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

Optimal. Leaf size=59 \[ \frac {x \left (a+b x^n\right )^p \left (\frac {b x^n}{a}+1\right )^{-p} F_1\left (\frac {1}{n};-p,2;1+\frac {1}{n};-\frac {b x^n}{a},-\frac {d x^n}{c}\right )}{c^2} \]

[Out]

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

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Rubi [A]  time = 0.03, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {430, 429} \[ \frac {x \left (a+b x^n\right )^p \left (\frac {b x^n}{a}+1\right )^{-p} F_1\left (\frac {1}{n};-p,2;1+\frac {1}{n};-\frac {b x^n}{a},-\frac {d x^n}{c}\right )}{c^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^n)^p/(c + d*x^n)^2,x]

[Out]

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

Rubi steps

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

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Mathematica [B]  time = 0.34, size = 180, normalized size = 3.05 \[ \frac {a c (n+1) x \left (a+b x^n\right )^p F_1\left (\frac {1}{n};-p,2;1+\frac {1}{n};-\frac {b x^n}{a},-\frac {d x^n}{c}\right )}{\left (c+d x^n\right )^2 \left (b c n p x^n F_1\left (1+\frac {1}{n};1-p,2;2+\frac {1}{n};-\frac {b x^n}{a},-\frac {d x^n}{c}\right )-2 a d n x^n F_1\left (1+\frac {1}{n};-p,3;2+\frac {1}{n};-\frac {b x^n}{a},-\frac {d x^n}{c}\right )+a c (n+1) F_1\left (\frac {1}{n};-p,2;1+\frac {1}{n};-\frac {b x^n}{a},-\frac {d x^n}{c}\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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fricas [F]  time = 0.90, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x^{n} + a\right )}^{p}}{d^{2} x^{2 \, n} + 2 \, c d x^{n} + c^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.90, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \,x^{n}+a \right )^{p}}{\left (d \,x^{n}+c \right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (a+b\,x^n\right )}^p}{{\left (c+d\,x^n\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^n)^p/(c + d*x^n)^2,x)

[Out]

int((a + b*x^n)^p/(c + d*x^n)^2, x)

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sympy [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: HeuristicGCDFailed} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: HeuristicGCDFailed

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