3.948 \(\int (a+b x)^n (c+d x)^p \, dx\)

Optimal. Leaf size=61 \[ -\frac{(a+b x)^{n+1} (c+d x)^{p+1} \, _2F_1\left (1,n+p+2;p+2;\frac{b (c+d x)}{b c-a d}\right )}{(p+1) (b c-a d)} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0651789, antiderivative size = 74, normalized size of antiderivative = 1.21, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{(a+b x)^{n+1} (c+d x)^p \left (\frac{b (c+d x)}{b c-a d}\right )^{-p} \, _2F_1\left (n+1,-p;n+2;-\frac{d (a+b x)}{b c-a d}\right )}{b (n+1)} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^n*(c + d*x)^p,x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 13.9201, size = 56, normalized size = 0.92 \[ \frac{\left (\frac{b \left (- c - d x\right )}{a d - b c}\right )^{- p} \left (a + b x\right )^{n + 1} \left (c + d x\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, n + 1 \\ n + 2 \end{matrix}\middle |{\frac{d \left (a + b x\right )}{a d - b c}} \right )}}{b \left (n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0946055, size = 73, normalized size = 1.2 \[ \frac{(a+b x)^n (c+d x)^{p+1} \left (\frac{d (a+b x)}{a d-b c}\right )^{-n} \, _2F_1\left (-n,p+1;p+2;\frac{b (c+d x)}{b c-a d}\right )}{d (p+1)} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [F]  time = 0.121, size = 0, normalized size = 0. \[ \int \left ( bx+a \right ) ^{n} \left ( dx+c \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^n*(d*x+c)^p,x)

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x + a)^n*(d*x + c)^p, x)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x + a)^n*(d*x + c)^p, x)