3.2775 \(\int \frac{(c x)^{-1+\frac{5 n}{2}}}{\sqrt{a+b x^n}} \, dx\)

Optimal. Leaf size=137 \[ \frac{3 a^2 x^{-5 n/2} (c x)^{5 n/2} \tanh ^{-1}\left (\frac{\sqrt{b} x^{n/2}}{\sqrt{a+b x^n}}\right )}{4 b^{5/2} c n}-\frac{3 a x^{-2 n} (c x)^{5 n/2} \sqrt{a+b x^n}}{4 b^2 c n}+\frac{x^{-n} (c x)^{5 n/2} \sqrt{a+b x^n}}{2 b c n} \]

[Out]

(-3*a*(c*x)^((5*n)/2)*Sqrt[a + b*x^n])/(4*b^2*c*n*x^(2*n)) + ((c*x)^((5*n)/2)*Sq
rt[a + b*x^n])/(2*b*c*n*x^n) + (3*a^2*(c*x)^((5*n)/2)*ArcTanh[(Sqrt[b]*x^(n/2))/
Sqrt[a + b*x^n]])/(4*b^(5/2)*c*n*x^((5*n)/2))

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Rubi [A]  time = 0.158779, antiderivative size = 137, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174 \[ \frac{3 a^2 x^{-5 n/2} (c x)^{5 n/2} \tanh ^{-1}\left (\frac{\sqrt{b} x^{n/2}}{\sqrt{a+b x^n}}\right )}{4 b^{5/2} c n}-\frac{3 a x^{-2 n} (c x)^{5 n/2} \sqrt{a+b x^n}}{4 b^2 c n}+\frac{x^{-n} (c x)^{5 n/2} \sqrt{a+b x^n}}{2 b c n} \]

Antiderivative was successfully verified.

[In]  Int[(c*x)^(-1 + (5*n)/2)/Sqrt[a + b*x^n],x]

[Out]

(-3*a*(c*x)^((5*n)/2)*Sqrt[a + b*x^n])/(4*b^2*c*n*x^(2*n)) + ((c*x)^((5*n)/2)*Sq
rt[a + b*x^n])/(2*b*c*n*x^n) + (3*a^2*(c*x)^((5*n)/2)*ArcTanh[(Sqrt[b]*x^(n/2))/
Sqrt[a + b*x^n]])/(4*b^(5/2)*c*n*x^((5*n)/2))

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Rubi in Sympy [A]  time = 23.199, size = 150, normalized size = 1.09 \[ \frac{a^{2} x^{- n} \left (c x\right )^{\frac{5 n}{2}}}{2 b c n \left (a + b x^{n}\right )^{\frac{3}{2}} \left (- \frac{b x^{n}}{a + b x^{n}} + 1\right )^{2}} - \frac{3 a^{2} x^{- 2 n} \left (c x\right )^{\frac{5 n}{2}}}{4 b^{2} c n \sqrt{a + b x^{n}} \left (- \frac{b x^{n}}{a + b x^{n}} + 1\right )} + \frac{3 a^{2} x^{- \frac{5 n}{2}} \left (c x\right )^{\frac{5 n}{2}} \operatorname{atanh}{\left (\frac{\sqrt{b} x^{\frac{n}{2}}}{\sqrt{a + b x^{n}}} \right )}}{4 b^{\frac{5}{2}} c n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x)**(-1+5/2*n)/(a+b*x**n)**(1/2),x)

[Out]

a**2*x**(-n)*(c*x)**(5*n/2)/(2*b*c*n*(a + b*x**n)**(3/2)*(-b*x**n/(a + b*x**n) +
1)**2) - 3*a**2*x**(-2*n)*(c*x)**(5*n/2)/(4*b**2*c*n*sqrt(a + b*x**n)*(-b*x**n/
(a + b*x**n) + 1)) + 3*a**2*x**(-5*n/2)*(c*x)**(5*n/2)*atanh(sqrt(b)*x**(n/2)/sq
rt(a + b*x**n))/(4*b**(5/2)*c*n)

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Mathematica [A]  time = 0.105287, size = 99, normalized size = 0.72 \[ \frac{x^{-5 n/2} (c x)^{5 n/2} \left (3 a^2 \log \left (\sqrt{b} \sqrt{a+b x^n}+b x^{n/2}\right )+\sqrt{b} x^{n/2} \sqrt{a+b x^n} \left (2 b x^n-3 a\right )\right )}{4 b^{5/2} c n} \]

Antiderivative was successfully verified.

[In]  Integrate[(c*x)^(-1 + (5*n)/2)/Sqrt[a + b*x^n],x]

[Out]

((c*x)^((5*n)/2)*(Sqrt[b]*x^(n/2)*Sqrt[a + b*x^n]*(-3*a + 2*b*x^n) + 3*a^2*Log[b
*x^(n/2) + Sqrt[b]*Sqrt[a + b*x^n]]))/(4*b^(5/2)*c*n*x^((5*n)/2))

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Maple [F]  time = 0.072, size = 0, normalized size = 0. \[ \int{1 \left ( cx \right ) ^{-1+{\frac{5\,n}{2}}}{\frac{1}{\sqrt{a+b{x}^{n}}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x)^(-1+5/2*n)/(a+b*x^n)^(1/2),x)

[Out]

int((c*x)^(-1+5/2*n)/(a+b*x^n)^(1/2),x)

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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)^(5/2*n - 1)/sqrt(b*x^n + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.249425, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, a^{2} c^{\frac{5}{2} \, n - 1} \log \left (-2 \, \sqrt{b x^{n} + a} b x^{\frac{1}{2} \, n} - 2 \, b^{\frac{3}{2}} x^{n} - a \sqrt{b}\right ) + 2 \,{\left (2 \, b^{\frac{3}{2}} c^{\frac{5}{2} \, n - 1} x^{\frac{3}{2} \, n} - 3 \, a \sqrt{b} c^{\frac{5}{2} \, n - 1} x^{\frac{1}{2} \, n}\right )} \sqrt{b x^{n} + a}}{8 \, b^{\frac{5}{2}} n}, \frac{3 \, a^{2} c^{\frac{5}{2} \, n - 1} \arctan \left (\frac{\sqrt{-b} x^{\frac{1}{2} \, n}}{\sqrt{b x^{n} + a}}\right ) +{\left (2 \, \sqrt{-b} b c^{\frac{5}{2} \, n - 1} x^{\frac{3}{2} \, n} - 3 \, a \sqrt{-b} c^{\frac{5}{2} \, n - 1} x^{\frac{1}{2} \, n}\right )} \sqrt{b x^{n} + a}}{4 \, \sqrt{-b} b^{2} n}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x)^(5/2*n - 1)/sqrt(b*x^n + a),x, algorithm="fricas")

[Out]

[1/8*(3*a^2*c^(5/2*n - 1)*log(-2*sqrt(b*x^n + a)*b*x^(1/2*n) - 2*b^(3/2)*x^n - a
*sqrt(b)) + 2*(2*b^(3/2)*c^(5/2*n - 1)*x^(3/2*n) - 3*a*sqrt(b)*c^(5/2*n - 1)*x^(
1/2*n))*sqrt(b*x^n + a))/(b^(5/2)*n), 1/4*(3*a^2*c^(5/2*n - 1)*arctan(sqrt(-b)*x
^(1/2*n)/sqrt(b*x^n + a)) + (2*sqrt(-b)*b*c^(5/2*n - 1)*x^(3/2*n) - 3*a*sqrt(-b)
*c^(5/2*n - 1)*x^(1/2*n))*sqrt(b*x^n + a))/(sqrt(-b)*b^2*n)]

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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+5/2*n)/(a+b*x**n)**(1/2),x)

[Out]

Exception raised: TypeError

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (c x\right )^{\frac{5}{2} \, n - 1}}{\sqrt{b x^{n} + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x)^(5/2*n - 1)/sqrt(b*x^n + a),x, algorithm="giac")

[Out]

integrate((c*x)^(5/2*n - 1)/sqrt(b*x^n + a), x)