Optimal. Leaf size=102 \[ \frac{c x^{n+1} (e x)^m (2 A d+B c)}{m+n+1}+\frac{d x^{2 n+1} (e x)^m (A d+2 B c)}{m+2 n+1}+\frac{A c^2 (e x)^{m+1}}{e (m+1)}+\frac{B d^2 x^{3 n+1} (e x)^m}{m+3 n+1} \]
[Out]
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Rubi [A] time = 0.187333, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{c x^{n+1} (e x)^m (2 A d+B c)}{m+n+1}+\frac{d x^{2 n+1} (e x)^m (A d+2 B c)}{m+2 n+1}+\frac{A c^2 (e x)^{m+1}}{e (m+1)}+\frac{B d^2 x^{3 n+1} (e x)^m}{m+3 n+1} \]
Antiderivative was successfully verified.
[In] Int[(e*x)^m*(A + B*x^n)*(c + d*x^n)^2,x]
[Out]
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Rubi in Sympy [A] time = 33.3986, size = 119, normalized size = 1.17 \[ \frac{A c^{2} \left (e x\right )^{m + 1}}{e \left (m + 1\right )} + \frac{B d^{2} x^{3 n} \left (e x\right )^{- 3 n} \left (e x\right )^{m + 3 n + 1}}{e \left (m + 3 n + 1\right )} + \frac{c x^{n} \left (e x\right )^{- n} \left (e x\right )^{m + n + 1} \left (2 A d + B c\right )}{e \left (m + n + 1\right )} + \frac{d x^{- m} x^{m + 2 n + 1} \left (e x\right )^{m} \left (A d + 2 B c\right )}{m + 2 n + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x)**m*(A+B*x**n)*(c+d*x**n)**2,x)
[Out]
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Mathematica [A] time = 0.148942, size = 78, normalized size = 0.76 \[ x (e x)^m \left (\frac{d x^{2 n} (A d+2 B c)}{m+2 n+1}+\frac{c x^n (2 A d+B c)}{m+n+1}+\frac{A c^2}{m+1}+\frac{B d^2 x^{3 n}}{m+3 n+1}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(e*x)^m*(A + B*x^n)*(c + d*x^n)^2,x]
[Out]
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Maple [C] time = 0.084, size = 732, normalized size = 7.2 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x)^m*(A+B*x^n)*(c+d*x^n)^2,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((B*x^n + A)*(d*x^n + c)^2*(e*x)^m,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.23835, size = 711, normalized size = 6.97 \[ \frac{{\left (B d^{2} m^{3} + 3 \, B d^{2} m^{2} + 3 \, B d^{2} m + B d^{2} + 2 \,{\left (B d^{2} m + B d^{2}\right )} n^{2} + 3 \,{\left (B d^{2} m^{2} + 2 \, B d^{2} m + B d^{2}\right )} n\right )} x x^{3 \, n} e^{\left (m \log \left (e\right ) + m \log \left (x\right )\right )} +{\left ({\left (2 \, B c d + A d^{2}\right )} m^{3} + 2 \, B c d + A d^{2} + 3 \,{\left (2 \, B c d + A d^{2}\right )} m^{2} + 3 \,{\left (2 \, B c d + A d^{2} +{\left (2 \, B c d + A d^{2}\right )} m\right )} n^{2} + 3 \,{\left (2 \, B c d + A d^{2}\right )} m + 4 \,{\left (2 \, B c d + A d^{2} +{\left (2 \, B c d + A d^{2}\right )} m^{2} + 2 \,{\left (2 \, B c d + A d^{2}\right )} m\right )} n\right )} x x^{2 \, n} e^{\left (m \log \left (e\right ) + m \log \left (x\right )\right )} +{\left ({\left (B c^{2} + 2 \, A c d\right )} m^{3} + B c^{2} + 2 \, A c d + 3 \,{\left (B c^{2} + 2 \, A c d\right )} m^{2} + 6 \,{\left (B c^{2} + 2 \, A c d +{\left (B c^{2} + 2 \, A c d\right )} m\right )} n^{2} + 3 \,{\left (B c^{2} + 2 \, A c d\right )} m + 5 \,{\left (B c^{2} + 2 \, A c d +{\left (B c^{2} + 2 \, A c d\right )} m^{2} + 2 \,{\left (B c^{2} + 2 \, A c d\right )} m\right )} n\right )} x x^{n} e^{\left (m \log \left (e\right ) + m \log \left (x\right )\right )} +{\left (A c^{2} m^{3} + 6 \, A c^{2} n^{3} + 3 \, A c^{2} m^{2} + 3 \, A c^{2} m + A c^{2} + 11 \,{\left (A c^{2} m + A c^{2}\right )} n^{2} + 6 \,{\left (A c^{2} m^{2} + 2 \, A c^{2} m + A c^{2}\right )} n\right )} x e^{\left (m \log \left (e\right ) + m \log \left (x\right )\right )}}{m^{4} + 6 \,{\left (m + 1\right )} n^{3} + 4 \, m^{3} + 11 \,{\left (m^{2} + 2 \, m + 1\right )} n^{2} + 6 \, m^{2} + 6 \,{\left (m^{3} + 3 \, m^{2} + 3 \, m + 1\right )} n + 4 \, m + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^n + A)*(d*x^n + c)^2*(e*x)^m,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((e*x)**m*(A+B*x**n)*(c+d*x**n)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.214565, size = 1, normalized size = 0.01 \[ \mathit{Done} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^n + A)*(d*x^n + c)^2*(e*x)^m,x, algorithm="giac")
[Out]