3.1.17 \(\int (e x)^m (a+b x^2) (A+B x^2) (c+d x^2)^3 \, dx\) [17]

Optimal. Leaf size=189 \[ \frac {a A c^3 (e x)^{1+m}}{e (1+m)}+\frac {c^2 (A b c+a B c+3 a A d) (e x)^{3+m}}{e^3 (3+m)}+\frac {c (3 a d (B c+A d)+b c (B c+3 A d)) (e x)^{5+m}}{e^5 (5+m)}+\frac {d (3 b c (B c+A d)+a d (3 B c+A d)) (e x)^{7+m}}{e^7 (7+m)}+\frac {d^2 (3 b B c+A b d+a B d) (e x)^{9+m}}{e^9 (9+m)}+\frac {b B d^3 (e x)^{11+m}}{e^{11} (11+m)} \]

[Out]

a*A*c^3*(e*x)^(1+m)/e/(1+m)+c^2*(3*A*a*d+A*b*c+B*a*c)*(e*x)^(3+m)/e^3/(3+m)+c*(3*a*d*(A*d+B*c)+b*c*(3*A*d+B*c)
)*(e*x)^(5+m)/e^5/(5+m)+d*(3*b*c*(A*d+B*c)+a*d*(A*d+3*B*c))*(e*x)^(7+m)/e^7/(7+m)+d^2*(A*b*d+B*a*d+3*B*b*c)*(e
*x)^(9+m)/e^9/(9+m)+b*B*d^3*(e*x)^(11+m)/e^11/(11+m)

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Rubi [A]
time = 0.12, antiderivative size = 189, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.034, Rules used = {584} \begin {gather*} \frac {c^2 (e x)^{m+3} (3 a A d+a B c+A b c)}{e^3 (m+3)}+\frac {d^2 (e x)^{m+9} (a B d+A b d+3 b B c)}{e^9 (m+9)}+\frac {d (e x)^{m+7} (a d (A d+3 B c)+3 b c (A d+B c))}{e^7 (m+7)}+\frac {c (e x)^{m+5} (3 a d (A d+B c)+b c (3 A d+B c))}{e^5 (m+5)}+\frac {a A c^3 (e x)^{m+1}}{e (m+1)}+\frac {b B d^3 (e x)^{m+11}}{e^{11} (m+11)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a*A*c^3*(e*x)^(1 + m))/(e*(1 + m)) + (c^2*(A*b*c + a*B*c + 3*a*A*d)*(e*x)^(3 + m))/(e^3*(3 + m)) + (c*(3*a*d*
(B*c + A*d) + b*c*(B*c + 3*A*d))*(e*x)^(5 + m))/(e^5*(5 + m)) + (d*(3*b*c*(B*c + A*d) + a*d*(3*B*c + A*d))*(e*
x)^(7 + m))/(e^7*(7 + m)) + (d^2*(3*b*B*c + A*b*d + a*B*d)*(e*x)^(9 + m))/(e^9*(9 + m)) + (b*B*d^3*(e*x)^(11 +
m))/(e^11*(11 + m))

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^2\right ) \left (A+B x^2\right ) \left (c+d x^2\right )^3 \, dx &=\int \left (a A c^3 (e x)^m+\frac {c^2 (A b c+a B c+3 a A d) (e x)^{2+m}}{e^2}+\frac {c (3 a d (B c+A d)+b c (B c+3 A d)) (e x)^{4+m}}{e^4}+\frac {d (3 b c (B c+A d)+a d (3 B c+A d)) (e x)^{6+m}}{e^6}+\frac {d^2 (3 b B c+A b d+a B d) (e x)^{8+m}}{e^8}+\frac {b B d^3 (e x)^{10+m}}{e^{10}}\right ) \, dx\\ &=\frac {a A c^3 (e x)^{1+m}}{e (1+m)}+\frac {c^2 (A b c+a B c+3 a A d) (e x)^{3+m}}{e^3 (3+m)}+\frac {c (3 a d (B c+A d)+b c (B c+3 A d)) (e x)^{5+m}}{e^5 (5+m)}+\frac {d (3 b c (B c+A d)+a d (3 B c+A d)) (e x)^{7+m}}{e^7 (7+m)}+\frac {d^2 (3 b B c+A b d+a B d) (e x)^{9+m}}{e^9 (9+m)}+\frac {b B d^3 (e x)^{11+m}}{e^{11} (11+m)}\\ \end {align*}

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Mathematica [A]
time = 0.44, size = 151, normalized size = 0.80 \begin {gather*} x (e x)^m \left (\frac {a A c^3}{1+m}+\frac {c^2 (A b c+a B c+3 a A d) x^2}{3+m}+\frac {c (3 a d (B c+A d)+b c (B c+3 A d)) x^4}{5+m}+\frac {d (3 b c (B c+A d)+a d (3 B c+A d)) x^6}{7+m}+\frac {d^2 (3 b B c+A b d+a B d) x^8}{9+m}+\frac {b B d^3 x^{10}}{11+m}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

x*(e*x)^m*((a*A*c^3)/(1 + m) + (c^2*(A*b*c + a*B*c + 3*a*A*d)*x^2)/(3 + m) + (c*(3*a*d*(B*c + A*d) + b*c*(B*c
+ 3*A*d))*x^4)/(5 + m) + (d*(3*b*c*(B*c + A*d) + a*d*(3*B*c + A*d))*x^6)/(7 + m) + (d^2*(3*b*B*c + A*b*d + a*B
*d)*x^8)/(9 + m) + (b*B*d^3*x^10)/(11 + m))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1228\) vs. \(2(189)=378\).
time = 0.11, size = 1229, normalized size = 6.50 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x)^m*(b*x^2+a)*(B*x^2+A)*(d*x^2+c)^3,x,method=_RETURNVERBOSE)

[Out]

x*(B*b*d^3*m^5*x^10+25*B*b*d^3*m^4*x^10+A*b*d^3*m^5*x^8+B*a*d^3*m^5*x^8+3*B*b*c*d^2*m^5*x^8+230*B*b*d^3*m^3*x^
10+27*A*b*d^3*m^4*x^8+27*B*a*d^3*m^4*x^8+81*B*b*c*d^2*m^4*x^8+950*B*b*d^3*m^2*x^10+A*a*d^3*m^5*x^6+3*A*b*c*d^2
*m^5*x^6+262*A*b*d^3*m^3*x^8+3*B*a*c*d^2*m^5*x^6+262*B*a*d^3*m^3*x^8+3*B*b*c^2*d*m^5*x^6+786*B*b*c*d^2*m^3*x^8
+1689*B*b*d^3*m*x^10+29*A*a*d^3*m^4*x^6+87*A*b*c*d^2*m^4*x^6+1122*A*b*d^3*m^2*x^8+87*B*a*c*d^2*m^4*x^6+1122*B*
a*d^3*m^2*x^8+87*B*b*c^2*d*m^4*x^6+3366*B*b*c*d^2*m^2*x^8+945*B*b*d^3*x^10+3*A*a*c*d^2*m^5*x^4+302*A*a*d^3*m^3
*x^6+3*A*b*c^2*d*m^5*x^4+906*A*b*c*d^2*m^3*x^6+2041*A*b*d^3*m*x^8+3*B*a*c^2*d*m^5*x^4+906*B*a*c*d^2*m^3*x^6+20
41*B*a*d^3*m*x^8+B*b*c^3*m^5*x^4+906*B*b*c^2*d*m^3*x^6+6123*B*b*c*d^2*m*x^8+93*A*a*c*d^2*m^4*x^4+1366*A*a*d^3*
m^2*x^6+93*A*b*c^2*d*m^4*x^4+4098*A*b*c*d^2*m^2*x^6+1155*A*b*d^3*x^8+93*B*a*c^2*d*m^4*x^4+4098*B*a*c*d^2*m^2*x
^6+1155*B*a*d^3*x^8+31*B*b*c^3*m^4*x^4+4098*B*b*c^2*d*m^2*x^6+3465*B*b*c*d^2*x^8+3*A*a*c^2*d*m^5*x^2+1050*A*a*
c*d^2*m^3*x^4+2577*A*a*d^3*m*x^6+A*b*c^3*m^5*x^2+1050*A*b*c^2*d*m^3*x^4+7731*A*b*c*d^2*m*x^6+B*a*c^3*m^5*x^2+1
050*B*a*c^2*d*m^3*x^4+7731*B*a*c*d^2*m*x^6+350*B*b*c^3*m^3*x^4+7731*B*b*c^2*d*m*x^6+99*A*a*c^2*d*m^4*x^2+5190*
A*a*c*d^2*m^2*x^4+1485*A*a*d^3*x^6+33*A*b*c^3*m^4*x^2+5190*A*b*c^2*d*m^2*x^4+4455*A*b*c*d^2*x^6+33*B*a*c^3*m^4
*x^2+5190*B*a*c^2*d*m^2*x^4+4455*B*a*c*d^2*x^6+1730*B*b*c^3*m^2*x^4+4455*B*b*c^2*d*x^6+A*a*c^3*m^5+1218*A*a*c^
2*d*m^3*x^2+10467*A*a*c*d^2*m*x^4+406*A*b*c^3*m^3*x^2+10467*A*b*c^2*d*m*x^4+406*B*a*c^3*m^3*x^2+10467*B*a*c^2*
d*m*x^4+3489*B*b*c^3*m*x^4+35*A*a*c^3*m^4+6786*A*a*c^2*d*m^2*x^2+6237*A*a*c*d^2*x^4+2262*A*b*c^3*m^2*x^2+6237*
A*b*c^2*d*x^4+2262*B*a*c^3*m^2*x^2+6237*B*a*c^2*d*x^4+2079*B*b*c^3*x^4+470*A*a*c^3*m^3+16059*A*a*c^2*d*m*x^2+5
353*A*b*c^3*m*x^2+5353*B*a*c^3*m*x^2+3010*A*a*c^3*m^2+10395*A*a*c^2*d*x^2+3465*A*b*c^3*x^2+3465*B*a*c^3*x^2+91
29*A*a*c^3*m+10395*A*a*c^3)*(e*x)^m/(11+m)/(9+m)/(7+m)/(5+m)/(3+m)/(1+m)

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Maxima [A]
time = 0.31, size = 353, normalized size = 1.87 \begin {gather*} \frac {B b d^{3} x^{11} e^{\left (m \log \left (x\right ) + m\right )}}{m + 11} + \frac {3 \, B b c d^{2} x^{9} e^{\left (m \log \left (x\right ) + m\right )}}{m + 9} + \frac {B a d^{3} x^{9} e^{\left (m \log \left (x\right ) + m\right )}}{m + 9} + \frac {A b d^{3} x^{9} e^{\left (m \log \left (x\right ) + m\right )}}{m + 9} + \frac {3 \, B b c^{2} d x^{7} e^{\left (m \log \left (x\right ) + m\right )}}{m + 7} + \frac {3 \, B a c d^{2} x^{7} e^{\left (m \log \left (x\right ) + m\right )}}{m + 7} + \frac {3 \, A b c d^{2} x^{7} e^{\left (m \log \left (x\right ) + m\right )}}{m + 7} + \frac {A a d^{3} x^{7} e^{\left (m \log \left (x\right ) + m\right )}}{m + 7} + \frac {B b c^{3} x^{5} e^{\left (m \log \left (x\right ) + m\right )}}{m + 5} + \frac {3 \, B a c^{2} d x^{5} e^{\left (m \log \left (x\right ) + m\right )}}{m + 5} + \frac {3 \, A b c^{2} d x^{5} e^{\left (m \log \left (x\right ) + m\right )}}{m + 5} + \frac {3 \, A a c d^{2} x^{5} e^{\left (m \log \left (x\right ) + m\right )}}{m + 5} + \frac {B a c^{3} x^{3} e^{\left (m \log \left (x\right ) + m\right )}}{m + 3} + \frac {A b c^{3} x^{3} e^{\left (m \log \left (x\right ) + m\right )}}{m + 3} + \frac {3 \, A a c^{2} d x^{3} e^{\left (m \log \left (x\right ) + m\right )}}{m + 3} + \frac {\left (x e\right )^{m + 1} A a c^{3} e^{\left (-1\right )}}{m + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

B*b*d^3*x^11*e^(m*log(x) + m)/(m + 11) + 3*B*b*c*d^2*x^9*e^(m*log(x) + m)/(m + 9) + B*a*d^3*x^9*e^(m*log(x) +
m)/(m + 9) + A*b*d^3*x^9*e^(m*log(x) + m)/(m + 9) + 3*B*b*c^2*d*x^7*e^(m*log(x) + m)/(m + 7) + 3*B*a*c*d^2*x^7
*e^(m*log(x) + m)/(m + 7) + 3*A*b*c*d^2*x^7*e^(m*log(x) + m)/(m + 7) + A*a*d^3*x^7*e^(m*log(x) + m)/(m + 7) +
B*b*c^3*x^5*e^(m*log(x) + m)/(m + 5) + 3*B*a*c^2*d*x^5*e^(m*log(x) + m)/(m + 5) + 3*A*b*c^2*d*x^5*e^(m*log(x)
+ m)/(m + 5) + 3*A*a*c*d^2*x^5*e^(m*log(x) + m)/(m + 5) + B*a*c^3*x^3*e^(m*log(x) + m)/(m + 3) + A*b*c^3*x^3*e
^(m*log(x) + m)/(m + 3) + 3*A*a*c^2*d*x^3*e^(m*log(x) + m)/(m + 3) + (x*e)^(m + 1)*A*a*c^3*e^(-1)/(m + 1)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 838 vs. \(2 (189) = 378\).
time = 1.11, size = 838, normalized size = 4.43 \begin {gather*} \frac {{\left ({\left (B b d^{3} m^{5} + 25 \, B b d^{3} m^{4} + 230 \, B b d^{3} m^{3} + 950 \, B b d^{3} m^{2} + 1689 \, B b d^{3} m + 945 \, B b d^{3}\right )} x^{11} + {\left ({\left (3 \, B b c d^{2} + {\left (B a + A b\right )} d^{3}\right )} m^{5} + 3465 \, B b c d^{2} + 27 \, {\left (3 \, B b c d^{2} + {\left (B a + A b\right )} d^{3}\right )} m^{4} + 1155 \, {\left (B a + A b\right )} d^{3} + 262 \, {\left (3 \, B b c d^{2} + {\left (B a + A b\right )} d^{3}\right )} m^{3} + 1122 \, {\left (3 \, B b c d^{2} + {\left (B a + A b\right )} d^{3}\right )} m^{2} + 2041 \, {\left (3 \, B b c d^{2} + {\left (B a + A b\right )} d^{3}\right )} m\right )} x^{9} + {\left ({\left (3 \, B b c^{2} d + A a d^{3} + 3 \, {\left (B a + A b\right )} c d^{2}\right )} m^{5} + 4455 \, B b c^{2} d + 1485 \, A a d^{3} + 29 \, {\left (3 \, B b c^{2} d + A a d^{3} + 3 \, {\left (B a + A b\right )} c d^{2}\right )} m^{4} + 4455 \, {\left (B a + A b\right )} c d^{2} + 302 \, {\left (3 \, B b c^{2} d + A a d^{3} + 3 \, {\left (B a + A b\right )} c d^{2}\right )} m^{3} + 1366 \, {\left (3 \, B b c^{2} d + A a d^{3} + 3 \, {\left (B a + A b\right )} c d^{2}\right )} m^{2} + 2577 \, {\left (3 \, B b c^{2} d + A a d^{3} + 3 \, {\left (B a + A b\right )} c d^{2}\right )} m\right )} x^{7} + {\left ({\left (B b c^{3} + 3 \, A a c d^{2} + 3 \, {\left (B a + A b\right )} c^{2} d\right )} m^{5} + 2079 \, B b c^{3} + 6237 \, A a c d^{2} + 31 \, {\left (B b c^{3} + 3 \, A a c d^{2} + 3 \, {\left (B a + A b\right )} c^{2} d\right )} m^{4} + 6237 \, {\left (B a + A b\right )} c^{2} d + 350 \, {\left (B b c^{3} + 3 \, A a c d^{2} + 3 \, {\left (B a + A b\right )} c^{2} d\right )} m^{3} + 1730 \, {\left (B b c^{3} + 3 \, A a c d^{2} + 3 \, {\left (B a + A b\right )} c^{2} d\right )} m^{2} + 3489 \, {\left (B b c^{3} + 3 \, A a c d^{2} + 3 \, {\left (B a + A b\right )} c^{2} d\right )} m\right )} x^{5} + {\left ({\left (3 \, A a c^{2} d + {\left (B a + A b\right )} c^{3}\right )} m^{5} + 10395 \, A a c^{2} d + 33 \, {\left (3 \, A a c^{2} d + {\left (B a + A b\right )} c^{3}\right )} m^{4} + 3465 \, {\left (B a + A b\right )} c^{3} + 406 \, {\left (3 \, A a c^{2} d + {\left (B a + A b\right )} c^{3}\right )} m^{3} + 2262 \, {\left (3 \, A a c^{2} d + {\left (B a + A b\right )} c^{3}\right )} m^{2} + 5353 \, {\left (3 \, A a c^{2} d + {\left (B a + A b\right )} c^{3}\right )} m\right )} x^{3} + {\left (A a c^{3} m^{5} + 35 \, A a c^{3} m^{4} + 470 \, A a c^{3} m^{3} + 3010 \, A a c^{3} m^{2} + 9129 \, A a c^{3} m + 10395 \, A a c^{3}\right )} x\right )} \left (x e\right )^{m}}{m^{6} + 36 \, m^{5} + 505 \, m^{4} + 3480 \, m^{3} + 12139 \, m^{2} + 19524 \, m + 10395} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((B*b*d^3*m^5 + 25*B*b*d^3*m^4 + 230*B*b*d^3*m^3 + 950*B*b*d^3*m^2 + 1689*B*b*d^3*m + 945*B*b*d^3)*x^11 + ((3*
B*b*c*d^2 + (B*a + A*b)*d^3)*m^5 + 3465*B*b*c*d^2 + 27*(3*B*b*c*d^2 + (B*a + A*b)*d^3)*m^4 + 1155*(B*a + A*b)*
d^3 + 262*(3*B*b*c*d^2 + (B*a + A*b)*d^3)*m^3 + 1122*(3*B*b*c*d^2 + (B*a + A*b)*d^3)*m^2 + 2041*(3*B*b*c*d^2 +
(B*a + A*b)*d^3)*m)*x^9 + ((3*B*b*c^2*d + A*a*d^3 + 3*(B*a + A*b)*c*d^2)*m^5 + 4455*B*b*c^2*d + 1485*A*a*d^3
+ 29*(3*B*b*c^2*d + A*a*d^3 + 3*(B*a + A*b)*c*d^2)*m^4 + 4455*(B*a + A*b)*c*d^2 + 302*(3*B*b*c^2*d + A*a*d^3 +
3*(B*a + A*b)*c*d^2)*m^3 + 1366*(3*B*b*c^2*d + A*a*d^3 + 3*(B*a + A*b)*c*d^2)*m^2 + 2577*(3*B*b*c^2*d + A*a*d
^3 + 3*(B*a + A*b)*c*d^2)*m)*x^7 + ((B*b*c^3 + 3*A*a*c*d^2 + 3*(B*a + A*b)*c^2*d)*m^5 + 2079*B*b*c^3 + 6237*A*
a*c*d^2 + 31*(B*b*c^3 + 3*A*a*c*d^2 + 3*(B*a + A*b)*c^2*d)*m^4 + 6237*(B*a + A*b)*c^2*d + 350*(B*b*c^3 + 3*A*a
*c*d^2 + 3*(B*a + A*b)*c^2*d)*m^3 + 1730*(B*b*c^3 + 3*A*a*c*d^2 + 3*(B*a + A*b)*c^2*d)*m^2 + 3489*(B*b*c^3 + 3
*A*a*c*d^2 + 3*(B*a + A*b)*c^2*d)*m)*x^5 + ((3*A*a*c^2*d + (B*a + A*b)*c^3)*m^5 + 10395*A*a*c^2*d + 33*(3*A*a*
c^2*d + (B*a + A*b)*c^3)*m^4 + 3465*(B*a + A*b)*c^3 + 406*(3*A*a*c^2*d + (B*a + A*b)*c^3)*m^3 + 2262*(3*A*a*c^
2*d + (B*a + A*b)*c^3)*m^2 + 5353*(3*A*a*c^2*d + (B*a + A*b)*c^3)*m)*x^3 + (A*a*c^3*m^5 + 35*A*a*c^3*m^4 + 470
*A*a*c^3*m^3 + 3010*A*a*c^3*m^2 + 9129*A*a*c^3*m + 10395*A*a*c^3)*x)*(x*e)^m/(m^6 + 36*m^5 + 505*m^4 + 3480*m^
3 + 12139*m^2 + 19524*m + 10395)

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 5992 vs. \(2 (184) = 368\).
time = 1.01, size = 5992, normalized size = 31.70 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise(((-A*a*c**3/(10*x**10) - 3*A*a*c**2*d/(8*x**8) - A*a*c*d**2/(2*x**6) - A*a*d**3/(4*x**4) - A*b*c**3/
(8*x**8) - A*b*c**2*d/(2*x**6) - 3*A*b*c*d**2/(4*x**4) - A*b*d**3/(2*x**2) - B*a*c**3/(8*x**8) - B*a*c**2*d/(2
*x**6) - 3*B*a*c*d**2/(4*x**4) - B*a*d**3/(2*x**2) - B*b*c**3/(6*x**6) - 3*B*b*c**2*d/(4*x**4) - 3*B*b*c*d**2/
(2*x**2) + B*b*d**3*log(x))/e**11, Eq(m, -11)), ((-A*a*c**3/(8*x**8) - A*a*c**2*d/(2*x**6) - 3*A*a*c*d**2/(4*x
**4) - A*a*d**3/(2*x**2) - A*b*c**3/(6*x**6) - 3*A*b*c**2*d/(4*x**4) - 3*A*b*c*d**2/(2*x**2) + A*b*d**3*log(x)
- B*a*c**3/(6*x**6) - 3*B*a*c**2*d/(4*x**4) - 3*B*a*c*d**2/(2*x**2) + B*a*d**3*log(x) - B*b*c**3/(4*x**4) - 3
*B*b*c**2*d/(2*x**2) + 3*B*b*c*d**2*log(x) + B*b*d**3*x**2/2)/e**9, Eq(m, -9)), ((-A*a*c**3/(6*x**6) - 3*A*a*c
**2*d/(4*x**4) - 3*A*a*c*d**2/(2*x**2) + A*a*d**3*log(x) - A*b*c**3/(4*x**4) - 3*A*b*c**2*d/(2*x**2) + 3*A*b*c
*d**2*log(x) + A*b*d**3*x**2/2 - B*a*c**3/(4*x**4) - 3*B*a*c**2*d/(2*x**2) + 3*B*a*c*d**2*log(x) + B*a*d**3*x*
*2/2 - B*b*c**3/(2*x**2) + 3*B*b*c**2*d*log(x) + 3*B*b*c*d**2*x**2/2 + B*b*d**3*x**4/4)/e**7, Eq(m, -7)), ((-A
*a*c**3/(4*x**4) - 3*A*a*c**2*d/(2*x**2) + 3*A*a*c*d**2*log(x) + A*a*d**3*x**2/2 - A*b*c**3/(2*x**2) + 3*A*b*c
**2*d*log(x) + 3*A*b*c*d**2*x**2/2 + A*b*d**3*x**4/4 - B*a*c**3/(2*x**2) + 3*B*a*c**2*d*log(x) + 3*B*a*c*d**2*
x**2/2 + B*a*d**3*x**4/4 + B*b*c**3*log(x) + 3*B*b*c**2*d*x**2/2 + 3*B*b*c*d**2*x**4/4 + B*b*d**3*x**6/6)/e**5
, Eq(m, -5)), ((-A*a*c**3/(2*x**2) + 3*A*a*c**2*d*log(x) + 3*A*a*c*d**2*x**2/2 + A*a*d**3*x**4/4 + A*b*c**3*lo
g(x) + 3*A*b*c**2*d*x**2/2 + 3*A*b*c*d**2*x**4/4 + A*b*d**3*x**6/6 + B*a*c**3*log(x) + 3*B*a*c**2*d*x**2/2 + 3
*B*a*c*d**2*x**4/4 + B*a*d**3*x**6/6 + B*b*c**3*x**2/2 + 3*B*b*c**2*d*x**4/4 + B*b*c*d**2*x**6/2 + B*b*d**3*x*
*8/8)/e**3, Eq(m, -3)), ((A*a*c**3*log(x) + 3*A*a*c**2*d*x**2/2 + 3*A*a*c*d**2*x**4/4 + A*a*d**3*x**6/6 + A*b*
c**3*x**2/2 + 3*A*b*c**2*d*x**4/4 + A*b*c*d**2*x**6/2 + A*b*d**3*x**8/8 + B*a*c**3*x**2/2 + 3*B*a*c**2*d*x**4/
4 + B*a*c*d**2*x**6/2 + B*a*d**3*x**8/8 + B*b*c**3*x**4/4 + B*b*c**2*d*x**6/2 + 3*B*b*c*d**2*x**8/8 + B*b*d**3
*x**10/10)/e, Eq(m, -1)), (A*a*c**3*m**5*x*(e*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 1952
4*m + 10395) + 35*A*a*c**3*m**4*x*(e*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 103
95) + 470*A*a*c**3*m**3*x*(e*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 30
10*A*a*c**3*m**2*x*(e*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 9129*A*a*
c**3*m*x*(e*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 10395*A*a*c**3*x*(e
*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 3*A*a*c**2*d*m**5*x**3*(e*x)**
m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 99*A*a*c**2*d*m**4*x**3*(e*x)**m/(m
**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 1218*A*a*c**2*d*m**3*x**3*(e*x)**m/(m**
6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 6786*A*a*c**2*d*m**2*x**3*(e*x)**m/(m**6
+ 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 16059*A*a*c**2*d*m*x**3*(e*x)**m/(m**6 + 36
*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 10395*A*a*c**2*d*x**3*(e*x)**m/(m**6 + 36*m**5
+ 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 3*A*a*c*d**2*m**5*x**5*(e*x)**m/(m**6 + 36*m**5 + 505
*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 93*A*a*c*d**2*m**4*x**5*(e*x)**m/(m**6 + 36*m**5 + 505*m**
4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 1050*A*a*c*d**2*m**3*x**5*(e*x)**m/(m**6 + 36*m**5 + 505*m**4
+ 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 5190*A*a*c*d**2*m**2*x**5*(e*x)**m/(m**6 + 36*m**5 + 505*m**4 +
3480*m**3 + 12139*m**2 + 19524*m + 10395) + 10467*A*a*c*d**2*m*x**5*(e*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480
*m**3 + 12139*m**2 + 19524*m + 10395) + 6237*A*a*c*d**2*x**5*(e*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 +
12139*m**2 + 19524*m + 10395) + A*a*d**3*m**5*x**7*(e*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m*
*2 + 19524*m + 10395) + 29*A*a*d**3*m**4*x**7*(e*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 1
9524*m + 10395) + 302*A*a*d**3*m**3*x**7*(e*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*
m + 10395) + 1366*A*a*d**3*m**2*x**7*(e*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m +
10395) + 2577*A*a*d**3*m*x**7*(e*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395)
+ 1485*A*a*d**3*x**7*(e*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + A*b*c**
3*m**5*x**3*(e*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 33*A*b*c**3*m**4
*x**3*(e*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 406*A*b*c**3*m**3*x**3
*(e*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 2262*A*b*c**3*m**2*x**3*(e*
x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + ...

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1708 vs. \(2 (189) = 378\).
time = 1.83, size = 1708, normalized size = 9.04 \begin {gather*} \frac {B b d^{3} m^{5} x^{11} x^{m} e^{m} + 25 \, B b d^{3} m^{4} x^{11} x^{m} e^{m} + 3 \, B b c d^{2} m^{5} x^{9} x^{m} e^{m} + B a d^{3} m^{5} x^{9} x^{m} e^{m} + A b d^{3} m^{5} x^{9} x^{m} e^{m} + 230 \, B b d^{3} m^{3} x^{11} x^{m} e^{m} + 81 \, B b c d^{2} m^{4} x^{9} x^{m} e^{m} + 27 \, B a d^{3} m^{4} x^{9} x^{m} e^{m} + 27 \, A b d^{3} m^{4} x^{9} x^{m} e^{m} + 950 \, B b d^{3} m^{2} x^{11} x^{m} e^{m} + 3 \, B b c^{2} d m^{5} x^{7} x^{m} e^{m} + 3 \, B a c d^{2} m^{5} x^{7} x^{m} e^{m} + 3 \, A b c d^{2} m^{5} x^{7} x^{m} e^{m} + A a d^{3} m^{5} x^{7} x^{m} e^{m} + 786 \, B b c d^{2} m^{3} x^{9} x^{m} e^{m} + 262 \, B a d^{3} m^{3} x^{9} x^{m} e^{m} + 262 \, A b d^{3} m^{3} x^{9} x^{m} e^{m} + 1689 \, B b d^{3} m x^{11} x^{m} e^{m} + 87 \, B b c^{2} d m^{4} x^{7} x^{m} e^{m} + 87 \, B a c d^{2} m^{4} x^{7} x^{m} e^{m} + 87 \, A b c d^{2} m^{4} x^{7} x^{m} e^{m} + 29 \, A a d^{3} m^{4} x^{7} x^{m} e^{m} + 3366 \, B b c d^{2} m^{2} x^{9} x^{m} e^{m} + 1122 \, B a d^{3} m^{2} x^{9} x^{m} e^{m} + 1122 \, A b d^{3} m^{2} x^{9} x^{m} e^{m} + 945 \, B b d^{3} x^{11} x^{m} e^{m} + B b c^{3} m^{5} x^{5} x^{m} e^{m} + 3 \, B a c^{2} d m^{5} x^{5} x^{m} e^{m} + 3 \, A b c^{2} d m^{5} x^{5} x^{m} e^{m} + 3 \, A a c d^{2} m^{5} x^{5} x^{m} e^{m} + 906 \, B b c^{2} d m^{3} x^{7} x^{m} e^{m} + 906 \, B a c d^{2} m^{3} x^{7} x^{m} e^{m} + 906 \, A b c d^{2} m^{3} x^{7} x^{m} e^{m} + 302 \, A a d^{3} m^{3} x^{7} x^{m} e^{m} + 6123 \, B b c d^{2} m x^{9} x^{m} e^{m} + 2041 \, B a d^{3} m x^{9} x^{m} e^{m} + 2041 \, A b d^{3} m x^{9} x^{m} e^{m} + 31 \, B b c^{3} m^{4} x^{5} x^{m} e^{m} + 93 \, B a c^{2} d m^{4} x^{5} x^{m} e^{m} + 93 \, A b c^{2} d m^{4} x^{5} x^{m} e^{m} + 93 \, A a c d^{2} m^{4} x^{5} x^{m} e^{m} + 4098 \, B b c^{2} d m^{2} x^{7} x^{m} e^{m} + 4098 \, B a c d^{2} m^{2} x^{7} x^{m} e^{m} + 4098 \, A b c d^{2} m^{2} x^{7} x^{m} e^{m} + 1366 \, A a d^{3} m^{2} x^{7} x^{m} e^{m} + 3465 \, B b c d^{2} x^{9} x^{m} e^{m} + 1155 \, B a d^{3} x^{9} x^{m} e^{m} + 1155 \, A b d^{3} x^{9} x^{m} e^{m} + B a c^{3} m^{5} x^{3} x^{m} e^{m} + A b c^{3} m^{5} x^{3} x^{m} e^{m} + 3 \, A a c^{2} d m^{5} x^{3} x^{m} e^{m} + 350 \, B b c^{3} m^{3} x^{5} x^{m} e^{m} + 1050 \, B a c^{2} d m^{3} x^{5} x^{m} e^{m} + 1050 \, A b c^{2} d m^{3} x^{5} x^{m} e^{m} + 1050 \, A a c d^{2} m^{3} x^{5} x^{m} e^{m} + 7731 \, B b c^{2} d m x^{7} x^{m} e^{m} + 7731 \, B a c d^{2} m x^{7} x^{m} e^{m} + 7731 \, A b c d^{2} m x^{7} x^{m} e^{m} + 2577 \, A a d^{3} m x^{7} x^{m} e^{m} + 33 \, B a c^{3} m^{4} x^{3} x^{m} e^{m} + 33 \, A b c^{3} m^{4} x^{3} x^{m} e^{m} + 99 \, A a c^{2} d m^{4} x^{3} x^{m} e^{m} + 1730 \, B b c^{3} m^{2} x^{5} x^{m} e^{m} + 5190 \, B a c^{2} d m^{2} x^{5} x^{m} e^{m} + 5190 \, A b c^{2} d m^{2} x^{5} x^{m} e^{m} + 5190 \, A a c d^{2} m^{2} x^{5} x^{m} e^{m} + 4455 \, B b c^{2} d x^{7} x^{m} e^{m} + 4455 \, B a c d^{2} x^{7} x^{m} e^{m} + 4455 \, A b c d^{2} x^{7} x^{m} e^{m} + 1485 \, A a d^{3} x^{7} x^{m} e^{m} + A a c^{3} m^{5} x x^{m} e^{m} + 406 \, B a c^{3} m^{3} x^{3} x^{m} e^{m} + 406 \, A b c^{3} m^{3} x^{3} x^{m} e^{m} + 1218 \, A a c^{2} d m^{3} x^{3} x^{m} e^{m} + 3489 \, B b c^{3} m x^{5} x^{m} e^{m} + 10467 \, B a c^{2} d m x^{5} x^{m} e^{m} + 10467 \, A b c^{2} d m x^{5} x^{m} e^{m} + 10467 \, A a c d^{2} m x^{5} x^{m} e^{m} + 35 \, A a c^{3} m^{4} x x^{m} e^{m} + 2262 \, B a c^{3} m^{2} x^{3} x^{m} e^{m} + 2262 \, A b c^{3} m^{2} x^{3} x^{m} e^{m} + 6786 \, A a c^{2} d m^{2} x^{3} x^{m} e^{m} + 2079 \, B b c^{3} x^{5} x^{m} e^{m} + 6237 \, B a c^{2} d x^{5} x^{m} e^{m} + 6237 \, A b c^{2} d x^{5} x^{m} e^{m} + 6237 \, A a c d^{2} x^{5} x^{m} e^{m} + 470 \, A a c^{3} m^{3} x x^{m} e^{m} + 5353 \, B a c^{3} m x^{3} x^{m} e^{m} + 5353 \, A b c^{3} m x^{3} x^{m} e^{m} + 16059 \, A a c^{2} d m x^{3} x^{m} e^{m} + 3010 \, A a c^{3} m^{2} x x^{m} e^{m} + 3465 \, B a c^{3} x^{3} x^{m} e^{m} + 3465 \, A b c^{3} x^{3} x^{m} e^{m} + 10395 \, A a c^{2} d x^{3} x^{m} e^{m} + 9129 \, A a c^{3} m x x^{m} e^{m} + 10395 \, A a c^{3} x x^{m} e^{m}}{m^{6} + 36 \, m^{5} + 505 \, m^{4} + 3480 \, m^{3} + 12139 \, m^{2} + 19524 \, m + 10395} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(B*b*d^3*m^5*x^11*x^m*e^m + 25*B*b*d^3*m^4*x^11*x^m*e^m + 3*B*b*c*d^2*m^5*x^9*x^m*e^m + B*a*d^3*m^5*x^9*x^m*e^
m + A*b*d^3*m^5*x^9*x^m*e^m + 230*B*b*d^3*m^3*x^11*x^m*e^m + 81*B*b*c*d^2*m^4*x^9*x^m*e^m + 27*B*a*d^3*m^4*x^9
*x^m*e^m + 27*A*b*d^3*m^4*x^9*x^m*e^m + 950*B*b*d^3*m^2*x^11*x^m*e^m + 3*B*b*c^2*d*m^5*x^7*x^m*e^m + 3*B*a*c*d
^2*m^5*x^7*x^m*e^m + 3*A*b*c*d^2*m^5*x^7*x^m*e^m + A*a*d^3*m^5*x^7*x^m*e^m + 786*B*b*c*d^2*m^3*x^9*x^m*e^m + 2
62*B*a*d^3*m^3*x^9*x^m*e^m + 262*A*b*d^3*m^3*x^9*x^m*e^m + 1689*B*b*d^3*m*x^11*x^m*e^m + 87*B*b*c^2*d*m^4*x^7*
x^m*e^m + 87*B*a*c*d^2*m^4*x^7*x^m*e^m + 87*A*b*c*d^2*m^4*x^7*x^m*e^m + 29*A*a*d^3*m^4*x^7*x^m*e^m + 3366*B*b*
c*d^2*m^2*x^9*x^m*e^m + 1122*B*a*d^3*m^2*x^9*x^m*e^m + 1122*A*b*d^3*m^2*x^9*x^m*e^m + 945*B*b*d^3*x^11*x^m*e^m
+ B*b*c^3*m^5*x^5*x^m*e^m + 3*B*a*c^2*d*m^5*x^5*x^m*e^m + 3*A*b*c^2*d*m^5*x^5*x^m*e^m + 3*A*a*c*d^2*m^5*x^5*x
^m*e^m + 906*B*b*c^2*d*m^3*x^7*x^m*e^m + 906*B*a*c*d^2*m^3*x^7*x^m*e^m + 906*A*b*c*d^2*m^3*x^7*x^m*e^m + 302*A
*a*d^3*m^3*x^7*x^m*e^m + 6123*B*b*c*d^2*m*x^9*x^m*e^m + 2041*B*a*d^3*m*x^9*x^m*e^m + 2041*A*b*d^3*m*x^9*x^m*e^
m + 31*B*b*c^3*m^4*x^5*x^m*e^m + 93*B*a*c^2*d*m^4*x^5*x^m*e^m + 93*A*b*c^2*d*m^4*x^5*x^m*e^m + 93*A*a*c*d^2*m^
4*x^5*x^m*e^m + 4098*B*b*c^2*d*m^2*x^7*x^m*e^m + 4098*B*a*c*d^2*m^2*x^7*x^m*e^m + 4098*A*b*c*d^2*m^2*x^7*x^m*e
^m + 1366*A*a*d^3*m^2*x^7*x^m*e^m + 3465*B*b*c*d^2*x^9*x^m*e^m + 1155*B*a*d^3*x^9*x^m*e^m + 1155*A*b*d^3*x^9*x
^m*e^m + B*a*c^3*m^5*x^3*x^m*e^m + A*b*c^3*m^5*x^3*x^m*e^m + 3*A*a*c^2*d*m^5*x^3*x^m*e^m + 350*B*b*c^3*m^3*x^5
*x^m*e^m + 1050*B*a*c^2*d*m^3*x^5*x^m*e^m + 1050*A*b*c^2*d*m^3*x^5*x^m*e^m + 1050*A*a*c*d^2*m^3*x^5*x^m*e^m +
7731*B*b*c^2*d*m*x^7*x^m*e^m + 7731*B*a*c*d^2*m*x^7*x^m*e^m + 7731*A*b*c*d^2*m*x^7*x^m*e^m + 2577*A*a*d^3*m*x^
7*x^m*e^m + 33*B*a*c^3*m^4*x^3*x^m*e^m + 33*A*b*c^3*m^4*x^3*x^m*e^m + 99*A*a*c^2*d*m^4*x^3*x^m*e^m + 1730*B*b*
c^3*m^2*x^5*x^m*e^m + 5190*B*a*c^2*d*m^2*x^5*x^m*e^m + 5190*A*b*c^2*d*m^2*x^5*x^m*e^m + 5190*A*a*c*d^2*m^2*x^5
*x^m*e^m + 4455*B*b*c^2*d*x^7*x^m*e^m + 4455*B*a*c*d^2*x^7*x^m*e^m + 4455*A*b*c*d^2*x^7*x^m*e^m + 1485*A*a*d^3
*x^7*x^m*e^m + A*a*c^3*m^5*x*x^m*e^m + 406*B*a*c^3*m^3*x^3*x^m*e^m + 406*A*b*c^3*m^3*x^3*x^m*e^m + 1218*A*a*c^
2*d*m^3*x^3*x^m*e^m + 3489*B*b*c^3*m*x^5*x^m*e^m + 10467*B*a*c^2*d*m*x^5*x^m*e^m + 10467*A*b*c^2*d*m*x^5*x^m*e
^m + 10467*A*a*c*d^2*m*x^5*x^m*e^m + 35*A*a*c^3*m^4*x*x^m*e^m + 2262*B*a*c^3*m^2*x^3*x^m*e^m + 2262*A*b*c^3*m^
2*x^3*x^m*e^m + 6786*A*a*c^2*d*m^2*x^3*x^m*e^m + 2079*B*b*c^3*x^5*x^m*e^m + 6237*B*a*c^2*d*x^5*x^m*e^m + 6237*
A*b*c^2*d*x^5*x^m*e^m + 6237*A*a*c*d^2*x^5*x^m*e^m + 470*A*a*c^3*m^3*x*x^m*e^m + 5353*B*a*c^3*m*x^3*x^m*e^m +
5353*A*b*c^3*m*x^3*x^m*e^m + 16059*A*a*c^2*d*m*x^3*x^m*e^m + 3010*A*a*c^3*m^2*x*x^m*e^m + 3465*B*a*c^3*x^3*x^m
*e^m + 3465*A*b*c^3*x^3*x^m*e^m + 10395*A*a*c^2*d*x^3*x^m*e^m + 9129*A*a*c^3*m*x*x^m*e^m + 10395*A*a*c^3*x*x^m
*e^m)/(m^6 + 36*m^5 + 505*m^4 + 3480*m^3 + 12139*m^2 + 19524*m + 10395)

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Mupad [B]
time = 1.43, size = 469, normalized size = 2.48 \begin {gather*} \frac {c^2\,x^3\,{\left (e\,x\right )}^m\,\left (3\,A\,a\,d+A\,b\,c+B\,a\,c\right )\,\left (m^5+33\,m^4+406\,m^3+2262\,m^2+5353\,m+3465\right )}{m^6+36\,m^5+505\,m^4+3480\,m^3+12139\,m^2+19524\,m+10395}+\frac {d^2\,x^9\,{\left (e\,x\right )}^m\,\left (A\,b\,d+B\,a\,d+3\,B\,b\,c\right )\,\left (m^5+27\,m^4+262\,m^3+1122\,m^2+2041\,m+1155\right )}{m^6+36\,m^5+505\,m^4+3480\,m^3+12139\,m^2+19524\,m+10395}+\frac {c\,x^5\,{\left (e\,x\right )}^m\,\left (3\,A\,a\,d^2+B\,b\,c^2+3\,A\,b\,c\,d+3\,B\,a\,c\,d\right )\,\left (m^5+31\,m^4+350\,m^3+1730\,m^2+3489\,m+2079\right )}{m^6+36\,m^5+505\,m^4+3480\,m^3+12139\,m^2+19524\,m+10395}+\frac {d\,x^7\,{\left (e\,x\right )}^m\,\left (A\,a\,d^2+3\,B\,b\,c^2+3\,A\,b\,c\,d+3\,B\,a\,c\,d\right )\,\left (m^5+29\,m^4+302\,m^3+1366\,m^2+2577\,m+1485\right )}{m^6+36\,m^5+505\,m^4+3480\,m^3+12139\,m^2+19524\,m+10395}+\frac {B\,b\,d^3\,x^{11}\,{\left (e\,x\right )}^m\,\left (m^5+25\,m^4+230\,m^3+950\,m^2+1689\,m+945\right )}{m^6+36\,m^5+505\,m^4+3480\,m^3+12139\,m^2+19524\,m+10395}+\frac {A\,a\,c^3\,x\,{\left (e\,x\right )}^m\,\left (m^5+35\,m^4+470\,m^3+3010\,m^2+9129\,m+10395\right )}{m^6+36\,m^5+505\,m^4+3480\,m^3+12139\,m^2+19524\,m+10395} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(c^2*x^3*(e*x)^m*(3*A*a*d + A*b*c + B*a*c)*(5353*m + 2262*m^2 + 406*m^3 + 33*m^4 + m^5 + 3465))/(19524*m + 121
39*m^2 + 3480*m^3 + 505*m^4 + 36*m^5 + m^6 + 10395) + (d^2*x^9*(e*x)^m*(A*b*d + B*a*d + 3*B*b*c)*(2041*m + 112
2*m^2 + 262*m^3 + 27*m^4 + m^5 + 1155))/(19524*m + 12139*m^2 + 3480*m^3 + 505*m^4 + 36*m^5 + m^6 + 10395) + (c
*x^5*(e*x)^m*(3*A*a*d^2 + B*b*c^2 + 3*A*b*c*d + 3*B*a*c*d)*(3489*m + 1730*m^2 + 350*m^3 + 31*m^4 + m^5 + 2079)
)/(19524*m + 12139*m^2 + 3480*m^3 + 505*m^4 + 36*m^5 + m^6 + 10395) + (d*x^7*(e*x)^m*(A*a*d^2 + 3*B*b*c^2 + 3*
A*b*c*d + 3*B*a*c*d)*(2577*m + 1366*m^2 + 302*m^3 + 29*m^4 + m^5 + 1485))/(19524*m + 12139*m^2 + 3480*m^3 + 50
5*m^4 + 36*m^5 + m^6 + 10395) + (B*b*d^3*x^11*(e*x)^m*(1689*m + 950*m^2 + 230*m^3 + 25*m^4 + m^5 + 945))/(1952
4*m + 12139*m^2 + 3480*m^3 + 505*m^4 + 36*m^5 + m^6 + 10395) + (A*a*c^3*x*(e*x)^m*(9129*m + 3010*m^2 + 470*m^3
+ 35*m^4 + m^5 + 10395))/(19524*m + 12139*m^2 + 3480*m^3 + 505*m^4 + 36*m^5 + m^6 + 10395)

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