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

Optimal. Leaf size=322 \[ \frac{(e x)^{m+1} \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right ) (b c (a B (m+1)-A b (m-2 n+1)) (a d (m+1)-b c (m-n+1))-a d (A b (m+1)-a B (m+2 n+1)) (b c (m+1)-a d (m+n+1)))}{2 a^3 b^3 e (m+1) n^2}+\frac{d (e x)^{m+1} (A b (m+1)-a B (m+2 n+1)) (b c (m+1)-a d (m+n+1))}{2 a^2 b^3 e (m+1) n^2}+\frac{(e x)^{m+1} (b c-a d) \left (c (a B (m+1)-A b (m-2 n+1))-d x^n (A b (m+1)-a B (m+2 n+1))\right )}{2 a^2 b^2 e n^2 \left (a+b x^n\right )}+\frac{(e x)^{m+1} (A b-a B) \left (c+d x^n\right )^2}{2 a b e n \left (a+b x^n\right )^2} \]

[Out]

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

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Rubi [A]  time = 1.44701, 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 \[ \frac{(e x)^{m+1} \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right ) (b c (a B (m+1)-A b (m-2 n+1)) (a d (m+1)-b c (m-n+1))-a d (A b (m+1)-a B (m+2 n+1)) (b c (m+1)-a d (m+n+1)))}{2 a^3 b^3 e (m+1) n^2}+\frac{d (e x)^{m+1} (A b (m+1)-a B (m+2 n+1)) (b c (m+1)-a d (m+n+1))}{2 a^2 b^3 e (m+1) n^2}+\frac{(e x)^{m+1} (b c-a d) \left (c (a B (m+1)-A b (m-2 n+1))-d x^n (A b (m+1)-a B (m+2 n+1))\right )}{2 a^2 b^2 e n^2 \left (a+b x^n\right )}+\frac{(e x)^{m+1} (A b-a B) \left (c+d x^n\right )^2}{2 a b e n \left (a+b x^n\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 127.055, size = 298, normalized size = 0.93 \[ \frac{\left (e x\right )^{m + 1} \left (c + d x^{n}\right )^{2} \left (A b - B a\right )}{2 a b e n \left (a + b x^{n}\right )^{2}} + \frac{\left (e x\right )^{m + 1} \left (a d - b c\right ) \left (c \left (- 2 A b n + \left (m + 1\right ) \left (A b - B a\right )\right ) + d x^{n} \left (- 2 A b n + \left (A b - B a\right ) \left (m + 2 n + 1\right )\right )\right )}{2 a^{2} b^{2} e n^{2} \left (a + b x^{n}\right )} - \frac{d \left (e x\right )^{m + 1} \left (- 2 A b n + \left (A b - B a\right ) \left (m + 2 n + 1\right )\right ) \left (b c n + \left (a d - b c\right ) \left (m + n + 1\right )\right )}{2 a^{2} b^{3} e n^{2} \left (m + 1\right )} + \frac{\left (e x\right )^{m + 1} \left (a d \left (- 2 A b n + \left (A b - B a\right ) \left (m + 2 n + 1\right )\right ) \left (b c n + \left (a d - b c\right ) \left (m + n + 1\right )\right ) - b c \left (- 2 A b n + \left (m + 1\right ) \left (A b - B a\right )\right ) \left (b c n + \left (m + 1\right ) \left (a d - b c\right )\right )\right ){{}_{2}F_{1}\left (\begin{matrix} 1, \frac{m + 1}{n} \\ \frac{m + n + 1}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{2 a^{3} b^{3} e n^{2} \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [B]  time = 1.80852, 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)*(c + d*x^n)^2)/(a + b*x^n)^3,x]

[Out]

(x*(e*x)^m*(a^2*A*b^3*c^2*(1 + m)*n - a^3*b^2*B*c^2*(1 + m)*n - 2*a^3*A*b^2*c*d*
(1 + m)*n + 2*a^4*b*B*c*d*(1 + m)*n + a^4*A*b*d^2*(1 + m)*n - a^5*B*d^2*(1 + m)*
n - a*A*b^3*c^2*(1 + m)*(a + b*x^n) + a^2*b^2*B*c^2*(1 + m)*(a + b*x^n) + 2*a^2*
A*b^2*c*d*(1 + m)*(a + b*x^n) - 2*a^3*b*B*c*d*(1 + m)*(a + b*x^n) - a^3*A*b*d^2*
(1 + m)*(a + b*x^n) + a^4*B*d^2*(1 + m)*(a + b*x^n) - a*A*b^3*c^2*m*(1 + m)*(a +
 b*x^n) + a^2*b^2*B*c^2*m*(1 + m)*(a + b*x^n) + 2*a^2*A*b^2*c*d*m*(1 + m)*(a + b
*x^n) - 2*a^3*b*B*c*d*m*(1 + m)*(a + b*x^n) - a^3*A*b*d^2*m*(1 + m)*(a + b*x^n)
+ a^4*B*d^2*m*(1 + m)*(a + b*x^n) + 2*a*A*b^3*c^2*(1 + m)*n*(a + b*x^n) - 4*a^3*
b*B*c*d*(1 + m)*n*(a + b*x^n) - 2*a^3*A*b*d^2*(1 + m)*n*(a + b*x^n) + 4*a^4*B*d^
2*(1 + m)*n*(a + b*x^n) + 2*a^3*B*d^2*n^2*(a + b*x^n)^2 + A*b^3*c^2*(a + b*x^n)^
2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)] - a*b^2*B*c^2*(a
+ b*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)] - 2*a*A*
b^2*c*d*(a + b*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a
)] + 2*a^2*b*B*c*d*(a + b*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n,
-((b*x^n)/a)] + a^2*A*b*d^2*(a + b*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m
 + n)/n, -((b*x^n)/a)] - a^3*B*d^2*(a + b*x^n)^2*Hypergeometric2F1[1, (1 + m)/n,
 (1 + m + n)/n, -((b*x^n)/a)] + 2*A*b^3*c^2*m*(a + b*x^n)^2*Hypergeometric2F1[1,
 (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)] - 2*a*b^2*B*c^2*m*(a + b*x^n)^2*Hyperge
ometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)] - 4*a*A*b^2*c*d*m*(a + b*x
^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)] + 4*a^2*b*B*c
*d*m*(a + b*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)]
+ 2*a^2*A*b*d^2*m*(a + b*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -
((b*x^n)/a)] - 2*a^3*B*d^2*m*(a + b*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 +
m + n)/n, -((b*x^n)/a)] + A*b^3*c^2*m^2*(a + b*x^n)^2*Hypergeometric2F1[1, (1 +
m)/n, (1 + m + n)/n, -((b*x^n)/a)] - a*b^2*B*c^2*m^2*(a + b*x^n)^2*Hypergeometri
c2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)] - 2*a*A*b^2*c*d*m^2*(a + b*x^n)^
2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)] + 2*a^2*b*B*c*d*m
^2*(a + b*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)] +
a^2*A*b*d^2*m^2*(a + b*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((
b*x^n)/a)] - a^3*B*d^2*m^2*(a + b*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m
+ n)/n, -((b*x^n)/a)] - 3*A*b^3*c^2*n*(a + b*x^n)^2*Hypergeometric2F1[1, (1 + m)
/n, (1 + m + n)/n, -((b*x^n)/a)] + a*b^2*B*c^2*n*(a + b*x^n)^2*Hypergeometric2F1
[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)] + 2*a*A*b^2*c*d*n*(a + b*x^n)^2*Hype
rgeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)] + 2*a^2*b*B*c*d*n*(a +
b*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)] + a^2*A*b*
d^2*n*(a + b*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)]
 - 3*a^3*B*d^2*n*(a + b*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -(
(b*x^n)/a)] - 3*A*b^3*c^2*m*n*(a + b*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 +
 m + n)/n, -((b*x^n)/a)] + a*b^2*B*c^2*m*n*(a + b*x^n)^2*Hypergeometric2F1[1, (1
 + m)/n, (1 + m + n)/n, -((b*x^n)/a)] + 2*a*A*b^2*c*d*m*n*(a + b*x^n)^2*Hypergeo
metric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)] + 2*a^2*b*B*c*d*m*n*(a + b*
x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)] + a^2*A*b*d^
2*m*n*(a + b*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)]
 - 3*a^3*B*d^2*m*n*(a + b*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n,
-((b*x^n)/a)] + 2*A*b^3*c^2*n^2*(a + b*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1
 + m + n)/n, -((b*x^n)/a)] - 2*a^3*B*d^2*n^2*(a + b*x^n)^2*Hypergeometric2F1[1,
(1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)]))/(2*a^3*b^3*(1 + m)*n^2*(a + b*x^n)^2)

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Maple [F]  time = 0.097, size = 0, normalized size = 0. \[ \int{\frac{ \left ( ex \right ) ^{m} \left ( A+B{x}^{n} \right ) \left ( c+d{x}^{n} \right ) ^{2}}{ \left ( a+b{x}^{n} \right ) ^{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((B*d^2*x^(3*n) + A*c^2 + (2*B*c*d + A*d^2)*x^(2*n) + (B*c^2 + 2*A*c*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. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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