3.4.28 \(\int (a+b x^n)^p (c+d x^n)^{-2-\frac {1}{n}-p} \, dx\) [328]

Optimal. Leaf size=193 \[ -\frac {b x \left (a+b x^n\right )^{1+p} \left (c+d x^n\right )^{-1-\frac {1}{n}-p}}{a (b c-a d) n (1+p)}+\frac {(b c+(b c-a d) n (1+p)) x \left (a+b x^n\right )^{1+p} \left (\frac {c \left (a+b x^n\right )}{a \left (c+d x^n\right )}\right )^{-1-p} \left (c+d x^n\right )^{-1-\frac {1}{n}-p} \, _2F_1\left (\frac {1}{n},-1-p;1+\frac {1}{n};-\frac {(b c-a d) x^n}{a \left (c+d x^n\right )}\right )}{a c (b c-a d) n (1+p)} \]

[Out]

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

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Rubi [A]
time = 0.06, 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 = {390, 388} \begin {gather*} \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)} \end {gather*}

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 388

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[x*((a + b*x^n)^p/(c*(c*((a
+ b*x^n)/(a*(c + d*x^n))))^p*(c + d*x^n)^(1/n + p)))*Hypergeometric2F1[1/n, -p, 1 + 1/n, (-(b*c - a*d))*(x^n/(
a*(c + d*x^n)))], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[n*(p + q + 1) + 1, 0]

Rule 390

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] && Eq
Q[n*(p + q + 2) + 1, 0] && (LtQ[p, -1] ||  !LtQ[q, -1]) && NeQ[p, -1]

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*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(1414\) vs. \(2(193)=386\).
time = 37.64, size = 1414, normalized size = 7.33 \begin {gather*} \frac {c^4 (1+n) (1+2 n) (1+3 n) x \left (a+b x^n\right )^{3+p} \left (c+d x^n\right )^{-2-\frac {1}{n}-p} \left (1+\frac {d x^n}{c}\right ) \Gamma \left (2+\frac {1}{n}\right ) \Gamma (-p) \left (\, _2F_1\left (1,-p;1+\frac {1}{n};\frac {(b c-a d) x^n}{c \left (a+b x^n\right )}\right )+\frac {d n x^n \left (\frac {c \, _2F_1\left (1,-p;2+\frac {1}{n};\frac {(b c-a d) x^n}{c \left (a+b x^n\right )}\right )}{1+n}+\frac {(b c-a d) x^n \Gamma \left (1+\frac {1}{n}\right ) \Gamma (1-p) \, _2F_1\left (2,1-p;3+\frac {1}{n};\frac {(b c-a d) x^n}{c \left (a+b x^n\right )}\right )}{(1+2 n) \left (a+b x^n\right ) \Gamma \left (2+\frac {1}{n}\right ) \Gamma (-p)}\right )}{c^2}\right )}{-c d (1+3 n) (1+n+n p) x^n \left (a+b x^n\right )^2 \left (c^2 (1+n) (1+2 n) \left (a+b x^n\right ) \Gamma \left (2+\frac {1}{n}\right ) \Gamma (-p) \, _2F_1\left (1,-p;1+\frac {1}{n};\frac {(b c-a d) x^n}{c \left (a+b x^n\right )}\right )+d n x^n \left (c (1+2 n) \left (a+b x^n\right ) \Gamma \left (2+\frac {1}{n}\right ) \Gamma (-p) \, _2F_1\left (1,-p;2+\frac {1}{n};\frac {(b c-a d) x^n}{c \left (a+b x^n\right )}\right )+(b c-a d) (1+n) x^n \Gamma \left (1+\frac {1}{n}\right ) \Gamma (1-p) \, _2F_1\left (2,1-p;3+\frac {1}{n};\frac {(b c-a d) x^n}{c \left (a+b x^n\right )}\right )\right )\right )+b c n (1+3 n) p x^n \left (a+b x^n\right ) \left (c+d x^n\right ) \left (c^2 (1+n) (1+2 n) \left (a+b x^n\right ) \Gamma \left (2+\frac {1}{n}\right ) \Gamma (-p) \, _2F_1\left (1,-p;1+\frac {1}{n};\frac {(b c-a d) x^n}{c \left (a+b x^n\right )}\right )+d n x^n \left (c (1+2 n) \left (a+b x^n\right ) \Gamma \left (2+\frac {1}{n}\right ) \Gamma (-p) \, _2F_1\left (1,-p;2+\frac {1}{n};\frac {(b c-a d) x^n}{c \left (a+b x^n\right )}\right )+(b c-a d) (1+n) x^n \Gamma \left (1+\frac {1}{n}\right ) \Gamma (1-p) \, _2F_1\left (2,1-p;3+\frac {1}{n};\frac {(b c-a d) x^n}{c \left (a+b x^n\right )}\right )\right )\right )+c (1+3 n) \left (a+b x^n\right )^2 \left (c+d x^n\right ) \left (c^2 (1+n) (1+2 n) \left (a+b x^n\right ) \Gamma \left (2+\frac {1}{n}\right ) \Gamma (-p) \, _2F_1\left (1,-p;1+\frac {1}{n};\frac {(b c-a d) x^n}{c \left (a+b x^n\right )}\right )+d n x^n \left (c (1+2 n) \left (a+b x^n\right ) \Gamma \left (2+\frac {1}{n}\right ) \Gamma (-p) \, _2F_1\left (1,-p;2+\frac {1}{n};\frac {(b c-a d) x^n}{c \left (a+b x^n\right )}\right )+(b c-a d) (1+n) x^n \Gamma \left (1+\frac {1}{n}\right ) \Gamma (1-p) \, _2F_1\left (2,1-p;3+\frac {1}{n};\frac {(b c-a d) x^n}{c \left (a+b x^n\right )}\right )\right )\right )+n^2 x^n \left (c+d x^n\right ) \left (a c^2 (-b c+a d) (1+2 n) (1+3 n) p \left (a+b x^n\right ) \Gamma \left (2+\frac {1}{n}\right ) \Gamma (-p) \, _2F_1\left (2,1-p;2+\frac {1}{n};\frac {(b c-a d) x^n}{c \left (a+b x^n\right )}\right )+c d (1+3 n) \left (a+b x^n\right )^2 \left (c (1+2 n) \left (a+b x^n\right ) \Gamma \left (2+\frac {1}{n}\right ) \Gamma (-p) \, _2F_1\left (1,-p;2+\frac {1}{n};\frac {(b c-a d) x^n}{c \left (a+b x^n\right )}\right )+(b c-a d) (1+n) x^n \Gamma \left (1+\frac {1}{n}\right ) \Gamma (1-p) \, _2F_1\left (2,1-p;3+\frac {1}{n};\frac {(b c-a d) x^n}{c \left (a+b x^n\right )}\right )\right )-d (b c-a d) x^n \left (b c (1+n) (1+3 n) x^n \left (a+b x^n\right ) \Gamma \left (1+\frac {1}{n}\right ) \Gamma (1-p) \, _2F_1\left (2,1-p;3+\frac {1}{n};\frac {(b c-a d) x^n}{c \left (a+b x^n\right )}\right )-c (1+n) (1+3 n) \left (a+b x^n\right )^2 \Gamma \left (1+\frac {1}{n}\right ) \Gamma (1-p) \, _2F_1\left (2,1-p;3+\frac {1}{n};\frac {(b c-a d) x^n}{c \left (a+b x^n\right )}\right )+a c n (1+3 n) p \left (a+b x^n\right ) \Gamma \left (2+\frac {1}{n}\right ) \Gamma (-p) \, _2F_1\left (2,1-p;3+\frac {1}{n};\frac {(b c-a d) x^n}{c \left (a+b x^n\right )}\right )-2 a (-b c+a d) n (1+n) (-1+p) x^n \Gamma \left (1+\frac {1}{n}\right ) \Gamma (1-p) \, _2F_1\left (3,2-p;4+\frac {1}{n};\frac {(b c-a d) x^n}{c \left (a+b x^n\right )}\right )\right )\right )} \end {gather*}

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 = 0.12, size = 0, normalized size = 0.00 \[\int \left (a +b \,x^{n}\right )^{p} \left (c +d \,x^{n}\right )^{-2-\frac {1}{n}-p}\, dx\]

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (a+b\,x^n\right )}^p}{{\left (c+d\,x^n\right )}^{p+\frac {1}{n}+2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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