Optimal. Leaf size=92 \[ -\frac{3 d^2 (b c-a d)}{5 b^4 (a+b x)^5}-\frac{d (b c-a d)^2}{2 b^4 (a+b x)^6}-\frac{(b c-a d)^3}{7 b^4 (a+b x)^7}-\frac{d^3}{4 b^4 (a+b x)^4} \]
[Out]
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Rubi [A] time = 0.122575, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ -\frac{3 d^2 (b c-a d)}{5 b^4 (a+b x)^5}-\frac{d (b c-a d)^2}{2 b^4 (a+b x)^6}-\frac{(b c-a d)^3}{7 b^4 (a+b x)^7}-\frac{d^3}{4 b^4 (a+b x)^4} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x)^3/(a + b*x)^8,x]
[Out]
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Rubi in Sympy [A] time = 26.435, size = 80, normalized size = 0.87 \[ - \frac{d^{3}}{4 b^{4} \left (a + b x\right )^{4}} + \frac{3 d^{2} \left (a d - b c\right )}{5 b^{4} \left (a + b x\right )^{5}} - \frac{d \left (a d - b c\right )^{2}}{2 b^{4} \left (a + b x\right )^{6}} + \frac{\left (a d - b c\right )^{3}}{7 b^{4} \left (a + b x\right )^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x+c)**3/(b*x+a)**8,x)
[Out]
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Mathematica [A] time = 0.0506232, size = 97, normalized size = 1.05 \[ -\frac{a^3 d^3+a^2 b d^2 (4 c+7 d x)+a b^2 d \left (10 c^2+28 c d x+21 d^2 x^2\right )+b^3 \left (20 c^3+70 c^2 d x+84 c d^2 x^2+35 d^3 x^3\right )}{140 b^4 (a+b x)^7} \]
Antiderivative was successfully verified.
[In] Integrate[(c + d*x)^3/(a + b*x)^8,x]
[Out]
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Maple [A] time = 0.009, size = 122, normalized size = 1.3 \[{\frac{3\,{d}^{2} \left ( ad-bc \right ) }{5\,{b}^{4} \left ( bx+a \right ) ^{5}}}-{\frac{d \left ({a}^{2}{d}^{2}-2\,abcd+{b}^{2}{c}^{2} \right ) }{2\,{b}^{4} \left ( bx+a \right ) ^{6}}}-{\frac{-{a}^{3}{d}^{3}+3\,{a}^{2}bc{d}^{2}-3\,a{b}^{2}{c}^{2}d+{b}^{3}{c}^{3}}{7\,{b}^{4} \left ( bx+a \right ) ^{7}}}-{\frac{{d}^{3}}{4\,{b}^{4} \left ( bx+a \right ) ^{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x+c)^3/(b*x+a)^8,x)
[Out]
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Maxima [A] time = 1.35788, size = 246, normalized size = 2.67 \[ -\frac{35 \, b^{3} d^{3} x^{3} + 20 \, b^{3} c^{3} + 10 \, a b^{2} c^{2} d + 4 \, a^{2} b c d^{2} + a^{3} d^{3} + 21 \,{\left (4 \, b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{2} + 7 \,{\left (10 \, b^{3} c^{2} d + 4 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x}{140 \,{\left (b^{11} x^{7} + 7 \, a b^{10} x^{6} + 21 \, a^{2} b^{9} x^{5} + 35 \, a^{3} b^{8} x^{4} + 35 \, a^{4} b^{7} x^{3} + 21 \, a^{5} b^{6} x^{2} + 7 \, a^{6} b^{5} x + a^{7} b^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)^3/(b*x + a)^8,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.220522, size = 246, normalized size = 2.67 \[ -\frac{35 \, b^{3} d^{3} x^{3} + 20 \, b^{3} c^{3} + 10 \, a b^{2} c^{2} d + 4 \, a^{2} b c d^{2} + a^{3} d^{3} + 21 \,{\left (4 \, b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{2} + 7 \,{\left (10 \, b^{3} c^{2} d + 4 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x}{140 \,{\left (b^{11} x^{7} + 7 \, a b^{10} x^{6} + 21 \, a^{2} b^{9} x^{5} + 35 \, a^{3} b^{8} x^{4} + 35 \, a^{4} b^{7} x^{3} + 21 \, a^{5} b^{6} x^{2} + 7 \, a^{6} b^{5} x + a^{7} b^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)^3/(b*x + a)^8,x, algorithm="fricas")
[Out]
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Sympy [A] time = 11.9931, size = 194, normalized size = 2.11 \[ - \frac{a^{3} d^{3} + 4 a^{2} b c d^{2} + 10 a b^{2} c^{2} d + 20 b^{3} c^{3} + 35 b^{3} d^{3} x^{3} + x^{2} \left (21 a b^{2} d^{3} + 84 b^{3} c d^{2}\right ) + x \left (7 a^{2} b d^{3} + 28 a b^{2} c d^{2} + 70 b^{3} c^{2} d\right )}{140 a^{7} b^{4} + 980 a^{6} b^{5} x + 2940 a^{5} b^{6} x^{2} + 4900 a^{4} b^{7} x^{3} + 4900 a^{3} b^{8} x^{4} + 2940 a^{2} b^{9} x^{5} + 980 a b^{10} x^{6} + 140 b^{11} x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x+c)**3/(b*x+a)**8,x)
[Out]
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GIAC/XCAS [A] time = 0.22187, size = 154, normalized size = 1.67 \[ -\frac{35 \, b^{3} d^{3} x^{3} + 84 \, b^{3} c d^{2} x^{2} + 21 \, a b^{2} d^{3} x^{2} + 70 \, b^{3} c^{2} d x + 28 \, a b^{2} c d^{2} x + 7 \, a^{2} b d^{3} x + 20 \, b^{3} c^{3} + 10 \, a b^{2} c^{2} d + 4 \, a^{2} b c d^{2} + a^{3} d^{3}}{140 \,{\left (b x + a\right )}^{7} b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)^3/(b*x + a)^8,x, algorithm="giac")
[Out]