3.2776 \(\int \frac{(c x)^{-1+\frac{3 n}{2}}}{\sqrt{a+b x^n}} \, dx\)

Optimal. Leaf size=91 \[ \frac{x^{-n} (c x)^{3 n/2} \sqrt{a+b x^n}}{b c n}-\frac{a x^{-3 n/2} (c x)^{3 n/2} \tanh ^{-1}\left (\frac{\sqrt{b} x^{n/2}}{\sqrt{a+b x^n}}\right )}{b^{3/2} c n} \]

[Out]

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

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Rubi [A]  time = 0.114104, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174 \[ \frac{x^{-n} (c x)^{3 n/2} \sqrt{a+b x^n}}{b c n}-\frac{a x^{-3 n/2} (c x)^{3 n/2} \tanh ^{-1}\left (\frac{\sqrt{b} x^{n/2}}{\sqrt{a+b x^n}}\right )}{b^{3/2} c n} \]

Antiderivative was successfully verified.

[In]  Int[(c*x)^(-1 + (3*n)/2)/Sqrt[a + b*x^n],x]

[Out]

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

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Rubi in Sympy [A]  time = 14.31, size = 88, normalized size = 0.97 \[ \frac{a x^{- n} \left (c x\right )^{\frac{3 n}{2}}}{b c n \sqrt{a + b x^{n}} \left (- \frac{b x^{n}}{a + b x^{n}} + 1\right )} - \frac{a x^{- \frac{3 n}{2}} \left (c x\right )^{\frac{3 n}{2}} \operatorname{atanh}{\left (\frac{\sqrt{b} x^{\frac{n}{2}}}{\sqrt{a + b x^{n}}} \right )}}{b^{\frac{3}{2}} c n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x)**(-1+3/2*n)/(a+b*x**n)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.0690079, size = 84, normalized size = 0.92 \[ \frac{x^{-3 n/2} (c x)^{3 n/2} \left (\sqrt{b} x^{n/2} \sqrt{a+b x^n}-a \log \left (\sqrt{b} \sqrt{a+b x^n}+b x^{n/2}\right )\right )}{b^{3/2} c n} \]

Antiderivative was successfully verified.

[In]  Integrate[(c*x)^(-1 + (3*n)/2)/Sqrt[a + b*x^n],x]

[Out]

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

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Maple [F]  time = 0.075, size = 0, normalized size = 0. \[ \int{1 \left ( cx \right ) ^{-1+{\frac{3\,n}{2}}}{\frac{1}{\sqrt{a+b{x}^{n}}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x)^(-1+3/2*n)/(a+b*x^n)^(1/2),x)

[Out]

int((c*x)^(-1+3/2*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)^(3/2*n - 1)/sqrt(b*x^n + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.246567, size = 1, normalized size = 0.01 \[ \left [\frac{2 \, \sqrt{b x^{n} + a} \sqrt{b} c^{\frac{3}{2} \, n - 1} x^{\frac{1}{2} \, n} + a c^{\frac{3}{2} \, n - 1} \log \left (2 \, \sqrt{b x^{n} + a} b x^{\frac{1}{2} \, n} - 2 \, b^{\frac{3}{2}} x^{n} - a \sqrt{b}\right )}{2 \, b^{\frac{3}{2}} n}, \frac{\sqrt{b x^{n} + a} \sqrt{-b} c^{\frac{3}{2} \, n - 1} x^{\frac{1}{2} \, n} - a c^{\frac{3}{2} \, n - 1} \arctan \left (\frac{\sqrt{-b} x^{\frac{1}{2} \, n}}{\sqrt{b x^{n} + a}}\right )}{\sqrt{-b} b n}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x)^(3/2*n - 1)/sqrt(b*x^n + a),x, algorithm="fricas")

[Out]

[1/2*(2*sqrt(b*x^n + a)*sqrt(b)*c^(3/2*n - 1)*x^(1/2*n) + a*c^(3/2*n - 1)*log(2*
sqrt(b*x^n + a)*b*x^(1/2*n) - 2*b^(3/2)*x^n - a*sqrt(b)))/(b^(3/2)*n), (sqrt(b*x
^n + a)*sqrt(-b)*c^(3/2*n - 1)*x^(1/2*n) - a*c^(3/2*n - 1)*arctan(sqrt(-b)*x^(1/
2*n)/sqrt(b*x^n + a)))/(sqrt(-b)*b*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+3/2*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. \[ \int \frac{\left (c x\right )^{\frac{3}{2} \, n - 1}}{\sqrt{b x^{n} + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x)^(3/2*n - 1)/sqrt(b*x^n + a),x, algorithm="giac")

[Out]

integrate((c*x)^(3/2*n - 1)/sqrt(b*x^n + a), x)