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

Optimal. Leaf size=98 \[ \frac{d x \left (a+b x^n\right )^{p+1}}{b (n p+n+1)}-\frac{x \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} (a d-b c (n p+n+1)) \, _2F_1\left (\frac{1}{n},-p;1+\frac{1}{n};-\frac{b x^n}{a}\right )}{b (n p+n+1)} \]

[Out]

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

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Rubi [A]  time = 0.119854, antiderivative size = 89, normalized size of antiderivative = 0.91, number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ x \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} \left (c-\frac{a d}{b n p+b n+b}\right ) \, _2F_1\left (\frac{1}{n},-p;1+\frac{1}{n};-\frac{b x^n}{a}\right )+\frac{d x \left (a+b x^n\right )^{p+1}}{b (n p+n+1)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 11.0367, size = 80, normalized size = 0.82 \[ \frac{d x \left (a + b x^{n}\right )^{p + 1}}{b \left (n p + n + 1\right )} - \frac{x \left (1 + \frac{b x^{n}}{a}\right )^{- p} \left (a + b x^{n}\right )^{p} \left (a d - b c \left (n \left (p + 1\right ) + 1\right )\right ){{}_{2}F_{1}\left (\begin{matrix} - p, \frac{1}{n} \\ 1 + \frac{1}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{b \left (n \left (p + 1\right ) + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.102702, size = 85, normalized size = 0.87 \[ \frac{x \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} \left (c (n+1) \, _2F_1\left (\frac{1}{n},-p;1+\frac{1}{n};-\frac{b x^n}{a}\right )+d x^n \, _2F_1\left (1+\frac{1}{n},-p;2+\frac{1}{n};-\frac{b x^n}{a}\right )\right )}{n+1} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.135, size = 0, normalized size = 0. \[ \int \left ( a+b{x}^{n} \right ) ^{p} \left ( c+d{x}^{n} \right ) \, dx \]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError