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 - 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))

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Rubi [A]  time = 0.0360545, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, 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 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 )^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 = 0.073117, 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))

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Maple [F]  time = 0.577, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b{x}^{n} \right ) ^{2} \left ( c+d{x}^{n} \right ) ^{-3-{n}^{-1}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x^{n} + a\right )}^{2}{\left (d x^{n} + c\right )}^{-\frac{1}{n} - 3}\,{d x} \end{align*}

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)

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Fricas [A]  time = 1.58541, size = 468, normalized size = 4.03 \begin{align*} \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}}} \end{align*}

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))

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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)**2*(c+d*x**n)**(-3-1/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)^2*(c+d*x^n)^(-3-1/n),x, algorithm="giac")

[Out]

Exception raised: TypeError