3.3085 \(\int \frac{(a+b x)^m (c+d x)^{-4-m}}{(e+f x)^2} \, dx\)

Optimal. Leaf size=648 \[ -\frac{d (a+b x)^{m+1} (c+d x)^{-m-2} \left (a^2 d^2 f^2 \left (m^2+7 m+12\right )-2 a b d f \left (c f \left (m^2+6 m+10\right )+d e (m+2)\right )+b^2 \left (-\left (-c^2 f^2 \left (m^2+5 m+6\right )-2 c d e f (m+4)+2 d^2 e^2\right )\right )\right )}{(m+2) (m+3) (b c-a d)^2 (b e-a f) (d e-c f)^3}-\frac{d (a+b x)^{m+1} (c+d x)^{-m-1} \left (a^3 d^3 f^3 \left (m^3+9 m^2+26 m+24\right )-a^2 b d^2 f^2 (m+3) \left (c f \left (3 m^2+15 m+20\right )+d e (3 m+4)\right )-a b^2 d f \left (-c^2 f^2 \left (3 m^3+21 m^2+50 m+44\right )-2 c d e f \left (3 m^2+15 m+16\right )+2 d^2 e^2 (m+2)\right )+b^3 \left (-\left (c^3 f^3 \left (m^3+6 m^2+11 m+6\right )+c^2 d e f^2 \left (3 m^2+17 m+26\right )-2 c d^2 e^2 f (m+5)+2 d^3 e^3\right )\right )\right )}{(m+1) (m+2) (m+3) (b c-a d)^3 (b e-a f) (d e-c f)^4}+\frac{f^3 (a+b x)^{m+1} (c+d x)^{-m-1} (a d f (m+4)-b (c f m+4 d e)) \, _2F_1\left (1,m+1;m+2;\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{(m+1) (b e-a f)^2 (d e-c f)^4}+\frac{d (a+b x)^{m+1} (c+d x)^{-m-3}}{(m+3) (e+f x) (b c-a d) (d e-c f)}-\frac{f (a+b x)^{m+1} (c+d x)^{-m-2} (a d f (m+4)-b (c f (m+3)+d e))}{(m+3) (e+f x) (b c-a d) (b e-a f) (d e-c f)^2} \]

[Out]

-((d*(a^2*d^2*f^2*(12 + 7*m + m^2) - b^2*(2*d^2*e^2 - 2*c*d*e*f*(4 + m) - c^2*f^
2*(6 + 5*m + m^2)) - 2*a*b*d*f*(d*e*(2 + m) + c*f*(10 + 6*m + m^2)))*(a + b*x)^(
1 + m)*(c + d*x)^(-2 - m))/((b*c - a*d)^2*(b*e - a*f)*(d*e - c*f)^3*(2 + m)*(3 +
m))) - (d*(a^3*d^3*f^3*(24 + 26*m + 9*m^2 + m^3) - a^2*b*d^2*f^2*(3 + m)*(d*e*(
4 + 3*m) + c*f*(20 + 15*m + 3*m^2)) - b^3*(2*d^3*e^3 - 2*c*d^2*e^2*f*(5 + m) + c
^2*d*e*f^2*(26 + 17*m + 3*m^2) + c^3*f^3*(6 + 11*m + 6*m^2 + m^3)) - a*b^2*d*f*(
2*d^2*e^2*(2 + m) - 2*c*d*e*f*(16 + 15*m + 3*m^2) - c^2*f^2*(44 + 50*m + 21*m^2
+ 3*m^3)))*(a + b*x)^(1 + m)*(c + d*x)^(-1 - m))/((b*c - a*d)^3*(b*e - a*f)*(d*e
- c*f)^4*(1 + m)*(2 + m)*(3 + m)) + (d*(a + b*x)^(1 + m)*(c + d*x)^(-3 - m))/((
b*c - a*d)*(d*e - c*f)*(3 + m)*(e + f*x)) - (f*(a*d*f*(4 + m) - b*(d*e + c*f*(3
+ m)))*(a + b*x)^(1 + m)*(c + d*x)^(-2 - m))/((b*c - a*d)*(b*e - a*f)*(d*e - c*f
)^2*(3 + m)*(e + f*x)) + (f^3*(a*d*f*(4 + m) - b*(4*d*e + c*f*m))*(a + b*x)^(1 +
m)*(c + d*x)^(-1 - m)*Hypergeometric2F1[1, 1 + m, 2 + m, ((d*e - c*f)*(a + b*x)
)/((b*e - a*f)*(c + d*x))])/((b*e - a*f)^2*(d*e - c*f)^4*(1 + m))

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Rubi [A]  time = 4.49424, antiderivative size = 646, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ -\frac{d (a+b x)^{m+1} (c+d x)^{-m-2} \left (a^2 d^2 f^2 \left (m^2+7 m+12\right )-2 a b d f \left (c f \left (m^2+6 m+10\right )+d e (m+2)\right )+b^2 \left (-\left (-c^2 f^2 \left (m^2+5 m+6\right )-2 c d e f (m+4)+2 d^2 e^2\right )\right )\right )}{(m+2) (m+3) (b c-a d)^2 (b e-a f) (d e-c f)^3}-\frac{d (a+b x)^{m+1} (c+d x)^{-m-1} \left (a^3 d^3 f^3 \left (m^3+9 m^2+26 m+24\right )-a^2 b d^2 f^2 (m+3) \left (c f \left (3 m^2+15 m+20\right )+d e (3 m+4)\right )-a b^2 d f \left (-c^2 f^2 \left (3 m^3+21 m^2+50 m+44\right )-2 c d e f \left (3 m^2+15 m+16\right )+2 d^2 e^2 (m+2)\right )+b^3 \left (-\left (c^3 f^3 \left (m^3+6 m^2+11 m+6\right )+c^2 d e f^2 \left (3 m^2+17 m+26\right )-2 c d^2 e^2 f (m+5)+2 d^3 e^3\right )\right )\right )}{(m+1) (m+2) (m+3) (b c-a d)^3 (b e-a f) (d e-c f)^4}+\frac{f^3 (a+b x)^{m+1} (c+d x)^{-m-1} (a d f (m+4)-b (c f m+4 d e)) \, _2F_1\left (1,m+1;m+2;\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{(m+1) (b e-a f)^2 (d e-c f)^4}+\frac{d (a+b x)^{m+1} (c+d x)^{-m-3}}{(m+3) (e+f x) (b c-a d) (d e-c f)}+\frac{f (a+b x)^{m+1} (c+d x)^{-m-2} (-a d f (m+4)+b c f (m+3)+b d e)}{(m+3) (e+f x) (b c-a d) (b e-a f) (d e-c f)^2} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^m*(c + d*x)^(-4 - m))/(e + f*x)^2,x]

[Out]

-((d*(a^2*d^2*f^2*(12 + 7*m + m^2) - b^2*(2*d^2*e^2 - 2*c*d*e*f*(4 + m) - c^2*f^
2*(6 + 5*m + m^2)) - 2*a*b*d*f*(d*e*(2 + m) + c*f*(10 + 6*m + m^2)))*(a + b*x)^(
1 + m)*(c + d*x)^(-2 - m))/((b*c - a*d)^2*(b*e - a*f)*(d*e - c*f)^3*(2 + m)*(3 +
m))) - (d*(a^3*d^3*f^3*(24 + 26*m + 9*m^2 + m^3) - a^2*b*d^2*f^2*(3 + m)*(d*e*(
4 + 3*m) + c*f*(20 + 15*m + 3*m^2)) - b^3*(2*d^3*e^3 - 2*c*d^2*e^2*f*(5 + m) + c
^2*d*e*f^2*(26 + 17*m + 3*m^2) + c^3*f^3*(6 + 11*m + 6*m^2 + m^3)) - a*b^2*d*f*(
2*d^2*e^2*(2 + m) - 2*c*d*e*f*(16 + 15*m + 3*m^2) - c^2*f^2*(44 + 50*m + 21*m^2
+ 3*m^3)))*(a + b*x)^(1 + m)*(c + d*x)^(-1 - m))/((b*c - a*d)^3*(b*e - a*f)*(d*e
- c*f)^4*(1 + m)*(2 + m)*(3 + m)) + (d*(a + b*x)^(1 + m)*(c + d*x)^(-3 - m))/((
b*c - a*d)*(d*e - c*f)*(3 + m)*(e + f*x)) + (f*(b*d*e + b*c*f*(3 + m) - a*d*f*(4
+ m))*(a + b*x)^(1 + m)*(c + d*x)^(-2 - m))/((b*c - a*d)*(b*e - a*f)*(d*e - c*f
)^2*(3 + m)*(e + f*x)) + (f^3*(a*d*f*(4 + m) - b*(4*d*e + c*f*m))*(a + b*x)^(1 +
m)*(c + d*x)^(-1 - m)*Hypergeometric2F1[1, 1 + m, 2 + m, ((d*e - c*f)*(a + b*x)
)/((b*e - a*f)*(c + d*x))])/((b*e - a*f)^2*(d*e - c*f)^4*(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)**(-4-m)/(f*x+e)**2,x)

[Out]

Timed out

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Mathematica [C]  time = 39.3011, size = 64249, normalized size = 99.15 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[((a + b*x)^m*(c + d*x)^(-4 - m))/(e + f*x)^2,x]

[Out]

Result too large to show

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Maple [F]  time = 0.13, size = 0, normalized size = 0. \[ \int{\frac{ \left ( bx+a \right ) ^{m} \left ( dx+c \right ) ^{-4-m}}{ \left ( fx+e \right ) ^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^m*(d*x+c)^(-4-m)/(f*x+e)^2,x)

[Out]

int((b*x+a)^m*(d*x+c)^(-4-m)/(f*x+e)^2,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m - 4}}{{\left (f x + e\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^m*(d*x + c)^(-m - 4)/(f*x + e)^2,x, algorithm="maxima")

[Out]

integrate((b*x + a)^m*(d*x + c)^(-m - 4)/(f*x + e)^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^m*(d*x + c)^(-m - 4)/(f*x + e)^2,x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m - 4}}{{\left (f x + e\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^m*(d*x + c)^(-m - 4)/(f*x + e)^2,x, algorithm="giac")

[Out]

integrate((b*x + a)^m*(d*x + c)^(-m - 4)/(f*x + e)^2, x)