3.3116 \(\int \frac{(a+b x)^m (c+d x)^{2-m}}{(e+f x)^7} \, dx\)

Optimal. Leaf size=541 \[ -\frac{f (a+b x)^{m+1} (c+d x)^{3-m} \left (a^2 d^2 f^2 \left (m^2+5 m+6\right )-2 a b d f \left (d e (7 m+12)-c f \left (-m^2+2 m+6\right )\right )+b^2 \left (c^2 f^2 \left (m^2-9 m+20\right )-2 c d e f (26-7 m)+38 d^2 e^2\right )\right )}{120 (e+f x)^4 (b e-a f)^3 (d e-c f)^3}+\frac{(b c-a d)^3 (a+b x)^{m+1} (c+d x)^{-m-1} \left (-a^3 d^3 f^3 \left (m^3+6 m^2+11 m+6\right )+3 a^2 b d^2 f^2 \left (m^2+3 m+2\right ) (6 d e-c f (3-m))-3 a b^2 d f (m+1) \left (c^2 f^2 \left (m^2-7 m+12\right )-12 c d e f (3-m)+30 d^2 e^2\right )+b^3 \left (-c^3 f^3 \left (-m^3+12 m^2-47 m+60\right )+18 c^2 d e f^2 \left (m^2-7 m+12\right )-90 c d^2 e^2 f (3-m)+120 d^3 e^3\right )\right ) \, _2F_1\left (4,m+1;m+2;\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{120 (m+1) (b e-a f)^7 (d e-c f)^3}-\frac{f (a+b x)^{m+1} (c+d x)^{3-m} (b (8 d e-c f (5-m))-a d f (m+3))}{30 (e+f x)^5 (b e-a f)^2 (d e-c f)^2}-\frac{f (a+b x)^{m+1} (c+d x)^{3-m}}{6 (e+f x)^6 (b e-a f) (d e-c f)} \]

[Out]

-(f*(a + b*x)^(1 + m)*(c + d*x)^(3 - m))/(6*(b*e - a*f)*(d*e - c*f)*(e + f*x)^6)
- (f*(b*(8*d*e - c*f*(5 - m)) - a*d*f*(3 + m))*(a + b*x)^(1 + m)*(c + d*x)^(3 -
m))/(30*(b*e - a*f)^2*(d*e - c*f)^2*(e + f*x)^5) - (f*(a^2*d^2*f^2*(6 + 5*m + m
^2) - 2*a*b*d*f*(d*e*(12 + 7*m) - c*f*(6 + 2*m - m^2)) + b^2*(38*d^2*e^2 - 2*c*d
*e*f*(26 - 7*m) + c^2*f^2*(20 - 9*m + m^2)))*(a + b*x)^(1 + m)*(c + d*x)^(3 - m)
)/(120*(b*e - a*f)^3*(d*e - c*f)^3*(e + f*x)^4) + ((b*c - a*d)^3*(3*a^2*b*d^2*f^
2*(6*d*e - c*f*(3 - m))*(2 + 3*m + m^2) - a^3*d^3*f^3*(6 + 11*m + 6*m^2 + m^3) -
3*a*b^2*d*f*(1 + m)*(30*d^2*e^2 - 12*c*d*e*f*(3 - m) + c^2*f^2*(12 - 7*m + m^2)
) + b^3*(120*d^3*e^3 - 90*c*d^2*e^2*f*(3 - m) + 18*c^2*d*e*f^2*(12 - 7*m + m^2)
- c^3*f^3*(60 - 47*m + 12*m^2 - m^3)))*(a + b*x)^(1 + m)*(c + d*x)^(-1 - m)*Hype
rgeometric2F1[4, 1 + m, 2 + m, ((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x))])
/(120*(b*e - a*f)^7*(d*e - c*f)^3*(1 + m))

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Rubi [A]  time = 2.87726, antiderivative size = 540, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\frac{f (a+b x)^{m+1} (c+d x)^{3-m} \left (a^2 d^2 f^2 \left (m^2+5 m+6\right )-2 a b d f \left (d e (7 m+12)-c f \left (-m^2+2 m+6\right )\right )+b^2 \left (c^2 f^2 \left (m^2-9 m+20\right )-2 c d e f (26-7 m)+38 d^2 e^2\right )\right )}{120 (e+f x)^4 (b e-a f)^3 (d e-c f)^3}+\frac{(b c-a d)^3 (a+b x)^{m+1} (c+d x)^{-m-1} \left (-a^3 d^3 f^3 \left (m^3+6 m^2+11 m+6\right )+3 a^2 b d^2 f^2 \left (m^2+3 m+2\right ) (6 d e-c f (3-m))-3 a b^2 d f (m+1) \left (c^2 f^2 \left (m^2-7 m+12\right )-12 c d e f (3-m)+30 d^2 e^2\right )+b^3 \left (-c^3 f^3 \left (-m^3+12 m^2-47 m+60\right )+18 c^2 d e f^2 \left (m^2-7 m+12\right )-90 c d^2 e^2 f (3-m)+120 d^3 e^3\right )\right ) \, _2F_1\left (4,m+1;m+2;\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{120 (m+1) (b e-a f)^7 (d e-c f)^3}-\frac{f (a+b x)^{m+1} (c+d x)^{3-m} (-a d f (m+3)-b c f (5-m)+8 b d e)}{30 (e+f x)^5 (b e-a f)^2 (d e-c f)^2}-\frac{f (a+b x)^{m+1} (c+d x)^{3-m}}{6 (e+f x)^6 (b e-a f) (d e-c f)} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^m*(c + d*x)^(2 - m))/(e + f*x)^7,x]

[Out]

-(f*(a + b*x)^(1 + m)*(c + d*x)^(3 - m))/(6*(b*e - a*f)*(d*e - c*f)*(e + f*x)^6)
- (f*(8*b*d*e - b*c*f*(5 - m) - a*d*f*(3 + m))*(a + b*x)^(1 + m)*(c + d*x)^(3 -
m))/(30*(b*e - a*f)^2*(d*e - c*f)^2*(e + f*x)^5) - (f*(a^2*d^2*f^2*(6 + 5*m + m
^2) - 2*a*b*d*f*(d*e*(12 + 7*m) - c*f*(6 + 2*m - m^2)) + b^2*(38*d^2*e^2 - 2*c*d
*e*f*(26 - 7*m) + c^2*f^2*(20 - 9*m + m^2)))*(a + b*x)^(1 + m)*(c + d*x)^(3 - m)
)/(120*(b*e - a*f)^3*(d*e - c*f)^3*(e + f*x)^4) + ((b*c - a*d)^3*(3*a^2*b*d^2*f^
2*(6*d*e - c*f*(3 - m))*(2 + 3*m + m^2) - a^3*d^3*f^3*(6 + 11*m + 6*m^2 + m^3) -
3*a*b^2*d*f*(1 + m)*(30*d^2*e^2 - 12*c*d*e*f*(3 - m) + c^2*f^2*(12 - 7*m + m^2)
) + b^3*(120*d^3*e^3 - 90*c*d^2*e^2*f*(3 - m) + 18*c^2*d*e*f^2*(12 - 7*m + m^2)
- c^3*f^3*(60 - 47*m + 12*m^2 - m^3)))*(a + b*x)^(1 + m)*(c + d*x)^(-1 - m)*Hype
rgeometric2F1[4, 1 + m, 2 + m, ((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x))])
/(120*(b*e - a*f)^7*(d*e - c*f)^3*(1 + 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)**7,x)

[Out]

Timed out

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Mathematica [C]  time = 34.6289, size = 79140, normalized size = 146.28 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[((a + b*x)^m*(c + d*x)^(2 - m))/(e + f*x)^7,x]

[Out]

Result too large to show

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Maple [F]  time = 0.836, size = 0, normalized size = 0. \[ \int{\frac{ \left ( bx+a \right ) ^{m} \left ( dx+c \right ) ^{2-m}}{ \left ( fx+e \right ) ^{7}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^m*(d*x+c)^(2-m)/(f*x+e)^7,x)

[Out]

int((b*x+a)^m*(d*x+c)^(2-m)/(f*x+e)^7,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m + 2}}{{\left (f x + e\right )}^{7}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^m*(d*x + c)^(-m + 2)/(f*x + e)^7,x, algorithm="maxima")

[Out]

integrate((b*x + a)^m*(d*x + c)^(-m + 2)/(f*x + e)^7, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m + 2}}{f^{7} x^{7} + 7 \, e f^{6} x^{6} + 21 \, e^{2} f^{5} x^{5} + 35 \, e^{3} f^{4} x^{4} + 35 \, e^{4} f^{3} x^{3} + 21 \, e^{5} f^{2} x^{2} + 7 \, e^{6} f x + e^{7}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^m*(d*x + c)^(-m + 2)/(f*x + e)^7,x, algorithm="fricas")

[Out]

integral((b*x + a)^m*(d*x + c)^(-m + 2)/(f^7*x^7 + 7*e*f^6*x^6 + 21*e^2*f^5*x^5
+ 35*e^3*f^4*x^4 + 35*e^4*f^3*x^3 + 21*e^5*f^2*x^2 + 7*e^6*f*x + e^7), 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)**7,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m + 2}}{{\left (f x + e\right )}^{7}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^m*(d*x + c)^(-m + 2)/(f*x + e)^7,x, algorithm="giac")

[Out]

integrate((b*x + a)^m*(d*x + c)^(-m + 2)/(f*x + e)^7, x)