3.64 \(\int \frac {(a+c x^{2 n})^p}{(d+e x^n)^2} \, dx\)

Optimal. Leaf size=261 \[ \frac {x \left (a+c x^{2 n}\right )^p \left (\frac {c x^{2 n}}{a}+1\right )^{-p} F_1\left (\frac {1}{2 n};-p,2;\frac {1}{2} \left (2+\frac {1}{n}\right );-\frac {c x^{2 n}}{a},\frac {e^2 x^{2 n}}{d^2}\right )}{d^2}+\frac {e^2 x^{2 n+1} \left (a+c x^{2 n}\right )^p \left (\frac {c x^{2 n}}{a}+1\right )^{-p} F_1\left (\frac {1}{2} \left (2+\frac {1}{n}\right );-p,2;\frac {1}{2} \left (4+\frac {1}{n}\right );-\frac {c x^{2 n}}{a},\frac {e^2 x^{2 n}}{d^2}\right )}{d^4 (2 n+1)}-\frac {2 e x^{n+1} \left (a+c x^{2 n}\right )^p \left (\frac {c x^{2 n}}{a}+1\right )^{-p} F_1\left (\frac {n+1}{2 n};-p,2;\frac {1}{2} \left (3+\frac {1}{n}\right );-\frac {c x^{2 n}}{a},\frac {e^2 x^{2 n}}{d^2}\right )}{d^3 (n+1)} \]

[Out]

e^2*x^(1+2*n)*(a+c*x^(2*n))^p*AppellF1(1+1/2/n,2,-p,2+1/2/n,e^2*x^(2*n)/d^2,-c*x^(2*n)/a)/d^4/(1+2*n)/((1+c*x^
(2*n)/a)^p)+x*(a+c*x^(2*n))^p*AppellF1(1/2/n,2,-p,1+1/2/n,e^2*x^(2*n)/d^2,-c*x^(2*n)/a)/d^2/((1+c*x^(2*n)/a)^p
)-2*e*x^(1+n)*(a+c*x^(2*n))^p*AppellF1(1/2*(1+n)/n,2,-p,3/2+1/2/n,e^2*x^(2*n)/d^2,-c*x^(2*n)/a)/d^3/(1+n)/((1+
c*x^(2*n)/a)^p)

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Rubi [A]  time = 0.24, antiderivative size = 261, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {1438, 430, 429, 511, 510} \[ -\frac {2 e x^{n+1} \left (a+c x^{2 n}\right )^p \left (\frac {c x^{2 n}}{a}+1\right )^{-p} F_1\left (\frac {n+1}{2 n};-p,2;\frac {1}{2} \left (3+\frac {1}{n}\right );-\frac {c x^{2 n}}{a},\frac {e^2 x^{2 n}}{d^2}\right )}{d^3 (n+1)}+\frac {e^2 x^{2 n+1} \left (a+c x^{2 n}\right )^p \left (\frac {c x^{2 n}}{a}+1\right )^{-p} F_1\left (\frac {1}{2} \left (2+\frac {1}{n}\right );-p,2;\frac {1}{2} \left (4+\frac {1}{n}\right );-\frac {c x^{2 n}}{a},\frac {e^2 x^{2 n}}{d^2}\right )}{d^4 (2 n+1)}+\frac {x \left (a+c x^{2 n}\right )^p \left (\frac {c x^{2 n}}{a}+1\right )^{-p} F_1\left (\frac {1}{2 n};-p,2;\frac {1}{2} \left (2+\frac {1}{n}\right );-\frac {c x^{2 n}}{a},\frac {e^2 x^{2 n}}{d^2}\right )}{d^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + c*x^(2*n))^p/(d + e*x^n)^2,x]

[Out]

(e^2*x^(1 + 2*n)*(a + c*x^(2*n))^p*AppellF1[(2 + n^(-1))/2, -p, 2, (4 + n^(-1))/2, -((c*x^(2*n))/a), (e^2*x^(2
*n))/d^2])/(d^4*(1 + 2*n)*(1 + (c*x^(2*n))/a)^p) + (x*(a + c*x^(2*n))^p*AppellF1[1/(2*n), -p, 2, (2 + n^(-1))/
2, -((c*x^(2*n))/a), (e^2*x^(2*n))/d^2])/(d^2*(1 + (c*x^(2*n))/a)^p) - (2*e*x^(1 + n)*(a + c*x^(2*n))^p*Appell
F1[(1 + n)/(2*n), -p, 2, (3 + n^(-1))/2, -((c*x^(2*n))/a), (e^2*x^(2*n))/d^2])/(d^3*(1 + n)*(1 + (c*x^(2*n))/a
)^p)

Rule 429

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^p*c^q*x*AppellF1[1/n, -p,
-q, 1 + 1/n, -((b*x^n)/a), -((d*x^n)/c)], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n
, -1] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rule 430

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[(a^IntPart[p]*(a + b*x^n)^F
racPart[p])/(1 + (b*x^n)/a)^FracPart[p], Int[(1 + (b*x^n)/a)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n,
p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n, -1] &&  !(IntegerQ[p] || GtQ[a, 0])

Rule 510

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(a^p*c^q
*(e*x)^(m + 1)*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, -((b*x^n)/a), -((d*x^n)/c)])/(e*(m + 1)), x] /; Free
Q[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] && (IntegerQ[p] || GtQ[a
, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rule 511

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[(a^IntPa
rt[p]*(a + b*x^n)^FracPart[p])/(1 + (b*x^n)/a)^FracPart[p], Int[(e*x)^m*(1 + (b*x^n)/a)^p*(c + d*x^n)^q, x], x
] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] &&  !(IntegerQ[
p] || GtQ[a, 0])

Rule 1438

Int[((d_) + (e_.)*(x_)^(n_))^(q_)*((a_) + (c_.)*(x_)^(n2_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + c*x^(2
*n))^p, (d/(d^2 - e^2*x^(2*n)) - (e*x^n)/(d^2 - e^2*x^(2*n)))^(-q), x], x] /; FreeQ[{a, c, d, e, n, p}, x] &&
EqQ[n2, 2*n] && NeQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && ILtQ[q, 0]

Rubi steps

\begin {align*} \int \frac {\left (a+c x^{2 n}\right )^p}{\left (d+e x^n\right )^2} \, dx &=\int \left (\frac {d^2 \left (a+c x^{2 n}\right )^p}{\left (d^2-e^2 x^{2 n}\right )^2}-\frac {2 d e x^n \left (a+c x^{2 n}\right )^p}{\left (-d^2+e^2 x^{2 n}\right )^2}+\frac {e^2 x^{2 n} \left (a+c x^{2 n}\right )^p}{\left (-d^2+e^2 x^{2 n}\right )^2}\right ) \, dx\\ &=d^2 \int \frac {\left (a+c x^{2 n}\right )^p}{\left (d^2-e^2 x^{2 n}\right )^2} \, dx-(2 d e) \int \frac {x^n \left (a+c x^{2 n}\right )^p}{\left (-d^2+e^2 x^{2 n}\right )^2} \, dx+e^2 \int \frac {x^{2 n} \left (a+c x^{2 n}\right )^p}{\left (-d^2+e^2 x^{2 n}\right )^2} \, dx\\ &=\left (d^2 \left (a+c x^{2 n}\right )^p \left (1+\frac {c x^{2 n}}{a}\right )^{-p}\right ) \int \frac {\left (1+\frac {c x^{2 n}}{a}\right )^p}{\left (d^2-e^2 x^{2 n}\right )^2} \, dx-\left (2 d e \left (a+c x^{2 n}\right )^p \left (1+\frac {c x^{2 n}}{a}\right )^{-p}\right ) \int \frac {x^n \left (1+\frac {c x^{2 n}}{a}\right )^p}{\left (-d^2+e^2 x^{2 n}\right )^2} \, dx+\left (e^2 \left (a+c x^{2 n}\right )^p \left (1+\frac {c x^{2 n}}{a}\right )^{-p}\right ) \int \frac {x^{2 n} \left (1+\frac {c x^{2 n}}{a}\right )^p}{\left (-d^2+e^2 x^{2 n}\right )^2} \, dx\\ &=\frac {e^2 x^{1+2 n} \left (a+c x^{2 n}\right )^p \left (1+\frac {c x^{2 n}}{a}\right )^{-p} F_1\left (\frac {1}{2} \left (2+\frac {1}{n}\right );-p,2;\frac {1}{2} \left (4+\frac {1}{n}\right );-\frac {c x^{2 n}}{a},\frac {e^2 x^{2 n}}{d^2}\right )}{d^4 (1+2 n)}+\frac {x \left (a+c x^{2 n}\right )^p \left (1+\frac {c x^{2 n}}{a}\right )^{-p} F_1\left (\frac {1}{2 n};-p,2;\frac {1}{2} \left (2+\frac {1}{n}\right );-\frac {c x^{2 n}}{a},\frac {e^2 x^{2 n}}{d^2}\right )}{d^2}-\frac {2 e x^{1+n} \left (a+c x^{2 n}\right )^p \left (1+\frac {c x^{2 n}}{a}\right )^{-p} F_1\left (\frac {1+n}{2 n};-p,2;\frac {1}{2} \left (3+\frac {1}{n}\right );-\frac {c x^{2 n}}{a},\frac {e^2 x^{2 n}}{d^2}\right )}{d^3 (1+n)}\\ \end {align*}

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Mathematica [F]  time = 0.10, size = 0, normalized size = 0.00 \[ \int \frac {\left (a+c x^{2 n}\right )^p}{\left (d+e x^n\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(a + c*x^(2*n))^p/(d + e*x^n)^2,x]

[Out]

Integrate[(a + c*x^(2*n))^p/(d + e*x^n)^2, x]

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fricas [F]  time = 1.39, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (c x^{2 \, n} + a\right )}^{p}}{e^{2} x^{2 \, n} + 2 \, d e x^{n} + d^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+c*x^(2*n))^p/(d+e*x^n)^2,x, algorithm="fricas")

[Out]

integral((c*x^(2*n) + a)^p/(e^2*x^(2*n) + 2*d*e*x^n + d^2), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (c x^{2 \, n} + a\right )}^{p}}{{\left (e x^{n} + d\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+c*x^(2*n))^p/(d+e*x^n)^2,x, algorithm="giac")

[Out]

integrate((c*x^(2*n) + a)^p/(e*x^n + d)^2, x)

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maple [F]  time = 0.10, size = 0, normalized size = 0.00 \[ \int \frac {\left (c \,x^{2 n}+a \right )^{p}}{\left (e \,x^{n}+d \right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+c*x^(2*n))^p/(d+e*x^n)^2,x, algorithm="maxima")

[Out]

integrate((c*x^(2*n) + a)^p/(e*x^n + d)^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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