Optimal. Leaf size=65 \[ -\frac{2 d (b c-a d)}{5 b^3 (a+b x)^5}-\frac{(b c-a d)^2}{6 b^3 (a+b x)^6}-\frac{d^2}{4 b^3 (a+b x)^4} \]
[Out]
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Rubi [A] time = 0.102345, 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 \[ -\frac{2 d (b c-a d)}{5 b^3 (a+b x)^5}-\frac{(b c-a d)^2}{6 b^3 (a+b x)^6}-\frac{d^2}{4 b^3 (a+b x)^4} \]
Antiderivative was successfully verified.
[In] Int[(a*c + (b*c + a*d)*x + b*d*x^2)^2/(a + b*x)^9,x]
[Out]
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Rubi in Sympy [A] time = 25.2377, size = 56, normalized size = 0.86 \[ - \frac{d^{2}}{4 b^{3} \left (a + b x\right )^{4}} + \frac{2 d \left (a d - b c\right )}{5 b^{3} \left (a + b x\right )^{5}} - \frac{\left (a d - b c\right )^{2}}{6 b^{3} \left (a + b x\right )^{6}} \]
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)**9,x)
[Out]
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Mathematica [A] time = 0.0388239, size = 58, normalized size = 0.89 \[ -\frac{a^2 d^2+2 a b d (2 c+3 d x)+b^2 \left (10 c^2+24 c d x+15 d^2 x^2\right )}{60 b^3 (a+b x)^6} \]
Antiderivative was successfully verified.
[In] Integrate[(a*c + (b*c + a*d)*x + b*d*x^2)^2/(a + b*x)^9,x]
[Out]
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Maple [A] time = 0.008, size = 71, normalized size = 1.1 \[{\frac{ \left ( 2\,ad-2\,bc \right ) d}{5\,{b}^{3} \left ( bx+a \right ) ^{5}}}-{\frac{{a}^{2}{d}^{2}-2\,cabd+{b}^{2}{c}^{2}}{6\,{b}^{3} \left ( bx+a \right ) ^{6}}}-{\frac{{d}^{2}}{4\,{b}^{3} \left ( bx+a \right ) ^{4}}} \]
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)^9,x)
[Out]
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Maxima [A] time = 0.740962, size = 162, normalized size = 2.49 \[ -\frac{15 \, b^{2} d^{2} x^{2} + 10 \, b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2} + 6 \,{\left (4 \, b^{2} c d + a b d^{2}\right )} x}{60 \,{\left (b^{9} x^{6} + 6 \, a b^{8} x^{5} + 15 \, a^{2} b^{7} x^{4} + 20 \, a^{3} b^{6} x^{3} + 15 \, a^{4} b^{5} x^{2} + 6 \, a^{5} b^{4} x + a^{6} b^{3}\right )}} \]
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)^9,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.212494, size = 162, normalized size = 2.49 \[ -\frac{15 \, b^{2} d^{2} x^{2} + 10 \, b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2} + 6 \,{\left (4 \, b^{2} c d + a b d^{2}\right )} x}{60 \,{\left (b^{9} x^{6} + 6 \, a b^{8} x^{5} + 15 \, a^{2} b^{7} x^{4} + 20 \, a^{3} b^{6} x^{3} + 15 \, a^{4} b^{5} x^{2} + 6 \, a^{5} b^{4} x + a^{6} b^{3}\right )}} \]
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)^9,x, algorithm="fricas")
[Out]
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Sympy [A] time = 5.33839, size = 128, normalized size = 1.97 \[ - \frac{a^{2} d^{2} + 4 a b c d + 10 b^{2} c^{2} + 15 b^{2} d^{2} x^{2} + x \left (6 a b d^{2} + 24 b^{2} c d\right )}{60 a^{6} b^{3} + 360 a^{5} b^{4} x + 900 a^{4} b^{5} x^{2} + 1200 a^{3} b^{6} x^{3} + 900 a^{2} b^{7} x^{4} + 360 a b^{8} x^{5} + 60 b^{9} x^{6}} \]
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)**9,x)
[Out]
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GIAC/XCAS [A] time = 0.210768, size = 82, normalized size = 1.26 \[ -\frac{15 \, b^{2} d^{2} x^{2} + 24 \, b^{2} c d x + 6 \, a b d^{2} x + 10 \, b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}}{60 \,{\left (b x + a\right )}^{6} b^{3}} \]
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)^9,x, algorithm="giac")
[Out]