3.328 \(\int (a+b x^n)^p (c+d x^n)^{-2-\frac{1}{n}-p} \, dx\)
Optimal. Leaf size=193 \[ \frac{x \left (a+b x^n\right )^{p+1} \left (c+d x^n\right )^{-\frac{1}{n}-p-1} (n (p+1) (b c-a d)+b c) \left (\frac{c \left (a+b x^n\right )}{a \left (c+d x^n\right )}\right )^{-p-1} \, _2F_1\left (\frac{1}{n},-p-1;1+\frac{1}{n};-\frac{(b c-a d) x^n}{a \left (d x^n+c\right )}\right )}{a c n (p+1) (b c-a d)}-\frac{b x \left (a+b x^n\right )^{p+1} \left (c+d x^n\right )^{-\frac{1}{n}-p-1}}{a n (p+1) (b c-a d)} \]
[Out]
-((b*x*(a + b*x^n)^(1 + p)*(c + d*x^n)^(-1 - n^(-1) - p))/(a*(b*c - a*d)*n*(1 + p))) + ((b*c + (b*c - a*d)*n*(
1 + p))*x*(a + b*x^n)^(1 + p)*((c*(a + b*x^n))/(a*(c + d*x^n)))^(-1 - p)*(c + d*x^n)^(-1 - n^(-1) - p)*Hyperge
ometric2F1[n^(-1), -1 - p, 1 + n^(-1), -(((b*c - a*d)*x^n)/(a*(c + d*x^n)))])/(a*c*(b*c - a*d)*n*(1 + p))
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Rubi [A] time = 0.0812199, antiderivative size = 179, normalized size of antiderivative =
0.93, number of steps used = 2, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules
used = {382, 380} \[ \frac{x \left (a+b x^n\right )^{p+1} \left (c+d x^n\right )^{-\frac{1}{n}-p-1} \left (\frac{b}{n (p+1) (b c-a d)}+\frac{1}{c}\right ) \left (\frac{c \left (a+b x^n\right )}{a \left (c+d x^n\right )}\right )^{-p-1} \, _2F_1\left (\frac{1}{n},-p-1;1+\frac{1}{n};-\frac{(b c-a d) x^n}{a \left (d x^n+c\right )}\right )}{a}-\frac{b x \left (a+b x^n\right )^{p+1} \left (c+d x^n\right )^{-\frac{1}{n}-p-1}}{a n (p+1) (b c-a d)} \]
Antiderivative was successfully verified.
[In]
Int[(a + b*x^n)^p*(c + d*x^n)^(-2 - n^(-1) - p),x]
[Out]
-((b*x*(a + b*x^n)^(1 + p)*(c + d*x^n)^(-1 - n^(-1) - p))/(a*(b*c - a*d)*n*(1 + p))) + ((c^(-1) + b/((b*c - a*
d)*n*(1 + p)))*x*(a + b*x^n)^(1 + p)*((c*(a + b*x^n))/(a*(c + d*x^n)))^(-1 - p)*(c + d*x^n)^(-1 - n^(-1) - p)*
Hypergeometric2F1[n^(-1), -1 - p, 1 + n^(-1), -(((b*c - a*d)*x^n)/(a*(c + d*x^n)))])/a
Rule 382
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Simp[(b*x*(a + b*x^n)^(p + 1)*(
c + d*x^n)^(q + 1))/(a*n*(p + 1)*(b*c - a*d)), x] + Dist[(b*c + n*(p + 1)*(b*c - a*d))/(a*n*(p + 1)*(b*c - a*d
)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n, q}, x] && NeQ[b*c - a*d, 0] && EqQ[
n*(p + q + 2) + 1, 0] && (LtQ[p, -1] || !LtQ[q, -1]) && NeQ[p, -1]
Rule 380
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(x*(a + b*x^n)^p*Hypergeome
tric2F1[1/n, -p, 1 + 1/n, -(((b*c - a*d)*x^n)/(a*(c + d*x^n)))])/(c*((c*(a + b*x^n))/(a*(c + d*x^n)))^p*(c + d
*x^n)^(1/n + p)), x] /; FreeQ[{a, b, c, d, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[n*(p + q + 1) + 1, 0]
Rubi steps
\begin{align*} \int \left (a+b x^n\right )^p \left (c+d x^n\right )^{-2-\frac{1}{n}-p} \, dx &=-\frac{b x \left (a+b x^n\right )^{1+p} \left (c+d x^n\right )^{-1-\frac{1}{n}-p}}{a (b c-a d) n (1+p)}+\frac{\left (1+\frac{b c}{(b c-a d) n (1+p)}\right ) \int \left (a+b x^n\right )^{1+p} \left (c+d x^n\right )^{-2-\frac{1}{n}-p} \, dx}{a}\\ &=-\frac{b x \left (a+b x^n\right )^{1+p} \left (c+d x^n\right )^{-1-\frac{1}{n}-p}}{a (b c-a d) n (1+p)}+\frac{\left (1+\frac{b c}{(b c-a d) n (1+p)}\right ) x \left (a+b x^n\right )^{1+p} \left (\frac{c \left (a+b x^n\right )}{a \left (c+d x^n\right )}\right )^{-1-p} \left (c+d x^n\right )^{-1-\frac{1}{n}-p} \, _2F_1\left (\frac{1}{n},-1-p;1+\frac{1}{n};-\frac{(b c-a d) x^n}{a \left (c+d x^n\right )}\right )}{a c}\\ \end{align*}
Mathematica [B] time = 45.8204, size = 1414, normalized size = 7.33 \[ \text{result too large to display} \]
Warning: Unable to verify antiderivative.
[In]
Integrate[(a + b*x^n)^p*(c + d*x^n)^(-2 - n^(-1) - p),x]
[Out]
(c^4*(1 + n)*(1 + 2*n)*(1 + 3*n)*x*(a + b*x^n)^(3 + p)*(c + d*x^n)^(-2 - n^(-1) - p)*(1 + (d*x^n)/c)*Gamma[2 +
n^(-1)]*Gamma[-p]*(Hypergeometric2F1[1, -p, 1 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))] + (d*n*x^n*((c*Hyp
ergeometric2F1[1, -p, 2 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))])/(1 + n) + ((b*c - a*d)*x^n*Gamma[1 + n^(
-1)]*Gamma[1 - p]*Hypergeometric2F1[2, 1 - p, 3 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))])/((1 + 2*n)*(a +
b*x^n)*Gamma[2 + n^(-1)]*Gamma[-p])))/c^2))/(-(c*d*(1 + 3*n)*(1 + n + n*p)*x^n*(a + b*x^n)^2*(c^2*(1 + n)*(1 +
2*n)*(a + b*x^n)*Gamma[2 + n^(-1)]*Gamma[-p]*Hypergeometric2F1[1, -p, 1 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b
*x^n))] + d*n*x^n*(c*(1 + 2*n)*(a + b*x^n)*Gamma[2 + n^(-1)]*Gamma[-p]*Hypergeometric2F1[1, -p, 2 + n^(-1), ((
b*c - a*d)*x^n)/(c*(a + b*x^n))] + (b*c - a*d)*(1 + n)*x^n*Gamma[1 + n^(-1)]*Gamma[1 - p]*Hypergeometric2F1[2,
1 - p, 3 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))]))) + b*c*n*(1 + 3*n)*p*x^n*(a + b*x^n)*(c + d*x^n)*(c^2
*(1 + n)*(1 + 2*n)*(a + b*x^n)*Gamma[2 + n^(-1)]*Gamma[-p]*Hypergeometric2F1[1, -p, 1 + n^(-1), ((b*c - a*d)*x
^n)/(c*(a + b*x^n))] + d*n*x^n*(c*(1 + 2*n)*(a + b*x^n)*Gamma[2 + n^(-1)]*Gamma[-p]*Hypergeometric2F1[1, -p, 2
+ n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))] + (b*c - a*d)*(1 + n)*x^n*Gamma[1 + n^(-1)]*Gamma[1 - p]*Hyperge
ometric2F1[2, 1 - p, 3 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))])) + c*(1 + 3*n)*(a + b*x^n)^2*(c + d*x^n)*
(c^2*(1 + n)*(1 + 2*n)*(a + b*x^n)*Gamma[2 + n^(-1)]*Gamma[-p]*Hypergeometric2F1[1, -p, 1 + n^(-1), ((b*c - a*
d)*x^n)/(c*(a + b*x^n))] + d*n*x^n*(c*(1 + 2*n)*(a + b*x^n)*Gamma[2 + n^(-1)]*Gamma[-p]*Hypergeometric2F1[1, -
p, 2 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))] + (b*c - a*d)*(1 + n)*x^n*Gamma[1 + n^(-1)]*Gamma[1 - p]*Hyp
ergeometric2F1[2, 1 - p, 3 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))])) + n^2*x^n*(c + d*x^n)*(a*c^2*(-(b*c)
+ a*d)*(1 + 2*n)*(1 + 3*n)*p*(a + b*x^n)*Gamma[2 + n^(-1)]*Gamma[-p]*Hypergeometric2F1[2, 1 - p, 2 + n^(-1),
((b*c - a*d)*x^n)/(c*(a + b*x^n))] + c*d*(1 + 3*n)*(a + b*x^n)^2*(c*(1 + 2*n)*(a + b*x^n)*Gamma[2 + n^(-1)]*Ga
mma[-p]*Hypergeometric2F1[1, -p, 2 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))] + (b*c - a*d)*(1 + n)*x^n*Gamm
a[1 + n^(-1)]*Gamma[1 - p]*Hypergeometric2F1[2, 1 - p, 3 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))]) - d*(b*
c - a*d)*x^n*(b*c*(1 + n)*(1 + 3*n)*x^n*(a + b*x^n)*Gamma[1 + n^(-1)]*Gamma[1 - p]*Hypergeometric2F1[2, 1 - p,
3 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))] - c*(1 + n)*(1 + 3*n)*(a + b*x^n)^2*Gamma[1 + n^(-1)]*Gamma[1
- p]*Hypergeometric2F1[2, 1 - p, 3 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))] + a*c*n*(1 + 3*n)*p*(a + b*x^n
)*Gamma[2 + n^(-1)]*Gamma[-p]*Hypergeometric2F1[2, 1 - p, 3 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))] - 2*a
*(-(b*c) + a*d)*n*(1 + n)*(-1 + p)*x^n*Gamma[1 + n^(-1)]*Gamma[1 - p]*Hypergeometric2F1[3, 2 - p, 4 + n^(-1),
((b*c - a*d)*x^n)/(c*(a + b*x^n))])))
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Maple [F] time = 0.793, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b{x}^{n} \right ) ^{p} \left ( c+d{x}^{n} \right ) ^{-2-{n}^{-1}-p}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a+b*x^n)^p*(c+d*x^n)^(-2-1/n-p),x)
[Out]
int((a+b*x^n)^p*(c+d*x^n)^(-2-1/n-p),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x^{n} + a\right )}^{p}{\left (d x^{n} + c\right )}^{-p - \frac{1}{n} - 2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*x^n)^p*(c+d*x^n)^(-2-1/n-p),x, algorithm="maxima")
[Out]
integrate((b*x^n + a)^p*(d*x^n + c)^(-p - 1/n - 2), x)
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b x^{n} + a\right )}^{p}}{{\left (d x^{n} + c\right )}^{\frac{n p + 2 \, n + 1}{n}}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*x^n)^p*(c+d*x^n)^(-2-1/n-p),x, algorithm="fricas")
[Out]
integral((b*x^n + a)^p/(d*x^n + c)^((n*p + 2*n + 1)/n), x)
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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)**p*(c+d*x**n)**(-2-1/n-p),x)
[Out]
Timed out
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x^{n} + a\right )}^{p}{\left (d x^{n} + c\right )}^{-p - \frac{1}{n} - 2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*x^n)^p*(c+d*x^n)^(-2-1/n-p),x, algorithm="giac")
[Out]
integrate((b*x^n + a)^p*(d*x^n + c)^(-p - 1/n - 2), x)