3.312 \(\int (a+b x^n)^p (c+d x^n)^q \, dx\)

Optimal. Leaf size=81 \[ x \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} \left (c+d x^n\right )^q \left (\frac{d x^n}{c}+1\right )^{-q} F_1\left (\frac{1}{n};-p,-q;1+\frac{1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right ) \]

[Out]

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

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Rubi [A]  time = 0.052638, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {430, 429} \[ x \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} \left (c+d x^n\right )^q \left (\frac{d x^n}{c}+1\right )^{-q} F_1\left (\frac{1}{n};-p,-q;1+\frac{1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^n)^p*(c + d*x^n)^q,x]

[Out]

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

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

Mathematica [B]  time = 0.393458, size = 190, normalized size = 2.35 \[ \frac{a c (n+1) x \left (a+b x^n\right )^p \left (c+d x^n\right )^q F_1\left (\frac{1}{n};-p,-q;1+\frac{1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right )}{b c n p x^n F_1\left (1+\frac{1}{n};1-p,-q;2+\frac{1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right )+a d n q x^n F_1\left (1+\frac{1}{n};-p,1-q;2+\frac{1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right )+a c (n+1) F_1\left (\frac{1}{n};-p,-q;1+\frac{1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right )} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [F]  time = 0.816, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b{x}^{n} \right ) ^{p} \left ( c+d{x}^{n} \right ) ^{q}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((a+b*x^n)^p*(c+d*x^n)^q,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 )}^{q}\,{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)^q,x, algorithm="maxima")

[Out]

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

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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 )}^{q}\,{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)^q,x, algorithm="giac")

[Out]

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