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

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Rubi [A]  time = 0.03, antiderivative size = 57, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.07, 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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fricas [A]  time = 1.27, size = 108, normalized size = 1.89 \[ \frac {{\left (b d x x^{2 \, n} + a c x + {\left (b c + a d\right )} x x^{n}\right )} {\left (d x^{n} + c\right )}^{\frac {a d - {\left (b c - a d\right )} n}{{\left (b c - a d\right )} n}}}{{\left (b x^{n} + a\right )}^{\frac {b c + {\left (b c - a d\right )} n}{{\left (b c - a d\right )} n}} a c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((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)

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maple [F]  time = 0.90, size = 0, normalized size = 0.00 \[ \int \left (b \,x^{n}+a \right )^{\frac {a d n -\left (n +1\right ) b c}{\left (-a d +b c \right ) n}} \left (d \,x^{n}+c \right )^{\frac {a d n -b c n +a d}{-a d n +b c n}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((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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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{{\left (a+b\,x^n\right )}^{\frac {a\,d\,n-b\,c\,\left (n+1\right )}{n\,\left (a\,d-b\,c\right )}}\,{\left (c+d\,x^n\right )}^{\frac {a\,d+a\,d\,n-b\,c\,n}{a\,d\,n-b\,c\,n}}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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)**((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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