3.200 \(\int \frac{\left (c+d x^n\right )^3}{a+b x^n} \, dx\)

Optimal. Leaf size=173 \[ \frac{d x \left (a^2 d^2 \left (2 n^2+3 n+1\right )-a b c d \left (6 n^2+7 n+2\right )+b^2 c^2 \left (6 n^2+4 n+1\right )\right )}{b^3 (n+1) (2 n+1)}+\frac{x (b c-a d)^3 \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a b^3}-\frac{d x \left (c+d x^n\right ) (a d (2 n+1)-b (4 c n+c))}{b^2 (n+1) (2 n+1)}+\frac{d x \left (c+d x^n\right )^2}{b (2 n+1)} \]

[Out]

(d*(a^2*d^2*(1 + 3*n + 2*n^2) + b^2*c^2*(1 + 4*n + 6*n^2) - a*b*c*d*(2 + 7*n + 6
*n^2))*x)/(b^3*(1 + n)*(1 + 2*n)) - (d*(a*d*(1 + 2*n) - b*(c + 4*c*n))*x*(c + d*
x^n))/(b^2*(1 + n)*(1 + 2*n)) + (d*x*(c + d*x^n)^2)/(b*(1 + 2*n)) + ((b*c - a*d)
^3*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), -((b*x^n)/a)])/(a*b^3)

_______________________________________________________________________________________

Rubi [A]  time = 0.614258, antiderivative size = 173, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ \frac{d x \left (a^2 d^2 \left (2 n^2+3 n+1\right )-a b c d \left (6 n^2+7 n+2\right )+b^2 c^2 \left (6 n^2+4 n+1\right )\right )}{b^3 (n+1) (2 n+1)}+\frac{x (b c-a d)^3 \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a b^3}-\frac{d x \left (c+d x^n\right ) (a d (2 n+1)-b (4 c n+c))}{b^2 (n+1) (2 n+1)}+\frac{d x \left (c+d x^n\right )^2}{b (2 n+1)} \]

Antiderivative was successfully verified.

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

[Out]

(d*(a^2*d^2*(1 + 3*n + 2*n^2) + b^2*c^2*(1 + 4*n + 6*n^2) - a*b*c*d*(2 + 7*n + 6
*n^2))*x)/(b^3*(1 + n)*(1 + 2*n)) - (d*(a*d*(1 + 2*n) - b*(c + 4*c*n))*x*(c + d*
x^n))/(b^2*(1 + n)*(1 + 2*n)) + (d*x*(c + d*x^n)^2)/(b*(1 + 2*n)) + ((b*c - a*d)
^3*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), -((b*x^n)/a)])/(a*b^3)

_______________________________________________________________________________________

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)**3/(a+b*x**n),x)

[Out]

Timed out

_______________________________________________________________________________________

Mathematica [A]  time = 0.50718, size = 104, normalized size = 0.6 \[ x \left (\frac{(b c-a d)^3 \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a b^3}+\frac{(a d-b c)^3}{a b^3}+\frac{d^2 x^n (3 b c-a d)}{b^2 (n+1)}+\frac{c^3}{a}+\frac{d^3 x^{2 n}}{2 b n+b}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [F]  time = 0.068, size = 0, normalized size = 0. \[ \int{\frac{ \left ( c+d{x}^{n} \right ) ^{3}}{a+b{x}^{n}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c+d*x^n)^3/(a+b*x^n),x)

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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)**3/(a+b*x**n),x)

[Out]

Exception raised: TypeError

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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