3.324 \(\int \left (a+x^2\right )^2 \left (x+\sqrt{a+x^2}\right )^n \, dx\)

Optimal. Leaf size=164 \[ -\frac{a^5 \left (\sqrt{a+x^2}+x\right )^{n-5}}{32 (5-n)}-\frac{5 a^4 \left (\sqrt{a+x^2}+x\right )^{n-3}}{32 (3-n)}-\frac{5 a^3 \left (\sqrt{a+x^2}+x\right )^{n-1}}{16 (1-n)}+\frac{5 a^2 \left (\sqrt{a+x^2}+x\right )^{n+1}}{16 (n+1)}+\frac{5 a \left (\sqrt{a+x^2}+x\right )^{n+3}}{32 (n+3)}+\frac{\left (\sqrt{a+x^2}+x\right )^{n+5}}{32 (n+5)} \]

[Out]

-(a^5*(x + Sqrt[a + x^2])^(-5 + n))/(32*(5 - n)) - (5*a^4*(x + Sqrt[a + x^2])^(-
3 + n))/(32*(3 - n)) - (5*a^3*(x + Sqrt[a + x^2])^(-1 + n))/(16*(1 - n)) + (5*a^
2*(x + Sqrt[a + x^2])^(1 + n))/(16*(1 + n)) + (5*a*(x + Sqrt[a + x^2])^(3 + n))/
(32*(3 + n)) + (x + Sqrt[a + x^2])^(5 + n)/(32*(5 + n))

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Rubi [A]  time = 0.205213, antiderivative size = 164, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095 \[ -\frac{a^5 \left (\sqrt{a+x^2}+x\right )^{n-5}}{32 (5-n)}-\frac{5 a^4 \left (\sqrt{a+x^2}+x\right )^{n-3}}{32 (3-n)}-\frac{5 a^3 \left (\sqrt{a+x^2}+x\right )^{n-1}}{16 (1-n)}+\frac{5 a^2 \left (\sqrt{a+x^2}+x\right )^{n+1}}{16 (n+1)}+\frac{5 a \left (\sqrt{a+x^2}+x\right )^{n+3}}{32 (n+3)}+\frac{\left (\sqrt{a+x^2}+x\right )^{n+5}}{32 (n+5)} \]

Antiderivative was successfully verified.

[In]  Int[(a + x^2)^2*(x + Sqrt[a + x^2])^n,x]

[Out]

-(a^5*(x + Sqrt[a + x^2])^(-5 + n))/(32*(5 - n)) - (5*a^4*(x + Sqrt[a + x^2])^(-
3 + n))/(32*(3 - n)) - (5*a^3*(x + Sqrt[a + x^2])^(-1 + n))/(16*(1 - n)) + (5*a^
2*(x + Sqrt[a + x^2])^(1 + n))/(16*(1 + n)) + (5*a*(x + Sqrt[a + x^2])^(3 + n))/
(32*(3 + n)) + (x + Sqrt[a + x^2])^(5 + n)/(32*(5 + n))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{\int ^{x + \sqrt{a + x^{2}}} \frac{x^{n} \left (a + x^{2}\right )^{5}}{x^{6}}\, dx}{32} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((x**2+a)**2*(x+(x**2+a)**(1/2))**n,x)

[Out]

Integral(x**n*(a + x**2)**5/x**6, (x, x + sqrt(a + x**2)))/32

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Mathematica [B]  time = 2.75552, size = 338, normalized size = 2.06 \[ \frac{1}{2} \left (\sqrt{a+x^2}+x\right )^n \left (-\frac{2 a^2 \left (x-n \sqrt{a+x^2}\right )}{n^2-1}+\frac{4 a \sqrt{a+x^2} \left (2 a^3 n+a^2 (n-3) n x \left ((n-3) x-2 \sqrt{a+x^2}\right )+a \left (n^2-4 n+3\right ) x^3 \left ((3 n+1) \sqrt{a+x^2}+(5 n+3) x\right )+4 \left (n^3-3 n^2-n+3\right ) x^5 \left (\sqrt{a+x^2}+x\right )\right )}{(n-3) (n-1) (n+1) (n+3) \left (\sqrt{a+x^2}+x\right )^2 \left (x \left (\sqrt{a+x^2}+x\right )+a\right )}+\frac{1}{16} \left (\frac{a^5}{(n-5) \left (\sqrt{a+x^2}+x\right )^5}-\frac{3 a^4}{(n-3) \left (\sqrt{a+x^2}+x\right )^3}+\frac{2 a^3}{(n-1) \left (\sqrt{a+x^2}+x\right )}+\frac{2 a^2 \left (\sqrt{a+x^2}+x\right )}{n+1}-\frac{3 a \left (\sqrt{a+x^2}+x\right )^3}{n+3}+\frac{\left (\sqrt{a+x^2}+x\right )^5}{n+5}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(a + x^2)^2*(x + Sqrt[a + x^2])^n,x]

[Out]

((x + Sqrt[a + x^2])^n*((-2*a^2*(x - n*Sqrt[a + x^2]))/(-1 + n^2) + (a^5/((-5 +
n)*(x + Sqrt[a + x^2])^5) - (3*a^4)/((-3 + n)*(x + Sqrt[a + x^2])^3) + (2*a^3)/(
(-1 + n)*(x + Sqrt[a + x^2])) + (2*a^2*(x + Sqrt[a + x^2]))/(1 + n) - (3*a*(x +
Sqrt[a + x^2])^3)/(3 + n) + (x + Sqrt[a + x^2])^5/(5 + n))/16 + (4*a*Sqrt[a + x^
2]*(2*a^3*n + a^2*(-3 + n)*n*x*((-3 + n)*x - 2*Sqrt[a + x^2]) + 4*(3 - n - 3*n^2
+ n^3)*x^5*(x + Sqrt[a + x^2]) + a*(3 - 4*n + n^2)*x^3*((3 + 5*n)*x + (1 + 3*n)
*Sqrt[a + x^2])))/((-3 + n)*(-1 + n)*(1 + n)*(3 + n)*(x + Sqrt[a + x^2])^2*(a +
x*(x + Sqrt[a + x^2])))))/2

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Maple [C]  time = 0.103, size = 216, normalized size = 1.3 \[{\frac{{2}^{n}{x}^{5+n}}{5+n}{\mbox{$_3$F$_2$}(-{\frac{n}{2}},-{\frac{5}{2}}-{\frac{n}{2}},{\frac{1}{2}}-{\frac{n}{2}};\,1-n,-{\frac{3}{2}}-{\frac{n}{2}};\,-{\frac{a}{{x}^{2}}})}}+{\frac{{2}^{1+n}a{x}^{3+n}}{3+n}{\mbox{$_3$F$_2$}(-{\frac{n}{2}},-{\frac{3}{2}}-{\frac{n}{2}},{\frac{1}{2}}-{\frac{n}{2}};\,1-n,-{\frac{1}{2}}-{\frac{n}{2}};\,-{\frac{a}{{x}^{2}}})}}+{\frac{n}{4\,\sqrt{\pi }}{a}^{{\frac{5}{2}}+{\frac{n}{2}}} \left ( 8\,{\frac{\sqrt{\pi }{x}^{1+n}{a}^{-1/2-n/2}}{ \left ( 1+n \right ) n \left ( -2+2\,n \right ) } \left ({\frac{an}{{x}^{2}}}+n-1 \right ) \left ( \sqrt{{\frac{a}{{x}^{2}}}+1}+1 \right ) ^{-1+n}}+4\,{\frac{\sqrt{\pi }{x}^{1+n}{a}^{-1/2-n/2}}{ \left ( 1+n \right ) n}\sqrt{{\frac{a}{{x}^{2}}}+1} \left ( \sqrt{{\frac{a}{{x}^{2}}}+1}+1 \right ) ^{-1+n}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^2+a)^2*(x+(x^2+a)^(1/2))^n,x)

[Out]

2^n/(5+n)*x^(5+n)*hypergeom([-1/2*n,-5/2-1/2*n,1/2-1/2*n],[1-n,-3/2-1/2*n],-a/x^
2)+2^(1+n)*a/(3+n)*x^(3+n)*hypergeom([-1/2*n,-3/2-1/2*n,1/2-1/2*n],[1-n,-1/2-1/2
*n],-a/x^2)+1/4*a^(5/2+1/2*n)/Pi^(1/2)*n*(8*Pi^(1/2)/(1+n)/n*x^(1+n)*a^(-1/2-1/2
*n)*(a/x^2*n+n-1)/(-2+2*n)*((a/x^2+1)^(1/2)+1)^(-1+n)+4*Pi^(1/2)/(1+n)/n*x^(1+n)
*a^(-1/2-1/2*n)*(a/x^2+1)^(1/2)*((a/x^2+1)^(1/2)+1)^(-1+n))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (x^{2} + a\right )}^{2}{\left (x + \sqrt{x^{2} + a}\right )}^{n}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 + a)^2*(x + sqrt(x^2 + a))^n,x, algorithm="maxima")

[Out]

integrate((x^2 + a)^2*(x + sqrt(x^2 + a))^n, x)

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Fricas [A]  time = 0.313923, size = 213, normalized size = 1.3 \[ -\frac{{\left (5 \,{\left (n^{4} - 10 \, n^{2} + 9\right )} x^{5} + 10 \,{\left (a n^{4} - 16 \, a n^{2} + 15 \, a\right )} x^{3} + 5 \,{\left (a^{2} n^{4} - 22 \, a^{2} n^{2} + 45 \, a^{2}\right )} x -{\left (a^{2} n^{5} - 30 \, a^{2} n^{3} +{\left (n^{5} - 10 \, n^{3} + 9 \, n\right )} x^{4} + 149 \, a^{2} n + 2 \,{\left (a n^{5} - 20 \, a n^{3} + 19 \, a n\right )} x^{2}\right )} \sqrt{x^{2} + a}\right )}{\left (x + \sqrt{x^{2} + a}\right )}^{n}}{n^{6} - 35 \, n^{4} + 259 \, n^{2} - 225} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 + a)^2*(x + sqrt(x^2 + a))^n,x, algorithm="fricas")

[Out]

-(5*(n^4 - 10*n^2 + 9)*x^5 + 10*(a*n^4 - 16*a*n^2 + 15*a)*x^3 + 5*(a^2*n^4 - 22*
a^2*n^2 + 45*a^2)*x - (a^2*n^5 - 30*a^2*n^3 + (n^5 - 10*n^3 + 9*n)*x^4 + 149*a^2
*n + 2*(a*n^5 - 20*a*n^3 + 19*a*n)*x^2)*sqrt(x^2 + a))*(x + sqrt(x^2 + a))^n/(n^
6 - 35*n^4 + 259*n^2 - 225)

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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**2+a)**2*(x+(x**2+a)**(1/2))**n,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (x^{2} + a\right )}^{2}{\left (x + \sqrt{x^{2} + a}\right )}^{n}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 + a)^2*(x + sqrt(x^2 + a))^n,x, algorithm="giac")

[Out]

integrate((x^2 + a)^2*(x + sqrt(x^2 + a))^n, x)