Optimal. Leaf size=119 \[ \frac{(a+b x) (b f-a g) \left (a^2+2 a b x+b^2 x^2\right )^p (d+e x)^{-2 p-1}}{(2 p+1) (b d-a e)^2}-\frac{\left (a^2+2 a b x+b^2 x^2\right )^{p+1} (e f-d g) (d+e x)^{-2 (p+1)}}{2 (p+1) (b d-a e)^2} \]
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Rubi [A] time = 0.195857, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.086 \[ \frac{(a+b x) (b f-a g) \left (a^2+2 a b x+b^2 x^2\right )^p (d+e x)^{-2 p-1}}{(2 p+1) (b d-a e)^2}-\frac{\left (a^2+2 a b x+b^2 x^2\right )^{p+1} (e f-d g) (d+e x)^{-2 (p+1)}}{2 (p+1) (b d-a e)^2} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^(-3 - 2*p)*(f + g*x)*(a^2 + 2*a*b*x + b^2*x^2)^p,x]
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Rubi in Sympy [A] time = 42.2037, size = 112, normalized size = 0.94 \[ - \frac{\left (2 a + 2 b x\right ) \left (d + e x\right )^{- 2 p - 1} \left (a g - b f\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{p}}{2 \left (2 p + 1\right ) \left (a e - b d\right )^{2}} + \frac{\left (d + e x\right )^{- 2 p - 2} \left (d g - e f\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{p + 1}}{2 \left (p + 1\right ) \left (a e - b d\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**(-3-2*p)*(g*x+f)*(b**2*x**2+2*a*b*x+a**2)**p,x)
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Mathematica [A] time = 0.351372, size = 97, normalized size = 0.82 \[ \frac{(a+b x) \left ((a+b x)^2\right )^p (d+e x)^{-2 (p+1)} (b (2 d f (p+1)+d g (2 p+1) x+e f x)-a (d g+e (2 f p+f+2 g (p+1) x)))}{2 (p+1) (2 p+1) (b d-a e)^2} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^(-3 - 2*p)*(f + g*x)*(a^2 + 2*a*b*x + b^2*x^2)^p,x]
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Maple [A] time = 0.013, size = 174, normalized size = 1.5 \[ -{\frac{ \left ( bx+a \right ) \left ( ex+d \right ) ^{-2-2\,p} \left ( 2\,aegpx-2\,bdgpx+2\,aefp+2\,aegx-2\,bdfp-bdgx-befx+adg+aef-2\,bdf \right ) \left ({b}^{2}{x}^{2}+2\,abx+{a}^{2} \right ) ^{p}}{4\,{a}^{2}{e}^{2}{p}^{2}-8\,abde{p}^{2}+4\,{b}^{2}{d}^{2}{p}^{2}+6\,{a}^{2}{e}^{2}p-12\,abdep+6\,{b}^{2}{d}^{2}p+2\,{a}^{2}{e}^{2}-4\,abde+2\,{b}^{2}{d}^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^(-3-2*p)*(g*x+f)*(b^2*x^2+2*a*b*x+a^2)^p,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (g x + f\right )}{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p}{\left (e x + d\right )}^{-2 \, p - 3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((g*x + f)*(b^2*x^2 + 2*a*b*x + a^2)^p*(e*x + d)^(-2*p - 3),x, algorithm="maxima")
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Fricas [A] time = 0.316951, size = 470, normalized size = 3.95 \[ -\frac{{\left (a^{2} d^{2} g -{\left (b^{2} e^{2} f + 2 \,{\left (b^{2} d e - a b e^{2}\right )} g p +{\left (b^{2} d e - 2 \, a b e^{2}\right )} g\right )} x^{3} - 2 \,{\left (a b d^{2} - a^{2} d e\right )} f p -{\left (3 \, b^{2} d e f +{\left (b^{2} d^{2} - 2 \, a b d e - 2 \, a^{2} e^{2}\right )} g + 2 \,{\left ({\left (b^{2} d e - a b e^{2}\right )} f +{\left (b^{2} d^{2} - a^{2} e^{2}\right )} g\right )} p\right )} x^{2} -{\left (2 \, a b d^{2} - a^{2} d e\right )} f +{\left (3 \, a^{2} d e g -{\left (2 \, b^{2} d^{2} + 2 \, a b d e - a^{2} e^{2}\right )} f - 2 \,{\left ({\left (b^{2} d^{2} - a^{2} e^{2}\right )} f +{\left (a b d^{2} - a^{2} d e\right )} g\right )} p\right )} x\right )}{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p}{\left (e x + d\right )}^{-2 \, p - 3}}{2 \,{\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2} + 2 \,{\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\right )} p^{2} + 3 \,{\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\right )} p\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((g*x + f)*(b^2*x^2 + 2*a*b*x + a^2)^p*(e*x + d)^(-2*p - 3),x, algorithm="fricas")
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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((e*x+d)**(-3-2*p)*(g*x+f)*(b**2*x**2+2*a*b*x+a**2)**p,x)
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GIAC/XCAS [A] time = 0.293, size = 1, normalized size = 0.01 \[ \mathit{Done} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((g*x + f)*(b^2*x^2 + 2*a*b*x + a^2)^p*(e*x + d)^(-2*p - 3),x, algorithm="giac")
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