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

Optimal. Leaf size=154 \[ -\frac{\sqrt{\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1} F_1\left (-\frac{2}{n};\frac{3}{2},\frac{3}{2};-\frac{2-n}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{2 a x^2 \sqrt{a+b x^n+c x^{2 n}}} \]

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

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Rubi [A]  time = 0.460117, antiderivative size = 154, 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 \[ -\frac{\sqrt{\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1} F_1\left (-\frac{2}{n};\frac{3}{2},\frac{3}{2};-\frac{2-n}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{2 a x^2 \sqrt{a+b x^n+c x^{2 n}}} \]

Antiderivative was successfully verified.

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

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Rubi in Sympy [A]  time = 41.4012, size = 131, normalized size = 0.85 \[ - \frac{\sqrt{a + b x^{n} + c x^{2 n}} \operatorname{appellf_{1}}{\left (- \frac{2}{n},\frac{3}{2},\frac{3}{2},\frac{n - 2}{n},- \frac{2 c x^{n}}{b - \sqrt{- 4 a c + b^{2}}},- \frac{2 c x^{n}}{b + \sqrt{- 4 a c + b^{2}}} \right )}}{2 a^{2} x^{2} \sqrt{\frac{2 c x^{n}}{b - \sqrt{- 4 a c + b^{2}}} + 1} \sqrt{\frac{2 c x^{n}}{b + \sqrt{- 4 a c + b^{2}}} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(a + b*x**n + c*x**(2*n))*appellf1(-2/n, 3/2, 3/2, (n - 2)/n, -2*c*x**n/(b
- sqrt(-4*a*c + b**2)), -2*c*x**n/(b + sqrt(-4*a*c + b**2)))/(2*a**2*x**2*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 = 6.20746, size = 2221, normalized size = 14.42 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[1/(x^3*(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^2*Sqrt[a + b*x^n + c*x^(2*n)]
) + (32*a*b*c*(-1 + n)*x^(-2 + n)*(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)*(b + Sqrt[b^
2 - 4*a*c] + 2*c*x^n)*AppellF1[(-2 + n)/n, 1/2, 1/2, 2 - 2/n, (-2*c*x^n)/(b + Sq
rt[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])*(-2 + n)*n*(a + x^n*(b + c*x^n))^(3/2)*((
b + Sqrt[b^2 - 4*a*c])*n*x^n*AppellF1[2 - 2/n, 1/2, 3/2, 3 - 2/n, (-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 - 2/n, 3/2, 1/2, 3 - 2/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c
]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])] - 8*a*(-1 + n)*AppellF1[(-2 + n)/n, 1/2,
1/2, 2 - 2/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*
a*c])])) + (2*a*b^2*(-2 + n)*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^n)*(b + Sqrt[b^2 -
4*a*c] + 2*c*x^n)*AppellF1[-2/n, 1/2, 1/2, (-2 + 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^2*(a + x^n*(b + c*x^n))^(3/2)*(-4*a*(-2 + n)*A
ppellF1[-2/n, 1/2, 1/2, (-2 + 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[(-2 + n)/n
, 1/2, 3/2, 2 - 2/n, (-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[(-2 + n)/n, 3/2, 1/2, 2 - 2/n, (
-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])]))) - (8*a
^2*c*(-2 + n)*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^n)*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^
n)*AppellF1[-2/n, 1/2, 1/2, (-2 + 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 + Sq
rt[b^2 - 4*a*c])*x^2*(a + x^n*(b + c*x^n))^(3/2)*(-4*a*(-2 + n)*AppellF1[-2/n, 1
/2, 1/2, (-2 + 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[(-2 + n)/n, 1/2, 3/2, 2 -
2/n, (-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[(-2 + n)/n, 3/2, 1/2, 2 - 2/n, (-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*(-2 + n)*(
-b + Sqrt[b^2 - 4*a*c] - 2*c*x^n)*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)*AppellF1[-2/
n, 1/2, 1/2, (-2 + n)/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqr
t[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^2*(a + x^n*(b + c*x^n))^(3/2)*(-4*a*(-2 + n)*AppellF1[-2/n, 1/2, 1/2, (-2
+ 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[(-2 + n)/n, 1/2, 3/2, 2 - 2/n, (-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[(-2 + n)/n, 3/2, 1/2, 2 - 2/n, (-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*(-2 + n)*(-b + Sqrt[b^
2 - 4*a*c] - 2*c*x^n)*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)*AppellF1[-2/n, 1/2, 1/2,
(-2 + 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^2*(a +
x^n*(b + c*x^n))^(3/2)*(-4*a*(-2 + n)*AppellF1[-2/n, 1/2, 1/2, (-2 + 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[(-2 + n)/n, 1/2, 3/2, 2 - 2/n, (-2*c*x^n)/(b + Sqr
t[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])] + (b - Sqrt[b^2 - 4*a*c])*A
ppellF1[(-2 + n)/n, 3/2, 1/2, 2 - 2/n, (-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. \[ \int{\frac{1}{{x}^{3}} \left ( a+b{x}^{n}+c{x}^{2\,n} \right ) ^{-{\frac{3}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/((c*x^(2*n) + b*x^n + a)^(3/2)*x^3), 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^3),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**3/(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. \[ \int \frac{1}{{\left (c x^{2 \, n} + b x^{n} + a\right )}^{\frac{3}{2}} x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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