Optimal. Leaf size=648 \[ -\frac{d (a+b x)^{m+1} (c+d x)^{-m-2} \left (a^2 d^2 f^2 \left (m^2+7 m+12\right )-2 a b d f \left (c f \left (m^2+6 m+10\right )+d e (m+2)\right )+b^2 \left (-\left (-c^2 f^2 \left (m^2+5 m+6\right )-2 c d e f (m+4)+2 d^2 e^2\right )\right )\right )}{(m+2) (m+3) (b c-a d)^2 (b e-a f) (d e-c f)^3}-\frac{d (a+b x)^{m+1} (c+d x)^{-m-1} \left (a^3 d^3 f^3 \left (m^3+9 m^2+26 m+24\right )-a^2 b d^2 f^2 (m+3) \left (c f \left (3 m^2+15 m+20\right )+d e (3 m+4)\right )-a b^2 d f \left (-c^2 f^2 \left (3 m^3+21 m^2+50 m+44\right )-2 c d e f \left (3 m^2+15 m+16\right )+2 d^2 e^2 (m+2)\right )+b^3 \left (-\left (c^3 f^3 \left (m^3+6 m^2+11 m+6\right )+c^2 d e f^2 \left (3 m^2+17 m+26\right )-2 c d^2 e^2 f (m+5)+2 d^3 e^3\right )\right )\right )}{(m+1) (m+2) (m+3) (b c-a d)^3 (b e-a f) (d e-c f)^4}+\frac{f^3 (a+b x)^{m+1} (c+d x)^{-m-1} (a d f (m+4)-b (c f m+4 d e)) \, _2F_1\left (1,m+1;m+2;\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{(m+1) (b e-a f)^2 (d e-c f)^4}+\frac{d (a+b x)^{m+1} (c+d x)^{-m-3}}{(m+3) (e+f x) (b c-a d) (d e-c f)}-\frac{f (a+b x)^{m+1} (c+d x)^{-m-2} (a d f (m+4)-b (c f (m+3)+d e))}{(m+3) (e+f x) (b c-a d) (b e-a f) (d e-c f)^2} \]
[Out]
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Rubi [A] time = 4.49424, antiderivative size = 646, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ -\frac{d (a+b x)^{m+1} (c+d x)^{-m-2} \left (a^2 d^2 f^2 \left (m^2+7 m+12\right )-2 a b d f \left (c f \left (m^2+6 m+10\right )+d e (m+2)\right )+b^2 \left (-\left (-c^2 f^2 \left (m^2+5 m+6\right )-2 c d e f (m+4)+2 d^2 e^2\right )\right )\right )}{(m+2) (m+3) (b c-a d)^2 (b e-a f) (d e-c f)^3}-\frac{d (a+b x)^{m+1} (c+d x)^{-m-1} \left (a^3 d^3 f^3 \left (m^3+9 m^2+26 m+24\right )-a^2 b d^2 f^2 (m+3) \left (c f \left (3 m^2+15 m+20\right )+d e (3 m+4)\right )-a b^2 d f \left (-c^2 f^2 \left (3 m^3+21 m^2+50 m+44\right )-2 c d e f \left (3 m^2+15 m+16\right )+2 d^2 e^2 (m+2)\right )+b^3 \left (-\left (c^3 f^3 \left (m^3+6 m^2+11 m+6\right )+c^2 d e f^2 \left (3 m^2+17 m+26\right )-2 c d^2 e^2 f (m+5)+2 d^3 e^3\right )\right )\right )}{(m+1) (m+2) (m+3) (b c-a d)^3 (b e-a f) (d e-c f)^4}+\frac{f^3 (a+b x)^{m+1} (c+d x)^{-m-1} (a d f (m+4)-b (c f m+4 d e)) \, _2F_1\left (1,m+1;m+2;\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{(m+1) (b e-a f)^2 (d e-c f)^4}+\frac{d (a+b x)^{m+1} (c+d x)^{-m-3}}{(m+3) (e+f x) (b c-a d) (d e-c f)}+\frac{f (a+b x)^{m+1} (c+d x)^{-m-2} (-a d f (m+4)+b c f (m+3)+b d e)}{(m+3) (e+f x) (b c-a d) (b e-a f) (d e-c f)^2} \]
Antiderivative was successfully verified.
[In] Int[((a + b*x)^m*(c + d*x)^(-4 - m))/(e + f*x)^2,x]
[Out]
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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)**(-4-m)/(f*x+e)**2,x)
[Out]
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Mathematica [C] time = 39.3011, size = 64249, normalized size = 99.15 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
[In] Integrate[((a + b*x)^m*(c + d*x)^(-4 - m))/(e + f*x)^2,x]
[Out]
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Maple [F] time = 0.13, size = 0, normalized size = 0. \[ \int{\frac{ \left ( bx+a \right ) ^{m} \left ( dx+c \right ) ^{-4-m}}{ \left ( fx+e \right ) ^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^m*(d*x+c)^(-4-m)/(f*x+e)^2,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m - 4}}{{\left (f x + e\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^m*(d*x + c)^(-m - 4)/(f*x + e)^2,x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m - 4}}{f^{2} x^{2} + 2 \, e f x + e^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^m*(d*x + c)^(-m - 4)/(f*x + e)^2,x, algorithm="fricas")
[Out]
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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)**(-4-m)/(f*x+e)**2,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m - 4}}{{\left (f x + e\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^m*(d*x + c)^(-m - 4)/(f*x + e)^2,x, algorithm="giac")
[Out]