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

Optimal. Leaf size=93 \[ \frac{x \left (a+b x^n\right )^p \left (c+d x^n\right )^{-\frac{1}{n}-p} \left (\frac{c \left (a+b x^n\right )}{a \left (c+d x^n\right )}\right )^{-p} \, _2F_1\left (\frac{1}{n},-p;1+\frac{1}{n};-\frac{(b c-a d) x^n}{a \left (d x^n+c\right )}\right )}{c} \]

[Out]

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

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Rubi [A]  time = 0.0219041, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.036, Rules used = {380} \[ \frac{x \left (a+b x^n\right )^p \left (c+d x^n\right )^{-\frac{1}{n}-p} \left (\frac{c \left (a+b x^n\right )}{a \left (c+d x^n\right )}\right )^{-p} \, _2F_1\left (\frac{1}{n},-p;1+\frac{1}{n};-\frac{(b c-a d) x^n}{a \left (d x^n+c\right )}\right )}{c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 380

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(x*(a + b*x^n)^p*Hypergeome
tric2F1[1/n, -p, 1 + 1/n, -(((b*c - a*d)*x^n)/(a*(c + d*x^n)))])/(c*((c*(a + b*x^n))/(a*(c + d*x^n)))^p*(c + d
*x^n)^(1/n + p)), x] /; FreeQ[{a, b, c, d, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[n*(p + q + 1) + 1, 0]

Rubi steps

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

Mathematica [A]  time = 0.067432, size = 94, normalized size = 1.01 \[ \frac{x \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} \left (c+d x^n\right )^{-\frac{n p+1}{n}} \left (\frac{d x^n}{c}+1\right )^p \, _2F_1\left (\frac{1}{n},-p;1+\frac{1}{n};\frac{(a d-b c) x^n}{a \left (d x^n+c\right )}\right )}{c} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [F]  time = 0.773, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b{x}^{n} \right ) ^{p} \left ( c+d{x}^{n} \right ) ^{-1-{n}^{-1}-p}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x^{n} + a\right )}^{p}{\left (d x^{n} + c\right )}^{-p - \frac{1}{n} - 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b x^{n} + a\right )}^{p}}{{\left (d x^{n} + c\right )}^{\frac{n p + n + 1}{n}}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x^{n} + a\right )}^{p}{\left (d x^{n} + c\right )}^{-p - \frac{1}{n} - 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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