Optimal. Leaf size=92 \[ -\frac{3 d^2 (b c-a d)}{4 b^4 (a+b x)^4}-\frac{3 d (b c-a d)^2}{5 b^4 (a+b x)^5}-\frac{(b c-a d)^3}{6 b^4 (a+b x)^6}-\frac{d^3}{3 b^4 (a+b x)^3} \]
[Out]
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Rubi [A] time = 0.152482, 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 \[ -\frac{3 d^2 (b c-a d)}{4 b^4 (a+b x)^4}-\frac{3 d (b c-a d)^2}{5 b^4 (a+b x)^5}-\frac{(b c-a d)^3}{6 b^4 (a+b x)^6}-\frac{d^3}{3 b^4 (a+b x)^3} \]
Antiderivative was successfully verified.
[In] Int[(a*c + (b*c + a*d)*x + b*d*x^2)^3/(a + b*x)^10,x]
[Out]
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Rubi in Sympy [A] time = 34.2609, size = 82, normalized size = 0.89 \[ - \frac{d^{3}}{3 b^{4} \left (a + b x\right )^{3}} + \frac{3 d^{2} \left (a d - b c\right )}{4 b^{4} \left (a + b x\right )^{4}} - \frac{3 d \left (a d - b c\right )^{2}}{5 b^{4} \left (a + b x\right )^{5}} + \frac{\left (a d - b c\right )^{3}}{6 b^{4} \left (a + b x\right )^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a*c+(a*d+b*c)*x+b*d*x**2)**3/(b*x+a)**10,x)
[Out]
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Mathematica [A] time = 0.0585383, size = 97, normalized size = 1.05 \[ -\frac{a^3 d^3+3 a^2 b d^2 (c+2 d x)+3 a b^2 d \left (2 c^2+6 c d x+5 d^2 x^2\right )+b^3 \left (10 c^3+36 c^2 d x+45 c d^2 x^2+20 d^3 x^3\right )}{60 b^4 (a+b x)^6} \]
Antiderivative was successfully verified.
[In] Integrate[(a*c + (b*c + a*d)*x + b*d*x^2)^3/(a + b*x)^10,x]
[Out]
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Maple [A] time = 0.009, size = 122, normalized size = 1.3 \[ -{\frac{3\,d \left ({a}^{2}{d}^{2}-2\,cabd+{b}^{2}{c}^{2} \right ) }{5\,{b}^{4} \left ( bx+a \right ) ^{5}}}-{\frac{-{a}^{3}{d}^{3}+3\,{a}^{2}c{d}^{2}b-3\,a{c}^{2}d{b}^{2}+{c}^{3}{b}^{3}}{6\,{b}^{4} \left ( bx+a \right ) ^{6}}}+{\frac{3\,{d}^{2} \left ( ad-bc \right ) }{4\,{b}^{4} \left ( bx+a \right ) ^{4}}}-{\frac{{d}^{3}}{3\,{b}^{4} \left ( bx+a \right ) ^{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a*c+(a*d+b*c)*x+x^2*b*d)^3/(b*x+a)^10,x)
[Out]
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Maxima [A] time = 0.740714, size = 231, normalized size = 2.51 \[ -\frac{20 \, b^{3} d^{3} x^{3} + 10 \, b^{3} c^{3} + 6 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} + a^{3} d^{3} + 15 \,{\left (3 \, b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{2} + 6 \,{\left (6 \, b^{3} c^{2} d + 3 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x}{60 \,{\left (b^{10} x^{6} + 6 \, a b^{9} x^{5} + 15 \, a^{2} b^{8} x^{4} + 20 \, a^{3} b^{7} x^{3} + 15 \, a^{4} b^{6} x^{2} + 6 \, a^{5} b^{5} x + a^{6} b^{4}\right )}} \]
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)^10,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.200863, size = 231, normalized size = 2.51 \[ -\frac{20 \, b^{3} d^{3} x^{3} + 10 \, b^{3} c^{3} + 6 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} + a^{3} d^{3} + 15 \,{\left (3 \, b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{2} + 6 \,{\left (6 \, b^{3} c^{2} d + 3 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x}{60 \,{\left (b^{10} x^{6} + 6 \, a b^{9} x^{5} + 15 \, a^{2} b^{8} x^{4} + 20 \, a^{3} b^{7} x^{3} + 15 \, a^{4} b^{6} x^{2} + 6 \, a^{5} b^{5} x + a^{6} b^{4}\right )}} \]
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)^10,x, algorithm="fricas")
[Out]
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Sympy [A] time = 16.8223, size = 182, normalized size = 1.98 \[ - \frac{a^{3} d^{3} + 3 a^{2} b c d^{2} + 6 a b^{2} c^{2} d + 10 b^{3} c^{3} + 20 b^{3} d^{3} x^{3} + x^{2} \left (15 a b^{2} d^{3} + 45 b^{3} c d^{2}\right ) + x \left (6 a^{2} b d^{3} + 18 a b^{2} c d^{2} + 36 b^{3} c^{2} d\right )}{60 a^{6} b^{4} + 360 a^{5} b^{5} x + 900 a^{4} b^{6} x^{2} + 1200 a^{3} b^{7} x^{3} + 900 a^{2} b^{8} x^{4} + 360 a b^{9} x^{5} + 60 b^{10} x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a*c+(a*d+b*c)*x+b*d*x**2)**3/(b*x+a)**10,x)
[Out]
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GIAC/XCAS [A] time = 0.210641, size = 154, normalized size = 1.67 \[ -\frac{20 \, b^{3} d^{3} x^{3} + 45 \, b^{3} c d^{2} x^{2} + 15 \, a b^{2} d^{3} x^{2} + 36 \, b^{3} c^{2} d x + 18 \, a b^{2} c d^{2} x + 6 \, a^{2} b d^{3} x + 10 \, b^{3} c^{3} + 6 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} + a^{3} d^{3}}{60 \,{\left (b x + a\right )}^{6} b^{4}} \]
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)^10,x, algorithm="giac")
[Out]