Optimal. Leaf size=38 \[ -\frac{b c-a d}{4 b^2 (a+b x)^4}-\frac{d}{3 b^2 (a+b x)^3} \]
[Out]
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Rubi [A] time = 0.0480787, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ -\frac{b c-a d}{4 b^2 (a+b x)^4}-\frac{d}{3 b^2 (a+b x)^3} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x)/(a + b*x)^5,x]
[Out]
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Rubi in Sympy [A] time = 8.54797, size = 31, normalized size = 0.82 \[ - \frac{d}{3 b^{2} \left (a + b x\right )^{3}} + \frac{a d - b c}{4 b^{2} \left (a + b x\right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x+c)/(b*x+a)**5,x)
[Out]
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Mathematica [A] time = 0.0149042, size = 27, normalized size = 0.71 \[ -\frac{a d+3 b c+4 b d x}{12 b^2 (a+b x)^4} \]
Antiderivative was successfully verified.
[In] Integrate[(c + d*x)/(a + b*x)^5,x]
[Out]
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Maple [A] time = 0.009, size = 35, normalized size = 0.9 \[ -{\frac{-ad+bc}{4\,{b}^{2} \left ( bx+a \right ) ^{4}}}-{\frac{d}{3\,{b}^{2} \left ( bx+a \right ) ^{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x+c)/(b*x+a)^5,x)
[Out]
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Maxima [A] time = 1.34367, size = 82, normalized size = 2.16 \[ -\frac{4 \, b d x + 3 \, b c + a d}{12 \,{\left (b^{6} x^{4} + 4 \, a b^{5} x^{3} + 6 \, a^{2} b^{4} x^{2} + 4 \, a^{3} b^{3} x + a^{4} b^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)/(b*x + a)^5,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.196378, size = 82, normalized size = 2.16 \[ -\frac{4 \, b d x + 3 \, b c + a d}{12 \,{\left (b^{6} x^{4} + 4 \, a b^{5} x^{3} + 6 \, a^{2} b^{4} x^{2} + 4 \, a^{3} b^{3} x + a^{4} b^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)/(b*x + a)^5,x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.22688, size = 65, normalized size = 1.71 \[ - \frac{a d + 3 b c + 4 b d x}{12 a^{4} b^{2} + 48 a^{3} b^{3} x + 72 a^{2} b^{4} x^{2} + 48 a b^{5} x^{3} + 12 b^{6} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x+c)/(b*x+a)**5,x)
[Out]
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GIAC/XCAS [A] time = 0.216161, size = 55, normalized size = 1.45 \[ -\frac{c}{4 \,{\left (b x + a\right )}^{4} b} - \frac{d}{3 \,{\left (b x + a\right )}^{3} b^{2}} + \frac{a d}{4 \,{\left (b x + a\right )}^{4} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)/(b*x + a)^5,x, algorithm="giac")
[Out]