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

Optimal. Leaf size=59 \[ \frac{x \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} F_1\left (\frac{1}{n};-p,1;1+\frac{1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right )}{c} \]

[Out]

(x*(a + b*x^n)^p*AppellF1[n^(-1), -p, 1, 1 + n^(-1), -((b*x^n)/a), -((d*x^n)/c)])/(c*(1 + (b*x^n)/a)^p)

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Rubi [A]  time = 0.0289819, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {430, 429} \[ \frac{x \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} F_1\left (\frac{1}{n};-p,1;1+\frac{1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right )}{c} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^n)^p/(c + d*x^n),x]

[Out]

(x*(a + b*x^n)^p*AppellF1[n^(-1), -p, 1, 1 + n^(-1), -((b*x^n)/a), -((d*x^n)/c)])/(c*(1 + (b*x^n)/a)^p)

Rule 430

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[(a^IntPart[p]*(a + b*x^n)^F
racPart[p])/(1 + (b*x^n)/a)^FracPart[p], Int[(1 + (b*x^n)/a)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n,
p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n, -1] &&  !(IntegerQ[p] || GtQ[a, 0])

Rule 429

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^p*c^q*x*AppellF1[1/n, -p,
-q, 1 + 1/n, -((b*x^n)/a), -((d*x^n)/c)], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n
, -1] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rubi steps

\begin{align*} \int \frac{\left (a+b x^n\right )^p}{c+d x^n} \, dx &=\left (\left (a+b x^n\right )^p \left (1+\frac{b x^n}{a}\right )^{-p}\right ) \int \frac{\left (1+\frac{b x^n}{a}\right )^p}{c+d x^n} \, dx\\ &=\frac{x \left (a+b x^n\right )^p \left (1+\frac{b x^n}{a}\right )^{-p} F_1\left (\frac{1}{n};-p,1;1+\frac{1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right )}{c}\\ \end{align*}

Mathematica [B]  time = 0.255761, size = 180, normalized size = 3.05 \[ \frac{a c (n+1) x \left (a+b x^n\right )^p F_1\left (\frac{1}{n};-p,1;1+\frac{1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right )}{\left (c+d x^n\right ) \left (b c n p x^n F_1\left (1+\frac{1}{n};1-p,1;2+\frac{1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right )-a d n x^n F_1\left (1+\frac{1}{n};-p,2;2+\frac{1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right )+a c (n+1) F_1\left (\frac{1}{n};-p,1;1+\frac{1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(a*c*(1 + n)*x*(a + b*x^n)^p*AppellF1[n^(-1), -p, 1, 1 + n^(-1), -((b*x^n)/a), -((d*x^n)/c)])/((c + d*x^n)*(b*
c*n*p*x^n*AppellF1[1 + n^(-1), 1 - p, 1, 2 + n^(-1), -((b*x^n)/a), -((d*x^n)/c)] - a*d*n*x^n*AppellF1[1 + n^(-
1), -p, 2, 2 + n^(-1), -((b*x^n)/a), -((d*x^n)/c)] + a*c*(1 + n)*AppellF1[n^(-1), -p, 1, 1 + n^(-1), -((b*x^n)
/a), -((d*x^n)/c)]))

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Maple [F]  time = 0.711, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b{x}^{n} \right ) ^{p}}{c+d{x}^{n}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*x**n)**p/(c + d*x**n), x)

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

[Out]

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