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

Optimal. Leaf size=310 \[ -\frac{d x \left (c+d x^n\right ) \left (-a^2 d^2 \left (6 n^2+5 n+1\right )+2 a b c d (3 n+1)^2-b^2 c^2 \left (18 n^2+7 n+1\right )\right )}{b^3 (n+1) (2 n+1) (3 n+1)}-\frac{d x \left (a^3 d^3 \left (6 n^3+11 n^2+6 n+1\right )-a^2 b c d^2 \left (24 n^3+38 n^2+19 n+3\right )+a b^2 c^2 d \left (36 n^3+45 n^2+20 n+3\right )-b^3 c^3 \left (24 n^3+18 n^2+7 n+1\right )\right )}{b^4 (n+1) (2 n+1) (3 n+1)}+\frac{x (b c-a d)^4 \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a b^4}-\frac{d x \left (c+d x^n\right )^2 (a d (3 n+1)-b (6 c n+c))}{b^2 \left (6 n^2+5 n+1\right )}+\frac{d x \left (c+d x^n\right )^3}{b (3 n+1)} \]

[Out]

-((d*(a^3*d^3*(1 + 6*n + 11*n^2 + 6*n^3) - b^3*c^3*(1 + 7*n + 18*n^2 + 24*n^3) -
a^2*b*c*d^2*(3 + 19*n + 38*n^2 + 24*n^3) + a*b^2*c^2*d*(3 + 20*n + 45*n^2 + 36*
n^3))*x)/(b^4*(1 + n)*(1 + 2*n)*(1 + 3*n))) - (d*(2*a*b*c*d*(1 + 3*n)^2 - a^2*d^
2*(1 + 5*n + 6*n^2) - b^2*c^2*(1 + 7*n + 18*n^2))*x*(c + d*x^n))/(b^3*(1 + n)*(1
+ 2*n)*(1 + 3*n)) - (d*(a*d*(1 + 3*n) - b*(c + 6*c*n))*x*(c + d*x^n)^2)/(b^2*(1
+ 5*n + 6*n^2)) + (d*x*(c + d*x^n)^3)/(b*(1 + 3*n)) + ((b*c - a*d)^4*x*Hypergeo
metric2F1[1, n^(-1), 1 + n^(-1), -((b*x^n)/a)])/(a*b^4)

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Rubi [A]  time = 1.12801, antiderivative size = 310, 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 \[ -\frac{d x \left (c+d x^n\right ) \left (-a^2 d^2 \left (6 n^2+5 n+1\right )+2 a b c d (3 n+1)^2-b^2 c^2 \left (18 n^2+7 n+1\right )\right )}{b^3 (n+1) (2 n+1) (3 n+1)}-\frac{d x \left (a^3 d^3 \left (6 n^3+11 n^2+6 n+1\right )-a^2 b c d^2 \left (24 n^3+38 n^2+19 n+3\right )+a b^2 c^2 d \left (36 n^3+45 n^2+20 n+3\right )-b^3 c^3 \left (24 n^3+18 n^2+7 n+1\right )\right )}{b^4 (n+1) (2 n+1) (3 n+1)}+\frac{x (b c-a d)^4 \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a b^4}-\frac{d x \left (c+d x^n\right )^2 (a d (3 n+1)-b (6 c n+c))}{b^2 \left (6 n^2+5 n+1\right )}+\frac{d x \left (c+d x^n\right )^3}{b (3 n+1)} \]

Antiderivative was successfully verified.

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

[Out]

-((d*(a^3*d^3*(1 + 6*n + 11*n^2 + 6*n^3) - b^3*c^3*(1 + 7*n + 18*n^2 + 24*n^3) -
a^2*b*c*d^2*(3 + 19*n + 38*n^2 + 24*n^3) + a*b^2*c^2*d*(3 + 20*n + 45*n^2 + 36*
n^3))*x)/(b^4*(1 + n)*(1 + 2*n)*(1 + 3*n))) - (d*(2*a*b*c*d*(1 + 3*n)^2 - a^2*d^
2*(1 + 5*n + 6*n^2) - b^2*c^2*(1 + 7*n + 18*n^2))*x*(c + d*x^n))/(b^3*(1 + n)*(1
+ 2*n)*(1 + 3*n)) - (d*(a*d*(1 + 3*n) - b*(c + 6*c*n))*x*(c + d*x^n)^2)/(b^2*(1
+ 5*n + 6*n^2)) + (d*x*(c + d*x^n)^3)/(b*(1 + 3*n)) + ((b*c - a*d)^4*x*Hypergeo
metric2F1[1, n^(-1), 1 + n^(-1), -((b*x^n)/a)])/(a*b^4)

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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),x)

[Out]

Timed out

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Mathematica [A]  time = 0.185418, size = 146, normalized size = 0.47 \[ x \left (\frac{d^2 x^n \left (a^2 d^2-4 a b c d+6 b^2 c^2\right )}{b^3 (n+1)}+\frac{(b c-a d)^4 \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a b^4}-\frac{(b c-a d)^4}{a b^4}+\frac{d^3 x^{2 n} (4 b c-a d)}{b^2 (2 n+1)}+\frac{c^4}{a}+\frac{d^4 x^{3 n}}{3 b n+b}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.072, size = 0, normalized size = 0. \[ \int{\frac{ \left ( c+d{x}^{n} \right ) ^{4}}{a+b{x}^{n}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c+d*x^n)^4/(a+b*x^n),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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