3.330 \(\int (a+b x^n)^2 (c+d x^n)^{-4-\frac{1}{n}} \, dx\)

Optimal. Leaf size=327 \[ -\frac{2 a^2 n^2 x \left (c+d x^n\right )^{-1/n} (3 a d n-b (3 c n+c))}{c^4 (n+1) (2 n+1) (3 n+1) (b c-a d)}-\frac{x \left (a+b x^n\right )^2 \left (c+d x^n\right )^{-\frac{1}{n}-2} (3 a d n-b (3 c n+c))}{c^2 \left (6 n^2+5 n+1\right ) (b c-a d)}-\frac{2 a n x \left (a+b x^n\right ) \left (c+d x^n\right )^{-\frac{1}{n}-1} (3 a d n-b (3 c n+c))}{c^3 (n+1) (2 n+1) (3 n+1) (b c-a d)}-\frac{x \left (a+b x^n\right )^3 \left (c+d x^n\right )^{-\frac{1}{n}-3} (3 a d n-b (3 c n+c))}{3 a c n (3 n+1) (b c-a d)}-\frac{b x \left (a+b x^n\right )^3 \left (c+d x^n\right )^{-\frac{1}{n}-3}}{3 a n (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.184194, antiderivative size = 327, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {382, 378, 191} \[ -\frac{2 a^2 n^2 x \left (c+d x^n\right )^{-1/n} (3 a d n-b (3 c n+c))}{c^4 (n+1) (2 n+1) (3 n+1) (b c-a d)}-\frac{x \left (a+b x^n\right )^2 \left (c+d x^n\right )^{-\frac{1}{n}-2} (3 a d n-b (3 c n+c))}{c^2 \left (6 n^2+5 n+1\right ) (b c-a d)}-\frac{2 a n x \left (a+b x^n\right ) \left (c+d x^n\right )^{-\frac{1}{n}-1} (3 a d n-b (3 c n+c))}{c^3 (n+1) (2 n+1) (3 n+1) (b c-a d)}-\frac{x \left (a+b x^n\right )^3 \left (c+d x^n\right )^{-\frac{1}{n}-3} (3 a d n-b (3 c n+c))}{3 a c n (3 n+1) (b c-a d)}-\frac{b x \left (a+b x^n\right )^3 \left (c+d x^n\right )^{-\frac{1}{n}-3}}{3 a n (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^n)^2*(c + d*x^n)^(-4 - n^(-1)),x]

[Out]

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

Rule 382

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Simp[(b*x*(a + b*x^n)^(p + 1)*(
c + d*x^n)^(q + 1))/(a*n*(p + 1)*(b*c - a*d)), x] + Dist[(b*c + n*(p + 1)*(b*c - a*d))/(a*n*(p + 1)*(b*c - a*d
)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n, q}, x] && NeQ[b*c - a*d, 0] && EqQ[
n*(p + q + 2) + 1, 0] && (LtQ[p, -1] ||  !LtQ[q, -1]) && NeQ[p, -1]

Rule 378

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1)*(c
+ d*x^n)^q)/(a*n*(p + 1)), x] - Dist[(c*q)/(a*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 1), x], x] /
; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[n*(p + q + 1) + 1, 0] && GtQ[q, 0] && NeQ[p, -1]

Rule 191

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^(p + 1))/a, x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rubi steps

\begin{align*} \int \left (a+b x^n\right )^2 \left (c+d x^n\right )^{-4-\frac{1}{n}} \, dx &=-\frac{b x \left (a+b x^n\right )^3 \left (c+d x^n\right )^{-3-\frac{1}{n}}}{3 a (b c-a d) n}+\frac{\left (3+\frac{b c}{b c n-a d n}\right ) \int \left (a+b x^n\right )^3 \left (c+d x^n\right )^{-4-\frac{1}{n}} \, dx}{3 a}\\ &=-\frac{b x \left (a+b x^n\right )^3 \left (c+d x^n\right )^{-3-\frac{1}{n}}}{3 a (b c-a d) n}+\frac{\left (3+\frac{b c}{b c n-a d n}\right ) x \left (a+b x^n\right )^3 \left (c+d x^n\right )^{-3-\frac{1}{n}}}{3 a c (1+3 n)}+\frac{\left (n \left (3+\frac{b c}{b c n-a d n}\right )\right ) \int \left (a+b x^n\right )^2 \left (c+d x^n\right )^{-3-\frac{1}{n}} \, dx}{c (1+3 n)}\\ &=-\frac{b x \left (a+b x^n\right )^3 \left (c+d x^n\right )^{-3-\frac{1}{n}}}{3 a (b c-a d) n}+\frac{\left (3+\frac{b c}{b c n-a d n}\right ) x \left (a+b x^n\right )^3 \left (c+d x^n\right )^{-3-\frac{1}{n}}}{3 a c (1+3 n)}+\frac{n \left (3+\frac{b c}{b c n-a d n}\right ) x \left (a+b x^n\right )^2 \left (c+d x^n\right )^{-2-\frac{1}{n}}}{c^2 \left (1+5 n+6 n^2\right )}+\frac{\left (2 a n^2 \left (3+\frac{b c}{b c n-a d n}\right )\right ) \int \left (a+b x^n\right ) \left (c+d x^n\right )^{-2-\frac{1}{n}} \, dx}{c^2 \left (1+5 n+6 n^2\right )}\\ &=-\frac{b x \left (a+b x^n\right )^3 \left (c+d x^n\right )^{-3-\frac{1}{n}}}{3 a (b c-a d) n}+\frac{\left (3+\frac{b c}{b c n-a d n}\right ) x \left (a+b x^n\right )^3 \left (c+d x^n\right )^{-3-\frac{1}{n}}}{3 a c (1+3 n)}+\frac{n \left (3+\frac{b c}{b c n-a d n}\right ) x \left (a+b x^n\right )^2 \left (c+d x^n\right )^{-2-\frac{1}{n}}}{c^2 \left (1+5 n+6 n^2\right )}+\frac{2 a n^2 \left (3+\frac{b c}{b c n-a d n}\right ) x \left (a+b x^n\right ) \left (c+d x^n\right )^{-1-\frac{1}{n}}}{c^3 (1+n) \left (1+5 n+6 n^2\right )}+\frac{\left (2 a^2 n^3 \left (3+\frac{b c}{b c n-a d n}\right )\right ) \int \left (c+d x^n\right )^{-1-\frac{1}{n}} \, dx}{c^3 (1+n) \left (1+5 n+6 n^2\right )}\\ &=-\frac{b x \left (a+b x^n\right )^3 \left (c+d x^n\right )^{-3-\frac{1}{n}}}{3 a (b c-a d) n}+\frac{\left (3+\frac{b c}{b c n-a d n}\right ) x \left (a+b x^n\right )^3 \left (c+d x^n\right )^{-3-\frac{1}{n}}}{3 a c (1+3 n)}+\frac{n \left (3+\frac{b c}{b c n-a d n}\right ) x \left (a+b x^n\right )^2 \left (c+d x^n\right )^{-2-\frac{1}{n}}}{c^2 \left (1+5 n+6 n^2\right )}+\frac{2 a n^2 \left (3+\frac{b c}{b c n-a d n}\right ) x \left (a+b x^n\right ) \left (c+d x^n\right )^{-1-\frac{1}{n}}}{c^3 (1+n) \left (1+5 n+6 n^2\right )}+\frac{2 a^2 n^3 \left (3+\frac{b c}{b c n-a d n}\right ) x \left (c+d x^n\right )^{-1/n}}{c^4 (1+n) \left (1+5 n+6 n^2\right )}\\ \end{align*}

Mathematica [C]  time = 0.198272, size = 153, normalized size = 0.47 \[ \frac{x \left (c+d x^n\right )^{-1/n} \left (\frac{d x^n}{c}+1\right )^{\frac{1}{n}} \left ((n+1) \left (a^2 (2 n+1) \, _2F_1\left (4+\frac{1}{n},\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right )+b^2 x^{2 n} \, _2F_1\left (2+\frac{1}{n},4+\frac{1}{n};3+\frac{1}{n};-\frac{d x^n}{c}\right )\right )+2 a b (2 n+1) x^n \, _2F_1\left (1+\frac{1}{n},4+\frac{1}{n};2+\frac{1}{n};-\frac{d x^n}{c}\right )\right )}{c^4 (n+1) (2 n+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^n)^2*(c + d*x^n)^(-4 - n^(-1)),x]

[Out]

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

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Maple [F]  time = 0.586, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b{x}^{n} \right ) ^{2} \left ( c+d{x}^{n} \right ) ^{-4-{n}^{-1}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^n)^2*(c+d*x^n)^(-4-1/n),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.65607, size = 819, normalized size = 2.5 \begin{align*} \frac{{\left (6 \, a^{2} d^{4} n^{3} + b^{2} c^{2} d^{2} n +{\left (b^{2} c^{2} d^{2} + 4 \, a b c d^{3}\right )} n^{2}\right )} x x^{4 \, n} +{\left (24 \, a^{2} c d^{3} n^{3} + b^{2} c^{3} d + 2 \,{\left (2 \, b^{2} c^{3} d + 8 \, a b c^{2} d^{2} + 3 \, a^{2} c d^{3}\right )} n^{2} +{\left (5 \, b^{2} c^{3} d + 4 \, a b c^{2} d^{2}\right )} n\right )} x x^{3 \, n} +{\left (36 \, a^{2} c^{2} d^{2} n^{3} + b^{2} c^{4} + 2 \, a b c^{3} d + 3 \,{\left (b^{2} c^{4} + 8 \, a b c^{3} d + 7 \, a^{2} c^{2} d^{2}\right )} n^{2} +{\left (4 \, b^{2} c^{4} + 14 \, a b c^{3} d + 3 \, a^{2} c^{2} d^{2}\right )} n\right )} x x^{2 \, n} +{\left (24 \, a^{2} c^{3} d n^{3} + 2 \, a b c^{4} + a^{2} c^{3} d + 2 \,{\left (6 \, a b c^{4} + 13 \, a^{2} c^{3} d\right )} n^{2} +{\left (10 \, a b c^{4} + 9 \, a^{2} c^{3} d\right )} n\right )} x x^{n} +{\left (6 \, a^{2} c^{4} n^{3} + 11 \, a^{2} c^{4} n^{2} + 6 \, a^{2} c^{4} n + a^{2} c^{4}\right )} x}{{\left (6 \, c^{4} n^{3} + 11 \, c^{4} n^{2} + 6 \, c^{4} n + c^{4}\right )}{\left (d x^{n} + c\right )}^{\frac{4 \, n + 1}{n}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^n)^2*(c+d*x^n)^(-4-1/n),x, algorithm="fricas")

[Out]

((6*a^2*d^4*n^3 + b^2*c^2*d^2*n + (b^2*c^2*d^2 + 4*a*b*c*d^3)*n^2)*x*x^(4*n) + (24*a^2*c*d^3*n^3 + b^2*c^3*d +
2*(2*b^2*c^3*d + 8*a*b*c^2*d^2 + 3*a^2*c*d^3)*n^2 + (5*b^2*c^3*d + 4*a*b*c^2*d^2)*n)*x*x^(3*n) + (36*a^2*c^2*
d^2*n^3 + b^2*c^4 + 2*a*b*c^3*d + 3*(b^2*c^4 + 8*a*b*c^3*d + 7*a^2*c^2*d^2)*n^2 + (4*b^2*c^4 + 14*a*b*c^3*d +
3*a^2*c^2*d^2)*n)*x*x^(2*n) + (24*a^2*c^3*d*n^3 + 2*a*b*c^4 + a^2*c^3*d + 2*(6*a*b*c^4 + 13*a^2*c^3*d)*n^2 + (
10*a*b*c^4 + 9*a^2*c^3*d)*n)*x*x^n + (6*a^2*c^4*n^3 + 11*a^2*c^4*n^2 + 6*a^2*c^4*n + a^2*c^4)*x)/((6*c^4*n^3 +
11*c^4*n^2 + 6*c^4*n + c^4)*(d*x^n + c)^((4*n + 1)/n))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^n)^2*(c+d*x^n)^(-4-1/n),x, algorithm="giac")

[Out]

Exception raised: TypeError