3.7 \(\int \frac{(e x)^m (A+B x^n) (c+d x^n)}{(a+b x^n)^3} \, dx\)

Optimal. Leaf size=228 \[ -\frac{(e x)^{m+1} \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right ) (b c (m-n+1) (a B (m+1)-A b (m-2 n+1))+a d (m+1) (A b (m-n+1)-a B (m+n+1)))}{2 a^3 b^2 e (m+1) n^2}-\frac{(e x)^{m+1} (A b (b c (m-2 n+1)-a d (m-n+1))-a B (b c (m+1)-a d (m+n+1)))}{2 a^2 b^2 e n^2 \left (a+b x^n\right )}+\frac{(e x)^{m+1} (A b-a B) \left (c+d x^n\right )}{2 a b e n \left (a+b x^n\right )^2} \]

[Out]

-((A*b*(b*c*(1 + m - 2*n) - a*d*(1 + m - n)) - a*B*(b*c*(1 + m) - a*d*(1 + m + n)))*(e*x)^(1 + m))/(2*a^2*b^2*
e*n^2*(a + b*x^n)) + ((A*b - a*B)*(e*x)^(1 + m)*(c + d*x^n))/(2*a*b*e*n*(a + b*x^n)^2) - ((b*c*(a*B*(1 + m) -
A*b*(1 + m - 2*n))*(1 + m - n) + a*d*(1 + m)*(A*b*(1 + m - n) - a*B*(1 + m + n)))*(e*x)^(1 + m)*Hypergeometric
2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)])/(2*a^3*b^2*e*(1 + m)*n^2)

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Rubi [A]  time = 0.272938, antiderivative size = 228, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103, Rules used = {594, 457, 364} \[ -\frac{(e x)^{m+1} \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right ) (b c (m-n+1) (a B (m+1)-A b (m-2 n+1))+a d (m+1) (A b (m-n+1)-a B (m+n+1)))}{2 a^3 b^2 e (m+1) n^2}-\frac{(e x)^{m+1} (A b (b c (m-2 n+1)-a d (m-n+1))-a B (b c (m+1)-a d (m+n+1)))}{2 a^2 b^2 e n^2 \left (a+b x^n\right )}+\frac{(e x)^{m+1} (A b-a B) \left (c+d x^n\right )}{2 a b e n \left (a+b x^n\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[((e*x)^m*(A + B*x^n)*(c + d*x^n))/(a + b*x^n)^3,x]

[Out]

-((A*b*(b*c*(1 + m - 2*n) - a*d*(1 + m - n)) - a*B*(b*c*(1 + m) - a*d*(1 + m + n)))*(e*x)^(1 + m))/(2*a^2*b^2*
e*n^2*(a + b*x^n)) + ((A*b - a*B)*(e*x)^(1 + m)*(c + d*x^n))/(2*a*b*e*n*(a + b*x^n)^2) - ((b*c*(a*B*(1 + m) -
A*b*(1 + m - 2*n))*(1 + m - n) + a*d*(1 + m)*(A*b*(1 + m - n) - a*B*(1 + m + n)))*(e*x)^(1 + m)*Hypergeometric
2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)])/(2*a^3*b^2*e*(1 + m)*n^2)

Rule 594

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)/(a*b*g*n*(p + 1)), x] + Dist[
1/(a*b*n*(p + 1)), Int[(g*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 1)*Simp[c*(b*e*n*(p + 1) + (b*e - a*f)*(m
+ 1)) + d*(b*e*n*(p + 1) + (b*e - a*f)*(m + n*q + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x]
&& LtQ[p, -1] && GtQ[q, 0] &&  !(EqQ[q, 1] && SimplerQ[b*c - a*d, b*e - a*f])

Rule 457

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> -Simp[((b*c - a*d
)*(e*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*b*e*n*(p + 1)), x] - Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*b
*n*(p + 1)), Int[(e*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] &
& LtQ[p, -1] && (( !IntegerQ[p + 1/2] && NeQ[p, -5/4]) ||  !RationalQ[m] || (IGtQ[n, 0] && ILtQ[p + 1/2, 0] &&
LeQ[-1, m, -(n*(p + 1))]))

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])

Rubi steps

\begin{align*} \int \frac{(e x)^m \left (A+B x^n\right ) \left (c+d x^n\right )}{\left (a+b x^n\right )^3} \, dx &=\frac{(A b-a B) (e x)^{1+m} \left (c+d x^n\right )}{2 a b e n \left (a+b x^n\right )^2}-\frac{\int \frac{(e x)^m \left (-c (a B (1+m)-A b (1+m-2 n))+d (A b (1+m-n)-a B (1+m+n)) x^n\right )}{\left (a+b x^n\right )^2} \, dx}{2 a b n}\\ &=-\frac{(A b (b c (1+m-2 n)-a d (1+m-n))-a B (b c (1+m)-a d (1+m+n))) (e x)^{1+m}}{2 a^2 b^2 e n^2 \left (a+b x^n\right )}+\frac{(A b-a B) (e x)^{1+m} \left (c+d x^n\right )}{2 a b e n \left (a+b x^n\right )^2}-\frac{(b c (a B (1+m)-A b (1+m-2 n)) (1+m-n)+a d (1+m) (A b (1+m-n)-a B (1+m+n))) \int \frac{(e x)^m}{a+b x^n} \, dx}{2 a^2 b^2 n^2}\\ &=-\frac{(A b (b c (1+m-2 n)-a d (1+m-n))-a B (b c (1+m)-a d (1+m+n))) (e x)^{1+m}}{2 a^2 b^2 e n^2 \left (a+b x^n\right )}+\frac{(A b-a B) (e x)^{1+m} \left (c+d x^n\right )}{2 a b e n \left (a+b x^n\right )^2}-\frac{(b c (a B (1+m)-A b (1+m-2 n)) (1+m-n)+a d (1+m) (A b (1+m-n)-a B (1+m+n))) (e x)^{1+m} \, _2F_1\left (1,\frac{1+m}{n};\frac{1+m+n}{n};-\frac{b x^n}{a}\right )}{2 a^3 b^2 e (1+m) n^2}\\ \end{align*}

Mathematica [A]  time = 0.161867, size = 136, normalized size = 0.6 \[ \frac{x (e x)^m \left (a^2 B d \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right )+a (-2 a B d+A b d+b B c) \, _2F_1\left (2,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right )+(A b-a B) (b c-a d) \, _2F_1\left (3,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right )\right )}{a^3 b^2 (m+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[((e*x)^m*(A + B*x^n)*(c + d*x^n))/(a + b*x^n)^3,x]

[Out]

(x*(e*x)^m*(a^2*B*d*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)] + a*(b*B*c + A*b*d - 2*a*B*d)
*Hypergeometric2F1[2, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)] + (A*b - a*B)*(b*c - a*d)*Hypergeometric2F1[3, (
1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)]))/(a^3*b^2*(1 + m))

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Maple [F]  time = 0.354, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( ex \right ) ^{m} \left ( A+B{x}^{n} \right ) \left ( c+d{x}^{n} \right ) }{ \left ( a+b{x}^{n} \right ) ^{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x)^m*(A+B*x^n)*(c+d*x^n)/(a+b*x^n)^3,x)

[Out]

int((e*x)^m*(A+B*x^n)*(c+d*x^n)/(a+b*x^n)^3,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\left ({\left ({\left (m^{2} - m{\left (3 \, n - 2\right )} + 2 \, n^{2} - 3 \, n + 1\right )} b^{2} c e^{m} -{\left (m^{2} - m{\left (n - 2\right )} - n + 1\right )} a b d e^{m}\right )} A -{\left ({\left (m^{2} - m{\left (n - 2\right )} - n + 1\right )} a b c e^{m} -{\left (m^{2} + m{\left (n + 2\right )} + n + 1\right )} a^{2} d e^{m}\right )} B\right )} \int \frac{x^{m}}{2 \,{\left (a^{2} b^{3} n^{2} x^{n} + a^{3} b^{2} n^{2}\right )}}\,{d x} + \frac{{\left ({\left (a^{2} b d e^{m}{\left (m - n + 1\right )} - a b^{2} c e^{m}{\left (m - 3 \, n + 1\right )}\right )} A -{\left (a^{3} d e^{m}{\left (m + n + 1\right )} - a^{2} b c e^{m}{\left (m - n + 1\right )}\right )} B\right )} x x^{m} -{\left ({\left (b^{3} c e^{m}{\left (m - 2 \, n + 1\right )} - a b^{2} d e^{m}{\left (m + 1\right )}\right )} A +{\left (a^{2} b d e^{m}{\left (m + 2 \, n + 1\right )} - a b^{2} c e^{m}{\left (m + 1\right )}\right )} B\right )} x e^{\left (m \log \left (x\right ) + n \log \left (x\right )\right )}}{2 \,{\left (a^{2} b^{4} n^{2} x^{2 \, n} + 2 \, a^{3} b^{3} n^{2} x^{n} + a^{4} b^{2} n^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^m*(A+B*x^n)*(c+d*x^n)/(a+b*x^n)^3,x, algorithm="maxima")

[Out]

(((m^2 - m*(3*n - 2) + 2*n^2 - 3*n + 1)*b^2*c*e^m - (m^2 - m*(n - 2) - n + 1)*a*b*d*e^m)*A - ((m^2 - m*(n - 2)
- n + 1)*a*b*c*e^m - (m^2 + m*(n + 2) + n + 1)*a^2*d*e^m)*B)*integrate(1/2*x^m/(a^2*b^3*n^2*x^n + a^3*b^2*n^2
), x) + 1/2*(((a^2*b*d*e^m*(m - n + 1) - a*b^2*c*e^m*(m - 3*n + 1))*A - (a^3*d*e^m*(m + n + 1) - a^2*b*c*e^m*(
m - n + 1))*B)*x*x^m - ((b^3*c*e^m*(m - 2*n + 1) - a*b^2*d*e^m*(m + 1))*A + (a^2*b*d*e^m*(m + 2*n + 1) - a*b^2
*c*e^m*(m + 1))*B)*x*e^(m*log(x) + n*log(x)))/(a^2*b^4*n^2*x^(2*n) + 2*a^3*b^3*n^2*x^n + a^4*b^2*n^2)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (B d x^{2 \, n} + A c +{\left (B c + A d\right )} x^{n}\right )} \left (e x\right )^{m}}{b^{3} x^{3 \, n} + 3 \, a b^{2} x^{2 \, n} + 3 \, a^{2} b x^{n} + a^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^m*(A+B*x^n)*(c+d*x^n)/(a+b*x^n)^3,x, algorithm="fricas")

[Out]

integral((B*d*x^(2*n) + A*c + (B*c + A*d)*x^n)*(e*x)^m/(b^3*x^(3*n) + 3*a*b^2*x^(2*n) + 3*a^2*b*x^n + a^3), 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)*(c+d*x**n)/(a+b*x**n)**3,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 (d x^{n} + c\right )} \left (e x\right )^{m}}{{\left (b x^{n} + a\right )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^m*(A+B*x^n)*(c+d*x^n)/(a+b*x^n)^3,x, algorithm="giac")

[Out]

integrate((B*x^n + A)*(d*x^n + c)*(e*x)^m/(b*x^n + a)^3, x)