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

Optimal. Leaf size=78 \[ \frac{x (b c-a d (1-2 n)) \, _2F_1\left (2,\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right )}{2 c^3 d n}-\frac{x (b c-a d)}{2 c d n \left (c+d x^n\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.0310969, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {385, 245} \[ \frac{x (b c-a d (1-2 n)) \, _2F_1\left (2,\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right )}{2 c^3 d n}-\frac{x (b c-a d)}{2 c d n \left (c+d x^n\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^n)/(c + d*x^n)^3,x]

[Out]

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

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 245

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*x*Hypergeometric2F1[-p, 1/n, 1/n + 1, -((b*x^n)/a)],
x] /; FreeQ[{a, b, n, p}, x] &&  !IGtQ[p, 0] &&  !IntegerQ[1/n] &&  !ILtQ[Simplify[1/n + p], 0] && (IntegerQ[p
] || GtQ[a, 0])

Rubi steps

\begin{align*} \int \frac{a+b x^n}{\left (c+d x^n\right )^3} \, dx &=-\frac{(b c-a d) x}{2 c d n \left (c+d x^n\right )^2}+\frac{(b c-a d (1-2 n)) \int \frac{1}{\left (c+d x^n\right )^2} \, dx}{2 c d n}\\ &=-\frac{(b c-a d) x}{2 c d n \left (c+d x^n\right )^2}+\frac{(b c-a d (1-2 n)) x \, _2F_1\left (2,\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right )}{2 c^3 d n}\\ \end{align*}

Mathematica [A]  time = 0.0421899, size = 58, normalized size = 0.74 \[ \frac{x \left (\frac{b}{\left (c+d x^n\right )^2}-\frac{(a d (2 n-1)+b c) \, _2F_1\left (3,\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right )}{c^3}\right )}{d-2 d n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.373, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b{x}^{n}}{ \left ( c+d{x}^{n} \right ) ^{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\left ({\left (2 \, n^{2} - 3 \, n + 1\right )} a d + b c{\left (n - 1\right )}\right )} \int \frac{1}{2 \,{\left (c^{2} d^{2} n^{2} x^{n} + c^{3} d n^{2}\right )}}\,{d x} + \frac{{\left (a d^{2}{\left (2 \, n - 1\right )} + b c d\right )} x x^{n} +{\left (a c d{\left (3 \, n - 1\right )} - b c^{2}{\left (n - 1\right )}\right )} x}{2 \,{\left (c^{2} d^{3} n^{2} x^{2 \, n} + 2 \, c^{3} d^{2} n^{2} x^{n} + c^{4} d n^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((2*n^2 - 3*n + 1)*a*d + b*c*(n - 1))*integrate(1/2/(c^2*d^2*n^2*x^n + c^3*d*n^2), x) + 1/2*((a*d^2*(2*n - 1)
+ b*c*d)*x*x^n + (a*c*d*(3*n - 1) - b*c^2*(n - 1))*x)/(c^2*d^3*n^2*x^(2*n) + 2*c^3*d^2*n^2*x^n + c^4*d*n^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*x^n + a)/(d^3*x^(3*n) + 3*c*d^2*x^(2*n) + 3*c^2*d*x^n + c^3), x)

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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,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b x^{n} + a}{{\left (d x^{n} + c\right )}^{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,x, algorithm="giac")

[Out]

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