3.3096 \(\int (a+b x)^m (c+d x)^{1-m} (e+f x)^3 \, dx\)

Optimal. Leaf size=445 \[ \frac{f (a+b x)^{m+1} (c+d x)^{2-m} \left (a^2 d^2 f^2 \left (m^2-7 m+12\right )-a b d f \left (15 d e (3-m)-c f \left (-2 m^2+2 m+9\right )\right )-3 b d f x (a d f (4-m)-b (7 d e-c f (m+3)))+b^2 \left (c^2 f^2 \left (m^2+5 m+6\right )-15 c d e f (m+2)+48 d^2 e^2\right )\right )}{60 b^3 d^3}-\frac{(b c-a d) (a+b x)^{m+1} (c+d x)^{-m} \left (\frac{b (c+d x)}{b c-a d}\right )^m \left (a^3 d^3 f^3 \left (-m^3+9 m^2-26 m+24\right )-3 a^2 b d^2 f^2 \left (m^2-5 m+6\right ) (5 d e-c f (m+1))+3 a b^2 d f (2-m) \left (c^2 f^2 \left (m^2+3 m+2\right )-10 c d e f (m+1)+20 d^2 e^2\right )+b^3 \left (-\left (-c^3 f^3 \left (m^3+6 m^2+11 m+6\right )+15 c^2 d e f^2 \left (m^2+3 m+2\right )-60 c d^2 e^2 f (m+1)+60 d^3 e^3\right )\right )\right ) \, _2F_1\left (m-1,m+1;m+2;-\frac{d (a+b x)}{b c-a d}\right )}{60 b^5 d^3 (m+1)}+\frac{f (e+f x)^2 (a+b x)^{m+1} (c+d x)^{2-m}}{5 b d} \]

[Out]

(f*(a + b*x)^(1 + m)*(c + d*x)^(2 - m)*(e + f*x)^2)/(5*b*d) + (f*(a + b*x)^(1 +
m)*(c + d*x)^(2 - m)*(a^2*d^2*f^2*(12 - 7*m + m^2) - a*b*d*f*(15*d*e*(3 - m) - c
*f*(9 + 2*m - 2*m^2)) + b^2*(48*d^2*e^2 - 15*c*d*e*f*(2 + m) + c^2*f^2*(6 + 5*m
+ m^2)) - 3*b*d*f*(a*d*f*(4 - m) - b*(7*d*e - c*f*(3 + m)))*x))/(60*b^3*d^3) - (
(b*c - a*d)*(a^3*d^3*f^3*(24 - 26*m + 9*m^2 - m^3) - 3*a^2*b*d^2*f^2*(6 - 5*m +
m^2)*(5*d*e - c*f*(1 + m)) + 3*a*b^2*d*f*(2 - m)*(20*d^2*e^2 - 10*c*d*e*f*(1 + m
) + c^2*f^2*(2 + 3*m + m^2)) - b^3*(60*d^3*e^3 - 60*c*d^2*e^2*f*(1 + m) + 15*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[-1 + m, 1 + m, 2 + m, -((d*(a +
b*x))/(b*c - a*d))])/(60*b^5*d^3*(1 + m)*(c + d*x)^m)

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Rubi [A]  time = 1.30579, antiderivative size = 444, 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 \[ \frac{f (a+b x)^{m+1} (c+d x)^{2-m} \left (a^2 d^2 f^2 \left (m^2-7 m+12\right )-a b d f \left (15 d e (3-m)-c f \left (-2 m^2+2 m+9\right )\right )+3 b d f x (-a d f (4-m)-b c f (m+3)+7 b d e)+b^2 \left (c^2 f^2 \left (m^2+5 m+6\right )-15 c d e f (m+2)+48 d^2 e^2\right )\right )}{60 b^3 d^3}-\frac{(b c-a d) (a+b x)^{m+1} (c+d x)^{-m} \left (\frac{b (c+d x)}{b c-a d}\right )^m \left (a^3 d^3 f^3 \left (-m^3+9 m^2-26 m+24\right )-3 a^2 b d^2 f^2 \left (m^2-5 m+6\right ) (5 d e-c f (m+1))+3 a b^2 d f (2-m) \left (c^2 f^2 \left (m^2+3 m+2\right )-10 c d e f (m+1)+20 d^2 e^2\right )+b^3 \left (-\left (-c^3 f^3 \left (m^3+6 m^2+11 m+6\right )+15 c^2 d e f^2 \left (m^2+3 m+2\right )-60 c d^2 e^2 f (m+1)+60 d^3 e^3\right )\right )\right ) \, _2F_1\left (m-1,m+1;m+2;-\frac{d (a+b x)}{b c-a d}\right )}{60 b^5 d^3 (m+1)}+\frac{f (e+f x)^2 (a+b x)^{m+1} (c+d x)^{2-m}}{5 b d} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^m*(c + d*x)^(1 - m)*(e + f*x)^3,x]

[Out]

(f*(a + b*x)^(1 + m)*(c + d*x)^(2 - m)*(e + f*x)^2)/(5*b*d) + (f*(a + b*x)^(1 +
m)*(c + d*x)^(2 - m)*(a^2*d^2*f^2*(12 - 7*m + m^2) - a*b*d*f*(15*d*e*(3 - m) - c
*f*(9 + 2*m - 2*m^2)) + b^2*(48*d^2*e^2 - 15*c*d*e*f*(2 + m) + c^2*f^2*(6 + 5*m
+ m^2)) + 3*b*d*f*(7*b*d*e - a*d*f*(4 - m) - b*c*f*(3 + m))*x))/(60*b^3*d^3) - (
(b*c - a*d)*(a^3*d^3*f^3*(24 - 26*m + 9*m^2 - m^3) - 3*a^2*b*d^2*f^2*(6 - 5*m +
m^2)*(5*d*e - c*f*(1 + m)) + 3*a*b^2*d*f*(2 - m)*(20*d^2*e^2 - 10*c*d*e*f*(1 + m
) + c^2*f^2*(2 + 3*m + m^2)) - b^3*(60*d^3*e^3 - 60*c*d^2*e^2*f*(1 + m) + 15*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[-1 + m, 1 + m, 2 + m, -((d*(a +
b*x))/(b*c - a*d))])/(60*b^5*d^3*(1 + m)*(c + d*x)^m)

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Rubi in Sympy [A]  time = 138.72, size = 564, normalized size = 1.27 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**m*(d*x+c)**(1-m)*(f*x+e)**3,x)

[Out]

f*(a + b*x)**(m + 1)*(c + d*x)**(-m + 2)*(e + f*x)**2/(5*b*d) - f*(a + b*x)**(m
+ 1)*(c + d*x)**(-m + 2)*(-a*d*f*(-m + 3)*(-7*b*d*e + f*(a*d*(-m + 4) + b*c*(m +
3))) - b*c*f*(m + 2)*(-7*b*d*e + f*(a*d*(-m + 4) + b*c*(m + 3))) + 3*b*d*f*x*(-
7*b*d*e + f*(a*d*(-m + 4) + b*c*(m + 3))) + 4*b*d*(-5*b*d*e**2 + e*(-7*b*d*e + f
*(a*d*(-m + 4) + b*c*(m + 3))) + f*(2*a*c*f + e*(a*d*(-m + 2) + b*c*(m + 1)))))/
(60*b**3*d**3) + (b*(-c - d*x)/(a*d - b*c))**m*(a + b*x)**(m + 1)*(c + d*x)**(-m
)*(a*d - b*c)*(a**2*d**2*f**2*(-m + 2)*(-m + 3)*(-7*b*d*e + f*(a*d*(-m + 4) + b*
c*(m + 3))) - 2*a*b*d*f*(-m + 2)*(-c*f*(m + 1)*(-7*b*d*e + f*(a*d*(-m + 4) + b*c
*(m + 3))) + 2*d*(-5*b*d*e**2 + e*(-7*b*d*e + f*(a*d*(-m + 4) + b*c*(m + 3))) +
f*(2*a*c*f + e*(a*d*(-m + 2) + b*c*(m + 1))))) + b**2*(c**2*f**2*(m + 1)*(m + 2)
*(-7*b*d*e + f*(a*d*(-m + 4) + b*c*(m + 3))) - 4*c*d*f*(m + 1)*(-5*b*d*e**2 + e*
(-7*b*d*e + f*(a*d*(-m + 4) + b*c*(m + 3))) + f*(2*a*c*f + e*(a*d*(-m + 2) + b*c
*(m + 1)))) + 12*d**2*e*(-5*b*d*e**2 + f*(2*a*c*f + e*(a*d*(-m + 2) + b*c*(m + 1
))))))*hyper((m - 1, m + 1), (m + 2,), d*(a + b*x)/(a*d - b*c))/(60*b**5*d**3*(m
+ 1))

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Mathematica [C]  time = 3.346, size = 461, normalized size = 1.04 \[ \frac{1}{4} (a+b x)^m (c+d x)^{1-m} \left (\frac{18 a c e^2 f x^2 F_1\left (2;-m,m-1;3;-\frac{b x}{a},-\frac{d x}{c}\right )}{3 a c F_1\left (2;-m,m-1;3;-\frac{b x}{a},-\frac{d x}{c}\right )+b c m x F_1\left (3;1-m,m-1;4;-\frac{b x}{a},-\frac{d x}{c}\right )-a d (m-1) x F_1\left (3;-m,m;4;-\frac{b x}{a},-\frac{d x}{c}\right )}+\frac{16 a c e f^2 x^3 F_1\left (3;-m,m-1;4;-\frac{b x}{a},-\frac{d x}{c}\right )}{4 a c F_1\left (3;-m,m-1;4;-\frac{b x}{a},-\frac{d x}{c}\right )+b c m x F_1\left (4;1-m,m-1;5;-\frac{b x}{a},-\frac{d x}{c}\right )-a d (m-1) x F_1\left (4;-m,m;5;-\frac{b x}{a},-\frac{d x}{c}\right )}+\frac{5 a c f^3 x^4 F_1\left (4;-m,m-1;5;-\frac{b x}{a},-\frac{d x}{c}\right )}{5 a c F_1\left (4;-m,m-1;5;-\frac{b x}{a},-\frac{d x}{c}\right )+b c m x F_1\left (5;1-m,m-1;6;-\frac{b x}{a},-\frac{d x}{c}\right )-a d (m-1) x F_1\left (5;-m,m;6;-\frac{b x}{a},-\frac{d x}{c}\right )}-\frac{4 e^3 (c+d x) \left (\frac{d (a+b x)}{a d-b c}\right )^{-m} \, _2F_1\left (2-m,-m;3-m;\frac{b (c+d x)}{b c-a d}\right )}{d (m-2)}\right ) \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(a + b*x)^m*(c + d*x)^(1 - m)*(e + f*x)^3,x]

[Out]

((a + b*x)^m*(c + d*x)^(1 - m)*((18*a*c*e^2*f*x^2*AppellF1[2, -m, -1 + m, 3, -((
b*x)/a), -((d*x)/c)])/(3*a*c*AppellF1[2, -m, -1 + m, 3, -((b*x)/a), -((d*x)/c)]
+ b*c*m*x*AppellF1[3, 1 - m, -1 + m, 4, -((b*x)/a), -((d*x)/c)] - a*d*(-1 + m)*x
*AppellF1[3, -m, m, 4, -((b*x)/a), -((d*x)/c)]) + (16*a*c*e*f^2*x^3*AppellF1[3,
-m, -1 + m, 4, -((b*x)/a), -((d*x)/c)])/(4*a*c*AppellF1[3, -m, -1 + m, 4, -((b*x
)/a), -((d*x)/c)] + b*c*m*x*AppellF1[4, 1 - m, -1 + m, 5, -((b*x)/a), -((d*x)/c)
] - a*d*(-1 + m)*x*AppellF1[4, -m, m, 5, -((b*x)/a), -((d*x)/c)]) + (5*a*c*f^3*x
^4*AppellF1[4, -m, -1 + m, 5, -((b*x)/a), -((d*x)/c)])/(5*a*c*AppellF1[4, -m, -1
+ m, 5, -((b*x)/a), -((d*x)/c)] + b*c*m*x*AppellF1[5, 1 - m, -1 + m, 6, -((b*x)
/a), -((d*x)/c)] - a*d*(-1 + m)*x*AppellF1[5, -m, m, 6, -((b*x)/a), -((d*x)/c)])
- (4*e^3*(c + d*x)*Hypergeometric2F1[2 - m, -m, 3 - m, (b*(c + d*x))/(b*c - a*d
)])/(d*(-2 + 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. \[ \int \left ( bx+a \right ) ^{m} \left ( dx+c \right ) ^{1-m} \left ( fx+e \right ) ^{3}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^m*(d*x+c)^(1-m)*(f*x+e)^3,x)

[Out]

int((b*x+a)^m*(d*x+c)^(1-m)*(f*x+e)^3,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (f x + e\right )}^{3}{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x + e)^3*(b*x + a)^m*(d*x + c)^(-m + 1),x, algorithm="maxima")

[Out]

integrate((f*x + e)^3*(b*x + a)^m*(d*x + c)^(-m + 1), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (f^{3} x^{3} + 3 \, e f^{2} x^{2} + 3 \, e^{2} f x + e^{3}\right )}{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m + 1}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x + e)^3*(b*x + a)^m*(d*x + c)^(-m + 1),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 + 1
), 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)**(1-m)*(f*x+e)**3,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (f x + e\right )}^{3}{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x + e)^3*(b*x + a)^m*(d*x + c)^(-m + 1),x, algorithm="giac")

[Out]

integrate((f*x + e)^3*(b*x + a)^m*(d*x + c)^(-m + 1), x)