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

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

[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))

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Rubi [A]  time = 0.0859997, antiderivative size = 178, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, 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 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 )^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*}

Mathematica [A]  time = 0.128798, size = 218, normalized size = 1.22 \[ \frac{x \left (c+d x^n\right )^{-\frac{1}{n}-3} \left (3 a^2 b c x^n \left (c^2 \left (6 n^2+5 n+1\right )+2 c d n (3 n+1) x^n+2 d^2 n^2 x^{2 n}\right )+a^3 \left (3 c^2 d n \left (6 n^2+5 n+1\right ) x^n+c^3 \left (6 n^3+11 n^2+6 n+1\right )+6 c d^2 n^2 (3 n+1) x^{2 n}+6 d^3 n^3 x^{3 n}\right )+3 a b^2 c^2 (n+1) x^{2 n} \left (3 c n+c+d n x^n\right )+b^3 c^3 \left (2 n^2+3 n+1\right ) x^{3 n}\right )}{c^4 (n+1) (2 n+1) (3 n+1)} \]

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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Maple [F]  time = 0.594, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b{x}^{n} \right ) ^{3} \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)^3*(c+d*x^n)^(-4-1/n),x)

[Out]

int((a+b*x^n)^3*(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 )}^{3}{\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)^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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Fricas [B]  time = 1.66309, size = 971, normalized size = 5.46 \begin{align*} \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}}} \end{align*}

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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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)**3*(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)^3*(c+d*x^n)^(-4-1/n),x, algorithm="giac")

[Out]

Exception raised: TypeError