3.35 \(\int \frac{(e x)^m \left (A+B x^n\right )}{\left (a+b x^n\right )^3 \left (c+d x^n\right )^2} \, dx\)

Optimal. Leaf size=567 \[ \frac{d (e x)^{m+1} \left (A \left (-2 a^2 d^2 n+a b c d (m-6 n+1)-b^2 c^2 (m-2 n+1)\right )+a B c (b c (m+1)-a d (m-6 n+1))\right )}{2 a^2 c e n^2 (b c-a d)^3 \left (c+d x^n\right )}+\frac{(e x)^{m+1} (A b (a d (m-5 n+1)-b c (m-2 n+1))+a B (b c (m+1)-a d (m-3 n+1)))}{2 a^2 e n^2 (b c-a d)^2 \left (a+b x^n\right ) \left (c+d x^n\right )}+\frac{b (e x)^{m+1} \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right ) \left (A b \left (a^2 d^2 \left (m^2+m (2-7 n)+12 n^2-7 n+1\right )-2 a b c d \left (m^2+m (2-5 n)+4 n^2-5 n+1\right )+b^2 c^2 \left (m^2+m (2-3 n)+2 n^2-3 n+1\right )\right )+a B \left (-a^2 d^2 \left (m^2+m (2-5 n)+6 n^2-5 n+1\right )+2 a b c d (m+1) (m-3 n+1)-b^2 c^2 (m+1) (m-n+1)\right )\right )}{2 a^3 e (m+1) n^2 (b c-a d)^4}+\frac{d^2 (e x)^{m+1} \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right ) (a d (B c (m+1)-A d (m-n+1))+b c (A d (m-4 n+1)-B c (m-3 n+1)))}{c^2 e (m+1) n (b c-a d)^4}+\frac{(e x)^{m+1} (A b-a B)}{2 a e n (b c-a d) \left (a+b x^n\right )^2 \left (c+d x^n\right )} \]

[Out]

(d*(a*B*c*(b*c*(1 + m) - a*d*(1 + m - 6*n)) + A*(a*b*c*d*(1 + m - 6*n) - b^2*c^2
*(1 + m - 2*n) - 2*a^2*d^2*n))*(e*x)^(1 + m))/(2*a^2*c*(b*c - a*d)^3*e*n^2*(c +
d*x^n)) + ((A*b - a*B)*(e*x)^(1 + m))/(2*a*(b*c - a*d)*e*n*(a + b*x^n)^2*(c + d*
x^n)) + ((a*B*(b*c*(1 + m) - a*d*(1 + m - 3*n)) + A*b*(a*d*(1 + m - 5*n) - b*c*(
1 + m - 2*n)))*(e*x)^(1 + m))/(2*a^2*(b*c - a*d)^2*e*n^2*(a + b*x^n)*(c + d*x^n)
) + (b*(a*B*(2*a*b*c*d*(1 + m)*(1 + m - 3*n) - b^2*c^2*(1 + m)*(1 + m - n) - a^2
*d^2*(1 + m^2 + m*(2 - 5*n) - 5*n + 6*n^2)) + A*b*(b^2*c^2*(1 + m^2 + m*(2 - 3*n
) - 3*n + 2*n^2) - 2*a*b*c*d*(1 + m^2 + m*(2 - 5*n) - 5*n + 4*n^2) + a^2*d^2*(1
+ m^2 + m*(2 - 7*n) - 7*n + 12*n^2)))*(e*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)
/n, (1 + m + n)/n, -((b*x^n)/a)])/(2*a^3*(b*c - a*d)^4*e*(1 + m)*n^2) + (d^2*(b*
c*(A*d*(1 + m - 4*n) - B*c*(1 + m - 3*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)^4*e*(1 + m)*n)

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Rubi [A]  time = 6.17036, antiderivative size = 567, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097 \[ \frac{d (e x)^{m+1} \left (A \left (-2 a^2 d^2 n+a b c d (m-6 n+1)-b^2 c^2 (m-2 n+1)\right )+a B c (b c (m+1)-a d (m-6 n+1))\right )}{2 a^2 c e n^2 (b c-a d)^3 \left (c+d x^n\right )}+\frac{(e x)^{m+1} (A b (a d (m-5 n+1)-b c (m-2 n+1))+a B (b c (m+1)-a d (m-3 n+1)))}{2 a^2 e n^2 (b c-a d)^2 \left (a+b x^n\right ) \left (c+d x^n\right )}+\frac{b (e x)^{m+1} \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right ) \left (A b \left (a^2 d^2 \left (m^2+m (2-7 n)+12 n^2-7 n+1\right )-2 a b c d \left (m^2+m (2-5 n)+4 n^2-5 n+1\right )+b^2 c^2 \left (m^2+m (2-3 n)+2 n^2-3 n+1\right )\right )+a B \left (-a^2 d^2 \left (m^2+m (2-5 n)+6 n^2-5 n+1\right )+2 a b c d (m+1) (m-3 n+1)-b^2 c^2 (m+1) (m-n+1)\right )\right )}{2 a^3 e (m+1) n^2 (b c-a d)^4}+\frac{d^2 (e x)^{m+1} \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right ) (a d (B c (m+1)-A d (m-n+1))+b c (A d (m-4 n+1)-B c (m-3 n+1)))}{c^2 e (m+1) n (b c-a d)^4}+\frac{(e x)^{m+1} (A b-a B)}{2 a e n (b c-a d) \left (a+b x^n\right )^2 \left (c+d x^n\right )} \]

Antiderivative was successfully verified.

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

[Out]

(d*(a*B*c*(b*c*(1 + m) - a*d*(1 + m - 6*n)) + A*(a*b*c*d*(1 + m - 6*n) - b^2*c^2
*(1 + m - 2*n) - 2*a^2*d^2*n))*(e*x)^(1 + m))/(2*a^2*c*(b*c - a*d)^3*e*n^2*(c +
d*x^n)) + ((A*b - a*B)*(e*x)^(1 + m))/(2*a*(b*c - a*d)*e*n*(a + b*x^n)^2*(c + d*
x^n)) + ((a*B*(b*c*(1 + m) - a*d*(1 + m - 3*n)) + A*b*(a*d*(1 + m - 5*n) - b*c*(
1 + m - 2*n)))*(e*x)^(1 + m))/(2*a^2*(b*c - a*d)^2*e*n^2*(a + b*x^n)*(c + d*x^n)
) + (b*(a*B*(2*a*b*c*d*(1 + m)*(1 + m - 3*n) - b^2*c^2*(1 + m)*(1 + m - n) - a^2
*d^2*(1 + m^2 + m*(2 - 5*n) - 5*n + 6*n^2)) + A*b*(b^2*c^2*(1 + m^2 + m*(2 - 3*n
) - 3*n + 2*n^2) - 2*a*b*c*d*(1 + m^2 + m*(2 - 5*n) - 5*n + 4*n^2) + a^2*d^2*(1
+ m^2 + m*(2 - 7*n) - 7*n + 12*n^2)))*(e*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)
/n, (1 + m + n)/n, -((b*x^n)/a)])/(2*a^3*(b*c - a*d)^4*e*(1 + m)*n^2) + (d^2*(b*
c*(A*d*(1 + m - 4*n) - B*c*(1 + m - 3*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)^4*e*(1 + m)*n)

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x)**m*(A+B*x**n)/(a+b*x**n)**3/(c+d*x**n)**2,x)

[Out]

Timed out

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Mathematica [B]  time = 5.04946, size = 2176, normalized size = 3.84 \[ \text{Result too large to show} \]

Antiderivative was successfully verified.

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

[Out]

(x*(e*x)^m*(2*a^3*c*d^2*(b*c - a*d)*(B*c - A*d)*(1 + m)*n*(a + b*x^n)^2 + a^2*b*
(A*b - a*B)*c^2*(b*c - a*d)^2*(1 + m)*n*(c + d*x^n) + a*b*c^2*(-(b*c) + a*d)*(1
+ m)*(a*B*(-(b*c*(1 + m)) + a*d*(1 + m - 4*n)) + A*b*(-(a*d*(1 + m - 6*n)) + b*c
*(1 + m - 2*n)))*(a + b*x^n)*(c + d*x^n) + A*b^4*c^4*(a + b*x^n)^2*(c + d*x^n)*H
ypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)] - a*b^3*B*c^4*(a + b
*x^n)^2*(c + d*x^n)*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)]
- 2*a*A*b^3*c^3*d*(a + b*x^n)^2*(c + d*x^n)*Hypergeometric2F1[1, (1 + m)/n, (1
+ m + n)/n, -((b*x^n)/a)] + 2*a^2*b^2*B*c^3*d*(a + b*x^n)^2*(c + d*x^n)*Hypergeo
metric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)] + a^2*A*b^2*c^2*d^2*(a + b*
x^n)^2*(c + d*x^n)*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)]
- a^3*b*B*c^2*d^2*(a + b*x^n)^2*(c + d*x^n)*Hypergeometric2F1[1, (1 + m)/n, (1 +
m + n)/n, -((b*x^n)/a)] + 2*A*b^4*c^4*m*(a + b*x^n)^2*(c + d*x^n)*Hypergeometri
c2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)] - 2*a*b^3*B*c^4*m*(a + b*x^n)^2*
(c + d*x^n)*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)] - 4*a*A
*b^3*c^3*d*m*(a + b*x^n)^2*(c + d*x^n)*Hypergeometric2F1[1, (1 + m)/n, (1 + m +
n)/n, -((b*x^n)/a)] + 4*a^2*b^2*B*c^3*d*m*(a + b*x^n)^2*(c + d*x^n)*Hypergeometr
ic2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)] + 2*a^2*A*b^2*c^2*d^2*m*(a + b*
x^n)^2*(c + d*x^n)*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)]
- 2*a^3*b*B*c^2*d^2*m*(a + b*x^n)^2*(c + d*x^n)*Hypergeometric2F1[1, (1 + m)/n,
(1 + m + n)/n, -((b*x^n)/a)] + A*b^4*c^4*m^2*(a + b*x^n)^2*(c + d*x^n)*Hypergeom
etric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)] - a*b^3*B*c^4*m^2*(a + b*x^n
)^2*(c + d*x^n)*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)] - 2
*a*A*b^3*c^3*d*m^2*(a + b*x^n)^2*(c + d*x^n)*Hypergeometric2F1[1, (1 + m)/n, (1
+ m + n)/n, -((b*x^n)/a)] + 2*a^2*b^2*B*c^3*d*m^2*(a + b*x^n)^2*(c + d*x^n)*Hype
rgeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)] + a^2*A*b^2*c^2*d^2*m^2
*(a + b*x^n)^2*(c + d*x^n)*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x
^n)/a)] - a^3*b*B*c^2*d^2*m^2*(a + b*x^n)^2*(c + d*x^n)*Hypergeometric2F1[1, (1
+ m)/n, (1 + m + n)/n, -((b*x^n)/a)] - 3*A*b^4*c^4*n*(a + b*x^n)^2*(c + d*x^n)*H
ypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)] + a*b^3*B*c^4*n*(a +
b*x^n)^2*(c + d*x^n)*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a
)] + 10*a*A*b^3*c^3*d*n*(a + b*x^n)^2*(c + d*x^n)*Hypergeometric2F1[1, (1 + m)/n
, (1 + m + n)/n, -((b*x^n)/a)] - 6*a^2*b^2*B*c^3*d*n*(a + b*x^n)^2*(c + d*x^n)*H
ypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)] - 7*a^2*A*b^2*c^2*d^
2*n*(a + b*x^n)^2*(c + d*x^n)*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((
b*x^n)/a)] + 5*a^3*b*B*c^2*d^2*n*(a + b*x^n)^2*(c + d*x^n)*Hypergeometric2F1[1,
(1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)] - 3*A*b^4*c^4*m*n*(a + b*x^n)^2*(c + d*x
^n)*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)] + a*b^3*B*c^4*m
*n*(a + b*x^n)^2*(c + d*x^n)*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b
*x^n)/a)] + 10*a*A*b^3*c^3*d*m*n*(a + b*x^n)^2*(c + d*x^n)*Hypergeometric2F1[1,
(1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)] - 6*a^2*b^2*B*c^3*d*m*n*(a + b*x^n)^2*(c
+ d*x^n)*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)] - 7*a^2*A
*b^2*c^2*d^2*m*n*(a + b*x^n)^2*(c + d*x^n)*Hypergeometric2F1[1, (1 + m)/n, (1 +
m + n)/n, -((b*x^n)/a)] + 5*a^3*b*B*c^2*d^2*m*n*(a + b*x^n)^2*(c + d*x^n)*Hyperg
eometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)] + 2*A*b^4*c^4*n^2*(a + b*
x^n)^2*(c + d*x^n)*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)]
- 8*a*A*b^3*c^3*d*n^2*(a + b*x^n)^2*(c + d*x^n)*Hypergeometric2F1[1, (1 + m)/n,
(1 + m + n)/n, -((b*x^n)/a)] + 12*a^2*A*b^2*c^2*d^2*n^2*(a + b*x^n)^2*(c + d*x^n
)*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)] - 6*a^3*b*B*c^2*d
^2*n^2*(a + b*x^n)^2*(c + d*x^n)*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n,
-((b*x^n)/a)] + 2*a^3*d^2*n*(b*c*(A*d*(1 + m - 4*n) - B*c*(1 + m - 3*n)) + a*d*(
B*c*(1 + m) + A*d*(-1 - m + n)))*(a + b*x^n)^2*(c + d*x^n)*Hypergeometric2F1[1,
(1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)]))/(2*a^3*c^2*(b*c - a*d)^4*(1 + m)*n^2*(
a + b*x^n)^2*(c + d*x^n))

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Maple [F]  time = 0.156, size = 0, normalized size = 0. \[ \int{\frac{ \left ( ex \right ) ^{m} \left ( A+B{x}^{n} \right ) }{ \left ( a+b{x}^{n} \right ) ^{3} \left ( c+d{x}^{n} \right ) ^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int((e*x)^m*(A+B*x^n)/(a+b*x^n)^3/(c+d*x^n)^2,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^n + A)*(e*x)^m/((b*x^n + a)^3*(d*x^n + c)^2),x, algorithm="maxima")

[Out]

(((m^2 - m*(3*n - 2) + 2*n^2 - 3*n + 1)*b^4*c^2*e^m - 2*(m^2 - m*(5*n - 2) + 4*n
^2 - 5*n + 1)*a*b^3*c*d*e^m + (m^2 - m*(7*n - 2) + 12*n^2 - 7*n + 1)*a^2*b^2*d^2
*e^m)*A - ((m^2 - m*(n - 2) - n + 1)*a*b^3*c^2*e^m - 2*(m^2 - m*(3*n - 2) - 3*n
+ 1)*a^2*b^2*c*d*e^m + (m^2 - m*(5*n - 2) + 6*n^2 - 5*n + 1)*a^3*b*d^2*e^m)*B)*i
ntegrate(1/2*x^m/(a^3*b^4*c^4*n^2 - 4*a^4*b^3*c^3*d*n^2 + 6*a^5*b^2*c^2*d^2*n^2
- 4*a^6*b*c*d^3*n^2 + a^7*d^4*n^2 + (a^2*b^5*c^4*n^2 - 4*a^3*b^4*c^3*d*n^2 + 6*a
^4*b^3*c^2*d^2*n^2 - 4*a^5*b^2*c*d^3*n^2 + a^6*b*d^4*n^2)*x^n), x) - ((a*d^4*e^m
*(m - n + 1) - b*c*d^3*e^m*(m - 4*n + 1))*A + (b*c^2*d^2*e^m*(m - 3*n + 1) - a*c
*d^3*e^m*(m + 1))*B)*integrate(x^m/(b^4*c^6*n - 4*a*b^3*c^5*d*n + 6*a^2*b^2*c^4*
d^2*n - 4*a^3*b*c^3*d^3*n + a^4*c^2*d^4*n + (b^4*c^5*d*n - 4*a*b^3*c^4*d^2*n + 6
*a^2*b^2*c^3*d^3*n - 4*a^3*b*c^2*d^4*n + a^4*c*d^5*n)*x^n), x) - 1/2*(((a*b^3*c^
3*e^m*(m - 3*n + 1) - a^2*b^2*c^2*d*e^m*(m - 7*n + 1) + 2*a^4*d^3*e^m*n)*A - (a^
2*b^2*c^3*e^m*(m - n + 1) - a^3*b*c^2*d*e^m*(m - 5*n + 1) + 2*a^4*c*d^2*e^m*n)*B
)*x*x^m + ((b^4*c^2*d*e^m*(m - 2*n + 1) - a*b^3*c*d^2*e^m*(m - 6*n + 1) + 2*a^2*
b^2*d^3*e^m*n)*A + (a^2*b^2*c*d^2*e^m*(m - 6*n + 1) - a*b^3*c^2*d*e^m*(m + 1))*B
)*x*e^(m*log(x) + 2*n*log(x)) + ((b^4*c^3*e^m*(m - 2*n + 1) - a^2*b^2*c*d^2*e^m*
(m - 7*n + 1) + 3*a*b^3*c^2*d*e^m*n + 4*a^3*b*d^3*e^m*n)*A + (a^3*b*c*d^2*e^m*(m
- 9*n + 1) - a*b^3*c^3*e^m*(m + 1) - 3*a^2*b^2*c^2*d*e^m*n)*B)*x*e^(m*log(x) +
n*log(x)))/(a^4*b^3*c^5*n^2 - 3*a^5*b^2*c^4*d*n^2 + 3*a^6*b*c^3*d^2*n^2 - a^7*c^
2*d^3*n^2 + (a^2*b^5*c^4*d*n^2 - 3*a^3*b^4*c^3*d^2*n^2 + 3*a^4*b^3*c^2*d^3*n^2 -
a^5*b^2*c*d^4*n^2)*x^(3*n) + (a^2*b^5*c^5*n^2 - a^3*b^4*c^4*d*n^2 - 3*a^4*b^3*c
^3*d^2*n^2 + 5*a^5*b^2*c^2*d^3*n^2 - 2*a^6*b*c*d^4*n^2)*x^(2*n) + (2*a^3*b^4*c^5
*n^2 - 5*a^4*b^3*c^4*d*n^2 + 3*a^5*b^2*c^3*d^2*n^2 + a^6*b*c^2*d^3*n^2 - a^7*c*d
^4*n^2)*x^n)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (B x^{n} + A\right )} \left (e x\right )^{m}}{b^{3} d^{2} x^{5 \, n} + a^{3} c^{2} +{\left (2 \, b^{3} c d + 3 \, a b^{2} d^{2}\right )} x^{4 \, n} +{\left (b^{3} c^{2} + 6 \, a b^{2} c d + 3 \, a^{2} b d^{2}\right )} x^{3 \, n} +{\left (3 \, a b^{2} c^{2} + 6 \, a^{2} b c d + a^{3} d^{2}\right )} x^{2 \, n} +{\left (3 \, a^{2} b c^{2} + 2 \, a^{3} c d\right )} x^{n}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^n + A)*(e*x)^m/((b*x^n + a)^3*(d*x^n + c)^2),x, algorithm="fricas")

[Out]

integral((B*x^n + A)*(e*x)^m/(b^3*d^2*x^(5*n) + a^3*c^2 + (2*b^3*c*d + 3*a*b^2*d
^2)*x^(4*n) + (b^3*c^2 + 6*a*b^2*c*d + 3*a^2*b*d^2)*x^(3*n) + (3*a*b^2*c^2 + 6*a
^2*b*c*d + a^3*d^2)*x^(2*n) + (3*a^2*b*c^2 + 2*a^3*c*d)*x^n), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x)**m*(A+B*x**n)/(a+b*x**n)**3/(c+d*x**n)**2,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x^{n} + A\right )} \left (e x\right )^{m}}{{\left (b x^{n} + a\right )}^{3}{\left (d x^{n} + c\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^n + A)*(e*x)^m/((b*x^n + a)^3*(d*x^n + c)^2),x, algorithm="giac")

[Out]

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