3.42 \(\int \frac{\left (d+e x^n\right )^3}{a+c x^{2 n}} \, dx\)

Optimal. Leaf size=141 \[ \frac{e x^{n+1} \left (3 c d^2-a e^2\right ) \, _2F_1\left (1,\frac{n+1}{2 n};\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{a c (n+1)}+\frac{d x \left (c d^2-3 a e^2\right ) \, _2F_1\left (1,\frac{1}{2 n};\frac{1}{2} \left (2+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{a c}+\frac{3 d e^2 x}{c}+\frac{e^3 x^{n+1}}{c (n+1)} \]

[Out]

(3*d*e^2*x)/c + (e^3*x^(1 + n))/(c*(1 + n)) + (d*(c*d^2 - 3*a*e^2)*x*Hypergeomet
ric2F1[1, 1/(2*n), (2 + n^(-1))/2, -((c*x^(2*n))/a)])/(a*c) + (e*(3*c*d^2 - a*e^
2)*x^(1 + n)*Hypergeometric2F1[1, (1 + n)/(2*n), (3 + n^(-1))/2, -((c*x^(2*n))/a
)])/(a*c*(1 + n))

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Rubi [A]  time = 0.288941, antiderivative size = 141, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19 \[ \frac{e x^{n+1} \left (3 c d^2-a e^2\right ) \, _2F_1\left (1,\frac{n+1}{2 n};\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{a c (n+1)}+\frac{d x \left (c d^2-3 a e^2\right ) \, _2F_1\left (1,\frac{1}{2 n};\frac{1}{2} \left (2+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{a c}+\frac{3 d e^2 x}{c}+\frac{e^3 x^{n+1}}{c (n+1)} \]

Antiderivative was successfully verified.

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

[Out]

(3*d*e^2*x)/c + (e^3*x^(1 + n))/(c*(1 + n)) + (d*(c*d^2 - 3*a*e^2)*x*Hypergeomet
ric2F1[1, 1/(2*n), (2 + n^(-1))/2, -((c*x^(2*n))/a)])/(a*c) + (e*(3*c*d^2 - a*e^
2)*x^(1 + n)*Hypergeometric2F1[1, (1 + n)/(2*n), (3 + n^(-1))/2, -((c*x^(2*n))/a
)])/(a*c*(1 + n))

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Rubi in Sympy [A]  time = 27.7449, size = 151, normalized size = 1.07 \[ \frac{d^{3} x{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{1}{2 n} \\ \frac{n + \frac{1}{2}}{n} \end{matrix}\middle |{- \frac{c x^{2 n}}{a}} \right )}}{a} + \frac{3 d^{2} e x^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{n + 1}{2 n} \\ \frac{3 n + 1}{2 n} \end{matrix}\middle |{- \frac{c x^{2 n}}{a}} \right )}}{a \left (n + 1\right )} + \frac{3 d e^{2} x^{2 n + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{n + \frac{1}{2}}{n} \\ 2 + \frac{1}{2 n} \end{matrix}\middle |{- \frac{c x^{2 n}}{a}} \right )}}{a \left (2 n + 1\right )} + \frac{e^{3} x^{3 n + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{3 n + 1}{2 n} \\ \frac{5 n + 1}{2 n} \end{matrix}\middle |{- \frac{c x^{2 n}}{a}} \right )}}{a \left (3 n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d**3*x*hyper((1, 1/(2*n)), ((n + 1/2)/n,), -c*x**(2*n)/a)/a + 3*d**2*e*x**(n + 1
)*hyper((1, (n + 1)/(2*n)), ((3*n + 1)/(2*n),), -c*x**(2*n)/a)/(a*(n + 1)) + 3*d
*e**2*x**(2*n + 1)*hyper((1, (n + 1/2)/n), (2 + 1/(2*n),), -c*x**(2*n)/a)/(a*(2*
n + 1)) + e**3*x**(3*n + 1)*hyper((1, (3*n + 1)/(2*n)), ((5*n + 1)/(2*n),), -c*x
**(2*n)/a)/(a*(3*n + 1))

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Mathematica [A]  time = 0.188088, size = 128, normalized size = 0.91 \[ \frac{d (n+1) x \left (c d^2-3 a e^2\right ) \, _2F_1\left (1,\frac{1}{2 n};1+\frac{1}{2 n};-\frac{c x^{2 n}}{a}\right )+e x \left (x^n \left (3 c d^2-a e^2\right ) \, _2F_1\left (1,\frac{n+1}{2 n};\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )+a e \left (3 d (n+1)+e x^n\right )\right )}{a c (n+1)} \]

Antiderivative was successfully verified.

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

[Out]

(d*(c*d^2 - 3*a*e^2)*(1 + n)*x*Hypergeometric2F1[1, 1/(2*n), 1 + 1/(2*n), -((c*x
^(2*n))/a)] + e*x*(a*e*(3*d*(1 + n) + e*x^n) + (3*c*d^2 - a*e^2)*x^n*Hypergeomet
ric2F1[1, (1 + n)/(2*n), (3 + n^(-1))/2, -((c*x^(2*n))/a)]))/(a*c*(1 + n))

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Maple [F]  time = 0.105, size = 0, normalized size = 0. \[ \int{\frac{ \left ( d+e{x}^{n} \right ) ^{3}}{a+c{x}^{2\,n}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(3*d*e^2*(n + 1)*x + e^3*x*x^n)/(c*(n + 1)) - integrate(-(c*d^3 - 3*a*d*e^2 + (3
*c*d^2*e - a*e^3)*x^n)/(c^2*x^(2*n) + a*c), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{e^{3} x^{3 \, n} + 3 \, d e^{2} x^{2 \, n} + 3 \, d^{2} e x^{n} + d^{3}}{c x^{2 \, n} + a}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((e^3*x^(3*n) + 3*d*e^2*x^(2*n) + 3*d^2*e*x^n + d^3)/(c*x^(2*n) + a), x)

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Sympy [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((d+e*x**n)**3/(a+c*x**(2*n)),x)

[Out]

Exception raised: TypeError

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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