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

Optimal. Leaf size=394 \[ -\frac{(e x)^{m+1} (b c-a d)^2 \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right ) (A b (b c (m-n+1)-a d (m+2 n+1))-a B (b c (m+1)-a d (m+3 n+1)))}{a^2 b^4 e (m+1) n}-\frac{d (e x)^{m+1} \left (A b \left (a^2 d^2 (m+2 n+1)-3 a b c d (m+n+1)+3 b^2 c^2 (m+1)\right )-a B \left (a^2 d^2 (m+3 n+1)-3 a b c d (m+2 n+1)+3 b^2 c^2 (m+n+1)\right )\right )}{a b^4 e (m+1) n}-\frac{d^2 x^{n+1} (e x)^m (A b (3 b c (m+n+1)-a d (m+2 n+1))-a B (3 b c (m+2 n+1)-a d (m+3 n+1)))}{a b^3 n (m+n+1)}-\frac{d^3 x^{2 n+1} (e x)^m (A b (m+2 n+1)-a B (m+3 n+1))}{a b^2 n (m+2 n+1)}+\frac{(e x)^{m+1} (A b-a B) \left (c+d x^n\right )^3}{a b e n \left (a+b x^n\right )} \]

[Out]

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

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Rubi [A]  time = 2.70145, antiderivative size = 389, normalized size of antiderivative = 0.99, number of steps used = 8, number of rules used = 5, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.161 \[ -\frac{(e x)^{m+1} (b c-a d)^2 \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right ) (A b (b c (m-n+1)-a d (m+2 n+1))-a B (b c (m+1)-a d (m+3 n+1)))}{a^2 b^4 e (m+1) n}-\frac{d (e x)^{m+1} \left (A b \left (a^2 d^2 (m+2 n+1)-3 a b c d (m+n+1)+3 b^2 c^2 (m+1)\right )-a B \left (a^2 d^2 (m+3 n+1)-3 a b c d (m+2 n+1)+3 b^2 c^2 (m+n+1)\right )\right )}{a b^4 e (m+1) n}-\frac{d^2 x^{n+1} (e x)^m (A b (3 b c (m+n+1)-a d (m+2 n+1))-a B (3 b c (m+2 n+1)-a d (m+3 n+1)))}{a b^3 n (m+n+1)}+\frac{(e x)^{m+1} (A b-a B) \left (c+d x^n\right )^3}{a b e n \left (a+b x^n\right )}-\frac{d^3 x^{2 n+1} (e x)^m \left (A-\frac{a B (m+3 n+1)}{b (m+2 n+1)}\right )}{a b n} \]

Antiderivative was successfully verified.

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

[Out]

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

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

[Out]

Timed out

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Mathematica [A]  time = 4.24765, size = 345, normalized size = 0.88 \[ \frac{x (e x)^m \left (-\frac{(b c-a d)^2 \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right ) (A b (b c (m-n+1)-a d (m+2 n+1))+a B (a d (m+3 n+1)-b c (m+1)))}{a^2 (m+1) n}+a^2 B d^3 \left (\frac{a}{a n+b n x^n}+\frac{3}{m+1}\right )+\frac{b^3 c^2 (-3 a A d-a B c+A b c)}{a n \left (a+b x^n\right )}+b^2 d \left (A d \left (3 c \left (\frac{a}{a n+b n x^n}+\frac{1}{m+1}\right )+\frac{d x^n}{m+n+1}\right )+B \left (3 c^2 \left (\frac{a}{a n+b n x^n}+\frac{1}{m+1}\right )+\frac{3 c d x^n}{m+n+1}+\frac{d^2 x^{2 n}}{m+2 n+1}\right )\right )+a b d^2 \left (A d \left (-\frac{a}{a n+b n x^n}-\frac{2}{m+1}\right )+B \left (-\frac{3 a c}{a n+b n x^n}-\frac{6 c}{m+1}-\frac{2 d x^n}{m+n+1}\right )\right )\right )}{b^4} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.088, size = 0, normalized size = 0. \[ \int{\frac{ \left ( ex \right ) ^{m} \left ( A+B{x}^{n} \right ) \left ( c+d{x}^{n} \right ) ^{3}}{ \left ( a+b{x}^{n} \right ) ^{2}}}\, dx \]

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)^2,x)

[Out]

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

[Out]

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

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

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)^2,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^2*x^(2*n) + 2*a*b*x^n + a^2)
, 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)**3/(a+b*x**n)**2,x)

[Out]

Timed out

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

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)^2,x, algorithm="giac")

[Out]

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