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

Optimal. Leaf size=228 \[ -\frac{(e x)^{m+1} \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right ) (b c (m-n+1) (a B (m+1)-A b (m-2 n+1))+a d (m+1) (A b (m-n+1)-a B (m+n+1)))}{2 a^3 b^2 e (m+1) n^2}-\frac{(e x)^{m+1} (A b (b c (m-2 n+1)-a d (m-n+1))-a B (b c (m+1)-a d (m+n+1)))}{2 a^2 b^2 e n^2 \left (a+b x^n\right )}+\frac{(e x)^{m+1} (A b-a B) \left (c+d x^n\right )}{2 a b e n \left (a+b x^n\right )^2} \]

[Out]

-((A*b*(b*c*(1 + m - 2*n) - a*d*(1 + m - n)) - a*B*(b*c*(1 + m) - a*d*(1 + m + n
)))*(e*x)^(1 + m))/(2*a^2*b^2*e*n^2*(a + b*x^n)) + ((A*b - a*B)*(e*x)^(1 + m)*(c
+ d*x^n))/(2*a*b*e*n*(a + b*x^n)^2) - ((b*c*(a*B*(1 + m) - A*b*(1 + m - 2*n))*(
1 + m - n) + a*d*(1 + m)*(A*b*(1 + m - n) - a*B*(1 + m + n)))*(e*x)^(1 + m)*Hype
rgeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)])/(2*a^3*b^2*e*(1 + m)*n
^2)

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Rubi [A]  time = 0.778994, antiderivative size = 228, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103 \[ -\frac{(e x)^{m+1} \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right ) (b c (m-n+1) (a B (m+1)-A b (m-2 n+1))+a d (m+1) (A b (m-n+1)-a B (m+n+1)))}{2 a^3 b^2 e (m+1) n^2}-\frac{(e x)^{m+1} (A b (b c (m-2 n+1)-a d (m-n+1))-a B (b c (m+1)-a d (m+n+1)))}{2 a^2 b^2 e n^2 \left (a+b x^n\right )}+\frac{(e x)^{m+1} (A b-a B) \left (c+d x^n\right )}{2 a b e n \left (a+b x^n\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

-((A*b*(b*c*(1 + m - 2*n) - a*d*(1 + m - n)) - a*B*(b*c*(1 + m) - a*d*(1 + m + n
)))*(e*x)^(1 + m))/(2*a^2*b^2*e*n^2*(a + b*x^n)) + ((A*b - a*B)*(e*x)^(1 + m)*(c
+ d*x^n))/(2*a*b*e*n*(a + b*x^n)^2) - ((b*c*(a*B*(1 + m) - A*b*(1 + m - 2*n))*(
1 + m - n) + a*d*(1 + m)*(A*b*(1 + m - n) - a*B*(1 + m + n)))*(e*x)^(1 + m)*Hype
rgeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)])/(2*a^3*b^2*e*(1 + m)*n
^2)

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Rubi in Sympy [A]  time = 58.7431, size = 199, normalized size = 0.87 \[ \frac{\left (e x\right )^{m + 1} \left (c + d x^{n}\right ) \left (A b - B a\right )}{2 a b e n \left (a + b x^{n}\right )^{2}} + \frac{\left (e x\right )^{m + 1} \left (a d \left (- 2 A b n + \left (A b - B a\right ) \left (m + n + 1\right )\right ) - b c \left (- 2 A b n + \left (m + 1\right ) \left (A b - B a\right )\right )\right )}{2 a^{2} b^{2} e n^{2} \left (a + b x^{n}\right )} - \frac{\left (e x\right )^{m + 1} \left (a d \left (m + 1\right ) \left (- 2 A b n + \left (A b - B a\right ) \left (m + n + 1\right )\right ) - b c \left (- 2 A b n + \left (m + 1\right ) \left (A b - B a\right )\right ) \left (m - n + 1\right )\right ){{}_{2}F_{1}\left (\begin{matrix} 1, \frac{m + 1}{n} \\ \frac{m + n + 1}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{2 a^{3} b^{2} e n^{2} \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [B]  time = 0.989523, size = 1153, normalized size = 5.06 \[ \frac{x (e x)^m \left (B d (m+1) n a^4-b B c (m+1) n a^3-A b d (m+1) n a^3-B d (m+1) \left (b x^n+a\right ) a^3-B d m (m+1) \left (b x^n+a\right ) a^3-2 B d (m+1) n \left (b x^n+a\right ) a^3+A b^2 c (m+1) n a^2+b B c (m+1) \left (b x^n+a\right ) a^2+A b d (m+1) \left (b x^n+a\right ) a^2+b B c m (m+1) \left (b x^n+a\right ) a^2+A b d m (m+1) \left (b x^n+a\right ) a^2+B d m^2 \left (b x^n+a\right )^2 \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right ) a^2+B d \left (b x^n+a\right )^2 \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right ) a^2+2 B d m \left (b x^n+a\right )^2 \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right ) a^2+B d n \left (b x^n+a\right )^2 \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right ) a^2+B d m n \left (b x^n+a\right )^2 \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right ) a^2-A b^2 c (m+1) \left (b x^n+a\right ) a-A b^2 c m (m+1) \left (b x^n+a\right ) a+2 A b^2 c (m+1) n \left (b x^n+a\right ) a-b B c m^2 \left (b x^n+a\right )^2 \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right ) a-A b d m^2 \left (b x^n+a\right )^2 \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right ) a-b B c \left (b x^n+a\right )^2 \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right ) a-A b d \left (b x^n+a\right )^2 \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right ) a-2 b B c m \left (b x^n+a\right )^2 \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right ) a-2 A b d m \left (b x^n+a\right )^2 \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right ) a+b B c n \left (b x^n+a\right )^2 \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right ) a+A b d n \left (b x^n+a\right )^2 \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right ) a+b B c m n \left (b x^n+a\right )^2 \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right ) a+A b d m n \left (b x^n+a\right )^2 \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right ) a+A b^2 c m^2 \left (b x^n+a\right )^2 \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right )+2 A b^2 c n^2 \left (b x^n+a\right )^2 \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right )+A b^2 c \left (b x^n+a\right )^2 \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right )+2 A b^2 c m \left (b x^n+a\right )^2 \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right )-3 A b^2 c n \left (b x^n+a\right )^2 \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right )-3 A b^2 c m n \left (b x^n+a\right )^2 \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right )\right )}{2 a^3 b^2 (m+1) n^2 \left (b x^n+a\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

(x*(e*x)^m*(a^2*A*b^2*c*(1 + m)*n - a^3*b*B*c*(1 + m)*n - a^3*A*b*d*(1 + m)*n +
a^4*B*d*(1 + m)*n - a*A*b^2*c*(1 + m)*(a + b*x^n) + a^2*b*B*c*(1 + m)*(a + b*x^n
) + a^2*A*b*d*(1 + m)*(a + b*x^n) - a^3*B*d*(1 + m)*(a + b*x^n) - a*A*b^2*c*m*(1
+ m)*(a + b*x^n) + a^2*b*B*c*m*(1 + m)*(a + b*x^n) + a^2*A*b*d*m*(1 + m)*(a + b
*x^n) - a^3*B*d*m*(1 + m)*(a + b*x^n) + 2*a*A*b^2*c*(1 + m)*n*(a + b*x^n) - 2*a^
3*B*d*(1 + m)*n*(a + b*x^n) + A*b^2*c*(a + b*x^n)^2*Hypergeometric2F1[1, (1 + m)
/n, (1 + m + n)/n, -((b*x^n)/a)] - a*b*B*c*(a + b*x^n)^2*Hypergeometric2F1[1, (1
+ m)/n, (1 + m + n)/n, -((b*x^n)/a)] - a*A*b*d*(a + b*x^n)^2*Hypergeometric2F1[
1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)] + a^2*B*d*(a + b*x^n)^2*Hypergeometri
c2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)] + 2*A*b^2*c*m*(a + b*x^n)^2*Hype
rgeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)] - 2*a*b*B*c*m*(a + b*x^
n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)] - 2*a*A*b*d*m*
(a + b*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)] + 2*a
^2*B*d*m*(a + b*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/
a)] + A*b^2*c*m^2*(a + b*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -
((b*x^n)/a)] - a*b*B*c*m^2*(a + b*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m
+ n)/n, -((b*x^n)/a)] - a*A*b*d*m^2*(a + b*x^n)^2*Hypergeometric2F1[1, (1 + m)/n
, (1 + m + n)/n, -((b*x^n)/a)] + a^2*B*d*m^2*(a + b*x^n)^2*Hypergeometric2F1[1,
(1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)] - 3*A*b^2*c*n*(a + b*x^n)^2*Hypergeometr
ic2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)] + a*b*B*c*n*(a + b*x^n)^2*Hyper
geometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)] + a*A*b*d*n*(a + b*x^n)^
2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)] + a^2*B*d*n*(a +
b*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)] - 3*A*b^2*
c*m*n*(a + b*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)]
+ a*b*B*c*m*n*(a + b*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b
*x^n)/a)] + a*A*b*d*m*n*(a + b*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n
)/n, -((b*x^n)/a)] + a^2*B*d*m*n*(a + b*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (
1 + m + n)/n, -((b*x^n)/a)] + 2*A*b^2*c*n^2*(a + b*x^n)^2*Hypergeometric2F1[1, (
1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)]))/(2*a^3*b^2*(1 + m)*n^2*(a + b*x^n)^2)

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Maple [F]  time = 0.095, size = 0, normalized size = 0. \[ \int{\frac{ \left ( ex \right ) ^{m} \left ( A+B{x}^{n} \right ) \left ( c+d{x}^{n} \right ) }{ \left ( a+b{x}^{n} \right ) ^{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[{\left ({\left ({\left (m^{2} - m{\left (3 \, n - 2\right )} + 2 \, n^{2} - 3 \, n + 1\right )} b^{2} c e^{m} -{\left (m^{2} - m{\left (n - 2\right )} - n + 1\right )} a b d e^{m}\right )} A -{\left ({\left (m^{2} - m{\left (n - 2\right )} - n + 1\right )} a b c e^{m} -{\left (m^{2} + m{\left (n + 2\right )} + n + 1\right )} a^{2} d e^{m}\right )} B\right )} \int \frac{x^{m}}{2 \,{\left (a^{2} b^{3} n^{2} x^{n} + a^{3} b^{2} n^{2}\right )}}\,{d x} + \frac{{\left ({\left (a^{2} b d e^{m}{\left (m - n + 1\right )} - a b^{2} c e^{m}{\left (m - 3 \, n + 1\right )}\right )} A -{\left (a^{3} d e^{m}{\left (m + n + 1\right )} - a^{2} b c e^{m}{\left (m - n + 1\right )}\right )} B\right )} x x^{m} -{\left ({\left (b^{3} c e^{m}{\left (m - 2 \, n + 1\right )} - a b^{2} d e^{m}{\left (m + 1\right )}\right )} A +{\left (a^{2} b d e^{m}{\left (m + 2 \, n + 1\right )} - a b^{2} c e^{m}{\left (m + 1\right )}\right )} B\right )} x e^{\left (m \log \left (x\right ) + n \log \left (x\right )\right )}}{2 \,{\left (a^{2} b^{4} n^{2} x^{2 \, n} + 2 \, a^{3} b^{3} n^{2} x^{n} + a^{4} b^{2} n^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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