Optimal. Leaf size=106 \[ -\frac{a B e^2-2 A c d e+3 B c d^2}{3 e^4 (d+e x)^3}+\frac{\left (a e^2+c d^2\right ) (B d-A e)}{4 e^4 (d+e x)^4}+\frac{c (3 B d-A e)}{2 e^4 (d+e x)^2}-\frac{B c}{e^4 (d+e x)} \]
[Out]
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Rubi [A] time = 0.172787, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05 \[ -\frac{a B e^2-2 A c d e+3 B c d^2}{3 e^4 (d+e x)^3}+\frac{\left (a e^2+c d^2\right ) (B d-A e)}{4 e^4 (d+e x)^4}+\frac{c (3 B d-A e)}{2 e^4 (d+e x)^2}-\frac{B c}{e^4 (d+e x)} \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*(a + c*x^2))/(d + e*x)^5,x]
[Out]
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Rubi in Sympy [A] time = 28.4649, size = 99, normalized size = 0.93 \[ - \frac{B c}{e^{4} \left (d + e x\right )} - \frac{c \left (A e - 3 B d\right )}{2 e^{4} \left (d + e x\right )^{2}} - \frac{- 2 A c d e + B a e^{2} + 3 B c d^{2}}{3 e^{4} \left (d + e x\right )^{3}} - \frac{\left (A e - B d\right ) \left (a e^{2} + c d^{2}\right )}{4 e^{4} \left (d + e x\right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(c*x**2+a)/(e*x+d)**5,x)
[Out]
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Mathematica [A] time = 0.0795001, size = 87, normalized size = 0.82 \[ -\frac{3 a A e^3+a B e^2 (d+4 e x)+A c e \left (d^2+4 d e x+6 e^2 x^2\right )+3 B c \left (d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right )}{12 e^4 (d+e x)^4} \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*(a + c*x^2))/(d + e*x)^5,x]
[Out]
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Maple [A] time = 0.009, size = 110, normalized size = 1. \[ -{\frac{aA{e}^{3}+Ac{d}^{2}e-aBd{e}^{2}-Bc{d}^{3}}{4\,{e}^{4} \left ( ex+d \right ) ^{4}}}-{\frac{-2\,Acde+aB{e}^{2}+3\,Bc{d}^{2}}{3\,{e}^{4} \left ( ex+d \right ) ^{3}}}-{\frac{c \left ( Ae-3\,Bd \right ) }{2\,{e}^{4} \left ( ex+d \right ) ^{2}}}-{\frac{Bc}{{e}^{4} \left ( ex+d \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(c*x^2+a)/(e*x+d)^5,x)
[Out]
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Maxima [A] time = 0.707898, size = 178, normalized size = 1.68 \[ -\frac{12 \, B c e^{3} x^{3} + 3 \, B c d^{3} + A c d^{2} e + B a d e^{2} + 3 \, A a e^{3} + 6 \,{\left (3 \, B c d e^{2} + A c e^{3}\right )} x^{2} + 4 \,{\left (3 \, B c d^{2} e + A c d e^{2} + B a e^{3}\right )} x}{12 \,{\left (e^{8} x^{4} + 4 \, d e^{7} x^{3} + 6 \, d^{2} e^{6} x^{2} + 4 \, d^{3} e^{5} x + d^{4} e^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)*(B*x + A)/(e*x + d)^5,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.270358, size = 178, normalized size = 1.68 \[ -\frac{12 \, B c e^{3} x^{3} + 3 \, B c d^{3} + A c d^{2} e + B a d e^{2} + 3 \, A a e^{3} + 6 \,{\left (3 \, B c d e^{2} + A c e^{3}\right )} x^{2} + 4 \,{\left (3 \, B c d^{2} e + A c d e^{2} + B a e^{3}\right )} x}{12 \,{\left (e^{8} x^{4} + 4 \, d e^{7} x^{3} + 6 \, d^{2} e^{6} x^{2} + 4 \, d^{3} e^{5} x + d^{4} e^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)*(B*x + A)/(e*x + d)^5,x, algorithm="fricas")
[Out]
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Sympy [A] time = 21.3971, size = 148, normalized size = 1.4 \[ - \frac{3 A a e^{3} + A c d^{2} e + B a d e^{2} + 3 B c d^{3} + 12 B c e^{3} x^{3} + x^{2} \left (6 A c e^{3} + 18 B c d e^{2}\right ) + x \left (4 A c d e^{2} + 4 B a e^{3} + 12 B c d^{2} e\right )}{12 d^{4} e^{4} + 48 d^{3} e^{5} x + 72 d^{2} e^{6} x^{2} + 48 d e^{7} x^{3} + 12 e^{8} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*(c*x**2+a)/(e*x+d)**5,x)
[Out]
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GIAC/XCAS [A] time = 0.286195, size = 212, normalized size = 2. \[ -\frac{1}{12} \,{\left (\frac{12 \, B c e^{8}}{x e + d} - \frac{18 \, B c d e^{8}}{{\left (x e + d\right )}^{2}} + \frac{12 \, B c d^{2} e^{8}}{{\left (x e + d\right )}^{3}} - \frac{3 \, B c d^{3} e^{8}}{{\left (x e + d\right )}^{4}} + \frac{6 \, A c e^{9}}{{\left (x e + d\right )}^{2}} - \frac{8 \, A c d e^{9}}{{\left (x e + d\right )}^{3}} + \frac{3 \, A c d^{2} e^{9}}{{\left (x e + d\right )}^{4}} + \frac{4 \, B a e^{10}}{{\left (x e + d\right )}^{3}} - \frac{3 \, B a d e^{10}}{{\left (x e + d\right )}^{4}} + \frac{3 \, A a e^{11}}{{\left (x e + d\right )}^{4}}\right )} e^{\left (-12\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)*(B*x + A)/(e*x + d)^5,x, algorithm="giac")
[Out]