3.427 \(\int \sqrt{a x^j+b x^n} \, dx\)

Optimal. Leaf size=87 \[ \frac{2 x \sqrt{a x^j+b x^n} \, _2F_1\left (-\frac{1}{2},\frac{n+2}{2 (j-n)};\frac{n+2}{2 j-2 n}+1;-\frac{a x^{j-n}}{b}\right )}{(n+2) \sqrt{\frac{a x^{j-n}}{b}+1}} \]

[Out]

(2*x*Sqrt[a*x^j + b*x^n]*Hypergeometric2F1[-1/2, (2 + n)/(2*(j - n)), 1 + (2 + n
)/(2*j - 2*n), -((a*x^(j - n))/b)])/((2 + n)*Sqrt[1 + (a*x^(j - n))/b])

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Rubi [A]  time = 0.136193, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{2 x \sqrt{a x^j+b x^n} \, _2F_1\left (-\frac{1}{2},\frac{n+2}{2 (j-n)};\frac{n+2}{2 j-2 n}+1;-\frac{a x^{j-n}}{b}\right )}{(n+2) \sqrt{\frac{a x^{j-n}}{b}+1}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a*x^j + b*x^n],x]

[Out]

(2*x*Sqrt[a*x^j + b*x^n]*Hypergeometric2F1[-1/2, (2 + n)/(2*(j - n)), 1 + (2 + n
)/(2*j - 2*n), -((a*x^(j - n))/b)])/((2 + n)*Sqrt[1 + (a*x^(j - n))/b])

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Rubi in Sympy [A]  time = 12.0182, size = 75, normalized size = 0.86 \[ \frac{2 x^{- \frac{n}{2}} x^{\frac{n}{2} + 1} \sqrt{a x^{j} + b x^{n}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{n + 2}{2 \left (j - n\right )} \\ \frac{j - \frac{n}{2} + 1}{j - n} \end{matrix}\middle |{- \frac{a x^{j - n}}{b}} \right )}}{\left (n + 2\right ) \sqrt{\frac{a x^{j - n}}{b} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.236125, size = 134, normalized size = 1.54 \[ \frac{2 x \left (a (j-n) x^j \sqrt{\frac{a x^{j-n}}{b}+1} \, _2F_1\left (\frac{1}{2},\frac{2 j-n+2}{2 j-2 n};\frac{4 j-3 n+2}{2 j-2 n};-\frac{a x^{j-n}}{b}\right )-(2 j-n+2) \left (a x^j+b x^n\right )\right )}{(n+2) (-2 j+n-2) \sqrt{a x^j+b x^n}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a*x^j + b*x^n],x]

[Out]

(2*x*(-((2 + 2*j - n)*(a*x^j + b*x^n)) + a*(j - n)*x^j*Sqrt[1 + (a*x^(j - n))/b]
*Hypergeometric2F1[1/2, (2 + 2*j - n)/(2*j - 2*n), (2 + 4*j - 3*n)/(2*j - 2*n),
-((a*x^(j - n))/b)]))/((2 + n)*(-2 - 2*j + n)*Sqrt[a*x^j + b*x^n])

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Maple [F]  time = 0.09, size = 0, normalized size = 0. \[ \int \sqrt{a{x}^{j}+b{x}^{n}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a*x^j+b*x^n)^(1/2),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{a x^{j} + b x^{n}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(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(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. \[ \int \sqrt{a x^{j} + b x^{n}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt(a*x**j + b*x**n), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{a x^{j} + b x^{n}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(a*x^j + b*x^n), x)