Optimal. Leaf size=65 \[ -\frac{d (b c-a d)}{2 b^3 (a+b x)^4}-\frac{(b c-a d)^2}{5 b^3 (a+b x)^5}-\frac{d^2}{3 b^3 (a+b x)^3} \]
[Out]
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Rubi [A] time = 0.100897, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069 \[ -\frac{d (b c-a d)}{2 b^3 (a+b x)^4}-\frac{(b c-a d)^2}{5 b^3 (a+b x)^5}-\frac{d^2}{3 b^3 (a+b x)^3} \]
Antiderivative was successfully verified.
[In] Int[(a*c + (b*c + a*d)*x + b*d*x^2)^2/(a + b*x)^8,x]
[Out]
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Rubi in Sympy [A] time = 24.5805, size = 54, normalized size = 0.83 \[ - \frac{d^{2}}{3 b^{3} \left (a + b x\right )^{3}} + \frac{d \left (a d - b c\right )}{2 b^{3} \left (a + b x\right )^{4}} - \frac{\left (a d - b c\right )^{2}}{5 b^{3} \left (a + b x\right )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a*c+(a*d+b*c)*x+b*d*x**2)**2/(b*x+a)**8,x)
[Out]
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Mathematica [A] time = 0.0459684, size = 57, normalized size = 0.88 \[ -\frac{a^2 d^2+a b d (3 c+5 d x)+b^2 \left (6 c^2+15 c d x+10 d^2 x^2\right )}{30 b^3 (a+b x)^5} \]
Antiderivative was successfully verified.
[In] Integrate[(a*c + (b*c + a*d)*x + b*d*x^2)^2/(a + b*x)^8,x]
[Out]
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Maple [A] time = 0.008, size = 71, normalized size = 1.1 \[ -{\frac{{a}^{2}{d}^{2}-2\,cabd+{b}^{2}{c}^{2}}{5\,{b}^{3} \left ( bx+a \right ) ^{5}}}+{\frac{ \left ( ad-bc \right ) d}{2\,{b}^{3} \left ( bx+a \right ) ^{4}}}-{\frac{{d}^{2}}{3\,{b}^{3} \left ( bx+a \right ) ^{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a*c+(a*d+b*c)*x+x^2*b*d)^2/(b*x+a)^8,x)
[Out]
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Maxima [A] time = 0.741436, size = 147, normalized size = 2.26 \[ -\frac{10 \, b^{2} d^{2} x^{2} + 6 \, b^{2} c^{2} + 3 \, a b c d + a^{2} d^{2} + 5 \,{\left (3 \, b^{2} c d + a b d^{2}\right )} x}{30 \,{\left (b^{8} x^{5} + 5 \, a b^{7} x^{4} + 10 \, a^{2} b^{6} x^{3} + 10 \, a^{3} b^{5} x^{2} + 5 \, a^{4} b^{4} x + a^{5} b^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*d*x^2 + a*c + (b*c + a*d)*x)^2/(b*x + a)^8,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.227226, size = 147, normalized size = 2.26 \[ -\frac{10 \, b^{2} d^{2} x^{2} + 6 \, b^{2} c^{2} + 3 \, a b c d + a^{2} d^{2} + 5 \,{\left (3 \, b^{2} c d + a b d^{2}\right )} x}{30 \,{\left (b^{8} x^{5} + 5 \, a b^{7} x^{4} + 10 \, a^{2} b^{6} x^{3} + 10 \, a^{3} b^{5} x^{2} + 5 \, a^{4} b^{4} x + a^{5} b^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*d*x^2 + a*c + (b*c + a*d)*x)^2/(b*x + a)^8,x, algorithm="fricas")
[Out]
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Sympy [A] time = 4.55562, size = 116, normalized size = 1.78 \[ - \frac{a^{2} d^{2} + 3 a b c d + 6 b^{2} c^{2} + 10 b^{2} d^{2} x^{2} + x \left (5 a b d^{2} + 15 b^{2} c d\right )}{30 a^{5} b^{3} + 150 a^{4} b^{4} x + 300 a^{3} b^{5} x^{2} + 300 a^{2} b^{6} x^{3} + 150 a b^{7} x^{4} + 30 b^{8} x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a*c+(a*d+b*c)*x+b*d*x**2)**2/(b*x+a)**8,x)
[Out]
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GIAC/XCAS [A] time = 0.209281, size = 82, normalized size = 1.26 \[ -\frac{10 \, b^{2} d^{2} x^{2} + 15 \, b^{2} c d x + 5 \, a b d^{2} x + 6 \, b^{2} c^{2} + 3 \, a b c d + a^{2} d^{2}}{30 \,{\left (b x + a\right )}^{5} b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*d*x^2 + a*c + (b*c + a*d)*x)^2/(b*x + a)^8,x, algorithm="giac")
[Out]