3.322 \(\int (a+b x^n)^2 (c+d x^n)^{-3-\frac {1}{n}} \, dx\)

Optimal. Leaf size=116 \[ \frac {2 a^2 n^2 x \left (c+d x^n\right )^{-1/n}}{c^3 (n+1) (2 n+1)}+\frac {2 a n x \left (a+b x^n\right ) \left (c+d x^n\right )^{-\frac {1}{n}-1}}{c^2 (n+1) (2 n+1)}+\frac {x \left (a+b x^n\right )^2 \left (c+d x^n\right )^{-\frac {1}{n}-2}}{c (2 n+1)} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.04, antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {378, 191} \[ \frac {2 a^2 n^2 x \left (c+d x^n\right )^{-1/n}}{c^3 (n+1) (2 n+1)}+\frac {2 a n x \left (a+b x^n\right ) \left (c+d x^n\right )^{-\frac {1}{n}-1}}{c^2 (n+1) (2 n+1)}+\frac {x \left (a+b x^n\right )^2 \left (c+d x^n\right )^{-\frac {1}{n}-2}}{c (2 n+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.08, size = 113, normalized size = 0.97 \[ \frac {x \left (c+d x^n\right )^{-\frac {1}{n}-2} \left (a^2 \left (c^2 \left (2 n^2+3 n+1\right )+2 c d n (2 n+1) x^n+2 d^2 n^2 x^{2 n}\right )+2 a b c x^n \left (2 c n+c+d n x^n\right )+b^2 c^2 (n+1) x^{2 n}\right )}{c^3 (n+1) (2 n+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 1.23, size = 231, normalized size = 1.99 \[ \frac {{\left (2 \, a^{2} d^{3} n^{2} + b^{2} c^{2} d + {\left (b^{2} c^{2} d + 2 \, a b c d^{2}\right )} n\right )} x x^{3 \, n} + {\left (6 \, a^{2} c d^{2} n^{2} + b^{2} c^{3} + 2 \, a b c^{2} d + {\left (b^{2} c^{3} + 6 \, a b c^{2} d + 2 \, a^{2} c d^{2}\right )} n\right )} x x^{2 \, n} + {\left (6 \, a^{2} c^{2} d n^{2} + 2 \, a b c^{3} + a^{2} c^{2} d + {\left (4 \, a b c^{3} + 5 \, a^{2} c^{2} d\right )} n\right )} x x^{n} + {\left (2 \, a^{2} c^{3} n^{2} + 3 \, a^{2} c^{3} n + a^{2} c^{3}\right )} x}{{\left (2 \, c^{3} n^{2} + 3 \, c^{3} n + c^{3}\right )} {\left (d x^{n} + c\right )}^{\frac {3 \, n + 1}{n}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^n)^2*(c+d*x^n)^(-3-1/n),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^n)^2*(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,x):;OUTPUT:Unab
le to divide, perhaps due to rounding error%%%{8,[1,0,4,3,1,3,2,0]%%%}+%%%{12,[1,0,4,2,1,3,2,0]%%%}+%%%{6,[1,0
,4,1,1,3,2,0]%%%}+%%%{1,[1,0,4,0,1,3,2,0]%%%}+%%%{8,[0,0,4,3,2,2,0,2]%%%}+%%%{8,[0,0,4,2,2,2,0,2]%%%}+%%%{8,[0
,0,4,2,1,3,1,1]%%%}+%%%{-4,[0,0,4,2,0,4,2,0]%%%}+%%%{2,[0,0,4,1,2,2,0,2]%%%}+%%%{8,[0,0,4,1,1,3,1,1]%%%}+%%%{-
4,[0,0,4,1,0,4,2,0]%%%}+%%%{2,[0,0,4,0,1,3,1,1]%%%}+%%%{-1,[0,0,4,0,0,4,2,0]%%%} / %%%{8,[0,0,5,3,2,3,0,0]%%%}
+%%%{12,[0,0,5,2,2,3,0,0]%%%}+%%%{6,[0,0,5,1,2,3,0,0]%%%}+%%%{1,[0,0,5,0,2,3,0,0]%%%} Error: Bad Argument Valu
e

________________________________________________________________________________________

maple [F]  time = 0.80, size = 0, normalized size = 0.00 \[ \int \left (b \,x^{n}+a \right )^{2} \left (d \,x^{n}+c \right )^{-\frac {1}{n}-3}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b x^{n} + a\right )}^{2} {\left (d x^{n} + c\right )}^{-\frac {1}{n} - 3}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^n)^2*(c+d*x^n)^(-3-1/n),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+b\,x^n\right )}^2}{{\left (c+d\,x^n\right )}^{\frac {1}{n}+3}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________