3.32 \(\int (a+b x)^m (A+B x) (c+d x)^n \, dx\)

Optimal. Leaf size=141 \[ \frac{(a+b x)^{m+1} (c+d x)^n \left (\frac{b (c+d x)}{b c-a d}\right )^{-n} (A b d (m+n+2)-B (a d (n+1)+b c (m+1))) \, _2F_1\left (m+1,-n;m+2;-\frac{d (a+b x)}{b c-a d}\right )}{b^2 d (m+1) (m+n+2)}+\frac{B (a+b x)^{m+1} (c+d x)^{n+1}}{b d (m+n+2)} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.229908, antiderivative size = 141, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ \frac{(a+b x)^{m+1} (c+d x)^n \left (\frac{b (c+d x)}{b c-a d}\right )^{-n} (A b d (m+n+2)-B (a d (n+1)+b c (m+1))) \, _2F_1\left (m+1,-n;m+2;-\frac{d (a+b x)}{b c-a d}\right )}{b^2 d (m+1) (m+n+2)}+\frac{B (a+b x)^{m+1} (c+d x)^{n+1}}{b d (m+n+2)} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 32.1858, size = 117, normalized size = 0.83 \[ \frac{B \left (a + b x\right )^{m + 1} \left (c + d x\right )^{n + 1}}{b d \left (m + n + 2\right )} - \frac{\left (\frac{b \left (- c - d x\right )}{a d - b c}\right )^{- n} \left (a + b x\right )^{m + 1} \left (c + d x\right )^{n} \left (- A b d \left (m + n + 2\right ) + B \left (a d \left (n + 1\right ) + b c \left (m + 1\right )\right )\right ){{}_{2}F_{1}\left (\begin{matrix} - n, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{d \left (a + b x\right )}{a d - b c}} \right )}}{b^{2} d \left (m + 1\right ) \left (m + n + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [C]  time = 0.510487, size = 202, normalized size = 1.43 \[ (a+b x)^m (c+d x)^n \left (\frac{A (c+d x) \left (\frac{d (a+b x)}{a d-b c}\right )^{-m} \, _2F_1\left (-m,n+1;n+2;\frac{b (c+d x)}{b c-a d}\right )}{d (n+1)}+\frac{3 a B c x^2 F_1\left (2;-m,-n;3;-\frac{b x}{a},-\frac{d x}{c}\right )}{6 a c F_1\left (2;-m,-n;3;-\frac{b x}{a},-\frac{d x}{c}\right )+2 b c m x F_1\left (3;1-m,-n;4;-\frac{b x}{a},-\frac{d x}{c}\right )+2 a d n x F_1\left (3;-m,1-n;4;-\frac{b x}{a},-\frac{d x}{c}\right )}\right ) \]

Warning: Unable to verify antiderivative.

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

[Out]

(a + b*x)^m*(c + d*x)^n*((3*a*B*c*x^2*AppellF1[2, -m, -n, 3, -((b*x)/a), -((d*x)
/c)])/(6*a*c*AppellF1[2, -m, -n, 3, -((b*x)/a), -((d*x)/c)] + 2*b*c*m*x*AppellF1
[3, 1 - m, -n, 4, -((b*x)/a), -((d*x)/c)] + 2*a*d*n*x*AppellF1[3, -m, 1 - n, 4,
-((b*x)/a), -((d*x)/c)]) + (A*(c + d*x)*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 = 0.082, size = 0, normalized size = 0. \[ \int \left ( bx+a \right ) ^{m} \left ( Bx+A \right ) \left ( dx+c \right ) ^{n}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((B*x + A)*(b*x + a)^m*(d*x + c)^n, x)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((B*x + A)*(b*x + a)^m*(d*x + c)^n, 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)**m*(B*x+A)*(d*x+c)**n,x)

[Out]

Timed out

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((B*x + A)*(b*x + a)^m*(d*x + c)^n, x)