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

Optimal. Leaf size=270 \[ \frac{d x^{n+1} (e x)^m \left (a^2 B d^2-a b d (A d+3 B c)+3 b^2 c (A d+B c)\right )}{b^3 (m+n+1)}-\frac{(e x)^{m+1} \left (a^3 B d^3-a^2 b d^2 (A d+3 B c)+3 a b^2 c d (A d+B c)+b^3 \left (-c^2\right ) (3 A d+B c)\right )}{b^4 e (m+1)}+\frac{(e x)^{m+1} (A b-a B) (b c-a d)^3 \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right )}{a b^4 e (m+1)}+\frac{d^2 x^{2 n+1} (e x)^m (-a B d+A b d+3 b B c)}{b^2 (m+2 n+1)}+\frac{B d^3 x^{3 n+1} (e x)^m}{b (m+3 n+1)} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.956674, antiderivative size = 270, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 4, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.129 \[ \frac{d x^{n+1} (e x)^m \left (a^2 B d^2-a b d (A d+3 B c)+3 b^2 c (A d+B c)\right )}{b^3 (m+n+1)}-\frac{(e x)^{m+1} \left (a^3 B d^3-a^2 b d^2 (A d+3 B c)+3 a b^2 c d (A d+B c)+b^3 \left (-c^2\right ) (3 A d+B c)\right )}{b^4 e (m+1)}+\frac{(e x)^{m+1} (A b-a B) (b c-a d)^3 \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right )}{a b^4 e (m+1)}+\frac{d^2 x^{2 n+1} (e x)^m (-a B d+A b d+3 b B c)}{b^2 (m+2 n+1)}+\frac{B d^3 x^{3 n+1} (e x)^m}{b (m+3 n+1)} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 89.9018, size = 286, normalized size = 1.06 \[ \frac{A c^{3} \left (e x\right )^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{m + 1}{n} \\ \frac{m + n + 1}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{a e \left (m + 1\right )} + \frac{B d^{3} x^{- m} x^{m + 4 n + 1} \left (e x\right )^{m}{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{m + 4 n + 1}{n} \\ \frac{m + 5 n + 1}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{a \left (m + 4 n + 1\right )} + \frac{c^{2} x^{n} \left (e x\right )^{- n} \left (e x\right )^{m + n + 1} \left (3 A d + B c\right ){{}_{2}F_{1}\left (\begin{matrix} 1, \frac{m + n + 1}{n} \\ \frac{m + 2 n + 1}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{a e \left (m + n + 1\right )} + \frac{3 c d x^{2 n} \left (e x\right )^{- 2 n} \left (e x\right )^{m + 2 n + 1} \left (A d + B c\right ){{}_{2}F_{1}\left (\begin{matrix} 1, \frac{m + 2 n + 1}{n} \\ \frac{m + 3 n + 1}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{a e \left (m + 2 n + 1\right )} + \frac{d^{2} x^{3 n} \left (e x\right )^{- 3 n} \left (e x\right )^{m + 3 n + 1} \left (A d + 3 B c\right ){{}_{2}F_{1}\left (\begin{matrix} 1, \frac{m + 3 n + 1}{n} \\ \frac{m + 4 n + 1}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{a e \left (m + 3 n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.649985, size = 212, normalized size = 0.79 \[ x (e x)^m \left (\frac{d x^n \left (a^2 B d^2-a b d (A d+3 B c)+3 b^2 c (A d+B c)\right )}{b^3 (m+n+1)}+\frac{(a B-A b) (a d-b c)^3 \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right )}{a b^4 (m+1)}-\frac{(a B-A b) (a d-b c)^3}{a b^4 (m+1)}+\frac{d^2 x^{2 n} (-a B d+A b d+3 b B c)}{b^2 (m+2 n+1)}+\frac{A c^3}{a m+a}+\frac{B d^3 x^{3 n}}{b m+3 b n+b}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [F]  time = 0.076, size = 0, normalized size = 0. \[ \int{\frac{ \left ( ex \right ) ^{m} \left ( A+B{x}^{n} \right ) \left ( c+d{x}^{n} \right ) ^{3}}{a+b{x}^{n}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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)^3*(e*x)^m/(b*x^n + a),x, algorithm="maxima")

[Out]

((b^4*c^3*e^m - 3*a*b^3*c^2*d*e^m + 3*a^2*b^2*c*d^2*e^m - a^3*b*d^3*e^m)*A - (a*
b^3*c^3*e^m - 3*a^2*b^2*c^2*d*e^m + 3*a^3*b*c*d^2*e^m - a^4*d^3*e^m)*B)*integrat
e(x^m/(b^5*x^n + a*b^4), x) + ((m^3 + 3*m^2*(n + 1) + (2*n^2 + 6*n + 3)*m + 2*n^
2 + 3*n + 1)*B*b^3*d^3*e^m*x*e^(m*log(x) + 3*n*log(x)) + ((3*(m^3 + 3*m^2*(2*n +
1) + 6*n^3 + (11*n^2 + 12*n + 3)*m + 11*n^2 + 6*n + 1)*b^3*c^2*d*e^m - 3*(m^3 +
3*m^2*(2*n + 1) + 6*n^3 + (11*n^2 + 12*n + 3)*m + 11*n^2 + 6*n + 1)*a*b^2*c*d^2
*e^m + (m^3 + 3*m^2*(2*n + 1) + 6*n^3 + (11*n^2 + 12*n + 3)*m + 11*n^2 + 6*n + 1
)*a^2*b*d^3*e^m)*A + ((m^3 + 3*m^2*(2*n + 1) + 6*n^3 + (11*n^2 + 12*n + 3)*m + 1
1*n^2 + 6*n + 1)*b^3*c^3*e^m - 3*(m^3 + 3*m^2*(2*n + 1) + 6*n^3 + (11*n^2 + 12*n
+ 3)*m + 11*n^2 + 6*n + 1)*a*b^2*c^2*d*e^m + 3*(m^3 + 3*m^2*(2*n + 1) + 6*n^3 +
(11*n^2 + 12*n + 3)*m + 11*n^2 + 6*n + 1)*a^2*b*c*d^2*e^m - (m^3 + 3*m^2*(2*n +
1) + 6*n^3 + (11*n^2 + 12*n + 3)*m + 11*n^2 + 6*n + 1)*a^3*d^3*e^m)*B)*x*x^m +
((m^3 + m^2*(4*n + 3) + (3*n^2 + 8*n + 3)*m + 3*n^2 + 4*n + 1)*A*b^3*d^3*e^m + (
3*(m^3 + m^2*(4*n + 3) + (3*n^2 + 8*n + 3)*m + 3*n^2 + 4*n + 1)*b^3*c*d^2*e^m -
(m^3 + m^2*(4*n + 3) + (3*n^2 + 8*n + 3)*m + 3*n^2 + 4*n + 1)*a*b^2*d^3*e^m)*B)*
x*e^(m*log(x) + 2*n*log(x)) + ((3*(m^3 + m^2*(5*n + 3) + (6*n^2 + 10*n + 3)*m +
6*n^2 + 5*n + 1)*b^3*c*d^2*e^m - (m^3 + m^2*(5*n + 3) + (6*n^2 + 10*n + 3)*m + 6
*n^2 + 5*n + 1)*a*b^2*d^3*e^m)*A + (3*(m^3 + m^2*(5*n + 3) + (6*n^2 + 10*n + 3)*
m + 6*n^2 + 5*n + 1)*b^3*c^2*d*e^m - 3*(m^3 + m^2*(5*n + 3) + (6*n^2 + 10*n + 3)
*m + 6*n^2 + 5*n + 1)*a*b^2*c*d^2*e^m + (m^3 + m^2*(5*n + 3) + (6*n^2 + 10*n + 3
)*m + 6*n^2 + 5*n + 1)*a^2*b*d^3*e^m)*B)*x*e^(m*log(x) + n*log(x)))/((m^4 + 2*m^
3*(3*n + 2) + (11*n^2 + 18*n + 6)*m^2 + 6*n^3 + 2*(3*n^3 + 11*n^2 + 9*n + 2)*m +
11*n^2 + 6*n + 1)*b^4)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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