3.206 \(\int \frac{\left (c+d x^n\right )^4}{\left (a+b x^n\right )^2} \, dx\)

Optimal. Leaf size=341 \[ -\frac{x (b c-a d)^3 (b c (1-n)-a d (3 n+1)) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a^2 b^4 n}-\frac{d x \left (c+d x^n\right ) \left (a^2 d^2 \left (6 n^2+5 n+1\right )-2 a b c d \left (5 n^2+4 n+1\right )+b^2 c^2 \left (2 n^2+3 n+1\right )\right )}{a b^3 n (n+1) (2 n+1)}-\frac{d x \left (-a^3 d^3 \left (6 n^3+11 n^2+6 n+1\right )+a^2 b c d^2 \left (16 n^3+26 n^2+15 n+3\right )-a b^2 c^2 d \left (12 n^3+17 n^2+12 n+3\right )+b^3 c^3 \left (2 n^2+3 n+1\right )\right )}{a b^4 n (n+1) (2 n+1)}+\frac{d x \left (c+d x^n\right )^2 (a d (3 n+1)-b (2 c n+c))}{a b^2 n (2 n+1)}+\frac{x (b c-a d) \left (c+d x^n\right )^3}{a b n \left (a+b x^n\right )} \]

[Out]

-((d*(b^3*c^3*(1 + 3*n + 2*n^2) - a^3*d^3*(1 + 6*n + 11*n^2 + 6*n^3) - a*b^2*c^2
*d*(3 + 12*n + 17*n^2 + 12*n^3) + a^2*b*c*d^2*(3 + 15*n + 26*n^2 + 16*n^3))*x)/(
a*b^4*n*(1 + n)*(1 + 2*n))) - (d*(b^2*c^2*(1 + 3*n + 2*n^2) - 2*a*b*c*d*(1 + 4*n
+ 5*n^2) + a^2*d^2*(1 + 5*n + 6*n^2))*x*(c + d*x^n))/(a*b^3*n*(1 + n)*(1 + 2*n)
) + (d*(a*d*(1 + 3*n) - b*(c + 2*c*n))*x*(c + d*x^n)^2)/(a*b^2*n*(1 + 2*n)) + ((
b*c - a*d)*x*(c + d*x^n)^3)/(a*b*n*(a + b*x^n)) - ((b*c - a*d)^3*(b*c*(1 - n) -
a*d*(1 + 3*n))*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), -((b*x^n)/a)])/(a^2*b^
4*n)

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Rubi [A]  time = 1.2872, antiderivative size = 341, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ -\frac{x (b c-a d)^3 (b c (1-n)-a d (3 n+1)) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a^2 b^4 n}-\frac{d x \left (c+d x^n\right ) \left (a^2 d^2 \left (6 n^2+5 n+1\right )-2 a b c d \left (5 n^2+4 n+1\right )+b^2 c^2 \left (2 n^2+3 n+1\right )\right )}{a b^3 n (n+1) (2 n+1)}-\frac{d x \left (-a^3 d^3 \left (6 n^3+11 n^2+6 n+1\right )+a^2 b c d^2 \left (16 n^3+26 n^2+15 n+3\right )-a b^2 c^2 d \left (12 n^3+17 n^2+12 n+3\right )+b^3 c^3 \left (2 n^2+3 n+1\right )\right )}{a b^4 n (n+1) (2 n+1)}+\frac{d x \left (c+d x^n\right )^2 (a d (3 n+1)-b (2 c n+c))}{a b^2 n (2 n+1)}+\frac{x (b c-a d) \left (c+d x^n\right )^3}{a b n \left (a+b x^n\right )} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x^n)^4/(a + b*x^n)^2,x]

[Out]

-((d*(b^3*c^3*(1 + 3*n + 2*n^2) - a^3*d^3*(1 + 6*n + 11*n^2 + 6*n^3) - a*b^2*c^2
*d*(3 + 12*n + 17*n^2 + 12*n^3) + a^2*b*c*d^2*(3 + 15*n + 26*n^2 + 16*n^3))*x)/(
a*b^4*n*(1 + n)*(1 + 2*n))) - (d*(b^2*c^2*(1 + 3*n + 2*n^2) - 2*a*b*c*d*(1 + 4*n
+ 5*n^2) + a^2*d^2*(1 + 5*n + 6*n^2))*x*(c + d*x^n))/(a*b^3*n*(1 + n)*(1 + 2*n)
) + (d*(a*d*(1 + 3*n) - b*(c + 2*c*n))*x*(c + d*x^n)^2)/(a*b^2*n*(1 + 2*n)) + ((
b*c - a*d)*x*(c + d*x^n)^3)/(a*b*n*(a + b*x^n)) - ((b*c - a*d)^3*(b*c*(1 - n) -
a*d*(1 + 3*n))*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), -((b*x^n)/a)])/(a^2*b^
4*n)

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c+d*x**n)**4/(a+b*x**n)**2,x)

[Out]

Timed out

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Mathematica [A]  time = 0.337356, size = 217, normalized size = 0.64 \[ \frac{x \left (\frac{(b c-a d)^3 (a d (3 n+1)+b c (n-1)) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a^2 n}+\frac{(a d-b c)^3 (a d (3 n+1)+b c (n-1))}{a^2 n}+\frac{-a^4 d^4+4 a^3 b c d^3-6 a^2 b^2 c^2 d^2+4 a b^3 c^3 d+b^4 c^4 (n-1)}{a^2 n}+\frac{2 b d^3 x^n (2 b c-a d)}{n+1}+\frac{(b c-a d)^4}{a n \left (a+b x^n\right )}+\frac{b^2 d^4 x^{2 n}}{2 n+1}\right )}{b^4} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x^n)^4/(a + b*x^n)^2,x]

[Out]

(x*((4*a*b^3*c^3*d - 6*a^2*b^2*c^2*d^2 + 4*a^3*b*c*d^3 - a^4*d^4 + b^4*c^4*(-1 +
n))/(a^2*n) + ((-(b*c) + a*d)^3*(b*c*(-1 + n) + a*d*(1 + 3*n)))/(a^2*n) + (2*b*
d^3*(2*b*c - a*d)*x^n)/(1 + n) + (b^2*d^4*x^(2*n))/(1 + 2*n) + (b*c - a*d)^4/(a*
n*(a + b*x^n)) + ((b*c - a*d)^3*(b*c*(-1 + n) + a*d*(1 + 3*n))*Hypergeometric2F1
[1, n^(-1), 1 + n^(-1), -((b*x^n)/a)])/(a^2*n)))/b^4

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Maple [F]  time = 0.078, size = 0, normalized size = 0. \[ \int{\frac{ \left ( c+d{x}^{n} \right ) ^{4}}{ \left ( a+b{x}^{n} \right ) ^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c+d*x^n)^4/(a+b*x^n)^2,x)

[Out]

int((c+d*x^n)^4/(a+b*x^n)^2,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -{\left (a^{4} d^{4}{\left (3 \, n + 1\right )} - 4 \, a^{3} b c d^{3}{\left (2 \, n + 1\right )} + 6 \, a^{2} b^{2} c^{2} d^{2}{\left (n + 1\right )} - b^{4} c^{4}{\left (n - 1\right )} - 4 \, a b^{3} c^{3} d\right )} \int \frac{1}{a b^{5} n x^{n} + a^{2} b^{4} n}\,{d x} + \frac{{\left (n^{2} + n\right )} a b^{3} d^{4} x x^{3 \, n} +{\left (4 \,{\left (2 \, n^{2} + n\right )} a b^{3} c d^{3} -{\left (3 \, n^{2} + n\right )} a^{2} b^{2} d^{4}\right )} x x^{2 \, n} +{\left (6 \,{\left (2 \, n^{3} + 3 \, n^{2} + n\right )} a b^{3} c^{2} d^{2} - 4 \,{\left (4 \, n^{3} + 4 \, n^{2} + n\right )} a^{2} b^{2} c d^{3} +{\left (6 \, n^{3} + 5 \, n^{2} + n\right )} a^{3} b d^{4}\right )} x x^{n} +{\left ({\left (2 \, n^{2} + 3 \, n + 1\right )} b^{4} c^{4} - 4 \,{\left (2 \, n^{2} + 3 \, n + 1\right )} a b^{3} c^{3} d + 6 \,{\left (2 \, n^{3} + 5 \, n^{2} + 4 \, n + 1\right )} a^{2} b^{2} c^{2} d^{2} - 4 \,{\left (4 \, n^{3} + 8 \, n^{2} + 5 \, n + 1\right )} a^{3} b c d^{3} +{\left (6 \, n^{3} + 11 \, n^{2} + 6 \, n + 1\right )} a^{4} d^{4}\right )} x}{{\left (2 \, n^{3} + 3 \, n^{2} + n\right )} a b^{5} x^{n} +{\left (2 \, n^{3} + 3 \, n^{2} + n\right )} a^{2} b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^n + c)^4/(b*x^n + a)^2,x, algorithm="maxima")

[Out]

-(a^4*d^4*(3*n + 1) - 4*a^3*b*c*d^3*(2*n + 1) + 6*a^2*b^2*c^2*d^2*(n + 1) - b^4*
c^4*(n - 1) - 4*a*b^3*c^3*d)*integrate(1/(a*b^5*n*x^n + a^2*b^4*n), x) + ((n^2 +
n)*a*b^3*d^4*x*x^(3*n) + (4*(2*n^2 + n)*a*b^3*c*d^3 - (3*n^2 + n)*a^2*b^2*d^4)*
x*x^(2*n) + (6*(2*n^3 + 3*n^2 + n)*a*b^3*c^2*d^2 - 4*(4*n^3 + 4*n^2 + n)*a^2*b^2
*c*d^3 + (6*n^3 + 5*n^2 + n)*a^3*b*d^4)*x*x^n + ((2*n^2 + 3*n + 1)*b^4*c^4 - 4*(
2*n^2 + 3*n + 1)*a*b^3*c^3*d + 6*(2*n^3 + 5*n^2 + 4*n + 1)*a^2*b^2*c^2*d^2 - 4*(
4*n^3 + 8*n^2 + 5*n + 1)*a^3*b*c*d^3 + (6*n^3 + 11*n^2 + 6*n + 1)*a^4*d^4)*x)/((
2*n^3 + 3*n^2 + n)*a*b^5*x^n + (2*n^3 + 3*n^2 + n)*a^2*b^4)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{d^{4} x^{4 \, n} + 4 \, c d^{3} x^{3 \, n} + 6 \, c^{2} d^{2} x^{2 \, n} + 4 \, c^{3} d x^{n} + c^{4}}{b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^n + c)^4/(b*x^n + a)^2,x, algorithm="fricas")

[Out]

integral((d^4*x^(4*n) + 4*c*d^3*x^(3*n) + 6*c^2*d^2*x^(2*n) + 4*c^3*d*x^n + c^4)
/(b^2*x^(2*n) + 2*a*b*x^n + a^2), x)

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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((c+d*x**n)**4/(a+b*x**n)**2,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x^{n} + c\right )}^{4}}{{\left (b x^{n} + a\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^n + c)^4/(b*x^n + a)^2,x, algorithm="giac")

[Out]

integrate((d*x^n + c)^4/(b*x^n + a)^2, x)