3.547 \(\int (d x)^{-1-2 n (1+p)} \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^p \, dx\)

Optimal. Leaf size=117 \[ \frac{(d x)^{-2 n (p+1)} \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{p+1}}{2 a^2 d n (p+1) (2 p+1)}-\frac{\left (a+b x^n\right ) (d x)^{-2 n (p+1)} \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^p}{a d n (2 p+1)} \]

[Out]

-(((a + b*x^n)*(a^2 + 2*a*b*x^n + b^2*x^(2*n))^p)/(a*d*n*(1 + 2*p)*(d*x)^(2*n*(1
+ p)))) + (a^2 + 2*a*b*x^n + b^2*x^(2*n))^(1 + p)/(2*a^2*d*n*(1 + p)*(1 + 2*p)*
(d*x)^(2*n*(1 + p)))

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Rubi [A]  time = 0.146146, antiderivative size = 124, normalized size of antiderivative = 1.06, number of steps used = 3, number of rules used = 3, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.086 \[ \frac{\left (\frac{b x^n}{a}+1\right )^2 (d x)^{-2 n (p+1)} \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^p}{2 d n \left (2 p^2+3 p+1\right )}-\frac{\left (\frac{b x^n}{a}+1\right ) (d x)^{-2 n (p+1)} \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^p}{d n (2 p+1)} \]

Antiderivative was successfully verified.

[In]  Int[(d*x)^(-1 - 2*n*(1 + p))*(a^2 + 2*a*b*x^n + b^2*x^(2*n))^p,x]

[Out]

-(((1 + (b*x^n)/a)*(a^2 + 2*a*b*x^n + b^2*x^(2*n))^p)/(d*n*(1 + 2*p)*(d*x)^(2*n*
(1 + p)))) + ((1 + (b*x^n)/a)^2*(a^2 + 2*a*b*x^n + b^2*x^(2*n))^p)/(2*d*n*(1 + 3
*p + 2*p^2)*(d*x)^(2*n*(1 + p)))

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Rubi in Sympy [A]  time = 13.515, size = 104, normalized size = 0.89 \[ - \frac{\left (d x\right )^{- 2 n \left (p + 1\right )} \left (2 a + 2 b x^{n}\right ) \left (a^{2} + 2 a b x^{n} + b^{2} x^{2 n}\right )^{p}}{2 a d n \left (2 p + 1\right )} + \frac{\left (d x\right )^{- 2 n \left (p + 1\right )} \left (a^{2} + 2 a b x^{n} + b^{2} x^{2 n}\right )^{p + 1}}{2 a^{2} d n \left (p + 1\right ) \left (2 p + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x)**(-1-2*n*(1+p))*(a**2+2*a*b*x**n+b**2*x**(2*n))**p,x)

[Out]

-(d*x)**(-2*n*(p + 1))*(2*a + 2*b*x**n)*(a**2 + 2*a*b*x**n + b**2*x**(2*n))**p/(
2*a*d*n*(2*p + 1)) + (d*x)**(-2*n*(p + 1))*(a**2 + 2*a*b*x**n + b**2*x**(2*n))**
(p + 1)/(2*a**2*d*n*(p + 1)*(2*p + 1))

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Mathematica [C]  time = 0.103026, size = 75, normalized size = 0.64 \[ -\frac{x (d x)^{-2 n (p+1)-1} \left (\left (a+b x^n\right )^2\right )^p \left (\frac{b x^n}{a}+1\right )^{-2 p} \, _2F_1\left (-2 p,-2 (p+1);1-2 (p+1);-\frac{b x^n}{a}\right )}{2 n (p+1)} \]

Antiderivative was successfully verified.

[In]  Integrate[(d*x)^(-1 - 2*n*(1 + p))*(a^2 + 2*a*b*x^n + b^2*x^(2*n))^p,x]

[Out]

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

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Maple [F]  time = 0.084, size = 0, normalized size = 0. \[ \int \left ( dx \right ) ^{-1-2\,n \left ( 1+p \right ) } \left ({a}^{2}+2\,ab{x}^{n}+{b}^{2}{x}^{2\,n} \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x)^(-1-2*n*(1+p))*(a^2+2*a*b*x^n+b^2*x^(2*n))^p,x)

[Out]

int((d*x)^(-1-2*n*(1+p))*(a^2+2*a*b*x^n+b^2*x^(2*n))^p,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}\right )}^{p} \left (d x\right )^{-2 \, n{\left (p + 1\right )} - 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^(2*n) + 2*a*b*x^n + a^2)^p*(d*x)^(-2*n*(p + 1) - 1),x, algorithm="maxima")

[Out]

integrate((b^2*x^(2*n) + 2*a*b*x^n + a^2)^p*(d*x)^(-2*n*(p + 1) - 1), x)

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Fricas [A]  time = 0.277254, size = 223, normalized size = 1.91 \[ -\frac{{\left (2 \, a b p x x^{n} e^{\left (-{\left (2 \, n p + 2 \, n + 1\right )} \log \left (d\right ) -{\left (2 \, n p + 2 \, n + 1\right )} \log \left (x\right )\right )} - b^{2} x x^{2 \, n} e^{\left (-{\left (2 \, n p + 2 \, n + 1\right )} \log \left (d\right ) -{\left (2 \, n p + 2 \, n + 1\right )} \log \left (x\right )\right )} +{\left (2 \, a^{2} p + a^{2}\right )} x e^{\left (-{\left (2 \, n p + 2 \, n + 1\right )} \log \left (d\right ) -{\left (2 \, n p + 2 \, n + 1\right )} \log \left (x\right )\right )}\right )}{\left (b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}\right )}^{p}}{2 \,{\left (2 \, a^{2} n p^{2} + 3 \, a^{2} n p + a^{2} n\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^(2*n) + 2*a*b*x^n + a^2)^p*(d*x)^(-2*n*(p + 1) - 1),x, algorithm="fricas")

[Out]

-1/2*(2*a*b*p*x*x^n*e^(-(2*n*p + 2*n + 1)*log(d) - (2*n*p + 2*n + 1)*log(x)) - b
^2*x*x^(2*n)*e^(-(2*n*p + 2*n + 1)*log(d) - (2*n*p + 2*n + 1)*log(x)) + (2*a^2*p
+ a^2)*x*e^(-(2*n*p + 2*n + 1)*log(d) - (2*n*p + 2*n + 1)*log(x)))*(b^2*x^(2*n)
+ 2*a*b*x^n + a^2)^p/(2*a^2*n*p^2 + 3*a^2*n*p + a^2*n)

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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((d*x)**(-1-2*n*(1+p))*(a**2+2*a*b*x**n+b**2*x**(2*n))**p,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}\right )}^{p} \left (d x\right )^{-2 \, n{\left (p + 1\right )} - 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^(2*n) + 2*a*b*x^n + a^2)^p*(d*x)^(-2*n*(p + 1) - 1),x, algorithm="giac")

[Out]

integrate((b^2*x^(2*n) + 2*a*b*x^n + a^2)^p*(d*x)^(-2*n*(p + 1) - 1), x)