3.3107 \(\int (a+b x)^m (c+d x)^{2-m} (e+f x)^2 \, dx\)

Optimal. Leaf size=262 \[ \frac{(b c-a d)^2 (a+b x)^{m+1} (c+d x)^{-m} \left (\frac{b (c+d x)}{b c-a d}\right )^m \left (a^2 d^2 f^2 \left (m^2-7 m+12\right )-2 a b d f (3-m) (5 d e-c f (m+1))+b^2 \left (c^2 f^2 \left (m^2+3 m+2\right )-10 c d e f (m+1)+20 d^2 e^2\right )\right ) \, _2F_1\left (m-2,m+1;m+2;-\frac{d (a+b x)}{b c-a d}\right )}{20 b^5 d^2 (m+1)}-\frac{f (a+b x)^{m+1} (c+d x)^{3-m} (a d f (4-m)-b (6 d e-c f (m+2)))}{20 b^2 d^2}+\frac{f (e+f x) (a+b x)^{m+1} (c+d x)^{3-m}}{5 b d} \]

[Out]

-(f*(a*d*f*(4 - m) - b*(6*d*e - c*f*(2 + m)))*(a + b*x)^(1 + m)*(c + d*x)^(3 - m
))/(20*b^2*d^2) + (f*(a + b*x)^(1 + m)*(c + d*x)^(3 - m)*(e + f*x))/(5*b*d) + ((
b*c - a*d)^2*(a^2*d^2*f^2*(12 - 7*m + m^2) - 2*a*b*d*f*(3 - m)*(5*d*e - c*f*(1 +
m)) + b^2*(20*d^2*e^2 - 10*c*d*e*f*(1 + m) + c^2*f^2*(2 + 3*m + m^2)))*(a + b*x
)^(1 + m)*((b*(c + d*x))/(b*c - a*d))^m*Hypergeometric2F1[-2 + m, 1 + m, 2 + m,
-((d*(a + b*x))/(b*c - a*d))])/(20*b^5*d^2*(1 + m)*(c + d*x)^m)

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Rubi [A]  time = 0.634567, antiderivative size = 261, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{(b c-a d)^2 (a+b x)^{m+1} (c+d x)^{-m} \left (\frac{b (c+d x)}{b c-a d}\right )^m \left (a^2 d^2 f^2 \left (m^2-7 m+12\right )-2 a b d f (3-m) (5 d e-c f (m+1))+b^2 \left (c^2 f^2 \left (m^2+3 m+2\right )-10 c d e f (m+1)+20 d^2 e^2\right )\right ) \, _2F_1\left (m-2,m+1;m+2;-\frac{d (a+b x)}{b c-a d}\right )}{20 b^5 d^2 (m+1)}+\frac{f (a+b x)^{m+1} (c+d x)^{3-m} (-a d f (4-m)-b c f (m+2)+6 b d e)}{20 b^2 d^2}+\frac{f (e+f x) (a+b x)^{m+1} (c+d x)^{3-m}}{5 b d} \]

Antiderivative was successfully verified.

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

[Out]

(f*(6*b*d*e - a*d*f*(4 - m) - b*c*f*(2 + m))*(a + b*x)^(1 + m)*(c + d*x)^(3 - m)
)/(20*b^2*d^2) + (f*(a + b*x)^(1 + m)*(c + d*x)^(3 - m)*(e + f*x))/(5*b*d) + ((b
*c - a*d)^2*(a^2*d^2*f^2*(12 - 7*m + m^2) - 2*a*b*d*f*(3 - m)*(5*d*e - c*f*(1 +
m)) + b^2*(20*d^2*e^2 - 10*c*d*e*f*(1 + m) + c^2*f^2*(2 + 3*m + m^2)))*(a + b*x)
^(1 + m)*((b*(c + d*x))/(b*c - a*d))^m*Hypergeometric2F1[-2 + m, 1 + m, 2 + m, -
((d*(a + b*x))/(b*c - a*d))])/(20*b^5*d^2*(1 + m)*(c + d*x)^m)

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Rubi in Sympy [A]  time = 62.4157, size = 223, normalized size = 0.85 \[ \frac{f \left (a + b x\right )^{m + 1} \left (c + d x\right )^{- m + 3} \left (e + f x\right )}{5 b d} - \frac{f \left (a + b x\right )^{m + 1} \left (c + d x\right )^{- m + 3} \left (- 6 b d e + f \left (a d \left (- m + 4\right ) + b c \left (m + 2\right )\right )\right )}{20 b^{2} d^{2}} - \frac{\left (\frac{b \left (- c - d x\right )}{a d - b c}\right )^{m} \left (a + b x\right )^{m + 1} \left (c + d x\right )^{- m} \left (a d - b c\right )^{2} \left (4 b d \left (- 5 b d e^{2} + f \left (a c f + e \left (a d \left (- m + 3\right ) + b c \left (m + 1\right )\right )\right )\right ) - f \left (a d \left (- m + 3\right ) + b c \left (m + 1\right )\right ) \left (- 6 b d e + f \left (a d \left (- m + 4\right ) + b c \left (m + 2\right )\right )\right )\right ){{}_{2}F_{1}\left (\begin{matrix} m - 2, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{d \left (a + b x\right )}{a d - b c}} \right )}}{20 b^{5} d^{2} \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

f*(a + b*x)**(m + 1)*(c + d*x)**(-m + 3)*(e + f*x)/(5*b*d) - f*(a + b*x)**(m + 1
)*(c + d*x)**(-m + 3)*(-6*b*d*e + f*(a*d*(-m + 4) + b*c*(m + 2)))/(20*b**2*d**2)
- (b*(-c - d*x)/(a*d - b*c))**m*(a + b*x)**(m + 1)*(c + d*x)**(-m)*(a*d - b*c)*
*2*(4*b*d*(-5*b*d*e**2 + f*(a*c*f + e*(a*d*(-m + 3) + b*c*(m + 1)))) - f*(a*d*(-
m + 3) + b*c*(m + 1))*(-6*b*d*e + f*(a*d*(-m + 4) + b*c*(m + 2))))*hyper((m - 2,
m + 1), (m + 2,), d*(a + b*x)/(a*d - b*c))/(20*b**5*d**2*(m + 1))

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Mathematica [C]  time = 0.894274, size = 340, normalized size = 1.3 \[ \frac{1}{3} (a+b x)^m (c+d x)^{2-m} \left (\frac{9 a c e f x^2 F_1\left (2;-m,m-2;3;-\frac{b x}{a},-\frac{d x}{c}\right )}{3 a c F_1\left (2;-m,m-2;3;-\frac{b x}{a},-\frac{d x}{c}\right )+b c m x F_1\left (3;1-m,m-2;4;-\frac{b x}{a},-\frac{d x}{c}\right )-a d (m-2) x F_1\left (3;-m,m-1;4;-\frac{b x}{a},-\frac{d x}{c}\right )}+\frac{4 a c f^2 x^3 F_1\left (3;-m,m-2;4;-\frac{b x}{a},-\frac{d x}{c}\right )}{4 a c F_1\left (3;-m,m-2;4;-\frac{b x}{a},-\frac{d x}{c}\right )+b c m x F_1\left (4;1-m,m-2;5;-\frac{b x}{a},-\frac{d x}{c}\right )-a d (m-2) x F_1\left (4;-m,m-1;5;-\frac{b x}{a},-\frac{d x}{c}\right )}-\frac{3 e^2 (c+d x) \left (\frac{d (a+b x)}{a d-b c}\right )^{-m} \, _2F_1\left (3-m,-m;4-m;\frac{b (c+d x)}{b c-a d}\right )}{d (m-3)}\right ) \]

Warning: Unable to verify antiderivative.

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

[Out]

((a + b*x)^m*(c + d*x)^(2 - m)*((9*a*c*e*f*x^2*AppellF1[2, -m, -2 + m, 3, -((b*x
)/a), -((d*x)/c)])/(3*a*c*AppellF1[2, -m, -2 + m, 3, -((b*x)/a), -((d*x)/c)] + b
*c*m*x*AppellF1[3, 1 - m, -2 + m, 4, -((b*x)/a), -((d*x)/c)] - a*d*(-2 + m)*x*Ap
pellF1[3, -m, -1 + m, 4, -((b*x)/a), -((d*x)/c)]) + (4*a*c*f^2*x^3*AppellF1[3, -
m, -2 + m, 4, -((b*x)/a), -((d*x)/c)])/(4*a*c*AppellF1[3, -m, -2 + m, 4, -((b*x)
/a), -((d*x)/c)] + b*c*m*x*AppellF1[4, 1 - m, -2 + m, 5, -((b*x)/a), -((d*x)/c)]
- a*d*(-2 + m)*x*AppellF1[4, -m, -1 + m, 5, -((b*x)/a), -((d*x)/c)]) - (3*e^2*(
c + d*x)*Hypergeometric2F1[3 - m, -m, 4 - m, (b*(c + d*x))/(b*c - a*d)])/(d*(-3
+ m)*((d*(a + b*x))/(-(b*c) + a*d))^m)))/3

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Maple [F]  time = 0.098, size = 0, normalized size = 0. \[ \int \left ( bx+a \right ) ^{m} \left ( dx+c \right ) ^{2-m} \left ( fx+e \right ) ^{2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((f^2*x^2 + 2*e*f*x + e^2)*(b*x + a)^m*(d*x + c)^(-m + 2), 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)**(2-m)*(f*x+e)**2,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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