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

Optimal. Leaf size=482 \[ -\frac{d (e x)^{m+1} \left (a^2 d (B c (m+1)-A d (m-2 n+1))-a b c (m-6 n+1) (B c-A d)-2 A b^2 c^2 n\right )}{2 a c^2 e n^2 (b c-a d)^3 \left (c+d x^n\right )}+\frac{d (e x)^{m+1} \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right ) \left (-a^2 d^2 (m-n+1) (B c (m+1)-A d (m-2 n+1))+2 a b c d \left (B c (m+1) (m-3 n+1)-A d \left (m^2+m (2-5 n)+4 n^2-5 n+1\right )\right )+b^2 c^2 (m-3 n+1) (A d (m-4 n+1)-B c (m-2 n+1))\right )}{2 c^3 e (m+1) n^2 (b c-a d)^4}+\frac{b^2 (e x)^{m+1} \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right ) (A b (a d (m-4 n+1)-b c (m-n+1))+a B (b c (m+1)-a d (m-3 n+1)))}{a^2 e (m+1) n (b c-a d)^4}+\frac{d (e x)^{m+1} (a A d-3 a B c+2 A b c)}{2 a c e n (b c-a d)^2 \left (c+d x^n\right )^2}+\frac{(e x)^{m+1} (A b-a B)}{a e n (b c-a d) \left (a+b x^n\right ) \left (c+d x^n\right )^2} \]

[Out]

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

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Rubi [A]  time = 5.18268, antiderivative size = 482, 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 \[ -\frac{d (e x)^{m+1} \left (a^2 d (B c (m+1)-A d (m-2 n+1))-a b c (m-6 n+1) (B c-A d)-2 A b^2 c^2 n\right )}{2 a c^2 e n^2 (b c-a d)^3 \left (c+d x^n\right )}+\frac{d (e x)^{m+1} \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right ) \left (-a^2 d^2 (m-n+1) (B c (m+1)-A d (m-2 n+1))+2 a b c d \left (B c (m+1) (m-3 n+1)-A d \left (m^2+m (2-5 n)+4 n^2-5 n+1\right )\right )+b^2 c^2 (m-3 n+1) (A d (m-4 n+1)-B c (m-2 n+1))\right )}{2 c^3 e (m+1) n^2 (b c-a d)^4}+\frac{b^2 (e x)^{m+1} \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right ) (A b (a d (m-4 n+1)-b c (m-n+1))+a B (b c (m+1)-a d (m-3 n+1)))}{a^2 e (m+1) n (b c-a d)^4}+\frac{d (e x)^{m+1} (a A d-3 a B c+2 A b c)}{2 a c e n (b c-a d)^2 \left (c+d x^n\right )^2}+\frac{(e x)^{m+1} (A b-a B)}{a e n (b c-a d) \left (a+b x^n\right ) \left (c+d x^n\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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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)**2/(c+d*x**n)**3,x)

[Out]

Timed out

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Mathematica [B]  time = 4.5155, size = 2178, normalized size = 4.52 \[ \text{Result too large to show} \]

Antiderivative was successfully verified.

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

[Out]

(x*(e*x)^m*(-(a^2*c^2*d*(b*c - a*d)^2*(B*c - A*d)*(1 + m)*n*(a + b*x^n)) + a^2*c
*d*(-(b*c) + a*d)*(1 + m)*(b*c*(A*d*(1 + m - 6*n) - B*c*(1 + m - 4*n)) + a*d*(B*
c*(1 + m) - A*d*(1 + m - 2*n)))*(a + b*x^n)*(c + d*x^n) + 2*a*b^2*(-(A*b) + a*B)
*c^3*(-(b*c) + a*d)*(1 + m)*n*(c + d*x^n)^2 + 2*b^2*c^3*(a*B*(b*c*(1 + m) - a*d*
(1 + m - 3*n)) + A*b*(a*d*(1 + m - 4*n) - b*c*(1 + m - n)))*n*(a + b*x^n)*(c + d
*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)] - a^2*b^2*B
*c^3*d*(a + b*x^n)*(c + d*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n,
-((d*x^n)/c)] + a^2*A*b^2*c^2*d^2*(a + b*x^n)*(c + d*x^n)^2*Hypergeometric2F1[1,
(1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)] + 2*a^3*b*B*c^2*d^2*(a + b*x^n)*(c + d*
x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)] - 2*a^3*A*b*
c*d^3*(a + b*x^n)*(c + d*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -
((d*x^n)/c)] - a^4*B*c*d^3*(a + b*x^n)*(c + d*x^n)^2*Hypergeometric2F1[1, (1 + m
)/n, (1 + m + n)/n, -((d*x^n)/c)] + a^4*A*d^4*(a + b*x^n)*(c + d*x^n)^2*Hypergeo
metric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)] - 2*a^2*b^2*B*c^3*d*m*(a +
b*x^n)*(c + d*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)
] + 2*a^2*A*b^2*c^2*d^2*m*(a + b*x^n)*(c + d*x^n)^2*Hypergeometric2F1[1, (1 + m)
/n, (1 + m + n)/n, -((d*x^n)/c)] + 4*a^3*b*B*c^2*d^2*m*(a + b*x^n)*(c + d*x^n)^2
*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)] - 4*a^3*A*b*c*d^3*
m*(a + b*x^n)*(c + d*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*
x^n)/c)] - 2*a^4*B*c*d^3*m*(a + b*x^n)*(c + d*x^n)^2*Hypergeometric2F1[1, (1 + m
)/n, (1 + m + n)/n, -((d*x^n)/c)] + 2*a^4*A*d^4*m*(a + b*x^n)*(c + d*x^n)^2*Hype
rgeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)] - a^2*b^2*B*c^3*d*m^2*(
a + b*x^n)*(c + d*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n
)/c)] + a^2*A*b^2*c^2*d^2*m^2*(a + b*x^n)*(c + d*x^n)^2*Hypergeometric2F1[1, (1
+ m)/n, (1 + m + n)/n, -((d*x^n)/c)] + 2*a^3*b*B*c^2*d^2*m^2*(a + b*x^n)*(c + d*
x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)] - 2*a^3*A*b*
c*d^3*m^2*(a + b*x^n)*(c + d*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/
n, -((d*x^n)/c)] - a^4*B*c*d^3*m^2*(a + b*x^n)*(c + d*x^n)^2*Hypergeometric2F1[1
, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)] + a^4*A*d^4*m^2*(a + b*x^n)*(c + d*x^n
)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)] + 5*a^2*b^2*B*c
^3*d*n*(a + b*x^n)*(c + d*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n,
-((d*x^n)/c)] - 7*a^2*A*b^2*c^2*d^2*n*(a + b*x^n)*(c + d*x^n)^2*Hypergeometric2F
1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)] - 6*a^3*b*B*c^2*d^2*n*(a + b*x^n)*(
c + d*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)] + 10*a
^3*A*b*c*d^3*n*(a + b*x^n)*(c + d*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m
+ n)/n, -((d*x^n)/c)] + a^4*B*c*d^3*n*(a + b*x^n)*(c + d*x^n)^2*Hypergeometric2F
1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)] - 3*a^4*A*d^4*n*(a + b*x^n)*(c + d*
x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)] + 5*a^2*b^2*
B*c^3*d*m*n*(a + b*x^n)*(c + d*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n
)/n, -((d*x^n)/c)] - 7*a^2*A*b^2*c^2*d^2*m*n*(a + b*x^n)*(c + d*x^n)^2*Hypergeom
etric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)] - 6*a^3*b*B*c^2*d^2*m*n*(a +
b*x^n)*(c + d*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c
)] + 10*a^3*A*b*c*d^3*m*n*(a + b*x^n)*(c + d*x^n)^2*Hypergeometric2F1[1, (1 + m)
/n, (1 + m + n)/n, -((d*x^n)/c)] + a^4*B*c*d^3*m*n*(a + b*x^n)*(c + d*x^n)^2*Hyp
ergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)] - 3*a^4*A*d^4*m*n*(a +
b*x^n)*(c + d*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c
)] - 6*a^2*b^2*B*c^3*d*n^2*(a + b*x^n)*(c + d*x^n)^2*Hypergeometric2F1[1, (1 + m
)/n, (1 + m + n)/n, -((d*x^n)/c)] + 12*a^2*A*b^2*c^2*d^2*n^2*(a + b*x^n)*(c + d*
x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)] - 8*a^3*A*b*
c*d^3*n^2*(a + b*x^n)*(c + d*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/
n, -((d*x^n)/c)] + 2*a^4*A*d^4*n^2*(a + b*x^n)*(c + d*x^n)^2*Hypergeometric2F1[1
, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)]))/(2*a^2*c^3*(b*c - a*d)^4*(1 + m)*n^2
*(a + b*x^n)*(c + d*x^n)^2)

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

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)^3,x)

[Out]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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