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

Optimal. Leaf size=386 \[ -\frac{b (e x)^{m+1} \left (3 a^2 d^2 (A d (m+1)-B c (m+n+1))-3 a b c d (A d (m+n+1)-B c (m+2 n+1))+b^2 c^2 (A d (m+2 n+1)-B c (m+3 n+1))\right )}{c d^4 e (m+1) n}-\frac{b^2 x^{n+1} (e x)^m (3 a d (A d (m+n+1)-B c (m+2 n+1))-b c (A d (m+2 n+1)-B c (m+3 n+1)))}{c d^3 n (m+n+1)}+\frac{(e x)^{m+1} (b c-a d)^2 \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right ) (a d (B c (m+1)-A d (m-n+1))+b c (A d (m+2 n+1)-B c (m+3 n+1)))}{c^2 d^4 e (m+1) n}-\frac{(e x)^{m+1} \left (a+b x^n\right )^3 (B c-A d)}{c d e n \left (c+d x^n\right )}-\frac{b^3 x^{2 n+1} (e x)^m (A d (m+2 n+1)-B c (m+3 n+1))}{c d^2 n (m+2 n+1)} \]

[Out]

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

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Rubi [A]  time = 2.74812, antiderivative size = 381, 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 \[ -\frac{b (e x)^{m+1} \left (3 a^2 d^2 (A d (m+1)-B c (m+n+1))-3 a b c d (A d (m+n+1)-B c (m+2 n+1))+b^2 c^2 (A d (m+2 n+1)-B c (m+3 n+1))\right )}{c d^4 e (m+1) n}-\frac{b^2 x^{n+1} (e x)^m (3 a d (A d (m+n+1)-B c (m+2 n+1))-b c (A d (m+2 n+1)-B c (m+3 n+1)))}{c d^3 n (m+n+1)}+\frac{(e x)^{m+1} (b c-a d)^2 \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right ) (a d (B c (m+1)-A d (m-n+1))+b c (A d (m+2 n+1)-B c (m+3 n+1)))}{c^2 d^4 e (m+1) n}-\frac{(e x)^{m+1} \left (a+b x^n\right )^3 (B c-A d)}{c d e n \left (c+d x^n\right )}-\frac{b^3 x^{2 n+1} (e x)^m \left (A-\frac{B c (m+3 n+1)}{d (m+2 n+1)}\right )}{c d n} \]

Antiderivative was successfully verified.

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

[Out]

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

[Out]

Timed out

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Mathematica [A]  time = 3.45025, size = 365, normalized size = 0.95 \[ x (e x)^m \left (-\frac{a^3 B c-a^3 A d}{c^2 d n+c d^2 n x^n}+\frac{3 a^2 b \left (-A d (m+1)+B c (m+n+1)+B d n x^n\right )}{d^2 (m+1) n \left (c+d x^n\right )}+\frac{3 a b^2 \left (A d \left (\frac{c}{c n+d n x^n}+\frac{1}{m+1}\right )+B \left (-\frac{c^2}{c n+d n x^n}-\frac{2 c}{m+1}+\frac{d x^n}{m+n+1}\right )\right )}{d^3}-\frac{(b c-a d)^2 \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right ) (a d (A d (m-n+1)-B c (m+1))+b c (B c (m+3 n+1)-A d (m+2 n+1)))}{c^2 d^4 (m+1) n}+\frac{b^3 \left (A d \left (-\frac{c^2}{c n+d n x^n}-\frac{2 c}{m+1}+\frac{d x^n}{m+n+1}\right )+B \left (\frac{c^3}{c n+d n x^n}+\frac{3 c^2}{m+1}-\frac{2 c d x^n}{m+n+1}+\frac{d^2 x^{2 n}}{m+2 n+1}\right )\right )}{d^4}\right ) \]

Antiderivative was successfully verified.

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

[Out]

x*(e*x)^m*((3*a^2*b*(-(A*d*(1 + m)) + B*c*(1 + m + n) + B*d*n*x^n))/(d^2*(1 + m)
*n*(c + d*x^n)) - (a^3*B*c - a^3*A*d)/(c^2*d*n + c*d^2*n*x^n) + (3*a*b^2*(A*d*((
1 + m)^(-1) + c/(c*n + d*n*x^n)) + B*((-2*c)/(1 + m) + (d*x^n)/(1 + m + n) - c^2
/(c*n + d*n*x^n))))/d^3 + (b^3*(A*d*((-2*c)/(1 + m) + (d*x^n)/(1 + m + n) - c^2/
(c*n + d*n*x^n)) + B*((3*c^2)/(1 + m) - (2*c*d*x^n)/(1 + m + n) + (d^2*x^(2*n))/
(1 + m + 2*n) + c^3/(c*n + d*n*x^n))))/d^4 - ((b*c - a*d)^2*(a*d*(-(B*c*(1 + m))
+ A*d*(1 + m - n)) + b*c*(-(A*d*(1 + m + 2*n)) + B*c*(1 + m + 3*n)))*Hypergeome
tric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)])/(c^2*d^4*(1 + m)*n))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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