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

Optimal. Leaf size=62 \[ \frac{2 x^{-j/2} (c x)^{j/2} \tanh ^{-1}\left (\frac{\sqrt{a} x^{j/2}}{\sqrt{a x^j+b x^n}}\right )}{\sqrt{a} c (j-n)} \]

[Out]

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

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Rubi [A]  time = 0.187209, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{2 x^{-j/2} (c x)^{j/2} \tanh ^{-1}\left (\frac{\sqrt{a} x^{j/2}}{\sqrt{a x^j+b x^n}}\right )}{\sqrt{a} c (j-n)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 16.8545, size = 48, normalized size = 0.77 \[ \frac{2 x^{- \frac{j}{2}} \left (c x\right )^{\frac{j}{2}} \operatorname{atanh}{\left (\frac{\sqrt{a} x^{\frac{j}{2}}}{\sqrt{a x^{j} + b x^{n}}} \right )}}{\sqrt{a} c \left (j - n\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.1581, size = 98, normalized size = 1.58 \[ \frac{2 \sqrt{b} (c x)^{j/2} x^{\frac{n-j}{2}} \sqrt{\frac{a x^{j-n}}{b}+1} \sinh ^{-1}\left (\frac{\sqrt{a} x^{\frac{j-n}{2}}}{\sqrt{b}}\right )}{\sqrt{a} c (j-n) \sqrt{a x^j+b x^n}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[b]*x^((-j + n)/2)*(c*x)^(j/2)*Sqrt[1 + (a*x^(j - n))/b]*ArcSinh[(Sqrt[a]
*x^((j - n)/2))/Sqrt[b]])/(Sqrt[a]*c*(j - n)*Sqrt[a*x^j + b*x^n])

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Maple [F]  time = 0.114, size = 0, normalized size = 0. \[ \int{1 \left ( cx \right ) ^{-1+{\frac{j}{2}}}{\frac{1}{\sqrt{a{x}^{j}+b{x}^{n}}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [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)^(1/2*j - 1)/sqrt(a*x^j + b*x^n),x, algorithm="fricas")

[Out]

Exception raised: TypeError

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((c*x)**(j/2 - 1)/sqrt(a*x**j + b*x**n), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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