3.1041 \(\int \frac{x^{-1+2 n} (a+b x^n)}{c+d x^n} \, dx\)

Optimal. Leaf size=60 \[ -\frac{x^n (b c-a d)}{d^2 n}+\frac{c (b c-a d) \log \left (c+d x^n\right )}{d^3 n}+\frac{b x^{2 n}}{2 d n} \]

[Out]

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

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Rubi [A]  time = 0.0540822, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {446, 77} \[ -\frac{x^n (b c-a d)}{d^2 n}+\frac{c (b c-a d) \log \left (c+d x^n\right )}{d^3 n}+\frac{b x^{2 n}}{2 d n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 446

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

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

Mathematica [A]  time = 0.0469774, size = 50, normalized size = 0.83 \[ \frac{d x^n \left (2 a d-2 b c+b d x^n\right )+2 c (b c-a d) \log \left (c+d x^n\right )}{2 d^3 n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.02, size = 87, normalized size = 1.5 \begin{align*}{\frac{{{\rm e}^{n\ln \left ( x \right ) }}a}{dn}}-{\frac{b{{\rm e}^{n\ln \left ( x \right ) }}c}{{d}^{2}n}}+{\frac{b \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{2\,dn}}-{\frac{c\ln \left ( c+d{{\rm e}^{n\ln \left ( x \right ) }} \right ) a}{{d}^{2}n}}+{\frac{{c}^{2}\ln \left ( c+d{{\rm e}^{n\ln \left ( x \right ) }} \right ) b}{{d}^{3}n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.96418, size = 112, normalized size = 1.87 \begin{align*} a{\left (\frac{x^{n}}{d n} - \frac{c \log \left (\frac{d x^{n} + c}{d}\right )}{d^{2} n}\right )} + \frac{1}{2} \, b{\left (\frac{2 \, c^{2} \log \left (\frac{d x^{n} + c}{d}\right )}{d^{3} n} + \frac{d x^{2 \, n} - 2 \, c x^{n}}{d^{2} n}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.04103, size = 119, normalized size = 1.98 \begin{align*} \frac{b d^{2} x^{2 \, n} - 2 \,{\left (b c d - a d^{2}\right )} x^{n} + 2 \,{\left (b c^{2} - a c d\right )} \log \left (d x^{n} + c\right )}{2 \, d^{3} n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 36.535, size = 105, normalized size = 1.75 \begin{align*} \begin{cases} \frac{\left (a + b\right ) \log{\left (x \right )}}{c} & \text{for}\: d = 0 \wedge n = 0 \\\frac{\left (a + b\right ) \log{\left (x \right )}}{c + d} & \text{for}\: n = 0 \\\frac{\frac{a x^{2 n}}{2 n} + \frac{b x^{3 n}}{3 n}}{c} & \text{for}\: d = 0 \\- \frac{a c \log{\left (\frac{c}{d} + x^{n} \right )}}{d^{2} n} + \frac{a x^{n}}{d n} + \frac{b c^{2} \log{\left (\frac{c}{d} + x^{n} \right )}}{d^{3} n} - \frac{b c x^{n}}{d^{2} n} + \frac{b x^{2 n}}{2 d n} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise(((a + b)*log(x)/c, Eq(d, 0) & Eq(n, 0)), ((a + b)*log(x)/(c + d), Eq(n, 0)), ((a*x**(2*n)/(2*n) + b*
x**(3*n)/(3*n))/c, Eq(d, 0)), (-a*c*log(c/d + x**n)/(d**2*n) + a*x**n/(d*n) + b*c**2*log(c/d + x**n)/(d**3*n)
- b*c*x**n/(d**2*n) + b*x**(2*n)/(2*d*n), True))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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