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

Optimal. Leaf size=315 \[ \frac{b (e x)^{m+1} \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right ) (A b (a d (m-3 n+1)-b c (m-n+1))+a B (b c (m+1)-a d (m-2 n+1)))}{a^2 e (m+1) n (b c-a d)^3}-\frac{d (e x)^{m+1} \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right ) (a d (B c (m+1)-A d (m-n+1))+b c (A d (m-3 n+1)-B c (m-2 n+1)))}{c^2 e (m+1) n (b c-a d)^3}+\frac{d (e x)^{m+1} (a A d-2 a B c+A b c)}{a c e n (b c-a d)^2 \left (c+d x^n\right )}+\frac{(e x)^{m+1} (A b-a B)}{a e n (b c-a d) \left (a+b x^n\right ) \left (c+d x^n\right )} \]

[Out]

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

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Rubi [A]  time = 1.09656, antiderivative size = 315, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097, Rules used = {595, 597, 364} \[ \frac{b (e x)^{m+1} \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right ) (A b (a d (m-3 n+1)-b c (m-n+1))+a B (b c (m+1)-a d (m-2 n+1)))}{a^2 e (m+1) n (b c-a d)^3}-\frac{d (e x)^{m+1} \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right ) (a d (B c (m+1)-A d (m-n+1))+b c (A d (m-3 n+1)-B c (m-2 n+1)))}{c^2 e (m+1) n (b c-a d)^3}+\frac{d (e x)^{m+1} (a A d-2 a B c+A b c)}{a c e n (b c-a d)^2 \left (c+d x^n\right )}+\frac{(e x)^{m+1} (A b-a B)}{a e n (b c-a d) \left (a+b x^n\right ) \left (c+d x^n\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.316594, size = 209, normalized size = 0.66 \[ \frac{x (e x)^m \left (\frac{b (a B-A b) (a d-b c) \, _2F_1\left (2,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right )}{a^2}-\frac{d (b c-a d) (B c-A d) \, _2F_1\left (2,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right )}{c^2}+\frac{b (a B d-2 A b d+b B c) \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right )}{a}-\frac{d (a B d-2 A b d+b B c) \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right )}{c}\right )}{(m+1) (b c-a d)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.716, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( ex \right ) ^{m} \left ( A+B{x}^{n} \right ) }{ \left ( a+b{x}^{n} \right ) ^{2} \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)/(a+b*x^n)^2/(c+d*x^n)^2,x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\left ({\left (b^{3} c e^{m}{\left (m - n + 1\right )} - a b^{2} d e^{m}{\left (m - 3 \, n + 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 )} \int -\frac{x^{m}}{a^{2} b^{3} c^{3} n - 3 \, a^{3} b^{2} c^{2} d n + 3 \, a^{4} b c d^{2} n - a^{5} d^{3} n +{\left (a b^{4} c^{3} n - 3 \, a^{2} b^{3} c^{2} d n + 3 \, a^{3} b^{2} c d^{2} n - a^{4} b d^{3} n\right )} x^{n}}\,{d x} -{\left ({\left (a d^{3} e^{m}{\left (m - n + 1\right )} - b c d^{2} e^{m}{\left (m - 3 \, n + 1\right )}\right )} A +{\left (b c^{2} d e^{m}{\left (m - 2 \, n + 1\right )} - a c d^{2} e^{m}{\left (m + 1\right )}\right )} B\right )} \int -\frac{x^{m}}{b^{3} c^{5} n - 3 \, a b^{2} c^{4} d n + 3 \, a^{2} b c^{3} d^{2} n - a^{3} c^{2} d^{3} n +{\left (b^{3} c^{4} d n - 3 \, a b^{2} c^{3} d^{2} n + 3 \, a^{2} b c^{2} d^{3} n - a^{3} c d^{4} n\right )} x^{n}}\,{d x} + \frac{{\left ({\left (b^{2} c^{2} e^{m} + a^{2} d^{2} e^{m}\right )} A -{\left (a b c^{2} e^{m} + a^{2} c d e^{m}\right )} B\right )} x x^{m} -{\left (2 \, B a b c d e^{m} -{\left (b^{2} c d e^{m} + a b d^{2} e^{m}\right )} A\right )} x e^{\left (m \log \left (x\right ) + n \log \left (x\right )\right )}}{a^{2} b^{2} c^{4} n - 2 \, a^{3} b c^{3} d n + a^{4} c^{2} d^{2} n +{\left (a b^{3} c^{3} d n - 2 \, a^{2} b^{2} c^{2} d^{2} n + a^{3} b c d^{3} n\right )} x^{2 \, n} +{\left (a b^{3} c^{4} n - a^{2} b^{2} c^{3} d n - a^{3} b c^{2} d^{2} n + a^{4} c d^{3} n\right )} x^{n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((b^3*c*e^m*(m - n + 1) - a*b^2*d*e^m*(m - 3*n + 1))*A + (a^2*b*d*e^m*(m - 2*n + 1) - a*b^2*c*e^m*(m + 1))*B)*
integrate(-x^m/(a^2*b^3*c^3*n - 3*a^3*b^2*c^2*d*n + 3*a^4*b*c*d^2*n - a^5*d^3*n + (a*b^4*c^3*n - 3*a^2*b^3*c^2
*d*n + 3*a^3*b^2*c*d^2*n - a^4*b*d^3*n)*x^n), x) - ((a*d^3*e^m*(m - n + 1) - b*c*d^2*e^m*(m - 3*n + 1))*A + (b
*c^2*d*e^m*(m - 2*n + 1) - a*c*d^2*e^m*(m + 1))*B)*integrate(-x^m/(b^3*c^5*n - 3*a*b^2*c^4*d*n + 3*a^2*b*c^3*d
^2*n - a^3*c^2*d^3*n + (b^3*c^4*d*n - 3*a*b^2*c^3*d^2*n + 3*a^2*b*c^2*d^3*n - a^3*c*d^4*n)*x^n), x) + (((b^2*c
^2*e^m + a^2*d^2*e^m)*A - (a*b*c^2*e^m + a^2*c*d*e^m)*B)*x*x^m - (2*B*a*b*c*d*e^m - (b^2*c*d*e^m + a*b*d^2*e^m
)*A)*x*e^(m*log(x) + n*log(x)))/(a^2*b^2*c^4*n - 2*a^3*b*c^3*d*n + a^4*c^2*d^2*n + (a*b^3*c^3*d*n - 2*a^2*b^2*
c^2*d^2*n + a^3*b*c*d^3*n)*x^(2*n) + (a*b^3*c^4*n - a^2*b^2*c^3*d*n - a^3*b*c^2*d^2*n + a^4*c*d^3*n)*x^n)

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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 (e x\right )^{m}}{b^{2} d^{2} x^{4 \, n} + a^{2} c^{2} + 2 \,{\left (b^{2} c d + a b d^{2}\right )} x^{3 \, n} +{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{2 \, n} + 2 \,{\left (a b c^{2} + a^{2} c d\right )} x^{n}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((B*x^n + A)*(e*x)^m/(b^2*d^2*x^(4*n) + a^2*c^2 + 2*(b^2*c*d + a*b*d^2)*x^(3*n) + (b^2*c^2 + 4*a*b*c*d
+ a^2*d^2)*x^(2*n) + 2*(a*b*c^2 + a^2*c*d)*x^n), 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)/(a+b*x**n)**2/(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 (e x\right )^{m}}{{\left (b x^{n} + a\right )}^{2}{\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)/(a+b*x^n)^2/(c+d*x^n)^2,x, algorithm="giac")

[Out]

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