3.1039 \(\int \frac{x^{-1+2 n} (a+b x^n)^3}{c+d x^n} \, dx\)

Optimal. Leaf size=130 \[ \frac{b x^{2 n} \left (3 a^2 d^2-3 a b c d+b^2 c^2\right )}{2 d^3 n}-\frac{b^2 x^{3 n} (b c-3 a d)}{3 d^2 n}-\frac{x^n (b c-a d)^3}{d^4 n}+\frac{c (b c-a d)^3 \log \left (c+d x^n\right )}{d^5 n}+\frac{b^3 x^{4 n}}{4 d n} \]

[Out]

-(((b*c - a*d)^3*x^n)/(d^4*n)) + (b*(b^2*c^2 - 3*a*b*c*d + 3*a^2*d^2)*x^(2*n))/(2*d^3*n) - (b^2*(b*c - 3*a*d)*
x^(3*n))/(3*d^2*n) + (b^3*x^(4*n))/(4*d*n) + (c*(b*c - a*d)^3*Log[c + d*x^n])/(d^5*n)

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Rubi [A]  time = 0.138426, antiderivative size = 130, 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, 77} \[ \frac{b x^{2 n} \left (3 a^2 d^2-3 a b c d+b^2 c^2\right )}{2 d^3 n}-\frac{b^2 x^{3 n} (b c-3 a d)}{3 d^2 n}-\frac{x^n (b c-a d)^3}{d^4 n}+\frac{c (b c-a d)^3 \log \left (c+d x^n\right )}{d^5 n}+\frac{b^3 x^{4 n}}{4 d n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(((b*c - a*d)^3*x^n)/(d^4*n)) + (b*(b^2*c^2 - 3*a*b*c*d + 3*a^2*d^2)*x^(2*n))/(2*d^3*n) - (b^2*(b*c - 3*a*d)*
x^(3*n))/(3*d^2*n) + (b^3*x^(4*n))/(4*d*n) + (c*(b*c - a*d)^3*Log[c + d*x^n])/(d^5*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 )^3}{c+d x^n} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x (a+b x)^3}{c+d x} \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{(-b c+a d)^3}{d^4}+\frac{b \left (b^2 c^2-3 a b c d+3 a^2 d^2\right ) x}{d^3}-\frac{b^2 (b c-3 a d) x^2}{d^2}+\frac{b^3 x^3}{d}+\frac{c (b c-a d)^3}{d^4 (c+d x)}\right ) \, dx,x,x^n\right )}{n}\\ &=-\frac{(b c-a d)^3 x^n}{d^4 n}+\frac{b \left (b^2 c^2-3 a b c d+3 a^2 d^2\right ) x^{2 n}}{2 d^3 n}-\frac{b^2 (b c-3 a d) x^{3 n}}{3 d^2 n}+\frac{b^3 x^{4 n}}{4 d n}+\frac{c (b c-a d)^3 \log \left (c+d x^n\right )}{d^5 n}\\ \end{align*}

Mathematica [A]  time = 0.195873, size = 115, normalized size = 0.88 \[ \frac{6 b d^2 x^{2 n} \left (3 a^2 d^2-3 a b c d+b^2 c^2\right )-4 b^2 d^3 x^{3 n} (b c-3 a d)+12 d x^n (a d-b c)^3+12 c (b c-a d)^3 \log \left (c+d x^n\right )+3 b^3 d^4 x^{4 n}}{12 d^5 n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(12*d*(-(b*c) + a*d)^3*x^n + 6*b*d^2*(b^2*c^2 - 3*a*b*c*d + 3*a^2*d^2)*x^(2*n) - 4*b^2*d^3*(b*c - 3*a*d)*x^(3*
n) + 3*b^3*d^4*x^(4*n) + 12*c*(b*c - a*d)^3*Log[c + d*x^n])/(12*d^5*n)

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Maple [B]  time = 0.028, size = 284, normalized size = 2.2 \begin{align*}{\frac{{{\rm e}^{n\ln \left ( x \right ) }}{a}^{3}}{dn}}-3\,{\frac{{{\rm e}^{n\ln \left ( x \right ) }}{a}^{2}bc}{{d}^{2}n}}+3\,{\frac{{{\rm e}^{n\ln \left ( x \right ) }}a{b}^{2}{c}^{2}}{{d}^{3}n}}-{\frac{{{\rm e}^{n\ln \left ( x \right ) }}{b}^{3}{c}^{3}}{{d}^{4}n}}+{\frac{{b}^{3} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{4}}{4\,dn}}+{\frac{3\,b \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}{a}^{2}}{2\,dn}}-{\frac{3\,{b}^{2} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}ac}{2\,{d}^{2}n}}+{\frac{{b}^{3} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}{c}^{2}}{2\,{d}^{3}n}}+{\frac{{b}^{2} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}a}{dn}}-{\frac{{b}^{3} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}c}{3\,{d}^{2}n}}-{\frac{c\ln \left ( c+d{{\rm e}^{n\ln \left ( x \right ) }} \right ){a}^{3}}{{d}^{2}n}}+3\,{\frac{{c}^{2}\ln \left ( c+d{{\rm e}^{n\ln \left ( x \right ) }} \right ){a}^{2}b}{{d}^{3}n}}-3\,{\frac{{c}^{3}\ln \left ( c+d{{\rm e}^{n\ln \left ( x \right ) }} \right ) a{b}^{2}}{{d}^{4}n}}+{\frac{{c}^{4}\ln \left ( c+d{{\rm e}^{n\ln \left ( x \right ) }} \right ){b}^{3}}{{d}^{5}n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.967582, size = 312, normalized size = 2.4 \begin{align*} a^{3}{\left (\frac{x^{n}}{d n} - \frac{c \log \left (\frac{d x^{n} + c}{d}\right )}{d^{2} n}\right )} + \frac{1}{12} \, b^{3}{\left (\frac{12 \, c^{4} \log \left (\frac{d x^{n} + c}{d}\right )}{d^{5} n} + \frac{3 \, d^{3} x^{4 \, n} - 4 \, c d^{2} x^{3 \, n} + 6 \, c^{2} d x^{2 \, n} - 12 \, c^{3} x^{n}}{d^{4} n}\right )} - \frac{1}{2} \, a b^{2}{\left (\frac{6 \, c^{3} \log \left (\frac{d x^{n} + c}{d}\right )}{d^{4} n} - \frac{2 \, d^{2} x^{3 \, n} - 3 \, c d x^{2 \, n} + 6 \, c^{2} x^{n}}{d^{3} n}\right )} + \frac{3}{2} \, a^{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)^3/(c+d*x^n),x, algorithm="maxima")

[Out]

a^3*(x^n/(d*n) - c*log((d*x^n + c)/d)/(d^2*n)) + 1/12*b^3*(12*c^4*log((d*x^n + c)/d)/(d^5*n) + (3*d^3*x^(4*n)
- 4*c*d^2*x^(3*n) + 6*c^2*d*x^(2*n) - 12*c^3*x^n)/(d^4*n)) - 1/2*a*b^2*(6*c^3*log((d*x^n + c)/d)/(d^4*n) - (2*
d^2*x^(3*n) - 3*c*d*x^(2*n) + 6*c^2*x^n)/(d^3*n)) + 3/2*a^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 = 1.04555, size = 363, normalized size = 2.79 \begin{align*} \frac{3 \, b^{3} d^{4} x^{4 \, n} - 4 \,{\left (b^{3} c d^{3} - 3 \, a b^{2} d^{4}\right )} x^{3 \, n} + 6 \,{\left (b^{3} c^{2} d^{2} - 3 \, a b^{2} c d^{3} + 3 \, a^{2} b d^{4}\right )} x^{2 \, n} - 12 \,{\left (b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3} - a^{3} d^{4}\right )} x^{n} + 12 \,{\left (b^{3} c^{4} - 3 \, a b^{2} c^{3} d + 3 \, a^{2} b c^{2} d^{2} - a^{3} c d^{3}\right )} \log \left (d x^{n} + c\right )}{12 \, d^{5} n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(3*b^3*d^4*x^(4*n) - 4*(b^3*c*d^3 - 3*a*b^2*d^4)*x^(3*n) + 6*(b^3*c^2*d^2 - 3*a*b^2*c*d^3 + 3*a^2*b*d^4)*
x^(2*n) - 12*(b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 - a^3*d^4)*x^n + 12*(b^3*c^4 - 3*a*b^2*c^3*d + 3*a^2
*b*c^2*d^2 - a^3*c*d^3)*log(d*x^n + c))/(d^5*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+2*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{{\left (b x^{n} + a\right )}^{3} 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)^3/(c+d*x^n),x, algorithm="giac")

[Out]

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