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

Optimal. Leaf size=149 \[ \frac{a x^2 \sqrt{a+b x^n+c x^{2 n}} F_1\left (\frac{2}{n};-\frac{3}{2},-\frac{3}{2};\frac{n+2}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{2 \sqrt{\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1}} \]

[Out]

(a*x^2*Sqrt[a + b*x^n + c*x^(2*n)]*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*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])])

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Rubi [A]  time = 0.351895, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{a x^2 \sqrt{a+b x^n+c x^{2 n}} F_1\left (\frac{2}{n};-\frac{3}{2},-\frac{3}{2};\frac{n+2}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{2 \sqrt{\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1}} \]

Antiderivative was successfully verified.

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

[Out]

(a*x^2*Sqrt[a + b*x^n + c*x^(2*n)]*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*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])])

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Rubi in Sympy [A]  time = 30.011, size = 129, normalized size = 0.87 \[ \frac{a x^{2} \sqrt{a + b x^{n} + c x^{2 n}} \operatorname{appellf_{1}}{\left (\frac{2}{n},- \frac{3}{2},- \frac{3}{2},\frac{n + 2}{n},- \frac{2 c x^{n}}{b - \sqrt{- 4 a c + b^{2}}},- \frac{2 c x^{n}}{b + \sqrt{- 4 a c + b^{2}}} \right )}}{2 \sqrt{\frac{2 c x^{n}}{b - \sqrt{- 4 a c + b^{2}}} + 1} \sqrt{\frac{2 c x^{n}}{b + \sqrt{- 4 a c + b^{2}}} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*x**2*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*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 = 6.38399, size = 3165, normalized size = 21.24 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

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

[Out]

Sqrt[a + b*x^n + c*x^(2*n)]*(((16*a*c + 48*a*c*n + 3*b^2*n^2 + 32*a*c*n^2)*x^2)/
(8*c*(1 + n)*(2 + n)*(2 + 3*n)) + (b*(4 + 7*n)*x^(2 + n))/(4*(1 + n)*(2 + 3*n))
+ (c*x^(2 + 2*n))/(2 + 3*n)) - (24*a^3*b*n^2*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 + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])])/((b -
Sqrt[b^2 - 4*a*c])*(b + Sqrt[b^2 - 4*a*c])*(2 + n)^2*(2 + 3*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 + Sq
rt[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 + S
qrt[b^2 - 4*a*c])])) + (6*a^2*b^3*n^2*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 + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])])/(c*(b - Sqrt
[b^2 - 4*a*c])*(b + Sqrt[b^2 - 4*a*c])*(2 + n)^2*(2 + 3*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])])) - (18*a^3*b*n^3*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 + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])])/((b - Sqrt[b^2 - 4
*a*c])*(b + Sqrt[b^2 - 4*a*c])*(2 + n)^2*(2 + 3*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])])) + (3*a^2*b^3*n^3*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])])/(2*c*(b - Sqrt[b^2 - 4*a*
c])*(b + Sqrt[b^2 - 4*a*c])*(2 + n)^2*(2 + 3*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]
)])) - (6*a^4*n^2*x^2*(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 - Sqrt[b^2 - 4*a*c])*(b + Sqrt[b^2 - 4
*a*c])*(1 + n)*(2 + 3*n)*(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 + S
qrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])]))) + (3*a^3*b^2*n^2*x^2*(
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])])/(2*c*(b - Sqrt[b^2 - 4*a*c])*(b + Sqrt[b^2 - 4*a*c])*(1 + n)*(2 +
3*n)*(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])]))) - (6*a^4*n^3*x^2*(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 - Sq
rt[b^2 - 4*a*c])*(b + Sqrt[b^2 - 4*a*c])*(1 + n)*(2 + 3*n)*(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])])))

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Maple [F]  time = 0.074, size = 0, normalized size = 0. \[ \int x \left ( a+b{x}^{n}+c{x}^{2\,n} \right ) ^{{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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