3.33 \(\int (a+b x)^m (c+d x)^n \left (A+B x+C x^2\right ) \, dx\)

Optimal. Leaf size=268 \[ -\frac{(a+b x)^{m+1} (c+d x)^n \left (\frac{b (c+d x)}{b c-a d}\right )^{-n} \, _2F_1\left (m+1,-n;m+2;-\frac{d (a+b x)}{b c-a d}\right ) \left (d (m+n+2) \left (a^2 C d (n+1)+a b c C (m+2)-A b^2 d (m+n+3)\right )-(a d (n+1)+b c (m+1)) (a C d (m+2 n+4)+b (c C (m+2)-B d (m+n+3)))\right )}{b^3 d^2 (m+1) (m+n+2) (m+n+3)}-\frac{(a+b x)^{m+1} (c+d x)^{n+1} (a C d (m+2 n+4)+b (c C (m+2)-B d (m+n+3)))}{b^2 d^2 (m+n+2) (m+n+3)}+\frac{C (a+b x)^{m+2} (c+d x)^{n+1}}{b^2 d (m+n+3)} \]

[Out]

-(((a*C*d*(4 + m + 2*n) + b*(c*C*(2 + m) - B*d*(3 + m + n)))*(a + b*x)^(1 + m)*(
c + d*x)^(1 + n))/(b^2*d^2*(2 + m + n)*(3 + m + n))) + (C*(a + b*x)^(2 + m)*(c +
d*x)^(1 + n))/(b^2*d*(3 + m + n)) - ((d*(2 + m + n)*(a*b*c*C*(2 + m) + a^2*C*d*
(1 + n) - A*b^2*d*(3 + m + n)) - (b*c*(1 + m) + a*d*(1 + n))*(a*C*d*(4 + m + 2*n
) + b*(c*C*(2 + m) - B*d*(3 + m + n))))*(a + b*x)^(1 + m)*(c + d*x)^n*Hypergeome
tric2F1[1 + m, -n, 2 + m, -((d*(a + b*x))/(b*c - a*d))])/(b^3*d^2*(1 + m)*(2 + m
+ n)*(3 + m + n)*((b*(c + d*x))/(b*c - a*d))^n)

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Rubi [A]  time = 0.719828, antiderivative size = 266, normalized size of antiderivative = 0.99, number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16 \[ -\frac{(a+b x)^{m+1} (c+d x)^n \left (\frac{b (c+d x)}{b c-a d}\right )^{-n} \, _2F_1\left (m+1,-n;m+2;-\frac{d (a+b x)}{b c-a d}\right ) \left (d (m+n+2) \left (a^2 C d (n+1)+a b c C (m+2)-A b^2 d (m+n+3)\right )-(a d (n+1)+b c (m+1)) (a C d (m+2 n+4)-b B d (m+n+3)+b c C (m+2))\right )}{b^3 d^2 (m+1) (m+n+2) (m+n+3)}-\frac{(a+b x)^{m+1} (c+d x)^{n+1} (a C d (m+2 n+4)-b B d (m+n+3)+b c C (m+2))}{b^2 d^2 (m+n+2) (m+n+3)}+\frac{C (a+b x)^{m+2} (c+d x)^{n+1}}{b^2 d (m+n+3)} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^m*(c + d*x)^n*(A + B*x + C*x^2),x]

[Out]

-(((b*c*C*(2 + m) - b*B*d*(3 + m + n) + a*C*d*(4 + m + 2*n))*(a + b*x)^(1 + m)*(
c + d*x)^(1 + n))/(b^2*d^2*(2 + m + n)*(3 + m + n))) + (C*(a + b*x)^(2 + m)*(c +
d*x)^(1 + n))/(b^2*d*(3 + m + n)) - ((d*(2 + m + n)*(a*b*c*C*(2 + m) + a^2*C*d*
(1 + n) - A*b^2*d*(3 + m + n)) - (b*c*(1 + m) + a*d*(1 + n))*(b*c*C*(2 + m) - b*
B*d*(3 + m + n) + a*C*d*(4 + m + 2*n)))*(a + b*x)^(1 + m)*(c + d*x)^n*Hypergeome
tric2F1[1 + m, -n, 2 + m, -((d*(a + b*x))/(b*c - a*d))])/(b^3*d^2*(1 + m)*(2 + m
+ n)*(3 + m + n)*((b*(c + d*x))/(b*c - a*d))^n)

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Rubi in Sympy [A]  time = 92.3149, size = 230, normalized size = 0.86 \[ \frac{C \left (\frac{b \left (- c - d x\right )}{a d - b c}\right )^{- n} \left (a + b x\right )^{m + 1} \left (c + d x\right )^{n} \left (a d - b c\right )^{2}{{}_{2}F_{1}\left (\begin{matrix} - n - 2, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{d \left (a + b x\right )}{a d - b c}} \right )}}{b^{3} d^{2} \left (m + 1\right )} + \frac{\left (\frac{b \left (- c - d x\right )}{a d - b c}\right )^{- n} \left (a + b x\right )^{m + 1} \left (c + d x\right )^{n} \left (A d^{2} - B c d + C c^{2}\right ){{}_{2}F_{1}\left (\begin{matrix} - n, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{d \left (a + b x\right )}{a d - b c}} \right )}}{b d^{2} \left (m + 1\right )} - \frac{\left (\frac{b \left (- c - d x\right )}{a d - b c}\right )^{- n} \left (a + b x\right )^{m + 1} \left (c + d x\right )^{n} \left (B d - 2 C c\right ) \left (a d - b c\right ){{}_{2}F_{1}\left (\begin{matrix} - n - 1, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{d \left (a + b x\right )}{a d - b c}} \right )}}{b^{2} d^{2} \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**m*(d*x+c)**n*(C*x**2+B*x+A),x)

[Out]

C*(b*(-c - d*x)/(a*d - b*c))**(-n)*(a + b*x)**(m + 1)*(c + d*x)**n*(a*d - b*c)**
2*hyper((-n - 2, m + 1), (m + 2,), d*(a + b*x)/(a*d - b*c))/(b**3*d**2*(m + 1))
+ (b*(-c - d*x)/(a*d - b*c))**(-n)*(a + b*x)**(m + 1)*(c + d*x)**n*(A*d**2 - B*c
*d + C*c**2)*hyper((-n, m + 1), (m + 2,), d*(a + b*x)/(a*d - b*c))/(b*d**2*(m +
1)) - (b*(-c - d*x)/(a*d - b*c))**(-n)*(a + b*x)**(m + 1)*(c + d*x)**n*(B*d - 2*
C*c)*(a*d - b*c)*hyper((-n - 1, m + 1), (m + 2,), d*(a + b*x)/(a*d - b*c))/(b**2
*d**2*(m + 1))

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Mathematica [C]  time = 0.970823, size = 327, normalized size = 1.22 \[ \frac{1}{3} (a+b x)^m (c+d x)^n \left (\frac{3 A (c+d x) \left (\frac{d (a+b x)}{a d-b c}\right )^{-m} \, _2F_1\left (-m,n+1;n+2;\frac{b (c+d x)}{b c-a d}\right )}{d (n+1)}+\frac{9 a B c x^2 F_1\left (2;-m,-n;3;-\frac{b x}{a},-\frac{d x}{c}\right )}{6 a c F_1\left (2;-m,-n;3;-\frac{b x}{a},-\frac{d x}{c}\right )+2 b c m x F_1\left (3;1-m,-n;4;-\frac{b x}{a},-\frac{d x}{c}\right )+2 a d n x F_1\left (3;-m,1-n;4;-\frac{b x}{a},-\frac{d x}{c}\right )}+\frac{4 a c C x^3 F_1\left (3;-m,-n;4;-\frac{b x}{a},-\frac{d x}{c}\right )}{4 a c F_1\left (3;-m,-n;4;-\frac{b x}{a},-\frac{d x}{c}\right )+b c m x F_1\left (4;1-m,-n;5;-\frac{b x}{a},-\frac{d x}{c}\right )+a d n x F_1\left (4;-m,1-n;5;-\frac{b x}{a},-\frac{d x}{c}\right )}\right ) \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(a + b*x)^m*(c + d*x)^n*(A + B*x + C*x^2),x]

[Out]

((a + b*x)^m*(c + d*x)^n*((9*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*AppellF
1[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)]) + (4*a*c*C*x^3*AppellF1[3, -m, -n, 4, -((b*x)/a), -((d
*x)/c)])/(4*a*c*AppellF1[3, -m, -n, 4, -((b*x)/a), -((d*x)/c)] + b*c*m*x*AppellF
1[4, 1 - m, -n, 5, -((b*x)/a), -((d*x)/c)] + a*d*n*x*AppellF1[4, -m, 1 - n, 5, -
((b*x)/a), -((d*x)/c)]) + (3*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)))/3

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Maple [F]  time = 0.092, size = 0, normalized size = 0. \[ \int \left ( bx+a \right ) ^{m} \left ( dx+c \right ) ^{n} \left ( C{x}^{2}+Bx+A \right ) \, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^m*(d*x+c)^n*(C*x^2+B*x+A),x)

[Out]

int((b*x+a)^m*(d*x+c)^n*(C*x^2+B*x+A),x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((C*x^2 + B*x + A)*(b*x + a)^m*(d*x + c)^n, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((C*x^2 + B*x + A)*(b*x + a)^m*(d*x + c)^n, x)

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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*(d*x+c)**n*(C*x**2+B*x+A),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((C*x^2 + B*x + A)*(b*x + a)^m*(d*x + c)^n, x)