3.2774 \(\int \frac{(c x)^{-1+\frac{7 n}{2}}}{\sqrt{a+b x^n}} \, dx\)

Optimal. Leaf size=178 \[ -\frac{5 a^3 x^{-7 n/2} (c x)^{7 n/2} \tanh ^{-1}\left (\frac{\sqrt{b} x^{n/2}}{\sqrt{a+b x^n}}\right )}{8 b^{7/2} c n}+\frac{5 a^2 x^{-3 n} (c x)^{7 n/2} \sqrt{a+b x^n}}{8 b^3 c n}-\frac{5 a x^{-2 n} (c x)^{7 n/2} \sqrt{a+b x^n}}{12 b^2 c n}+\frac{x^{-n} (c x)^{7 n/2} \sqrt{a+b x^n}}{3 b c n} \]

[Out]

(5*a^2*(c*x)^((7*n)/2)*Sqrt[a + b*x^n])/(8*b^3*c*n*x^(3*n)) - (5*a*(c*x)^((7*n)/
2)*Sqrt[a + b*x^n])/(12*b^2*c*n*x^(2*n)) + ((c*x)^((7*n)/2)*Sqrt[a + b*x^n])/(3*
b*c*n*x^n) - (5*a^3*(c*x)^((7*n)/2)*ArcTanh[(Sqrt[b]*x^(n/2))/Sqrt[a + b*x^n]])/
(8*b^(7/2)*c*n*x^((7*n)/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.226015, antiderivative size = 178, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174 \[ -\frac{5 a^3 x^{-7 n/2} (c x)^{7 n/2} \tanh ^{-1}\left (\frac{\sqrt{b} x^{n/2}}{\sqrt{a+b x^n}}\right )}{8 b^{7/2} c n}+\frac{5 a^2 x^{-3 n} (c x)^{7 n/2} \sqrt{a+b x^n}}{8 b^3 c n}-\frac{5 a x^{-2 n} (c x)^{7 n/2} \sqrt{a+b x^n}}{12 b^2 c n}+\frac{x^{-n} (c x)^{7 n/2} \sqrt{a+b x^n}}{3 b c n} \]

Antiderivative was successfully verified.

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

[Out]

(5*a^2*(c*x)^((7*n)/2)*Sqrt[a + b*x^n])/(8*b^3*c*n*x^(3*n)) - (5*a*(c*x)^((7*n)/
2)*Sqrt[a + b*x^n])/(12*b^2*c*n*x^(2*n)) + ((c*x)^((7*n)/2)*Sqrt[a + b*x^n])/(3*
b*c*n*x^n) - (5*a^3*(c*x)^((7*n)/2)*ArcTanh[(Sqrt[b]*x^(n/2))/Sqrt[a + b*x^n]])/
(8*b^(7/2)*c*n*x^((7*n)/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 30.9365, size = 202, normalized size = 1.13 \[ \frac{a^{3} x^{- n} \left (c x\right )^{\frac{7 n}{2}}}{3 b c n \left (a + b x^{n}\right )^{\frac{5}{2}} \left (- \frac{b x^{n}}{a + b x^{n}} + 1\right )^{3}} - \frac{5 a^{3} x^{- 2 n} \left (c x\right )^{\frac{7 n}{2}}}{12 b^{2} c n \left (a + b x^{n}\right )^{\frac{3}{2}} \left (- \frac{b x^{n}}{a + b x^{n}} + 1\right )^{2}} + \frac{5 a^{3} x^{- 3 n} \left (c x\right )^{\frac{7 n}{2}}}{8 b^{3} c n \sqrt{a + b x^{n}} \left (- \frac{b x^{n}}{a + b x^{n}} + 1\right )} - \frac{5 a^{3} x^{- \frac{7 n}{2}} \left (c x\right )^{\frac{7 n}{2}} \operatorname{atanh}{\left (\frac{\sqrt{b} x^{\frac{n}{2}}}{\sqrt{a + b x^{n}}} \right )}}{8 b^{\frac{7}{2}} c n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**3*x**(-n)*(c*x)**(7*n/2)/(3*b*c*n*(a + b*x**n)**(5/2)*(-b*x**n/(a + b*x**n) +
1)**3) - 5*a**3*x**(-2*n)*(c*x)**(7*n/2)/(12*b**2*c*n*(a + b*x**n)**(3/2)*(-b*x
**n/(a + b*x**n) + 1)**2) + 5*a**3*x**(-3*n)*(c*x)**(7*n/2)/(8*b**3*c*n*sqrt(a +
b*x**n)*(-b*x**n/(a + b*x**n) + 1)) - 5*a**3*x**(-7*n/2)*(c*x)**(7*n/2)*atanh(s
qrt(b)*x**(n/2)/sqrt(a + b*x**n))/(8*b**(7/2)*c*n)

_______________________________________________________________________________________

Mathematica [A]  time = 0.156697, size = 112, normalized size = 0.63 \[ \frac{x^{-7 n/2} (c x)^{7 n/2} \left (\sqrt{b} x^{n/2} \sqrt{a+b x^n} \left (15 a^2-10 a b x^n+8 b^2 x^{2 n}\right )-15 a^3 \log \left (\sqrt{b} \sqrt{a+b x^n}+b x^{n/2}\right )\right )}{24 b^{7/2} c n} \]

Antiderivative was successfully verified.

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

[Out]

((c*x)^((7*n)/2)*(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)*c*n*x^((
7*n)/2))

_______________________________________________________________________________________

Maple [F]  time = 0.073, size = 0, normalized size = 0. \[ \int{1 \left ( cx \right ) ^{-1+{\frac{7\,n}{2}}}{\frac{1}{\sqrt{a+b{x}^{n}}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.263089, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, a^{3} c^{\frac{7}{2} \, n - 1} \log \left (2 \, \sqrt{b x^{n} + a} b x^{\frac{1}{2} \, n} - 2 \, b^{\frac{3}{2}} x^{n} - a \sqrt{b}\right ) + 2 \,{\left (8 \, b^{\frac{5}{2}} c^{\frac{7}{2} \, n - 1} x^{\frac{5}{2} \, n} - 10 \, a b^{\frac{3}{2}} c^{\frac{7}{2} \, n - 1} x^{\frac{3}{2} \, n} + 15 \, a^{2} \sqrt{b} c^{\frac{7}{2} \, n - 1} x^{\frac{1}{2} \, n}\right )} \sqrt{b x^{n} + a}}{48 \, b^{\frac{7}{2}} n}, -\frac{15 \, a^{3} c^{\frac{7}{2} \, n - 1} \arctan \left (\frac{\sqrt{-b} x^{\frac{1}{2} \, n}}{\sqrt{b x^{n} + a}}\right ) -{\left (8 \, \sqrt{-b} b^{2} c^{\frac{7}{2} \, n - 1} x^{\frac{5}{2} \, n} - 10 \, a \sqrt{-b} b c^{\frac{7}{2} \, n - 1} x^{\frac{3}{2} \, n} + 15 \, a^{2} \sqrt{-b} c^{\frac{7}{2} \, n - 1} x^{\frac{1}{2} \, n}\right )} \sqrt{b x^{n} + a}}{24 \, \sqrt{-b} b^{3} n}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/48*(15*a^3*c^(7/2*n - 1)*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)*c^(7/2*n - 1)*x^(5/2*n) - 10*a*b^(3/2)*c^(7/2*n - 1)*x
^(3/2*n) + 15*a^2*sqrt(b)*c^(7/2*n - 1)*x^(1/2*n))*sqrt(b*x^n + a))/(b^(7/2)*n),
-1/24*(15*a^3*c^(7/2*n - 1)*arctan(sqrt(-b)*x^(1/2*n)/sqrt(b*x^n + a)) - (8*sqr
t(-b)*b^2*c^(7/2*n - 1)*x^(5/2*n) - 10*a*sqrt(-b)*b*c^(7/2*n - 1)*x^(3/2*n) + 15
*a^2*sqrt(-b)*c^(7/2*n - 1)*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((c*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. \[ \int \frac{\left (c x\right )^{\frac{7}{2} \, n - 1}}{\sqrt{b x^{n} + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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