3.429 \(\int \frac{1}{\left (a x^j+b x^n\right )^{3/2}} \, dx\)

Optimal. Leaf size=101 \[ \frac{2 x^{1-n} \sqrt{\frac{a x^{j-n}}{b}+1} \, _2F_1\left (\frac{3}{2},\frac{1-\frac{3 n}{2}}{j-n};\frac{1-\frac{3 n}{2}}{j-n}+1;-\frac{a x^{j-n}}{b}\right )}{b (2-3 n) \sqrt{a x^j+b x^n}} \]

[Out]

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

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Rubi [A]  time = 0.142158, antiderivative size = 101, 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 \[ \frac{2 x^{1-n} \sqrt{\frac{a x^{j-n}}{b}+1} \, _2F_1\left (\frac{3}{2},\frac{1-\frac{3 n}{2}}{j-n};\frac{1-\frac{3 n}{2}}{j-n}+1;-\frac{a x^{j-n}}{b}\right )}{b (2-3 n) \sqrt{a x^j+b x^n}} \]

Antiderivative was successfully verified.

[In]  Int[(a*x^j + b*x^n)^(-3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 13.6764, size = 82, normalized size = 0.81 \[ \frac{2 x^{- \frac{n}{2}} x^{- \frac{3 n}{2} + 1} \sqrt{a x^{j} + b x^{n}}{{}_{2}F_{1}\left (\begin{matrix} \frac{3}{2}, \frac{- \frac{3 n}{2} + 1}{j - n} \\ \frac{j - \frac{5 n}{2} + 1}{j - n} \end{matrix}\middle |{- \frac{a x^{j - n}}{b}} \right )}}{b^{2} \left (- 3 n + 2\right ) \sqrt{\frac{a x^{j - n}}{b} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*x**(-n/2)*x**(-3*n/2 + 1)*sqrt(a*x**j + b*x**n)*hyper((3/2, (-3*n/2 + 1)/(j -
n)), ((j - 5*n/2 + 1)/(j - n),), -a*x**(j - n)/b)/(b**2*(-3*n + 2)*sqrt(a*x**(j
- n)/b + 1))

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Mathematica [A]  time = 0.13877, size = 104, normalized size = 1.03 \[ \frac{2 x^{1-j} \left (\sqrt{\frac{a x^{j-n}}{b}+1} \, _2F_1\left (\frac{1}{2},-\frac{2 j+n-2}{2 (j-n)};\frac{2-3 n}{2 j-2 n};-\frac{a x^{j-n}}{b}\right )-1\right )}{a (j-n) \sqrt{a x^j+b x^n}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.058, size = 0, normalized size = 0. \[ \int \left ( a{x}^{j}+b{x}^{n} \right ) ^{-{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a*x^j+b*x^n)^(3/2),x)

[Out]

int(1/(a*x^j+b*x^n)^(3/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (a x^{j} + b x^{n}\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*x^j + b*x^n)^(-3/2),x, algorithm="maxima")

[Out]

integrate((a*x^j + b*x^n)^(-3/2), 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((a*x^j + b*x^n)^(-3/2),x, algorithm="fricas")

[Out]

Exception raised: TypeError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a*x**j+b*x**n)**(3/2),x)

[Out]

Integral((a*x**j + b*x**n)**(-3/2), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (a x^{j} + b x^{n}\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((a*x^j + b*x^n)^(-3/2), x)