3.3134 \(\int (a+b x)^m (c+d x)^n \, dx\)

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

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0740607, 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 \[ \frac{(a+b x)^{m+1} (c+d x)^n \left (\frac{b (c+d x)}{b c-a d}\right )^{-n} \, _2F_1\left (m+1,-n;m+2;-\frac{d (a+b x)}{b c-a d}\right )}{b (m+1)} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^m*(c + d*x)^n,x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 14.4892, size = 56, normalized size = 0.92 \[ \frac{\left (\frac{b \left (- c - d x\right )}{a d - b c}\right )^{- n} \left (a + b x\right )^{m + 1} \left (c + d x\right )^{n}{{}_{2}F_{1}\left (\begin{matrix} - n, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{d \left (a + b x\right )}{a d - b c}} \right )}}{b \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0828212, size = 73, normalized size = 1.2 \[ \frac{(a+b x)^m (c+d x)^{n+1} \left (\frac{d (a+b x)}{a d-b c}\right )^{-m} \, _2F_1\left (-m,n+1;n+2;\frac{b (c+d x)}{b c-a d}\right )}{d (n+1)} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [F]  time = 0.137, size = 0, normalized size = 0. \[ \int \left ( bx+a \right ) ^{m} \left ( dx+c \right ) ^{n}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^m*(d*x+c)^n,x)

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x + a)^m*(d*x + c)^n, x)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((b*x + a)^m*(d*x + c)^n, x)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x + a)^m*(d*x + c)^n, x)