3.1.8 \(\int (e x)^m (a+b x^n)^3 (A+B x^n) (c+d x^n)^2 \, dx\) [8]

Optimal. Leaf size=318 \[ \frac {a^2 c (3 A b c+a B c+2 a A d) x^{1+n} (e x)^m}{1+m+n}+\frac {a \left (a B c (3 b c+2 a d)+A \left (3 b^2 c^2+6 a b c d+a^2 d^2\right )\right ) x^{1+2 n} (e x)^m}{1+m+2 n}+\frac {\left (a B \left (3 b^2 c^2+6 a b c d+a^2 d^2\right )+A b \left (b^2 c^2+6 a b c d+3 a^2 d^2\right )\right ) x^{1+3 n} (e x)^m}{1+m+3 n}+\frac {b \left (3 a^2 B d^2+3 a b d (2 B c+A d)+b^2 c (B c+2 A d)\right ) x^{1+4 n} (e x)^m}{1+m+4 n}+\frac {b^2 d (2 b B c+A b d+3 a B d) x^{1+5 n} (e x)^m}{1+m+5 n}+\frac {b^3 B d^2 x^{1+6 n} (e x)^m}{1+m+6 n}+\frac {a^3 A c^2 (e x)^{1+m}}{e (1+m)} \]

[Out]

a^2*c*(2*A*a*d+3*A*b*c+B*a*c)*x^(1+n)*(e*x)^m/(1+m+n)+a*(a*B*c*(2*a*d+3*b*c)+A*(a^2*d^2+6*a*b*c*d+3*b^2*c^2))*
x^(1+2*n)*(e*x)^m/(1+m+2*n)+(a*B*(a^2*d^2+6*a*b*c*d+3*b^2*c^2)+A*b*(3*a^2*d^2+6*a*b*c*d+b^2*c^2))*x^(1+3*n)*(e
*x)^m/(1+m+3*n)+b*(3*a^2*B*d^2+3*a*b*d*(A*d+2*B*c)+b^2*c*(2*A*d+B*c))*x^(1+4*n)*(e*x)^m/(1+m+4*n)+b^2*d*(A*b*d
+3*B*a*d+2*B*b*c)*x^(1+5*n)*(e*x)^m/(1+m+5*n)+b^3*B*d^2*x^(1+6*n)*(e*x)^m/(1+m+6*n)+a^3*A*c^2*(e*x)^(1+m)/e/(1
+m)

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Rubi [A]
time = 0.28, antiderivative size = 318, normalized size of antiderivative = 1.00, number of steps used = 14, 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 {a^3 A c^2 (e x)^{m+1}}{e (m+1)}+\frac {a x^{2 n+1} (e x)^m \left (A \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+a B c (2 a d+3 b c)\right )}{m+2 n+1}+\frac {x^{3 n+1} (e x)^m \left (A b \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+a B \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )\right )}{m+3 n+1}+\frac {b x^{4 n+1} (e x)^m \left (3 a^2 B d^2+3 a b d (A d+2 B c)+b^2 c (2 A d+B c)\right )}{m+4 n+1}+\frac {a^2 c x^{n+1} (e x)^m (2 a A d+a B c+3 A b c)}{m+n+1}+\frac {b^2 d x^{5 n+1} (e x)^m (3 a B d+A b d+2 b B c)}{m+5 n+1}+\frac {b^3 B d^2 x^{6 n+1} (e x)^m}{m+6 n+1} \end {gather*}

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]

(a^2*c*(3*A*b*c + a*B*c + 2*a*A*d)*x^(1 + n)*(e*x)^m)/(1 + m + n) + (a*(a*B*c*(3*b*c + 2*a*d) + A*(3*b^2*c^2 +
6*a*b*c*d + a^2*d^2))*x^(1 + 2*n)*(e*x)^m)/(1 + m + 2*n) + ((a*B*(3*b^2*c^2 + 6*a*b*c*d + a^2*d^2) + A*b*(b^2
*c^2 + 6*a*b*c*d + 3*a^2*d^2))*x^(1 + 3*n)*(e*x)^m)/(1 + m + 3*n) + (b*(3*a^2*B*d^2 + 3*a*b*d*(2*B*c + A*d) +
b^2*c*(B*c + 2*A*d))*x^(1 + 4*n)*(e*x)^m)/(1 + m + 4*n) + (b^2*d*(2*b*B*c + A*b*d + 3*a*B*d)*x^(1 + 5*n)*(e*x)
^m)/(1 + m + 5*n) + (b^3*B*d^2*x^(1 + 6*n)*(e*x)^m)/(1 + m + 6*n) + (a^3*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 )^3 \left (A+B x^n\right ) \left (c+d x^n\right )^2 \, dx &=\int \left (a^3 A c^2 (e x)^m+a^2 c (3 A b c+a B c+2 a A d) x^n (e x)^m+a \left (a B c (3 b c+2 a d)+A \left (3 b^2 c^2+6 a b c d+a^2 d^2\right )\right ) x^{2 n} (e x)^m+\left (a B \left (3 b^2 c^2+6 a b c d+a^2 d^2\right )+A b \left (b^2 c^2+6 a b c d+3 a^2 d^2\right )\right ) x^{3 n} (e x)^m+b \left (3 a^2 B d^2+3 a b d (2 B c+A d)+b^2 c (B c+2 A d)\right ) x^{4 n} (e x)^m+b^2 d (2 b B c+A b d+3 a B d) x^{5 n} (e x)^m+b^3 B d^2 x^{6 n} (e x)^m\right ) \, dx\\ &=\frac {a^3 A c^2 (e x)^{1+m}}{e (1+m)}+\left (b^3 B d^2\right ) \int x^{6 n} (e x)^m \, dx+\left (a^2 c (3 A b c+a B c+2 a A d)\right ) \int x^n (e x)^m \, dx+\left (b^2 d (2 b B c+A b d+3 a B d)\right ) \int x^{5 n} (e x)^m \, dx+\left (b \left (3 a^2 B d^2+3 a b d (2 B c+A d)+b^2 c (B c+2 A d)\right )\right ) \int x^{4 n} (e x)^m \, dx+\left (a \left (a B c (3 b c+2 a d)+A \left (3 b^2 c^2+6 a b c d+a^2 d^2\right )\right )\right ) \int x^{2 n} (e x)^m \, dx+\left (a B \left (3 b^2 c^2+6 a b c d+a^2 d^2\right )+A b \left (b^2 c^2+6 a b c d+3 a^2 d^2\right )\right ) \int x^{3 n} (e x)^m \, dx\\ &=\frac {a^3 A c^2 (e x)^{1+m}}{e (1+m)}+\left (b^3 B d^2 x^{-m} (e x)^m\right ) \int x^{m+6 n} \, dx+\left (a^2 c (3 A b c+a B c+2 a A d) x^{-m} (e x)^m\right ) \int x^{m+n} \, dx+\left (b^2 d (2 b B c+A b d+3 a B d) x^{-m} (e x)^m\right ) \int x^{m+5 n} \, dx+\left (b \left (3 a^2 B d^2+3 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+4 n} \, dx+\left (a \left (a B c (3 b c+2 a d)+A \left (3 b^2 c^2+6 a b c d+a^2 d^2\right )\right ) x^{-m} (e x)^m\right ) \int x^{m+2 n} \, dx+\left (\left (a B \left (3 b^2 c^2+6 a b c d+a^2 d^2\right )+A b \left (b^2 c^2+6 a b c d+3 a^2 d^2\right )\right ) x^{-m} (e x)^m\right ) \int x^{m+3 n} \, dx\\ &=\frac {a^2 c (3 A b c+a B c+2 a A d) x^{1+n} (e x)^m}{1+m+n}+\frac {a \left (a B c (3 b c+2 a d)+A \left (3 b^2 c^2+6 a b c d+a^2 d^2\right )\right ) x^{1+2 n} (e x)^m}{1+m+2 n}+\frac {\left (a B \left (3 b^2 c^2+6 a b c d+a^2 d^2\right )+A b \left (b^2 c^2+6 a b c d+3 a^2 d^2\right )\right ) x^{1+3 n} (e x)^m}{1+m+3 n}+\frac {b \left (3 a^2 B d^2+3 a b d (2 B c+A d)+b^2 c (B c+2 A d)\right ) x^{1+4 n} (e x)^m}{1+m+4 n}+\frac {b^2 d (2 b B c+A b d+3 a B d) x^{1+5 n} (e x)^m}{1+m+5 n}+\frac {b^3 B d^2 x^{1+6 n} (e x)^m}{1+m+6 n}+\frac {a^3 A c^2 (e x)^{1+m}}{e (1+m)}\\ \end {align*}

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Mathematica [A]
time = 1.34, size = 273, normalized size = 0.86 \begin {gather*} x (e x)^m \left (\frac {a^3 A c^2}{1+m}+\frac {a^2 c (3 A b c+a B c+2 a A d) x^n}{1+m+n}+\frac {a \left (a B c (3 b c+2 a d)+A \left (3 b^2 c^2+6 a b c d+a^2 d^2\right )\right ) x^{2 n}}{1+m+2 n}+\frac {\left (a B \left (3 b^2 c^2+6 a b c d+a^2 d^2\right )+A b \left (b^2 c^2+6 a b c d+3 a^2 d^2\right )\right ) x^{3 n}}{1+m+3 n}+\frac {b \left (3 a^2 B d^2+3 a b d (2 B c+A d)+b^2 c (B c+2 A d)\right ) x^{4 n}}{1+m+4 n}+\frac {b^2 d (2 b B c+A b d+3 a B d) x^{5 n}}{1+m+5 n}+\frac {b^3 B d^2 x^{6 n}}{1+m+6 n}\right ) \end {gather*}

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*((a^3*A*c^2)/(1 + m) + (a^2*c*(3*A*b*c + a*B*c + 2*a*A*d)*x^n)/(1 + m + n) + (a*(a*B*c*(3*b*c + 2*a*
d) + A*(3*b^2*c^2 + 6*a*b*c*d + a^2*d^2))*x^(2*n))/(1 + m + 2*n) + ((a*B*(3*b^2*c^2 + 6*a*b*c*d + a^2*d^2) + A
*b*(b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2))*x^(3*n))/(1 + m + 3*n) + (b*(3*a^2*B*d^2 + 3*a*b*d*(2*B*c + A*d) + b^2*c
*(B*c + 2*A*d))*x^(4*n))/(1 + m + 4*n) + (b^2*d*(2*b*B*c + A*b*d + 3*a*B*d)*x^(5*n))/(1 + m + 5*n) + (b^3*B*d^
2*x^(6*n))/(1 + m + 6*n))

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.50, size = 11389, normalized size = 35.81

method	result	size
risch	\(\text {Expression too large to display}\)	\(11389\)

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,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. 702 vs. \(2 (324) = 648\).
time = 0.38, size = 702, normalized size = 2.21 \begin {gather*} \frac {\left (x e\right )^{m + 1} A a^{3} c^{2} e^{\left (-1\right )}}{m + 1} + \frac {B b^{3} d^{2} x e^{\left (m \log \left (x\right ) + 6 \, n \log \left (x\right ) + m\right )}}{m + 6 \, n + 1} + \frac {2 \, B b^{3} c d x e^{\left (m \log \left (x\right ) + 5 \, n \log \left (x\right ) + m\right )}}{m + 5 \, n + 1} + \frac {3 \, B a 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 {A b^{3} d^{2} x e^{\left (m \log \left (x\right ) + 5 \, n \log \left (x\right ) + m\right )}}{m + 5 \, n + 1} + \frac {B b^{3} c^{2} x e^{\left (m \log \left (x\right ) + 4 \, n \log \left (x\right ) + m\right )}}{m + 4 \, n + 1} + \frac {6 \, B a 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 \, A b^{3} c d x e^{\left (m \log \left (x\right ) + 4 \, n \log \left (x\right ) + m\right )}}{m + 4 \, n + 1} + \frac {3 \, B a^{2} b d^{2} x e^{\left (m \log \left (x\right ) + 4 \, n \log \left (x\right ) + m\right )}}{m + 4 \, n + 1} + \frac {3 \, A 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 {3 \, B a 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 {A b^{3} c^{2} x e^{\left (m \log \left (x\right ) + 3 \, n \log \left (x\right ) + m\right )}}{m + 3 \, n + 1} + \frac {6 \, B a^{2} b c d x e^{\left (m \log \left (x\right ) + 3 \, n \log \left (x\right ) + m\right )}}{m + 3 \, n + 1} + \frac {6 \, A 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^{3} d^{2} x e^{\left (m \log \left (x\right ) + 3 \, n \log \left (x\right ) + m\right )}}{m + 3 \, n + 1} + \frac {3 \, A a^{2} b d^{2} x e^{\left (m \log \left (x\right ) + 3 \, n \log \left (x\right ) + m\right )}}{m + 3 \, n + 1} + \frac {3 \, B a^{2} b c^{2} x e^{\left (m \log \left (x\right ) + 2 \, n \log \left (x\right ) + m\right )}}{m + 2 \, n + 1} + \frac {3 \, A 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^{3} c d x e^{\left (m \log \left (x\right ) + 2 \, n \log \left (x\right ) + m\right )}}{m + 2 \, n + 1} + \frac {6 \, A a^{2} 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^{3} 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^{3} c^{2} x e^{\left (m \log \left (x\right ) + n \log \left (x\right ) + m\right )}}{m + n + 1} + \frac {3 \, A a^{2} 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^{3} 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)^3*(A+B*x^n)*(c+d*x^n)^2,x, algorithm="maxima")

[Out]

(x*e)^(m + 1)*A*a^3*c^2*e^(-1)/(m + 1) + B*b^3*d^2*x*e^(m*log(x) + 6*n*log(x) + m)/(m + 6*n + 1) + 2*B*b^3*c*d
*x*e^(m*log(x) + 5*n*log(x) + m)/(m + 5*n + 1) + 3*B*a*b^2*d^2*x*e^(m*log(x) + 5*n*log(x) + m)/(m + 5*n + 1) +
A*b^3*d^2*x*e^(m*log(x) + 5*n*log(x) + m)/(m + 5*n + 1) + B*b^3*c^2*x*e^(m*log(x) + 4*n*log(x) + m)/(m + 4*n
+ 1) + 6*B*a*b^2*c*d*x*e^(m*log(x) + 4*n*log(x) + m)/(m + 4*n + 1) + 2*A*b^3*c*d*x*e^(m*log(x) + 4*n*log(x) +
m)/(m + 4*n + 1) + 3*B*a^2*b*d^2*x*e^(m*log(x) + 4*n*log(x) + m)/(m + 4*n + 1) + 3*A*a*b^2*d^2*x*e^(m*log(x) +
4*n*log(x) + m)/(m + 4*n + 1) + 3*B*a*b^2*c^2*x*e^(m*log(x) + 3*n*log(x) + m)/(m + 3*n + 1) + A*b^3*c^2*x*e^(
m*log(x) + 3*n*log(x) + m)/(m + 3*n + 1) + 6*B*a^2*b*c*d*x*e^(m*log(x) + 3*n*log(x) + m)/(m + 3*n + 1) + 6*A*a
*b^2*c*d*x*e^(m*log(x) + 3*n*log(x) + m)/(m + 3*n + 1) + B*a^3*d^2*x*e^(m*log(x) + 3*n*log(x) + m)/(m + 3*n +
1) + 3*A*a^2*b*d^2*x*e^(m*log(x) + 3*n*log(x) + m)/(m + 3*n + 1) + 3*B*a^2*b*c^2*x*e^(m*log(x) + 2*n*log(x) +
m)/(m + 2*n + 1) + 3*A*a*b^2*c^2*x*e^(m*log(x) + 2*n*log(x) + m)/(m + 2*n + 1) + 2*B*a^3*c*d*x*e^(m*log(x) + 2
*n*log(x) + m)/(m + 2*n + 1) + 6*A*a^2*b*c*d*x*e^(m*log(x) + 2*n*log(x) + m)/(m + 2*n + 1) + A*a^3*d^2*x*e^(m*
log(x) + 2*n*log(x) + m)/(m + 2*n + 1) + B*a^3*c^2*x*e^(m*log(x) + n*log(x) + m)/(m + n + 1) + 3*A*a^2*b*c^2*x
*e^(m*log(x) + n*log(x) + m)/(m + n + 1) + 2*A*a^3*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. 6617 vs. \(2 (324) = 648\).
time = 1.17, size = 6617, normalized size = 20.81 \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)^3*(A+B*x^n)*(c+d*x^n)^2,x, algorithm="fricas")

[Out]

((B*b^3*d^2*m^6 + 6*B*b^3*d^2*m^5 + 15*B*b^3*d^2*m^4 + 20*B*b^3*d^2*m^3 + 15*B*b^3*d^2*m^2 + 6*B*b^3*d^2*m + B
*b^3*d^2 + 120*(B*b^3*d^2*m + B*b^3*d^2)*n^5 + 274*(B*b^3*d^2*m^2 + 2*B*b^3*d^2*m + B*b^3*d^2)*n^4 + 225*(B*b^
3*d^2*m^3 + 3*B*b^3*d^2*m^2 + 3*B*b^3*d^2*m + B*b^3*d^2)*n^3 + 85*(B*b^3*d^2*m^4 + 4*B*b^3*d^2*m^3 + 6*B*b^3*d
^2*m^2 + 4*B*b^3*d^2*m + B*b^3*d^2)*n^2 + 15*(B*b^3*d^2*m^5 + 5*B*b^3*d^2*m^4 + 10*B*b^3*d^2*m^3 + 10*B*b^3*d^
2*m^2 + 5*B*b^3*d^2*m + B*b^3*d^2)*n)*x*x^(6*n)*e^(m*log(x) + m) + ((2*B*b^3*c*d + (3*B*a*b^2 + A*b^3)*d^2)*m^
6 + 2*B*b^3*c*d + 6*(2*B*b^3*c*d + (3*B*a*b^2 + A*b^3)*d^2)*m^5 + 144*(2*B*b^3*c*d + (3*B*a*b^2 + A*b^3)*d^2 +
(2*B*b^3*c*d + (3*B*a*b^2 + A*b^3)*d^2)*m)*n^5 + 15*(2*B*b^3*c*d + (3*B*a*b^2 + A*b^3)*d^2)*m^4 + 324*(2*B*b^
3*c*d + (3*B*a*b^2 + A*b^3)*d^2 + (2*B*b^3*c*d + (3*B*a*b^2 + A*b^3)*d^2)*m^2 + 2*(2*B*b^3*c*d + (3*B*a*b^2 +
A*b^3)*d^2)*m)*n^4 + 20*(2*B*b^3*c*d + (3*B*a*b^2 + A*b^3)*d^2)*m^3 + 260*(2*B*b^3*c*d + (2*B*b^3*c*d + (3*B*a
*b^2 + A*b^3)*d^2)*m^3 + (3*B*a*b^2 + A*b^3)*d^2 + 3*(2*B*b^3*c*d + (3*B*a*b^2 + A*b^3)*d^2)*m^2 + 3*(2*B*b^3*
c*d + (3*B*a*b^2 + A*b^3)*d^2)*m)*n^3 + (3*B*a*b^2 + A*b^3)*d^2 + 15*(2*B*b^3*c*d + (3*B*a*b^2 + A*b^3)*d^2)*m
^2 + 95*(2*B*b^3*c*d + (2*B*b^3*c*d + (3*B*a*b^2 + A*b^3)*d^2)*m^4 + 4*(2*B*b^3*c*d + (3*B*a*b^2 + A*b^3)*d^2)
*m^3 + (3*B*a*b^2 + A*b^3)*d^2 + 6*(2*B*b^3*c*d + (3*B*a*b^2 + A*b^3)*d^2)*m^2 + 4*(2*B*b^3*c*d + (3*B*a*b^2 +
A*b^3)*d^2)*m)*n^2 + 6*(2*B*b^3*c*d + (3*B*a*b^2 + A*b^3)*d^2)*m + 16*(2*B*b^3*c*d + (2*B*b^3*c*d + (3*B*a*b^
2 + A*b^3)*d^2)*m^5 + 5*(2*B*b^3*c*d + (3*B*a*b^2 + A*b^3)*d^2)*m^4 + 10*(2*B*b^3*c*d + (3*B*a*b^2 + A*b^3)*d^
2)*m^3 + (3*B*a*b^2 + A*b^3)*d^2 + 10*(2*B*b^3*c*d + (3*B*a*b^2 + A*b^3)*d^2)*m^2 + 5*(2*B*b^3*c*d + (3*B*a*b^
2 + A*b^3)*d^2)*m)*n)*x*x^(5*n)*e^(m*log(x) + m) + ((B*b^3*c^2 + 2*(3*B*a*b^2 + A*b^3)*c*d + 3*(B*a^2*b + A*a*
b^2)*d^2)*m^6 + B*b^3*c^2 + 6*(B*b^3*c^2 + 2*(3*B*a*b^2 + A*b^3)*c*d + 3*(B*a^2*b + A*a*b^2)*d^2)*m^5 + 180*(B
*b^3*c^2 + 2*(3*B*a*b^2 + A*b^3)*c*d + 3*(B*a^2*b + A*a*b^2)*d^2 + (B*b^3*c^2 + 2*(3*B*a*b^2 + A*b^3)*c*d + 3*
(B*a^2*b + A*a*b^2)*d^2)*m)*n^5 + 15*(B*b^3*c^2 + 2*(3*B*a*b^2 + A*b^3)*c*d + 3*(B*a^2*b + A*a*b^2)*d^2)*m^4 +
396*(B*b^3*c^2 + 2*(3*B*a*b^2 + A*b^3)*c*d + 3*(B*a^2*b + A*a*b^2)*d^2 + (B*b^3*c^2 + 2*(3*B*a*b^2 + A*b^3)*c
*d + 3*(B*a^2*b + A*a*b^2)*d^2)*m^2 + 2*(B*b^3*c^2 + 2*(3*B*a*b^2 + A*b^3)*c*d + 3*(B*a^2*b + A*a*b^2)*d^2)*m)
*n^4 + 20*(B*b^3*c^2 + 2*(3*B*a*b^2 + A*b^3)*c*d + 3*(B*a^2*b + A*a*b^2)*d^2)*m^3 + 307*(B*b^3*c^2 + (B*b^3*c^
2 + 2*(3*B*a*b^2 + A*b^3)*c*d + 3*(B*a^2*b + A*a*b^2)*d^2)*m^3 + 2*(3*B*a*b^2 + A*b^3)*c*d + 3*(B*a^2*b + A*a*
b^2)*d^2 + 3*(B*b^3*c^2 + 2*(3*B*a*b^2 + A*b^3)*c*d + 3*(B*a^2*b + A*a*b^2)*d^2)*m^2 + 3*(B*b^3*c^2 + 2*(3*B*a
*b^2 + A*b^3)*c*d + 3*(B*a^2*b + A*a*b^2)*d^2)*m)*n^3 + 2*(3*B*a*b^2 + A*b^3)*c*d + 3*(B*a^2*b + A*a*b^2)*d^2
+ 15*(B*b^3*c^2 + 2*(3*B*a*b^2 + A*b^3)*c*d + 3*(B*a^2*b + A*a*b^2)*d^2)*m^2 + 107*(B*b^3*c^2 + (B*b^3*c^2 + 2
*(3*B*a*b^2 + A*b^3)*c*d + 3*(B*a^2*b + A*a*b^2)*d^2)*m^4 + 4*(B*b^3*c^2 + 2*(3*B*a*b^2 + A*b^3)*c*d + 3*(B*a^
2*b + A*a*b^2)*d^2)*m^3 + 2*(3*B*a*b^2 + A*b^3)*c*d + 3*(B*a^2*b + A*a*b^2)*d^2 + 6*(B*b^3*c^2 + 2*(3*B*a*b^2
+ A*b^3)*c*d + 3*(B*a^2*b + A*a*b^2)*d^2)*m^2 + 4*(B*b^3*c^2 + 2*(3*B*a*b^2 + A*b^3)*c*d + 3*(B*a^2*b + A*a*b^
2)*d^2)*m)*n^2 + 6*(B*b^3*c^2 + 2*(3*B*a*b^2 + A*b^3)*c*d + 3*(B*a^2*b + A*a*b^2)*d^2)*m + 17*(B*b^3*c^2 + (B*
b^3*c^2 + 2*(3*B*a*b^2 + A*b^3)*c*d + 3*(B*a^2*b + A*a*b^2)*d^2)*m^5 + 5*(B*b^3*c^2 + 2*(3*B*a*b^2 + A*b^3)*c*
d + 3*(B*a^2*b + A*a*b^2)*d^2)*m^4 + 10*(B*b^3*c^2 + 2*(3*B*a*b^2 + A*b^3)*c*d + 3*(B*a^2*b + A*a*b^2)*d^2)*m^
3 + 2*(3*B*a*b^2 + A*b^3)*c*d + 3*(B*a^2*b + A*a*b^2)*d^2 + 10*(B*b^3*c^2 + 2*(3*B*a*b^2 + A*b^3)*c*d + 3*(B*a
^2*b + A*a*b^2)*d^2)*m^2 + 5*(B*b^3*c^2 + 2*(3*B*a*b^2 + A*b^3)*c*d + 3*(B*a^2*b + A*a*b^2)*d^2)*m)*n)*x*x^(4*
n)*e^(m*log(x) + m) + (((3*B*a*b^2 + A*b^3)*c^2 + 6*(B*a^2*b + A*a*b^2)*c*d + (B*a^3 + 3*A*a^2*b)*d^2)*m^6 + 6
*((3*B*a*b^2 + A*b^3)*c^2 + 6*(B*a^2*b + A*a*b^2)*c*d + (B*a^3 + 3*A*a^2*b)*d^2)*m^5 + 240*((3*B*a*b^2 + A*b^3
)*c^2 + 6*(B*a^2*b + A*a*b^2)*c*d + (B*a^3 + 3*A*a^2*b)*d^2 + ((3*B*a*b^2 + A*b^3)*c^2 + 6*(B*a^2*b + A*a*b^2)
*c*d + (B*a^3 + 3*A*a^2*b)*d^2)*m)*n^5 + 15*((3*B*a*b^2 + A*b^3)*c^2 + 6*(B*a^2*b + A*a*b^2)*c*d + (B*a^3 + 3*
A*a^2*b)*d^2)*m^4 + 508*((3*B*a*b^2 + A*b^3)*c^2 + 6*(B*a^2*b + A*a*b^2)*c*d + (B*a^3 + 3*A*a^2*b)*d^2 + ((3*B
*a*b^2 + A*b^3)*c^2 + 6*(B*a^2*b + A*a*b^2)*c*d + (B*a^3 + 3*A*a^2*b)*d^2)*m^2 + 2*((3*B*a*b^2 + A*b^3)*c^2 +
6*(B*a^2*b + A*a*b^2)*c*d + (B*a^3 + 3*A*a^2*b)*d^2)*m)*n^4 + 20*((3*B*a*b^2 + A*b^3)*c^2 + 6*(B*a^2*b + A*a*b
^2)*c*d + (B*a^3 + 3*A*a^2*b)*d^2)*m^3 + 372*(((3*B*a*b^2 + A*b^3)*c^2 + 6*(B*a^2*b + A*a*b^2)*c*d + (B*a^3 +
3*A*a^2*b)*d^2)*m^3 + (3*B*a*b^2 + A*b^3)*c^2 + 6*(B*a^2*b + A*a*b^2)*c*d + (B*a^3 + 3*A*a^2*b)*d^2 + 3*((3*B*
a*b^2 + A*b^3)*c^2 + 6*(B*a^2*b + A*a*b^2)*c*d + (B*a^3 + 3*A*a^2*b)*d^2)*m^2 + 3*((3*B*a*b^2 + A*b^3)*c^2 + 6
*(B*a^2*b + A*a*b^2)*c*d + (B*a^3 + 3*A*a^2*b)*...

________________________________________________________________________________________

Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 15358 vs. \(2 (324) = 648\).
time = 1.72, size = 15358, normalized size = 48.30 \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)^3*(A+B*x^n)*(c+d*x^n)^2,x, algorithm="giac")

[Out]

(B*b^3*d^2*m^6*x*x^m*x^(6*n)*e^m + 15*B*b^3*d^2*m^5*n*x*x^m*x^(6*n)*e^m + 85*B*b^3*d^2*m^4*n^2*x*x^m*x^(6*n)*e
^m + 225*B*b^3*d^2*m^3*n^3*x*x^m*x^(6*n)*e^m + 274*B*b^3*d^2*m^2*n^4*x*x^m*x^(6*n)*e^m + 120*B*b^3*d^2*m*n^5*x
*x^m*x^(6*n)*e^m + 2*B*b^3*c*d*m^6*x*x^m*x^(5*n)*e^m + 3*B*a*b^2*d^2*m^6*x*x^m*x^(5*n)*e^m + A*b^3*d^2*m^6*x*x
^m*x^(5*n)*e^m + 32*B*b^3*c*d*m^5*n*x*x^m*x^(5*n)*e^m + 48*B*a*b^2*d^2*m^5*n*x*x^m*x^(5*n)*e^m + 16*A*b^3*d^2*
m^5*n*x*x^m*x^(5*n)*e^m + 190*B*b^3*c*d*m^4*n^2*x*x^m*x^(5*n)*e^m + 285*B*a*b^2*d^2*m^4*n^2*x*x^m*x^(5*n)*e^m
+ 95*A*b^3*d^2*m^4*n^2*x*x^m*x^(5*n)*e^m + 520*B*b^3*c*d*m^3*n^3*x*x^m*x^(5*n)*e^m + 780*B*a*b^2*d^2*m^3*n^3*x
*x^m*x^(5*n)*e^m + 260*A*b^3*d^2*m^3*n^3*x*x^m*x^(5*n)*e^m + 648*B*b^3*c*d*m^2*n^4*x*x^m*x^(5*n)*e^m + 972*B*a
*b^2*d^2*m^2*n^4*x*x^m*x^(5*n)*e^m + 324*A*b^3*d^2*m^2*n^4*x*x^m*x^(5*n)*e^m + 288*B*b^3*c*d*m*n^5*x*x^m*x^(5*
n)*e^m + 432*B*a*b^2*d^2*m*n^5*x*x^m*x^(5*n)*e^m + 144*A*b^3*d^2*m*n^5*x*x^m*x^(5*n)*e^m + B*b^3*c^2*m^6*x*x^m
*x^(4*n)*e^m + 6*B*a*b^2*c*d*m^6*x*x^m*x^(4*n)*e^m + 2*A*b^3*c*d*m^6*x*x^m*x^(4*n)*e^m + 3*B*a^2*b*d^2*m^6*x*x
^m*x^(4*n)*e^m + 3*A*a*b^2*d^2*m^6*x*x^m*x^(4*n)*e^m + 17*B*b^3*c^2*m^5*n*x*x^m*x^(4*n)*e^m + 102*B*a*b^2*c*d*
m^5*n*x*x^m*x^(4*n)*e^m + 34*A*b^3*c*d*m^5*n*x*x^m*x^(4*n)*e^m + 51*B*a^2*b*d^2*m^5*n*x*x^m*x^(4*n)*e^m + 51*A
*a*b^2*d^2*m^5*n*x*x^m*x^(4*n)*e^m + 107*B*b^3*c^2*m^4*n^2*x*x^m*x^(4*n)*e^m + 642*B*a*b^2*c*d*m^4*n^2*x*x^m*x
^(4*n)*e^m + 214*A*b^3*c*d*m^4*n^2*x*x^m*x^(4*n)*e^m + 321*B*a^2*b*d^2*m^4*n^2*x*x^m*x^(4*n)*e^m + 321*A*a*b^2
*d^2*m^4*n^2*x*x^m*x^(4*n)*e^m + 307*B*b^3*c^2*m^3*n^3*x*x^m*x^(4*n)*e^m + 1842*B*a*b^2*c*d*m^3*n^3*x*x^m*x^(4
*n)*e^m + 614*A*b^3*c*d*m^3*n^3*x*x^m*x^(4*n)*e^m + 921*B*a^2*b*d^2*m^3*n^3*x*x^m*x^(4*n)*e^m + 921*A*a*b^2*d^
2*m^3*n^3*x*x^m*x^(4*n)*e^m + 396*B*b^3*c^2*m^2*n^4*x*x^m*x^(4*n)*e^m + 2376*B*a*b^2*c*d*m^2*n^4*x*x^m*x^(4*n)
*e^m + 792*A*b^3*c*d*m^2*n^4*x*x^m*x^(4*n)*e^m + 1188*B*a^2*b*d^2*m^2*n^4*x*x^m*x^(4*n)*e^m + 1188*A*a*b^2*d^2
*m^2*n^4*x*x^m*x^(4*n)*e^m + 180*B*b^3*c^2*m*n^5*x*x^m*x^(4*n)*e^m + 1080*B*a*b^2*c*d*m*n^5*x*x^m*x^(4*n)*e^m
+ 360*A*b^3*c*d*m*n^5*x*x^m*x^(4*n)*e^m + 540*B*a^2*b*d^2*m*n^5*x*x^m*x^(4*n)*e^m + 540*A*a*b^2*d^2*m*n^5*x*x^
m*x^(4*n)*e^m + 3*B*a*b^2*c^2*m^6*x*x^m*x^(3*n)*e^m + A*b^3*c^2*m^6*x*x^m*x^(3*n)*e^m + 6*B*a^2*b*c*d*m^6*x*x^
m*x^(3*n)*e^m + 6*A*a*b^2*c*d*m^6*x*x^m*x^(3*n)*e^m + B*a^3*d^2*m^6*x*x^m*x^(3*n)*e^m + 3*A*a^2*b*d^2*m^6*x*x^
m*x^(3*n)*e^m + 54*B*a*b^2*c^2*m^5*n*x*x^m*x^(3*n)*e^m + 18*A*b^3*c^2*m^5*n*x*x^m*x^(3*n)*e^m + 108*B*a^2*b*c*
d*m^5*n*x*x^m*x^(3*n)*e^m + 108*A*a*b^2*c*d*m^5*n*x*x^m*x^(3*n)*e^m + 18*B*a^3*d^2*m^5*n*x*x^m*x^(3*n)*e^m + 5
4*A*a^2*b*d^2*m^5*n*x*x^m*x^(3*n)*e^m + 363*B*a*b^2*c^2*m^4*n^2*x*x^m*x^(3*n)*e^m + 121*A*b^3*c^2*m^4*n^2*x*x^
m*x^(3*n)*e^m + 726*B*a^2*b*c*d*m^4*n^2*x*x^m*x^(3*n)*e^m + 726*A*a*b^2*c*d*m^4*n^2*x*x^m*x^(3*n)*e^m + 121*B*
a^3*d^2*m^4*n^2*x*x^m*x^(3*n)*e^m + 363*A*a^2*b*d^2*m^4*n^2*x*x^m*x^(3*n)*e^m + 1116*B*a*b^2*c^2*m^3*n^3*x*x^m
*x^(3*n)*e^m + 372*A*b^3*c^2*m^3*n^3*x*x^m*x^(3*n)*e^m + 2232*B*a^2*b*c*d*m^3*n^3*x*x^m*x^(3*n)*e^m + 2232*A*a
*b^2*c*d*m^3*n^3*x*x^m*x^(3*n)*e^m + 372*B*a^3*d^2*m^3*n^3*x*x^m*x^(3*n)*e^m + 1116*A*a^2*b*d^2*m^3*n^3*x*x^m*
x^(3*n)*e^m + 1524*B*a*b^2*c^2*m^2*n^4*x*x^m*x^(3*n)*e^m + 508*A*b^3*c^2*m^2*n^4*x*x^m*x^(3*n)*e^m + 3048*B*a^
2*b*c*d*m^2*n^4*x*x^m*x^(3*n)*e^m + 3048*A*a*b^2*c*d*m^2*n^4*x*x^m*x^(3*n)*e^m + 508*B*a^3*d^2*m^2*n^4*x*x^m*x
^(3*n)*e^m + 1524*A*a^2*b*d^2*m^2*n^4*x*x^m*x^(3*n)*e^m + 720*B*a*b^2*c^2*m*n^5*x*x^m*x^(3*n)*e^m + 240*A*b^3*
c^2*m*n^5*x*x^m*x^(3*n)*e^m + 1440*B*a^2*b*c*d*m*n^5*x*x^m*x^(3*n)*e^m + 1440*A*a*b^2*c*d*m*n^5*x*x^m*x^(3*n)*
e^m + 240*B*a^3*d^2*m*n^5*x*x^m*x^(3*n)*e^m + 720*A*a^2*b*d^2*m*n^5*x*x^m*x^(3*n)*e^m + 3*B*a^2*b*c^2*m^6*x*x^
m*x^(2*n)*e^m + 3*A*a*b^2*c^2*m^6*x*x^m*x^(2*n)*e^m + 2*B*a^3*c*d*m^6*x*x^m*x^(2*n)*e^m + 6*A*a^2*b*c*d*m^6*x*
x^m*x^(2*n)*e^m + A*a^3*d^2*m^6*x*x^m*x^(2*n)*e^m + 57*B*a^2*b*c^2*m^5*n*x*x^m*x^(2*n)*e^m + 57*A*a*b^2*c^2*m^
5*n*x*x^m*x^(2*n)*e^m + 38*B*a^3*c*d*m^5*n*x*x^m*x^(2*n)*e^m + 114*A*a^2*b*c*d*m^5*n*x*x^m*x^(2*n)*e^m + 19*A*
a^3*d^2*m^5*n*x*x^m*x^(2*n)*e^m + 411*B*a^2*b*c^2*m^4*n^2*x*x^m*x^(2*n)*e^m + 411*A*a*b^2*c^2*m^4*n^2*x*x^m*x^
(2*n)*e^m + 274*B*a^3*c*d*m^4*n^2*x*x^m*x^(2*n)*e^m + 822*A*a^2*b*c*d*m^4*n^2*x*x^m*x^(2*n)*e^m + 137*A*a^3*d^
2*m^4*n^2*x*x^m*x^(2*n)*e^m + 1383*B*a^2*b*c^2*m^3*n^3*x*x^m*x^(2*n)*e^m + 1383*A*a*b^2*c^2*m^3*n^3*x*x^m*x^(2
*n)*e^m + 922*B*a^3*c*d*m^3*n^3*x*x^m*x^(2*n)*e^m + 2766*A*a^2*b*c*d*m^3*n^3*x*x^m*x^(2*n)*e^m + 461*A*a^3*d^2
*m^3*n^3*x*x^m*x^(2*n)*e^m + 2106*B*a^2*b*c^2*m^2*n^4*x*x^m*x^(2*n)*e^m + 2106*A*a*b^2*c^2*m^2*n^4*x*x^m*x^(2*
n)*e^m + 1404*B*a^3*c*d*m^2*n^4*x*x^m*x^(2*n)*e^m + 4212*A*a^2*b*c*d*m^2*n^4*x*x^m*x^(2*n)*e^m + 702*A*a^3*d^2
*m^2*n^4*x*x^m*x^(2*n)*e^m + 1080*B*a^2*b*c^2*m*n^5*x*x^m*x^(2*n)*e^m + 1080*A*a*b^2*c^2*m*n^5*x*x^m*x^(2*n)*e
^m + 720*B*a^3*c*d*m*n^5*x*x^m*x^(2*n)*e^m + 2160*A*a^2*b*c*d*m*n^5*x*x^m*x^(2*n)*e^m + 360*A*a^3*d^2*m*n^5*x*
x^m*x^(2*n)*e^m + B*a^3*c^2*m^6*x*x^m*x^n*e^m +...

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Mupad [B]
time = 6.35, size = 1882, normalized size = 5.92 \begin {gather*} \frac {x\,x^{3\,n}\,{\left (e\,x\right )}^m\,\left (B\,a^3\,d^2+6\,B\,a^2\,b\,c\,d+3\,A\,a^2\,b\,d^2+3\,B\,a\,b^2\,c^2+6\,A\,a\,b^2\,c\,d+A\,b^3\,c^2\right )\,\left (m^5+18\,m^4\,n+5\,m^4+121\,m^3\,n^2+72\,m^3\,n+10\,m^3+372\,m^2\,n^3+363\,m^2\,n^2+108\,m^2\,n+10\,m^2+508\,m\,n^4+744\,m\,n^3+363\,m\,n^2+72\,m\,n+5\,m+240\,n^5+508\,n^4+372\,n^3+121\,n^2+18\,n+1\right )}{m^6+21\,m^5\,n+6\,m^5+175\,m^4\,n^2+105\,m^4\,n+15\,m^4+735\,m^3\,n^3+700\,m^3\,n^2+210\,m^3\,n+20\,m^3+1624\,m^2\,n^4+2205\,m^2\,n^3+1050\,m^2\,n^2+210\,m^2\,n+15\,m^2+1764\,m\,n^5+3248\,m\,n^4+2205\,m\,n^3+700\,m\,n^2+105\,m\,n+6\,m+720\,n^6+1764\,n^5+1624\,n^4+735\,n^3+175\,n^2+21\,n+1}+\frac {A\,a^3\,c^2\,x\,{\left (e\,x\right )}^m}{m+1}+\frac {a\,x\,x^{2\,n}\,{\left (e\,x\right )}^m\,\left (2\,B\,a^2\,c\,d+A\,a^2\,d^2+3\,B\,a\,b\,c^2+6\,A\,a\,b\,c\,d+3\,A\,b^2\,c^2\right )\,\left (m^5+19\,m^4\,n+5\,m^4+137\,m^3\,n^2+76\,m^3\,n+10\,m^3+461\,m^2\,n^3+411\,m^2\,n^2+114\,m^2\,n+10\,m^2+702\,m\,n^4+922\,m\,n^3+411\,m\,n^2+76\,m\,n+5\,m+360\,n^5+702\,n^4+461\,n^3+137\,n^2+19\,n+1\right )}{m^6+21\,m^5\,n+6\,m^5+175\,m^4\,n^2+105\,m^4\,n+15\,m^4+735\,m^3\,n^3+700\,m^3\,n^2+210\,m^3\,n+20\,m^3+1624\,m^2\,n^4+2205\,m^2\,n^3+1050\,m^2\,n^2+210\,m^2\,n+15\,m^2+1764\,m\,n^5+3248\,m\,n^4+2205\,m\,n^3+700\,m\,n^2+105\,m\,n+6\,m+720\,n^6+1764\,n^5+1624\,n^4+735\,n^3+175\,n^2+21\,n+1}+\frac {b\,x\,x^{4\,n}\,{\left (e\,x\right )}^m\,\left (3\,B\,a^2\,d^2+6\,B\,a\,b\,c\,d+3\,A\,a\,b\,d^2+B\,b^2\,c^2+2\,A\,b^2\,c\,d\right )\,\left (m^5+17\,m^4\,n+5\,m^4+107\,m^3\,n^2+68\,m^3\,n+10\,m^3+307\,m^2\,n^3+321\,m^2\,n^2+102\,m^2\,n+10\,m^2+396\,m\,n^4+614\,m\,n^3+321\,m\,n^2+68\,m\,n+5\,m+180\,n^5+396\,n^4+307\,n^3+107\,n^2+17\,n+1\right )}{m^6+21\,m^5\,n+6\,m^5+175\,m^4\,n^2+105\,m^4\,n+15\,m^4+735\,m^3\,n^3+700\,m^3\,n^2+210\,m^3\,n+20\,m^3+1624\,m^2\,n^4+2205\,m^2\,n^3+1050\,m^2\,n^2+210\,m^2\,n+15\,m^2+1764\,m\,n^5+3248\,m\,n^4+2205\,m\,n^3+700\,m\,n^2+105\,m\,n+6\,m+720\,n^6+1764\,n^5+1624\,n^4+735\,n^3+175\,n^2+21\,n+1}+\frac {a^2\,c\,x\,x^n\,{\left (e\,x\right )}^m\,\left (2\,A\,a\,d+3\,A\,b\,c+B\,a\,c\right )\,\left (m^5+20\,m^4\,n+5\,m^4+155\,m^3\,n^2+80\,m^3\,n+10\,m^3+580\,m^2\,n^3+465\,m^2\,n^2+120\,m^2\,n+10\,m^2+1044\,m\,n^4+1160\,m\,n^3+465\,m\,n^2+80\,m\,n+5\,m+720\,n^5+1044\,n^4+580\,n^3+155\,n^2+20\,n+1\right )}{m^6+21\,m^5\,n+6\,m^5+175\,m^4\,n^2+105\,m^4\,n+15\,m^4+735\,m^3\,n^3+700\,m^3\,n^2+210\,m^3\,n+20\,m^3+1624\,m^2\,n^4+2205\,m^2\,n^3+1050\,m^2\,n^2+210\,m^2\,n+15\,m^2+1764\,m\,n^5+3248\,m\,n^4+2205\,m\,n^3+700\,m\,n^2+105\,m\,n+6\,m+720\,n^6+1764\,n^5+1624\,n^4+735\,n^3+175\,n^2+21\,n+1}+\frac {b^2\,d\,x\,x^{5\,n}\,{\left (e\,x\right )}^m\,\left (A\,b\,d+3\,B\,a\,d+2\,B\,b\,c\right )\,\left (m^5+16\,m^4\,n+5\,m^4+95\,m^3\,n^2+64\,m^3\,n+10\,m^3+260\,m^2\,n^3+285\,m^2\,n^2+96\,m^2\,n+10\,m^2+324\,m\,n^4+520\,m\,n^3+285\,m\,n^2+64\,m\,n+5\,m+144\,n^5+324\,n^4+260\,n^3+95\,n^2+16\,n+1\right )}{m^6+21\,m^5\,n+6\,m^5+175\,m^4\,n^2+105\,m^4\,n+15\,m^4+735\,m^3\,n^3+700\,m^3\,n^2+210\,m^3\,n+20\,m^3+1624\,m^2\,n^4+2205\,m^2\,n^3+1050\,m^2\,n^2+210\,m^2\,n+15\,m^2+1764\,m\,n^5+3248\,m\,n^4+2205\,m\,n^3+700\,m\,n^2+105\,m\,n+6\,m+720\,n^6+1764\,n^5+1624\,n^4+735\,n^3+175\,n^2+21\,n+1}+\frac {B\,b^3\,d^2\,x\,x^{6\,n}\,{\left (e\,x\right )}^m\,\left (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\right )}{m^6+21\,m^5\,n+6\,m^5+175\,m^4\,n^2+105\,m^4\,n+15\,m^4+735\,m^3\,n^3+700\,m^3\,n^2+210\,m^3\,n+20\,m^3+1624\,m^2\,n^4+2205\,m^2\,n^3+1050\,m^2\,n^2+210\,m^2\,n+15\,m^2+1764\,m\,n^5+3248\,m\,n^4+2205\,m\,n^3+700\,m\,n^2+105\,m\,n+6\,m+720\,n^6+1764\,n^5+1624\,n^4+735\,n^3+175\,n^2+21\,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)^3*(c + d*x^n)^2,x)

[Out]

(x*x^(3*n)*(e*x)^m*(A*b^3*c^2 + B*a^3*d^2 + 3*A*a^2*b*d^2 + 3*B*a*b^2*c^2 + 6*A*a*b^2*c*d + 6*B*a^2*b*c*d)*(5*
m + 18*n + 72*m*n + 363*m*n^2 + 108*m^2*n + 744*m*n^3 + 72*m^3*n + 508*m*n^4 + 18*m^4*n + 10*m^2 + 10*m^3 + 5*
m^4 + m^5 + 121*n^2 + 372*n^3 + 508*n^4 + 240*n^5 + 363*m^2*n^2 + 372*m^2*n^3 + 121*m^3*n^2 + 1))/(6*m + 21*n
+ 105*m*n + 700*m*n^2 + 210*m^2*n + 2205*m*n^3 + 210*m^3*n + 3248*m*n^4 + 105*m^4*n + 1764*m*n^5 + 21*m^5*n +
15*m^2 + 20*m^3 + 15*m^4 + 6*m^5 + m^6 + 175*n^2 + 735*n^3 + 1624*n^4 + 1764*n^5 + 720*n^6 + 1050*m^2*n^2 + 22
05*m^2*n^3 + 700*m^3*n^2 + 1624*m^2*n^4 + 735*m^3*n^3 + 175*m^4*n^2 + 1) + (A*a^3*c^2*x*(e*x)^m)/(m + 1) + (a*
x*x^(2*n)*(e*x)^m*(A*a^2*d^2 + 3*A*b^2*c^2 + 3*B*a*b*c^2 + 2*B*a^2*c*d + 6*A*a*b*c*d)*(5*m + 19*n + 76*m*n + 4
11*m*n^2 + 114*m^2*n + 922*m*n^3 + 76*m^3*n + 702*m*n^4 + 19*m^4*n + 10*m^2 + 10*m^3 + 5*m^4 + m^5 + 137*n^2 +
461*n^3 + 702*n^4 + 360*n^5 + 411*m^2*n^2 + 461*m^2*n^3 + 137*m^3*n^2 + 1))/(6*m + 21*n + 105*m*n + 700*m*n^2
+ 210*m^2*n + 2205*m*n^3 + 210*m^3*n + 3248*m*n^4 + 105*m^4*n + 1764*m*n^5 + 21*m^5*n + 15*m^2 + 20*m^3 + 15*
m^4 + 6*m^5 + m^6 + 175*n^2 + 735*n^3 + 1624*n^4 + 1764*n^5 + 720*n^6 + 1050*m^2*n^2 + 2205*m^2*n^3 + 700*m^3*
n^2 + 1624*m^2*n^4 + 735*m^3*n^3 + 175*m^4*n^2 + 1) + (b*x*x^(4*n)*(e*x)^m*(3*B*a^2*d^2 + B*b^2*c^2 + 3*A*a*b*
d^2 + 2*A*b^2*c*d + 6*B*a*b*c*d)*(5*m + 17*n + 68*m*n + 321*m*n^2 + 102*m^2*n + 614*m*n^3 + 68*m^3*n + 396*m*n
^4 + 17*m^4*n + 10*m^2 + 10*m^3 + 5*m^4 + m^5 + 107*n^2 + 307*n^3 + 396*n^4 + 180*n^5 + 321*m^2*n^2 + 307*m^2*
n^3 + 107*m^3*n^2 + 1))/(6*m + 21*n + 105*m*n + 700*m*n^2 + 210*m^2*n + 2205*m*n^3 + 210*m^3*n + 3248*m*n^4 +
105*m^4*n + 1764*m*n^5 + 21*m^5*n + 15*m^2 + 20*m^3 + 15*m^4 + 6*m^5 + m^6 + 175*n^2 + 735*n^3 + 1624*n^4 + 17
64*n^5 + 720*n^6 + 1050*m^2*n^2 + 2205*m^2*n^3 + 700*m^3*n^2 + 1624*m^2*n^4 + 735*m^3*n^3 + 175*m^4*n^2 + 1) +
(a^2*c*x*x^n*(e*x)^m*(2*A*a*d + 3*A*b*c + B*a*c)*(5*m + 20*n + 80*m*n + 465*m*n^2 + 120*m^2*n + 1160*m*n^3 +
80*m^3*n + 1044*m*n^4 + 20*m^4*n + 10*m^2 + 10*m^3 + 5*m^4 + m^5 + 155*n^2 + 580*n^3 + 1044*n^4 + 720*n^5 + 46
5*m^2*n^2 + 580*m^2*n^3 + 155*m^3*n^2 + 1))/(6*m + 21*n + 105*m*n + 700*m*n^2 + 210*m^2*n + 2205*m*n^3 + 210*m
^3*n + 3248*m*n^4 + 105*m^4*n + 1764*m*n^5 + 21*m^5*n + 15*m^2 + 20*m^3 + 15*m^4 + 6*m^5 + m^6 + 175*n^2 + 735
*n^3 + 1624*n^4 + 1764*n^5 + 720*n^6 + 1050*m^2*n^2 + 2205*m^2*n^3 + 700*m^3*n^2 + 1624*m^2*n^4 + 735*m^3*n^3
+ 175*m^4*n^2 + 1) + (b^2*d*x*x^(5*n)*(e*x)^m*(A*b*d + 3*B*a*d + 2*B*b*c)*(5*m + 16*n + 64*m*n + 285*m*n^2 + 9
6*m^2*n + 520*m*n^3 + 64*m^3*n + 324*m*n^4 + 16*m^4*n + 10*m^2 + 10*m^3 + 5*m^4 + m^5 + 95*n^2 + 260*n^3 + 324
*n^4 + 144*n^5 + 285*m^2*n^2 + 260*m^2*n^3 + 95*m^3*n^2 + 1))/(6*m + 21*n + 105*m*n + 700*m*n^2 + 210*m^2*n +
2205*m*n^3 + 210*m^3*n + 3248*m*n^4 + 105*m^4*n + 1764*m*n^5 + 21*m^5*n + 15*m^2 + 20*m^3 + 15*m^4 + 6*m^5 + m
^6 + 175*n^2 + 735*n^3 + 1624*n^4 + 1764*n^5 + 720*n^6 + 1050*m^2*n^2 + 2205*m^2*n^3 + 700*m^3*n^2 + 1624*m^2*
n^4 + 735*m^3*n^3 + 175*m^4*n^2 + 1) + (B*b^3*d^2*x*x^(6*n)*(e*x)^m*(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))/(6*m + 21*n + 105*m*n + 700*m*n^2 + 210*m^2*n + 2205*m
*n^3 + 210*m^3*n + 3248*m*n^4 + 105*m^4*n + 1764*m*n^5 + 21*m^5*n + 15*m^2 + 20*m^3 + 15*m^4 + 6*m^5 + m^6 + 1
75*n^2 + 735*n^3 + 1624*n^4 + 1764*n^5 + 720*n^6 + 1050*m^2*n^2 + 2205*m^2*n^3 + 700*m^3*n^2 + 1624*m^2*n^4 +
735*m^3*n^3 + 175*m^4*n^2 + 1)

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