3.85 \(\int \left (d+e x^n\right ) \sqrt{a+b x^n+c x^{2 n}} \, dx\)

Optimal. Leaf size=292 \[ \frac{d x \sqrt{a+b x^n+c x^{2 n}} F_1\left (\frac{1}{n};-\frac{1}{2},-\frac{1}{2};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{\sqrt{\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1}}+\frac{e x^{n+1} \sqrt{a+b x^n+c x^{2 n}} F_1\left (1+\frac{1}{n};-\frac{1}{2},-\frac{1}{2};2+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{(n+1) \sqrt{\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1}} \]

[Out]

(e*x^(1 + n)*Sqrt[a + b*x^n + c*x^(2*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])])/(
(1 + n)*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])]) + (d*x*Sqrt[a + b*x^n + c*x^(2*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])])/(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 = 0.842371, antiderivative size = 292, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ \frac{d x \sqrt{a+b x^n+c x^{2 n}} F_1\left (\frac{1}{n};-\frac{1}{2},-\frac{1}{2};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{\sqrt{\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1}}+\frac{e x^{n+1} \sqrt{a+b x^n+c x^{2 n}} F_1\left (1+\frac{1}{n};-\frac{1}{2},-\frac{1}{2};2+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{(n+1) \sqrt{\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1}} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x^n)*Sqrt[a + b*x^n + c*x^(2*n)],x]

[Out]

(e*x^(1 + n)*Sqrt[a + b*x^n + c*x^(2*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])])/(
(1 + n)*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])]) + (d*x*Sqrt[a + b*x^n + c*x^(2*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])])/(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 = 80.1638, size = 262, normalized size = 0.9 \[ \frac{d x \sqrt{a + b x^{n} + c x^{2 n}} \operatorname{appellf_{1}}{\left (\frac{1}{n},- \frac{1}{2},- \frac{1}{2},1 + \frac{1}{n},- \frac{2 c x^{n}}{b - \sqrt{- 4 a c + b^{2}}},- \frac{2 c x^{n}}{b + \sqrt{- 4 a c + b^{2}}} \right )}}{\sqrt{\frac{2 c x^{n}}{b - \sqrt{- 4 a c + b^{2}}} + 1} \sqrt{\frac{2 c x^{n}}{b + \sqrt{- 4 a c + b^{2}}} + 1}} + \frac{e x^{n + 1} \sqrt{a + b x^{n} + c x^{2 n}} \operatorname{appellf_{1}}{\left (\frac{n + 1}{n},- \frac{1}{2},- \frac{1}{2},2 + \frac{1}{n},- \frac{2 c x^{n}}{b - \sqrt{- 4 a c + b^{2}}},- \frac{2 c x^{n}}{b + \sqrt{- 4 a c + b^{2}}} \right )}}{\left (n + 1\right ) \sqrt{\frac{2 c x^{n}}{b - \sqrt{- 4 a c + b^{2}}} + 1} \sqrt{\frac{2 c x^{n}}{b + \sqrt{- 4 a c + b^{2}}} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d*x*sqrt(a + b*x**n + c*x**(2*n))*appellf1(1/n, -1/2, -1/2, 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)) + e*x
**(n + 1)*sqrt(a + b*x**n + c*x**(2*n))*appellf1((n + 1)/n, -1/2, -1/2, 2 + 1/n,
-2*c*x**n/(b - sqrt(-4*a*c + b**2)), -2*c*x**n/(b + sqrt(-4*a*c + b**2)))/((n +
1)*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 = 6.25912, size = 3778, normalized size = 12.94 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(d + e*x^n)*Sqrt[a + b*x^n + c*x^(2*n)],x]

[Out]

Sqrt[a + b*x^n + c*x^(2*n)]*(((2*c*d + 4*c*d*n + b*e*n)*x)/(2*c*(1 + n)*(1 + 2*n
)) + (e*x^(1 + n))/(1 + 2*n)) - (2*a^2*b*d*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^(-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*(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])]))) - (4*a^3*e*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^(-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*(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 + Sq
rt[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])]))) + (2*a^2*b^2*e*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^(-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*(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])]))) - (4*a^2*b*d*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*(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])]))) - (4*a^3*e*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*(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])]))) + (a^2*b^2*e*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*(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])*Appell
F1[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])]))) - (4*a^3*d*n*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)*(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])])) + (2*a^3*b*e*n*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])])/(c*(b - Sqrt[b^2 - 4*a*c])*(
b + Sqrt[b^2 - 4*a*c])*(1 + 2*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])])
) - (8*a^3*d*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)*(a + x^n*(b + c*x^n))^(3/2)*((b + Sqrt[b^2 - 4*a*c])*n*x^n*App
ellF1[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])]))

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Maple [F]  time = 0.087, size = 0, normalized size = 0. \[ \int \left ( d+e{x}^{n} \right ) \sqrt{a+b{x}^{n}+c{x}^{2\,n}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(c*x^(2*n) + b*x^n + a)*(e*x^n + d), 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(sqrt(c*x^(2*n) + b*x^n + a)*(e*x^n + d),x, algorithm="fricas")

[Out]

Exception raised: TypeError

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \left (d + e x^{n}\right ) \sqrt{a + b x^{n} + c x^{2 n}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d+e*x**n)*(a+b*x**n+c*x**(2*n))**(1/2),x)

[Out]

Integral((d + e*x**n)*sqrt(a + b*x**n + c*x**(2*n)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(c*x^(2*n) + b*x^n + a)*(e*x^n + d), x)