Optimal. Leaf size=87 \[ \frac{a^4 x^{m+1}}{m+1}+\frac{8 a^3 b x^{m+\frac{3}{2}}}{2 m+3}+\frac{6 a^2 b^2 x^{m+2}}{m+2}+\frac{8 a b^3 x^{m+\frac{5}{2}}}{2 m+5}+\frac{b^4 x^{m+3}}{m+3} \]
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Rubi [A] time = 0.107431, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \frac{a^4 x^{m+1}}{m+1}+\frac{8 a^3 b x^{m+\frac{3}{2}}}{2 m+3}+\frac{6 a^2 b^2 x^{m+2}}{m+2}+\frac{8 a b^3 x^{m+\frac{5}{2}}}{2 m+5}+\frac{b^4 x^{m+3}}{m+3} \]
Antiderivative was successfully verified.
[In] Int[(a + b*Sqrt[x])^4*x^m,x]
[Out]
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Rubi in Sympy [A] time = 15.83, size = 76, normalized size = 0.87 \[ \frac{a^{4} x^{m + 1}}{m + 1} + \frac{8 a^{3} b x^{m + \frac{3}{2}}}{2 m + 3} + \frac{6 a^{2} b^{2} x^{m + 2}}{m + 2} + \frac{8 a b^{3} x^{m + \frac{5}{2}}}{2 m + 5} + \frac{b^{4} x^{m + 3}}{m + 3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**m*(a+b*x**(1/2))**4,x)
[Out]
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Mathematica [A] time = 0.0969382, size = 83, normalized size = 0.95 \[ 2 x^{m+1} \left (\frac{a^4}{2 m+2}+\frac{4 a^3 b \sqrt{x}}{2 m+3}+\frac{3 a^2 b^2 x}{m+2}+\frac{4 a b^3 x^{3/2}}{2 m+5}+\frac{b^4 x^2}{2 m+6}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*Sqrt[x])^4*x^m,x]
[Out]
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Maple [F] time = 0.028, size = 0, normalized size = 0. \[ \int{x}^{m} \left ( a+b\sqrt{x} \right ) ^{4}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^m*(a+b*x^(1/2))^4,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*sqrt(x) + a)^4*x^m,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.255215, size = 351, normalized size = 4.03 \[ \frac{{\left ({\left (4 \, b^{4} m^{4} + 28 \, b^{4} m^{3} + 71 \, b^{4} m^{2} + 77 \, b^{4} m + 30 \, b^{4}\right )} x^{3} + 6 \,{\left (4 \, a^{2} b^{2} m^{4} + 32 \, a^{2} b^{2} m^{3} + 91 \, a^{2} b^{2} m^{2} + 108 \, a^{2} b^{2} m + 45 \, a^{2} b^{2}\right )} x^{2} +{\left (4 \, a^{4} m^{4} + 36 \, a^{4} m^{3} + 119 \, a^{4} m^{2} + 171 \, a^{4} m + 90 \, a^{4}\right )} x + 8 \,{\left ({\left (2 \, a b^{3} m^{4} + 15 \, a b^{3} m^{3} + 40 \, a b^{3} m^{2} + 45 \, a b^{3} m + 18 \, a b^{3}\right )} x^{2} +{\left (2 \, a^{3} b m^{4} + 17 \, a^{3} b m^{3} + 52 \, a^{3} b m^{2} + 67 \, a^{3} b m + 30 \, a^{3} b\right )} x\right )} \sqrt{x}\right )} x^{m}}{4 \, m^{5} + 40 \, m^{4} + 155 \, m^{3} + 290 \, m^{2} + 261 \, m + 90} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*sqrt(x) + a)^4*x^m,x, algorithm="fricas")
[Out]
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**m*(a+b*x**(1/2))**4,x)
[Out]
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GIAC/XCAS [A] time = 0.276674, size = 150, normalized size = 1.72 \[ \frac{b^{4} x^{3} e^{\left (2 \, m{\rm ln}\left (\sqrt{x}\right )\right )}}{m + 3} + \frac{8 \, a b^{3} x^{\frac{5}{2}} e^{\left (2 \, m{\rm ln}\left (\sqrt{x}\right )\right )}}{2 \, m + 5} + \frac{6 \, a^{2} b^{2} x^{2} e^{\left (2 \, m{\rm ln}\left (\sqrt{x}\right )\right )}}{m + 2} + \frac{8 \, a^{3} b x^{\frac{3}{2}} e^{\left (2 \, m{\rm ln}\left (\sqrt{x}\right )\right )}}{2 \, m + 3} + \frac{a^{4} x e^{\left (2 \, m{\rm ln}\left (\sqrt{x}\right )\right )}}{m + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*sqrt(x) + a)^4*x^m,x, algorithm="giac")
[Out]