3.1050 \(\int \frac{x^{-1+3 n}}{(a+b x^n)^3 (c+d x^n)} \, dx\)

Optimal. Leaf size=120 \[ -\frac{a^2}{2 b^2 n (b c-a d) \left (a+b x^n\right )^2}+\frac{a (2 b c-a d)}{b^2 n (b c-a d)^2 \left (a+b x^n\right )}+\frac{c^2 \log \left (a+b x^n\right )}{n (b c-a d)^3}-\frac{c^2 \log \left (c+d x^n\right )}{n (b c-a d)^3} \]

[Out]

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

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Rubi [A]  time = 0.105352, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {446, 88} \[ -\frac{a^2}{2 b^2 n (b c-a d) \left (a+b x^n\right )^2}+\frac{a (2 b c-a d)}{b^2 n (b c-a d)^2 \left (a+b x^n\right )}+\frac{c^2 \log \left (a+b x^n\right )}{n (b c-a d)^3}-\frac{c^2 \log \left (c+d x^n\right )}{n (b c-a d)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-a^2/(2*b^2*(b*c - a*d)*n*(a + b*x^n)^2) + (a*(2*b*c - a*d))/(b^2*(b*c - a*d)^2*n*(a + b*x^n)) + (c^2*Log[a +
b*x^n])/((b*c - a*d)^3*n) - (c^2*Log[c + d*x^n])/((b*c - a*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 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps

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

Mathematica [A]  time = 0.139842, size = 112, normalized size = 0.93 \[ \frac{-\frac{a^2}{2 b^2 (b c-a d) \left (a+b x^n\right )^2}+\frac{a (2 b c-a d)}{b^2 (b c-a d)^2 \left (a+b x^n\right )}+\frac{c^2 \log \left (a+b x^n\right )}{(b c-a d)^3}-\frac{c^2 \log \left (c+d x^n\right )}{(b c-a d)^3}}{n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.054, size = 214, normalized size = 1.8 \begin{align*}{\frac{1}{ \left ( a+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}} \left ({\frac{ \left ( -ad+2\,bc \right ) a{{\rm e}^{n\ln \left ( x \right ) }}}{nb \left ({a}^{2}{d}^{2}-2\,abcd+{b}^{2}{c}^{2} \right ) }}+{\frac{{a}^{2} \left ( -ad+3\,bc \right ) }{ \left ( 2\,{a}^{2}{d}^{2}-4\,abcd+2\,{b}^{2}{c}^{2} \right ){b}^{2}n}} \right ) }+{\frac{{c}^{2}\ln \left ( c+d{{\rm e}^{n\ln \left ( x \right ) }} \right ) }{n \left ({a}^{3}{d}^{3}-3\,{a}^{2}bc{d}^{2}+3\,a{b}^{2}{c}^{2}d-{b}^{3}{c}^{3} \right ) }}-{\frac{{c}^{2}\ln \left ( a+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) }{n \left ({a}^{3}{d}^{3}-3\,{a}^{2}bc{d}^{2}+3\,a{b}^{2}{c}^{2}d-{b}^{3}{c}^{3} \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 0.972715, size = 354, normalized size = 2.95 \begin{align*} \frac{c^{2} \log \left (\frac{b x^{n} + a}{b}\right )}{b^{3} c^{3} n - 3 \, a b^{2} c^{2} d n + 3 \, a^{2} b c d^{2} n - a^{3} d^{3} n} - \frac{c^{2} \log \left (\frac{d x^{n} + c}{d}\right )}{b^{3} c^{3} n - 3 \, a b^{2} c^{2} d n + 3 \, a^{2} b c d^{2} n - a^{3} d^{3} n} + \frac{3 \, a^{2} b c - a^{3} d + 2 \,{\left (2 \, a b^{2} c - a^{2} b d\right )} x^{n}}{2 \,{\left (a^{2} b^{4} c^{2} n - 2 \, a^{3} b^{3} c d n + a^{4} b^{2} d^{2} n +{\left (b^{6} c^{2} n - 2 \, a b^{5} c d n + a^{2} b^{4} d^{2} n\right )} x^{2 \, n} + 2 \,{\left (a b^{5} c^{2} n - 2 \, a^{2} b^{4} c d n + a^{3} b^{3} d^{2} n\right )} x^{n}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c^2*log((b*x^n + a)/b)/(b^3*c^3*n - 3*a*b^2*c^2*d*n + 3*a^2*b*c*d^2*n - a^3*d^3*n) - c^2*log((d*x^n + c)/d)/(b
^3*c^3*n - 3*a*b^2*c^2*d*n + 3*a^2*b*c*d^2*n - a^3*d^3*n) + 1/2*(3*a^2*b*c - a^3*d + 2*(2*a*b^2*c - a^2*b*d)*x
^n)/(a^2*b^4*c^2*n - 2*a^3*b^3*c*d*n + a^4*b^2*d^2*n + (b^6*c^2*n - 2*a*b^5*c*d*n + a^2*b^4*d^2*n)*x^(2*n) + 2
*(a*b^5*c^2*n - 2*a^2*b^4*c*d*n + a^3*b^3*d^2*n)*x^n)

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Fricas [B]  time = 1.10577, size = 598, normalized size = 4.98 \begin{align*} \frac{3 \, a^{2} b^{2} c^{2} - 4 \, a^{3} b c d + a^{4} d^{2} + 2 \,{\left (2 \, a b^{3} c^{2} - 3 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x^{n} + 2 \,{\left (b^{4} c^{2} x^{2 \, n} + 2 \, a b^{3} c^{2} x^{n} + a^{2} b^{2} c^{2}\right )} \log \left (b x^{n} + a\right ) - 2 \,{\left (b^{4} c^{2} x^{2 \, n} + 2 \, a b^{3} c^{2} x^{n} + a^{2} b^{2} c^{2}\right )} \log \left (d x^{n} + c\right )}{2 \,{\left ({\left (b^{7} c^{3} - 3 \, a b^{6} c^{2} d + 3 \, a^{2} b^{5} c d^{2} - a^{3} b^{4} d^{3}\right )} n x^{2 \, n} + 2 \,{\left (a b^{6} c^{3} - 3 \, a^{2} b^{5} c^{2} d + 3 \, a^{3} b^{4} c d^{2} - a^{4} b^{3} d^{3}\right )} n x^{n} +{\left (a^{2} b^{5} c^{3} - 3 \, a^{3} b^{4} c^{2} d + 3 \, a^{4} b^{3} c d^{2} - a^{5} b^{2} d^{3}\right )} n\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(3*a^2*b^2*c^2 - 4*a^3*b*c*d + a^4*d^2 + 2*(2*a*b^3*c^2 - 3*a^2*b^2*c*d + a^3*b*d^2)*x^n + 2*(b^4*c^2*x^(2
*n) + 2*a*b^3*c^2*x^n + a^2*b^2*c^2)*log(b*x^n + a) - 2*(b^4*c^2*x^(2*n) + 2*a*b^3*c^2*x^n + a^2*b^2*c^2)*log(
d*x^n + c))/((b^7*c^3 - 3*a*b^6*c^2*d + 3*a^2*b^5*c*d^2 - a^3*b^4*d^3)*n*x^(2*n) + 2*(a*b^6*c^3 - 3*a^2*b^5*c^
2*d + 3*a^3*b^4*c*d^2 - a^4*b^3*d^3)*n*x^n + (a^2*b^5*c^3 - 3*a^3*b^4*c^2*d + 3*a^4*b^3*c*d^2 - a^5*b^2*d^3)*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(x**(-1+3*n)/(a+b*x**n)**3/(c+d*x**n),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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