3.2680 \(\int \frac{x^{-1+\frac{7 n}{2}}}{\sqrt{a+b x^n}} \, dx\)

Optimal. Leaf size=129 \[ -\frac{5 a^3 \tanh ^{-1}\left (\frac{\sqrt{b} x^{n/2}}{\sqrt{a+b x^n}}\right )}{8 b^{7/2} n}+\frac{5 a^2 x^{n/2} \sqrt{a+b x^n}}{8 b^3 n}-\frac{5 a x^{3 n/2} \sqrt{a+b x^n}}{12 b^2 n}+\frac{x^{5 n/2} \sqrt{a+b x^n}}{3 b n} \]

[Out]

(5*a^2*x^(n/2)*Sqrt[a + b*x^n])/(8*b^3*n) - (5*a*x^((3*n)/2)*Sqrt[a + b*x^n])/(1
2*b^2*n) + (x^((5*n)/2)*Sqrt[a + b*x^n])/(3*b*n) - (5*a^3*ArcTanh[(Sqrt[b]*x^(n/
2))/Sqrt[a + b*x^n]])/(8*b^(7/2)*n)

_______________________________________________________________________________________

Rubi [A]  time = 0.160444, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ -\frac{5 a^3 \tanh ^{-1}\left (\frac{\sqrt{b} x^{n/2}}{\sqrt{a+b x^n}}\right )}{8 b^{7/2} n}+\frac{5 a^2 x^{n/2} \sqrt{a+b x^n}}{8 b^3 n}-\frac{5 a x^{3 n/2} \sqrt{a+b x^n}}{12 b^2 n}+\frac{x^{5 n/2} \sqrt{a+b x^n}}{3 b n} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 + (7*n)/2)/Sqrt[a + b*x^n],x]

[Out]

(5*a^2*x^(n/2)*Sqrt[a + b*x^n])/(8*b^3*n) - (5*a*x^((3*n)/2)*Sqrt[a + b*x^n])/(1
2*b^2*n) + (x^((5*n)/2)*Sqrt[a + b*x^n])/(3*b*n) - (5*a^3*ArcTanh[(Sqrt[b]*x^(n/
2))/Sqrt[a + b*x^n]])/(8*b^(7/2)*n)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 20.6, size = 160, normalized size = 1.24 \[ \frac{a^{3} x^{\frac{5 n}{2}}}{3 b n \left (a + b x^{n}\right )^{\frac{5}{2}} \left (- \frac{b x^{n}}{a + b x^{n}} + 1\right )^{3}} - \frac{5 a^{3} x^{\frac{3 n}{2}}}{12 b^{2} n \left (a + b x^{n}\right )^{\frac{3}{2}} \left (- \frac{b x^{n}}{a + b x^{n}} + 1\right )^{2}} + \frac{5 a^{3} x^{\frac{n}{2}}}{8 b^{3} n \sqrt{a + b x^{n}} \left (- \frac{b x^{n}}{a + b x^{n}} + 1\right )} - \frac{5 a^{3} \operatorname{atanh}{\left (\frac{\sqrt{b} x^{\frac{n}{2}}}{\sqrt{a + b x^{n}}} \right )}}{8 b^{\frac{7}{2}} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(-1+7/2*n)/(a+b*x**n)**(1/2),x)

[Out]

a**3*x**(5*n/2)/(3*b*n*(a + b*x**n)**(5/2)*(-b*x**n/(a + b*x**n) + 1)**3) - 5*a*
*3*x**(3*n/2)/(12*b**2*n*(a + b*x**n)**(3/2)*(-b*x**n/(a + b*x**n) + 1)**2) + 5*
a**3*x**(n/2)/(8*b**3*n*sqrt(a + b*x**n)*(-b*x**n/(a + b*x**n) + 1)) - 5*a**3*at
anh(sqrt(b)*x**(n/2)/sqrt(a + b*x**n))/(8*b**(7/2)*n)

_______________________________________________________________________________________

Mathematica [A]  time = 0.121634, size = 93, normalized size = 0.72 \[ \frac{\sqrt{b} x^{n/2} \sqrt{a+b x^n} \left (15 a^2-10 a b x^n+8 b^2 x^{2 n}\right )-15 a^3 \log \left (\sqrt{b} \sqrt{a+b x^n}+b x^{n/2}\right )}{24 b^{7/2} n} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(-1 + (7*n)/2)/Sqrt[a + b*x^n],x]

[Out]

(Sqrt[b]*x^(n/2)*Sqrt[a + b*x^n]*(15*a^2 - 10*a*b*x^n + 8*b^2*x^(2*n)) - 15*a^3*
Log[b*x^(n/2) + Sqrt[b]*Sqrt[a + b*x^n]])/(24*b^(7/2)*n)

_______________________________________________________________________________________

Maple [A]  time = 0.088, size = 98, normalized size = 0.8 \[{\frac{1}{24\,{b}^{3}n}{{\rm e}^{{\frac{n\ln \left ( x \right ) }{2}}}} \left ( 8\, \left ({{\rm e}^{1/2\,n\ln \left ( x \right ) }} \right ) ^{4}{b}^{2}-10\,a \left ({{\rm e}^{1/2\,n\ln \left ( x \right ) }} \right ) ^{2}b+15\,{a}^{2} \right ) \sqrt{a+b \left ({{\rm e}^{{\frac{n\ln \left ( x \right ) }{2}}}} \right ) ^{2}}}-{\frac{5\,{a}^{3}}{8\,n}\ln \left ({{\rm e}^{{\frac{n\ln \left ( x \right ) }{2}}}}\sqrt{b}+\sqrt{a+b \left ({{\rm e}^{{\frac{n\ln \left ( x \right ) }{2}}}} \right ) ^{2}} \right ){b}^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1+7/2*n)/(a+b*x^n)^(1/2),x)

[Out]

1/24*exp(1/2*n*ln(x))*(8*exp(1/2*n*ln(x))^4*b^2-10*a*exp(1/2*n*ln(x))^2*b+15*a^2
)*(a+b*exp(1/2*n*ln(x))^2)^(1/2)/b^3/n-5/8*a^3/b^(7/2)/n*ln(exp(1/2*n*ln(x))*b^(
1/2)+(a+b*exp(1/2*n*ln(x))^2)^(1/2))

_______________________________________________________________________________________

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(x^(7/2*n - 1)/sqrt(b*x^n + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.265074, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, a^{3} \log \left (2 \, \sqrt{b x^{n} + a} b x^{\frac{1}{2} \, n} - 2 \, b^{\frac{3}{2}} x^{n} - a \sqrt{b}\right ) + 2 \,{\left (8 \, b^{\frac{5}{2}} x^{\frac{5}{2} \, n} - 10 \, a b^{\frac{3}{2}} x^{\frac{3}{2} \, n} + 15 \, a^{2} \sqrt{b} x^{\frac{1}{2} \, n}\right )} \sqrt{b x^{n} + a}}{48 \, b^{\frac{7}{2}} n}, -\frac{15 \, a^{3} \arctan \left (\frac{\sqrt{-b} x^{\frac{1}{2} \, n}}{\sqrt{b x^{n} + a}}\right ) -{\left (8 \, \sqrt{-b} b^{2} x^{\frac{5}{2} \, n} - 10 \, a \sqrt{-b} b x^{\frac{3}{2} \, n} + 15 \, a^{2} \sqrt{-b} x^{\frac{1}{2} \, n}\right )} \sqrt{b x^{n} + a}}{24 \, \sqrt{-b} b^{3} n}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(7/2*n - 1)/sqrt(b*x^n + a),x, algorithm="fricas")

[Out]

[1/48*(15*a^3*log(2*sqrt(b*x^n + a)*b*x^(1/2*n) - 2*b^(3/2)*x^n - a*sqrt(b)) + 2
*(8*b^(5/2)*x^(5/2*n) - 10*a*b^(3/2)*x^(3/2*n) + 15*a^2*sqrt(b)*x^(1/2*n))*sqrt(
b*x^n + a))/(b^(7/2)*n), -1/24*(15*a^3*arctan(sqrt(-b)*x^(1/2*n)/sqrt(b*x^n + a)
) - (8*sqrt(-b)*b^2*x^(5/2*n) - 10*a*sqrt(-b)*b*x^(3/2*n) + 15*a^2*sqrt(-b)*x^(1
/2*n))*sqrt(b*x^n + a))/(sqrt(-b)*b^3*n)]

_______________________________________________________________________________________

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**(-1+7/2*n)/(a+b*x**n)**(1/2),x)

[Out]

Exception raised: TypeError

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{\frac{7}{2} \, n - 1}}{\sqrt{b x^{n} + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(7/2*n - 1)/sqrt(b*x^n + a),x, algorithm="giac")

[Out]

integrate(x^(7/2*n - 1)/sqrt(b*x^n + a), x)