Optimal. Leaf size=51 \[ \frac{c \left (a^2+2 a b x+b^2 x^2\right )^{p+1} (d+e x)^{-2 (p+1)}}{2 (p+1) (b d-a e)} \]
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Rubi [A] time = 0.0893562, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.026 \[ \frac{c \left (a^2+2 a b x+b^2 x^2\right )^{p+1} (d+e x)^{-2 (p+1)}}{2 (p+1) (b d-a e)} \]
Antiderivative was successfully verified.
[In] Int[(a*c + b*c*x)*(d + e*x)^(-3 - 2*p)*(a^2 + 2*a*b*x + b^2*x^2)^p,x]
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Rubi in Sympy [A] time = 16.4198, size = 46, normalized size = 0.9 \[ - \frac{c \left (d + e x\right )^{- 2 p - 2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{p + 1}}{2 \left (p + 1\right ) \left (a e - b d\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*c*x+a*c)*(e*x+d)**(-3-2*p)*(b**2*x**2+2*a*b*x+a**2)**p,x)
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Mathematica [A] time = 0.138549, size = 42, normalized size = 0.82 \[ \frac{c \left ((a+b x)^2\right )^{p+1} (d+e x)^{-2 (p+1)}}{2 (p+1) (b d-a e)} \]
Antiderivative was successfully verified.
[In] Integrate[(a*c + b*c*x)*(d + e*x)^(-3 - 2*p)*(a^2 + 2*a*b*x + b^2*x^2)^p,x]
[Out]
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Maple [A] time = 0.007, size = 59, normalized size = 1.2 \[ -{\frac{ \left ( bx+a \right ) ^{2} \left ( ex+d \right ) ^{-2-2\,p}c \left ({b}^{2}{x}^{2}+2\,abx+{a}^{2} \right ) ^{p}}{2\,aep-2\,bdp+2\,ae-2\,bd}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*c*x+a*c)*(e*x+d)^(-3-2*p)*(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. \[ \int{\left (b c x + a c\right )}{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p}{\left (e x + d\right )}^{-2 \, p - 3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*c*x + a*c)*(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 = 0.313486, size = 134, normalized size = 2.63 \[ \frac{{\left (b^{2} c e x^{3} + a^{2} c d +{\left (b^{2} c d + 2 \, a b c e\right )} x^{2} +{\left (2 \, a b c d + a^{2} c e\right )} x\right )}{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p}{\left (e x + d\right )}^{-2 \, p - 3}}{2 \,{\left (b d - a e +{\left (b d - a e\right )} p\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*c*x + a*c)*(b^2*x^2 + 2*a*b*x + a^2)^p*(e*x + d)^(-2*p - 3),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*c*x+a*c)*(e*x+d)**(-3-2*p)*(b**2*x**2+2*a*b*x+a**2)**p,x)
[Out]
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GIAC/XCAS [A] time = 0.306423, size = 420, normalized size = 8.24 \[ \frac{b^{2} c x^{3} e^{\left (p{\rm ln}\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right ) - 2 \, p{\rm ln}\left (x e + d\right ) - 3 \,{\rm ln}\left (x e + d\right ) + 1\right )} + b^{2} c d x^{2} e^{\left (p{\rm ln}\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right ) - 2 \, p{\rm ln}\left (x e + d\right ) - 3 \,{\rm ln}\left (x e + d\right )\right )} + 2 \, a b c x^{2} e^{\left (p{\rm ln}\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right ) - 2 \, p{\rm ln}\left (x e + d\right ) - 3 \,{\rm ln}\left (x e + d\right ) + 1\right )} + 2 \, a b c d x e^{\left (p{\rm ln}\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right ) - 2 \, p{\rm ln}\left (x e + d\right ) - 3 \,{\rm ln}\left (x e + d\right )\right )} + a^{2} c x e^{\left (p{\rm ln}\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right ) - 2 \, p{\rm ln}\left (x e + d\right ) - 3 \,{\rm ln}\left (x e + d\right ) + 1\right )} + a^{2} c d e^{\left (p{\rm ln}\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right ) - 2 \, p{\rm ln}\left (x e + d\right ) - 3 \,{\rm ln}\left (x e + d\right )\right )}}{2 \,{\left (b d p - a p e + b d - a e\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*c*x + a*c)*(b^2*x^2 + 2*a*b*x + a^2)^p*(e*x + d)^(-2*p - 3),x, algorithm="giac")
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