3.215 \(\int \left (a+b x^n\right )^p \left (c+d x^n\right )^2 \, dx\)

Optimal. Leaf size=202 \[ -\frac{x \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} (b c (n p+n+1) (a d-b c (n (p+2)+1))-a d (a d (n+1)-b c (n (p+3)+1))) \, _2F_1\left (\frac{1}{n},-p;1+\frac{1}{n};-\frac{b x^n}{a}\right )}{b^2 (n p+n+1) (n (p+2)+1)}-\frac{d x \left (a+b x^n\right )^{p+1} (a d (n+1)-b c (n (p+3)+1))}{b^2 (n p+n+1) (n (p+2)+1)}+\frac{d x \left (c+d x^n\right ) \left (a+b x^n\right )^{p+1}}{b (n p+2 n+1)} \]

[Out]

-((d*(a*d*(1 + n) - b*c*(1 + n*(3 + p)))*x*(a + b*x^n)^(1 + p))/(b^2*(1 + n + n*
p)*(1 + n*(2 + p)))) + (d*x*(a + b*x^n)^(1 + p)*(c + d*x^n))/(b*(1 + 2*n + n*p))
- ((b*c*(1 + n + n*p)*(a*d - b*c*(1 + n*(2 + p))) - a*d*(a*d*(1 + n) - b*c*(1 +
n*(3 + p))))*x*(a + b*x^n)^p*Hypergeometric2F1[n^(-1), -p, 1 + n^(-1), -((b*x^n
)/a)])/(b^2*(1 + n + n*p)*(1 + n*(2 + p))*(1 + (b*x^n)/a)^p)

_______________________________________________________________________________________

Rubi [A]  time = 0.624457, antiderivative size = 197, normalized size of antiderivative = 0.98, number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ -\frac{d x \left (a+b x^n\right )^{p+1} (a d (n+1)-b (c n (p+3)+c))}{b^2 (n p+n+1) (n (p+2)+1)}-\frac{x \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} \left (c (a d-b (c n (p+2)+c))-\frac{a d (a d (n+1)-b (c n (p+3)+c))}{b (n p+n+1)}\right ) \, _2F_1\left (\frac{1}{n},-p;1+\frac{1}{n};-\frac{b x^n}{a}\right )}{b (n (p+2)+1)}+\frac{d x \left (c+d x^n\right ) \left (a+b x^n\right )^{p+1}}{b (n (p+2)+1)} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^n)^p*(c + d*x^n)^2,x]

[Out]

-((d*(a*d*(1 + n) - b*(c + c*n*(3 + p)))*x*(a + b*x^n)^(1 + p))/(b^2*(1 + n + n*
p)*(1 + n*(2 + p)))) + (d*x*(a + b*x^n)^(1 + p)*(c + d*x^n))/(b*(1 + n*(2 + p)))
- ((c*(a*d - b*(c + c*n*(2 + p))) - (a*d*(a*d*(1 + n) - b*(c + c*n*(3 + p))))/(
b*(1 + n + n*p)))*x*(a + b*x^n)^p*Hypergeometric2F1[n^(-1), -p, 1 + n^(-1), -((b
*x^n)/a)])/(b*(1 + n*(2 + p))*(1 + (b*x^n)/a)^p)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 36.2797, size = 173, normalized size = 0.86 \[ \frac{d x \left (a + b x^{n}\right )^{p + 1} \left (c + d x^{n}\right )}{b \left (n \left (p + 2\right ) + 1\right )} - \frac{d x \left (a + b x^{n}\right )^{p + 1} \left (a d \left (n + 1\right ) - b c \left (n \left (p + 3\right ) + 1\right )\right )}{b^{2} \left (n \left (p + 1\right ) + 1\right ) \left (n \left (p + 2\right ) + 1\right )} + \frac{x \left (1 + \frac{b x^{n}}{a}\right )^{- p} \left (a + b x^{n}\right )^{p} \left (a d \left (a d \left (n + 1\right ) - b c \left (n \left (p + 3\right ) + 1\right )\right ) - b c \left (a d - b c \left (n \left (p + 2\right ) + 1\right )\right ) \left (n \left (p + 1\right ) + 1\right )\right ){{}_{2}F_{1}\left (\begin{matrix} - p, \frac{1}{n} \\ 1 + \frac{1}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{b^{2} \left (n \left (p + 1\right ) + 1\right ) \left (n \left (p + 2\right ) + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b*x**n)**p*(c+d*x**n)**2,x)

[Out]

d*x*(a + b*x**n)**(p + 1)*(c + d*x**n)/(b*(n*(p + 2) + 1)) - d*x*(a + b*x**n)**(
p + 1)*(a*d*(n + 1) - b*c*(n*(p + 3) + 1))/(b**2*(n*(p + 1) + 1)*(n*(p + 2) + 1)
) + x*(1 + b*x**n/a)**(-p)*(a + b*x**n)**p*(a*d*(a*d*(n + 1) - b*c*(n*(p + 3) +
1)) - b*c*(a*d - b*c*(n*(p + 2) + 1))*(n*(p + 1) + 1))*hyper((-p, 1/n), (1 + 1/n
,), -b*x**n/a)/(b**2*(n*(p + 1) + 1)*(n*(p + 2) + 1))

_______________________________________________________________________________________

Mathematica [A]  time = 0.24786, size = 140, normalized size = 0.69 \[ \frac{x \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} \left ((n+1) \left (c^2 (2 n+1) \, _2F_1\left (\frac{1}{n},-p;1+\frac{1}{n};-\frac{b x^n}{a}\right )+d^2 x^{2 n} \, _2F_1\left (2+\frac{1}{n},-p;3+\frac{1}{n};-\frac{b x^n}{a}\right )\right )+2 c d (2 n+1) x^n \, _2F_1\left (1+\frac{1}{n},-p;2+\frac{1}{n};-\frac{b x^n}{a}\right )\right )}{(n+1) (2 n+1)} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^n)^p*(c + d*x^n)^2,x]

[Out]

(x*(a + b*x^n)^p*(2*c*d*(1 + 2*n)*x^n*Hypergeometric2F1[1 + n^(-1), -p, 2 + n^(-
1), -((b*x^n)/a)] + (1 + n)*(d^2*x^(2*n)*Hypergeometric2F1[2 + n^(-1), -p, 3 + n
^(-1), -((b*x^n)/a)] + c^2*(1 + 2*n)*Hypergeometric2F1[n^(-1), -p, 1 + n^(-1), -
((b*x^n)/a)])))/((1 + n)*(1 + 2*n)*(1 + (b*x^n)/a)^p)

_______________________________________________________________________________________

Maple [F]  time = 0.137, size = 0, normalized size = 0. \[ \int \left ( a+b{x}^{n} \right ) ^{p} \left ( c+d{x}^{n} \right ) ^{2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^n)^p*(c+d*x^n)^2,x)

[Out]

int((a+b*x^n)^p*(c+d*x^n)^2,x)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (d x^{n} + c\right )}^{2}{\left (b x^{n} + a\right )}^{p}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^n + c)^2*(b*x^n + a)^p,x, algorithm="maxima")

[Out]

integrate((d*x^n + c)^2*(b*x^n + a)^p, x)

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (d^{2} x^{2 \, n} + 2 \, c d x^{n} + c^{2}\right )}{\left (b x^{n} + a\right )}^{p}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^n + c)^2*(b*x^n + a)^p,x, algorithm="fricas")

[Out]

integral((d^2*x^(2*n) + 2*c*d*x^n + c^2)*(b*x^n + a)^p, x)

_______________________________________________________________________________________

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)**2,x)

[Out]

Exception raised: TypeError

_______________________________________________________________________________________

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)^2*(b*x^n + a)^p,x, algorithm="giac")

[Out]

Exception raised: TypeError