3.321 \(\int (a+b x^n)^3 (c+d x^n)^{-4-\frac {1}{n}} \, dx\)

Optimal. Leaf size=178 \[ \frac {6 a^3 n^3 x \left (c+d x^n\right )^{-1/n}}{c^4 (n+1) (2 n+1) (3 n+1)}+\frac {6 a^2 n^2 x \left (a+b x^n\right ) \left (c+d x^n\right )^{-\frac {1}{n}-1}}{c^3 (n+1) (2 n+1) (3 n+1)}+\frac {3 a n x \left (a+b x^n\right )^2 \left (c+d x^n\right )^{-\frac {1}{n}-2}}{c^2 \left (6 n^2+5 n+1\right )}+\frac {x \left (a+b x^n\right )^3 \left (c+d x^n\right )^{-\frac {1}{n}-3}}{c (3 n+1)} \]

[Out]

x*(a+b*x^n)^3*(c+d*x^n)^(-3-1/n)/c/(1+3*n)+3*a*n*x*(a+b*x^n)^2*(c+d*x^n)^(-2-1/n)/c^2/(6*n^2+5*n+1)+6*a^2*n^2*
x*(a+b*x^n)*(c+d*x^n)^(-1-1/n)/c^3/(6*n^3+11*n^2+6*n+1)+6*a^3*n^3*x/c^4/(6*n^3+11*n^2+6*n+1)/((c+d*x^n)^(1/n))

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Rubi [A]  time = 0.09, antiderivative size = 178, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {378, 191} \[ \frac {6 a^2 n^2 x \left (a+b x^n\right ) \left (c+d x^n\right )^{-\frac {1}{n}-1}}{c^3 (n+1) (2 n+1) (3 n+1)}+\frac {6 a^3 n^3 x \left (c+d x^n\right )^{-1/n}}{c^4 (n+1) (2 n+1) (3 n+1)}+\frac {3 a n x \left (a+b x^n\right )^2 \left (c+d x^n\right )^{-\frac {1}{n}-2}}{c^2 \left (6 n^2+5 n+1\right )}+\frac {x \left (a+b x^n\right )^3 \left (c+d x^n\right )^{-\frac {1}{n}-3}}{c (3 n+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]  time = 0.14, size = 218, normalized size = 1.22 \[ \frac {x \left (c+d x^n\right )^{-\frac {1}{n}-3} \left (a^3 \left (c^3 \left (6 n^3+11 n^2+6 n+1\right )+3 c^2 d n \left (6 n^2+5 n+1\right ) x^n+6 c d^2 n^2 (3 n+1) x^{2 n}+6 d^3 n^3 x^{3 n}\right )+3 a^2 b c x^n \left (c^2 \left (6 n^2+5 n+1\right )+2 c d n (3 n+1) x^n+2 d^2 n^2 x^{2 n}\right )+3 a b^2 c^2 (n+1) x^{2 n} \left (3 c n+c+d n x^n\right )+b^3 c^3 \left (2 n^2+3 n+1\right ) x^{3 n}\right )}{c^4 (n+1) (2 n+1) (3 n+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(c + d*x^n)^(-3 - n^(-1))*(b^3*c^3*(1 + 3*n + 2*n^2)*x^(3*n) + 3*a*b^2*c^2*(1 + n)*x^(2*n)*(c + 3*c*n + d*n
*x^n) + 3*a^2*b*c*x^n*(c^2*(1 + 5*n + 6*n^2) + 2*c*d*n*(1 + 3*n)*x^n + 2*d^2*n^2*x^(2*n)) + a^3*(c^3*(1 + 6*n
+ 11*n^2 + 6*n^3) + 3*c^2*d*n*(1 + 5*n + 6*n^2)*x^n + 6*c*d^2*n^2*(1 + 3*n)*x^(2*n) + 6*d^3*n^3*x^(3*n))))/(c^
4*(1 + n)*(1 + 2*n)*(1 + 3*n))

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fricas [B]  time = 0.80, size = 478, normalized size = 2.69 \[ \frac {{\left (6 \, a^{3} d^{4} n^{3} + b^{3} c^{3} d + {\left (2 \, b^{3} c^{3} d + 3 \, a b^{2} c^{2} d^{2} + 6 \, a^{2} b c d^{3}\right )} n^{2} + 3 \, {\left (b^{3} c^{3} d + a b^{2} c^{2} d^{2}\right )} n\right )} x x^{4 \, n} + {\left (24 \, a^{3} c d^{3} n^{3} + b^{3} c^{4} + 3 \, a b^{2} c^{3} d + 2 \, {\left (b^{3} c^{4} + 6 \, a b^{2} c^{3} d + 12 \, a^{2} b c^{2} d^{2} + 3 \, a^{3} c d^{3}\right )} n^{2} + 3 \, {\left (b^{3} c^{4} + 5 \, a b^{2} c^{3} d + 2 \, a^{2} b c^{2} d^{2}\right )} n\right )} x x^{3 \, n} + 3 \, {\left (12 \, a^{3} c^{2} d^{2} n^{3} + a b^{2} c^{4} + a^{2} b c^{3} d + {\left (3 \, a b^{2} c^{4} + 12 \, a^{2} b c^{3} d + 7 \, a^{3} c^{2} d^{2}\right )} n^{2} + {\left (4 \, a b^{2} c^{4} + 7 \, a^{2} b c^{3} d + a^{3} c^{2} d^{2}\right )} n\right )} x x^{2 \, n} + {\left (24 \, a^{3} c^{3} d n^{3} + 3 \, a^{2} b c^{4} + a^{3} c^{3} d + 2 \, {\left (9 \, a^{2} b c^{4} + 13 \, a^{3} c^{3} d\right )} n^{2} + 3 \, {\left (5 \, a^{2} b c^{4} + 3 \, a^{3} c^{3} d\right )} n\right )} x x^{n} + {\left (6 \, a^{3} c^{4} n^{3} + 11 \, a^{3} c^{4} n^{2} + 6 \, a^{3} c^{4} n + a^{3} c^{4}\right )} x}{{\left (6 \, c^{4} n^{3} + 11 \, c^{4} n^{2} + 6 \, c^{4} n + c^{4}\right )} {\left (d x^{n} + c\right )}^{\frac {4 \, n + 1}{n}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^n)^3*(c+d*x^n)^(-4-1/n),x, algorithm="fricas")

[Out]

((6*a^3*d^4*n^3 + b^3*c^3*d + (2*b^3*c^3*d + 3*a*b^2*c^2*d^2 + 6*a^2*b*c*d^3)*n^2 + 3*(b^3*c^3*d + a*b^2*c^2*d
^2)*n)*x*x^(4*n) + (24*a^3*c*d^3*n^3 + b^3*c^4 + 3*a*b^2*c^3*d + 2*(b^3*c^4 + 6*a*b^2*c^3*d + 12*a^2*b*c^2*d^2
+ 3*a^3*c*d^3)*n^2 + 3*(b^3*c^4 + 5*a*b^2*c^3*d + 2*a^2*b*c^2*d^2)*n)*x*x^(3*n) + 3*(12*a^3*c^2*d^2*n^3 + a*b
^2*c^4 + a^2*b*c^3*d + (3*a*b^2*c^4 + 12*a^2*b*c^3*d + 7*a^3*c^2*d^2)*n^2 + (4*a*b^2*c^4 + 7*a^2*b*c^3*d + a^3
*c^2*d^2)*n)*x*x^(2*n) + (24*a^3*c^3*d*n^3 + 3*a^2*b*c^4 + a^3*c^3*d + 2*(9*a^2*b*c^4 + 13*a^3*c^3*d)*n^2 + 3*
(5*a^2*b*c^4 + 3*a^3*c^3*d)*n)*x*x^n + (6*a^3*c^4*n^3 + 11*a^3*c^4*n^2 + 6*a^3*c^4*n + a^3*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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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)^3*(c+d*x^n)^(-4-1/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%%%{81,[2,0,6,4,2,4,3,0]%%%}+%%%{108,[2,0,6,3,2,4,3,0]%%%}+%%%{54,[
2,0,6,2,2,4,3,0]%%%}+%%%{12,[2,0,6,1,2,4,3,0]%%%}+%%%{1,[2,0,6,0,2,4,3,0]%%%}+%%%{243,[1,0,6,4,2,4,2,1]%%%}+%%
%{-81,[1,0,6,4,1,5,3,0]%%%}+%%%{324,[1,0,6,3,2,4,2,1]%%%}+%%%{-108,[1,0,6,3,1,5,3,0]%%%}+%%%{162,[1,0,6,2,2,4,
2,1]%%%}+%%%{-54,[1,0,6,2,1,5,3,0]%%%}+%%%{36,[1,0,6,1,2,4,2,1]%%%}+%%%{-12,[1,0,6,1,1,5,3,0]%%%}+%%%{3,[1,0,6
,0,2,4,2,1]%%%}+%%%{-1,[1,0,6,0,1,5,3,0]%%%}+%%%{81,[0,0,6,4,3,3,0,3]%%%}+%%%{81,[0,0,6,3,3,3,0,3]%%%}+%%%{81,
[0,0,6,3,2,4,1,2]%%%}+%%%{-81,[0,0,6,3,1,5,2,1]%%%}+%%%{27,[0,0,6,3,0,6,3,0]%%%}+%%%{27,[0,0,6,2,3,3,0,3]%%%}+
%%%{81,[0,0,6,2,2,4,1,2]%%%}+%%%{-81,[0,0,6,2,1,5,2,1]%%%}+%%%{27,[0,0,6,2,0,6,3,0]%%%}+%%%{3,[0,0,6,1,3,3,0,3
]%%%}+%%%{27,[0,0,6,1,2,4,1,2]%%%}+%%%{-27,[0,0,6,1,1,5,2,1]%%%}+%%%{9,[0,0,6,1,0,6,3,0]%%%}+%%%{3,[0,0,6,0,2,
4,1,2]%%%}+%%%{-3,[0,0,6,0,1,5,2,1]%%%}+%%%{1,[0,0,6,0,0,6,3,0]%%%} / %%%{81,[0,0,7,4,3,4,0,0]%%%}+%%%{108,[0,
0,7,3,3,4,0,0]%%%}+%%%{54,[0,0,7,2,3,4,0,0]%%%}+%%%{12,[0,0,7,1,3,4,0,0]%%%}+%%%{1,[0,0,7,0,3,4,0,0]%%%} Error
: Bad Argument Value

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maple [F]  time = 0.83, size = 0, normalized size = 0.00 \[ \int \left (b \,x^{n}+a \right )^{3} \left (d \,x^{n}+c \right )^{-\frac {1}{n}-4}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b x^{n} + a\right )}^{3} {\left (d x^{n} + c\right )}^{-\frac {1}{n} - 4}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^n)^3*(c+d*x^n)^(-4-1/n),x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+b\,x^n\right )}^3}{{\left (c+d\,x^n\right )}^{\frac {1}{n}+4}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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)**3*(c+d*x**n)**(-4-1/n),x)

[Out]

Timed out

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