Optimal. Leaf size=61 \[ d^5 \left (b^2-4 a c\right )^2 \log \left (a+b x+c x^2\right )+d^5 \left (b^2-4 a c\right ) (b+2 c x)^2+\frac{1}{2} d^5 (b+2 c x)^4 \]
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Rubi [A] time = 0.111041, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ d^5 \left (b^2-4 a c\right )^2 \log \left (a+b x+c x^2\right )+d^5 \left (b^2-4 a c\right ) (b+2 c x)^2+\frac{1}{2} d^5 (b+2 c x)^4 \]
Antiderivative was successfully verified.
[In] Int[(b*d + 2*c*d*x)^5/(a + b*x + c*x^2),x]
[Out]
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Rubi in Sympy [A] time = 32.1528, size = 58, normalized size = 0.95 \[ \frac{d^{5} \left (b + 2 c x\right )^{4}}{2} + d^{5} \left (b + 2 c x\right )^{2} \left (- 4 a c + b^{2}\right ) + d^{5} \left (- 4 a c + b^{2}\right )^{2} \log{\left (a + b x + c x^{2} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((2*c*d*x+b*d)**5/(c*x**2+b*x+a),x)
[Out]
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Mathematica [A] time = 0.0506072, size = 54, normalized size = 0.89 \[ d^5 \left (8 c x (b+c x) \left (c \left (c x^2-2 a\right )+b^2+b c x\right )+\left (b^2-4 a c\right )^2 \log (a+x (b+c x))\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(b*d + 2*c*d*x)^5/(a + b*x + c*x^2),x]
[Out]
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Maple [B] time = 0.004, size = 133, normalized size = 2.2 \[ 8\,{x}^{4}{c}^{4}{d}^{5}+16\,{x}^{3}b{c}^{3}{d}^{5}-16\,{x}^{2}a{c}^{3}{d}^{5}+16\,{x}^{2}{b}^{2}{c}^{2}{d}^{5}+16\,\ln \left ( c{x}^{2}+bx+a \right ){a}^{2}{c}^{2}{d}^{5}-8\,\ln \left ( c{x}^{2}+bx+a \right ) a{b}^{2}c{d}^{5}+\ln \left ( c{x}^{2}+bx+a \right ){b}^{4}{d}^{5}-16\,xab{c}^{2}{d}^{5}+8\,x{b}^{3}c{d}^{5} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((2*c*d*x+b*d)^5/(c*x^2+b*x+a),x)
[Out]
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Maxima [A] time = 0.700541, size = 134, normalized size = 2.2 \[ 8 \, c^{4} d^{5} x^{4} + 16 \, b c^{3} d^{5} x^{3} + 16 \,{\left (b^{2} c^{2} - a c^{3}\right )} d^{5} x^{2} + 8 \,{\left (b^{3} c - 2 \, a b c^{2}\right )} d^{5} x +{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} d^{5} \log \left (c x^{2} + b x + a\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*d*x + b*d)^5/(c*x^2 + b*x + a),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.203749, size = 134, normalized size = 2.2 \[ 8 \, c^{4} d^{5} x^{4} + 16 \, b c^{3} d^{5} x^{3} + 16 \,{\left (b^{2} c^{2} - a c^{3}\right )} d^{5} x^{2} + 8 \,{\left (b^{3} c - 2 \, a b c^{2}\right )} d^{5} x +{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} d^{5} \log \left (c x^{2} + b x + a\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*d*x + b*d)^5/(c*x^2 + b*x + a),x, algorithm="fricas")
[Out]
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Sympy [A] time = 3.02219, size = 99, normalized size = 1.62 \[ 16 b c^{3} d^{5} x^{3} + 8 c^{4} d^{5} x^{4} + d^{5} \left (4 a c - b^{2}\right )^{2} \log{\left (a + b x + c x^{2} \right )} + x^{2} \left (- 16 a c^{3} d^{5} + 16 b^{2} c^{2} d^{5}\right ) + x \left (- 16 a b c^{2} d^{5} + 8 b^{3} c d^{5}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*d*x+b*d)**5/(c*x**2+b*x+a),x)
[Out]
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GIAC/XCAS [A] time = 0.218121, size = 159, normalized size = 2.61 \[{\left (b^{4} d^{5} - 8 \, a b^{2} c d^{5} + 16 \, a^{2} c^{2} d^{5}\right )}{\rm ln}\left (c x^{2} + b x + a\right ) + \frac{8 \,{\left (c^{8} d^{5} x^{4} + 2 \, b c^{7} d^{5} x^{3} + 2 \, b^{2} c^{6} d^{5} x^{2} - 2 \, a c^{7} d^{5} x^{2} + b^{3} c^{5} d^{5} x - 2 \, a b c^{6} d^{5} x\right )}}{c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*d*x + b*d)^5/(c*x^2 + b*x + a),x, algorithm="giac")
[Out]