3.10 \(\int (e x)^m (a+b x^2) (A+B x^2) (c+d x^2)^2 \, dx\)
Optimal. Leaf size=144 \[ \frac {d (e x)^{m+7} (a B d+A b d+2 b B c)}{e^7 (m+7)}+\frac {(e x)^{m+5} (a d (A d+2 B c)+b c (2 A d+B c))}{e^5 (m+5)}+\frac {c (e x)^{m+3} (2 a A d+a B c+A b c)}{e^3 (m+3)}+\frac {a A c^2 (e x)^{m+1}}{e (m+1)}+\frac {b B d^2 (e x)^{m+9}}{e^9 (m+9)} \]
[Out]
a*A*c^2*(e*x)^(1+m)/e/(1+m)+c*(2*A*a*d+A*b*c+B*a*c)*(e*x)^(3+m)/e^3/(3+m)+(a*d*(A*d+2*B*c)+b*c*(2*A*d+B*c))*(e
*x)^(5+m)/e^5/(5+m)+d*(A*b*d+B*a*d+2*B*b*c)*(e*x)^(7+m)/e^7/(7+m)+b*B*d^2*(e*x)^(9+m)/e^9/(9+m)
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Rubi [A] time = 0.13, antiderivative size = 144, 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 =
{570} \[ \frac {c (e x)^{m+3} (2 a A d+a B c+A b c)}{e^3 (m+3)}+\frac {(e x)^{m+5} (a d (A d+2 B c)+b c (2 A d+B c))}{e^5 (m+5)}+\frac {d (e x)^{m+7} (a B d+A b d+2 b B c)}{e^7 (m+7)}+\frac {a A c^2 (e x)^{m+1}}{e (m+1)}+\frac {b B d^2 (e x)^{m+9}}{e^9 (m+9)} \]
Antiderivative was successfully verified.
[In]
Int[(e*x)^m*(a + b*x^2)*(A + B*x^2)*(c + d*x^2)^2,x]
[Out]
(a*A*c^2*(e*x)^(1 + m))/(e*(1 + m)) + (c*(A*b*c + a*B*c + 2*a*A*d)*(e*x)^(3 + m))/(e^3*(3 + m)) + ((a*d*(2*B*c
+ A*d) + b*c*(B*c + 2*A*d))*(e*x)^(5 + m))/(e^5*(5 + m)) + (d*(2*b*B*c + A*b*d + a*B*d)*(e*x)^(7 + m))/(e^7*(
7 + m)) + (b*B*d^2*(e*x)^(9 + m))/(e^9*(9 + m))
Rule 570
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 )^2 \, dx &=\int \left (a A c^2 (e x)^m+\frac {c (A b c+a B c+2 a A d) (e x)^{2+m}}{e^2}+\frac {(a d (2 B c+A d)+b c (B c+2 A d)) (e x)^{4+m}}{e^4}+\frac {d (2 b B c+A b d+a B d) (e x)^{6+m}}{e^6}+\frac {b B d^2 (e x)^{8+m}}{e^8}\right ) \, dx\\ &=\frac {a A c^2 (e x)^{1+m}}{e (1+m)}+\frac {c (A b c+a B c+2 a A d) (e x)^{3+m}}{e^3 (3+m)}+\frac {(a d (2 B c+A d)+b c (B c+2 A d)) (e x)^{5+m}}{e^5 (5+m)}+\frac {d (2 b B c+A b d+a B d) (e x)^{7+m}}{e^7 (7+m)}+\frac {b B d^2 (e x)^{9+m}}{e^9 (9+m)}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 113, normalized size = 0.78 \[ x (e x)^m \left (\frac {d x^6 (a B d+A b d+2 b B c)}{m+7}+\frac {x^4 (a d (A d+2 B c)+b c (2 A d+B c))}{m+5}+\frac {c x^2 (2 a A d+a B c+A b c)}{m+3}+\frac {a A c^2}{m+1}+\frac {b B d^2 x^8}{m+9}\right ) \]
Antiderivative was successfully verified.
[In]
Integrate[(e*x)^m*(a + b*x^2)*(A + B*x^2)*(c + d*x^2)^2,x]
[Out]
x*(e*x)^m*((a*A*c^2)/(1 + m) + (c*(A*b*c + a*B*c + 2*a*A*d)*x^2)/(3 + m) + ((a*d*(2*B*c + A*d) + b*c*(B*c + 2*
A*d))*x^4)/(5 + m) + (d*(2*b*B*c + A*b*d + a*B*d)*x^6)/(7 + m) + (b*B*d^2*x^8)/(9 + m))
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fricas [B] time = 0.73, size = 495, normalized size = 3.44 \[ \frac {{\left ({\left (B b d^{2} m^{4} + 16 \, B b d^{2} m^{3} + 86 \, B b d^{2} m^{2} + 176 \, B b d^{2} m + 105 \, B b d^{2}\right )} x^{9} + {\left ({\left (2 \, B b c d + {\left (B a + A b\right )} d^{2}\right )} m^{4} + 270 \, B b c d + 18 \, {\left (2 \, B b c d + {\left (B a + A b\right )} d^{2}\right )} m^{3} + 135 \, {\left (B a + A b\right )} d^{2} + 104 \, {\left (2 \, B b c d + {\left (B a + A b\right )} d^{2}\right )} m^{2} + 222 \, {\left (2 \, B b c d + {\left (B a + A b\right )} d^{2}\right )} m\right )} x^{7} + {\left ({\left (B b c^{2} + A a d^{2} + 2 \, {\left (B a + A b\right )} c d\right )} m^{4} + 189 \, B b c^{2} + 189 \, A a d^{2} + 20 \, {\left (B b c^{2} + A a d^{2} + 2 \, {\left (B a + A b\right )} c d\right )} m^{3} + 378 \, {\left (B a + A b\right )} c d + 130 \, {\left (B b c^{2} + A a d^{2} + 2 \, {\left (B a + A b\right )} c d\right )} m^{2} + 300 \, {\left (B b c^{2} + A a d^{2} + 2 \, {\left (B a + A b\right )} c d\right )} m\right )} x^{5} + {\left ({\left (2 \, A a c d + {\left (B a + A b\right )} c^{2}\right )} m^{4} + 630 \, A a c d + 22 \, {\left (2 \, A a c d + {\left (B a + A b\right )} c^{2}\right )} m^{3} + 315 \, {\left (B a + A b\right )} c^{2} + 164 \, {\left (2 \, A a c d + {\left (B a + A b\right )} c^{2}\right )} m^{2} + 458 \, {\left (2 \, A a c d + {\left (B a + A b\right )} c^{2}\right )} m\right )} x^{3} + {\left (A a c^{2} m^{4} + 24 \, A a c^{2} m^{3} + 206 \, A a c^{2} m^{2} + 744 \, A a c^{2} m + 945 \, A a c^{2}\right )} x\right )} \left (e x\right )^{m}}{m^{5} + 25 \, m^{4} + 230 \, m^{3} + 950 \, m^{2} + 1689 \, m + 945} \]
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)^2,x, algorithm="fricas")
[Out]
((B*b*d^2*m^4 + 16*B*b*d^2*m^3 + 86*B*b*d^2*m^2 + 176*B*b*d^2*m + 105*B*b*d^2)*x^9 + ((2*B*b*c*d + (B*a + A*b)
*d^2)*m^4 + 270*B*b*c*d + 18*(2*B*b*c*d + (B*a + A*b)*d^2)*m^3 + 135*(B*a + A*b)*d^2 + 104*(2*B*b*c*d + (B*a +
A*b)*d^2)*m^2 + 222*(2*B*b*c*d + (B*a + A*b)*d^2)*m)*x^7 + ((B*b*c^2 + A*a*d^2 + 2*(B*a + A*b)*c*d)*m^4 + 189
*B*b*c^2 + 189*A*a*d^2 + 20*(B*b*c^2 + A*a*d^2 + 2*(B*a + A*b)*c*d)*m^3 + 378*(B*a + A*b)*c*d + 130*(B*b*c^2 +
A*a*d^2 + 2*(B*a + A*b)*c*d)*m^2 + 300*(B*b*c^2 + A*a*d^2 + 2*(B*a + A*b)*c*d)*m)*x^5 + ((2*A*a*c*d + (B*a +
A*b)*c^2)*m^4 + 630*A*a*c*d + 22*(2*A*a*c*d + (B*a + A*b)*c^2)*m^3 + 315*(B*a + A*b)*c^2 + 164*(2*A*a*c*d + (B
*a + A*b)*c^2)*m^2 + 458*(2*A*a*c*d + (B*a + A*b)*c^2)*m)*x^3 + (A*a*c^2*m^4 + 24*A*a*c^2*m^3 + 206*A*a*c^2*m^
2 + 744*A*a*c^2*m + 945*A*a*c^2)*x)*(e*x)^m/(m^5 + 25*m^4 + 230*m^3 + 950*m^2 + 1689*m + 945)
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giac [B] time = 0.64, size = 1009, normalized size = 7.01 \[ \frac {B b d^{2} m^{4} x^{9} x^{m} e^{m} + 16 \, B b d^{2} m^{3} x^{9} x^{m} e^{m} + 2 \, B b c d m^{4} x^{7} x^{m} e^{m} + B a d^{2} m^{4} x^{7} x^{m} e^{m} + A b d^{2} m^{4} x^{7} x^{m} e^{m} + 86 \, B b d^{2} m^{2} x^{9} x^{m} e^{m} + 36 \, B b c d m^{3} x^{7} x^{m} e^{m} + 18 \, B a d^{2} m^{3} x^{7} x^{m} e^{m} + 18 \, A b d^{2} m^{3} x^{7} x^{m} e^{m} + 176 \, B b d^{2} m x^{9} x^{m} e^{m} + B b c^{2} m^{4} x^{5} x^{m} e^{m} + 2 \, B a c d m^{4} x^{5} x^{m} e^{m} + 2 \, A b c d m^{4} x^{5} x^{m} e^{m} + A a d^{2} m^{4} x^{5} x^{m} e^{m} + 208 \, B b c d m^{2} x^{7} x^{m} e^{m} + 104 \, B a d^{2} m^{2} x^{7} x^{m} e^{m} + 104 \, A b d^{2} m^{2} x^{7} x^{m} e^{m} + 105 \, B b d^{2} x^{9} x^{m} e^{m} + 20 \, B b c^{2} m^{3} x^{5} x^{m} e^{m} + 40 \, B a c d m^{3} x^{5} x^{m} e^{m} + 40 \, A b c d m^{3} x^{5} x^{m} e^{m} + 20 \, A a d^{2} m^{3} x^{5} x^{m} e^{m} + 444 \, B b c d m x^{7} x^{m} e^{m} + 222 \, B a d^{2} m x^{7} x^{m} e^{m} + 222 \, A b d^{2} m x^{7} x^{m} e^{m} + B a c^{2} m^{4} x^{3} x^{m} e^{m} + A b c^{2} m^{4} x^{3} x^{m} e^{m} + 2 \, A a c d m^{4} x^{3} x^{m} e^{m} + 130 \, B b c^{2} m^{2} x^{5} x^{m} e^{m} + 260 \, B a c d m^{2} x^{5} x^{m} e^{m} + 260 \, A b c d m^{2} x^{5} x^{m} e^{m} + 130 \, A a d^{2} m^{2} x^{5} x^{m} e^{m} + 270 \, B b c d x^{7} x^{m} e^{m} + 135 \, B a d^{2} x^{7} x^{m} e^{m} + 135 \, A b d^{2} x^{7} x^{m} e^{m} + 22 \, B a c^{2} m^{3} x^{3} x^{m} e^{m} + 22 \, A b c^{2} m^{3} x^{3} x^{m} e^{m} + 44 \, A a c d m^{3} x^{3} x^{m} e^{m} + 300 \, B b c^{2} m x^{5} x^{m} e^{m} + 600 \, B a c d m x^{5} x^{m} e^{m} + 600 \, A b c d m x^{5} x^{m} e^{m} + 300 \, A a d^{2} m x^{5} x^{m} e^{m} + A a c^{2} m^{4} x x^{m} e^{m} + 164 \, B a c^{2} m^{2} x^{3} x^{m} e^{m} + 164 \, A b c^{2} m^{2} x^{3} x^{m} e^{m} + 328 \, A a c d m^{2} x^{3} x^{m} e^{m} + 189 \, B b c^{2} x^{5} x^{m} e^{m} + 378 \, B a c d x^{5} x^{m} e^{m} + 378 \, A b c d x^{5} x^{m} e^{m} + 189 \, A a d^{2} x^{5} x^{m} e^{m} + 24 \, A a c^{2} m^{3} x x^{m} e^{m} + 458 \, B a c^{2} m x^{3} x^{m} e^{m} + 458 \, A b c^{2} m x^{3} x^{m} e^{m} + 916 \, A a c d m x^{3} x^{m} e^{m} + 206 \, A a c^{2} m^{2} x x^{m} e^{m} + 315 \, B a c^{2} x^{3} x^{m} e^{m} + 315 \, A b c^{2} x^{3} x^{m} e^{m} + 630 \, A a c d x^{3} x^{m} e^{m} + 744 \, A a c^{2} m x x^{m} e^{m} + 945 \, A a c^{2} x x^{m} e^{m}}{m^{5} + 25 \, m^{4} + 230 \, m^{3} + 950 \, m^{2} + 1689 \, m + 945} \]
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)^2,x, algorithm="giac")
[Out]
(B*b*d^2*m^4*x^9*x^m*e^m + 16*B*b*d^2*m^3*x^9*x^m*e^m + 2*B*b*c*d*m^4*x^7*x^m*e^m + B*a*d^2*m^4*x^7*x^m*e^m +
A*b*d^2*m^4*x^7*x^m*e^m + 86*B*b*d^2*m^2*x^9*x^m*e^m + 36*B*b*c*d*m^3*x^7*x^m*e^m + 18*B*a*d^2*m^3*x^7*x^m*e^m
+ 18*A*b*d^2*m^3*x^7*x^m*e^m + 176*B*b*d^2*m*x^9*x^m*e^m + B*b*c^2*m^4*x^5*x^m*e^m + 2*B*a*c*d*m^4*x^5*x^m*e^
m + 2*A*b*c*d*m^4*x^5*x^m*e^m + A*a*d^2*m^4*x^5*x^m*e^m + 208*B*b*c*d*m^2*x^7*x^m*e^m + 104*B*a*d^2*m^2*x^7*x^
m*e^m + 104*A*b*d^2*m^2*x^7*x^m*e^m + 105*B*b*d^2*x^9*x^m*e^m + 20*B*b*c^2*m^3*x^5*x^m*e^m + 40*B*a*c*d*m^3*x^
5*x^m*e^m + 40*A*b*c*d*m^3*x^5*x^m*e^m + 20*A*a*d^2*m^3*x^5*x^m*e^m + 444*B*b*c*d*m*x^7*x^m*e^m + 222*B*a*d^2*
m*x^7*x^m*e^m + 222*A*b*d^2*m*x^7*x^m*e^m + B*a*c^2*m^4*x^3*x^m*e^m + A*b*c^2*m^4*x^3*x^m*e^m + 2*A*a*c*d*m^4*
x^3*x^m*e^m + 130*B*b*c^2*m^2*x^5*x^m*e^m + 260*B*a*c*d*m^2*x^5*x^m*e^m + 260*A*b*c*d*m^2*x^5*x^m*e^m + 130*A*
a*d^2*m^2*x^5*x^m*e^m + 270*B*b*c*d*x^7*x^m*e^m + 135*B*a*d^2*x^7*x^m*e^m + 135*A*b*d^2*x^7*x^m*e^m + 22*B*a*c
^2*m^3*x^3*x^m*e^m + 22*A*b*c^2*m^3*x^3*x^m*e^m + 44*A*a*c*d*m^3*x^3*x^m*e^m + 300*B*b*c^2*m*x^5*x^m*e^m + 600
*B*a*c*d*m*x^5*x^m*e^m + 600*A*b*c*d*m*x^5*x^m*e^m + 300*A*a*d^2*m*x^5*x^m*e^m + A*a*c^2*m^4*x*x^m*e^m + 164*B
*a*c^2*m^2*x^3*x^m*e^m + 164*A*b*c^2*m^2*x^3*x^m*e^m + 328*A*a*c*d*m^2*x^3*x^m*e^m + 189*B*b*c^2*x^5*x^m*e^m +
378*B*a*c*d*x^5*x^m*e^m + 378*A*b*c*d*x^5*x^m*e^m + 189*A*a*d^2*x^5*x^m*e^m + 24*A*a*c^2*m^3*x*x^m*e^m + 458*
B*a*c^2*m*x^3*x^m*e^m + 458*A*b*c^2*m*x^3*x^m*e^m + 916*A*a*c*d*m*x^3*x^m*e^m + 206*A*a*c^2*m^2*x*x^m*e^m + 31
5*B*a*c^2*x^3*x^m*e^m + 315*A*b*c^2*x^3*x^m*e^m + 630*A*a*c*d*x^3*x^m*e^m + 744*A*a*c^2*m*x*x^m*e^m + 945*A*a*
c^2*x*x^m*e^m)/(m^5 + 25*m^4 + 230*m^3 + 950*m^2 + 1689*m + 945)
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maple [B] time = 0.01, size = 711, normalized size = 4.94 \[ \frac {\left (B b \,d^{2} m^{4} x^{8}+16 B b \,d^{2} m^{3} x^{8}+A b \,d^{2} m^{4} x^{6}+B a \,d^{2} m^{4} x^{6}+2 B b c d \,m^{4} x^{6}+86 B b \,d^{2} m^{2} x^{8}+18 A b \,d^{2} m^{3} x^{6}+18 B a \,d^{2} m^{3} x^{6}+36 B b c d \,m^{3} x^{6}+176 B b \,d^{2} m \,x^{8}+A a \,d^{2} m^{4} x^{4}+2 A b c d \,m^{4} x^{4}+104 A b \,d^{2} m^{2} x^{6}+2 B a c d \,m^{4} x^{4}+104 B a \,d^{2} m^{2} x^{6}+B b \,c^{2} m^{4} x^{4}+208 B b c d \,m^{2} x^{6}+105 b B \,d^{2} x^{8}+20 A a \,d^{2} m^{3} x^{4}+40 A b c d \,m^{3} x^{4}+222 A b \,d^{2} m \,x^{6}+40 B a c d \,m^{3} x^{4}+222 B a \,d^{2} m \,x^{6}+20 B b \,c^{2} m^{3} x^{4}+444 B b c d m \,x^{6}+2 A a c d \,m^{4} x^{2}+130 A a \,d^{2} m^{2} x^{4}+A b \,c^{2} m^{4} x^{2}+260 A b c d \,m^{2} x^{4}+135 A b \,d^{2} x^{6}+B a \,c^{2} m^{4} x^{2}+260 B a c d \,m^{2} x^{4}+135 B a \,d^{2} x^{6}+130 B b \,c^{2} m^{2} x^{4}+270 B b c d \,x^{6}+44 A a c d \,m^{3} x^{2}+300 A a \,d^{2} m \,x^{4}+22 A b \,c^{2} m^{3} x^{2}+600 A b c d m \,x^{4}+22 B a \,c^{2} m^{3} x^{2}+600 B a c d m \,x^{4}+300 B b \,c^{2} m \,x^{4}+A a \,c^{2} m^{4}+328 A a c d \,m^{2} x^{2}+189 A a \,d^{2} x^{4}+164 A b \,c^{2} m^{2} x^{2}+378 A b c d \,x^{4}+164 B a \,c^{2} m^{2} x^{2}+378 B a c d \,x^{4}+189 B b \,c^{2} x^{4}+24 A a \,c^{2} m^{3}+916 A a c d m \,x^{2}+458 A b \,c^{2} m \,x^{2}+458 B a \,c^{2} m \,x^{2}+206 A a \,c^{2} m^{2}+630 A a c d \,x^{2}+315 A b \,c^{2} x^{2}+315 B a \,c^{2} x^{2}+744 A a \,c^{2} m +945 a A \,c^{2}\right ) x \left (e x \right )^{m}}{\left (m +9\right ) \left (m +7\right ) \left (m +5\right ) \left (m +3\right ) \left (m +1\right )} \]
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)^2,x)
[Out]
x*(B*b*d^2*m^4*x^8+16*B*b*d^2*m^3*x^8+A*b*d^2*m^4*x^6+B*a*d^2*m^4*x^6+2*B*b*c*d*m^4*x^6+86*B*b*d^2*m^2*x^8+18*
A*b*d^2*m^3*x^6+18*B*a*d^2*m^3*x^6+36*B*b*c*d*m^3*x^6+176*B*b*d^2*m*x^8+A*a*d^2*m^4*x^4+2*A*b*c*d*m^4*x^4+104*
A*b*d^2*m^2*x^6+2*B*a*c*d*m^4*x^4+104*B*a*d^2*m^2*x^6+B*b*c^2*m^4*x^4+208*B*b*c*d*m^2*x^6+105*B*b*d^2*x^8+20*A
*a*d^2*m^3*x^4+40*A*b*c*d*m^3*x^4+222*A*b*d^2*m*x^6+40*B*a*c*d*m^3*x^4+222*B*a*d^2*m*x^6+20*B*b*c^2*m^3*x^4+44
4*B*b*c*d*m*x^6+2*A*a*c*d*m^4*x^2+130*A*a*d^2*m^2*x^4+A*b*c^2*m^4*x^2+260*A*b*c*d*m^2*x^4+135*A*b*d^2*x^6+B*a*
c^2*m^4*x^2+260*B*a*c*d*m^2*x^4+135*B*a*d^2*x^6+130*B*b*c^2*m^2*x^4+270*B*b*c*d*x^6+44*A*a*c*d*m^3*x^2+300*A*a
*d^2*m*x^4+22*A*b*c^2*m^3*x^2+600*A*b*c*d*m*x^4+22*B*a*c^2*m^3*x^2+600*B*a*c*d*m*x^4+300*B*b*c^2*m*x^4+A*a*c^2
*m^4+328*A*a*c*d*m^2*x^2+189*A*a*d^2*x^4+164*A*b*c^2*m^2*x^2+378*A*b*c*d*x^4+164*B*a*c^2*m^2*x^2+378*B*a*c*d*x
^4+189*B*b*c^2*x^4+24*A*a*c^2*m^3+916*A*a*c*d*m*x^2+458*A*b*c^2*m*x^2+458*B*a*c^2*m*x^2+206*A*a*c^2*m^2+630*A*
a*c*d*x^2+315*A*b*c^2*x^2+315*B*a*c^2*x^2+744*A*a*c^2*m+945*A*a*c^2)*(e*x)^m/(m+9)/(m+7)/(m+5)/(m+3)/(m+1)
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maxima [A] time = 1.80, size = 242, normalized size = 1.68 \[ \frac {B b d^{2} e^{m} x^{9} x^{m}}{m + 9} + \frac {2 \, B b c d e^{m} x^{7} x^{m}}{m + 7} + \frac {B a d^{2} e^{m} x^{7} x^{m}}{m + 7} + \frac {A b d^{2} e^{m} x^{7} x^{m}}{m + 7} + \frac {B b c^{2} e^{m} x^{5} x^{m}}{m + 5} + \frac {2 \, B a c d e^{m} x^{5} x^{m}}{m + 5} + \frac {2 \, A b c d e^{m} x^{5} x^{m}}{m + 5} + \frac {A a d^{2} e^{m} x^{5} x^{m}}{m + 5} + \frac {B a c^{2} e^{m} x^{3} x^{m}}{m + 3} + \frac {A b c^{2} e^{m} x^{3} x^{m}}{m + 3} + \frac {2 \, A a c d e^{m} x^{3} x^{m}}{m + 3} + \frac {\left (e x\right )^{m + 1} A a c^{2}}{e {\left (m + 1\right )}} \]
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)^2,x, algorithm="maxima")
[Out]
B*b*d^2*e^m*x^9*x^m/(m + 9) + 2*B*b*c*d*e^m*x^7*x^m/(m + 7) + B*a*d^2*e^m*x^7*x^m/(m + 7) + A*b*d^2*e^m*x^7*x^
m/(m + 7) + B*b*c^2*e^m*x^5*x^m/(m + 5) + 2*B*a*c*d*e^m*x^5*x^m/(m + 5) + 2*A*b*c*d*e^m*x^5*x^m/(m + 5) + A*a*
d^2*e^m*x^5*x^m/(m + 5) + B*a*c^2*e^m*x^3*x^m/(m + 3) + A*b*c^2*e^m*x^3*x^m/(m + 3) + 2*A*a*c*d*e^m*x^3*x^m/(m
+ 3) + (e*x)^(m + 1)*A*a*c^2/(e*(m + 1))
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mupad [B] time = 1.18, size = 305, normalized size = 2.12 \[ {\left (e\,x\right )}^m\,\left (\frac {x^5\,\left (A\,a\,d^2+B\,b\,c^2+2\,A\,b\,c\,d+2\,B\,a\,c\,d\right )\,\left (m^4+20\,m^3+130\,m^2+300\,m+189\right )}{m^5+25\,m^4+230\,m^3+950\,m^2+1689\,m+945}+\frac {c\,x^3\,\left (2\,A\,a\,d+A\,b\,c+B\,a\,c\right )\,\left (m^4+22\,m^3+164\,m^2+458\,m+315\right )}{m^5+25\,m^4+230\,m^3+950\,m^2+1689\,m+945}+\frac {d\,x^7\,\left (A\,b\,d+B\,a\,d+2\,B\,b\,c\right )\,\left (m^4+18\,m^3+104\,m^2+222\,m+135\right )}{m^5+25\,m^4+230\,m^3+950\,m^2+1689\,m+945}+\frac {A\,a\,c^2\,x\,\left (m^4+24\,m^3+206\,m^2+744\,m+945\right )}{m^5+25\,m^4+230\,m^3+950\,m^2+1689\,m+945}+\frac {B\,b\,d^2\,x^9\,\left (m^4+16\,m^3+86\,m^2+176\,m+105\right )}{m^5+25\,m^4+230\,m^3+950\,m^2+1689\,m+945}\right ) \]
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)^2,x)
[Out]
(e*x)^m*((x^5*(A*a*d^2 + B*b*c^2 + 2*A*b*c*d + 2*B*a*c*d)*(300*m + 130*m^2 + 20*m^3 + m^4 + 189))/(1689*m + 95
0*m^2 + 230*m^3 + 25*m^4 + m^5 + 945) + (c*x^3*(2*A*a*d + A*b*c + B*a*c)*(458*m + 164*m^2 + 22*m^3 + m^4 + 315
))/(1689*m + 950*m^2 + 230*m^3 + 25*m^4 + m^5 + 945) + (d*x^7*(A*b*d + B*a*d + 2*B*b*c)*(222*m + 104*m^2 + 18*
m^3 + m^4 + 135))/(1689*m + 950*m^2 + 230*m^3 + 25*m^4 + m^5 + 945) + (A*a*c^2*x*(744*m + 206*m^2 + 24*m^3 + m
^4 + 945))/(1689*m + 950*m^2 + 230*m^3 + 25*m^4 + m^5 + 945) + (B*b*d^2*x^9*(176*m + 86*m^2 + 16*m^3 + m^4 + 1
05))/(1689*m + 950*m^2 + 230*m^3 + 25*m^4 + m^5 + 945))
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sympy [A] time = 5.66, size = 3373, normalized size = 23.42 \[ \text {result too large to display} \]
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)**2,x)
[Out]
Piecewise(((-A*a*c**2/(8*x**8) - A*a*c*d/(3*x**6) - A*a*d**2/(4*x**4) - A*b*c**2/(6*x**6) - A*b*c*d/(2*x**4) -
A*b*d**2/(2*x**2) - B*a*c**2/(6*x**6) - B*a*c*d/(2*x**4) - B*a*d**2/(2*x**2) - B*b*c**2/(4*x**4) - B*b*c*d/x*
*2 + B*b*d**2*log(x))/e**9, Eq(m, -9)), ((-A*a*c**2/(6*x**6) - A*a*c*d/(2*x**4) - A*a*d**2/(2*x**2) - A*b*c**2
/(4*x**4) - A*b*c*d/x**2 + A*b*d**2*log(x) - B*a*c**2/(4*x**4) - B*a*c*d/x**2 + B*a*d**2*log(x) - B*b*c**2/(2*
x**2) + 2*B*b*c*d*log(x) + B*b*d**2*x**2/2)/e**7, Eq(m, -7)), ((-A*a*c**2/(4*x**4) - A*a*c*d/x**2 + A*a*d**2*l
og(x) - A*b*c**2/(2*x**2) + 2*A*b*c*d*log(x) + A*b*d**2*x**2/2 - B*a*c**2/(2*x**2) + 2*B*a*c*d*log(x) + B*a*d*
*2*x**2/2 + B*b*c**2*log(x) + B*b*c*d*x**2 + B*b*d**2*x**4/4)/e**5, Eq(m, -5)), ((-A*a*c**2/(2*x**2) + 2*A*a*c
*d*log(x) + A*a*d**2*x**2/2 + A*b*c**2*log(x) + A*b*c*d*x**2 + A*b*d**2*x**4/4 + B*a*c**2*log(x) + B*a*c*d*x**
2 + B*a*d**2*x**4/4 + B*b*c**2*x**2/2 + B*b*c*d*x**4/2 + B*b*d**2*x**6/6)/e**3, Eq(m, -3)), ((A*a*c**2*log(x)
+ A*a*c*d*x**2 + A*a*d**2*x**4/4 + A*b*c**2*x**2/2 + A*b*c*d*x**4/2 + A*b*d**2*x**6/6 + B*a*c**2*x**2/2 + B*a*
c*d*x**4/2 + B*a*d**2*x**6/6 + B*b*c**2*x**4/4 + B*b*c*d*x**6/3 + B*b*d**2*x**8/8)/e, Eq(m, -1)), (A*a*c**2*e*
*m*m**4*x*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 24*A*a*c**2*e**m*m**3*x*x**m/(m**5 + 25
*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 206*A*a*c**2*e**m*m**2*x*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m
**2 + 1689*m + 945) + 744*A*a*c**2*e**m*m*x*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 945*A
*a*c**2*e**m*x*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 2*A*a*c*d*e**m*m**4*x**3*x**m/(m**
5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 44*A*a*c*d*e**m*m**3*x**3*x**m/(m**5 + 25*m**4 + 230*m**3
+ 950*m**2 + 1689*m + 945) + 328*A*a*c*d*e**m*m**2*x**3*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m +
945) + 916*A*a*c*d*e**m*m*x**3*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 630*A*a*c*d*e**m*x
**3*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + A*a*d**2*e**m*m**4*x**5*x**m/(m**5 + 25*m**4
+ 230*m**3 + 950*m**2 + 1689*m + 945) + 20*A*a*d**2*e**m*m**3*x**5*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2
+ 1689*m + 945) + 130*A*a*d**2*e**m*m**2*x**5*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 300
*A*a*d**2*e**m*m*x**5*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 189*A*a*d**2*e**m*x**5*x**m
/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + A*b*c**2*e**m*m**4*x**3*x**m/(m**5 + 25*m**4 + 230*m*
*3 + 950*m**2 + 1689*m + 945) + 22*A*b*c**2*e**m*m**3*x**3*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m
+ 945) + 164*A*b*c**2*e**m*m**2*x**3*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 458*A*b*c**
2*e**m*m*x**3*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 315*A*b*c**2*e**m*x**3*x**m/(m**5 +
25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 2*A*b*c*d*e**m*m**4*x**5*x**m/(m**5 + 25*m**4 + 230*m**3 + 95
0*m**2 + 1689*m + 945) + 40*A*b*c*d*e**m*m**3*x**5*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945)
+ 260*A*b*c*d*e**m*m**2*x**5*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 600*A*b*c*d*e**m*m*x
**5*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 378*A*b*c*d*e**m*x**5*x**m/(m**5 + 25*m**4 +
230*m**3 + 950*m**2 + 1689*m + 945) + A*b*d**2*e**m*m**4*x**7*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 168
9*m + 945) + 18*A*b*d**2*e**m*m**3*x**7*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 104*A*b*d
**2*e**m*m**2*x**7*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 222*A*b*d**2*e**m*m*x**7*x**m/
(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 135*A*b*d**2*e**m*x**7*x**m/(m**5 + 25*m**4 + 230*m**3
+ 950*m**2 + 1689*m + 945) + B*a*c**2*e**m*m**4*x**3*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 94
5) + 22*B*a*c**2*e**m*m**3*x**3*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 164*B*a*c**2*e**m
*m**2*x**3*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 458*B*a*c**2*e**m*m*x**3*x**m/(m**5 +
25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 315*B*a*c**2*e**m*x**3*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m
**2 + 1689*m + 945) + 2*B*a*c*d*e**m*m**4*x**5*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 40
*B*a*c*d*e**m*m**3*x**5*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 260*B*a*c*d*e**m*m**2*x**
5*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 600*B*a*c*d*e**m*m*x**5*x**m/(m**5 + 25*m**4 +
230*m**3 + 950*m**2 + 1689*m + 945) + 378*B*a*c*d*e**m*x**5*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*
m + 945) + B*a*d**2*e**m*m**4*x**7*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 18*B*a*d**2*e*
*m*m**3*x**7*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 104*B*a*d**2*e**m*m**2*x**7*x**m/(m*
*5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 222*B*a*d**2*e**m*m*x**7*x**m/(m**5 + 25*m**4 + 230*m**3
+ 950*m**2 + 1689*m + 945) + 135*B*a*d**2*e**m*x**7*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945)
+ B*b*c**2*e**m*m**4*x**5*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 20*B*b*c**2*e**m*m**3*
x**5*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 130*B*b*c**2*e**m*m**2*x**5*x**m/(m**5 + 25*
m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 300*B*b*c**2*e**m*m*x**5*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m*
*2 + 1689*m + 945) + 189*B*b*c**2*e**m*x**5*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 2*B*b
*c*d*e**m*m**4*x**7*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 36*B*b*c*d*e**m*m**3*x**7*x**
m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 208*B*b*c*d*e**m*m**2*x**7*x**m/(m**5 + 25*m**4 + 23
0*m**3 + 950*m**2 + 1689*m + 945) + 444*B*b*c*d*e**m*m*x**7*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*
m + 945) + 270*B*b*c*d*e**m*x**7*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + B*b*d**2*e**m*m*
*4*x**9*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 16*B*b*d**2*e**m*m**3*x**9*x**m/(m**5 + 2
5*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 86*B*b*d**2*e**m*m**2*x**9*x**m/(m**5 + 25*m**4 + 230*m**3 + 95
0*m**2 + 1689*m + 945) + 176*B*b*d**2*e**m*m*x**9*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) +
105*B*b*d**2*e**m*x**9*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945), True))
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