3.3104 \(\int \frac{(a+b x)^m (c+d x)^{1-m}}{(e+f x)^5} \, dx\)

Optimal. Leaf size=311 \[ -\frac{(b c-a d)^2 (a+b x)^{m+1} (c+d x)^{-m-1} \left (-a^2 d^2 f^2 \left (m^2+3 m+2\right )+2 a b d f (m+1) (4 d e-c f (2-m))+b^2 \left (-\left (c^2 f^2 \left (m^2-5 m+6\right )-8 c d e f (2-m)+12 d^2 e^2\right )\right )\right ) \, _2F_1\left (3,m+1;m+2;\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{12 (m+1) (b e-a f)^5 (d e-c f)^2}-\frac{f (a+b x)^{m+1} (c+d x)^{2-m} (b (5 d e-c f (3-m))-a d f (m+2))}{12 (e+f x)^3 (b e-a f)^2 (d e-c f)^2}-\frac{f (a+b x)^{m+1} (c+d x)^{2-m}}{4 (e+f x)^4 (b e-a f) (d e-c f)} \]

[Out]

-(f*(a + b*x)^(1 + m)*(c + d*x)^(2 - m))/(4*(b*e - a*f)*(d*e - c*f)*(e + f*x)^4)
- (f*(b*(5*d*e - c*f*(3 - m)) - a*d*f*(2 + m))*(a + b*x)^(1 + m)*(c + d*x)^(2 -
m))/(12*(b*e - a*f)^2*(d*e - c*f)^2*(e + f*x)^3) - ((b*c - a*d)^2*(2*a*b*d*f*(4
*d*e - c*f*(2 - m))*(1 + m) - a^2*d^2*f^2*(2 + 3*m + m^2) - b^2*(12*d^2*e^2 - 8*
c*d*e*f*(2 - m) + c^2*f^2*(6 - 5*m + m^2)))*(a + b*x)^(1 + m)*(c + d*x)^(-1 - m)
*Hypergeometric2F1[3, 1 + m, 2 + m, ((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*
x))])/(12*(b*e - a*f)^5*(d*e - c*f)^2*(1 + m))

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Rubi [A]  time = 0.928158, antiderivative size = 310, 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 \[ -\frac{(b c-a d)^2 (a+b x)^{m+1} (c+d x)^{-m-1} \left (-a^2 d^2 f^2 \left (m^2+3 m+2\right )+2 a b d f (m+1) (4 d e-c f (2-m))+b^2 \left (-\left (c^2 f^2 \left (m^2-5 m+6\right )-8 c d e f (2-m)+12 d^2 e^2\right )\right )\right ) \, _2F_1\left (3,m+1;m+2;\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{12 (m+1) (b e-a f)^5 (d e-c f)^2}-\frac{f (a+b x)^{m+1} (c+d x)^{2-m} (-a d f (m+2)-b c f (3-m)+5 b d e)}{12 (e+f x)^3 (b e-a f)^2 (d e-c f)^2}-\frac{f (a+b x)^{m+1} (c+d x)^{2-m}}{4 (e+f x)^4 (b e-a f) (d e-c f)} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^m*(c + d*x)^(1 - m))/(e + f*x)^5,x]

[Out]

-(f*(a + b*x)^(1 + m)*(c + d*x)^(2 - m))/(4*(b*e - a*f)*(d*e - c*f)*(e + f*x)^4)
- (f*(5*b*d*e - b*c*f*(3 - m) - a*d*f*(2 + m))*(a + b*x)^(1 + m)*(c + d*x)^(2 -
m))/(12*(b*e - a*f)^2*(d*e - c*f)^2*(e + f*x)^3) - ((b*c - a*d)^2*(2*a*b*d*f*(4
*d*e - c*f*(2 - m))*(1 + m) - a^2*d^2*f^2*(2 + 3*m + m^2) - b^2*(12*d^2*e^2 - 8*
c*d*e*f*(2 - m) + c^2*f^2*(6 - 5*m + m^2)))*(a + b*x)^(1 + m)*(c + d*x)^(-1 - m)
*Hypergeometric2F1[3, 1 + m, 2 + m, ((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*
x))])/(12*(b*e - a*f)^5*(d*e - c*f)^2*(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)**(1-m)/(f*x+e)**5,x)

[Out]

Timed out

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Mathematica [C]  time = 22.6485, size = 61774, normalized size = 198.63 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[((a + b*x)^m*(c + d*x)^(1 - m))/(e + f*x)^5,x]

[Out]

Result too large to show

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Maple [F]  time = 0.301, size = 0, normalized size = 0. \[ \int{\frac{ \left ( bx+a \right ) ^{m} \left ( dx+c \right ) ^{1-m}}{ \left ( fx+e \right ) ^{5}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^m*(d*x+c)^(1-m)/(f*x+e)^5,x)

[Out]

int((b*x+a)^m*(d*x+c)^(1-m)/(f*x+e)^5,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m + 1}}{{\left (f x + e\right )}^{5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^m*(d*x + c)^(-m + 1)/(f*x + e)^5,x, algorithm="maxima")

[Out]

integrate((b*x + a)^m*(d*x + c)^(-m + 1)/(f*x + e)^5, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m + 1}}{f^{5} x^{5} + 5 \, e f^{4} x^{4} + 10 \, e^{2} f^{3} x^{3} + 10 \, e^{3} f^{2} x^{2} + 5 \, e^{4} f x + e^{5}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^m*(d*x + c)^(-m + 1)/(f*x + e)^5,x, algorithm="fricas")

[Out]

integral((b*x + a)^m*(d*x + c)^(-m + 1)/(f^5*x^5 + 5*e*f^4*x^4 + 10*e^2*f^3*x^3
+ 10*e^3*f^2*x^2 + 5*e^4*f*x + e^5), 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)**5,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m + 1}}{{\left (f x + e\right )}^{5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^m*(d*x + c)^(-m + 1)/(f*x + e)^5,x, algorithm="giac")

[Out]

integrate((b*x + a)^m*(d*x + c)^(-m + 1)/(f*x + e)^5, x)