3.3069 \(\int \frac{(a+b x)^m (c+d x)^{-2-m}}{(e+f x)^3} \, dx\)

Optimal. Leaf size=453 \[ \frac{f (a+b x)^{m+1} (c+d x)^{-m-1} \left (-a^2 d^2 f^2 \left (m^2+5 m+6\right )+2 a b d f (m+2) (c f m+3 d e)+b^2 \left (-\left (-c^2 f^2 (1-m) m+6 c d e f m+6 d^2 e^2\right )\right )\right ) \, _2F_1\left (1,m+1;m+2;\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{2 (m+1) (b e-a f)^3 (d e-c f)^3}+\frac{f (a+b x)^{m+1} (c+d x)^{-m} \left (a^2 d^2 f^2 \left (m^2+5 m+6\right )-a b d f \left (c f \left (2 m^2+5 m+3\right )+d e (5 m+9)\right )+b^2 \left (-c^2 f^2 \left (1-m^2\right )+5 c d e f (m+1)+2 d^2 e^2\right )\right )}{2 (m+1) (e+f x) (b c-a d) (b e-a f)^2 (d e-c f)^3}+\frac{d (a+b x)^{m+1} (c+d x)^{-m-1}}{(m+1) (e+f x)^2 (b c-a d) (d e-c f)}-\frac{f (a+b x)^{m+1} (c+d x)^{-m} (a d f (m+3)-b (c f (m+1)+2 d e))}{2 (m+1) (e+f x)^2 (b c-a d) (b e-a f) (d e-c f)^2} \]

[Out]

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

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Rubi [A]  time = 1.70044, antiderivative size = 452, 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 \[ \frac{f (a+b x)^{m+1} (c+d x)^{-m-1} \left (-a^2 d^2 f^2 \left (m^2+5 m+6\right )+2 a b d f (m+2) (c f m+3 d e)+b^2 \left (-\left (-c^2 f^2 (1-m) m+6 c d e f m+6 d^2 e^2\right )\right )\right ) \, _2F_1\left (1,m+1;m+2;\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{2 (m+1) (b e-a f)^3 (d e-c f)^3}+\frac{f (a+b x)^{m+1} (c+d x)^{-m} \left (a^2 d^2 f^2 \left (m^2+5 m+6\right )-a b d f \left (c f \left (2 m^2+5 m+3\right )+d e (5 m+9)\right )+b^2 \left (-c^2 f^2 \left (1-m^2\right )+5 c d e f (m+1)+2 d^2 e^2\right )\right )}{2 (m+1) (e+f x) (b c-a d) (b e-a f)^2 (d e-c f)^3}+\frac{d (a+b x)^{m+1} (c+d x)^{-m-1}}{(m+1) (e+f x)^2 (b c-a d) (d e-c f)}+\frac{f (a+b x)^{m+1} (c+d x)^{-m} (-a d f (m+3)+b c f (m+1)+2 b d e)}{2 (m+1) (e+f x)^2 (b c-a d) (b e-a f) (d e-c f)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

[Out]

Timed out

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Mathematica [C]  time = 25.2963, size = 57971, normalized size = 127.97 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

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

[Out]

Result too large to show

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Maple [F]  time = 0.135, size = 0, normalized size = 0. \[ \int{\frac{ \left ( bx+a \right ) ^{m} \left ( dx+c \right ) ^{-2-m}}{ \left ( fx+e \right ) ^{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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