\(\int (a+b x^n) (c+d x^n)^{-3-\frac {1}{n}} \, dx\) [331]

Optimal result

Integrand size = 23, antiderivative size = 127 \[ \int \left (a+b x^n\right ) \left (c+d x^n\right )^{-3-\frac {1}{n}} \, dx=-\frac {(b c-a d) x \left (c+d x^n\right )^{-2-\frac {1}{n}}}{c d (1+2 n)}+\frac {(b c+2 a d n) x \left (c+d x^n\right )^{-1-\frac {1}{n}}}{c^2 d (1+n) (1+2 n)}+\frac {n (b c+2 a d n) x \left (c+d x^n\right )^{-1/n}}{c^3 d (1+n) (1+2 n)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {393, 198, 197} \[ \int \left (a+b x^n\right ) \left (c+d x^n\right )^{-3-\frac {1}{n}} \, dx=\frac {n x \left (c+d x^n\right )^{-1/n} (2 a d n+b c)}{c^3 d (n+1) (2 n+1)}+\frac {x \left (c+d x^n\right )^{-\frac {1}{n}-1} (2 a d n+b c)}{c^2 d (n+1) (2 n+1)}-\frac {x (b c-a d) \left (c+d x^n\right )^{-\frac {1}{n}-2}}{c d (2 n+1)} \]

[In]

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

[Out]

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

Rule 197

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]

Rule 198

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

Rule 393

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

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

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 0.21 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.74 \[ \int \left (a+b x^n\right ) \left (c+d x^n\right )^{-3-\frac {1}{n}} \, dx=\frac {x \left (c+d x^n\right )^{-1/n} \left (1+\frac {d x^n}{c}\right )^{\frac {1}{n}} \left (b c \operatorname {Hypergeometric2F1}\left (2+\frac {1}{n},\frac {1}{n},1+\frac {1}{n},-\frac {d x^n}{c}\right )+(-b c+a d) \operatorname {Hypergeometric2F1}\left (3+\frac {1}{n},\frac {1}{n},1+\frac {1}{n},-\frac {d x^n}{c}\right )\right )}{c^3 d} \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(433\) vs. \(2(127)=254\).

Time = 4.40 (sec) , antiderivative size = 434, normalized size of antiderivative = 3.42

method	result	size
parallelrisch	\(\frac {2 x \,x^{3 n} \left (c +d \,x^{n}\right )^{-\frac {1+3 n}{n}} a \,d^{3} n^{2}+x \,x^{3 n} \left (c +d \,x^{n}\right )^{-\frac {1+3 n}{n}} b c \,d^{2} n +6 x \,x^{2 n} \left (c +d \,x^{n}\right )^{-\frac {1+3 n}{n}} a c \,d^{2} n^{2}+2 x \,x^{2 n} \left (c +d \,x^{n}\right )^{-\frac {1+3 n}{n}} a c \,d^{2} n +3 x \,x^{2 n} \left (c +d \,x^{n}\right )^{-\frac {1+3 n}{n}} b \,c^{2} d n +6 x \,x^{n} \left (c +d \,x^{n}\right )^{-\frac {1+3 n}{n}} a \,c^{2} d \,n^{2}+x \,x^{2 n} \left (c +d \,x^{n}\right )^{-\frac {1+3 n}{n}} b \,c^{2} d +5 x \,x^{n} \left (c +d \,x^{n}\right )^{-\frac {1+3 n}{n}} a \,c^{2} d n +2 x \,x^{n} \left (c +d \,x^{n}\right )^{-\frac {1+3 n}{n}} b \,c^{3} n +2 x \left (c +d \,x^{n}\right )^{-\frac {1+3 n}{n}} a \,c^{3} n^{2}+x \,x^{n} \left (c +d \,x^{n}\right )^{-\frac {1+3 n}{n}} a \,c^{2} d +x \,x^{n} \left (c +d \,x^{n}\right )^{-\frac {1+3 n}{n}} b \,c^{3}+3 x \left (c +d \,x^{n}\right )^{-\frac {1+3 n}{n}} a \,c^{3} n +x \left (c +d \,x^{n}\right )^{-\frac {1+3 n}{n}} a \,c^{3}}{c^{3} \left (2 n^{2}+3 n +1\right )}\)	\(434\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.36 \[ \int \left (a+b x^n\right ) \left (c+d x^n\right )^{-3-\frac {1}{n}} \, dx=\frac {{\left (2 \, a d^{3} n^{2} + b c d^{2} n\right )} x x^{3 \, n} + {\left (6 \, a c d^{2} n^{2} + b c^{2} d + {\left (3 \, b c^{2} d + 2 \, a c d^{2}\right )} n\right )} x x^{2 \, n} + {\left (6 \, a c^{2} d n^{2} + b c^{3} + a c^{2} d + {\left (2 \, b c^{3} + 5 \, a c^{2} d\right )} n\right )} x x^{n} + {\left (2 \, a c^{3} n^{2} + 3 \, a c^{3} n + a c^{3}\right )} x}{{\left (2 \, c^{3} n^{2} + 3 \, c^{3} n + c^{3}\right )} {\left (d x^{n} + c\right )}^{\frac {3 \, n + 1}{n}}} \]

[In]

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

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 959 vs. \(2 (105) = 210\).

Time = 2.48 (sec) , antiderivative size = 959, normalized size of antiderivative = 7.55 \[ \int \left (a+b x^n\right ) \left (c+d x^n\right )^{-3-\frac {1}{n}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

2*a*c**2*c**(1/n)*c**(-3 - 2/n)*n**2*x*gamma(1/n)/(c**2*n**3*(1 + d*x**n/c)**(1/n)*gamma(3 + 1/n) + 2*c*d*n**3
*x**n*(1 + d*x**n/c)**(1/n)*gamma(3 + 1/n) + d**2*n**3*x**(2*n)*(1 + d*x**n/c)**(1/n)*gamma(3 + 1/n)) + 3*a*c*
*2*c**(1/n)*c**(-3 - 2/n)*n*x*gamma(1/n)/(c**2*n**3*(1 + d*x**n/c)**(1/n)*gamma(3 + 1/n) + 2*c*d*n**3*x**n*(1
+ d*x**n/c)**(1/n)*gamma(3 + 1/n) + d**2*n**3*x**(2*n)*(1 + d*x**n/c)**(1/n)*gamma(3 + 1/n)) + a*c**2*c**(1/n)
*c**(-3 - 2/n)*x*gamma(1/n)/(c**2*n**3*(1 + d*x**n/c)**(1/n)*gamma(3 + 1/n) + 2*c*d*n**3*x**n*(1 + d*x**n/c)**
(1/n)*gamma(3 + 1/n) + d**2*n**3*x**(2*n)*(1 + d*x**n/c)**(1/n)*gamma(3 + 1/n)) + 4*a*c*c**(1/n)*c**(-3 - 2/n)
*d*n**2*x*x**n*gamma(1/n)/(c**2*n**3*(1 + d*x**n/c)**(1/n)*gamma(3 + 1/n) + 2*c*d*n**3*x**n*(1 + d*x**n/c)**(1
/n)*gamma(3 + 1/n) + d**2*n**3*x**(2*n)*(1 + d*x**n/c)**(1/n)*gamma(3 + 1/n)) + 2*a*c*c**(1/n)*c**(-3 - 2/n)*d
*n*x*x**n*gamma(1/n)/(c**2*n**3*(1 + d*x**n/c)**(1/n)*gamma(3 + 1/n) + 2*c*d*n**3*x**n*(1 + d*x**n/c)**(1/n)*g
amma(3 + 1/n) + d**2*n**3*x**(2*n)*(1 + d*x**n/c)**(1/n)*gamma(3 + 1/n)) + 2*a*c**(1/n)*c**(-3 - 2/n)*d**2*n**
2*x*x**(2*n)*gamma(1/n)/(c**2*n**3*(1 + d*x**n/c)**(1/n)*gamma(3 + 1/n) + 2*c*d*n**3*x**n*(1 + d*x**n/c)**(1/n
)*gamma(3 + 1/n) + d**2*n**3*x**(2*n)*(1 + d*x**n/c)**(1/n)*gamma(3 + 1/n)) + 2*b*c*c**(-3 - 1/n)*c**(1 + 1/n)
*n*(c/(d*x**n) + 1)**(-1 - 1/n)*gamma(1 + 1/n)/(c*d**(1 + 1/n)*n**2*gamma(3 + 1/n) + d*d**(1 + 1/n)*n**2*x**n*
gamma(3 + 1/n)) + b*c*c**(-3 - 1/n)*c**(1 + 1/n)*(c/(d*x**n) + 1)**(-1 - 1/n)*gamma(1 + 1/n)/(c*d**(1 + 1/n)*n
**2*gamma(3 + 1/n) + d*d**(1 + 1/n)*n**2*x**n*gamma(3 + 1/n)) + b*c**(-3 - 1/n)*c**(1 + 1/n)*d*n*x**n*(c/(d*x*
*n) + 1)**(-1 - 1/n)*gamma(1 + 1/n)/(c*d**(1 + 1/n)*n**2*gamma(3 + 1/n) + d*d**(1 + 1/n)*n**2*x**n*gamma(3 + 1
/n))

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-2)]

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

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Unable to divide, perhaps due to rounding error%%%{4,[0,0,2,2,1,1,0,1]%%%}+%%%{2,[0,0,2,1,1,1,0,1]%%%}+%%%{
2,[0,0,2,1,

Mupad [F(-1)]

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

[In]

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

[Out]

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