\(\int \frac {1}{(a+b x^n) (c+d x^n)^3} \, dx\) [304]

Optimal result

Integrand size = 19, antiderivative size = 210 \[ \int \frac {1}{\left (a+b x^n\right ) \left (c+d x^n\right )^3} \, dx=-\frac {d x}{2 c (b c-a d) n \left (c+d x^n\right )^2}-\frac {d (a d (1-2 n)-b (c-4 c n)) x}{2 c^2 (b c-a d)^2 n^2 \left (c+d x^n\right )}+\frac {b^3 x \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {b x^n}{a}\right )}{a (b c-a d)^3}-\frac {d \left (a^2 d^2 \left (1-3 n+2 n^2\right )-2 a b c d \left (1-4 n+3 n^2\right )+b^2 c^2 \left (1-5 n+6 n^2\right )\right ) x \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {d x^n}{c}\right )}{2 c^3 (b c-a d)^3 n^2} \]

[Out]

-1/2*d*x/c/(-a*d+b*c)/n/(c+d*x^n)^2-1/2*d*(a*d*(1-2*n)-b*(-4*c*n+c))*x/c^2/(-a*d+b*c)^2/n^2/(c+d*x^n)+b^3*x*hy
pergeom([1, 1/n],[1+1/n],-b*x^n/a)/a/(-a*d+b*c)^3-1/2*d*(a^2*d^2*(2*n^2-3*n+1)-2*a*b*c*d*(3*n^2-4*n+1)+b^2*c^2
*(6*n^2-5*n+1))*x*hypergeom([1, 1/n],[1+1/n],-d*x^n/c)/c^3/(-a*d+b*c)^3/n^2

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {425, 541, 536, 251} \[ \int \frac {1}{\left (a+b x^n\right ) \left (c+d x^n\right )^3} \, dx=-\frac {d x \left (a^2 d^2 \left (2 n^2-3 n+1\right )-2 a b c d \left (3 n^2-4 n+1\right )+b^2 c^2 \left (6 n^2-5 n+1\right )\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {d x^n}{c}\right )}{2 c^3 n^2 (b c-a d)^3}+\frac {b^3 x \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {b x^n}{a}\right )}{a (b c-a d)^3}+\frac {d x (b c (1-4 n)-a d (1-2 n))}{2 c^2 n^2 (b c-a d)^2 \left (c+d x^n\right )}-\frac {d x}{2 c n (b c-a d) \left (c+d x^n\right )^2} \]

[In]

Int[1/((a + b*x^n)*(c + d*x^n)^3),x]

[Out]

-1/2*(d*x)/(c*(b*c - a*d)*n*(c + d*x^n)^2) + (d*(b*c*(1 - 4*n) - a*d*(1 - 2*n))*x)/(2*c^2*(b*c - a*d)^2*n^2*(c
+ d*x^n)) + (b^3*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), -((b*x^n)/a)])/(a*(b*c - a*d)^3) - (d*(a^2*d^2*(1
- 3*n + 2*n^2) - 2*a*b*c*d*(1 - 4*n + 3*n^2) + b^2*c^2*(1 - 5*n + 6*n^2))*x*Hypergeometric2F1[1, n^(-1), 1 +
n^(-1), -((d*x^n)/c)])/(2*c^3*(b*c - a*d)^3*n^2)

Rule 251

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*x*Hypergeometric2F1[-p, 1/n, 1/n + 1, (-b)*(x^n/a)],
x] /; FreeQ[{a, b, n, p}, x] &&  !IGtQ[p, 0] &&  !IntegerQ[1/n] &&  !ILtQ[Simplify[1/n + p], 0] && (IntegerQ[p
] || GtQ[a, 0])

Rule 425

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[1/(a*n*(p + 1)*(b*c - a*d)), Int[(a + b*x^n)^(p + 1
)*(c + d*x^n)^q*Simp[b*c + n*(p + 1)*(b*c - a*d) + d*b*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c,
d, n, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &&  !( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomi
alQ[a, b, c, d, n, p, q, x]

Rule 536

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> Dist[(b*e - a*f
)/(b*c - a*d), Int[1/(a + b*x^n), x], x] - Dist[(d*e - c*f)/(b*c - a*d), Int[1/(c + d*x^n), x], x] /; FreeQ[{a
, b, c, d, e, f, n}, x]

Rule 541

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

Rubi steps \begin{align*} \text {integral}& = -\frac {d x}{2 c (b c-a d) n \left (c+d x^n\right )^2}+\frac {\int \frac {2 b c n+a (d-2 d n)+b d (1-2 n) x^n}{\left (a+b x^n\right ) \left (c+d x^n\right )^2} \, dx}{2 c (b c-a d) n} \\ & = -\frac {d x}{2 c (b c-a d) n \left (c+d x^n\right )^2}-\frac {d (a d (1-2 n)-b (c-4 c n)) x}{2 c^2 (b c-a d)^2 n^2 \left (c+d x^n\right )}+\frac {\int \frac {2 b^2 c^2 n^2+a^2 d^2 \left (1-3 n+2 n^2\right )-a b c d \left (1-5 n+4 n^2\right )-b d (b c (1-4 n)-a d (1-2 n)) (1-n) x^n}{\left (a+b x^n\right ) \left (c+d x^n\right )} \, dx}{2 c^2 (b c-a d)^2 n^2} \\ & = -\frac {d x}{2 c (b c-a d) n \left (c+d x^n\right )^2}-\frac {d (a d (1-2 n)-b (c-4 c n)) x}{2 c^2 (b c-a d)^2 n^2 \left (c+d x^n\right )}+\frac {b^3 \int \frac {1}{a+b x^n} \, dx}{(b c-a d)^3}-\frac {\left (d \left (a^2 d^2 \left (1-3 n+2 n^2\right )-2 a b c d \left (1-4 n+3 n^2\right )+b^2 c^2 \left (1-5 n+6 n^2\right )\right )\right ) \int \frac {1}{c+d x^n} \, dx}{2 c^2 (b c-a d)^3 n^2} \\ & = -\frac {d x}{2 c (b c-a d) n \left (c+d x^n\right )^2}-\frac {d (a d (1-2 n)-b (c-4 c n)) x}{2 c^2 (b c-a d)^2 n^2 \left (c+d x^n\right )}+\frac {b^3 x \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {b x^n}{a}\right )}{a (b c-a d)^3}-\frac {d \left (a^2 d^2 \left (1-3 n+2 n^2\right )-2 a b c d \left (1-4 n+3 n^2\right )+b^2 c^2 \left (1-5 n+6 n^2\right )\right ) x \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {d x^n}{c}\right )}{2 c^3 (b c-a d)^3 n^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.27 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (a+b x^n\right ) \left (c+d x^n\right )^3} \, dx=\frac {x \left (-a c^2 d (b c-a d)^2 n+a c d (b c-a d) (a d (-1+2 n)+b (c-4 c n)) \left (c+d x^n\right )+2 b^3 c^3 n^2 \left (c+d x^n\right )^2 \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {b x^n}{a}\right )-a d \left (a^2 d^2 \left (1-3 n+2 n^2\right )-2 a b c d \left (1-4 n+3 n^2\right )+b^2 c^2 \left (1-5 n+6 n^2\right )\right ) \left (c+d x^n\right )^2 \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {d x^n}{c}\right )\right )}{2 a c^3 (b c-a d)^3 n^2 \left (c+d x^n\right )^2} \]

[In]

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

[Out]

(x*(-(a*c^2*d*(b*c - a*d)^2*n) + a*c*d*(b*c - a*d)*(a*d*(-1 + 2*n) + b*(c - 4*c*n))*(c + d*x^n) + 2*b^3*c^3*n^
2*(c + d*x^n)^2*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), -((b*x^n)/a)] - a*d*(a^2*d^2*(1 - 3*n + 2*n^2) - 2*a*
b*c*d*(1 - 4*n + 3*n^2) + b^2*c^2*(1 - 5*n + 6*n^2))*(c + d*x^n)^2*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), -(
(d*x^n)/c)]))/(2*a*c^3*(b*c - a*d)^3*n^2*(c + d*x^n)^2)

Maple [F]

\[\int \frac {1}{\left (a +b \,x^{n}\right ) \left (c +d \,x^{n}\right )^{3}}d x\]

[In]

int(1/(a+b*x^n)/(c+d*x^n)^3,x)

[Out]

int(1/(a+b*x^n)/(c+d*x^n)^3,x)

Fricas [F]

\[ \int \frac {1}{\left (a+b x^n\right ) \left (c+d x^n\right )^3} \, dx=\int { \frac {1}{{\left (b x^{n} + a\right )} {\left (d x^{n} + c\right )}^{3}} \,d x } \]

[In]

integrate(1/(a+b*x^n)/(c+d*x^n)^3,x, algorithm="fricas")

[Out]

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

Sympy [F(-2)]

Exception generated. \[ \int \frac {1}{\left (a+b x^n\right ) \left (c+d x^n\right )^3} \, dx=\text {Exception raised: HeuristicGCDFailed} \]

[In]

integrate(1/(a+b*x**n)/(c+d*x**n)**3,x)

[Out]

Exception raised: HeuristicGCDFailed >> no luck

Maxima [F]

\[ \int \frac {1}{\left (a+b x^n\right ) \left (c+d x^n\right )^3} \, dx=\int { \frac {1}{{\left (b x^{n} + a\right )} {\left (d x^{n} + c\right )}^{3}} \,d x } \]

[In]

integrate(1/(a+b*x^n)/(c+d*x^n)^3,x, algorithm="maxima")

[Out]

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

Giac [F]

\[ \int \frac {1}{\left (a+b x^n\right ) \left (c+d x^n\right )^3} \, dx=\int { \frac {1}{{\left (b x^{n} + a\right )} {\left (d x^{n} + c\right )}^{3}} \,d x } \]

[In]

integrate(1/(a+b*x^n)/(c+d*x^n)^3,x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (a+b x^n\right ) \left (c+d x^n\right )^3} \, dx=\int \frac {1}{\left (a+b\,x^n\right )\,{\left (c+d\,x^n\right )}^3} \,d x \]

[In]

int(1/((a + b*x^n)*(c + d*x^n)^3),x)

[Out]

int(1/((a + b*x^n)*(c + d*x^n)^3), x)