Optimal. Leaf size=154 \[ -\frac{a^3 A}{5 x^5}-\frac{a^2 (a B+3 A b)}{4 x^4}-\frac{a \left (A \left (a c+b^2\right )+a b B\right )}{x^3}+3 c \log (x) \left (a B c+A b c+b^2 B\right )-\frac{3 a A c^2+6 a b B c+3 A b^2 c+b^3 B}{x}-\frac{A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )}{2 x^2}+c^2 x (A c+3 b B)+\frac{1}{2} B c^3 x^2 \]
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Rubi [A] time = 0.308818, antiderivative size = 154, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048 \[ -\frac{a^3 A}{5 x^5}-\frac{a^2 (a B+3 A b)}{4 x^4}-\frac{a \left (A \left (a c+b^2\right )+a b B\right )}{x^3}+3 c \log (x) \left (a B c+A b c+b^2 B\right )-\frac{3 a A c^2+6 a b B c+3 A b^2 c+b^3 B}{x}-\frac{A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )}{2 x^2}+c^2 x (A c+3 b B)+\frac{1}{2} B c^3 x^2 \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*(a + b*x + c*x^2)^3)/x^6,x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \frac{A a^{3}}{5 x^{5}} + B c^{3} \int x\, dx - \frac{a^{2} \left (3 A b + B a\right )}{4 x^{4}} - \frac{a \left (A a c + A b^{2} + B a b\right )}{x^{3}} + 3 c \left (A b c + B a c + B b^{2}\right ) \log{\left (x \right )} - \frac{3 A a c^{2} + 3 A b^{2} c + 6 B a b c + B b^{3}}{x} - \frac{3 A a b c + \frac{A b^{3}}{2} + \frac{3 B a^{2} c}{2} + \frac{3 B a b^{2}}{2}}{x^{2}} + \frac{c^{2} \left (A c + 3 B b\right ) \int A\, dx}{A} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(c*x**2+b*x+a)**3/x**6,x)
[Out]
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Mathematica [A] time = 0.206456, size = 161, normalized size = 1.05 \[ -\frac{a^3 (4 A+5 B x)+5 a^2 x \left (3 A b+4 A c x+4 b B x+6 B c x^2\right )+10 a x^2 \left (2 A \left (b^2+3 b c x+3 c^2 x^2\right )+3 b B x (b+4 c x)\right )-60 c x^5 \log (x) \left (a B c+A b c+b^2 B\right )+10 x^3 \left (A \left (b^3+6 b^2 c x-2 c^3 x^3\right )-B x \left (-2 b^3+6 b c^2 x^2+c^3 x^3\right )\right )}{20 x^5} \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*(a + b*x + c*x^2)^3)/x^6,x]
[Out]
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Maple [A] time = 0.012, size = 186, normalized size = 1.2 \[{\frac{B{c}^{3}{x}^{2}}{2}}+Ax{c}^{3}+3\,Bxb{c}^{2}+3\,A\ln \left ( x \right ) b{c}^{2}+3\,B\ln \left ( x \right ) a{c}^{2}+3\,B\ln \left ( x \right ){b}^{2}c-{\frac{3\,A{a}^{2}b}{4\,{x}^{4}}}-{\frac{B{a}^{3}}{4\,{x}^{4}}}-{\frac{A{a}^{2}c}{{x}^{3}}}-{\frac{a{b}^{2}A}{{x}^{3}}}-{\frac{B{a}^{2}b}{{x}^{3}}}-3\,{\frac{Aabc}{{x}^{2}}}-{\frac{A{b}^{3}}{2\,{x}^{2}}}-{\frac{3\,B{a}^{2}c}{2\,{x}^{2}}}-{\frac{3\,a{b}^{2}B}{2\,{x}^{2}}}-{\frac{A{a}^{3}}{5\,{x}^{5}}}-3\,{\frac{aA{c}^{2}}{x}}-3\,{\frac{A{b}^{2}c}{x}}-6\,{\frac{abBc}{x}}-{\frac{B{b}^{3}}{x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(c*x^2+b*x+a)^3/x^6,x)
[Out]
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Maxima [A] time = 0.691626, size = 220, normalized size = 1.43 \[ \frac{1}{2} \, B c^{3} x^{2} +{\left (3 \, B b c^{2} + A c^{3}\right )} x + 3 \,{\left (B b^{2} c +{\left (B a + A b\right )} c^{2}\right )} \log \left (x\right ) - \frac{20 \,{\left (B b^{3} + 3 \, A a c^{2} + 3 \,{\left (2 \, B a b + A b^{2}\right )} c\right )} x^{4} + 4 \, A a^{3} + 10 \,{\left (3 \, B a b^{2} + A b^{3} + 3 \,{\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{3} + 20 \,{\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{2} + 5 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x}{20 \, x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^3*(B*x + A)/x^6,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.283722, size = 227, normalized size = 1.47 \[ \frac{10 \, B c^{3} x^{7} + 20 \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{6} + 60 \,{\left (B b^{2} c +{\left (B a + A b\right )} c^{2}\right )} x^{5} \log \left (x\right ) - 20 \,{\left (B b^{3} + 3 \, A a c^{2} + 3 \,{\left (2 \, B a b + A b^{2}\right )} c\right )} x^{4} - 4 \, A a^{3} - 10 \,{\left (3 \, B a b^{2} + A b^{3} + 3 \,{\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{3} - 20 \,{\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{2} - 5 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x}{20 \, x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^3*(B*x + A)/x^6,x, algorithm="fricas")
[Out]
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Sympy [A] time = 41.4124, size = 175, normalized size = 1.14 \[ \frac{B c^{3} x^{2}}{2} + 3 c \left (A b c + B a c + B b^{2}\right ) \log{\left (x \right )} + x \left (A c^{3} + 3 B b c^{2}\right ) - \frac{4 A a^{3} + x^{4} \left (60 A a c^{2} + 60 A b^{2} c + 120 B a b c + 20 B b^{3}\right ) + x^{3} \left (60 A a b c + 10 A b^{3} + 30 B a^{2} c + 30 B a b^{2}\right ) + x^{2} \left (20 A a^{2} c + 20 A a b^{2} + 20 B a^{2} b\right ) + x \left (15 A a^{2} b + 5 B a^{3}\right )}{20 x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*(c*x**2+b*x+a)**3/x**6,x)
[Out]
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GIAC/XCAS [A] time = 0.269551, size = 219, normalized size = 1.42 \[ \frac{1}{2} \, B c^{3} x^{2} + 3 \, B b c^{2} x + A c^{3} x + 3 \,{\left (B b^{2} c + B a c^{2} + A b c^{2}\right )}{\rm ln}\left ({\left | x \right |}\right ) - \frac{20 \,{\left (B b^{3} + 6 \, B a b c + 3 \, A b^{2} c + 3 \, A a c^{2}\right )} x^{4} + 4 \, A a^{3} + 10 \,{\left (3 \, B a b^{2} + A b^{3} + 3 \, B a^{2} c + 6 \, A a b c\right )} x^{3} + 20 \,{\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{2} + 5 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x}{20 \, x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^3*(B*x + A)/x^6,x, algorithm="giac")
[Out]