Optimal. Leaf size=96 \[ -\frac{b^2 x (n p+n+1) \left (a-b x^{n/2}\right )^{p+1} \left (a+b x^{n/2}\right )^{p+1} \left (d x^n-\frac{a^2 d n (p+1)}{b^2 (n p+n+1)}\right )^{-\frac{n p+n+1}{n}}}{a^4 d n (p+1)} \]
[Out]
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Rubi [A] time = 0.361072, antiderivative size = 104, normalized size of antiderivative = 1.08, number of steps used = 2, number of rules used = 2, integrand size = 76, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.026 \[ -\frac{b^2 x (n p+n+1) \left (a^2-b^2 x^n\right ) \left (a-b x^{n/2}\right )^p \left (a+b x^{n/2}\right )^p \left (d x^n-\frac{a^2 d n (p+1)}{b^2 (n p+n+1)}\right )^{-\frac{n p+n+1}{n}}}{a^4 d n (p+1)} \]
Antiderivative was successfully verified.
[In] Int[(a - b*x^(n/2))^p*(a + b*x^(n/2))^p*((a^2*d*(1 + p))/(b^2*(1 + (-1 - 2*n - n*p)/n)) + d*x^n)^((-1 - 2*n - n*p)/n),x]
[Out]
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Rubi in Sympy [A] time = 51.4377, size = 102, normalized size = 1.06 \[ - \frac{b^{2} x \left (a - b x^{\frac{n}{2}}\right )^{p} \left (a + b x^{\frac{n}{2}}\right )^{p} \left (a^{2} - b^{2} x^{n}\right )^{- p} \left (a^{2} - b^{2} x^{n}\right )^{p + 1} \left (- \frac{a^{2} d n \left (p + 1\right )}{b^{2} \left (n p + n + 1\right )} + d x^{n}\right )^{- p - 1 - \frac{1}{n}} \left (n p + n + 1\right )}{a^{4} d n \left (p + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a-b*x**(1/2*n))**p*(a+b*x**(1/2*n))**p*(a**2*d*(1+p)/b**2/(1+(-n*p-2*n-1)/n)+d*x**n)**((-n*p-2*n-1)/n),x)
[Out]
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Mathematica [A] time = 0.92852, size = 0, normalized size = 0. \[ \int \left (a-b x^{n/2}\right )^p \left (a+b x^{n/2}\right )^p \left (\frac{a^2 d (1+p)}{b^2 \left (1+\frac{-1-2 n-n p}{n}\right )}+d x^n\right )^{\frac{-1-2 n-n p}{n}} \, dx \]
Verification is Not applicable to the result.
[In] Integrate[(a - b*x^(n/2))^p*(a + b*x^(n/2))^p*((a^2*d*(1 + p))/(b^2*(1 + (-1 - 2*n - n*p)/n)) + d*x^n)^((-1 - 2*n - n*p)/n),x]
[Out]
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Maple [F] time = 1.044, size = 0, normalized size = 0. \[ \int \left ( a-b{x}^{{\frac{n}{2}}} \right ) ^{p} \left ( a+b{x}^{{\frac{n}{2}}} \right ) ^{p} \left ({\frac{{a}^{2}d \left ( 1+p \right ) }{{b}^{2}} \left ( 1+{\frac{-np-2\,n-1}{n}} \right ) ^{-1}}+d{x}^{n} \right ) ^{{\frac{-np-2\,n-1}{n}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a-b*x^(1/2*n))^p*(a+b*x^(1/2*n))^p*(a^2*d*(1+p)/b^2/(1+(-n*p-2*n-1)/n)+d*x^n)^((-n*p-2*n-1)/n),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{\frac{1}{2} \, n} + a\right )}^{p}{\left (-b x^{\frac{1}{2} \, n} + a\right )}^{p}{\left (d x^{n} - \frac{a^{2} d{\left (p + 1\right )}}{b^{2}{\left (\frac{n p + 2 \, n + 1}{n} - 1\right )}}\right )}^{-\frac{n p + 2 \, n + 1}{n}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^(1/2*n) + a)^p*(-b*x^(1/2*n) + a)^p*(d*x^n - a^2*d*(p + 1)/(b^2*((n*p + 2*n + 1)/n - 1)))^(-(n*p + 2*n + 1)/n),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.28891, size = 243, normalized size = 2.53 \[ \frac{{\left ({\left (b^{4} n p + b^{4} n + b^{4}\right )} x x^{2 \, n} -{\left (2 \, a^{2} b^{2} n p + 2 \, a^{2} b^{2} n + a^{2} b^{2}\right )} x x^{n} +{\left (a^{4} n p + a^{4} n\right )} x\right )}{\left (b x^{\frac{1}{2} \, n} + a\right )}^{p}{\left (-b x^{\frac{1}{2} \, n} + a\right )}^{p}}{{\left (a^{4} n p + a^{4} n\right )} \left (-\frac{a^{2} d n p + a^{2} d n -{\left (b^{2} d n p + b^{2} d n + b^{2} d\right )} x^{n}}{b^{2} n p + b^{2} n + b^{2}}\right )^{\frac{n p + 2 \, n + 1}{n}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^(1/2*n) + a)^p*(-b*x^(1/2*n) + a)^p/(d*x^n - a^2*d*(p + 1)/(b^2*((n*p + 2*n + 1)/n - 1)))^((n*p + 2*n + 1)/n),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-b*x**(1/2*n))**p*(a+b*x**(1/2*n))**p*(a**2*d*(1+p)/b**2/(1+(-n*p-2*n-1)/n)+d*x**n)**((-n*p-2*n-1)/n),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{\frac{1}{2} \, n} + a\right )}^{p}{\left (-b x^{\frac{1}{2} \, n} + a\right )}^{p}}{{\left (d x^{n} - \frac{a^{2} d{\left (p + 1\right )}}{b^{2}{\left (\frac{n p + 2 \, n + 1}{n} - 1\right )}}\right )}^{\frac{n p + 2 \, n + 1}{n}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^(1/2*n) + a)^p*(-b*x^(1/2*n) + a)^p/(d*x^n - a^2*d*(p + 1)/(b^2*((n*p + 2*n + 1)/n - 1)))^((n*p + 2*n + 1)/n),x, algorithm="giac")
[Out]