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

Optimal. Leaf size=58 \[ \frac{x \left (a+b x^n\right ) \left (c+d x^n\right )^{-\frac{1}{n}-1}}{c (n+1)}+\frac{a n x \left (c+d x^n\right )^{-1/n}}{c^2 (n+1)} \]

[Out]

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

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Rubi [A]  time = 0.0136548, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {378, 191} \[ \frac{x \left (a+b x^n\right ) \left (c+d x^n\right )^{-\frac{1}{n}-1}}{c (n+1)}+\frac{a n x \left (c+d x^n\right )^{-1/n}}{c^2 (n+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [C]  time = 0.107201, size = 82, normalized size = 1.41 \[ \frac{x \left (c+d x^n\right )^{-\frac{n+1}{n}} \left (a (n+1) \left (c+d x^n\right ) \left (\frac{d x^n}{c}+1\right )^{\frac{1}{n}} \, _2F_1\left (2+\frac{1}{n},\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right )+b c x^n\right )}{c^2 (n+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.421, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b{x}^{n} \right ) \left ( c+d{x}^{n} \right ) ^{-2-{n}^{-1}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.61014, size = 177, normalized size = 3.05 \begin{align*} \frac{{\left (a d^{2} n + b c d\right )} x x^{2 \, n} +{\left (2 \, a c d n + b c^{2} + a c d\right )} x x^{n} +{\left (a c^{2} n + a c^{2}\right )} x}{{\left (c^{2} n + c^{2}\right )}{\left (d x^{n} + c\right )}^{\frac{2 \, n + 1}{n}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: TypeError