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

Optimal. Leaf size=99 \[ \frac{2 \sqrt{a} x^{j/2} (c x)^{-j/2} \tanh ^{-1}\left (\frac{\sqrt{a} x^{j/2}}{\sqrt{a x^j+b x^n}}\right )}{c (j-n)}-\frac{2 (c x)^{-j/2} \sqrt{a x^j+b x^n}}{c (j-n)} \]

[Out]

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

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Rubi [A]  time = 0.264335, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148 \[ \frac{2 \sqrt{a} x^{j/2} (c x)^{-j/2} \tanh ^{-1}\left (\frac{\sqrt{a} x^{j/2}}{\sqrt{a x^j+b x^n}}\right )}{c (j-n)}-\frac{2 (c x)^{-j/2} \sqrt{a x^j+b x^n}}{c (j-n)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 26.2271, size = 75, normalized size = 0.76 \[ \frac{2 \sqrt{a} x^{\frac{j}{2}} \left (c x\right )^{- \frac{j}{2}} \operatorname{atanh}{\left (\frac{\sqrt{a} x^{\frac{j}{2}}}{\sqrt{a x^{j} + b x^{n}}} \right )}}{c \left (j - n\right )} - \frac{2 \left (c x\right )^{- \frac{j}{2}} \sqrt{a x^{j} + b x^{n}}}{c \left (j - n\right )} \]

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

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Mathematica [A]  time = 0.0982943, size = 109, normalized size = 1.1 \[ -\frac{2 (c x)^{-j/2} \left (-\sqrt{a} \sqrt{b} x^{\frac{j+n}{2}} \sqrt{\frac{a x^{j-n}}{b}+1} \sinh ^{-1}\left (\frac{\sqrt{a} x^{\frac{j-n}{2}}}{\sqrt{b}}\right )+a x^j+b x^n\right )}{c (j-n) \sqrt{a x^j+b x^n}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.412, size = 0, normalized size = 0. \[ \int \left ( cx \right ) ^{-1-{\frac{j}{2}}}\sqrt{a{x}^{j}+b{x}^{n}}\, dx \]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: TypeError

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

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]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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