Optimal. Leaf size=77 \[ \frac{x (d x)^m \left (\frac{b \sqrt [3]{x}}{a}+1\right )^{-2 p} \left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^p \, _2F_1\left (3 (m+1),-2 p;3 m+4;-\frac{b \sqrt [3]{x}}{a}\right )}{m+1} \]
[Out]
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Rubi [A] time = 0.0837354, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{x (d x)^m \left (\frac{b \sqrt [3]{x}}{a}+1\right )^{-2 p} \left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^p \, _2F_1\left (3 (m+1),-2 p;3 m+4;-\frac{b \sqrt [3]{x}}{a}\right )}{m+1} \]
Antiderivative was successfully verified.
[In] Int[(a^2 + 2*a*b*x^(1/3) + b^2*x^(2/3))^p*(d*x)^m,x]
[Out]
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Rubi in Sympy [A] time = 26.5421, size = 75, normalized size = 0.97 \[ \frac{x^{- m} x^{m + 1} \left (d x\right )^{m} \left (1 + \frac{b \sqrt [3]{x}}{a}\right )^{- 2 p} \left (a^{2} + 2 a b \sqrt [3]{x} + b^{2} x^{\frac{2}{3}}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - 2 p, 3 m + 3 \\ 3 m + 4 \end{matrix}\middle |{- \frac{b \sqrt [3]{x}}{a}} \right )}}{m + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a**2+2*a*b*x**(1/3)+b**2*x**(2/3))**p*(d*x)**m,x)
[Out]
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Mathematica [A] time = 0.0964403, size = 68, normalized size = 0.88 \[ \frac{x (d x)^m \left (\left (a+b \sqrt [3]{x}\right )^2\right )^p \left (\frac{b \sqrt [3]{x}}{a}+1\right )^{-2 p} \, _2F_1\left (3 (m+1),-2 p;3 (m+1)+1;-\frac{b \sqrt [3]{x}}{a}\right )}{m+1} \]
Antiderivative was successfully verified.
[In] Integrate[(a^2 + 2*a*b*x^(1/3) + b^2*x^(2/3))^p*(d*x)^m,x]
[Out]
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Maple [F] time = 0.121, size = 0, normalized size = 0. \[ \int \left ({a}^{2}+2\,ab\sqrt [3]{x}+{b}^{2}{x}^{{\frac{2}{3}}} \right ) ^{p} \left ( dx \right ) ^{m}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a^2+2*a*b*x^(1/3)+b^2*x^(2/3))^p*(d*x)^m,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b^{2} x^{\frac{2}{3}} + 2 \, a b x^{\frac{1}{3}} + a^{2}\right )}^{p} \left (d x\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^(2/3) + 2*a*b*x^(1/3) + a^2)^p*(d*x)^m,x, algorithm="maxima")
[Out]
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Fricas [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((b^2*x^(2/3) + 2*a*b*x^(1/3) + a^2)^p*(d*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((a**2+2*a*b*x**(1/3)+b**2*x**(2/3))**p*(d*x)**m,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b^{2} x^{\frac{2}{3}} + 2 \, a b x^{\frac{1}{3}} + a^{2}\right )}^{p} \left (d x\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^(2/3) + 2*a*b*x^(1/3) + a^2)^p*(d*x)^m,x, algorithm="giac")
[Out]