Optimal. Leaf size=102 \[ -\frac{a \left (a^2 d+b^2 c\right ) (a+b x)^{n+1}}{b^4 (n+1)}+\frac{\left (3 a^2 d+b^2 c\right ) (a+b x)^{n+2}}{b^4 (n+2)}-\frac{3 a d (a+b x)^{n+3}}{b^4 (n+3)}+\frac{d (a+b x)^{n+4}}{b^4 (n+4)} \]
[Out]
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Rubi [A] time = 0.117719, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062 \[ -\frac{a \left (a^2 d+b^2 c\right ) (a+b x)^{n+1}}{b^4 (n+1)}+\frac{\left (3 a^2 d+b^2 c\right ) (a+b x)^{n+2}}{b^4 (n+2)}-\frac{3 a d (a+b x)^{n+3}}{b^4 (n+3)}+\frac{d (a+b x)^{n+4}}{b^4 (n+4)} \]
Antiderivative was successfully verified.
[In] Int[x*(a + b*x)^n*(c + d*x^2),x]
[Out]
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Rubi in Sympy [A] time = 23.1907, size = 90, normalized size = 0.88 \[ - \frac{3 a d \left (a + b x\right )^{n + 3}}{b^{4} \left (n + 3\right )} - \frac{a \left (a + b x\right )^{n + 1} \left (a^{2} d + b^{2} c\right )}{b^{4} \left (n + 1\right )} + \frac{d \left (a + b x\right )^{n + 4}}{b^{4} \left (n + 4\right )} + \frac{\left (a + b x\right )^{n + 2} \left (3 a^{2} d + b^{2} c\right )}{b^{4} \left (n + 2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*(b*x+a)**n*(d*x**2+c),x)
[Out]
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Mathematica [A] time = 0.113742, size = 109, normalized size = 1.07 \[ \frac{(a+b x)^{n+1} \left (-6 a^3 d+6 a^2 b d (n+1) x-a b^2 \left (c \left (n^2+7 n+12\right )+3 d \left (n^2+3 n+2\right ) x^2\right )+b^3 \left (n^2+4 n+3\right ) x \left (c (n+4)+d (n+2) x^2\right )\right )}{b^4 (n+1) (n+2) (n+3) (n+4)} \]
Antiderivative was successfully verified.
[In] Integrate[x*(a + b*x)^n*(c + d*x^2),x]
[Out]
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Maple [A] time = 0.008, size = 195, normalized size = 1.9 \[ -{\frac{ \left ( bx+a \right ) ^{1+n} \left ( -{b}^{3}d{n}^{3}{x}^{3}-6\,{b}^{3}d{n}^{2}{x}^{3}+3\,a{b}^{2}d{n}^{2}{x}^{2}-{b}^{3}c{n}^{3}x-11\,{b}^{3}dn{x}^{3}+9\,a{b}^{2}dn{x}^{2}-8\,{b}^{3}c{n}^{2}x-6\,d{x}^{3}{b}^{3}-6\,{a}^{2}bdnx+a{b}^{2}c{n}^{2}+6\,ad{x}^{2}{b}^{2}-19\,{b}^{3}cnx-6\,{a}^{2}bdx+7\,a{b}^{2}cn-12\,{b}^{3}cx+6\,{a}^{3}d+12\,a{b}^{2}c \right ) }{{b}^{4} \left ({n}^{4}+10\,{n}^{3}+35\,{n}^{2}+50\,n+24 \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*(b*x+a)^n*(d*x^2+c),x)
[Out]
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Maxima [A] time = 0.712687, size = 197, normalized size = 1.93 \[ \frac{{\left (b^{2}{\left (n + 1\right )} x^{2} + a b n x - a^{2}\right )}{\left (b x + a\right )}^{n} c}{{\left (n^{2} + 3 \, n + 2\right )} b^{2}} + \frac{{\left ({\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} b^{4} x^{4} +{\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a b^{3} x^{3} - 3 \,{\left (n^{2} + n\right )} a^{2} b^{2} x^{2} + 6 \, a^{3} b n x - 6 \, a^{4}\right )}{\left (b x + a\right )}^{n} d}{{\left (n^{4} + 10 \, n^{3} + 35 \, n^{2} + 50 \, n + 24\right )} b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)*(b*x + a)^n*x,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.278833, size = 338, normalized size = 3.31 \[ -\frac{{\left (a^{2} b^{2} c n^{2} + 7 \, a^{2} b^{2} c n + 12 \, a^{2} b^{2} c + 6 \, a^{4} d -{\left (b^{4} d n^{3} + 6 \, b^{4} d n^{2} + 11 \, b^{4} d n + 6 \, b^{4} d\right )} x^{4} -{\left (a b^{3} d n^{3} + 3 \, a b^{3} d n^{2} + 2 \, a b^{3} d n\right )} x^{3} -{\left (b^{4} c n^{3} + 12 \, b^{4} c +{\left (8 \, b^{4} c - 3 \, a^{2} b^{2} d\right )} n^{2} +{\left (19 \, b^{4} c - 3 \, a^{2} b^{2} d\right )} n\right )} x^{2} -{\left (a b^{3} c n^{3} + 7 \, a b^{3} c n^{2} + 6 \,{\left (2 \, a b^{3} c + a^{3} b d\right )} n\right )} x\right )}{\left (b x + a\right )}^{n}}{b^{4} n^{4} + 10 \, b^{4} n^{3} + 35 \, b^{4} n^{2} + 50 \, b^{4} n + 24 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)*(b*x + a)^n*x,x, algorithm="fricas")
[Out]
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Sympy [A] time = 7.68216, size = 2241, normalized size = 21.97 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*(b*x+a)**n*(d*x**2+c),x)
[Out]
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GIAC/XCAS [A] time = 0.279534, size = 610, normalized size = 5.98 \[ \frac{b^{4} d n^{3} x^{4} e^{\left (n{\rm ln}\left (b x + a\right )\right )} + a b^{3} d n^{3} x^{3} e^{\left (n{\rm ln}\left (b x + a\right )\right )} + 6 \, b^{4} d n^{2} x^{4} e^{\left (n{\rm ln}\left (b x + a\right )\right )} + b^{4} c n^{3} x^{2} e^{\left (n{\rm ln}\left (b x + a\right )\right )} + 3 \, a b^{3} d n^{2} x^{3} e^{\left (n{\rm ln}\left (b x + a\right )\right )} + 11 \, b^{4} d n x^{4} e^{\left (n{\rm ln}\left (b x + a\right )\right )} + a b^{3} c n^{3} x e^{\left (n{\rm ln}\left (b x + a\right )\right )} + 8 \, b^{4} c n^{2} x^{2} e^{\left (n{\rm ln}\left (b x + a\right )\right )} - 3 \, a^{2} b^{2} d n^{2} x^{2} e^{\left (n{\rm ln}\left (b x + a\right )\right )} + 2 \, a b^{3} d n x^{3} e^{\left (n{\rm ln}\left (b x + a\right )\right )} + 6 \, b^{4} d x^{4} e^{\left (n{\rm ln}\left (b x + a\right )\right )} + 7 \, a b^{3} c n^{2} x e^{\left (n{\rm ln}\left (b x + a\right )\right )} + 19 \, b^{4} c n x^{2} e^{\left (n{\rm ln}\left (b x + a\right )\right )} - 3 \, a^{2} b^{2} d n x^{2} e^{\left (n{\rm ln}\left (b x + a\right )\right )} - a^{2} b^{2} c n^{2} e^{\left (n{\rm ln}\left (b x + a\right )\right )} + 12 \, a b^{3} c n x e^{\left (n{\rm ln}\left (b x + a\right )\right )} + 6 \, a^{3} b d n x e^{\left (n{\rm ln}\left (b x + a\right )\right )} + 12 \, b^{4} c x^{2} e^{\left (n{\rm ln}\left (b x + a\right )\right )} - 7 \, a^{2} b^{2} c n e^{\left (n{\rm ln}\left (b x + a\right )\right )} - 12 \, a^{2} b^{2} c e^{\left (n{\rm ln}\left (b x + a\right )\right )} - 6 \, a^{4} d e^{\left (n{\rm ln}\left (b x + a\right )\right )}}{b^{4} n^{4} + 10 \, b^{4} n^{3} + 35 \, b^{4} n^{2} + 50 \, b^{4} n + 24 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)*(b*x + a)^n*x,x, algorithm="giac")
[Out]