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

Optimal. Leaf size=152 \[ -\frac{\sqrt{\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1} F_1\left (-\frac{1}{n};\frac{3}{2},\frac{3}{2};-\frac{1-n}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{a x \sqrt{a+b x^n+c x^{2 n}}} \]

[Out]

-((Sqrt[1 + (2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*x^n)/(b + Sqrt[b^2
- 4*a*c])]*AppellF1[-n^(-1), 3/2, 3/2, -((1 - n)/n), (-2*c*x^n)/(b - Sqrt[b^2 -
4*a*c]), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/(a*x*Sqrt[a + b*x^n + c*x^(2*n)]))

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Rubi [A]  time = 0.464031, antiderivative size = 152, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ -\frac{\sqrt{\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1} F_1\left (-\frac{1}{n};\frac{3}{2},\frac{3}{2};-\frac{1-n}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{a x \sqrt{a+b x^n+c x^{2 n}}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^2*(a + b*x^n + c*x^(2*n))^(3/2)),x]

[Out]

-((Sqrt[1 + (2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*x^n)/(b + Sqrt[b^2
- 4*a*c])]*AppellF1[-n^(-1), 3/2, 3/2, -((1 - n)/n), (-2*c*x^n)/(b - Sqrt[b^2 -
4*a*c]), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/(a*x*Sqrt[a + b*x^n + c*x^(2*n)]))

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Rubi in Sympy [A]  time = 43.0356, size = 128, normalized size = 0.84 \[ - \frac{\sqrt{a + b x^{n} + c x^{2 n}} \operatorname{appellf_{1}}{\left (- \frac{1}{n},\frac{3}{2},\frac{3}{2},\frac{n - 1}{n},- \frac{2 c x^{n}}{b - \sqrt{- 4 a c + b^{2}}},- \frac{2 c x^{n}}{b + \sqrt{- 4 a c + b^{2}}} \right )}}{a^{2} x \sqrt{\frac{2 c x^{n}}{b - \sqrt{- 4 a c + b^{2}}} + 1} \sqrt{\frac{2 c x^{n}}{b + \sqrt{- 4 a c + b^{2}}} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(a + b*x**n + c*x**(2*n))*appellf1(-1/n, 3/2, 3/2, (n - 1)/n, -2*c*x**n/(b
- sqrt(-4*a*c + b**2)), -2*c*x**n/(b + sqrt(-4*a*c + b**2)))/(a**2*x*sqrt(2*c*x*
*n/(b - sqrt(-4*a*c + b**2)) + 1)*sqrt(2*c*x**n/(b + sqrt(-4*a*c + b**2)) + 1))

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Mathematica [B]  time = 6.18849, size = 2225, normalized size = 14.64 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[1/(x^2*(a + b*x^n + c*x^(2*n))^(3/2)),x]

[Out]

(2*(-b^2 + 2*a*c - b*c*x^n))/(a*(-b^2 + 4*a*c)*n*x*Sqrt[a + b*x^n + c*x^(2*n)])
+ (8*a*b*c*(-1 + 2*n)*x^(-1 + n)*(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)*(b + Sqrt[b^2
- 4*a*c] + 2*c*x^n)*AppellF1[(-1 + n)/n, 1/2, 1/2, 2 - n^(-1), (-2*c*x^n)/(b +
Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])])/((-b^2 + 4*a*c)*(b - Sq
rt[b^2 - 4*a*c])*(b + Sqrt[b^2 - 4*a*c])*(-1 + n)*n*(a + x^n*(b + c*x^n))^(3/2)*
((b + Sqrt[b^2 - 4*a*c])*n*x^n*AppellF1[2 - n^(-1), 1/2, 3/2, 3 - n^(-1), (-2*c*
x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])] - (-b + Sqrt[b
^2 - 4*a*c])*n*x^n*AppellF1[2 - n^(-1), 3/2, 1/2, 3 - n^(-1), (-2*c*x^n)/(b + Sq
rt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])] + 4*a*(1 - 2*n)*AppellF1[(
-1 + n)/n, 1/2, 1/2, 2 - n^(-1), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(
-b + Sqrt[b^2 - 4*a*c])])) + (4*a*b^2*(-1 + n)*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^n
)*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)*AppellF1[-n^(-1), 1/2, 1/2, (-1 + n)/n, (-2*
c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])])/((-b^2 + 4*
a*c)*(b - Sqrt[b^2 - 4*a*c])*(b + Sqrt[b^2 - 4*a*c])*x*(a + x^n*(b + c*x^n))^(3/
2)*(-4*a*(-1 + n)*AppellF1[-n^(-1), 1/2, 1/2, (-1 + n)/n, (-2*c*x^n)/(b + Sqrt[b
^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])] + n*x^n*((b + Sqrt[b^2 - 4*a*c
])*AppellF1[(-1 + n)/n, 1/2, 3/2, 2 - n^(-1), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])
, (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])] + (b - Sqrt[b^2 - 4*a*c])*AppellF1[(-1 + n
)/n, 3/2, 1/2, 2 - n^(-1), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + S
qrt[b^2 - 4*a*c])]))) - (16*a^2*c*(-1 + n)*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^n)*(b
+ Sqrt[b^2 - 4*a*c] + 2*c*x^n)*AppellF1[-n^(-1), 1/2, 1/2, (-1 + n)/n, (-2*c*x^
n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])])/((-b^2 + 4*a*c)
*(b - Sqrt[b^2 - 4*a*c])*(b + Sqrt[b^2 - 4*a*c])*x*(a + x^n*(b + c*x^n))^(3/2)*(
-4*a*(-1 + n)*AppellF1[-n^(-1), 1/2, 1/2, (-1 + n)/n, (-2*c*x^n)/(b + Sqrt[b^2 -
4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])] + n*x^n*((b + Sqrt[b^2 - 4*a*c])*A
ppellF1[(-1 + n)/n, 1/2, 3/2, 2 - n^(-1), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2
*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])] + (b - Sqrt[b^2 - 4*a*c])*AppellF1[(-1 + n)/n,
3/2, 1/2, 2 - n^(-1), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[
b^2 - 4*a*c])]))) + (8*a*b^2*(-1 + n)*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^n)*(b + Sq
rt[b^2 - 4*a*c] + 2*c*x^n)*AppellF1[-n^(-1), 1/2, 1/2, (-1 + n)/n, (-2*c*x^n)/(b
+ Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])])/((-b^2 + 4*a*c)*(b -
Sqrt[b^2 - 4*a*c])*(b + Sqrt[b^2 - 4*a*c])*n*x*(a + x^n*(b + c*x^n))^(3/2)*(-4*
a*(-1 + n)*AppellF1[-n^(-1), 1/2, 1/2, (-1 + n)/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*
a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])] + n*x^n*((b + Sqrt[b^2 - 4*a*c])*Appe
llF1[(-1 + n)/n, 1/2, 3/2, 2 - n^(-1), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*
x^n)/(-b + Sqrt[b^2 - 4*a*c])] + (b - Sqrt[b^2 - 4*a*c])*AppellF1[(-1 + n)/n, 3/
2, 1/2, 2 - n^(-1), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2
- 4*a*c])]))) - (16*a^2*c*(-1 + n)*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^n)*(b + Sqrt
[b^2 - 4*a*c] + 2*c*x^n)*AppellF1[-n^(-1), 1/2, 1/2, (-1 + n)/n, (-2*c*x^n)/(b +
Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])])/((-b^2 + 4*a*c)*(b - S
qrt[b^2 - 4*a*c])*(b + Sqrt[b^2 - 4*a*c])*n*x*(a + x^n*(b + c*x^n))^(3/2)*(-4*a*
(-1 + n)*AppellF1[-n^(-1), 1/2, 1/2, (-1 + n)/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*
c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])] + n*x^n*((b + Sqrt[b^2 - 4*a*c])*Appell
F1[(-1 + n)/n, 1/2, 3/2, 2 - n^(-1), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^
n)/(-b + Sqrt[b^2 - 4*a*c])] + (b - Sqrt[b^2 - 4*a*c])*AppellF1[(-1 + n)/n, 3/2,
1/2, 2 - n^(-1), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 -
4*a*c])])))

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Maple [F]  time = 0.02, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{2}} \left ( a+b{x}^{n}+c{x}^{2\,n} \right ) ^{-{\frac{3}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^2/(a+b*x^n+c*x^(2*n))^(3/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^(2*n) + b*x^n + a)^(3/2)*x^2),x, algorithm="maxima")

[Out]

integrate(1/((c*x^(2*n) + b*x^n + a)^(3/2)*x^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(1/((c*x^(2*n) + b*x^n + a)^(3/2)*x^2),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(1/x**2/(a+b*x**n+c*x**(2*n))**(3/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^(2*n) + b*x^n + a)^(3/2)*x^2),x, algorithm="giac")

[Out]

integrate(1/((c*x^(2*n) + b*x^n + a)^(3/2)*x^2), x)