3.3115 \(\int \frac{(a+b x)^m (c+d x)^{2-m}}{(e+f x)^6} \, dx\)

Optimal. Leaf size=311 \[ -\frac{(b c-a d)^3 (a+b x)^{m+1} (c+d x)^{-m-1} \left (-a^2 d^2 f^2 \left (m^2+3 m+2\right )+2 a b d f (m+1) (5 d e-c f (3-m))+b^2 \left (-\left (c^2 f^2 \left (m^2-7 m+12\right )-10 c d e f (3-m)+20 d^2 e^2\right )\right )\right ) \, _2F_1\left (4,m+1;m+2;\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{20 (m+1) (b e-a f)^6 (d e-c f)^2}-\frac{f (a+b x)^{m+1} (c+d x)^{3-m} (b (6 d e-c f (4-m))-a d f (m+2))}{20 (e+f x)^4 (b e-a f)^2 (d e-c f)^2}-\frac{f (a+b x)^{m+1} (c+d x)^{3-m}}{5 (e+f x)^5 (b e-a f) (d e-c f)} \]

[Out]

-(f*(a + b*x)^(1 + m)*(c + d*x)^(3 - m))/(5*(b*e - a*f)*(d*e - c*f)*(e + f*x)^5)
- (f*(b*(6*d*e - c*f*(4 - m)) - a*d*f*(2 + m))*(a + b*x)^(1 + m)*(c + d*x)^(3 -
m))/(20*(b*e - a*f)^2*(d*e - c*f)^2*(e + f*x)^4) - ((b*c - a*d)^3*(2*a*b*d*f*(5
*d*e - c*f*(3 - m))*(1 + m) - a^2*d^2*f^2*(2 + 3*m + m^2) - b^2*(20*d^2*e^2 - 10
*c*d*e*f*(3 - m) + c^2*f^2*(12 - 7*m + m^2)))*(a + b*x)^(1 + m)*(c + d*x)^(-1 -
m)*Hypergeometric2F1[4, 1 + m, 2 + m, ((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c +
d*x))])/(20*(b*e - a*f)^6*(d*e - c*f)^2*(1 + m))

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Rubi [A]  time = 0.927891, 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 \[ -\frac{(b c-a d)^3 (a+b x)^{m+1} (c+d x)^{-m-1} \left (-a^2 d^2 f^2 \left (m^2+3 m+2\right )+2 a b d f (m+1) (5 d e-c f (3-m))+b^2 \left (-\left (c^2 f^2 \left (m^2-7 m+12\right )-10 c d e f (3-m)+20 d^2 e^2\right )\right )\right ) \, _2F_1\left (4,m+1;m+2;\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{20 (m+1) (b e-a f)^6 (d e-c f)^2}-\frac{f (a+b x)^{m+1} (c+d x)^{3-m} (-a d f (m+2)-b c f (4-m)+6 b d e)}{20 (e+f x)^4 (b e-a f)^2 (d e-c f)^2}-\frac{f (a+b x)^{m+1} (c+d x)^{3-m}}{5 (e+f x)^5 (b e-a f) (d e-c f)} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^m*(c + d*x)^(2 - m))/(e + f*x)^6,x]

[Out]

-(f*(a + b*x)^(1 + m)*(c + d*x)^(3 - m))/(5*(b*e - a*f)*(d*e - c*f)*(e + f*x)^5)
- (f*(6*b*d*e - b*c*f*(4 - m) - a*d*f*(2 + m))*(a + b*x)^(1 + m)*(c + d*x)^(3 -
m))/(20*(b*e - a*f)^2*(d*e - c*f)^2*(e + f*x)^4) - ((b*c - a*d)^3*(2*a*b*d*f*(5
*d*e - c*f*(3 - m))*(1 + m) - a^2*d^2*f^2*(2 + 3*m + m^2) - b^2*(20*d^2*e^2 - 10
*c*d*e*f*(3 - m) + c^2*f^2*(12 - 7*m + m^2)))*(a + b*x)^(1 + m)*(c + d*x)^(-1 -
m)*Hypergeometric2F1[4, 1 + m, 2 + m, ((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c +
d*x))])/(20*(b*e - a*f)^6*(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)**(2-m)/(f*x+e)**6,x)

[Out]

Timed out

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Mathematica [C]  time = 30.3431, size = 29088, normalized size = 93.53 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[((a + b*x)^m*(c + d*x)^(2 - m))/(e + f*x)^6,x]

[Out]

Result too large to show

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Maple [F]  time = 0.469, size = 0, normalized size = 0. \[ \int{\frac{ \left ( bx+a \right ) ^{m} \left ( dx+c \right ) ^{2-m}}{ \left ( fx+e \right ) ^{6}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^m*(d*x+c)^(2-m)/(f*x+e)^6,x)

[Out]

int((b*x+a)^m*(d*x+c)^(2-m)/(f*x+e)^6,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m + 2}}{{\left (f x + e\right )}^{6}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^m*(d*x + c)^(-m + 2)/(f*x + e)^6,x, algorithm="maxima")

[Out]

integrate((b*x + a)^m*(d*x + c)^(-m + 2)/(f*x + e)^6, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m + 2}}{f^{6} x^{6} + 6 \, e f^{5} x^{5} + 15 \, e^{2} f^{4} x^{4} + 20 \, e^{3} f^{3} x^{3} + 15 \, e^{4} f^{2} x^{2} + 6 \, e^{5} f x + e^{6}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^m*(d*x + c)^(-m + 2)/(f*x + e)^6,x, algorithm="fricas")

[Out]

integral((b*x + a)^m*(d*x + c)^(-m + 2)/(f^6*x^6 + 6*e*f^5*x^5 + 15*e^2*f^4*x^4
+ 20*e^3*f^3*x^3 + 15*e^4*f^2*x^2 + 6*e^5*f*x + e^6), 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)**6,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m + 2}}{{\left (f x + e\right )}^{6}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^m*(d*x + c)^(-m + 2)/(f*x + e)^6,x, algorithm="giac")

[Out]

integrate((b*x + a)^m*(d*x + c)^(-m + 2)/(f*x + e)^6, x)