3.3132 \(\int (a+b x)^m (c+d x)^n (e+f x)^2 \, dx\)

Optimal. Leaf size=259 \[ \frac{(a+b x)^{m+1} (c+d x)^{n+1} (f (a d (n+1)+b c (m+1)) (b d e (m+n+4)-f (a d (n+2)+b c (m+2)))+b d (m+n+2) (a f (c f+d e (n+1))+b e (c f (m+1)-d e (m+n+3)))) \, _2F_1\left (1,m+n+2;n+2;\frac{b (c+d x)}{b c-a d}\right )}{b^2 d^2 (n+1) (m+n+2) (m+n+3) (b c-a d)}+\frac{f (a+b x)^{m+1} (c+d x)^{n+1} (b d e (m+n+4)-f (a d (n+2)+b c (m+2)))}{b^2 d^2 (m+n+2) (m+n+3)}+\frac{f (e+f x) (a+b x)^{m+1} (c+d x)^{n+1}}{b d (m+n+3)} \]

[Out]

(f*(b*d*e*(4 + m + n) - f*(b*c*(2 + m) + a*d*(2 + n)))*(a + b*x)^(1 + m)*(c + d*
x)^(1 + n))/(b^2*d^2*(2 + m + n)*(3 + m + n)) + (f*(a + b*x)^(1 + m)*(c + d*x)^(
1 + n)*(e + f*x))/(b*d*(3 + m + n)) + ((f*(b*c*(1 + m) + a*d*(1 + n))*(b*d*e*(4
+ m + n) - f*(b*c*(2 + m) + a*d*(2 + n))) + b*d*(2 + m + n)*(a*f*(c*f + d*e*(1 +
n)) + b*e*(c*f*(1 + m) - d*e*(3 + m + n))))*(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^2*d^2*(b*
c - a*d)*(1 + n)*(2 + m + n)*(3 + m + n))

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Rubi [A]  time = 0.92876, antiderivative size = 272, normalized size of antiderivative = 1.05, number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{(a+b x)^{m+1} (c+d x)^n \left (\frac{b (c+d x)}{b c-a d}\right )^{-n} \left (b d (m+n+2) \left (b d e^2 (m+n+3)-f (a c f+a d e (n+1)+b c e (m+1))\right )-f (a d (n+1)+b c (m+1)) (b d e (m+n+4)-f (a d (n+2)+b c (m+2)))\right ) \, _2F_1\left (m+1,-n;m+2;-\frac{d (a+b x)}{b c-a d}\right )}{b^3 d^2 (m+1) (m+n+2) (m+n+3)}+\frac{f (a+b x)^{m+1} (c+d x)^{n+1} (b d e (m+n+4)-f (a d (n+2)+b c (m+2)))}{b^2 d^2 (m+n+2) (m+n+3)}+\frac{f (e+f x) (a+b x)^{m+1} (c+d x)^{n+1}}{b d (m+n+3)} \]

Antiderivative was successfully verified.

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

[Out]

(f*(b*d*e*(4 + m + n) - f*(b*c*(2 + m) + a*d*(2 + n)))*(a + b*x)^(1 + m)*(c + d*
x)^(1 + n))/(b^2*d^2*(2 + m + n)*(3 + m + n)) + (f*(a + b*x)^(1 + m)*(c + d*x)^(
1 + n)*(e + f*x))/(b*d*(3 + m + n)) + ((b*d*(2 + m + n)*(b*d*e^2*(3 + m + n) - f
*(a*c*f + b*c*e*(1 + m) + a*d*e*(1 + n))) - f*(b*c*(1 + m) + a*d*(1 + n))*(b*d*e
*(4 + m + n) - f*(b*c*(2 + m) + a*d*(2 + n))))*(a + b*x)^(1 + m)*(c + d*x)^n*Hyp
ergeometric2F1[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 = 99.0872, size = 248, normalized size = 0.96 \[ \frac{f \left (a + b x\right )^{m + 1} \left (c + d x\right )^{n + 1} \left (e + f x\right )}{b d \left (m + n + 3\right )} - \frac{f \left (a + b x\right )^{m + 1} \left (c + d x\right )^{n + 1} \left (- b d e \left (m + n + 4\right ) + f \left (a d \left (n + 2\right ) + b c \left (m + 2\right )\right )\right )}{b^{2} d^{2} \left (m + n + 2\right ) \left (m + n + 3\right )} + \frac{\left (\frac{b \left (- c - d x\right )}{a d - b c}\right )^{- n} \left (a + b x\right )^{m + 1} \left (c + d x\right )^{n} \left (- b d \left (- b d e^{2} \left (m + n + 3\right ) + f \left (a c f + e \left (a d \left (n + 1\right ) + b c \left (m + 1\right )\right )\right )\right ) \left (m + n + 2\right ) + f \left (a d \left (n + 1\right ) + b c \left (m + 1\right )\right ) \left (- b d e \left (m + n + 4\right ) + f \left (a d \left (n + 2\right ) + b c \left (m + 2\right )\right )\right )\right ){{}_{2}F_{1}\left (\begin{matrix} - n, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{d \left (a + b x\right )}{a d - b c}} \right )}}{b^{3} d^{2} \left (m + 1\right ) \left (m + n + 2\right ) \left (m + n + 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

f*(a + b*x)**(m + 1)*(c + d*x)**(n + 1)*(e + f*x)/(b*d*(m + n + 3)) - f*(a + b*x
)**(m + 1)*(c + d*x)**(n + 1)*(-b*d*e*(m + n + 4) + f*(a*d*(n + 2) + b*c*(m + 2)
))/(b**2*d**2*(m + n + 2)*(m + n + 3)) + (b*(-c - d*x)/(a*d - b*c))**(-n)*(a + b
*x)**(m + 1)*(c + d*x)**n*(-b*d*(-b*d*e**2*(m + n + 3) + f*(a*c*f + e*(a*d*(n +
1) + b*c*(m + 1))))*(m + n + 2) + f*(a*d*(n + 1) + b*c*(m + 1))*(-b*d*e*(m + n +
4) + f*(a*d*(n + 2) + b*c*(m + 2))))*hyper((-n, m + 1), (m + 2,), d*(a + b*x)/(
a*d - b*c))/(b**3*d**2*(m + 1)*(m + n + 2)*(m + n + 3))

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Mathematica [C]  time = 1.375, size = 330, normalized size = 1.27 \[ \frac{1}{3} (a+b x)^m (c+d x)^n \left (\frac{9 a c e f x^2 F_1\left (2;-m,-n;3;-\frac{b x}{a},-\frac{d x}{c}\right )}{3 a c F_1\left (2;-m,-n;3;-\frac{b x}{a},-\frac{d x}{c}\right )+b c m x F_1\left (3;1-m,-n;4;-\frac{b x}{a},-\frac{d x}{c}\right )+a d n x F_1\left (3;-m,1-n;4;-\frac{b x}{a},-\frac{d x}{c}\right )}+\frac{4 a c f^2 x^3 F_1\left (3;-m,-n;4;-\frac{b x}{a},-\frac{d x}{c}\right )}{4 a c F_1\left (3;-m,-n;4;-\frac{b x}{a},-\frac{d x}{c}\right )+b c m x F_1\left (4;1-m,-n;5;-\frac{b x}{a},-\frac{d x}{c}\right )+a d n x F_1\left (4;-m,1-n;5;-\frac{b x}{a},-\frac{d x}{c}\right )}+\frac{3 e^2 (c+d x) \left (\frac{d (a+b x)}{a d-b c}\right )^{-m} \, _2F_1\left (-m,n+1;n+2;\frac{b (c+d x)}{b c-a d}\right )}{d (n+1)}\right ) \]

Warning: Unable to verify antiderivative.

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

[Out]

((a + b*x)^m*(c + d*x)^n*((9*a*c*e*f*x^2*AppellF1[2, -m, -n, 3, -((b*x)/a), -((d
*x)/c)])/(3*a*c*AppellF1[2, -m, -n, 3, -((b*x)/a), -((d*x)/c)] + b*c*m*x*AppellF
1[3, 1 - m, -n, 4, -((b*x)/a), -((d*x)/c)] + a*d*n*x*AppellF1[3, -m, 1 - n, 4, -
((b*x)/a), -((d*x)/c)]) + (4*a*c*f^2*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*e^2*(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 = 0.098, size = 0, normalized size = 0. \[ \int \left ( bx+a \right ) ^{m} \left ( dx+c \right ) ^{n} \left ( fx+e \right ) ^{2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^m*(d*x+c)^n*(f*x+e)^2,x)

[Out]

int((b*x+a)^m*(d*x+c)^n*(f*x+e)^2,x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((f*x + e)^2*(b*x + a)^m*(d*x + c)^n, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((f^2*x^2 + 2*e*f*x + e^2)*(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*(f*x+e)**2,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((f*x + e)^2*(b*x + a)^m*(d*x + c)^n, x)