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

Optimal. Leaf size=63 \[ \frac{x^{m+1} \left (c x^2\right )^p (a+b x)^n \left (\frac{b x}{a}+1\right )^{-n} \, _2F_1\left (-n,m+2 p+1;m+2 p+2;-\frac{b x}{a}\right )}{m+2 p+1} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0490944, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{x^{m+1} \left (c x^2\right )^p (a+b x)^n \left (\frac{b x}{a}+1\right )^{-n} \, _2F_1\left (-n,m+2 p+1;m+2 p+2;-\frac{b x}{a}\right )}{m+2 p+1} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 17.1306, size = 61, normalized size = 0.97 \[ \frac{x^{- 2 p} x^{m + 2 p + 1} \left (c x^{2}\right )^{p} \left (1 + \frac{b x}{a}\right )^{- n} \left (a + b x\right )^{n}{{}_{2}F_{1}\left (\begin{matrix} - n, m + 2 p + 1 \\ m + 2 p + 2 \end{matrix}\middle |{- \frac{b x}{a}} \right )}}{m + 2 p + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0551248, size = 63, normalized size = 1. \[ \frac{x^{m+1} \left (c x^2\right )^p (a+b x)^n \left (\frac{b x}{a}+1\right )^{-n} \, _2F_1\left (-n,m+2 p+1;m+2 p+2;-\frac{b x}{a}\right )}{m+2 p+1} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [F]  time = 0.182, size = 0, normalized size = 0. \[ \int{x}^{m} \left ( c{x}^{2} \right ) ^{p} \left ( bx+a \right ) ^{n}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^m*(c*x^2)^p*(b*x+a)^n,x)

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**m*(c*x**2)**p*(b*x+a)**n,x)

[Out]

Integral(x**m*(c*x**2)**p*(a + b*x)**n, x)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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