm="fricas")

[Out]

integral((d*x^n + c)^((n - 1)/n)/(b^2*x^(2*n) + 2*a*b*x^n + a^2), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (d x^{n} + c\right )}^{-\frac {1}{n} + 1}}{{\left (b x^{n} + a\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c+d*x^n)^(1-1/n)/(a+b*x^n)^2,x, algorithm="giac")

[Out]

integrate((d*x^n + c)^(-1/n + 1)/(b*x^n + a)^2, x)

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maple [F]  time = 0.92, size = 0, normalized size = 0.00 \[ \int \frac {\left (d \,x^{n}+c \right )^{-\frac {1}{n}+1}}{\left (b \,x^{n}+a \right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x^n+c)^(-1/n+1)/(b*x^n+a)^2,x)

[Out]

int((d*x^n+c)^(-1/n+1)/(b*x^n+a)^2,x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (d x^{n} + c\right )}^{-\frac {1}{n} + 1}}{{\left (b x^{n} + a\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c+d*x^n)^(1-1/n)/(a+b*x^n)^2,x, algorithm="maxima")

[Out]

integrate((d*x^n + c)^(-1/n + 1)/(b*x^n + a)^2, x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (c+d\,x^n\right )}^{1-\frac {1}{n}}}{{\left (a+b\,x^n\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x^n)^(1 - 1/n)/(a + b*x^n)^2,x)

[Out]

int((c + d*x^n)^(1 - 1/n)/(a + b*x^n)^2, x)

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sympy [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: HeuristicGCDFailed} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c+d*x**n)**(1-1/n)/(a+b*x**n)**2,x)

[Out]

Exception raised: HeuristicGCDFailed

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