3.536 \(\int \frac{(d x)^m}{\left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2}} \, dx\)

Optimal. Leaf size=76 \[ \frac{(d x)^{m+1} \left (a+b x^n\right ) \, _2F_1\left (3,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right )}{a^3 d (m+1) \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}} \]

[Out]

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

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Rubi [A]  time = 0.0952113, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \frac{(d x)^{m+1} \left (a+b x^n\right ) \, _2F_1\left (3,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right )}{a^3 d (m+1) \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}} \]

Antiderivative was successfully verified.

[In]  Int[(d*x)^m/(a^2 + 2*a*b*x^n + b^2*x^(2*n))^(3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 16.3555, size = 68, normalized size = 0.89 \[ \frac{b \left (d x\right )^{m + 1} \sqrt{a^{2} + 2 a b x^{n} + b^{2} x^{2 n}}{{}_{2}F_{1}\left (\begin{matrix} 3, \frac{m + 1}{n} \\ \frac{m + n + 1}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{a^{3} d \left (m + 1\right ) \left (a b + b^{2} x^{n}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.19964, size = 119, normalized size = 1.57 \[ \frac{x (d x)^m \left (a+b x^n\right ) \left (a^2 n+\frac{\left (m^2+m (2-3 n)+2 n^2-3 n+1\right ) \left (a+b x^n\right )^2 \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right )}{m+1}-a (m-2 n+1) \left (a+b x^n\right )\right )}{2 a^3 n^2 \left (\left (a+b x^n\right )^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.057, size = 0, normalized size = 0. \[ \int{ \left ( dx \right ) ^{m} \left ({a}^{2}+2\,ab{x}^{n}+{b}^{2}{x}^{2\,n} \right ) ^{-{\frac{3}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x)^m/(a^2+2*a*b*x^n+b^2*x^(2*n))^(3/2),x)

[Out]

int((d*x)^m/(a^2+2*a*b*x^n+b^2*x^(2*n))^(3/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[{\left (m^{2} - m{\left (3 \, n - 2\right )} + 2 \, n^{2} - 3 \, n + 1\right )} d^{m} \int \frac{x^{m}}{2 \,{\left (a^{2} b n^{2} x^{n} + a^{3} n^{2}\right )}}\,{d x} - \frac{a d^{m}{\left (m - 3 \, n + 1\right )} x x^{m} + b d^{m}{\left (m - 2 \, n + 1\right )} x e^{\left (m \log \left (x\right ) + n \log \left (x\right )\right )}}{2 \,{\left (a^{2} b^{2} n^{2} x^{2 \, n} + 2 \, a^{3} b n^{2} x^{n} + a^{4} n^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(m^2 - m*(3*n - 2) + 2*n^2 - 3*n + 1)*d^m*integrate(1/2*x^m/(a^2*b*n^2*x^n + a^3
*n^2), x) - 1/2*(a*d^m*(m - 3*n + 1)*x*x^m + b*d^m*(m - 2*n + 1)*x*e^(m*log(x) +
n*log(x)))/(a^2*b^2*n^2*x^(2*n) + 2*a^3*b*n^2*x^n + a^4*n^2)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\left (d x\right )^{m}}{{\left (b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}\right )}^{\frac{3}{2}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x)^m/(b^2*x^(2*n) + 2*a*b*x^n + a^2)^(3/2),x, algorithm="fricas")

[Out]

integral((d*x)^m/(b^2*x^(2*n) + 2*a*b*x^n + a^2)^(3/2), x)

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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((d*x)**m/(a**2+2*a*b*x**n+b**2*x**(2*n))**(3/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((d*x)^m/(b^2*x^(2*n) + 2*a*b*x^n + a^2)^(3/2), x)