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

Optimal. Leaf size=127 \[ \frac{x \left (c+d x^n\right )^{-\frac{1}{n}-1} (2 a d n+b c)}{c^2 d (n+1) (2 n+1)}+\frac{n x \left (c+d x^n\right )^{-1/n} (2 a d n+b c)}{c^3 d (n+1) (2 n+1)}-\frac{x (b c-a d) \left (c+d x^n\right )^{-\frac{1}{n}-2}}{c d (2 n+1)} \]

[Out]

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

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Rubi [A]  time = 0.063562, antiderivative size = 127, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {385, 192, 191} \[ \frac{x \left (c+d x^n\right )^{-\frac{1}{n}-1} (2 a d n+b c)}{c^2 d (n+1) (2 n+1)}+\frac{n x \left (c+d x^n\right )^{-1/n} (2 a d n+b c)}{c^3 d (n+1) (2 n+1)}-\frac{x (b c-a d) \left (c+d x^n\right )^{-\frac{1}{n}-2}}{c d (2 n+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 385

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> -Simp[((b*c - a*d)*x*(a + b*x^n)^(p +
1))/(a*b*n*(p + 1)), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(a*b*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /
; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/n + p, 0])

Rule 192

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, n, p}, x] && ILtQ[Simplify[1/n + p + 1
], 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 )^{-3-\frac{1}{n}} \, dx &=-\frac{(b c-a d) x \left (c+d x^n\right )^{-2-\frac{1}{n}}}{c d (1+2 n)}+\frac{(b c+2 a d n) \int \left (c+d x^n\right )^{-2-\frac{1}{n}} \, dx}{c d (1+2 n)}\\ &=-\frac{(b c-a d) x \left (c+d x^n\right )^{-2-\frac{1}{n}}}{c d (1+2 n)}+\frac{(b c+2 a d n) x \left (c+d x^n\right )^{-1-\frac{1}{n}}}{c^2 d (1+n) (1+2 n)}+\frac{(n (b c+2 a d n)) \int \left (c+d x^n\right )^{-1-\frac{1}{n}} \, dx}{c^2 d (1+n) (1+2 n)}\\ &=-\frac{(b c-a d) x \left (c+d x^n\right )^{-2-\frac{1}{n}}}{c d (1+2 n)}+\frac{(b c+2 a d n) x \left (c+d x^n\right )^{-1-\frac{1}{n}}}{c^2 d (1+n) (1+2 n)}+\frac{n (b c+2 a d n) x \left (c+d x^n\right )^{-1/n}}{c^3 d (1+n) (1+2 n)}\\ \end{align*}

Mathematica [C]  time = 0.0969818, size = 96, normalized size = 0.76 \[ \frac{x \left (c+d x^n\right )^{-1/n} \left (\frac{d x^n}{c}+1\right )^{\frac{1}{n}} \left (a (n+1) \, _2F_1\left (3+\frac{1}{n},\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right )+b x^n \, _2F_1\left (1+\frac{1}{n},3+\frac{1}{n};2+\frac{1}{n};-\frac{d x^n}{c}\right )\right )}{c^3 (n+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(1 + (d*x^n)/c)^n^(-1)*(b*x^n*Hypergeometric2F1[1 + n^(-1), 3 + n^(-1), 2 + n^(-1), -((d*x^n)/c)] + a*(1 +
n)*Hypergeometric2F1[3 + n^(-1), n^(-1), 1 + n^(-1), -((d*x^n)/c)]))/(c^3*(1 + n)*(c + d*x^n)^n^(-1))

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Maple [F]  time = 0.449, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b{x}^{n} \right ) \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)*(c+d*x^n)^(-3-1/n),x)

[Out]

int((a+b*x^n)*(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 )}{\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)*(c+d*x^n)^(-3-1/n),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.61052, size = 360, normalized size = 2.83 \begin{align*} \frac{{\left (2 \, a d^{3} n^{2} + b c d^{2} n\right )} x x^{3 \, n} +{\left (6 \, a c d^{2} n^{2} + b c^{2} d +{\left (3 \, b c^{2} d + 2 \, a c d^{2}\right )} n\right )} x x^{2 \, n} +{\left (6 \, a c^{2} d n^{2} + b c^{3} + a c^{2} d +{\left (2 \, b c^{3} + 5 \, a c^{2} d\right )} n\right )} x x^{n} +{\left (2 \, a c^{3} n^{2} + 3 \, a c^{3} n + a 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)*(c+d*x^n)^(-3-1/n),x, algorithm="fricas")

[Out]

((2*a*d^3*n^2 + b*c*d^2*n)*x*x^(3*n) + (6*a*c*d^2*n^2 + b*c^2*d + (3*b*c^2*d + 2*a*c*d^2)*n)*x*x^(2*n) + (6*a*
c^2*d*n^2 + b*c^3 + a*c^2*d + (2*b*c^3 + 5*a*c^2*d)*n)*x*x^n + (2*a*c^3*n^2 + 3*a*c^3*n + a*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)*(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)*(c+d*x^n)^(-3-1/n),x, algorithm="giac")

[Out]

Exception raised: TypeError