3.36 \(\int \frac{(e x)^m \left (a+b x^n\right )^2 \left (A+B x^n\right )}{\left (c+d x^n\right )^3} \, dx\)
Optimal. Leaf size=322 \[ \frac{(e x)^{m+1} \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right ) (a d (b c (m+1)-a d (m-n+1)) (B c (m+1)-A d (m-2 n+1))-b c (a d (m+1)-b c (m+n+1)) (A d (m+1)-B c (m+2 n+1)))}{2 c^3 d^3 e (m+1) n^2}+\frac{b (e x)^{m+1} (a d (m+1)-b c (m+n+1)) (A d (m+1)-B c (m+2 n+1))}{2 c^2 d^3 e (m+1) n^2}-\frac{(e x)^{m+1} (b c-a d) \left (a (B c (m+1)-A d (m-2 n+1))-b x^n (A d (m+1)-B c (m+2 n+1))\right )}{2 c^2 d^2 e n^2 \left (c+d x^n\right )}-\frac{(e x)^{m+1} \left (a+b x^n\right )^2 (B c-A d)}{2 c d e n \left (c+d x^n\right )^2} \]
[Out]
(b*(a*d*(1 + m) - b*c*(1 + m + n))*(A*d*(1 + m) - B*c*(1 + m + 2*n))*(e*x)^(1 +
m))/(2*c^2*d^3*e*(1 + m)*n^2) - ((B*c - A*d)*(e*x)^(1 + m)*(a + b*x^n)^2)/(2*c*d
*e*n*(c + d*x^n)^2) - ((b*c - a*d)*(e*x)^(1 + m)*(a*(B*c*(1 + m) - A*d*(1 + m -
2*n)) - b*(A*d*(1 + m) - B*c*(1 + m + 2*n))*x^n))/(2*c^2*d^2*e*n^2*(c + d*x^n))
+ ((a*d*(B*c*(1 + m) - A*d*(1 + m - 2*n))*(b*c*(1 + m) - a*d*(1 + m - n)) - b*c*
(a*d*(1 + m) - b*c*(1 + m + n))*(A*d*(1 + m) - B*c*(1 + m + 2*n)))*(e*x)^(1 + m)
*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)])/(2*c^3*d^3*e*(1 +
m)*n^2)
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Rubi [A] time = 1.48365, antiderivative size = 322, normalized size of antiderivative = 1.,
number of steps used = 4, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097
\[ \frac{(e x)^{m+1} \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right ) (a d (b c (m+1)-a d (m-n+1)) (B c (m+1)-A d (m-2 n+1))-b c (a d (m+1)-b c (m+n+1)) (A d (m+1)-B c (m+2 n+1)))}{2 c^3 d^3 e (m+1) n^2}+\frac{b (e x)^{m+1} (a d (m+1)-b c (m+n+1)) (A d (m+1)-B c (m+2 n+1))}{2 c^2 d^3 e (m+1) n^2}-\frac{(e x)^{m+1} (b c-a d) \left (a (B c (m+1)-A d (m-2 n+1))-b x^n (A d (m+1)-B c (m+2 n+1))\right )}{2 c^2 d^2 e n^2 \left (c+d x^n\right )}-\frac{(e x)^{m+1} \left (a+b x^n\right )^2 (B c-A d)}{2 c d e n \left (c+d x^n\right )^2} \]
Antiderivative was successfully verified.
[In] Int[((e*x)^m*(a + b*x^n)^2*(A + B*x^n))/(c + d*x^n)^3,x]
[Out]
(b*(a*d*(1 + m) - b*c*(1 + m + n))*(A*d*(1 + m) - B*c*(1 + m + 2*n))*(e*x)^(1 +
m))/(2*c^2*d^3*e*(1 + m)*n^2) - ((B*c - A*d)*(e*x)^(1 + m)*(a + b*x^n)^2)/(2*c*d
*e*n*(c + d*x^n)^2) - ((b*c - a*d)*(e*x)^(1 + m)*(a*(B*c*(1 + m) - A*d*(1 + m -
2*n)) - b*(A*d*(1 + m) - B*c*(1 + m + 2*n))*x^n))/(2*c^2*d^2*e*n^2*(c + d*x^n))
+ ((a*d*(B*c*(1 + m) - A*d*(1 + m - 2*n))*(b*c*(1 + m) - a*d*(1 + m - n)) - b*c*
(a*d*(1 + m) - b*c*(1 + m + n))*(A*d*(1 + m) - B*c*(1 + m + 2*n)))*(e*x)^(1 + m)
*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)])/(2*c^3*d^3*e*(1 +
m)*n^2)
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Rubi in Sympy [A] time = 120.586, size = 298, normalized size = 0.93 \[ \frac{b \left (e x\right )^{m + 1} \left (- 2 A d n + \left (A d - B c\right ) \left (m + 2 n + 1\right )\right ) \left (- a d n + \left (a d - b c\right ) \left (m + n + 1\right )\right )}{2 c^{2} d^{3} e n^{2} \left (m + 1\right )} + \frac{\left (e x\right )^{m + 1} \left (a + b x^{n}\right )^{2} \left (A d - B c\right )}{2 c d e n \left (c + d x^{n}\right )^{2}} - \frac{\left (e x\right )^{m + 1} \left (a d - b c\right ) \left (a \left (- 2 A d n + \left (m + 1\right ) \left (A d - B c\right )\right ) + b x^{n} \left (- 2 A d n + \left (A d - B c\right ) \left (m + 2 n + 1\right )\right )\right )}{2 c^{2} d^{2} e n^{2} \left (c + d x^{n}\right )} + \frac{\left (e x\right )^{m + 1} \left (a d \left (- 2 A d n + \left (m + 1\right ) \left (A d - B c\right )\right ) \left (- a d n + \left (m + 1\right ) \left (a d - b c\right )\right ) - b c \left (- 2 A d n + \left (A d - B c\right ) \left (m + 2 n + 1\right )\right ) \left (- a d n + \left (a d - b c\right ) \left (m + n + 1\right )\right )\right ){{}_{2}F_{1}\left (\begin{matrix} 1, \frac{m + 1}{n} \\ \frac{m + n + 1}{n} \end{matrix}\middle |{- \frac{d x^{n}}{c}} \right )}}{2 c^{3} d^{3} e n^{2} \left (m + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x)**m*(a+b*x**n)**2*(A+B*x**n)/(c+d*x**n)**3,x)
[Out]
b*(e*x)**(m + 1)*(-2*A*d*n + (A*d - B*c)*(m + 2*n + 1))*(-a*d*n + (a*d - b*c)*(m
+ n + 1))/(2*c**2*d**3*e*n**2*(m + 1)) + (e*x)**(m + 1)*(a + b*x**n)**2*(A*d -
B*c)/(2*c*d*e*n*(c + d*x**n)**2) - (e*x)**(m + 1)*(a*d - b*c)*(a*(-2*A*d*n + (m
+ 1)*(A*d - B*c)) + b*x**n*(-2*A*d*n + (A*d - B*c)*(m + 2*n + 1)))/(2*c**2*d**2*
e*n**2*(c + d*x**n)) + (e*x)**(m + 1)*(a*d*(-2*A*d*n + (m + 1)*(A*d - B*c))*(-a*
d*n + (m + 1)*(a*d - b*c)) - b*c*(-2*A*d*n + (A*d - B*c)*(m + 2*n + 1))*(-a*d*n
+ (a*d - b*c)*(m + n + 1)))*hyper((1, (m + 1)/n), ((m + n + 1)/n,), -d*x**n/c)/(
2*c**3*d**3*e*n**2*(m + 1))
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Mathematica [B] time = 2.0036, size = 1924, normalized size = 5.98 \[ \text{result too large to display} \]
Antiderivative was successfully verified.
[In] Integrate[((e*x)^m*(a + b*x^n)^2*(A + B*x^n))/(c + d*x^n)^3,x]
[Out]
(x*(e*x)^m*(-(b^2*B*c^5*(1 + m)*n) + A*b^2*c^4*d*(1 + m)*n + 2*a*b*B*c^4*d*(1 +
m)*n - 2*a*A*b*c^3*d^2*(1 + m)*n - a^2*B*c^3*d^2*(1 + m)*n + a^2*A*c^2*d^3*(1 +
m)*n + b^2*B*c^4*(1 + m)*(c + d*x^n) - A*b^2*c^3*d*(1 + m)*(c + d*x^n) - 2*a*b*B
*c^3*d*(1 + m)*(c + d*x^n) + 2*a*A*b*c^2*d^2*(1 + m)*(c + d*x^n) + a^2*B*c^2*d^2
*(1 + m)*(c + d*x^n) - a^2*A*c*d^3*(1 + m)*(c + d*x^n) + b^2*B*c^4*m*(1 + m)*(c
+ d*x^n) - A*b^2*c^3*d*m*(1 + m)*(c + d*x^n) - 2*a*b*B*c^3*d*m*(1 + m)*(c + d*x^
n) + 2*a*A*b*c^2*d^2*m*(1 + m)*(c + d*x^n) + a^2*B*c^2*d^2*m*(1 + m)*(c + d*x^n)
- a^2*A*c*d^3*m*(1 + m)*(c + d*x^n) + 4*b^2*B*c^4*(1 + m)*n*(c + d*x^n) - 2*A*b
^2*c^3*d*(1 + m)*n*(c + d*x^n) - 4*a*b*B*c^3*d*(1 + m)*n*(c + d*x^n) + 2*a^2*A*c
*d^3*(1 + m)*n*(c + d*x^n) + 2*b^2*B*c^3*n^2*(c + d*x^n)^2 - b^2*B*c^3*(c + d*x^
n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)] + A*b^2*c^2*d*
(c + d*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)] + 2*a
*b*B*c^2*d*(c + d*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n
)/c)] - 2*a*A*b*c*d^2*(c + d*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/
n, -((d*x^n)/c)] - a^2*B*c*d^2*(c + d*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1
+ m + n)/n, -((d*x^n)/c)] + a^2*A*d^3*(c + d*x^n)^2*Hypergeometric2F1[1, (1 + m)
/n, (1 + m + n)/n, -((d*x^n)/c)] - 2*b^2*B*c^3*m*(c + d*x^n)^2*Hypergeometric2F1
[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)] + 2*A*b^2*c^2*d*m*(c + d*x^n)^2*Hype
rgeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)] + 4*a*b*B*c^2*d*m*(c +
d*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)] - 4*a*A*b*
c*d^2*m*(c + d*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c
)] - 2*a^2*B*c*d^2*m*(c + d*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n
, -((d*x^n)/c)] + 2*a^2*A*d^3*m*(c + d*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1
+ m + n)/n, -((d*x^n)/c)] - b^2*B*c^3*m^2*(c + d*x^n)^2*Hypergeometric2F1[1, (1
+ m)/n, (1 + m + n)/n, -((d*x^n)/c)] + A*b^2*c^2*d*m^2*(c + d*x^n)^2*Hypergeome
tric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)] + 2*a*b*B*c^2*d*m^2*(c + d*x^
n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)] - 2*a*A*b*c*d^
2*m^2*(c + d*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)]
- a^2*B*c*d^2*m^2*(c + d*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n,
-((d*x^n)/c)] + a^2*A*d^3*m^2*(c + d*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 +
m + n)/n, -((d*x^n)/c)] - 3*b^2*B*c^3*n*(c + d*x^n)^2*Hypergeometric2F1[1, (1 +
m)/n, (1 + m + n)/n, -((d*x^n)/c)] + A*b^2*c^2*d*n*(c + d*x^n)^2*Hypergeometric
2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)] + 2*a*b*B*c^2*d*n*(c + d*x^n)^2*H
ypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)] + 2*a*A*b*c*d^2*n*(c
+ d*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)] + a^2*B
*c*d^2*n*(c + d*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/
c)] - 3*a^2*A*d^3*n*(c + d*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n,
-((d*x^n)/c)] - 3*b^2*B*c^3*m*n*(c + d*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (
1 + m + n)/n, -((d*x^n)/c)] + A*b^2*c^2*d*m*n*(c + d*x^n)^2*Hypergeometric2F1[1,
(1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)] + 2*a*b*B*c^2*d*m*n*(c + d*x^n)^2*Hyper
geometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)] + 2*a*A*b*c*d^2*m*n*(c +
d*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)] + a^2*B*c
*d^2*m*n*(c + d*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/
c)] - 3*a^2*A*d^3*m*n*(c + d*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/
n, -((d*x^n)/c)] - 2*b^2*B*c^3*n^2*(c + d*x^n)^2*Hypergeometric2F1[1, (1 + m)/n,
(1 + m + n)/n, -((d*x^n)/c)] + 2*a^2*A*d^3*n^2*(c + d*x^n)^2*Hypergeometric2F1[
1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)]))/(2*c^3*d^3*(1 + m)*n^2*(c + d*x^n)^
2)
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Maple [F] time = 0.108, size = 0, normalized size = 0. \[ \int{\frac{ \left ( ex \right ) ^{m} \left ( a+b{x}^{n} \right ) ^{2} \left ( A+B{x}^{n} \right ) }{ \left ( c+d{x}^{n} \right ) ^{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x)^m*(a+b*x^n)^2*(A+B*x^n)/(c+d*x^n)^3,x)
[Out]
int((e*x)^m*(a+b*x^n)^2*(A+B*x^n)/(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)*(b*x^n + a)^2*(e*x)^m/(d*x^n + c)^3,x, algorithm="maxima")
[Out]
(((m^2 + m*(n + 2) + n + 1)*b^2*c^2*d*e^m - 2*(m^2 - m*(n - 2) - n + 1)*a*b*c*d^
2*e^m + (m^2 - m*(3*n - 2) + 2*n^2 - 3*n + 1)*a^2*d^3*e^m)*A - ((m^2 + m*(3*n +
2) + 2*n^2 + 3*n + 1)*b^2*c^3*e^m - 2*(m^2 + m*(n + 2) + n + 1)*a*b*c^2*d*e^m +
(m^2 - m*(n - 2) - n + 1)*a^2*c*d^2*e^m)*B)*integrate(1/2*x^m/(c^2*d^4*n^2*x^n +
c^3*d^3*n^2), x) + 1/2*(2*B*b^2*c^2*d^2*e^m*n^2*x*e^(m*log(x) + 2*n*log(x)) - (
((m^2 + m*(n + 2) + n + 1)*b^2*c^3*d*e^m - 2*(m^2 - m*(n - 2) - n + 1)*a*b*c^2*d
^2*e^m + (m^2 - m*(3*n - 2) - 3*n + 1)*a^2*c*d^3*e^m)*A - ((m^2 + m*(3*n + 2) +
2*n^2 + 3*n + 1)*b^2*c^4*e^m - 2*(m^2 + m*(n + 2) + n + 1)*a*b*c^3*d*e^m + (m^2
- m*(n - 2) - n + 1)*a^2*c^2*d^2*e^m)*B)*x*x^m - (((m^2 + 2*m*(n + 1) + 2*n + 1)
*b^2*c^2*d^2*e^m - 2*(m^2 + 2*m + 1)*a*b*c*d^3*e^m + (m^2 - 2*m*(n - 1) - 2*n +
1)*a^2*d^4*e^m)*A - ((m^2 + 2*m*(2*n + 1) + 4*n^2 + 4*n + 1)*b^2*c^3*d*e^m - 2*(
m^2 + 2*m*(n + 1) + 2*n + 1)*a*b*c^2*d^2*e^m + (m^2 + 2*m + 1)*a^2*c*d^3*e^m)*B)
*x*e^(m*log(x) + n*log(x)))/((m*n^2 + n^2)*c^2*d^5*x^(2*n) + 2*(m*n^2 + n^2)*c^3
*d^4*x^n + (m*n^2 + n^2)*c^4*d^3)
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (B b^{2} x^{3 \, n} + A a^{2} +{\left (2 \, B a b + A b^{2}\right )} x^{2 \, n} +{\left (B a^{2} + 2 \, A a b\right )} x^{n}\right )} \left (e x\right )^{m}}{d^{3} x^{3 \, n} + 3 \, c d^{2} x^{2 \, n} + 3 \, c^{2} d x^{n} + c^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^n + A)*(b*x^n + a)^2*(e*x)^m/(d*x^n + c)^3,x, algorithm="fricas")
[Out]
integral((B*b^2*x^(3*n) + A*a^2 + (2*B*a*b + A*b^2)*x^(2*n) + (B*a^2 + 2*A*a*b)*
x^n)*(e*x)^m/(d^3*x^(3*n) + 3*c*d^2*x^(2*n) + 3*c^2*d*x^n + c^3), 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)**2*(A+B*x**n)/(c+d*x**n)**3,x)
[Out]
Timed out
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x^{n} + A\right )}{\left (b x^{n} + a\right )}^{2} \left (e x\right )^{m}}{{\left (d x^{n} + c\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^n + A)*(b*x^n + a)^2*(e*x)^m/(d*x^n + c)^3,x, algorithm="giac")
[Out]
integrate((B*x^n + A)*(b*x^n + a)^2*(e*x)^m/(d*x^n + c)^3, x)