3.3173 \(\int \frac{(a+b x)^m}{(c+d x)^2 (e+f x)^2} \, dx\)

Optimal. Leaf size=281 \[ \frac{d^2 (a+b x)^{m+1} (2 a d f-b (c f (2-m)+d e m)) \, _2F_1\left (1,m+1;m+2;-\frac{d (a+b x)}{b c-a d}\right )}{(m+1) (b c-a d)^2 (d e-c f)^3}-\frac{f^2 (a+b x)^{m+1} (2 a d f-b (c f m+d e (2-m))) \, _2F_1\left (1,m+1;m+2;-\frac{f (a+b x)}{b e-a f}\right )}{(m+1) (b e-a f)^2 (d e-c f)^3}+\frac{f (a+b x)^{m+1} (-2 a d f+b c f+b d e)}{(e+f x) (b c-a d) (b e-a f) (d e-c f)^2}+\frac{d (a+b x)^{m+1}}{(c+d x) (e+f x) (b c-a d) (d e-c f)} \]

[Out]

(f*(b*d*e + b*c*f - 2*a*d*f)*(a + b*x)^(1 + m))/((b*c - a*d)*(b*e - a*f)*(d*e -
c*f)^2*(e + f*x)) + (d*(a + b*x)^(1 + m))/((b*c - a*d)*(d*e - c*f)*(c + d*x)*(e
+ f*x)) + (d^2*(2*a*d*f - b*(c*f*(2 - m) + d*e*m))*(a + b*x)^(1 + m)*Hypergeomet
ric2F1[1, 1 + m, 2 + m, -((d*(a + b*x))/(b*c - a*d))])/((b*c - a*d)^2*(d*e - c*f
)^3*(1 + m)) - (f^2*(2*a*d*f - b*(d*e*(2 - m) + c*f*m))*(a + b*x)^(1 + m)*Hyperg
eometric2F1[1, 1 + m, 2 + m, -((f*(a + b*x))/(b*e - a*f))])/((b*e - a*f)^2*(d*e
- c*f)^3*(1 + m))

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Rubi [A]  time = 1.2358, antiderivative size = 281, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{d^2 (a+b x)^{m+1} (2 a d f-b c f (2-m)-b d e m) \, _2F_1\left (1,m+1;m+2;-\frac{d (a+b x)}{b c-a d}\right )}{(m+1) (b c-a d)^2 (d e-c f)^3}-\frac{f^2 (a+b x)^{m+1} (2 a d f-b c f m-b d e (2-m)) \, _2F_1\left (1,m+1;m+2;-\frac{f (a+b x)}{b e-a f}\right )}{(m+1) (b e-a f)^2 (d e-c f)^3}+\frac{f (a+b x)^{m+1} (-2 a d f+b c f+b d e)}{(e+f x) (b c-a d) (b e-a f) (d e-c f)^2}+\frac{d (a+b x)^{m+1}}{(c+d x) (e+f x) (b c-a d) (d e-c f)} \]

Antiderivative was successfully verified.

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

[Out]

(f*(b*d*e + b*c*f - 2*a*d*f)*(a + b*x)^(1 + m))/((b*c - a*d)*(b*e - a*f)*(d*e -
c*f)^2*(e + f*x)) + (d*(a + b*x)^(1 + m))/((b*c - a*d)*(d*e - c*f)*(c + d*x)*(e
+ f*x)) + (d^2*(2*a*d*f - b*c*f*(2 - m) - b*d*e*m)*(a + b*x)^(1 + m)*Hypergeomet
ric2F1[1, 1 + m, 2 + m, -((d*(a + b*x))/(b*c - a*d))])/((b*c - a*d)^2*(d*e - c*f
)^3*(1 + m)) - (f^2*(2*a*d*f - b*d*e*(2 - m) - b*c*f*m)*(a + b*x)^(1 + m)*Hyperg
eometric2F1[1, 1 + m, 2 + m, -((f*(a + b*x))/(b*e - a*f))])/((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)**2/(f*x+e)**2,x)

[Out]

Timed out

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Mathematica [A]  time = 0.134376, size = 0, normalized size = 0. \[ \int \frac{(a+b x)^m}{(c+d x)^2 (e+f x)^2} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [F]  time = 0.125, size = 0, normalized size = 0. \[ \int{\frac{ \left ( bx+a \right ) ^{m}}{ \left ( dx+c \right ) ^{2} \left ( fx+e \right ) ^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((b*x + a)^m/(d^2*f^2*x^4 + c^2*e^2 + 2*(d^2*e*f + c*d*f^2)*x^3 + (d^2*e
^2 + 4*c*d*e*f + c^2*f^2)*x^2 + 2*(c*d*e^2 + c^2*e*f)*x), 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/(f*x+e)**2,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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