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

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 = 0.128784, 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 \[ x \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} \left (c+d x^n\right )^q \left (\frac{d x^n}{c}+1\right )^{-q} F_1\left (\frac{1}{n};-p,-q;1+\frac{1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right ) \]

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)

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Rubi in Sympy [A]  time = 21.6673, size = 63, normalized size = 0.78 \[ x \left (1 + \frac{b x^{n}}{a}\right )^{- p} \left (1 + \frac{d x^{n}}{c}\right )^{- q} \left (a + b x^{n}\right )^{p} \left (c + d x^{n}\right )^{q} \operatorname{appellf_{1}}{\left (\frac{1}{n},- p,- q,1 + \frac{1}{n},- \frac{b x^{n}}{a},- \frac{d x^{n}}{c} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b*x**n)**p*(c+d*x**n)**q,x)

[Out]

x*(1 + b*x**n/a)**(-p)*(1 + d*x**n/c)**(-q)*(a + b*x**n)**p*(c + d*x**n)**q*appe
llf1(1/n, -p, -q, 1 + 1/n, -b*x**n/a, -d*x**n/c)

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Mathematica [B]  time = 0.661915, 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 = 0.207, size = 0, normalized size = 0. \[ \int \left ( a+b{x}^{n} \right ) ^{p} \left ( c+d{x}^{n} \right ) ^{q}\, dx \]

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. \[ \int{\left (b x^{n} + a\right )}^{p}{\left (d x^{n} + c\right )}^{q}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^p*(d*x^n + c)^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. \[{\rm integral}\left ({\left (b x^{n} + a\right )}^{p}{\left (d x^{n} + c\right )}^{q}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^p*(d*x^n + c)^q,x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{n} + a\right )}^{p}{\left (d x^{n} + c\right )}^{q}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^p*(d*x^n + c)^q,x, algorithm="giac")

[Out]

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