3.1.9 \(\int (e x)^m (a+b x^n)^2 (A+B x^n) (c+d x^n)^2 \, dx\) [9]
Optimal. Leaf size=237 \[ \frac {a c (a B c+2 A (b c+a d)) x^{1+n} (e x)^m}{1+m+n}+\frac {\left (2 a B c (b c+a d)+A \left (b^2 c^2+4 a b c d+a^2 d^2\right )\right ) x^{1+2 n} (e x)^m}{1+m+2 n}+\frac {\left (a^2 B d^2+2 a b d (2 B c+A d)+b^2 c (B c+2 A d)\right ) x^{1+3 n} (e x)^m}{1+m+3 n}+\frac {b d (2 b B c+A b d+2 a B d) x^{1+4 n} (e x)^m}{1+m+4 n}+\frac {b^2 B d^2 x^{1+5 n} (e x)^m}{1+m+5 n}+\frac {a^2 A c^2 (e x)^{1+m}}{e (1+m)} \]
[Out]
a*c*(a*B*c+2*A*(a*d+b*c))*x^(1+n)*(e*x)^m/(1+m+n)+(2*a*B*c*(a*d+b*c)+A*(a^2*d^2+4*a*b*c*d+b^2*c^2))*x^(1+2*n)*
(e*x)^m/(1+m+2*n)+(a^2*B*d^2+2*a*b*d*(A*d+2*B*c)+b^2*c*(2*A*d+B*c))*x^(1+3*n)*(e*x)^m/(1+m+3*n)+b*d*(A*b*d+2*B
*a*d+2*B*b*c)*x^(1+4*n)*(e*x)^m/(1+m+4*n)+b^2*B*d^2*x^(1+5*n)*(e*x)^m/(1+m+5*n)+a^2*A*c^2*(e*x)^(1+m)/e/(1+m)
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Rubi [A]
time = 0.20, antiderivative size = 237, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 3, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {584, 20, 30}
\begin {gather*} \frac {x^{2 n+1} (e x)^m \left (A \left (a^2 d^2+4 a b c d+b^2 c^2\right )+2 a B c (a d+b c)\right )}{m+2 n+1}+\frac {x^{3 n+1} (e x)^m \left (a^2 B d^2+2 a b d (A d+2 B c)+b^2 c (2 A d+B c)\right )}{m+3 n+1}+\frac {a^2 A c^2 (e x)^{m+1}}{e (m+1)}+\frac {a c x^{n+1} (e x)^m (2 A (a d+b c)+a B c)}{m+n+1}+\frac {b d x^{4 n+1} (e x)^m (2 a B d+A b d+2 b B c)}{m+4 n+1}+\frac {b^2 B d^2 x^{5 n+1} (e x)^m}{m+5 n+1} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(e*x)^m*(a + b*x^n)^2*(A + B*x^n)*(c + d*x^n)^2,x]
[Out]
(a*c*(a*B*c + 2*A*(b*c + a*d))*x^(1 + n)*(e*x)^m)/(1 + m + n) + ((2*a*B*c*(b*c + a*d) + A*(b^2*c^2 + 4*a*b*c*d
+ a^2*d^2))*x^(1 + 2*n)*(e*x)^m)/(1 + m + 2*n) + ((a^2*B*d^2 + 2*a*b*d*(2*B*c + A*d) + b^2*c*(B*c + 2*A*d))*x
^(1 + 3*n)*(e*x)^m)/(1 + m + 3*n) + (b*d*(2*b*B*c + A*b*d + 2*a*B*d)*x^(1 + 4*n)*(e*x)^m)/(1 + m + 4*n) + (b^2
*B*d^2*x^(1 + 5*n)*(e*x)^m)/(1 + m + 5*n) + (a^2*A*c^2*(e*x)^(1 + m))/(e*(1 + m))
Rule 20
Int[(u_.)*((a_.)*(v_))^(m_)*((b_.)*(v_))^(n_), x_Symbol] :> Dist[b^IntPart[n]*((b*v)^FracPart[n]/(a^IntPart[n]
*(a*v)^FracPart[n])), Int[u*(a*v)^(m + n), x], x] /; FreeQ[{a, b, m, n}, x] && !IntegerQ[m] && !IntegerQ[n]
&& !IntegerQ[m + n]
Rule 30
Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]
Rule 584
Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_))^
(r_.), x_Symbol] :> Int[ExpandIntegrand[(g*x)^m*(a + b*x^n)^p*(c + d*x^n)^q*(e + f*x^n)^r, x], x] /; FreeQ[{a,
b, c, d, e, f, g, m, n}, x] && IGtQ[p, -2] && IGtQ[q, 0] && IGtQ[r, 0]
Rubi steps
\begin {align*} \int (e x)^m \left (a+b x^n\right )^2 \left (A+B x^n\right ) \left (c+d x^n\right )^2 \, dx &=\int \left (a^2 A c^2 (e x)^m+a c (a B c+2 A (b c+a d)) x^n (e x)^m+\left (2 a B c (b c+a d)+A \left (b^2 c^2+4 a b c d+a^2 d^2\right )\right ) x^{2 n} (e x)^m+\left (a^2 B d^2+2 a b d (2 B c+A d)+b^2 c (B c+2 A d)\right ) x^{3 n} (e x)^m+b d (2 b B c+A b d+2 a B d) x^{4 n} (e x)^m+b^2 B d^2 x^{5 n} (e x)^m\right ) \, dx\\ &=\frac {a^2 A c^2 (e x)^{1+m}}{e (1+m)}+\left (b^2 B d^2\right ) \int x^{5 n} (e x)^m \, dx+(b d (2 b B c+A b d+2 a B d)) \int x^{4 n} (e x)^m \, dx+(a c (a B c+2 A (b c+a d))) \int x^n (e x)^m \, dx+\left (a^2 B d^2+2 a b d (2 B c+A d)+b^2 c (B c+2 A d)\right ) \int x^{3 n} (e x)^m \, dx+\left (2 a B c (b c+a d)+A \left (b^2 c^2+4 a b c d+a^2 d^2\right )\right ) \int x^{2 n} (e x)^m \, dx\\ &=\frac {a^2 A c^2 (e x)^{1+m}}{e (1+m)}+\left (b^2 B d^2 x^{-m} (e x)^m\right ) \int x^{m+5 n} \, dx+\left (b d (2 b B c+A b d+2 a B d) x^{-m} (e x)^m\right ) \int x^{m+4 n} \, dx+\left (a c (a B c+2 A (b c+a d)) x^{-m} (e x)^m\right ) \int x^{m+n} \, dx+\left (\left (a^2 B d^2+2 a b d (2 B c+A d)+b^2 c (B c+2 A d)\right ) x^{-m} (e x)^m\right ) \int x^{m+3 n} \, dx+\left (\left (2 a B c (b c+a d)+A \left (b^2 c^2+4 a b c d+a^2 d^2\right )\right ) x^{-m} (e x)^m\right ) \int x^{m+2 n} \, dx\\ &=\frac {a c (a B c+2 A (b c+a d)) x^{1+n} (e x)^m}{1+m+n}+\frac {\left (2 a B c (b c+a d)+A \left (b^2 c^2+4 a b c d+a^2 d^2\right )\right ) x^{1+2 n} (e x)^m}{1+m+2 n}+\frac {\left (a^2 B d^2+2 a b d (2 B c+A d)+b^2 c (B c+2 A d)\right ) x^{1+3 n} (e x)^m}{1+m+3 n}+\frac {b d (2 b B c+A b d+2 a B d) x^{1+4 n} (e x)^m}{1+m+4 n}+\frac {b^2 B d^2 x^{1+5 n} (e x)^m}{1+m+5 n}+\frac {a^2 A c^2 (e x)^{1+m}}{e (1+m)}\\ \end {align*}
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Mathematica [A]
time = 0.60, size = 199, normalized size = 0.84 \begin {gather*} x (e x)^m \left (\frac {a^2 A c^2}{1+m}+\frac {a c (a B c+2 A (b c+a d)) x^n}{1+m+n}+\frac {\left (2 a B c (b c+a d)+A \left (b^2 c^2+4 a b c d+a^2 d^2\right )\right ) x^{2 n}}{1+m+2 n}+\frac {\left (a^2 B d^2+2 a b d (2 B c+A d)+b^2 c (B c+2 A d)\right ) x^{3 n}}{1+m+3 n}+\frac {b d (2 b B c+A b d+2 a B d) x^{4 n}}{1+m+4 n}+\frac {b^2 B d^2 x^{5 n}}{1+m+5 n}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(e*x)^m*(a + b*x^n)^2*(A + B*x^n)*(c + d*x^n)^2,x]
[Out]
x*(e*x)^m*((a^2*A*c^2)/(1 + m) + (a*c*(a*B*c + 2*A*(b*c + a*d))*x^n)/(1 + m + n) + ((2*a*B*c*(b*c + a*d) + A*(
b^2*c^2 + 4*a*b*c*d + a^2*d^2))*x^(2*n))/(1 + m + 2*n) + ((a^2*B*d^2 + 2*a*b*d*(2*B*c + A*d) + b^2*c*(B*c + 2*
A*d))*x^(3*n))/(1 + m + 3*n) + (b*d*(2*b*B*c + A*b*d + 2*a*B*d)*x^(4*n))/(1 + m + 4*n) + (b^2*B*d^2*x^(5*n))/(
1 + m + 5*n))
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.43, size = 5908, normalized size = 24.93
| | |
method |
result |
size |
| | |
risch |
\(\text {Expression too large to display}\) |
\(5908\) |
| | |
|
|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((e*x)^m*(a+b*x^n)^2*(A+B*x^n)*(c+d*x^n)^2,x,method=_RETURNVERBOSE)
[Out]
result too large to display
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 506 vs.
\(2 (242) = 484\).
time = 0.35, size = 506, normalized size = 2.14 \begin {gather*} \frac {\left (x e\right )^{m + 1} A a^{2} c^{2} e^{\left (-1\right )}}{m + 1} + \frac {B b^{2} d^{2} x e^{\left (m \log \left (x\right ) + 5 \, n \log \left (x\right ) + m\right )}}{m + 5 \, n + 1} + \frac {2 \, B b^{2} c d x e^{\left (m \log \left (x\right ) + 4 \, n \log \left (x\right ) + m\right )}}{m + 4 \, n + 1} + \frac {2 \, B a b d^{2} x e^{\left (m \log \left (x\right ) + 4 \, n \log \left (x\right ) + m\right )}}{m + 4 \, n + 1} + \frac {A b^{2} d^{2} x e^{\left (m \log \left (x\right ) + 4 \, n \log \left (x\right ) + m\right )}}{m + 4 \, n + 1} + \frac {B b^{2} c^{2} x e^{\left (m \log \left (x\right ) + 3 \, n \log \left (x\right ) + m\right )}}{m + 3 \, n + 1} + \frac {4 \, B a b c d x e^{\left (m \log \left (x\right ) + 3 \, n \log \left (x\right ) + m\right )}}{m + 3 \, n + 1} + \frac {2 \, A b^{2} c d x e^{\left (m \log \left (x\right ) + 3 \, n \log \left (x\right ) + m\right )}}{m + 3 \, n + 1} + \frac {B a^{2} d^{2} x e^{\left (m \log \left (x\right ) + 3 \, n \log \left (x\right ) + m\right )}}{m + 3 \, n + 1} + \frac {2 \, A a b d^{2} x e^{\left (m \log \left (x\right ) + 3 \, n \log \left (x\right ) + m\right )}}{m + 3 \, n + 1} + \frac {2 \, B a b c^{2} x e^{\left (m \log \left (x\right ) + 2 \, n \log \left (x\right ) + m\right )}}{m + 2 \, n + 1} + \frac {A b^{2} c^{2} x e^{\left (m \log \left (x\right ) + 2 \, n \log \left (x\right ) + m\right )}}{m + 2 \, n + 1} + \frac {2 \, B a^{2} c d x e^{\left (m \log \left (x\right ) + 2 \, n \log \left (x\right ) + m\right )}}{m + 2 \, n + 1} + \frac {4 \, A a b c d x e^{\left (m \log \left (x\right ) + 2 \, n \log \left (x\right ) + m\right )}}{m + 2 \, n + 1} + \frac {A a^{2} d^{2} x e^{\left (m \log \left (x\right ) + 2 \, n \log \left (x\right ) + m\right )}}{m + 2 \, n + 1} + \frac {B a^{2} c^{2} x e^{\left (m \log \left (x\right ) + n \log \left (x\right ) + m\right )}}{m + n + 1} + \frac {2 \, A a b c^{2} x e^{\left (m \log \left (x\right ) + n \log \left (x\right ) + m\right )}}{m + n + 1} + \frac {2 \, A a^{2} c d x e^{\left (m \log \left (x\right ) + n \log \left (x\right ) + m\right )}}{m + n + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x)^m*(a+b*x^n)^2*(A+B*x^n)*(c+d*x^n)^2,x, algorithm="maxima")
[Out]
(x*e)^(m + 1)*A*a^2*c^2*e^(-1)/(m + 1) + B*b^2*d^2*x*e^(m*log(x) + 5*n*log(x) + m)/(m + 5*n + 1) + 2*B*b^2*c*d
*x*e^(m*log(x) + 4*n*log(x) + m)/(m + 4*n + 1) + 2*B*a*b*d^2*x*e^(m*log(x) + 4*n*log(x) + m)/(m + 4*n + 1) + A
*b^2*d^2*x*e^(m*log(x) + 4*n*log(x) + m)/(m + 4*n + 1) + B*b^2*c^2*x*e^(m*log(x) + 3*n*log(x) + m)/(m + 3*n +
1) + 4*B*a*b*c*d*x*e^(m*log(x) + 3*n*log(x) + m)/(m + 3*n + 1) + 2*A*b^2*c*d*x*e^(m*log(x) + 3*n*log(x) + m)/(
m + 3*n + 1) + B*a^2*d^2*x*e^(m*log(x) + 3*n*log(x) + m)/(m + 3*n + 1) + 2*A*a*b*d^2*x*e^(m*log(x) + 3*n*log(x
) + m)/(m + 3*n + 1) + 2*B*a*b*c^2*x*e^(m*log(x) + 2*n*log(x) + m)/(m + 2*n + 1) + A*b^2*c^2*x*e^(m*log(x) + 2
*n*log(x) + m)/(m + 2*n + 1) + 2*B*a^2*c*d*x*e^(m*log(x) + 2*n*log(x) + m)/(m + 2*n + 1) + 4*A*a*b*c*d*x*e^(m*
log(x) + 2*n*log(x) + m)/(m + 2*n + 1) + A*a^2*d^2*x*e^(m*log(x) + 2*n*log(x) + m)/(m + 2*n + 1) + B*a^2*c^2*x
*e^(m*log(x) + n*log(x) + m)/(m + n + 1) + 2*A*a*b*c^2*x*e^(m*log(x) + n*log(x) + m)/(m + n + 1) + 2*A*a^2*c*d
*x*e^(m*log(x) + n*log(x) + m)/(m + n + 1)
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 3497 vs.
\(2 (242) = 484\).
time = 2.39, size = 3497, normalized size = 14.76 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x)^m*(a+b*x^n)^2*(A+B*x^n)*(c+d*x^n)^2,x, algorithm="fricas")
[Out]
((B*b^2*d^2*m^5 + 5*B*b^2*d^2*m^4 + 10*B*b^2*d^2*m^3 + 10*B*b^2*d^2*m^2 + 5*B*b^2*d^2*m + B*b^2*d^2 + 24*(B*b^
2*d^2*m + B*b^2*d^2)*n^4 + 50*(B*b^2*d^2*m^2 + 2*B*b^2*d^2*m + B*b^2*d^2)*n^3 + 35*(B*b^2*d^2*m^3 + 3*B*b^2*d^
2*m^2 + 3*B*b^2*d^2*m + B*b^2*d^2)*n^2 + 10*(B*b^2*d^2*m^4 + 4*B*b^2*d^2*m^3 + 6*B*b^2*d^2*m^2 + 4*B*b^2*d^2*m
+ B*b^2*d^2)*n)*x*x^(5*n)*e^(m*log(x) + m) + ((2*B*b^2*c*d + (2*B*a*b + A*b^2)*d^2)*m^5 + 2*B*b^2*c*d + 5*(2*
B*b^2*c*d + (2*B*a*b + A*b^2)*d^2)*m^4 + 30*(2*B*b^2*c*d + (2*B*a*b + A*b^2)*d^2 + (2*B*b^2*c*d + (2*B*a*b + A
*b^2)*d^2)*m)*n^4 + 10*(2*B*b^2*c*d + (2*B*a*b + A*b^2)*d^2)*m^3 + 61*(2*B*b^2*c*d + (2*B*a*b + A*b^2)*d^2 + (
2*B*b^2*c*d + (2*B*a*b + A*b^2)*d^2)*m^2 + 2*(2*B*b^2*c*d + (2*B*a*b + A*b^2)*d^2)*m)*n^3 + (2*B*a*b + A*b^2)*
d^2 + 10*(2*B*b^2*c*d + (2*B*a*b + A*b^2)*d^2)*m^2 + 41*(2*B*b^2*c*d + (2*B*b^2*c*d + (2*B*a*b + A*b^2)*d^2)*m
^3 + (2*B*a*b + A*b^2)*d^2 + 3*(2*B*b^2*c*d + (2*B*a*b + A*b^2)*d^2)*m^2 + 3*(2*B*b^2*c*d + (2*B*a*b + A*b^2)*
d^2)*m)*n^2 + 5*(2*B*b^2*c*d + (2*B*a*b + A*b^2)*d^2)*m + 11*(2*B*b^2*c*d + (2*B*b^2*c*d + (2*B*a*b + A*b^2)*d
^2)*m^4 + 4*(2*B*b^2*c*d + (2*B*a*b + A*b^2)*d^2)*m^3 + (2*B*a*b + A*b^2)*d^2 + 6*(2*B*b^2*c*d + (2*B*a*b + A*
b^2)*d^2)*m^2 + 4*(2*B*b^2*c*d + (2*B*a*b + A*b^2)*d^2)*m)*n)*x*x^(4*n)*e^(m*log(x) + m) + ((B*b^2*c^2 + 2*(2*
B*a*b + A*b^2)*c*d + (B*a^2 + 2*A*a*b)*d^2)*m^5 + B*b^2*c^2 + 5*(B*b^2*c^2 + 2*(2*B*a*b + A*b^2)*c*d + (B*a^2
+ 2*A*a*b)*d^2)*m^4 + 40*(B*b^2*c^2 + 2*(2*B*a*b + A*b^2)*c*d + (B*a^2 + 2*A*a*b)*d^2 + (B*b^2*c^2 + 2*(2*B*a*
b + A*b^2)*c*d + (B*a^2 + 2*A*a*b)*d^2)*m)*n^4 + 10*(B*b^2*c^2 + 2*(2*B*a*b + A*b^2)*c*d + (B*a^2 + 2*A*a*b)*d
^2)*m^3 + 78*(B*b^2*c^2 + 2*(2*B*a*b + A*b^2)*c*d + (B*a^2 + 2*A*a*b)*d^2 + (B*b^2*c^2 + 2*(2*B*a*b + A*b^2)*c
*d + (B*a^2 + 2*A*a*b)*d^2)*m^2 + 2*(B*b^2*c^2 + 2*(2*B*a*b + A*b^2)*c*d + (B*a^2 + 2*A*a*b)*d^2)*m)*n^3 + 2*(
2*B*a*b + A*b^2)*c*d + (B*a^2 + 2*A*a*b)*d^2 + 10*(B*b^2*c^2 + 2*(2*B*a*b + A*b^2)*c*d + (B*a^2 + 2*A*a*b)*d^2
)*m^2 + 49*(B*b^2*c^2 + (B*b^2*c^2 + 2*(2*B*a*b + A*b^2)*c*d + (B*a^2 + 2*A*a*b)*d^2)*m^3 + 2*(2*B*a*b + A*b^2
)*c*d + (B*a^2 + 2*A*a*b)*d^2 + 3*(B*b^2*c^2 + 2*(2*B*a*b + A*b^2)*c*d + (B*a^2 + 2*A*a*b)*d^2)*m^2 + 3*(B*b^2
*c^2 + 2*(2*B*a*b + A*b^2)*c*d + (B*a^2 + 2*A*a*b)*d^2)*m)*n^2 + 5*(B*b^2*c^2 + 2*(2*B*a*b + A*b^2)*c*d + (B*a
^2 + 2*A*a*b)*d^2)*m + 12*(B*b^2*c^2 + (B*b^2*c^2 + 2*(2*B*a*b + A*b^2)*c*d + (B*a^2 + 2*A*a*b)*d^2)*m^4 + 4*(
B*b^2*c^2 + 2*(2*B*a*b + A*b^2)*c*d + (B*a^2 + 2*A*a*b)*d^2)*m^3 + 2*(2*B*a*b + A*b^2)*c*d + (B*a^2 + 2*A*a*b)
*d^2 + 6*(B*b^2*c^2 + 2*(2*B*a*b + A*b^2)*c*d + (B*a^2 + 2*A*a*b)*d^2)*m^2 + 4*(B*b^2*c^2 + 2*(2*B*a*b + A*b^2
)*c*d + (B*a^2 + 2*A*a*b)*d^2)*m)*n)*x*x^(3*n)*e^(m*log(x) + m) + ((A*a^2*d^2 + (2*B*a*b + A*b^2)*c^2 + 2*(B*a
^2 + 2*A*a*b)*c*d)*m^5 + A*a^2*d^2 + 5*(A*a^2*d^2 + (2*B*a*b + A*b^2)*c^2 + 2*(B*a^2 + 2*A*a*b)*c*d)*m^4 + 60*
(A*a^2*d^2 + (2*B*a*b + A*b^2)*c^2 + 2*(B*a^2 + 2*A*a*b)*c*d + (A*a^2*d^2 + (2*B*a*b + A*b^2)*c^2 + 2*(B*a^2 +
2*A*a*b)*c*d)*m)*n^4 + 10*(A*a^2*d^2 + (2*B*a*b + A*b^2)*c^2 + 2*(B*a^2 + 2*A*a*b)*c*d)*m^3 + 107*(A*a^2*d^2
+ (2*B*a*b + A*b^2)*c^2 + 2*(B*a^2 + 2*A*a*b)*c*d + (A*a^2*d^2 + (2*B*a*b + A*b^2)*c^2 + 2*(B*a^2 + 2*A*a*b)*c
*d)*m^2 + 2*(A*a^2*d^2 + (2*B*a*b + A*b^2)*c^2 + 2*(B*a^2 + 2*A*a*b)*c*d)*m)*n^3 + (2*B*a*b + A*b^2)*c^2 + 2*(
B*a^2 + 2*A*a*b)*c*d + 10*(A*a^2*d^2 + (2*B*a*b + A*b^2)*c^2 + 2*(B*a^2 + 2*A*a*b)*c*d)*m^2 + 59*(A*a^2*d^2 +
(A*a^2*d^2 + (2*B*a*b + A*b^2)*c^2 + 2*(B*a^2 + 2*A*a*b)*c*d)*m^3 + (2*B*a*b + A*b^2)*c^2 + 2*(B*a^2 + 2*A*a*b
)*c*d + 3*(A*a^2*d^2 + (2*B*a*b + A*b^2)*c^2 + 2*(B*a^2 + 2*A*a*b)*c*d)*m^2 + 3*(A*a^2*d^2 + (2*B*a*b + A*b^2)
*c^2 + 2*(B*a^2 + 2*A*a*b)*c*d)*m)*n^2 + 5*(A*a^2*d^2 + (2*B*a*b + A*b^2)*c^2 + 2*(B*a^2 + 2*A*a*b)*c*d)*m + 1
3*(A*a^2*d^2 + (A*a^2*d^2 + (2*B*a*b + A*b^2)*c^2 + 2*(B*a^2 + 2*A*a*b)*c*d)*m^4 + 4*(A*a^2*d^2 + (2*B*a*b + A
*b^2)*c^2 + 2*(B*a^2 + 2*A*a*b)*c*d)*m^3 + (2*B*a*b + A*b^2)*c^2 + 2*(B*a^2 + 2*A*a*b)*c*d + 6*(A*a^2*d^2 + (2
*B*a*b + A*b^2)*c^2 + 2*(B*a^2 + 2*A*a*b)*c*d)*m^2 + 4*(A*a^2*d^2 + (2*B*a*b + A*b^2)*c^2 + 2*(B*a^2 + 2*A*a*b
)*c*d)*m)*n)*x*x^(2*n)*e^(m*log(x) + m) + ((2*A*a^2*c*d + (B*a^2 + 2*A*a*b)*c^2)*m^5 + 2*A*a^2*c*d + 5*(2*A*a^
2*c*d + (B*a^2 + 2*A*a*b)*c^2)*m^4 + 120*(2*A*a^2*c*d + (B*a^2 + 2*A*a*b)*c^2 + (2*A*a^2*c*d + (B*a^2 + 2*A*a*
b)*c^2)*m)*n^4 + 10*(2*A*a^2*c*d + (B*a^2 + 2*A*a*b)*c^2)*m^3 + 154*(2*A*a^2*c*d + (B*a^2 + 2*A*a*b)*c^2 + (2*
A*a^2*c*d + (B*a^2 + 2*A*a*b)*c^2)*m^2 + 2*(2*A*a^2*c*d + (B*a^2 + 2*A*a*b)*c^2)*m)*n^3 + (B*a^2 + 2*A*a*b)*c^
2 + 10*(2*A*a^2*c*d + (B*a^2 + 2*A*a*b)*c^2)*m^2 + 71*(2*A*a^2*c*d + (2*A*a^2*c*d + (B*a^2 + 2*A*a*b)*c^2)*m^3
+ (B*a^2 + 2*A*a*b)*c^2 + 3*(2*A*a^2*c*d + (B*a^2 + 2*A*a*b)*c^2)*m^2 + 3*(2*A*a^2*c*d + (B*a^2 + 2*A*a*b)*c^
2)*m)*n^2 + 5*(2*A*a^2*c*d + (B*a^2 + 2*A*a*b)*c^2)*m + 14*(2*A*a^2*c*d + (2*A*a^2*c*d + (B*a^2 + 2*A*a*b)*c^2
)*m^4 + 4*(2*A*a^2*c*d + (B*a^2 + 2*A*a*b)*c^2)*m^3 + (B*a^2 + 2*A*a*b)*c^2 + 6*(2*A*a^2*c*d + (B*a^2 + 2*A*a*
b)*c^2)*m^2 + 4*(2*A*a^2*c*d + (B*a^2 + 2*A*a*b...
________________________________________________________________________________________
Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 71580 vs.
\(2 (233) = 466\).
time = 189.29, size = 71580, normalized size = 302.03 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x)**m*(a+b*x**n)**2*(A+B*x**n)*(c+d*x**n)**2,x)
[Out]
Piecewise(((A + B)*(a + b)**2*(c + d)**2*log(x)/e, Eq(m, -1) & Eq(n, 0)), ((A*a**2*c**2*log(x) + 2*A*a**2*c*d*
x**n/n + A*a**2*d**2*x**(2*n)/(2*n) + 2*A*a*b*c**2*x**n/n + 2*A*a*b*c*d*x**(2*n)/n + 2*A*a*b*d**2*x**(3*n)/(3*
n) + A*b**2*c**2*x**(2*n)/(2*n) + 2*A*b**2*c*d*x**(3*n)/(3*n) + A*b**2*d**2*x**(4*n)/(4*n) + B*a**2*c**2*x**n/
n + B*a**2*c*d*x**(2*n)/n + B*a**2*d**2*x**(3*n)/(3*n) + B*a*b*c**2*x**(2*n)/n + 4*B*a*b*c*d*x**(3*n)/(3*n) +
B*a*b*d**2*x**(4*n)/(2*n) + B*b**2*c**2*x**(3*n)/(3*n) + B*b**2*c*d*x**(4*n)/(2*n) + B*b**2*d**2*x**(5*n)/(5*n
))/e, Eq(m, -1)), (A*a**2*c**2*Piecewise((-1/(5*n*(e*x)**(5*n)), Ne(n, 0)), (log(x), True))/e + 2*A*a**2*c*d*P
iecewise((-x**n/(4*n*(e*x)**(5*n)), Ne(n, 0)), (log(x), True))/e + A*a**2*d**2*Piecewise((-x**(2*n)/(3*n*(e*x)
**(5*n)), Ne(n, 0)), (log(x), True))/e + 2*A*a*b*c**2*Piecewise((-x**n/(4*n*(e*x)**(5*n)), Ne(n, 0)), (log(x),
True))/e + 4*A*a*b*c*d*Piecewise((-x**(2*n)/(3*n*(e*x)**(5*n)), Ne(n, 0)), (log(x), True))/e + 2*A*a*b*d**2*P
iecewise((-x**(3*n)/(2*n*(e*x)**(5*n)), Ne(n, 0)), (log(x), True))/e + A*b**2*c**2*Piecewise((-x**(2*n)/(3*n*(
e*x)**(5*n)), Ne(n, 0)), (log(x), True))/e + 2*A*b**2*c*d*Piecewise((-x**(3*n)/(2*n*(e*x)**(5*n)), Ne(n, 0)),
(log(x), True))/e + A*b**2*d**2*Piecewise((-x**(4*n)/(n*(e*x)**(5*n)), Ne(n, 0)), (log(x), True))/e + B*a**2*c
**2*Piecewise((-x**n/(4*n*(e*x)**(5*n)), Ne(n, 0)), (log(x), True))/e + 2*B*a**2*c*d*Piecewise((-x**(2*n)/(3*n
*(e*x)**(5*n)), Ne(n, 0)), (log(x), True))/e + B*a**2*d**2*Piecewise((-x**(3*n)/(2*n*(e*x)**(5*n)), Ne(n, 0)),
(log(x), True))/e + 2*B*a*b*c**2*Piecewise((-x**(2*n)/(3*n*(e*x)**(5*n)), Ne(n, 0)), (log(x), True))/e + 4*B*
a*b*c*d*Piecewise((-x**(3*n)/(2*n*(e*x)**(5*n)), Ne(n, 0)), (log(x), True))/e + 2*B*a*b*d**2*Piecewise((-x**(4
*n)/(n*(e*x)**(5*n)), Ne(n, 0)), (log(x), True))/e + B*b**2*c**2*Piecewise((-x**(3*n)/(2*n*(e*x)**(5*n)), Ne(n
, 0)), (log(x), True))/e + 2*B*b**2*c*d*Piecewise((-x**(4*n)/(n*(e*x)**(5*n)), Ne(n, 0)), (log(x), True))/e +
B*b**2*d**2*Piecewise((0, (Abs(x) < 1) & (1/Abs(x) < 1)), (log(x)/e**(5*n), Abs(x) < 1), (-log(1/x)/e**(5*n),
1/Abs(x) < 1), (-meijerg(((), (1, 1)), ((0, 0), ()), x)/e**(5*n) + meijerg(((1, 1), ()), ((), (0, 0)), x)/e**(
5*n), True))/e, Eq(m, -5*n - 1)), (A*a**2*c**2*Piecewise((-1/(4*n*(e*x)**(4*n)), Ne(n, 0)), (log(x), True))/e
+ 2*A*a**2*c*d*Piecewise((-x**n/(3*n*(e*x)**(4*n)), Ne(n, 0)), (log(x), True))/e + A*a**2*d**2*Piecewise((-x**
(2*n)/(2*n*(e*x)**(4*n)), Ne(n, 0)), (log(x), True))/e + 2*A*a*b*c**2*Piecewise((-x**n/(3*n*(e*x)**(4*n)), Ne(
n, 0)), (log(x), True))/e + 4*A*a*b*c*d*Piecewise((-x**(2*n)/(2*n*(e*x)**(4*n)), Ne(n, 0)), (log(x), True))/e
+ 2*A*a*b*d**2*Piecewise((-x**(3*n)/(n*(e*x)**(4*n)), Ne(n, 0)), (log(x), True))/e + A*b**2*c**2*Piecewise((-x
**(2*n)/(2*n*(e*x)**(4*n)), Ne(n, 0)), (log(x), True))/e + 2*A*b**2*c*d*Piecewise((-x**(3*n)/(n*(e*x)**(4*n)),
Ne(n, 0)), (log(x), True))/e + A*b**2*d**2*Piecewise((0, (Abs(x) < 1) & (1/Abs(x) < 1)), (log(x)/e**(4*n), Ab
s(x) < 1), (-log(1/x)/e**(4*n), 1/Abs(x) < 1), (-meijerg(((), (1, 1)), ((0, 0), ()), x)/e**(4*n) + meijerg(((1
, 1), ()), ((), (0, 0)), x)/e**(4*n), True))/e + B*a**2*c**2*Piecewise((-x**n/(3*n*(e*x)**(4*n)), Ne(n, 0)), (
log(x), True))/e + 2*B*a**2*c*d*Piecewise((-x**(2*n)/(2*n*(e*x)**(4*n)), Ne(n, 0)), (log(x), True))/e + B*a**2
*d**2*Piecewise((-x**(3*n)/(n*(e*x)**(4*n)), Ne(n, 0)), (log(x), True))/e + 2*B*a*b*c**2*Piecewise((-x**(2*n)/
(2*n*(e*x)**(4*n)), Ne(n, 0)), (log(x), True))/e + 4*B*a*b*c*d*Piecewise((-x**(3*n)/(n*(e*x)**(4*n)), Ne(n, 0)
), (log(x), True))/e + 2*B*a*b*d**2*Piecewise((0, (Abs(x) < 1) & (1/Abs(x) < 1)), (log(x)/e**(4*n), Abs(x) < 1
), (-log(1/x)/e**(4*n), 1/Abs(x) < 1), (-meijerg(((), (1, 1)), ((0, 0), ()), x)/e**(4*n) + meijerg(((1, 1), ()
), ((), (0, 0)), x)/e**(4*n), True))/e + B*b**2*c**2*Piecewise((-x**(3*n)/(n*(e*x)**(4*n)), Ne(n, 0)), (log(x)
, True))/e + 2*B*b**2*c*d*Piecewise((0, (Abs(x) < 1) & (1/Abs(x) < 1)), (log(x)/e**(4*n), Abs(x) < 1), (-log(1
/x)/e**(4*n), 1/Abs(x) < 1), (-meijerg(((), (1, 1)), ((0, 0), ()), x)/e**(4*n) + meijerg(((1, 1), ()), ((), (0
, 0)), x)/e**(4*n), True))/e + B*b**2*d**2*Piecewise((x**(5*n)/(n*(e*x)**(4*n)), Ne(n, 0)), (log(x), True))/e,
Eq(m, -4*n - 1)), (A*a**2*c**2*Piecewise((-1/(3*n*(e*x)**(3*n)), Ne(n, 0)), (log(x), True))/e + 2*A*a**2*c*d*
Piecewise((-x**n/(2*n*(e*x)**(3*n)), Ne(n, 0)), (log(x), True))/e + A*a**2*d**2*Piecewise((-x**(2*n)/(n*(e*x)*
*(3*n)), Ne(n, 0)), (log(x), True))/e + 2*A*a*b*c**2*Piecewise((-x**n/(2*n*(e*x)**(3*n)), Ne(n, 0)), (log(x),
True))/e + 4*A*a*b*c*d*Piecewise((-x**(2*n)/(n*(e*x)**(3*n)), Ne(n, 0)), (log(x), True))/e + 2*A*a*b*d**2*Piec
ewise((0, (Abs(x) < 1) & (1/Abs(x) < 1)), (log(x)/e**(3*n), Abs(x) < 1), (-log(1/x)/e**(3*n), 1/Abs(x) < 1), (
-meijerg(((), (1, 1)), ((0, 0), ()), x)/e**(3*n) + meijerg(((1, 1), ()), ((), (0, 0)), x)/e**(3*n), True))/e +
A*b**2*c**2*Piecewise((-x**(2*n)/(n*(e*x)**(3*n)), Ne(n, 0)), (log(x), True))/e + 2*A*b**2*c*d*Piecewise((0,
(Abs(x) < 1) & (1/Abs(x) < 1)), (log(x)/e**(3*n...
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 8103 vs.
\(2 (242) = 484\).
time = 1.98, size = 8103, normalized size = 34.19 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x)^m*(a+b*x^n)^2*(A+B*x^n)*(c+d*x^n)^2,x, algorithm="giac")
[Out]
(B*b^2*d^2*m^5*x*x^m*x^(5*n)*e^m + 10*B*b^2*d^2*m^4*n*x*x^m*x^(5*n)*e^m + 35*B*b^2*d^2*m^3*n^2*x*x^m*x^(5*n)*e
^m + 50*B*b^2*d^2*m^2*n^3*x*x^m*x^(5*n)*e^m + 24*B*b^2*d^2*m*n^4*x*x^m*x^(5*n)*e^m + 2*B*b^2*c*d*m^5*x*x^m*x^(
4*n)*e^m + 2*B*a*b*d^2*m^5*x*x^m*x^(4*n)*e^m + A*b^2*d^2*m^5*x*x^m*x^(4*n)*e^m + 22*B*b^2*c*d*m^4*n*x*x^m*x^(4
*n)*e^m + 22*B*a*b*d^2*m^4*n*x*x^m*x^(4*n)*e^m + 11*A*b^2*d^2*m^4*n*x*x^m*x^(4*n)*e^m + 82*B*b^2*c*d*m^3*n^2*x
*x^m*x^(4*n)*e^m + 82*B*a*b*d^2*m^3*n^2*x*x^m*x^(4*n)*e^m + 41*A*b^2*d^2*m^3*n^2*x*x^m*x^(4*n)*e^m + 122*B*b^2
*c*d*m^2*n^3*x*x^m*x^(4*n)*e^m + 122*B*a*b*d^2*m^2*n^3*x*x^m*x^(4*n)*e^m + 61*A*b^2*d^2*m^2*n^3*x*x^m*x^(4*n)*
e^m + 60*B*b^2*c*d*m*n^4*x*x^m*x^(4*n)*e^m + 60*B*a*b*d^2*m*n^4*x*x^m*x^(4*n)*e^m + 30*A*b^2*d^2*m*n^4*x*x^m*x
^(4*n)*e^m + B*b^2*c^2*m^5*x*x^m*x^(3*n)*e^m + 4*B*a*b*c*d*m^5*x*x^m*x^(3*n)*e^m + 2*A*b^2*c*d*m^5*x*x^m*x^(3*
n)*e^m + B*a^2*d^2*m^5*x*x^m*x^(3*n)*e^m + 2*A*a*b*d^2*m^5*x*x^m*x^(3*n)*e^m + 12*B*b^2*c^2*m^4*n*x*x^m*x^(3*n
)*e^m + 48*B*a*b*c*d*m^4*n*x*x^m*x^(3*n)*e^m + 24*A*b^2*c*d*m^4*n*x*x^m*x^(3*n)*e^m + 12*B*a^2*d^2*m^4*n*x*x^m
*x^(3*n)*e^m + 24*A*a*b*d^2*m^4*n*x*x^m*x^(3*n)*e^m + 49*B*b^2*c^2*m^3*n^2*x*x^m*x^(3*n)*e^m + 196*B*a*b*c*d*m
^3*n^2*x*x^m*x^(3*n)*e^m + 98*A*b^2*c*d*m^3*n^2*x*x^m*x^(3*n)*e^m + 49*B*a^2*d^2*m^3*n^2*x*x^m*x^(3*n)*e^m + 9
8*A*a*b*d^2*m^3*n^2*x*x^m*x^(3*n)*e^m + 78*B*b^2*c^2*m^2*n^3*x*x^m*x^(3*n)*e^m + 312*B*a*b*c*d*m^2*n^3*x*x^m*x
^(3*n)*e^m + 156*A*b^2*c*d*m^2*n^3*x*x^m*x^(3*n)*e^m + 78*B*a^2*d^2*m^2*n^3*x*x^m*x^(3*n)*e^m + 156*A*a*b*d^2*
m^2*n^3*x*x^m*x^(3*n)*e^m + 40*B*b^2*c^2*m*n^4*x*x^m*x^(3*n)*e^m + 160*B*a*b*c*d*m*n^4*x*x^m*x^(3*n)*e^m + 80*
A*b^2*c*d*m*n^4*x*x^m*x^(3*n)*e^m + 40*B*a^2*d^2*m*n^4*x*x^m*x^(3*n)*e^m + 80*A*a*b*d^2*m*n^4*x*x^m*x^(3*n)*e^
m + 2*B*a*b*c^2*m^5*x*x^m*x^(2*n)*e^m + A*b^2*c^2*m^5*x*x^m*x^(2*n)*e^m + 2*B*a^2*c*d*m^5*x*x^m*x^(2*n)*e^m +
4*A*a*b*c*d*m^5*x*x^m*x^(2*n)*e^m + A*a^2*d^2*m^5*x*x^m*x^(2*n)*e^m + 26*B*a*b*c^2*m^4*n*x*x^m*x^(2*n)*e^m + 1
3*A*b^2*c^2*m^4*n*x*x^m*x^(2*n)*e^m + 26*B*a^2*c*d*m^4*n*x*x^m*x^(2*n)*e^m + 52*A*a*b*c*d*m^4*n*x*x^m*x^(2*n)*
e^m + 13*A*a^2*d^2*m^4*n*x*x^m*x^(2*n)*e^m + 118*B*a*b*c^2*m^3*n^2*x*x^m*x^(2*n)*e^m + 59*A*b^2*c^2*m^3*n^2*x*
x^m*x^(2*n)*e^m + 118*B*a^2*c*d*m^3*n^2*x*x^m*x^(2*n)*e^m + 236*A*a*b*c*d*m^3*n^2*x*x^m*x^(2*n)*e^m + 59*A*a^2
*d^2*m^3*n^2*x*x^m*x^(2*n)*e^m + 214*B*a*b*c^2*m^2*n^3*x*x^m*x^(2*n)*e^m + 107*A*b^2*c^2*m^2*n^3*x*x^m*x^(2*n)
*e^m + 214*B*a^2*c*d*m^2*n^3*x*x^m*x^(2*n)*e^m + 428*A*a*b*c*d*m^2*n^3*x*x^m*x^(2*n)*e^m + 107*A*a^2*d^2*m^2*n
^3*x*x^m*x^(2*n)*e^m + 120*B*a*b*c^2*m*n^4*x*x^m*x^(2*n)*e^m + 60*A*b^2*c^2*m*n^4*x*x^m*x^(2*n)*e^m + 120*B*a^
2*c*d*m*n^4*x*x^m*x^(2*n)*e^m + 240*A*a*b*c*d*m*n^4*x*x^m*x^(2*n)*e^m + 60*A*a^2*d^2*m*n^4*x*x^m*x^(2*n)*e^m +
B*a^2*c^2*m^5*x*x^m*x^n*e^m + 2*A*a*b*c^2*m^5*x*x^m*x^n*e^m + 2*A*a^2*c*d*m^5*x*x^m*x^n*e^m + 14*B*a^2*c^2*m^
4*n*x*x^m*x^n*e^m + 28*A*a*b*c^2*m^4*n*x*x^m*x^n*e^m + 28*A*a^2*c*d*m^4*n*x*x^m*x^n*e^m + 71*B*a^2*c^2*m^3*n^2
*x*x^m*x^n*e^m + 142*A*a*b*c^2*m^3*n^2*x*x^m*x^n*e^m + 142*A*a^2*c*d*m^3*n^2*x*x^m*x^n*e^m + 154*B*a^2*c^2*m^2
*n^3*x*x^m*x^n*e^m + 308*A*a*b*c^2*m^2*n^3*x*x^m*x^n*e^m + 308*A*a^2*c*d*m^2*n^3*x*x^m*x^n*e^m + 120*B*a^2*c^2
*m*n^4*x*x^m*x^n*e^m + 240*A*a*b*c^2*m*n^4*x*x^m*x^n*e^m + 240*A*a^2*c*d*m*n^4*x*x^m*x^n*e^m + A*a^2*c^2*m^5*x
*x^m*e^m + 15*A*a^2*c^2*m^4*n*x*x^m*e^m + 85*A*a^2*c^2*m^3*n^2*x*x^m*e^m + 225*A*a^2*c^2*m^2*n^3*x*x^m*e^m + 2
74*A*a^2*c^2*m*n^4*x*x^m*e^m + 120*A*a^2*c^2*n^5*x*x^m*e^m + 5*B*b^2*d^2*m^4*x*x^m*x^(5*n)*e^m + 40*B*b^2*d^2*
m^3*n*x*x^m*x^(5*n)*e^m + 105*B*b^2*d^2*m^2*n^2*x*x^m*x^(5*n)*e^m + 100*B*b^2*d^2*m*n^3*x*x^m*x^(5*n)*e^m + 24
*B*b^2*d^2*n^4*x*x^m*x^(5*n)*e^m + 10*B*b^2*c*d*m^4*x*x^m*x^(4*n)*e^m + 10*B*a*b*d^2*m^4*x*x^m*x^(4*n)*e^m + 5
*A*b^2*d^2*m^4*x*x^m*x^(4*n)*e^m + 88*B*b^2*c*d*m^3*n*x*x^m*x^(4*n)*e^m + 88*B*a*b*d^2*m^3*n*x*x^m*x^(4*n)*e^m
+ 44*A*b^2*d^2*m^3*n*x*x^m*x^(4*n)*e^m + 246*B*b^2*c*d*m^2*n^2*x*x^m*x^(4*n)*e^m + 246*B*a*b*d^2*m^2*n^2*x*x^
m*x^(4*n)*e^m + 123*A*b^2*d^2*m^2*n^2*x*x^m*x^(4*n)*e^m + 244*B*b^2*c*d*m*n^3*x*x^m*x^(4*n)*e^m + 244*B*a*b*d^
2*m*n^3*x*x^m*x^(4*n)*e^m + 122*A*b^2*d^2*m*n^3*x*x^m*x^(4*n)*e^m + 60*B*b^2*c*d*n^4*x*x^m*x^(4*n)*e^m + 60*B*
a*b*d^2*n^4*x*x^m*x^(4*n)*e^m + 30*A*b^2*d^2*n^4*x*x^m*x^(4*n)*e^m + 5*B*b^2*c^2*m^4*x*x^m*x^(3*n)*e^m + 20*B*
a*b*c*d*m^4*x*x^m*x^(3*n)*e^m + 10*A*b^2*c*d*m^4*x*x^m*x^(3*n)*e^m + 5*B*a^2*d^2*m^4*x*x^m*x^(3*n)*e^m + 10*A*
a*b*d^2*m^4*x*x^m*x^(3*n)*e^m + 48*B*b^2*c^2*m^3*n*x*x^m*x^(3*n)*e^m + 192*B*a*b*c*d*m^3*n*x*x^m*x^(3*n)*e^m +
96*A*b^2*c*d*m^3*n*x*x^m*x^(3*n)*e^m + 48*B*a^2*d^2*m^3*n*x*x^m*x^(3*n)*e^m + 96*A*a*b*d^2*m^3*n*x*x^m*x^(3*n
)*e^m + 147*B*b^2*c^2*m^2*n^2*x*x^m*x^(3*n)*e^m + 588*B*a*b*c*d*m^2*n^2*x*x^m*x^(3*n)*e^m + 294*A*b^2*c*d*m^2*
n^2*x*x^m*x^(3*n)*e^m + 147*B*a^2*d^2*m^2*n^2*x*x^m*x^(3*n)*e^m + 294*A*a*b*d^2*m^2*n^2*x*x^m*x^(3*n)*e^m + 15
6*B*b^2*c^2*m*n^3*x*x^m*x^(3*n)*e^m + 624*B*a*b*c*d*m*n^3*x*x^m*x^(3*n)*e^m + 312*A*b^2*c*d*m*n^3*x*x^m*x^(3*n
)*e^m + 156*B*a^2*d^2*m*n^3*x*x^m*x^(3*n)*e^m +...
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Mupad [B]
time = 5.59, size = 1119, normalized size = 4.72 \begin {gather*} \frac {x\,x^{2\,n}\,{\left (e\,x\right )}^m\,\left (2\,B\,a^2\,c\,d+A\,a^2\,d^2+2\,B\,a\,b\,c^2+4\,A\,a\,b\,c\,d+A\,b^2\,c^2\right )\,\left (m^4+13\,m^3\,n+4\,m^3+59\,m^2\,n^2+39\,m^2\,n+6\,m^2+107\,m\,n^3+118\,m\,n^2+39\,m\,n+4\,m+60\,n^4+107\,n^3+59\,n^2+13\,n+1\right )}{m^5+15\,m^4\,n+5\,m^4+85\,m^3\,n^2+60\,m^3\,n+10\,m^3+225\,m^2\,n^3+255\,m^2\,n^2+90\,m^2\,n+10\,m^2+274\,m\,n^4+450\,m\,n^3+255\,m\,n^2+60\,m\,n+5\,m+120\,n^5+274\,n^4+225\,n^3+85\,n^2+15\,n+1}+\frac {x\,x^{3\,n}\,{\left (e\,x\right )}^m\,\left (B\,a^2\,d^2+4\,B\,a\,b\,c\,d+2\,A\,a\,b\,d^2+B\,b^2\,c^2+2\,A\,b^2\,c\,d\right )\,\left (m^4+12\,m^3\,n+4\,m^3+49\,m^2\,n^2+36\,m^2\,n+6\,m^2+78\,m\,n^3+98\,m\,n^2+36\,m\,n+4\,m+40\,n^4+78\,n^3+49\,n^2+12\,n+1\right )}{m^5+15\,m^4\,n+5\,m^4+85\,m^3\,n^2+60\,m^3\,n+10\,m^3+225\,m^2\,n^3+255\,m^2\,n^2+90\,m^2\,n+10\,m^2+274\,m\,n^4+450\,m\,n^3+255\,m\,n^2+60\,m\,n+5\,m+120\,n^5+274\,n^4+225\,n^3+85\,n^2+15\,n+1}+\frac {A\,a^2\,c^2\,x\,{\left (e\,x\right )}^m}{m+1}+\frac {b\,d\,x\,x^{4\,n}\,{\left (e\,x\right )}^m\,\left (A\,b\,d+2\,B\,a\,d+2\,B\,b\,c\right )\,\left (m^4+11\,m^3\,n+4\,m^3+41\,m^2\,n^2+33\,m^2\,n+6\,m^2+61\,m\,n^3+82\,m\,n^2+33\,m\,n+4\,m+30\,n^4+61\,n^3+41\,n^2+11\,n+1\right )}{m^5+15\,m^4\,n+5\,m^4+85\,m^3\,n^2+60\,m^3\,n+10\,m^3+225\,m^2\,n^3+255\,m^2\,n^2+90\,m^2\,n+10\,m^2+274\,m\,n^4+450\,m\,n^3+255\,m\,n^2+60\,m\,n+5\,m+120\,n^5+274\,n^4+225\,n^3+85\,n^2+15\,n+1}+\frac {B\,b^2\,d^2\,x\,x^{5\,n}\,{\left (e\,x\right )}^m\,\left (m^4+10\,m^3\,n+4\,m^3+35\,m^2\,n^2+30\,m^2\,n+6\,m^2+50\,m\,n^3+70\,m\,n^2+30\,m\,n+4\,m+24\,n^4+50\,n^3+35\,n^2+10\,n+1\right )}{m^5+15\,m^4\,n+5\,m^4+85\,m^3\,n^2+60\,m^3\,n+10\,m^3+225\,m^2\,n^3+255\,m^2\,n^2+90\,m^2\,n+10\,m^2+274\,m\,n^4+450\,m\,n^3+255\,m\,n^2+60\,m\,n+5\,m+120\,n^5+274\,n^4+225\,n^3+85\,n^2+15\,n+1}+\frac {a\,c\,x\,x^n\,{\left (e\,x\right )}^m\,\left (2\,A\,a\,d+2\,A\,b\,c+B\,a\,c\right )\,\left (m^4+14\,m^3\,n+4\,m^3+71\,m^2\,n^2+42\,m^2\,n+6\,m^2+154\,m\,n^3+142\,m\,n^2+42\,m\,n+4\,m+120\,n^4+154\,n^3+71\,n^2+14\,n+1\right )}{m^5+15\,m^4\,n+5\,m^4+85\,m^3\,n^2+60\,m^3\,n+10\,m^3+225\,m^2\,n^3+255\,m^2\,n^2+90\,m^2\,n+10\,m^2+274\,m\,n^4+450\,m\,n^3+255\,m\,n^2+60\,m\,n+5\,m+120\,n^5+274\,n^4+225\,n^3+85\,n^2+15\,n+1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((e*x)^m*(A + B*x^n)*(a + b*x^n)^2*(c + d*x^n)^2,x)
[Out]
(x*x^(2*n)*(e*x)^m*(A*a^2*d^2 + A*b^2*c^2 + 2*B*a*b*c^2 + 2*B*a^2*c*d + 4*A*a*b*c*d)*(4*m + 13*n + 39*m*n + 11
8*m*n^2 + 39*m^2*n + 107*m*n^3 + 13*m^3*n + 6*m^2 + 4*m^3 + m^4 + 59*n^2 + 107*n^3 + 60*n^4 + 59*m^2*n^2 + 1))
/(5*m + 15*n + 60*m*n + 255*m*n^2 + 90*m^2*n + 450*m*n^3 + 60*m^3*n + 274*m*n^4 + 15*m^4*n + 10*m^2 + 10*m^3 +
5*m^4 + m^5 + 85*n^2 + 225*n^3 + 274*n^4 + 120*n^5 + 255*m^2*n^2 + 225*m^2*n^3 + 85*m^3*n^2 + 1) + (x*x^(3*n)
*(e*x)^m*(B*a^2*d^2 + B*b^2*c^2 + 2*A*a*b*d^2 + 2*A*b^2*c*d + 4*B*a*b*c*d)*(4*m + 12*n + 36*m*n + 98*m*n^2 + 3
6*m^2*n + 78*m*n^3 + 12*m^3*n + 6*m^2 + 4*m^3 + m^4 + 49*n^2 + 78*n^3 + 40*n^4 + 49*m^2*n^2 + 1))/(5*m + 15*n
+ 60*m*n + 255*m*n^2 + 90*m^2*n + 450*m*n^3 + 60*m^3*n + 274*m*n^4 + 15*m^4*n + 10*m^2 + 10*m^3 + 5*m^4 + m^5
+ 85*n^2 + 225*n^3 + 274*n^4 + 120*n^5 + 255*m^2*n^2 + 225*m^2*n^3 + 85*m^3*n^2 + 1) + (A*a^2*c^2*x*(e*x)^m)/(
m + 1) + (b*d*x*x^(4*n)*(e*x)^m*(A*b*d + 2*B*a*d + 2*B*b*c)*(4*m + 11*n + 33*m*n + 82*m*n^2 + 33*m^2*n + 61*m*
n^3 + 11*m^3*n + 6*m^2 + 4*m^3 + m^4 + 41*n^2 + 61*n^3 + 30*n^4 + 41*m^2*n^2 + 1))/(5*m + 15*n + 60*m*n + 255*
m*n^2 + 90*m^2*n + 450*m*n^3 + 60*m^3*n + 274*m*n^4 + 15*m^4*n + 10*m^2 + 10*m^3 + 5*m^4 + m^5 + 85*n^2 + 225*
n^3 + 274*n^4 + 120*n^5 + 255*m^2*n^2 + 225*m^2*n^3 + 85*m^3*n^2 + 1) + (B*b^2*d^2*x*x^(5*n)*(e*x)^m*(4*m + 10
*n + 30*m*n + 70*m*n^2 + 30*m^2*n + 50*m*n^3 + 10*m^3*n + 6*m^2 + 4*m^3 + m^4 + 35*n^2 + 50*n^3 + 24*n^4 + 35*
m^2*n^2 + 1))/(5*m + 15*n + 60*m*n + 255*m*n^2 + 90*m^2*n + 450*m*n^3 + 60*m^3*n + 274*m*n^4 + 15*m^4*n + 10*m
^2 + 10*m^3 + 5*m^4 + m^5 + 85*n^2 + 225*n^3 + 274*n^4 + 120*n^5 + 255*m^2*n^2 + 225*m^2*n^3 + 85*m^3*n^2 + 1)
+ (a*c*x*x^n*(e*x)^m*(2*A*a*d + 2*A*b*c + B*a*c)*(4*m + 14*n + 42*m*n + 142*m*n^2 + 42*m^2*n + 154*m*n^3 + 14
*m^3*n + 6*m^2 + 4*m^3 + m^4 + 71*n^2 + 154*n^3 + 120*n^4 + 71*m^2*n^2 + 1))/(5*m + 15*n + 60*m*n + 255*m*n^2
+ 90*m^2*n + 450*m*n^3 + 60*m^3*n + 274*m*n^4 + 15*m^4*n + 10*m^2 + 10*m^3 + 5*m^4 + m^5 + 85*n^2 + 225*n^3 +
274*n^4 + 120*n^5 + 255*m^2*n^2 + 225*m^2*n^3 + 85*m^3*n^2 + 1)
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