3.3068 \(\int \frac{(a+b x)^m (c+d x)^{-2-m}}{(e+f x)^2} \, dx\)

Optimal. Leaf size=245 \[ \frac{f (a+b x)^{m+1} (c+d x)^{-m-1} (a d f (m+2)-b (c f m+2 d e)) \, _2F_1\left (1,m+1;m+2;\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{(m+1) (b e-a f)^2 (d e-c f)^2}+\frac{d (a+b x)^{m+1} (c+d x)^{-m-1}}{(m+1) (e+f x) (b c-a d) (d e-c f)}-\frac{f (a+b x)^{m+1} (c+d x)^{-m} (a d f (m+2)-b (c f (m+1)+d e))}{(m+1) (e+f x) (b c-a d) (b e-a f) (d e-c f)^2} \]

[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)) - (f*(a*d*f*(2 + m) - b*(d*e + c*f*(1 + m)))*(a + b*x)^(1 + m))/((b*c - a*d
)*(b*e - a*f)*(d*e - c*f)^2*(1 + m)*(c + d*x)^m*(e + f*x)) + (f*(a*d*f*(2 + m) -
b*(2*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)^2*(1 + m))

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Rubi [A]  time = 0.645786, antiderivative size = 243, normalized size of antiderivative = 0.99, number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{f (a+b x)^{m+1} (c+d x)^{-m-1} (a d f (m+2)-b (c f m+2 d e)) \, _2F_1\left (1,m+1;m+2;\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{(m+1) (b e-a f)^2 (d e-c f)^2}+\frac{d (a+b x)^{m+1} (c+d x)^{-m-1}}{(m+1) (e+f x) (b c-a d) (d e-c f)}+\frac{f (a+b x)^{m+1} (c+d x)^{-m} (-a d f (m+2)+b c f (m+1)+b d e)}{(m+1) (e+f x) (b c-a d) (b e-a f) (d e-c f)^2} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^m*(c + d*x)^(-2 - m))/(e + f*x)^2,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)) + (f*(b*d*e + b*c*f*(1 + m) - a*d*f*(2 + m))*(a + b*x)^(1 + m))/((b*c - a*d
)*(b*e - a*f)*(d*e - c*f)^2*(1 + m)*(c + d*x)^m*(e + f*x)) + (f*(a*d*f*(2 + m) -
b*(2*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)^2*(1 + m))

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Rubi in Sympy [A]  time = 115.884, size = 199, normalized size = 0.81 \[ - \frac{d \left (a + b x\right )^{m + 1} \left (c + d x\right )^{- m - 1} \left (- b c f - b d e + f \left (a d \left (m + 2\right ) - b c m\right )\right )}{\left (m + 1\right ) \left (a d - b c\right ) \left (a f - b e\right ) \left (c f - d e\right )^{2}} - \frac{f \left (a + b x\right )^{m + 1} \left (c + d x\right )^{- m - 1}}{\left (e + f x\right ) \left (a f - b e\right ) \left (c f - d e\right )} - \frac{f \left (a + b x\right )^{m} \left (c + d x\right )^{- m} \left (a d f m + 2 a d f - b c f m - 2 b d e\right ){{}_{2}F_{1}\left (\begin{matrix} - m, 1 \\ - m + 1 \end{matrix}\middle |{\frac{\left (- c - d x\right ) \left (- a f + b e\right )}{\left (a + b x\right ) \left (c f - d e\right )}} \right )}}{m \left (a f - b e\right ) \left (c f - d e\right )^{3}} \]

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)**2,x)

[Out]

-d*(a + b*x)**(m + 1)*(c + d*x)**(-m - 1)*(-b*c*f - b*d*e + f*(a*d*(m + 2) - b*c
*m))/((m + 1)*(a*d - b*c)*(a*f - b*e)*(c*f - d*e)**2) - f*(a + b*x)**(m + 1)*(c
+ d*x)**(-m - 1)/((e + f*x)*(a*f - b*e)*(c*f - d*e)) - f*(a + b*x)**m*(c + d*x)*
*(-m)*(a*d*f*m + 2*a*d*f - b*c*f*m - 2*b*d*e)*hyper((-m, 1), (-m + 1,), (-c - d*
x)*(-a*f + b*e)/((a + b*x)*(c*f - d*e)))/(m*(a*f - b*e)*(c*f - d*e)**3)

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Mathematica [C]  time = 21.4809, size = 21480, normalized size = 87.67 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

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

[Out]

Result too large to show

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Maple [F]  time = 0.101, size = 0, normalized size = 0. \[ \int{\frac{ \left ( bx+a \right ) ^{m} \left ( dx+c \right ) ^{-2-m}}{ \left ( fx+e \right ) ^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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