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

Optimal. Leaf size=57 \[ \frac{x \left (a+b x^n\right )^{-\frac{b c}{n (b c-a d)}} \left (c+d x^n\right )^{\frac{a d}{n (b c-a d)}}}{a c} \]

[Out]

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

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Rubi [A]  time = 0.0271107, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 69, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.014, Rules used = {381} \[ \frac{x \left (a+b x^n\right )^{-\frac{b c}{n (b c-a d)}} \left (c+d x^n\right )^{\frac{a d}{n (b c-a d)}}}{a c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 381

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(x*(a + b*x^n)^(p + 1)*(c +
d*x^n)^(q + 1))/(a*c), x] /; FreeQ[{a, b, c, d, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[n*(p + q + 2) + 1, 0
] && EqQ[a*d*(p + 1) + b*c*(q + 1), 0]

Rubi steps

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

Mathematica [A]  time = 0.0492145, size = 55, normalized size = 0.96 \[ \frac{x \left (a+b x^n\right )^{-\frac{b c}{b c n-a d n}} \left (c+d x^n\right )^{\frac{a d}{b c n-a d n}}}{a c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.817, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b{x}^{n} \right ) ^{{\frac{adn-bc \left ( 1+n \right ) }{ \left ( -ad+bc \right ) n}}} \left ( c+d{x}^{n} \right ) ^{{\frac{adn-bcn+ad}{-adn+bcn}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.71863, size = 209, normalized size = 3.67 \begin{align*} \frac{{\left (b d x x^{2 \, n} + a c x +{\left (b c + a d\right )} x x^{n}\right )}{\left (d x^{n} + c\right )}^{\frac{a d -{\left (b c - a d\right )} n}{{\left (b c - a d\right )} n}}}{{\left (b x^{n} + a\right )}^{\frac{b c +{\left (b c - a d\right )} n}{{\left (b c - a d\right )} n}} a c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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