Optimal. Leaf size=118 \[ \frac{c^2 (b c-a d)^2 \log \left (c+d x^n\right )}{d^5 n}-\frac{c x^n (b c-a d)^2}{d^4 n}+\frac{x^{2 n} (b c-a d)^2}{2 d^3 n}-\frac{b x^{3 n} (b c-2 a d)}{3 d^2 n}+\frac{b^2 x^{4 n}}{4 d n} \]
[Out]
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Rubi [A] time = 0.303166, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{c^2 (b c-a d)^2 \log \left (c+d x^n\right )}{d^5 n}-\frac{c x^n (b c-a d)^2}{d^4 n}+\frac{x^{2 n} (b c-a d)^2}{2 d^3 n}-\frac{b x^{3 n} (b c-2 a d)}{3 d^2 n}+\frac{b^2 x^{4 n}}{4 d n} \]
Antiderivative was successfully verified.
[In] Int[(x^(-1 + 3*n)*(a + b*x^n)^2)/(c + d*x^n),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{b^{2} x^{4 n}}{4 d n} + \frac{b x^{3 n} \left (2 a d - b c\right )}{3 d^{2} n} + \frac{c^{2} \left (a d - b c\right )^{2} \log{\left (c + d x^{n} \right )}}{d^{5} n} + \frac{\left (a d - b c\right )^{2} \int ^{x^{n}} x\, dx}{d^{3} n} - \frac{\left (a d - b c\right )^{2} \int ^{x^{n}} c\, dx}{d^{4} n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(-1+3*n)*(a+b*x**n)**2/(c+d*x**n),x)
[Out]
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Mathematica [A] time = 0.181159, size = 125, normalized size = 1.06 \[ \frac{d x^n \left (6 a^2 d^2 \left (d x^n-2 c\right )+4 a b d \left (6 c^2-3 c d x^n+2 d^2 x^{2 n}\right )+b^2 \left (-12 c^3+6 c^2 d x^n-4 c d^2 x^{2 n}+3 d^3 x^{3 n}\right )\right )+12 c^2 (b c-a d)^2 \log \left (c+d x^n\right )}{12 d^5 n} \]
Antiderivative was successfully verified.
[In] Integrate[(x^(-1 + 3*n)*(a + b*x^n)^2)/(c + d*x^n),x]
[Out]
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Maple [B] time = 0.04, size = 236, normalized size = 2. \[{\frac{{b}^{2} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{4}}{4\,dn}}+{\frac{ \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}{a}^{2}}{2\,dn}}-{\frac{ \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}cab}{{d}^{2}n}}+{\frac{{b}^{2} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}{c}^{2}}{2\,{d}^{3}n}}+{\frac{2\,b \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}a}{3\,dn}}-{\frac{{b}^{2} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}c}{3\,{d}^{2}n}}-{\frac{c{{\rm e}^{n\ln \left ( x \right ) }}{a}^{2}}{{d}^{2}n}}+2\,{\frac{{c}^{2}{{\rm e}^{n\ln \left ( x \right ) }}ab}{{d}^{3}n}}-{\frac{{c}^{3}{{\rm e}^{n\ln \left ( x \right ) }}{b}^{2}}{{d}^{4}n}}+{\frac{{c}^{2}\ln \left ( c+d{{\rm e}^{n\ln \left ( x \right ) }} \right ){a}^{2}}{{d}^{3}n}}-2\,{\frac{{c}^{3}\ln \left ( c+d{{\rm e}^{n\ln \left ( x \right ) }} \right ) ab}{{d}^{4}n}}+{\frac{{c}^{4}\ln \left ( c+d{{\rm e}^{n\ln \left ( x \right ) }} \right ){b}^{2}}{{d}^{5}n}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(-1+3*n)*(a+b*x^n)^2/(c+d*x^n),x)
[Out]
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Maxima [A] time = 1.38964, size = 259, normalized size = 2.19 \[ \frac{1}{12} \, b^{2}{\left (\frac{12 \, c^{4} \log \left (\frac{d x^{n} + c}{d}\right )}{d^{5} n} + \frac{3 \, d^{3} x^{4 \, n} - 4 \, c d^{2} x^{3 \, n} + 6 \, c^{2} d x^{2 \, n} - 12 \, c^{3} x^{n}}{d^{4} n}\right )} - \frac{1}{3} \, a b{\left (\frac{6 \, c^{3} \log \left (\frac{d x^{n} + c}{d}\right )}{d^{4} n} - \frac{2 \, d^{2} x^{3 \, n} - 3 \, c d x^{2 \, n} + 6 \, c^{2} x^{n}}{d^{3} n}\right )} + \frac{1}{2} \, a^{2}{\left (\frac{2 \, c^{2} \log \left (\frac{d x^{n} + c}{d}\right )}{d^{3} n} + \frac{d x^{2 \, n} - 2 \, c x^{n}}{d^{2} n}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^n + a)^2*x^(3*n - 1)/(d*x^n + c),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.235193, size = 197, normalized size = 1.67 \[ \frac{3 \, b^{2} d^{4} x^{4 \, n} - 4 \,{\left (b^{2} c d^{3} - 2 \, a b d^{4}\right )} x^{3 \, n} + 6 \,{\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} x^{2 \, n} - 12 \,{\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3}\right )} x^{n} + 12 \,{\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2}\right )} \log \left (d x^{n} + c\right )}{12 \, d^{5} n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^n + a)^2*x^(3*n - 1)/(d*x^n + c),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**(-1+3*n)*(a+b*x**n)**2/(c+d*x**n),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{n} + a\right )}^{2} x^{3 \, n - 1}}{d x^{n} + c}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^n + a)^2*x^(3*n - 1)/(d*x^n + c),x, algorithm="giac")
[Out]