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

Optimal. Leaf size=140 \[ \frac{a x \sqrt{a+b x^n+c x^{2 n}} F_1\left (\frac{1}{n};-\frac{3}{2},-\frac{3}{2};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{\sqrt{\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1}} \]

[Out]

(a*x*Sqrt[a + b*x^n + c*x^(2*n)]*AppellF1[n^(-1), -3/2, -3/2, 1 + n^(-1), (-2*c*
x^n)/(b - Sqrt[b^2 - 4*a*c]), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/(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 = 0.224714, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{a x \sqrt{a+b x^n+c x^{2 n}} F_1\left (\frac{1}{n};-\frac{3}{2},-\frac{3}{2};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{\sqrt{\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1}} \]

Antiderivative was successfully verified.

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

[Out]

(a*x*Sqrt[a + b*x^n + c*x^(2*n)]*AppellF1[n^(-1), -3/2, -3/2, 1 + n^(-1), (-2*c*
x^n)/(b - Sqrt[b^2 - 4*a*c]), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/(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 = 37.0902, size = 126, normalized size = 0.9 \[ \frac{a x \sqrt{a + b x^{n} + c x^{2 n}} \operatorname{appellf_{1}}{\left (\frac{1}{n},- \frac{3}{2},- \frac{3}{2},1 + \frac{1}{n},- \frac{2 c x^{n}}{b - \sqrt{- 4 a c + b^{2}}},- \frac{2 c x^{n}}{b + \sqrt{- 4 a c + b^{2}}} \right )}}{\sqrt{\frac{2 c x^{n}}{b - \sqrt{- 4 a c + b^{2}}} + 1} \sqrt{\frac{2 c x^{n}}{b + \sqrt{- 4 a c + b^{2}}} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*x*sqrt(a + b*x**n + c*x**(2*n))*appellf1(1/n, -3/2, -3/2, 1 + 1/n, -2*c*x**n/(
b - sqrt(-4*a*c + b**2)), -2*c*x**n/(b + sqrt(-4*a*c + b**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 = 6.32739, size = 3058, normalized size = 21.84 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

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

[Out]

Sqrt[a + b*x^n + c*x^(2*n)]*(((4*a*c + 24*a*c*n + 3*b^2*n^2 + 32*a*c*n^2)*x)/(4*
c*(1 + n)*(1 + 2*n)*(1 + 3*n)) + (b*(2 + 7*n)*x^(1 + n))/(2*(1 + 2*n)*(1 + 3*n))
+ (c*x^(1 + 2*n))/(1 + 3*n)) - (12*a^3*b*n^2*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^(-1), 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 - Sqrt[b^2 - 4*a*c])*(b + Sqrt[b^2 - 4*a*c])*(1 + n)^2*(1 + 3*n)*(a + x^n*(b
+ c*x^n))^(3/2)*(-4*(a + 2*a*n)*AppellF1[1 + n^(-1), 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])] + n*x^n*((b
+ Sqrt[b^2 - 4*a*c])*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]
)*AppellF1[2 + n^(-1), 3/2, 1/2, 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])]))) + (3*a^2*b^3*n^2*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^(-1), 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])])/(c*(b - Sqrt[b^2 - 4*a*c])*(b + Sqrt[b^2 - 4*a*c])*(1 + n)^2*(1 + 3*n
)*(a + x^n*(b + c*x^n))^(3/2)*(-4*(a + 2*a*n)*AppellF1[1 + n^(-1), 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])]
+ n*x^n*((b + Sqrt[b^2 - 4*a*c])*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])*AppellF1[2 + n^(-1), 3/2, 1/2, 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])]))) - (18*a^3*b*n^3*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^(-1), 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 - Sqrt[b^2 - 4*a*c])*(b + Sqrt[b^2 - 4*a*c])*(1 + n)^
2*(1 + 3*n)*(a + x^n*(b + c*x^n))^(3/2)*(-4*(a + 2*a*n)*AppellF1[1 + n^(-1), 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])] + n*x^n*((b + Sqrt[b^2 - 4*a*c])*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])*AppellF1[2 + n^(-1), 3/2, 1/2, 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])]))) + (3*a^2*b^3*n^3*
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)*Ap
pellF1[1 + n^(-1), 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])])/(2*c*(b - Sqrt[b^2 - 4*a*c])*(b + Sqrt[b^2 - 4
*a*c])*(1 + n)^2*(1 + 3*n)*(a + x^n*(b + c*x^n))^(3/2)*(-4*(a + 2*a*n)*AppellF1[
1 + n^(-1), 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])] + n*x^n*((b + Sqrt[b^2 - 4*a*c])*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])*AppellF1[2 + n^(-1), 3/2, 1/2, 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])]))) -
(12*a^4*n^2*x*(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^(-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])*(b + Sqrt[b^2 - 4*a*c
])*(1 + 2*n)*(1 + 3*n)*(a + x^n*(b + c*x^n))^(3/2)*((b + Sqrt[b^2 - 4*a*c])*n*x^
n*AppellF1[1 + n^(-1), 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])*n*x^n*AppellF1[1
+ n^(-1), 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])] - 4*a*(1 + n)*AppellF1[n^(-1), 1/2, 1/2, 1 + n^(-1), (-
2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])])) + (3*a^3
*b^2*n^2*x*(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)*A
ppellF1[n^(-1), 1/2, 1/2, 1 + n^(-1), (-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]
)*(1 + 2*n)*(1 + 3*n)*(a + x^n*(b + c*x^n))^(3/2)*((b + Sqrt[b^2 - 4*a*c])*n*x^n
*AppellF1[1 + n^(-1), 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])*n*x^n*AppellF1[1
+ n^(-1), 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])] - 4*a*(1 + n)*AppellF1[n^(-1), 1/2, 1/2, 1 + n^(-1), (-2
*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])])) - (24*a^4
*n^3*x*(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)*Appel
lF1[n^(-1), 1/2, 1/2, 1 + 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])*(b + Sqrt[b^2 - 4*a*c])*(1 +
2*n)*(1 + 3*n)*(a + x^n*(b + c*x^n))^(3/2)*((b + Sqrt[b^2 - 4*a*c])*n*x^n*Appel
lF1[1 + n^(-1), 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])*n*x^n*AppellF1[1 + n^(-
1), 3/2, 1/2, 2 + n^(-1), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sq
rt[b^2 - 4*a*c])] - 4*a*(1 + n)*AppellF1[n^(-1), 1/2, 1/2, 1 + 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 = 0.071, size = 0, normalized size = 0. \[ \int \left ( a+b{x}^{n}+c{x}^{2\,n} \right ) ^{{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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