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)^(1/n))/((c+d*x^n)^(1/n))

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Rubi [A]  time = 0.10, antiderivative size = 28, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.35, size = 28, normalized size = 1.00 \[ x \left (a+b x^n\right )^{-1/n} \left (c+d x^n\right )^{-1/n} \]

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))

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fricas [B]  time = 0.95, size = 61, normalized size = 2.18 \[ \frac {b d x x^{2 \, n} + a c x + {\left (b c + a d\right )} x x^{n}}{{\left (b x^{n} + a\right )}^{\frac {n + 1}{n}} {\left (d x^{n} + c\right )}^{\frac {n + 1}{n}}} \]

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))

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giac [B]  time = 0.42, size = 228, normalized size = 8.14 \[ b d x x^{2 \, n} e^{\left (-\frac {n \log \left (b x^{n} + a\right ) + \log \left (b x^{n} + a\right )}{n} - \frac {n \log \left (d x^{n} + c\right ) + \log \left (d x^{n} + c\right )}{n}\right )} + b c x x^{n} e^{\left (-\frac {n \log \left (b x^{n} + a\right ) + \log \left (b x^{n} + a\right )}{n} - \frac {n \log \left (d x^{n} + c\right ) + \log \left (d x^{n} + c\right )}{n}\right )} + a d x x^{n} e^{\left (-\frac {n \log \left (b x^{n} + a\right ) + \log \left (b x^{n} + a\right )}{n} - \frac {n \log \left (d x^{n} + c\right ) + \log \left (d x^{n} + c\right )}{n}\right )} + a c x e^{\left (-\frac {n \log \left (b x^{n} + a\right ) + \log \left (b x^{n} + a\right )}{n} - \frac {n \log \left (d x^{n} + c\right ) + \log \left (d x^{n} + c\right )}{n}\right )} \]

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)

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maple [F]  time = 1.08, size = 0, normalized size = 0.00 \[ \int \left (-b d \,x^{2 n}+a c \right ) \left (b \,x^{n}+a \right )^{\frac {-n -1}{n}} \left (d \,x^{n}+c \right )^{\frac {-n -1}{n}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {b d x^{2 \, n} - a c}{{\left (b x^{n} + a\right )}^{\frac {n + 1}{n}} {\left (d x^{n} + c\right )}^{\frac {n + 1}{n}}}\,{d x} \]

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)

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mupad [B]  time = 5.20, size = 95, normalized size = 3.39 \[ \frac {\frac {a\,c\,x}{{\left (a+b\,x^n\right )}^{\frac {n+1}{n}}}+\frac {x\,x^n\,\left (a\,d+b\,c\right )}{{\left (a+b\,x^n\right )}^{\frac {n+1}{n}}}+\frac {b\,d\,x\,x^{2\,n}}{{\left (a+b\,x^n\right )}^{\frac {n+1}{n}}}}{{\left (c+d\,x^n\right )}^{\frac {n+1}{n}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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

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