3.3095 \(\int (a+b x)^m (c+d x)^{1-m} (e+f x)^4 \, dx\)

Optimal. Leaf size=803 \[ \frac{f (a+b x)^{m+1} (e+f x)^3 (c+d x)^{2-m}}{6 b d}-\frac{f (a d f (5-m)-b (9 d e-c f (m+4))) (a+b x)^{m+1} (e+f x)^2 (c+d x)^{2-m}}{30 b^2 d^2}-\frac{f (a+b x)^{m+1} \left (-\left (312 d^3 e^3-24 c d^2 f (7 m+13) e^2+24 c^2 d f^2 \left (m^2+5 m+6\right ) e-c^3 f^3 \left (m^3+9 m^2+26 m+24\right )\right ) b^3+3 a d f \left (8 d^2 (20-7 m) e^2-8 c d f \left (-2 m^2+2 m+9\right ) e+c^2 f^2 \left (-m^3-2 m^2+7 m+12\right )\right ) b^2-3 a^2 d^2 f^2 (4-m) \left (8 d e (3-m)-c f \left (-m^2+m+4\right )\right ) b+3 d f (5 b d (a f (3 c f+d e (2-m))-b e (6 d e-c f (m+1)))-(a d f (4-m)-b (2 d e-c f (m+3))) (a d f (5-m)-b (9 d e-c f (m+4)))) x b+a^3 d^3 f^3 \left (-m^3+12 m^2-47 m+60\right )\right ) (c+d x)^{2-m}}{360 b^4 d^4}+\frac{(b c-a d) \left (\left (360 d^4 e^4-480 c d^3 f (m+1) e^3+180 c^2 d^2 f^2 \left (m^2+3 m+2\right ) e^2-24 c^3 d f^3 \left (m^3+6 m^2+11 m+6\right ) e+c^4 f^4 \left (m^4+10 m^3+35 m^2+50 m+24\right )\right ) b^4-4 a d f (2-m) \left (120 d^3 e^3-90 c d^2 f (m+1) e^2+18 c^2 d f^2 \left (m^2+3 m+2\right ) e-c^3 f^3 \left (m^3+6 m^2+11 m+6\right )\right ) b^3+6 a^2 d^2 f^2 \left (m^2-5 m+6\right ) \left (30 d^2 e^2-12 c d f (m+1) e+c^2 f^2 \left (m^2+3 m+2\right )\right ) b^2-4 a^3 d^3 f^3 \left (-m^3+9 m^2-26 m+24\right ) (6 d e-c f (m+1)) b+a^4 d^4 f^4 \left (m^4-14 m^3+71 m^2-154 m+120\right )\right ) (a+b x)^{m+1} \left (\frac{b (c+d x)}{b c-a d}\right )^m \, _2F_1\left (m-1,m+1;m+2;-\frac{d (a+b x)}{b c-a d}\right ) (c+d x)^{-m}}{360 b^6 d^4 (m+1)} \]

[Out]

-(f*(a*d*f*(5 - m) - b*(9*d*e - c*f*(4 + m)))*(a + b*x)^(1 + m)*(c + d*x)^(2 - m
)*(e + f*x)^2)/(30*b^2*d^2) + (f*(a + b*x)^(1 + m)*(c + d*x)^(2 - m)*(e + f*x)^3
)/(6*b*d) - (f*(a + b*x)^(1 + m)*(c + d*x)^(2 - m)*(a^3*d^3*f^3*(60 - 47*m + 12*
m^2 - m^3) - 3*a^2*b*d^2*f^2*(4 - m)*(8*d*e*(3 - m) - c*f*(4 + m - m^2)) + 3*a*b
^2*d*f*(8*d^2*e^2*(20 - 7*m) - 8*c*d*e*f*(9 + 2*m - 2*m^2) + c^2*f^2*(12 + 7*m -
2*m^2 - m^3)) - b^3*(312*d^3*e^3 - 24*c*d^2*e^2*f*(13 + 7*m) + 24*c^2*d*e*f^2*(
6 + 5*m + m^2) - c^3*f^3*(24 + 26*m + 9*m^2 + m^3)) + 3*b*d*f*(5*b*d*(a*f*(3*c*f
+ d*e*(2 - m)) - b*e*(6*d*e - c*f*(1 + m))) - (a*d*f*(4 - m) - b*(2*d*e - c*f*(
3 + m)))*(a*d*f*(5 - m) - b*(9*d*e - c*f*(4 + m))))*x))/(360*b^4*d^4) + ((b*c -
a*d)*(a^4*d^4*f^4*(120 - 154*m + 71*m^2 - 14*m^3 + m^4) - 4*a^3*b*d^3*f^3*(24 -
26*m + 9*m^2 - m^3)*(6*d*e - c*f*(1 + m)) + 6*a^2*b^2*d^2*f^2*(6 - 5*m + m^2)*(3
0*d^2*e^2 - 12*c*d*e*f*(1 + m) + c^2*f^2*(2 + 3*m + m^2)) - 4*a*b^3*d*f*(2 - m)*
(120*d^3*e^3 - 90*c*d^2*e^2*f*(1 + m) + 18*c^2*d*e*f^2*(2 + 3*m + m^2) - c^3*f^3
*(6 + 11*m + 6*m^2 + m^3)) + b^4*(360*d^4*e^4 - 480*c*d^3*e^3*f*(1 + m) + 180*c^
2*d^2*e^2*f^2*(2 + 3*m + m^2) - 24*c^3*d*e*f^3*(6 + 11*m + 6*m^2 + m^3) + c^4*f^
4*(24 + 50*m + 35*m^2 + 10*m^3 + m^4)))*(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))])/(
360*b^6*d^4*(1 + m)*(c + d*x)^m)

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Rubi [A]  time = 7.38695, antiderivative size = 799, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ \frac{f (a+b x)^{m+1} (e+f x)^3 (c+d x)^{2-m}}{6 b d}+\frac{f (9 b d e-a d f (5-m)-b c f (m+4)) (a+b x)^{m+1} (e+f x)^2 (c+d x)^{2-m}}{30 b^2 d^2}-\frac{f (a+b x)^{m+1} \left (-\left (312 d^3 e^3-24 c d^2 f (7 m+13) e^2+24 c^2 d f^2 \left (m^2+5 m+6\right ) e-c^3 f^3 \left (m^3+9 m^2+26 m+24\right )\right ) b^3+3 a d f \left (8 d^2 (20-7 m) e^2-8 c d f \left (-2 m^2+2 m+9\right ) e+c^2 f^2 \left (-m^3-2 m^2+7 m+12\right )\right ) b^2-3 a^2 d^2 f^2 (4-m) \left (8 d e (3-m)-c f \left (-m^2+m+4\right )\right ) b-3 d f ((2 b d e-a d f (4-m)-b c f (m+3)) (9 b d e-a d f (5-m)-b c f (m+4))-5 b d (a f (3 c f+d e (2-m))-b e (6 d e-c f (m+1)))) x b+a^3 d^3 f^3 \left (-m^3+12 m^2-47 m+60\right )\right ) (c+d x)^{2-m}}{360 b^4 d^4}+\frac{(b c-a d) \left (\left (360 d^4 e^4-480 c d^3 f (m+1) e^3+180 c^2 d^2 f^2 \left (m^2+3 m+2\right ) e^2-24 c^3 d f^3 \left (m^3+6 m^2+11 m+6\right ) e+c^4 f^4 \left (m^4+10 m^3+35 m^2+50 m+24\right )\right ) b^4-4 a d f (2-m) \left (120 d^3 e^3-90 c d^2 f (m+1) e^2+18 c^2 d f^2 \left (m^2+3 m+2\right ) e-c^3 f^3 \left (m^3+6 m^2+11 m+6\right )\right ) b^3+6 a^2 d^2 f^2 \left (m^2-5 m+6\right ) \left (30 d^2 e^2-12 c d f (m+1) e+c^2 f^2 \left (m^2+3 m+2\right )\right ) b^2-4 a^3 d^3 f^3 \left (-m^3+9 m^2-26 m+24\right ) (6 d e-c f (m+1)) b+a^4 d^4 f^4 \left (m^4-14 m^3+71 m^2-154 m+120\right )\right ) (a+b x)^{m+1} \left (\frac{b (c+d x)}{b c-a d}\right )^m \, _2F_1\left (m-1,m+1;m+2;-\frac{d (a+b x)}{b c-a d}\right ) (c+d x)^{-m}}{360 b^6 d^4 (m+1)} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^m*(c + d*x)^(1 - m)*(e + f*x)^4,x]

[Out]

(f*(9*b*d*e - a*d*f*(5 - m) - b*c*f*(4 + m))*(a + b*x)^(1 + m)*(c + d*x)^(2 - m)
*(e + f*x)^2)/(30*b^2*d^2) + (f*(a + b*x)^(1 + m)*(c + d*x)^(2 - m)*(e + f*x)^3)
/(6*b*d) - (f*(a + b*x)^(1 + m)*(c + d*x)^(2 - m)*(a^3*d^3*f^3*(60 - 47*m + 12*m
^2 - m^3) - 3*a^2*b*d^2*f^2*(4 - m)*(8*d*e*(3 - m) - c*f*(4 + m - m^2)) + 3*a*b^
2*d*f*(8*d^2*e^2*(20 - 7*m) - 8*c*d*e*f*(9 + 2*m - 2*m^2) + c^2*f^2*(12 + 7*m -
2*m^2 - m^3)) - b^3*(312*d^3*e^3 - 24*c*d^2*e^2*f*(13 + 7*m) + 24*c^2*d*e*f^2*(6
+ 5*m + m^2) - c^3*f^3*(24 + 26*m + 9*m^2 + m^3)) - 3*b*d*f*((2*b*d*e - a*d*f*(
4 - m) - b*c*f*(3 + m))*(9*b*d*e - a*d*f*(5 - m) - b*c*f*(4 + m)) - 5*b*d*(a*f*(
3*c*f + d*e*(2 - m)) - b*e*(6*d*e - c*f*(1 + m))))*x))/(360*b^4*d^4) + ((b*c - a
*d)*(a^4*d^4*f^4*(120 - 154*m + 71*m^2 - 14*m^3 + m^4) - 4*a^3*b*d^3*f^3*(24 - 2
6*m + 9*m^2 - m^3)*(6*d*e - c*f*(1 + m)) + 6*a^2*b^2*d^2*f^2*(6 - 5*m + m^2)*(30
*d^2*e^2 - 12*c*d*e*f*(1 + m) + c^2*f^2*(2 + 3*m + m^2)) - 4*a*b^3*d*f*(2 - m)*(
120*d^3*e^3 - 90*c*d^2*e^2*f*(1 + m) + 18*c^2*d*e*f^2*(2 + 3*m + m^2) - c^3*f^3*
(6 + 11*m + 6*m^2 + m^3)) + b^4*(360*d^4*e^4 - 480*c*d^3*e^3*f*(1 + m) + 180*c^2
*d^2*e^2*f^2*(2 + 3*m + m^2) - 24*c^3*d*e*f^3*(6 + 11*m + 6*m^2 + m^3) + c^4*f^4
*(24 + 50*m + 35*m^2 + 10*m^3 + m^4)))*(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))])/(3
60*b^6*d^4*(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)**(1-m)*(f*x+e)**4,x)

[Out]

Timed out

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Mathematica [C]  time = 7.24836, size = 676, normalized size = 0.84 \[ \frac{6 a c e^3 f x^2 (a+b x)^m (c+d x)^{1-m} F_1\left (2;-m,m-1;3;-\frac{b x}{a},-\frac{d x}{c}\right )}{3 a c F_1\left (2;-m,m-1;3;-\frac{b x}{a},-\frac{d x}{c}\right )+b c m x F_1\left (3;1-m,m-1;4;-\frac{b x}{a},-\frac{d x}{c}\right )-a d (m-1) x F_1\left (3;-m,m;4;-\frac{b x}{a},-\frac{d x}{c}\right )}+\frac{8 a c e^2 f^2 x^3 (a+b x)^m (c+d x)^{1-m} F_1\left (3;-m,m-1;4;-\frac{b x}{a},-\frac{d x}{c}\right )}{4 a c F_1\left (3;-m,m-1;4;-\frac{b x}{a},-\frac{d x}{c}\right )+b c m x F_1\left (4;1-m,m-1;5;-\frac{b x}{a},-\frac{d x}{c}\right )-a d (m-1) x F_1\left (4;-m,m;5;-\frac{b x}{a},-\frac{d x}{c}\right )}+\frac{5 a c e f^3 x^4 (a+b x)^m (c+d x)^{1-m} F_1\left (4;-m,m-1;5;-\frac{b x}{a},-\frac{d x}{c}\right )}{5 a c F_1\left (4;-m,m-1;5;-\frac{b x}{a},-\frac{d x}{c}\right )+b c m x F_1\left (5;1-m,m-1;6;-\frac{b x}{a},-\frac{d x}{c}\right )-a d (m-1) x F_1\left (5;-m,m;6;-\frac{b x}{a},-\frac{d x}{c}\right )}+\frac{6 a c f^4 x^5 (a+b x)^m (c+d x)^{1-m} F_1\left (5;-m,m-1;6;-\frac{b x}{a},-\frac{d x}{c}\right )}{5 \left (6 a c F_1\left (5;-m,m-1;6;-\frac{b x}{a},-\frac{d x}{c}\right )+b c m x F_1\left (6;1-m,m-1;7;-\frac{b x}{a},-\frac{d x}{c}\right )-a d (m-1) x F_1\left (6;-m,m;7;-\frac{b x}{a},-\frac{d x}{c}\right )\right )}-\frac{e^4 (c+d x)^{2-m} \left (a+\frac{b (c+d x)}{d}-\frac{b c}{d}\right )^m \left (\frac{b (c+d x)}{d \left (a-\frac{b c}{d}\right )}+1\right )^{-m} \, _2F_1\left (2-m,-m;3-m;-\frac{b (c+d x)}{\left (a-\frac{b c}{d}\right ) d}\right )}{d (m-2)} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(a + b*x)^m*(c + d*x)^(1 - m)*(e + f*x)^4,x]

[Out]

(6*a*c*e^3*f*x^2*(a + b*x)^m*(c + d*x)^(1 - m)*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*Ap
pellF1[3, -m, m, 4, -((b*x)/a), -((d*x)/c)]) + (8*a*c*e^2*f^2*x^3*(a + b*x)^m*(c
+ d*x)^(1 - m)*AppellF1[3, -m, -1 + m, 4, -((b*x)/a), -((d*x)/c)])/(4*a*c*Appel
lF1[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*e*f^3*x^4*(a + b*x)^m*(c + d*x)^(1 - m)*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)]) + (6*a*c*f^4*x^5*(a
+ b*x)^m*(c + d*x)^(1 - m)*AppellF1[5, -m, -1 + m, 6, -((b*x)/a), -((d*x)/c)])/
(5*(6*a*c*AppellF1[5, -m, -1 + m, 6, -((b*x)/a), -((d*x)/c)] + b*c*m*x*AppellF1[
6, 1 - m, -1 + m, 7, -((b*x)/a), -((d*x)/c)] - a*d*(-1 + m)*x*AppellF1[6, -m, m,
7, -((b*x)/a), -((d*x)/c)])) - (e^4*(c + d*x)^(2 - m)*(a - (b*c)/d + (b*(c + d*
x))/d)^m*Hypergeometric2F1[2 - m, -m, 3 - m, -((b*(c + d*x))/((a - (b*c)/d)*d))]
)/(d*(-2 + m)*(1 + (b*(c + d*x))/((a - (b*c)/d)*d))^m)

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Maple [F]  time = 0.11, size = 0, normalized size = 0. \[ \int \left ( bx+a \right ) ^{m} \left ( dx+c \right ) ^{1-m} \left ( fx+e \right ) ^{4}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^m*(d*x+c)^(1-m)*(f*x+e)^4,x)

[Out]

int((b*x+a)^m*(d*x+c)^(1-m)*(f*x+e)^4,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (f x + e\right )}^{4}{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x + e)^4*(b*x + a)^m*(d*x + c)^(-m + 1),x, algorithm="maxima")

[Out]

integrate((f*x + e)^4*(b*x + a)^m*(d*x + c)^(-m + 1), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (f^{4} x^{4} + 4 \, e f^{3} x^{3} + 6 \, e^{2} f^{2} x^{2} + 4 \, e^{3} f x + e^{4}\right )}{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m + 1}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x + e)^4*(b*x + a)^m*(d*x + c)^(-m + 1),x, algorithm="fricas")

[Out]

integral((f^4*x^4 + 4*e*f^3*x^3 + 6*e^2*f^2*x^2 + 4*e^3*f*x + e^4)*(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)**4,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (f x + e\right )}^{4}{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x + e)^4*(b*x + a)^m*(d*x + c)^(-m + 1),x, algorithm="giac")

[Out]

integrate((f*x + e)^4*(b*x + a)^m*(d*x + c)^(-m + 1), x)