3.3172 \(\int \frac{(a+b x)^m}{(c+d x) (e+f x)^2} \, dx\)

Optimal. Leaf size=187 \[ \frac{d^2 (a+b x)^{m+1} \, _2F_1\left (1,m+1;m+2;-\frac{d (a+b x)}{b c-a d}\right )}{(m+1) (b c-a d) (d e-c f)^2}+\frac{f (a+b x)^{m+1} (a d f-b (c f m+d e (1-m))) \, _2F_1\left (1,m+1;m+2;-\frac{f (a+b x)}{b e-a f}\right )}{(m+1) (b e-a f)^2 (d e-c f)^2}-\frac{f (a+b x)^{m+1}}{(e+f x) (b e-a f) (d e-c f)} \]

[Out]

-((f*(a + b*x)^(1 + m))/((b*e - a*f)*(d*e - c*f)*(e + f*x))) + (d^2*(a + b*x)^(1
+ m)*Hypergeometric2F1[1, 1 + m, 2 + m, -((d*(a + b*x))/(b*c - a*d))])/((b*c -
a*d)*(d*e - c*f)^2*(1 + m)) + (f*(a*d*f - b*(d*e*(1 - m) + c*f*m))*(a + b*x)^(1
+ m)*Hypergeometric2F1[1, 1 + m, 2 + m, -((f*(a + b*x))/(b*e - a*f))])/((b*e - a
*f)^2*(d*e - c*f)^2*(1 + m))

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Rubi [A]  time = 0.545902, antiderivative size = 187, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{d^2 (a+b x)^{m+1} \, _2F_1\left (1,m+1;m+2;-\frac{d (a+b x)}{b c-a d}\right )}{(m+1) (b c-a d) (d e-c f)^2}+\frac{f (a+b x)^{m+1} (a d f-b c f m-b d e (1-m)) \, _2F_1\left (1,m+1;m+2;-\frac{f (a+b x)}{b e-a f}\right )}{(m+1) (b e-a f)^2 (d e-c f)^2}-\frac{f (a+b x)^{m+1}}{(e+f x) (b e-a f) (d e-c f)} \]

Antiderivative was successfully verified.

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

[Out]

-((f*(a + b*x)^(1 + m))/((b*e - a*f)*(d*e - c*f)*(e + f*x))) + (d^2*(a + b*x)^(1
+ m)*Hypergeometric2F1[1, 1 + m, 2 + m, -((d*(a + b*x))/(b*c - a*d))])/((b*c -
a*d)*(d*e - c*f)^2*(1 + m)) + (f*(a*d*f - b*d*e*(1 - m) - b*c*f*m)*(a + b*x)^(1
+ m)*Hypergeometric2F1[1, 1 + m, 2 + m, -((f*(a + b*x))/(b*e - a*f))])/((b*e - a
*f)^2*(d*e - c*f)^2*(1 + m))

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Rubi in Sympy [A]  time = 145.93, size = 150, normalized size = 0.8 \[ - \frac{d^{2} \left (a + b x\right )^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{d \left (a + b x\right )}{a d - b c}} \right )}}{\left (m + 1\right ) \left (a d - b c\right ) \left (c f - d e\right )^{2}} + \frac{f \left (a + b x\right )^{m + 1} \left (a d f - b c f m + b d e m - b d e\right ){{}_{2}F_{1}\left (\begin{matrix} 1, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{f \left (a + b x\right )}{a f - b e}} \right )}}{\left (m + 1\right ) \left (a f - b e\right )^{2} \left (c f - d e\right )^{2}} - \frac{f \left (a + b x\right )^{m + 1}}{\left (e + f x\right ) \left (a f - b e\right ) \left (c f - d e\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0927666, size = 0, normalized size = 0. \[ \int \frac{(a+b x)^m}{(c+d x) (e+f x)^2} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [F]  time = 0.131, size = 0, normalized size = 0. \[ \int{\frac{ \left ( bx+a \right ) ^{m}}{ \left ( dx+c \right ) \left ( fx+e \right ) ^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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