Optimal. Leaf size=84 \[ a^3 c x+\frac{3 a^2 b c x^{n+1}}{n+1}+\frac{3 a b^2 c x^{2 n+1}}{2 n+1}+\frac{d \left (a+b x^n\right )^4}{4 b n}+\frac{b^3 c x^{3 n+1}}{3 n+1} \]
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Rubi [A] time = 0.110754, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ a^3 c x+\frac{3 a^2 b c x^{n+1}}{n+1}+\frac{3 a b^2 c x^{2 n+1}}{2 n+1}+\frac{d \left (a+b x^n\right )^4}{4 b n}+\frac{b^3 c x^{3 n+1}}{3 n+1} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x^(-1 + n))*(a + b*x^n)^3,x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{3 a^{2} b c x^{n + 1}}{n + 1} + \frac{3 a b^{2} c x^{2 n + 1}}{2 n + 1} + \frac{b^{3} c x^{3 n + 1}}{3 n + 1} + c \int a^{3}\, dx + \frac{d \left (a + b x^{n}\right )^{4}}{4 b n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c+d*x**(-1+n))*(a+b*x**n)**3,x)
[Out]
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Mathematica [A] time = 0.185538, size = 162, normalized size = 1.93 \[ \frac{4 a^3 \left (6 n^3+11 n^2+6 n+1\right ) \left (c n x+d x^n\right )+6 a^2 b \left (6 n^2+5 n+1\right ) x^n \left (2 c n x+d (n+1) x^n\right )+4 a b^2 \left (3 n^2+4 n+1\right ) x^{2 n} \left (3 c n x+d (2 n+1) x^n\right )+b^3 \left (2 n^2+3 n+1\right ) x^{3 n} \left (4 c n x+d (3 n+1) x^n\right )}{4 n (n+1) (2 n+1) (3 n+1)} \]
Antiderivative was successfully verified.
[In] Integrate[(c + d*x^(-1 + n))*(a + b*x^n)^3,x]
[Out]
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Maple [A] time = 0.029, size = 130, normalized size = 1.6 \[{a}^{3}cx+{\frac{{a}^{3}d{{\rm e}^{n\ln \left ( x \right ) }}}{n}}+{\frac{a{b}^{2}d \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}}{n}}+{\frac{{b}^{3}cx \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}}{1+3\,n}}+{\frac{{b}^{3}d \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{4}}{4\,n}}+{\frac{3\,{a}^{2}bd \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{2\,n}}+3\,{\frac{ac{b}^{2}x \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{1+2\,n}}+3\,{\frac{{a}^{2}bcx{{\rm e}^{n\ln \left ( x \right ) }}}{1+n}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c+d*x^(-1+n))*(a+b*x^n)^3,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^(n - 1) + c)*(b*x^n + a)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.227749, size = 412, normalized size = 4.9 \[ \frac{4 \,{\left (6 \, a^{3} c n^{4} + 11 \, a^{3} c n^{3} + 6 \, a^{3} c n^{2} + a^{3} c n\right )} x +{\left (6 \, b^{3} d n^{3} + 11 \, b^{3} d n^{2} + 6 \, b^{3} d n + b^{3} d\right )} x^{4 \, n} + 4 \,{\left (6 \, a b^{2} d n^{3} + 11 \, a b^{2} d n^{2} + 6 \, a b^{2} d n + a b^{2} d +{\left (2 \, b^{3} c n^{3} + 3 \, b^{3} c n^{2} + b^{3} c n\right )} x\right )} x^{3 \, n} + 6 \,{\left (6 \, a^{2} b d n^{3} + 11 \, a^{2} b d n^{2} + 6 \, a^{2} b d n + a^{2} b d + 2 \,{\left (3 \, a b^{2} c n^{3} + 4 \, a b^{2} c n^{2} + a b^{2} c n\right )} x\right )} x^{2 \, n} + 4 \,{\left (6 \, a^{3} d n^{3} + 11 \, a^{3} d n^{2} + 6 \, a^{3} d n + a^{3} d + 3 \,{\left (6 \, a^{2} b c n^{3} + 5 \, a^{2} b c n^{2} + a^{2} b c n\right )} x\right )} x^{n}}{4 \,{\left (6 \, n^{4} + 11 \, n^{3} + 6 \, n^{2} + n\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^(n - 1) + c)*(b*x^n + a)^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 10.984, size = 1251, normalized size = 14.89 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c+d*x**(-1+n))*(a+b*x**n)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.226922, size = 572, normalized size = 6.81 \[ \frac{24 \, a^{3} c n^{4} x + 8 \, b^{3} c n^{3} x e^{\left (3 \, n{\rm ln}\left (x\right )\right )} + 36 \, a b^{2} c n^{3} x e^{\left (2 \, n{\rm ln}\left (x\right )\right )} + 72 \, a^{2} b c n^{3} x e^{\left (n{\rm ln}\left (x\right )\right )} + 44 \, a^{3} c n^{3} x + 6 \, b^{3} d n^{3} e^{\left (4 \, n{\rm ln}\left (x\right )\right )} + 24 \, a b^{2} d n^{3} e^{\left (3 \, n{\rm ln}\left (x\right )\right )} + 12 \, b^{3} c n^{2} x e^{\left (3 \, n{\rm ln}\left (x\right )\right )} + 36 \, a^{2} b d n^{3} e^{\left (2 \, n{\rm ln}\left (x\right )\right )} + 48 \, a b^{2} c n^{2} x e^{\left (2 \, n{\rm ln}\left (x\right )\right )} + 24 \, a^{3} d n^{3} e^{\left (n{\rm ln}\left (x\right )\right )} + 60 \, a^{2} b c n^{2} x e^{\left (n{\rm ln}\left (x\right )\right )} + 24 \, a^{3} c n^{2} x + 11 \, b^{3} d n^{2} e^{\left (4 \, n{\rm ln}\left (x\right )\right )} + 44 \, a b^{2} d n^{2} e^{\left (3 \, n{\rm ln}\left (x\right )\right )} + 4 \, b^{3} c n x e^{\left (3 \, n{\rm ln}\left (x\right )\right )} + 66 \, a^{2} b d n^{2} e^{\left (2 \, n{\rm ln}\left (x\right )\right )} + 12 \, a b^{2} c n x e^{\left (2 \, n{\rm ln}\left (x\right )\right )} + 44 \, a^{3} d n^{2} e^{\left (n{\rm ln}\left (x\right )\right )} + 12 \, a^{2} b c n x e^{\left (n{\rm ln}\left (x\right )\right )} + 4 \, a^{3} c n x + 6 \, b^{3} d n e^{\left (4 \, n{\rm ln}\left (x\right )\right )} + 24 \, a b^{2} d n e^{\left (3 \, n{\rm ln}\left (x\right )\right )} + 36 \, a^{2} b d n e^{\left (2 \, n{\rm ln}\left (x\right )\right )} + 24 \, a^{3} d n e^{\left (n{\rm ln}\left (x\right )\right )} + b^{3} d e^{\left (4 \, n{\rm ln}\left (x\right )\right )} + 4 \, a b^{2} d e^{\left (3 \, n{\rm ln}\left (x\right )\right )} + 6 \, a^{2} b d e^{\left (2 \, n{\rm ln}\left (x\right )\right )} + 4 \, a^{3} d e^{\left (n{\rm ln}\left (x\right )\right )}}{4 \,{\left (6 \, n^{4} + 11 \, n^{3} + 6 \, n^{2} + n\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^(n - 1) + c)*(b*x^n + a)^3,x, algorithm="giac")
[Out]