3.3077 \(\int \frac{(a+b x)^m (c+d x)^{-3-m}}{(e+f x)^2} \, dx\)

Optimal. Leaf size=400 \[ -\frac{d (a+b x)^{m+1} (c+d x)^{-m-1} \left (a^2 d^2 f^2 \left (m^2+5 m+6\right )-a b d f \left (c f \left (2 m^2+8 m+9\right )+d e (2 m+3)\right )+b^2 \left (-\left (-c^2 f^2 \left (m^2+3 m+2\right )-c d e f (2 m+5)+d^2 e^2\right )\right )\right )}{(m+1) (m+2) (b c-a d)^2 (b e-a f) (d e-c f)^3}-\frac{f^2 (a+b x)^{m+1} (c+d x)^{-m-1} (a d f (m+3)-b (c f m+3 d e)) \, _2F_1\left (1,m+1;m+2;\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{(m+1) (b e-a f)^2 (d e-c f)^3}+\frac{d (a+b x)^{m+1} (c+d x)^{-m-2}}{(m+2) (e+f x) (b c-a d) (d e-c f)}-\frac{f (a+b x)^{m+1} (c+d x)^{-m-1} (a d f (m+3)-b (c f (m+2)+d e))}{(m+2) (e+f x) (b c-a d) (b e-a f) (d e-c f)^2} \]

[Out]

-((d*(a^2*d^2*f^2*(6 + 5*m + m^2) - b^2*(d^2*e^2 - c*d*e*f*(5 + 2*m) - c^2*f^2*(
2 + 3*m + m^2)) - a*b*d*f*(d*e*(3 + 2*m) + c*f*(9 + 8*m + 2*m^2)))*(a + b*x)^(1
+ m)*(c + d*x)^(-1 - m))/((b*c - a*d)^2*(b*e - a*f)*(d*e - c*f)^3*(1 + m)*(2 + m
))) + (d*(a + b*x)^(1 + m)*(c + d*x)^(-2 - m))/((b*c - a*d)*(d*e - c*f)*(2 + m)*
(e + f*x)) - (f*(a*d*f*(3 + m) - b*(d*e + c*f*(2 + m)))*(a + b*x)^(1 + m)*(c + d
*x)^(-1 - m))/((b*c - a*d)*(b*e - a*f)*(d*e - c*f)^2*(2 + m)*(e + f*x)) - (f^2*(
a*d*f*(3 + m) - b*(3*d*e + c*f*m))*(a + b*x)^(1 + m)*(c + d*x)^(-1 - m)*Hypergeo
metric2F1[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)^3*(1 + m))

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Rubi [A]  time = 1.73184, antiderivative size = 398, 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 \[ -\frac{d (a+b x)^{m+1} (c+d x)^{-m-1} \left (a^2 d^2 f^2 \left (m^2+5 m+6\right )-a b d f \left (c f \left (2 m^2+8 m+9\right )+d e (2 m+3)\right )+b^2 \left (-\left (-c^2 f^2 \left (m^2+3 m+2\right )-c d e f (2 m+5)+d^2 e^2\right )\right )\right )}{(m+1) (m+2) (b c-a d)^2 (b e-a f) (d e-c f)^3}-\frac{f^2 (a+b x)^{m+1} (c+d x)^{-m-1} (a d f (m+3)-b (c f m+3 d e)) \, _2F_1\left (1,m+1;m+2;\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{(m+1) (b e-a f)^2 (d e-c f)^3}+\frac{d (a+b x)^{m+1} (c+d x)^{-m-2}}{(m+2) (e+f x) (b c-a d) (d e-c f)}+\frac{f (a+b x)^{m+1} (c+d x)^{-m-1} (-a d f (m+3)+b c f (m+2)+b d e)}{(m+2) (e+f x) (b c-a d) (b e-a f) (d e-c f)^2} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^m*(c + d*x)^(-3 - m))/(e + f*x)^2,x]

[Out]

-((d*(a^2*d^2*f^2*(6 + 5*m + m^2) - b^2*(d^2*e^2 - c*d*e*f*(5 + 2*m) - c^2*f^2*(
2 + 3*m + m^2)) - a*b*d*f*(d*e*(3 + 2*m) + c*f*(9 + 8*m + 2*m^2)))*(a + b*x)^(1
+ m)*(c + d*x)^(-1 - m))/((b*c - a*d)^2*(b*e - a*f)*(d*e - c*f)^3*(1 + m)*(2 + m
))) + (d*(a + b*x)^(1 + m)*(c + d*x)^(-2 - m))/((b*c - a*d)*(d*e - c*f)*(2 + m)*
(e + f*x)) + (f*(b*d*e + b*c*f*(2 + m) - a*d*f*(3 + m))*(a + b*x)^(1 + m)*(c + d
*x)^(-1 - m))/((b*c - a*d)*(b*e - a*f)*(d*e - c*f)^2*(2 + m)*(e + f*x)) - (f^2*(
a*d*f*(3 + m) - b*(3*d*e + c*f*m))*(a + b*x)^(1 + m)*(c + d*x)^(-1 - m)*Hypergeo
metric2F1[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)^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)**(-3-m)/(f*x+e)**2,x)

[Out]

Timed out

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Mathematica [C]  time = 26.8199, size = 38673, normalized size = 96.68 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[((a + b*x)^m*(c + d*x)^(-3 - m))/(e + f*x)^2,x]

[Out]

Result too large to show

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Maple [F]  time = 180., size = 0, normalized size = 0. \[ \text{hanged} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^m*(d*x+c)^(-3-m)/(f*x+e)^2,x)

[Out]

int((b*x+a)^m*(d*x+c)^(-3-m)/(f*x+e)^2,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m - 3}}{{\left (f x + e\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^m*(d*x + c)^(-m - 3)/(f*x + e)^2,x, algorithm="maxima")

[Out]

integrate((b*x + a)^m*(d*x + c)^(-m - 3)/(f*x + e)^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^m*(d*x + c)^(-m - 3)/(f*x + e)^2,x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m - 3}}{{\left (f x + e\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^m*(d*x + c)^(-m - 3)/(f*x + e)^2,x, algorithm="giac")

[Out]

integrate((b*x + a)^m*(d*x + c)^(-m - 3)/(f*x + e)^2, x)