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

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 = 0.0845283, 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 \[ \frac{x \left (a+b x^n\right )^{-\frac{b c}{n (b c-a d)}} \left (c+d x^n\right )^{\frac{a d}{n (b c-a d)}}}{a c} \]

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))
)

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Rubi in Sympy [A]  time = 13.5218, size = 41, normalized size = 0.72 \[ \frac{x \left (a + b x^{n}\right )^{\frac{b c}{n \left (a d - b c\right )}} \left (c + d x^{n}\right )^{- \frac{a d}{n \left (a d - b c\right )}}}{a c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_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]

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

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Mathematica [C]  time = 1.93871, size = 461, normalized size = 8.09 \[ \frac{a c (n+1) x (a d-b c) \left (a+b x^n\right )^{\frac{a d n-b c (n+1)}{n (b c-a d)}} \left (c+d x^n\right )^{\frac{a d n+a d-b c n}{b c n-a d n}} F_1\left (\frac{1}{n};\frac{b c+b n c-a d n}{b c n-a d n},\frac{b c n-a d (n+1)}{(b c-a d) n};1+\frac{1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right )}{b c x^n (b c (n+1)-a d n) F_1\left (1+\frac{1}{n};\frac{b c+2 b n c-2 a d n}{b c n-a d n},\frac{b c n-a d (n+1)}{(b c-a d) n};2+\frac{1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right )-a \left (d x^n (a d (n+1)-b c n) F_1\left (1+\frac{1}{n};\frac{b c+b n c-a d n}{b c n-a d n},-\frac{a d+2 a n d-2 b c n}{b c n-a d n};2+\frac{1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right )+c (n+1) (b c-a d) F_1\left (\frac{1}{n};\frac{b c+b n c-a d n}{b c n-a d n},\frac{b c n-a d (n+1)}{(b c-a d) n};1+\frac{1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right )\right )} \]

Warning: Unable to verify antiderivative.

[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]

(a*c*(-(b*c) + a*d)*(1 + n)*x*(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))*AppellF1[n^(-1), (b*c + b*
c*n - a*d*n)/(b*c*n - a*d*n), (b*c*n - a*d*(1 + n))/((b*c - a*d)*n), 1 + n^(-1),
-((b*x^n)/a), -((d*x^n)/c)])/(b*c*(-(a*d*n) + b*c*(1 + n))*x^n*AppellF1[1 + n^(
-1), (b*c + 2*b*c*n - 2*a*d*n)/(b*c*n - a*d*n), (b*c*n - a*d*(1 + n))/((b*c - a*
d)*n), 2 + n^(-1), -((b*x^n)/a), -((d*x^n)/c)] - a*(d*(-(b*c*n) + a*d*(1 + n))*x
^n*AppellF1[1 + n^(-1), (b*c + b*c*n - a*d*n)/(b*c*n - a*d*n), -((a*d - 2*b*c*n
+ 2*a*d*n)/(b*c*n - a*d*n)), 2 + n^(-1), -((b*x^n)/a), -((d*x^n)/c)] + c*(b*c -
a*d)*(1 + n)*AppellF1[n^(-1), (b*c + b*c*n - a*d*n)/(b*c*n - a*d*n), (b*c*n - a*
d*(1 + n))/((b*c - a*d)*n), 1 + n^(-1), -((b*x^n)/a), -((d*x^n)/c)]))

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Maple [F]  time = 0.306, size = 0, normalized size = 0. \[ \int \left ( a+b{x}^{n} \right ) ^{{\frac{adn-bc \left ( 1+n \right ) }{ \left ( -ad+bc \right ) n}}} \left ( c+d{x}^{n} \right ) ^{{\frac{adn-bcn+ad}{-adn+bcn}}}\, dx \]

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. \[ \int{\left (b x^{n} + a\right )}^{-\frac{b c{\left (n + 1\right )} - a d n}{{\left (b c - a d\right )} n}}{\left (d x^{n} + c\right )}^{-\frac{b c n - a d n - a d}{b c n - a d n}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((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, algorithm="maxima")

[Out]

integrate((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 = 0.264862, size = 146, normalized size = 2.56 \[ \frac{{\left (b d x x^{2 \, n} + a c x +{\left (b c + a d\right )} x x^{n}\right )}{\left (d x^{n} + c\right )}^{\frac{a d -{\left (b c - a d\right )} n}{{\left (b c - a d\right )} n}}}{{\left (b x^{n} + a\right )}^{\frac{b c +{\left (b c - a d\right )} n}{{\left (b c - a d\right )} n}} a c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  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, 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. \[ \text{Timed out} \]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  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, 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)