Optimal. Leaf size=92 \[ -\frac{d^2 (b c-a d)}{2 b^4 (a+b x)^6}-\frac{3 d (b c-a d)^2}{7 b^4 (a+b x)^7}-\frac{(b c-a d)^3}{8 b^4 (a+b x)^8}-\frac{d^3}{5 b^4 (a+b x)^5} \]
[Out]
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Rubi [A] time = 0.148922, antiderivative size = 92, 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 \[ -\frac{d^2 (b c-a d)}{2 b^4 (a+b x)^6}-\frac{3 d (b c-a d)^2}{7 b^4 (a+b x)^7}-\frac{(b c-a d)^3}{8 b^4 (a+b x)^8}-\frac{d^3}{5 b^4 (a+b x)^5} \]
Antiderivative was successfully verified.
[In] Int[(a*c + (b*c + a*d)*x + b*d*x^2)^3/(a + b*x)^12,x]
[Out]
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Rubi in Sympy [A] time = 34.8638, size = 80, normalized size = 0.87 \[ - \frac{d^{3}}{5 b^{4} \left (a + b x\right )^{5}} + \frac{d^{2} \left (a d - b c\right )}{2 b^{4} \left (a + b x\right )^{6}} - \frac{3 d \left (a d - b c\right )^{2}}{7 b^{4} \left (a + b x\right )^{7}} + \frac{\left (a d - b c\right )^{3}}{8 b^{4} \left (a + b x\right )^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a*c+(a*d+b*c)*x+b*d*x**2)**3/(b*x+a)**12,x)
[Out]
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Mathematica [A] time = 0.0646926, size = 97, normalized size = 1.05 \[ -\frac{a^3 d^3+a^2 b d^2 (5 c+8 d x)+a b^2 d \left (15 c^2+40 c d x+28 d^2 x^2\right )+b^3 \left (35 c^3+120 c^2 d x+140 c d^2 x^2+56 d^3 x^3\right )}{280 b^4 (a+b x)^8} \]
Antiderivative was successfully verified.
[In] Integrate[(a*c + (b*c + a*d)*x + b*d*x^2)^3/(a + b*x)^12,x]
[Out]
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Maple [A] time = 0.008, size = 122, normalized size = 1.3 \[ -{\frac{{d}^{3}}{5\,{b}^{4} \left ( bx+a \right ) ^{5}}}+{\frac{{d}^{2} \left ( ad-bc \right ) }{2\,{b}^{4} \left ( bx+a \right ) ^{6}}}-{\frac{3\,d \left ({a}^{2}{d}^{2}-2\,cabd+{b}^{2}{c}^{2} \right ) }{7\,{b}^{4} \left ( bx+a \right ) ^{7}}}-{\frac{-{a}^{3}{d}^{3}+3\,{a}^{2}c{d}^{2}b-3\,a{c}^{2}d{b}^{2}+{c}^{3}{b}^{3}}{8\,{b}^{4} \left ( bx+a \right ) ^{8}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a*c+(a*d+b*c)*x+x^2*b*d)^3/(b*x+a)^12,x)
[Out]
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Maxima [A] time = 0.74917, size = 261, normalized size = 2.84 \[ -\frac{56 \, b^{3} d^{3} x^{3} + 35 \, b^{3} c^{3} + 15 \, a b^{2} c^{2} d + 5 \, a^{2} b c d^{2} + a^{3} d^{3} + 28 \,{\left (5 \, b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{2} + 8 \,{\left (15 \, b^{3} c^{2} d + 5 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x}{280 \,{\left (b^{12} x^{8} + 8 \, a b^{11} x^{7} + 28 \, a^{2} b^{10} x^{6} + 56 \, a^{3} b^{9} x^{5} + 70 \, a^{4} b^{8} x^{4} + 56 \, a^{5} b^{7} x^{3} + 28 \, a^{6} b^{6} x^{2} + 8 \, a^{7} b^{5} x + a^{8} b^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*d*x^2 + a*c + (b*c + a*d)*x)^3/(b*x + a)^12,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.200284, size = 261, normalized size = 2.84 \[ -\frac{56 \, b^{3} d^{3} x^{3} + 35 \, b^{3} c^{3} + 15 \, a b^{2} c^{2} d + 5 \, a^{2} b c d^{2} + a^{3} d^{3} + 28 \,{\left (5 \, b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{2} + 8 \,{\left (15 \, b^{3} c^{2} d + 5 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x}{280 \,{\left (b^{12} x^{8} + 8 \, a b^{11} x^{7} + 28 \, a^{2} b^{10} x^{6} + 56 \, a^{3} b^{9} x^{5} + 70 \, a^{4} b^{8} x^{4} + 56 \, a^{5} b^{7} x^{3} + 28 \, a^{6} b^{6} x^{2} + 8 \, a^{7} b^{5} x + a^{8} b^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*d*x^2 + a*c + (b*c + a*d)*x)^3/(b*x + a)^12,x, algorithm="fricas")
[Out]
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Sympy [A] time = 50.822, size = 206, normalized size = 2.24 \[ - \frac{a^{3} d^{3} + 5 a^{2} b c d^{2} + 15 a b^{2} c^{2} d + 35 b^{3} c^{3} + 56 b^{3} d^{3} x^{3} + x^{2} \left (28 a b^{2} d^{3} + 140 b^{3} c d^{2}\right ) + x \left (8 a^{2} b d^{3} + 40 a b^{2} c d^{2} + 120 b^{3} c^{2} d\right )}{280 a^{8} b^{4} + 2240 a^{7} b^{5} x + 7840 a^{6} b^{6} x^{2} + 15680 a^{5} b^{7} x^{3} + 19600 a^{4} b^{8} x^{4} + 15680 a^{3} b^{9} x^{5} + 7840 a^{2} b^{10} x^{6} + 2240 a b^{11} x^{7} + 280 b^{12} x^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a*c+(a*d+b*c)*x+b*d*x**2)**3/(b*x+a)**12,x)
[Out]
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GIAC/XCAS [A] time = 0.209289, size = 154, normalized size = 1.67 \[ -\frac{56 \, b^{3} d^{3} x^{3} + 140 \, b^{3} c d^{2} x^{2} + 28 \, a b^{2} d^{3} x^{2} + 120 \, b^{3} c^{2} d x + 40 \, a b^{2} c d^{2} x + 8 \, a^{2} b d^{3} x + 35 \, b^{3} c^{3} + 15 \, a b^{2} c^{2} d + 5 \, a^{2} b c d^{2} + a^{3} d^{3}}{280 \,{\left (b x + a\right )}^{8} b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*d*x^2 + a*c + (b*c + a*d)*x)^3/(b*x + a)^12,x, algorithm="giac")
[Out]