3.3106 \(\int (a+b x)^m (c+d x)^{2-m} (e+f x)^3 \, dx\)

Optimal. Leaf size=447 \[ \frac{f (a+b x)^{m+1} (c+d x)^{3-m} \left (a^2 d^2 f^2 \left (m^2-9 m+20\right )-2 a b d f \left (9 d e (4-m)-c f \left (-m^2+2 m+6\right )\right )-4 b d f x (a d f (5-m)-b (8 d e-c f (m+3)))+b^2 \left (c^2 f^2 \left (m^2+5 m+6\right )-18 c d e f (m+2)+70 d^2 e^2\right )\right )}{120 b^3 d^3}-\frac{(b c-a d)^2 (a+b x)^{m+1} (c+d x)^{-m} \left (\frac{b (c+d x)}{b c-a d}\right )^m \left (a^3 d^3 f^3 \left (-m^3+12 m^2-47 m+60\right )-3 a^2 b d^2 f^2 \left (m^2-7 m+12\right ) (6 d e-c f (m+1))+3 a b^2 d f (3-m) \left (c^2 f^2 \left (m^2+3 m+2\right )-12 c d e f (m+1)+30 d^2 e^2\right )+b^3 \left (-\left (-c^3 f^3 \left (m^3+6 m^2+11 m+6\right )+18 c^2 d e f^2 \left (m^2+3 m+2\right )-90 c d^2 e^2 f (m+1)+120 d^3 e^3\right )\right )\right ) \, _2F_1\left (m-2,m+1;m+2;-\frac{d (a+b x)}{b c-a d}\right )}{120 b^6 d^3 (m+1)}+\frac{f (e+f x)^2 (a+b x)^{m+1} (c+d x)^{3-m}}{6 b d} \]

[Out]

(f*(a + b*x)^(1 + m)*(c + d*x)^(3 - m)*(e + f*x)^2)/(6*b*d) + (f*(a + b*x)^(1 +
m)*(c + d*x)^(3 - m)*(a^2*d^2*f^2*(20 - 9*m + m^2) - 2*a*b*d*f*(9*d*e*(4 - m) -
c*f*(6 + 2*m - m^2)) + b^2*(70*d^2*e^2 - 18*c*d*e*f*(2 + m) + c^2*f^2*(6 + 5*m +
m^2)) - 4*b*d*f*(a*d*f*(5 - m) - b*(8*d*e - c*f*(3 + m)))*x))/(120*b^3*d^3) - (
(b*c - a*d)^2*(a^3*d^3*f^3*(60 - 47*m + 12*m^2 - m^3) - 3*a^2*b*d^2*f^2*(12 - 7*
m + m^2)*(6*d*e - c*f*(1 + m)) + 3*a*b^2*d*f*(3 - m)*(30*d^2*e^2 - 12*c*d*e*f*(1
+ m) + c^2*f^2*(2 + 3*m + m^2)) - b^3*(120*d^3*e^3 - 90*c*d^2*e^2*f*(1 + m) + 1
8*c^2*d*e*f^2*(2 + 3*m + m^2) - c^3*f^3*(6 + 11*m + 6*m^2 + m^3)))*(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))])/(120*b^6*d^3*(1 + m)*(c + d*x)^m)

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Rubi [A]  time = 1.36539, antiderivative size = 446, 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 \[ \frac{f (a+b x)^{m+1} (c+d x)^{3-m} \left (a^2 d^2 f^2 \left (m^2-9 m+20\right )-2 a b d f \left (9 d e (4-m)-c f \left (-m^2+2 m+6\right )\right )+4 b d f x (-a d f (5-m)-b c f (m+3)+8 b d e)+b^2 \left (c^2 f^2 \left (m^2+5 m+6\right )-18 c d e f (m+2)+70 d^2 e^2\right )\right )}{120 b^3 d^3}-\frac{(b c-a d)^2 (a+b x)^{m+1} (c+d x)^{-m} \left (\frac{b (c+d x)}{b c-a d}\right )^m \left (a^3 d^3 f^3 \left (-m^3+12 m^2-47 m+60\right )-3 a^2 b d^2 f^2 \left (m^2-7 m+12\right ) (6 d e-c f (m+1))+3 a b^2 d f (3-m) \left (c^2 f^2 \left (m^2+3 m+2\right )-12 c d e f (m+1)+30 d^2 e^2\right )+b^3 \left (-\left (-c^3 f^3 \left (m^3+6 m^2+11 m+6\right )+18 c^2 d e f^2 \left (m^2+3 m+2\right )-90 c d^2 e^2 f (m+1)+120 d^3 e^3\right )\right )\right ) \, _2F_1\left (m-2,m+1;m+2;-\frac{d (a+b x)}{b c-a d}\right )}{120 b^6 d^3 (m+1)}+\frac{f (e+f x)^2 (a+b x)^{m+1} (c+d x)^{3-m}}{6 b d} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^m*(c + d*x)^(2 - m)*(e + f*x)^3,x]

[Out]

(f*(a + b*x)^(1 + m)*(c + d*x)^(3 - m)*(e + f*x)^2)/(6*b*d) + (f*(a + b*x)^(1 +
m)*(c + d*x)^(3 - m)*(a^2*d^2*f^2*(20 - 9*m + m^2) - 2*a*b*d*f*(9*d*e*(4 - m) -
c*f*(6 + 2*m - m^2)) + b^2*(70*d^2*e^2 - 18*c*d*e*f*(2 + m) + c^2*f^2*(6 + 5*m +
m^2)) + 4*b*d*f*(8*b*d*e - a*d*f*(5 - m) - b*c*f*(3 + m))*x))/(120*b^3*d^3) - (
(b*c - a*d)^2*(a^3*d^3*f^3*(60 - 47*m + 12*m^2 - m^3) - 3*a^2*b*d^2*f^2*(12 - 7*
m + m^2)*(6*d*e - c*f*(1 + m)) + 3*a*b^2*d*f*(3 - m)*(30*d^2*e^2 - 12*c*d*e*f*(1
+ m) + c^2*f^2*(2 + 3*m + m^2)) - b^3*(120*d^3*e^3 - 90*c*d^2*e^2*f*(1 + m) + 1
8*c^2*d*e*f^2*(2 + 3*m + m^2) - c^3*f^3*(6 + 11*m + 6*m^2 + m^3)))*(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))])/(120*b^6*d^3*(1 + m)*(c + d*x)^m)

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

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)**3,x)

[Out]

Timed out

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Mathematica [C]  time = 4.29928, size = 467, normalized size = 1.04 \[ \frac{1}{4} (a+b x)^m (c+d x)^{2-m} \left (\frac{18 a c e^2 f x^2 F_1\left (2;-m,m-2;3;-\frac{b x}{a},-\frac{d x}{c}\right )}{3 a c F_1\left (2;-m,m-2;3;-\frac{b x}{a},-\frac{d x}{c}\right )+b c m x F_1\left (3;1-m,m-2;4;-\frac{b x}{a},-\frac{d x}{c}\right )-a d (m-2) x F_1\left (3;-m,m-1;4;-\frac{b x}{a},-\frac{d x}{c}\right )}+\frac{16 a c e f^2 x^3 F_1\left (3;-m,m-2;4;-\frac{b x}{a},-\frac{d x}{c}\right )}{4 a c F_1\left (3;-m,m-2;4;-\frac{b x}{a},-\frac{d x}{c}\right )+b c m x F_1\left (4;1-m,m-2;5;-\frac{b x}{a},-\frac{d x}{c}\right )-a d (m-2) x F_1\left (4;-m,m-1;5;-\frac{b x}{a},-\frac{d x}{c}\right )}+\frac{5 a c f^3 x^4 F_1\left (4;-m,m-2;5;-\frac{b x}{a},-\frac{d x}{c}\right )}{5 a c F_1\left (4;-m,m-2;5;-\frac{b x}{a},-\frac{d x}{c}\right )+b c m x F_1\left (5;1-m,m-2;6;-\frac{b x}{a},-\frac{d x}{c}\right )-a d (m-2) x F_1\left (5;-m,m-1;6;-\frac{b x}{a},-\frac{d x}{c}\right )}-\frac{4 e^3 (c+d x) \left (\frac{d (a+b x)}{a d-b c}\right )^{-m} \, _2F_1\left (3-m,-m;4-m;\frac{b (c+d x)}{b c-a d}\right )}{d (m-3)}\right ) \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(a + b*x)^m*(c + d*x)^(2 - m)*(e + f*x)^3,x]

[Out]

((a + b*x)^m*(c + d*x)^(2 - m)*((18*a*c*e^2*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
*AppellF1[3, -m, -1 + m, 4, -((b*x)/a), -((d*x)/c)]) + (16*a*c*e*f^2*x^3*AppellF
1[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)]) + (5
*a*c*f^3*x^4*AppellF1[4, -m, -2 + m, 5, -((b*x)/a), -((d*x)/c)])/(5*a*c*AppellF1
[4, -m, -2 + m, 5, -((b*x)/a), -((d*x)/c)] + b*c*m*x*AppellF1[5, 1 - m, -2 + m,
6, -((b*x)/a), -((d*x)/c)] - a*d*(-2 + m)*x*AppellF1[5, -m, -1 + m, 6, -((b*x)/a
), -((d*x)/c)]) - (4*e^3*(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)))/4

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Maple [F]  time = 0.099, size = 0, normalized size = 0. \[ \int \left ( bx+a \right ) ^{m} \left ( dx+c \right ) ^{2-m} \left ( fx+e \right ) ^{3}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^m*(d*x+c)^(2-m)*(f*x+e)^3,x)

[Out]

int((b*x+a)^m*(d*x+c)^(2-m)*(f*x+e)^3,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (f x + e\right )}^{3}{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m + 2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x + e)^3*(b*x + a)^m*(d*x + c)^(-m + 2),x, algorithm="maxima")

[Out]

integrate((f*x + e)^3*(b*x + a)^m*(d*x + c)^(-m + 2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (f^{3} x^{3} + 3 \, e f^{2} x^{2} + 3 \, e^{2} f x + e^{3}\right )}{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m + 2}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x + e)^3*(b*x + a)^m*(d*x + c)^(-m + 2),x, algorithm="fricas")

[Out]

integral((f^3*x^3 + 3*e*f^2*x^2 + 3*e^2*f*x + e^3)*(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)**3,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (f x + e\right )}^{3}{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m + 2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x + e)^3*(b*x + a)^m*(d*x + c)^(-m + 2),x, algorithm="giac")

[Out]

integrate((f*x + e)^3*(b*x + a)^m*(d*x + c)^(-m + 2), x)