Optimal. Leaf size=74 \[ -\frac{a (b c-a d) (a+b x)^{n+1}}{b^3 (n+1)}+\frac{(b c-2 a d) (a+b x)^{n+2}}{b^3 (n+2)}+\frac{d (a+b x)^{n+3}}{b^3 (n+3)} \]
[Out]
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Rubi [A] time = 0.0877288, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071 \[ -\frac{a (b c-a d) (a+b x)^{n+1}}{b^3 (n+1)}+\frac{(b c-2 a d) (a+b x)^{n+2}}{b^3 (n+2)}+\frac{d (a+b x)^{n+3}}{b^3 (n+3)} \]
Antiderivative was successfully verified.
[In] Int[x*(a + b*x)^n*(c + d*x),x]
[Out]
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Rubi in Sympy [A] time = 15.9591, size = 63, normalized size = 0.85 \[ \frac{a \left (a + b x\right )^{n + 1} \left (a d - b c\right )}{b^{3} \left (n + 1\right )} + \frac{d \left (a + b x\right )^{n + 3}}{b^{3} \left (n + 3\right )} - \frac{\left (a + b x\right )^{n + 2} \left (2 a d - b c\right )}{b^{3} \left (n + 2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*(b*x+a)**n*(d*x+c),x)
[Out]
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Mathematica [A] time = 0.0713217, size = 72, normalized size = 0.97 \[ \frac{(a+b x)^{n+1} \left (2 a^2 d-a b (c (n+3)+2 d (n+1) x)+b^2 (n+1) x (c (n+3)+d (n+2) x)\right )}{b^3 (n+1) (n+2) (n+3)} \]
Antiderivative was successfully verified.
[In] Integrate[x*(a + b*x)^n*(c + d*x),x]
[Out]
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Maple [A] time = 0.007, size = 114, normalized size = 1.5 \[{\frac{ \left ( bx+a \right ) ^{1+n} \left ({b}^{2}d{n}^{2}{x}^{2}+{b}^{2}c{n}^{2}x+3\,{b}^{2}dn{x}^{2}-2\,abdnx+4\,{b}^{2}cnx+2\,d{x}^{2}{b}^{2}-abcn-2\,abdx+3\,{b}^{2}cx+2\,{a}^{2}d-3\,abc \right ) }{{b}^{3} \left ({n}^{3}+6\,{n}^{2}+11\,n+6 \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*(b*x+a)^n*(d*x+c),x)
[Out]
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Maxima [A] time = 1.36563, size = 153, normalized size = 2.07 \[ \frac{{\left (b^{2}{\left (n + 1\right )} x^{2} + a b n x - a^{2}\right )}{\left (b x + a\right )}^{n} c}{{\left (n^{2} + 3 \, n + 2\right )} b^{2}} + \frac{{\left ({\left (n^{2} + 3 \, n + 2\right )} b^{3} x^{3} +{\left (n^{2} + n\right )} a b^{2} x^{2} - 2 \, a^{2} b n x + 2 \, a^{3}\right )}{\left (b x + a\right )}^{n} d}{{\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)*(b*x + a)^n*x,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.228417, size = 215, normalized size = 2.91 \[ -\frac{{\left (a^{2} b c n + 3 \, a^{2} b c - 2 \, a^{3} d -{\left (b^{3} d n^{2} + 3 \, b^{3} d n + 2 \, b^{3} d\right )} x^{3} -{\left (3 \, b^{3} c +{\left (b^{3} c + a b^{2} d\right )} n^{2} +{\left (4 \, b^{3} c + a b^{2} d\right )} n\right )} x^{2} -{\left (a b^{2} c n^{2} +{\left (3 \, a b^{2} c - 2 \, a^{2} b d\right )} n\right )} x\right )}{\left (b x + a\right )}^{n}}{b^{3} n^{3} + 6 \, b^{3} n^{2} + 11 \, b^{3} n + 6 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)*(b*x + a)^n*x,x, algorithm="fricas")
[Out]
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Sympy [A] time = 4.62855, size = 1090, normalized size = 14.73 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*(b*x+a)**n*(d*x+c),x)
[Out]
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GIAC/XCAS [A] time = 0.237385, size = 389, normalized size = 5.26 \[ \frac{b^{3} d n^{2} x^{3} e^{\left (n{\rm ln}\left (b x + a\right )\right )} + b^{3} c n^{2} x^{2} e^{\left (n{\rm ln}\left (b x + a\right )\right )} + a b^{2} d n^{2} x^{2} e^{\left (n{\rm ln}\left (b x + a\right )\right )} + 3 \, b^{3} d n x^{3} e^{\left (n{\rm ln}\left (b x + a\right )\right )} + a b^{2} c n^{2} x e^{\left (n{\rm ln}\left (b x + a\right )\right )} + 4 \, b^{3} c n x^{2} e^{\left (n{\rm ln}\left (b x + a\right )\right )} + a b^{2} d n x^{2} e^{\left (n{\rm ln}\left (b x + a\right )\right )} + 2 \, b^{3} d x^{3} e^{\left (n{\rm ln}\left (b x + a\right )\right )} + 3 \, a b^{2} c n x e^{\left (n{\rm ln}\left (b x + a\right )\right )} - 2 \, a^{2} b d n x e^{\left (n{\rm ln}\left (b x + a\right )\right )} + 3 \, b^{3} c x^{2} e^{\left (n{\rm ln}\left (b x + a\right )\right )} - a^{2} b c n e^{\left (n{\rm ln}\left (b x + a\right )\right )} - 3 \, a^{2} b c e^{\left (n{\rm ln}\left (b x + a\right )\right )} + 2 \, a^{3} d e^{\left (n{\rm ln}\left (b x + a\right )\right )}}{b^{3} n^{3} + 6 \, b^{3} n^{2} + 11 \, b^{3} n + 6 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)*(b*x + a)^n*x,x, algorithm="giac")
[Out]