3.44 \(\int \frac{(e x)^m (a+b x^n)^p (A+B x^n)}{(c+d x^n)^2} \, dx\)

Optimal. Leaf size=304 \[ -\frac{(e x)^{m+1} \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} (a d (B c (m+1)-A d (m-n+1))+b c (A d (m-n (1-p)+1)-B c (m+n p+1))) F_1\left (\frac{m+1}{n};-p,1;\frac{m+n+1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right )}{c^2 d e (m+1) n (b c-a d)}-\frac{b (e x)^{m+1} (m+n p+1) (B c-A d) \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+1}{n},-p;\frac{m+n+1}{n};-\frac{b x^n}{a}\right )}{c d e (m+1) n (b c-a d)}+\frac{(e x)^{m+1} (B c-A d) \left (a+b x^n\right )^{p+1}}{c e n (b c-a d) \left (c+d x^n\right )} \]

[Out]

((B*c - A*d)*(e*x)^(1 + m)*(a + b*x^n)^(1 + p))/(c*(b*c - a*d)*e*n*(c + d*x^n)) - ((a*d*(B*c*(1 + m) - A*d*(1
+ m - n)) + b*c*(A*d*(1 + m - n*(1 - p)) - B*c*(1 + m + n*p)))*(e*x)^(1 + m)*(a + b*x^n)^p*AppellF1[(1 + m)/n,
-p, 1, (1 + m + n)/n, -((b*x^n)/a), -((d*x^n)/c)])/(c^2*d*(b*c - a*d)*e*(1 + m)*n*(1 + (b*x^n)/a)^p) - (b*(B*
c - A*d)*(1 + m + n*p)*(e*x)^(1 + m)*(a + b*x^n)^p*Hypergeometric2F1[(1 + m)/n, -p, (1 + m + n)/n, -((b*x^n)/a
)])/(c*d*(b*c - a*d)*e*(1 + m)*n*(1 + (b*x^n)/a)^p)

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Rubi [A]  time = 0.536458, antiderivative size = 304, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.194, Rules used = {595, 597, 365, 364, 511, 510} \[ -\frac{(e x)^{m+1} \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} (a d (B c (m+1)-A d (m-n+1))+b c (A d (m-n (1-p)+1)-B c (m+n p+1))) F_1\left (\frac{m+1}{n};-p,1;\frac{m+n+1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right )}{c^2 d e (m+1) n (b c-a d)}-\frac{b (e x)^{m+1} (m+n p+1) (B c-A d) \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+1}{n},-p;\frac{m+n+1}{n};-\frac{b x^n}{a}\right )}{c d e (m+1) n (b c-a d)}+\frac{(e x)^{m+1} (B c-A d) \left (a+b x^n\right )^{p+1}}{c e n (b c-a d) \left (c+d x^n\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((B*c - A*d)*(e*x)^(1 + m)*(a + b*x^n)^(1 + p))/(c*(b*c - a*d)*e*n*(c + d*x^n)) - ((a*d*(B*c*(1 + m) - A*d*(1
+ m - n)) + b*c*(A*d*(1 + m - n*(1 - p)) - B*c*(1 + m + n*p)))*(e*x)^(1 + m)*(a + b*x^n)^p*AppellF1[(1 + m)/n,
-p, 1, (1 + m + n)/n, -((b*x^n)/a), -((d*x^n)/c)])/(c^2*d*(b*c - a*d)*e*(1 + m)*n*(1 + (b*x^n)/a)^p) - (b*(B*
c - A*d)*(1 + m + n*p)*(e*x)^(1 + m)*(a + b*x^n)^p*Hypergeometric2F1[(1 + m)/n, -p, (1 + m + n)/n, -((b*x^n)/a
)])/(c*d*(b*c - a*d)*e*(1 + m)*n*(1 + (b*x^n)/a)^p)

Rule 595

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_)*((e_) + (f_.)*(x_)^(n_)), x
_Symbol] :> -Simp[((b*e - a*f)*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*g*n*(b*c - a*d)*(p +
1)), x] + Dist[1/(a*n*(b*c - a*d)*(p + 1)), Int[(g*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f)*(
m + 1) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*f)*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d,
e, f, g, m, n, q}, x] && LtQ[p, -1]

Rule 597

Int[(((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((e_) + (f_.)*(x_)^(n_)))/((c_) + (d_.)*(x_)^(n_)), x_Sy
mbol] :> Int[ExpandIntegrand[((g*x)^m*(a + b*x^n)^p*(e + f*x^n))/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, e,
f, g, m, n, p}, x]

Rule 365

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a + b*x^n)^FracPart[p])
/(1 + (b*x^n)/a)^FracPart[p], Int[(c*x)^m*(1 + (b*x^n)/a)^p, x], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[
p, 0] &&  !(ILtQ[p, 0] || GtQ[a, 0])

Rule 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 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 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])

Rubi steps

\begin{align*} \int \frac{(e x)^m \left (a+b x^n\right )^p \left (A+B x^n\right )}{\left (c+d x^n\right )^2} \, dx &=\frac{(B c-A d) (e x)^{1+m} \left (a+b x^n\right )^{1+p}}{c (b c-a d) e n \left (c+d x^n\right )}+\frac{\int \frac{(e x)^m \left (a+b x^n\right )^p \left (-a (B c-A d) (1+m)+A (b c-a d) n-b (B c-A d) (1+m+n p) x^n\right )}{c+d x^n} \, dx}{c (b c-a d) n}\\ &=\frac{(B c-A d) (e x)^{1+m} \left (a+b x^n\right )^{1+p}}{c (b c-a d) e n \left (c+d x^n\right )}+\frac{\int \left (-\frac{b (B c-A d) (1+m+n p) (e x)^m \left (a+b x^n\right )^p}{d}+\frac{(d (-a (B c-A d) (1+m)+A (b c-a d) n)+b c (B c-A d) (1+m+n p)) (e x)^m \left (a+b x^n\right )^p}{d \left (c+d x^n\right )}\right ) \, dx}{c (b c-a d) n}\\ &=\frac{(B c-A d) (e x)^{1+m} \left (a+b x^n\right )^{1+p}}{c (b c-a d) e n \left (c+d x^n\right )}-\frac{(b (B c-A d) (1+m+n p)) \int (e x)^m \left (a+b x^n\right )^p \, dx}{c d (b c-a d) n}+\frac{(d (-a (B c-A d) (1+m)+A (b c-a d) n)+b c (B c-A d) (1+m+n p)) \int \frac{(e x)^m \left (a+b x^n\right )^p}{c+d x^n} \, dx}{c d (b c-a d) n}\\ &=\frac{(B c-A d) (e x)^{1+m} \left (a+b x^n\right )^{1+p}}{c (b c-a d) e n \left (c+d x^n\right )}-\frac{\left (b (B c-A d) (1+m+n p) \left (a+b x^n\right )^p \left (1+\frac{b x^n}{a}\right )^{-p}\right ) \int (e x)^m \left (1+\frac{b x^n}{a}\right )^p \, dx}{c d (b c-a d) n}+\frac{\left ((d (-a (B c-A d) (1+m)+A (b c-a d) n)+b c (B c-A d) (1+m+n p)) \left (a+b x^n\right )^p \left (1+\frac{b x^n}{a}\right )^{-p}\right ) \int \frac{(e x)^m \left (1+\frac{b x^n}{a}\right )^p}{c+d x^n} \, dx}{c d (b c-a d) n}\\ &=\frac{(B c-A d) (e x)^{1+m} \left (a+b x^n\right )^{1+p}}{c (b c-a d) e n \left (c+d x^n\right )}-\frac{(a d (B c-A d) (1+m)-A d (b c-a d) n-b c (B c-A d) (1+m+n p)) (e x)^{1+m} \left (a+b x^n\right )^p \left (1+\frac{b x^n}{a}\right )^{-p} F_1\left (\frac{1+m}{n};-p,1;\frac{1+m+n}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right )}{c^2 d (b c-a d) e (1+m) n}-\frac{b (B c-A d) (1+m+n p) (e x)^{1+m} \left (a+b x^n\right )^p \left (1+\frac{b x^n}{a}\right )^{-p} \, _2F_1\left (\frac{1+m}{n},-p;\frac{1+m+n}{n};-\frac{b x^n}{a}\right )}{c d (b c-a d) e (1+m) n}\\ \end{align*}

Mathematica [A]  time = 0.347816, size = 138, normalized size = 0.45 \[ \frac{x (e x)^m \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} \left (A (m+n+1) F_1\left (\frac{m+1}{n};-p,2;\frac{m+n+1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right )+B (m+1) x^n F_1\left (\frac{m+n+1}{n};-p,2;\frac{m+2 n+1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right )\right )}{c^2 (m+1) (m+n+1)} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [F]  time = 0.669, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( ex \right ) ^{m} \left ( a+b{x}^{n} \right ) ^{p} \left ( A+B{x}^{n} \right ) }{ \left ( c+d{x}^{n} \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (B x^{n} + A\right )}{\left (b x^{n} + a\right )}^{p} \left (e x\right )^{m}}{{\left (d x^{n} + c\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((B*x^n + A)*(b*x^n + a)^p*(e*x)^m/(d*x^n + c)^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 )}{\left (b x^{n} + a\right )}^{p} \left (e x\right )^{m}}{d^{2} x^{2 \, n} + 2 \, c d x^{n} + c^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (B x^{n} + A\right )}{\left (b x^{n} + a\right )}^{p} \left (e x\right )^{m}}{{\left (d x^{n} + c\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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