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

Optimal. Leaf size=28 \[ x \left (a+b x^n\right )^{-1/n} \left (c+d x^n\right )^{-1/n} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.101101, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 48, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.021, Rules used = {1898} \[ x \left (a+b x^n\right )^{-1/n} \left (c+d x^n\right )^{-1/n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 1898

Int[((a_) + (b_.)*(x_)^(n_.))^(p_.)*((c_) + (d_.)*(x_)^(n_.))^(p_.)*((e_) + (g_.)*(x_)^(n2_.)), x_Symbol] :> S
imp[(e*x*(a + b*x^n)^(p + 1)*(c + d*x^n)^(p + 1))/(a*c), x] /; FreeQ[{a, b, c, d, e, g, n, p}, x] && EqQ[n2, 2
*n] && EqQ[n*(p + 1) + 1, 0] && EqQ[a*c*g - b*d*e*(2*n*(p + 1) + 1), 0]

Rubi steps

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

Mathematica [A]  time = 0.306663, size = 28, normalized size = 1. \[ x \left (a+b x^n\right )^{-1/n} \left (c+d x^n\right )^{-1/n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [F]  time = 0.817, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b{x}^{n} \right ) ^{{\frac{-1-n}{n}}} \left ( c+d{x}^{n} \right ) ^{{\frac{-1-n}{n}}} \left ( ac-bd{x}^{2\,n} \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [B]  time = 2.00106, size = 128, normalized size = 4.57 \begin{align*} \frac{b d x x^{2 \, n} + a c x +{\left (b c + a d\right )} x x^{n}}{{\left (b x^{n} + a\right )}^{\frac{n + 1}{n}}{\left (d x^{n} + c\right )}^{\frac{n + 1}{n}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 1.11932, size = 308, normalized size = 11. \begin{align*} b d x x^{2 \, n} e^{\left (-\frac{n \log \left (b x^{n} + a\right ) + \log \left (b x^{n} + a\right )}{n} - \frac{n \log \left (d x^{n} + c\right ) + \log \left (d x^{n} + c\right )}{n}\right )} + b c x x^{n} e^{\left (-\frac{n \log \left (b x^{n} + a\right ) + \log \left (b x^{n} + a\right )}{n} - \frac{n \log \left (d x^{n} + c\right ) + \log \left (d x^{n} + c\right )}{n}\right )} + a d x x^{n} e^{\left (-\frac{n \log \left (b x^{n} + a\right ) + \log \left (b x^{n} + a\right )}{n} - \frac{n \log \left (d x^{n} + c\right ) + \log \left (d x^{n} + c\right )}{n}\right )} + a c x e^{\left (-\frac{n \log \left (b x^{n} + a\right ) + \log \left (b x^{n} + a\right )}{n} - \frac{n \log \left (d x^{n} + c\right ) + \log \left (d x^{n} + c\right )}{n}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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