3.442 \(\int x^{-1-n-(-1+p) q} \left (b x^n+a x^q\right )^p \, dx\)

Optimal. Leaf size=69 \[ \frac{b x^{-p q} \left (a+b x^{n-q}\right ) \left (a x^q+b x^n\right )^p \, _2F_1\left (2,p+1;p+2;\frac{b x^{n-q}}{a}+1\right )}{a^2 (p+1) (n-q)} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.150952, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{b x^{-p q} \left (a+b x^{n-q}\right ) \left (a x^q+b x^n\right )^p \, _2F_1\left (2,p+1;p+2;\frac{b x^{n-q}}{a}+1\right )}{a^2 (p+1) (n-q)} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 22.1997, size = 92, normalized size = 1.33 \[ \frac{x^{n - q} x^{- n - p q + q} \left (- \frac{b}{a}\right )^{\frac{2 n - p q + q \left (p - 1\right ) - q}{n - q}} \left (a + b x^{n - q}\right )^{- p} \left (a + b x^{n - q}\right )^{p + 1} \left (a x^{q} + b x^{n}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} 2, p + 1 \\ p + 2 \end{matrix}\middle |{1 + \frac{b x^{n - q}}{a}} \right )}}{b \left (n - q\right ) \left (p + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**(n - q)*x**(-n - p*q + q)*(-b/a)**((2*n - p*q + q*(p - 1) - q)/(n - q))*(a +
b*x**(n - q))**(-p)*(a + b*x**(n - q))**(p + 1)*(a*x**q + b*x**n)**p*hyper((2, p
+ 1), (p + 2,), 1 + b*x**(n - q)/a)/(b*(n - q)*(p + 1))

_______________________________________________________________________________________

Mathematica [A]  time = 0.140689, size = 82, normalized size = 1.19 \[ \frac{x^{-n-p q+q} \left (a x^q+b x^n\right )^p \left (\frac{a x^{q-n}}{b}+1\right )^{-p} \, _2F_1\left (1-p,-p;2-p;-\frac{a x^{q-n}}{b}\right )}{(p-1) (n-q)} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [F]  time = 0.343, size = 0, normalized size = 0. \[ \int{x}^{-1-n- \left ( -1+p \right ) q} \left ( b{x}^{n}+a{x}^{q} \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (b x^{n} + a x^{q}\right )}^{p} x^{-{\left (p - 1\right )} q - n - 1}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((b*x^n + a*x^q)^p*x^(-(p - 1)*q - n - 1), x)

_______________________________________________________________________________________

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

[Out]

Timed out

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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