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

Optimal. Leaf size=144 \[ \frac {a^2 A c (e x)^{1+m}}{e (1+m)}+\frac {a (2 A b c+a B c+a A d) (e x)^{3+m}}{e^3 (3+m)}+\frac {(a B (2 b c+a d)+A b (b c+2 a d)) (e x)^{5+m}}{e^5 (5+m)}+\frac {b (b B c+A b d+2 a B d) (e x)^{7+m}}{e^7 (7+m)}+\frac {b^2 B d (e x)^{9+m}}{e^9 (9+m)} \]

[Out]

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

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Rubi [A]
time = 0.08, 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 = {584} \begin {gather*} \frac {a^2 A c (e x)^{m+1}}{e (m+1)}+\frac {b (e x)^{m+7} (2 a B d+A b d+b B c)}{e^7 (m+7)}+\frac {(e x)^{m+5} (A b (2 a d+b c)+a B (a d+2 b c))}{e^5 (m+5)}+\frac {a (e x)^{m+3} (a A d+a B c+2 A b c)}{e^3 (m+3)}+\frac {b^2 B d (e x)^{m+9}}{e^9 (m+9)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]
time = 0.14, size = 113, normalized size = 0.78 \begin {gather*} x (e x)^m \left (\frac {a^2 A c}{1+m}+\frac {a (2 A b c+a B c+a A d) x^2}{3+m}+\frac {(a B (2 b c+a d)+A b (b c+2 a d)) x^4}{5+m}+\frac {b (b B c+A b d+2 a B d) x^6}{7+m}+\frac {b^2 B d x^8}{9+m}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(710\) vs. \(2(144)=288\).
time = 0.11, size = 711, normalized size = 4.94

method	result	size
gosper	\(\frac {x \left (B \,b^{2} d \,m^{4} x^{8}+16 B \,b^{2} d \,m^{3} x^{8}+A \,b^{2} d \,m^{4} x^{6}+2 B a b d \,m^{4} x^{6}+B \,b^{2} c \,m^{4} x^{6}+86 B \,b^{2} d \,m^{2} x^{8}+18 A \,b^{2} d \,m^{3} x^{6}+36 B a b d \,m^{3} x^{6}+18 B \,b^{2} c \,m^{3} x^{6}+176 m \,x^{8} B d \,b^{2}+2 A a b d \,m^{4} x^{4}+A \,b^{2} c \,m^{4} x^{4}+104 A \,b^{2} d \,m^{2} x^{6}+B \,a^{2} d \,m^{4} x^{4}+2 B a b c \,m^{4} x^{4}+208 B a b d \,m^{2} x^{6}+104 B \,b^{2} c \,m^{2} x^{6}+105 B d \,b^{2} x^{8}+40 A a b d \,m^{3} x^{4}+20 A \,b^{2} c \,m^{3} x^{4}+222 A \,b^{2} d \,x^{6} m +20 B \,a^{2} d \,m^{3} x^{4}+40 B a b c \,m^{3} x^{4}+444 B a b d \,x^{6} m +222 B \,b^{2} c \,x^{6} m +A \,a^{2} d \,m^{4} x^{2}+2 A a b c \,m^{4} x^{2}+260 A a b d \,m^{2} x^{4}+130 A \,b^{2} c \,m^{2} x^{4}+135 A \,b^{2} d \,x^{6}+B \,a^{2} c \,m^{4} x^{2}+130 B \,a^{2} d \,m^{2} x^{4}+260 B a b c \,m^{2} x^{4}+270 B a b d \,x^{6}+135 B \,b^{2} c \,x^{6}+22 A \,a^{2} d \,m^{3} x^{2}+44 A a b c \,m^{3} x^{2}+600 A a b d \,x^{4} m +300 A \,b^{2} c \,x^{4} m +22 B \,a^{2} c \,m^{3} x^{2}+300 B \,a^{2} d \,x^{4} m +600 B a b c \,x^{4} m +A \,a^{2} c \,m^{4}+164 A \,a^{2} d \,m^{2} x^{2}+328 A a b c \,m^{2} x^{2}+378 A a b d \,x^{4}+189 A \,b^{2} c \,x^{4}+164 B \,a^{2} c \,m^{2} x^{2}+189 B \,a^{2} d \,x^{4}+378 B a b c \,x^{4}+24 A \,a^{2} c \,m^{3}+458 A \,a^{2} d \,x^{2} m +916 A a b c \,x^{2} m +458 B \,a^{2} c \,x^{2} m +206 A \,a^{2} c \,m^{2}+315 A \,a^{2} d \,x^{2}+630 A a b c \,x^{2}+315 B \,a^{2} c \,x^{2}+744 A c \,a^{2} m +945 A c \,a^{2}\right ) \left (e x \right )^{m}}{\left (9+m \right ) \left (7+m \right ) \left (5+m \right ) \left (3+m \right ) \left (1+m \right )}\)	\(711\)
risch	\(\frac {x \left (B \,b^{2} d \,m^{4} x^{8}+16 B \,b^{2} d \,m^{3} x^{8}+A \,b^{2} d \,m^{4} x^{6}+2 B a b d \,m^{4} x^{6}+B \,b^{2} c \,m^{4} x^{6}+86 B \,b^{2} d \,m^{2} x^{8}+18 A \,b^{2} d \,m^{3} x^{6}+36 B a b d \,m^{3} x^{6}+18 B \,b^{2} c \,m^{3} x^{6}+176 m \,x^{8} B d \,b^{2}+2 A a b d \,m^{4} x^{4}+A \,b^{2} c \,m^{4} x^{4}+104 A \,b^{2} d \,m^{2} x^{6}+B \,a^{2} d \,m^{4} x^{4}+2 B a b c \,m^{4} x^{4}+208 B a b d \,m^{2} x^{6}+104 B \,b^{2} c \,m^{2} x^{6}+105 B d \,b^{2} x^{8}+40 A a b d \,m^{3} x^{4}+20 A \,b^{2} c \,m^{3} x^{4}+222 A \,b^{2} d \,x^{6} m +20 B \,a^{2} d \,m^{3} x^{4}+40 B a b c \,m^{3} x^{4}+444 B a b d \,x^{6} m +222 B \,b^{2} c \,x^{6} m +A \,a^{2} d \,m^{4} x^{2}+2 A a b c \,m^{4} x^{2}+260 A a b d \,m^{2} x^{4}+130 A \,b^{2} c \,m^{2} x^{4}+135 A \,b^{2} d \,x^{6}+B \,a^{2} c \,m^{4} x^{2}+130 B \,a^{2} d \,m^{2} x^{4}+260 B a b c \,m^{2} x^{4}+270 B a b d \,x^{6}+135 B \,b^{2} c \,x^{6}+22 A \,a^{2} d \,m^{3} x^{2}+44 A a b c \,m^{3} x^{2}+600 A a b d \,x^{4} m +300 A \,b^{2} c \,x^{4} m +22 B \,a^{2} c \,m^{3} x^{2}+300 B \,a^{2} d \,x^{4} m +600 B a b c \,x^{4} m +A \,a^{2} c \,m^{4}+164 A \,a^{2} d \,m^{2} x^{2}+328 A a b c \,m^{2} x^{2}+378 A a b d \,x^{4}+189 A \,b^{2} c \,x^{4}+164 B \,a^{2} c \,m^{2} x^{2}+189 B \,a^{2} d \,x^{4}+378 B a b c \,x^{4}+24 A \,a^{2} c \,m^{3}+458 A \,a^{2} d \,x^{2} m +916 A a b c \,x^{2} m +458 B \,a^{2} c \,x^{2} m +206 A \,a^{2} c \,m^{2}+315 A \,a^{2} d \,x^{2}+630 A a b c \,x^{2}+315 B \,a^{2} c \,x^{2}+744 A c \,a^{2} m +945 A c \,a^{2}\right ) \left (e x \right )^{m}}{\left (9+m \right ) \left (7+m \right ) \left (5+m \right ) \left (3+m \right ) \left (1+m \right )}\)	\(711\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*(B*b^2*d*m^4*x^8+16*B*b^2*d*m^3*x^8+A*b^2*d*m^4*x^6+2*B*a*b*d*m^4*x^6+B*b^2*c*m^4*x^6+86*B*b^2*d*m^2*x^8+18*
A*b^2*d*m^3*x^6+36*B*a*b*d*m^3*x^6+18*B*b^2*c*m^3*x^6+176*B*b^2*d*m*x^8+2*A*a*b*d*m^4*x^4+A*b^2*c*m^4*x^4+104*
A*b^2*d*m^2*x^6+B*a^2*d*m^4*x^4+2*B*a*b*c*m^4*x^4+208*B*a*b*d*m^2*x^6+104*B*b^2*c*m^2*x^6+105*B*b^2*d*x^8+40*A
*a*b*d*m^3*x^4+20*A*b^2*c*m^3*x^4+222*A*b^2*d*m*x^6+20*B*a^2*d*m^3*x^4+40*B*a*b*c*m^3*x^4+444*B*a*b*d*m*x^6+22
2*B*b^2*c*m*x^6+A*a^2*d*m^4*x^2+2*A*a*b*c*m^4*x^2+260*A*a*b*d*m^2*x^4+130*A*b^2*c*m^2*x^4+135*A*b^2*d*x^6+B*a^
2*c*m^4*x^2+130*B*a^2*d*m^2*x^4+260*B*a*b*c*m^2*x^4+270*B*a*b*d*x^6+135*B*b^2*c*x^6+22*A*a^2*d*m^3*x^2+44*A*a*
b*c*m^3*x^2+600*A*a*b*d*m*x^4+300*A*b^2*c*m*x^4+22*B*a^2*c*m^3*x^2+300*B*a^2*d*m*x^4+600*B*a*b*c*m*x^4+A*a^2*c
*m^4+164*A*a^2*d*m^2*x^2+328*A*a*b*c*m^2*x^2+378*A*a*b*d*x^4+189*A*b^2*c*x^4+164*B*a^2*c*m^2*x^2+189*B*a^2*d*x
^4+378*B*a*b*c*x^4+24*A*a^2*c*m^3+458*A*a^2*d*m*x^2+916*A*a*b*c*m*x^2+458*B*a^2*c*m*x^2+206*A*a^2*c*m^2+315*A*
a^2*d*x^2+630*A*a*b*c*x^2+315*B*a^2*c*x^2+744*A*a^2*c*m+945*A*a^2*c)*(e*x)^m/(9+m)/(7+m)/(5+m)/(3+m)/(1+m)

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Maxima [A]
time = 0.29, size = 253, normalized size = 1.76 \begin {gather*} \frac {B b^{2} d x^{9} e^{\left (m \log \left (x\right ) + m\right )}}{m + 9} + \frac {B b^{2} c x^{7} e^{\left (m \log \left (x\right ) + m\right )}}{m + 7} + \frac {2 \, B a b d x^{7} e^{\left (m \log \left (x\right ) + m\right )}}{m + 7} + \frac {A b^{2} d x^{7} e^{\left (m \log \left (x\right ) + m\right )}}{m + 7} + \frac {2 \, B a b c x^{5} e^{\left (m \log \left (x\right ) + m\right )}}{m + 5} + \frac {A b^{2} c x^{5} e^{\left (m \log \left (x\right ) + m\right )}}{m + 5} + \frac {B a^{2} d x^{5} e^{\left (m \log \left (x\right ) + m\right )}}{m + 5} + \frac {2 \, A a b d x^{5} e^{\left (m \log \left (x\right ) + m\right )}}{m + 5} + \frac {B a^{2} c x^{3} e^{\left (m \log \left (x\right ) + m\right )}}{m + 3} + \frac {2 \, A a b c x^{3} e^{\left (m \log \left (x\right ) + m\right )}}{m + 3} + \frac {A a^{2} d x^{3} e^{\left (m \log \left (x\right ) + m\right )}}{m + 3} + \frac {\left (x e\right )^{m + 1} A a^{2} c 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)^2*(B*x^2+A)*(d*x^2+c),x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 533 vs. \(2 (144) = 288\).
time = 1.85, size = 533, normalized size = 3.70 \begin {gather*} \frac {{\left ({\left (B b^{2} d m^{4} + 16 \, B b^{2} d m^{3} + 86 \, B b^{2} d m^{2} + 176 \, B b^{2} d m + 105 \, B b^{2} d\right )} x^{9} + {\left ({\left (B b^{2} c + {\left (2 \, B a b + A b^{2}\right )} d\right )} m^{4} + 135 \, B b^{2} c + 18 \, {\left (B b^{2} c + {\left (2 \, B a b + A b^{2}\right )} d\right )} m^{3} + 104 \, {\left (B b^{2} c + {\left (2 \, B a b + A b^{2}\right )} d\right )} m^{2} + 135 \, {\left (2 \, B a b + A b^{2}\right )} d + 222 \, {\left (B b^{2} c + {\left (2 \, B a b + A b^{2}\right )} d\right )} m\right )} x^{7} + {\left ({\left ({\left (2 \, B a b + A b^{2}\right )} c + {\left (B a^{2} + 2 \, A a b\right )} d\right )} m^{4} + 20 \, {\left ({\left (2 \, B a b + A b^{2}\right )} c + {\left (B a^{2} + 2 \, A a b\right )} d\right )} m^{3} + 130 \, {\left ({\left (2 \, B a b + A b^{2}\right )} c + {\left (B a^{2} + 2 \, A a b\right )} d\right )} m^{2} + 189 \, {\left (2 \, B a b + A b^{2}\right )} c + 189 \, {\left (B a^{2} + 2 \, A a b\right )} d + 300 \, {\left ({\left (2 \, B a b + A b^{2}\right )} c + {\left (B a^{2} + 2 \, A a b\right )} d\right )} m\right )} x^{5} + {\left ({\left (A a^{2} d + {\left (B a^{2} + 2 \, A a b\right )} c\right )} m^{4} + 315 \, A a^{2} d + 22 \, {\left (A a^{2} d + {\left (B a^{2} + 2 \, A a b\right )} c\right )} m^{3} + 164 \, {\left (A a^{2} d + {\left (B a^{2} + 2 \, A a b\right )} c\right )} m^{2} + 315 \, {\left (B a^{2} + 2 \, A a b\right )} c + 458 \, {\left (A a^{2} d + {\left (B a^{2} + 2 \, A a b\right )} c\right )} m\right )} x^{3} + {\left (A a^{2} c m^{4} + 24 \, A a^{2} c m^{3} + 206 \, A a^{2} c m^{2} + 744 \, A a^{2} c m + 945 \, A a^{2} c\right )} x\right )} \left (x e\right )^{m}}{m^{5} + 25 \, m^{4} + 230 \, m^{3} + 950 \, m^{2} + 1689 \, m + 945} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((B*b^2*d*m^4 + 16*B*b^2*d*m^3 + 86*B*b^2*d*m^2 + 176*B*b^2*d*m + 105*B*b^2*d)*x^9 + ((B*b^2*c + (2*B*a*b + A*
b^2)*d)*m^4 + 135*B*b^2*c + 18*(B*b^2*c + (2*B*a*b + A*b^2)*d)*m^3 + 104*(B*b^2*c + (2*B*a*b + A*b^2)*d)*m^2 +
135*(2*B*a*b + A*b^2)*d + 222*(B*b^2*c + (2*B*a*b + A*b^2)*d)*m)*x^7 + (((2*B*a*b + A*b^2)*c + (B*a^2 + 2*A*a
*b)*d)*m^4 + 20*((2*B*a*b + A*b^2)*c + (B*a^2 + 2*A*a*b)*d)*m^3 + 130*((2*B*a*b + A*b^2)*c + (B*a^2 + 2*A*a*b)
*d)*m^2 + 189*(2*B*a*b + A*b^2)*c + 189*(B*a^2 + 2*A*a*b)*d + 300*((2*B*a*b + A*b^2)*c + (B*a^2 + 2*A*a*b)*d)*
m)*x^5 + ((A*a^2*d + (B*a^2 + 2*A*a*b)*c)*m^4 + 315*A*a^2*d + 22*(A*a^2*d + (B*a^2 + 2*A*a*b)*c)*m^3 + 164*(A*
a^2*d + (B*a^2 + 2*A*a*b)*c)*m^2 + 315*(B*a^2 + 2*A*a*b)*c + 458*(A*a^2*d + (B*a^2 + 2*A*a*b)*c)*m)*x^3 + (A*a
^2*c*m^4 + 24*A*a^2*c*m^3 + 206*A*a^2*c*m^2 + 744*A*a^2*c*m + 945*A*a^2*c)*x)*(x*e)^m/(m^5 + 25*m^4 + 230*m^3
+ 950*m^2 + 1689*m + 945)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 3271 vs. \(2 (139) = 278\).
time = 0.69, size = 3271, normalized size = 22.72 \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)**2*(B*x**2+A)*(d*x**2+c),x)

[Out]

Piecewise(((-A*a**2*c/(8*x**8) - A*a**2*d/(6*x**6) - A*a*b*c/(3*x**6) - A*a*b*d/(2*x**4) - A*b**2*c/(4*x**4) -
A*b**2*d/(2*x**2) - B*a**2*c/(6*x**6) - B*a**2*d/(4*x**4) - B*a*b*c/(2*x**4) - B*a*b*d/x**2 - B*b**2*c/(2*x**
2) + B*b**2*d*log(x))/e**9, Eq(m, -9)), ((-A*a**2*c/(6*x**6) - A*a**2*d/(4*x**4) - A*a*b*c/(2*x**4) - A*a*b*d/
x**2 - A*b**2*c/(2*x**2) + A*b**2*d*log(x) - B*a**2*c/(4*x**4) - B*a**2*d/(2*x**2) - B*a*b*c/x**2 + 2*B*a*b*d*
log(x) + B*b**2*c*log(x) + B*b**2*d*x**2/2)/e**7, Eq(m, -7)), ((-A*a**2*c/(4*x**4) - A*a**2*d/(2*x**2) - A*a*b
*c/x**2 + 2*A*a*b*d*log(x) + A*b**2*c*log(x) + A*b**2*d*x**2/2 - B*a**2*c/(2*x**2) + B*a**2*d*log(x) + 2*B*a*b
*c*log(x) + B*a*b*d*x**2 + B*b**2*c*x**2/2 + B*b**2*d*x**4/4)/e**5, Eq(m, -5)), ((-A*a**2*c/(2*x**2) + A*a**2*
d*log(x) + 2*A*a*b*c*log(x) + A*a*b*d*x**2 + A*b**2*c*x**2/2 + A*b**2*d*x**4/4 + B*a**2*c*log(x) + B*a**2*d*x*
*2/2 + B*a*b*c*x**2 + B*a*b*d*x**4/2 + B*b**2*c*x**4/4 + B*b**2*d*x**6/6)/e**3, Eq(m, -3)), ((A*a**2*c*log(x)
+ A*a**2*d*x**2/2 + A*a*b*c*x**2 + A*a*b*d*x**4/2 + A*b**2*c*x**4/4 + A*b**2*d*x**6/6 + B*a**2*c*x**2/2 + B*a*
*2*d*x**4/4 + B*a*b*c*x**4/2 + B*a*b*d*x**6/3 + B*b**2*c*x**6/6 + B*b**2*d*x**8/8)/e, Eq(m, -1)), (A*a**2*c*m*
*4*x*(e*x)**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 24*A*a**2*c*m**3*x*(e*x)**m/(m**5 + 25*m
**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 206*A*a**2*c*m**2*x*(e*x)**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2
+ 1689*m + 945) + 744*A*a**2*c*m*x*(e*x)**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 945*A*a**
2*c*x*(e*x)**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + A*a**2*d*m**4*x**3*(e*x)**m/(m**5 + 25*
m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 22*A*a**2*d*m**3*x**3*(e*x)**m/(m**5 + 25*m**4 + 230*m**3 + 950*m
**2 + 1689*m + 945) + 164*A*a**2*d*m**2*x**3*(e*x)**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) +
458*A*a**2*d*m*x**3*(e*x)**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 315*A*a**2*d*x**3*(e*x)**
m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 2*A*a*b*c*m**4*x**3*(e*x)**m/(m**5 + 25*m**4 + 230*m
**3 + 950*m**2 + 1689*m + 945) + 44*A*a*b*c*m**3*x**3*(e*x)**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m
+ 945) + 328*A*a*b*c*m**2*x**3*(e*x)**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 916*A*a*b*c*m*
x**3*(e*x)**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 630*A*a*b*c*x**3*(e*x)**m/(m**5 + 25*m**
4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 2*A*a*b*d*m**4*x**5*(e*x)**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 +
1689*m + 945) + 40*A*a*b*d*m**3*x**5*(e*x)**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 260*A*a
*b*d*m**2*x**5*(e*x)**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 600*A*a*b*d*m*x**5*(e*x)**m/(m
**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 378*A*a*b*d*x**5*(e*x)**m/(m**5 + 25*m**4 + 230*m**3 + 9
50*m**2 + 1689*m + 945) + A*b**2*c*m**4*x**5*(e*x)**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) +
20*A*b**2*c*m**3*x**5*(e*x)**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 130*A*b**2*c*m**2*x**5*
(e*x)**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 300*A*b**2*c*m*x**5*(e*x)**m/(m**5 + 25*m**4
+ 230*m**3 + 950*m**2 + 1689*m + 945) + 189*A*b**2*c*x**5*(e*x)**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 168
9*m + 945) + A*b**2*d*m**4*x**7*(e*x)**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 18*A*b**2*d*m
**3*x**7*(e*x)**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 104*A*b**2*d*m**2*x**7*(e*x)**m/(m**
5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 222*A*b**2*d*m*x**7*(e*x)**m/(m**5 + 25*m**4 + 230*m**3 +
950*m**2 + 1689*m + 945) + 135*A*b**2*d*x**7*(e*x)**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) +
B*a**2*c*m**4*x**3*(e*x)**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 22*B*a**2*c*m**3*x**3*(e*x
)**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 164*B*a**2*c*m**2*x**3*(e*x)**m/(m**5 + 25*m**4 +
230*m**3 + 950*m**2 + 1689*m + 945) + 458*B*a**2*c*m*x**3*(e*x)**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 16
89*m + 945) + 315*B*a**2*c*x**3*(e*x)**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + B*a**2*d*m**4
*x**5*(e*x)**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 20*B*a**2*d*m**3*x**5*(e*x)**m/(m**5 +
25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 130*B*a**2*d*m**2*x**5*(e*x)**m/(m**5 + 25*m**4 + 230*m**3 + 9
50*m**2 + 1689*m + 945) + 300*B*a**2*d*m*x**5*(e*x)**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) +
189*B*a**2*d*x**5*(e*x)**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 2*B*a*b*c*m**4*x**5*(e*x)*
*m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 40*B*a*b*c*m**3*x**5*(e*x)**m/(m**5 + 25*m**4 + 230
*m**3 + 950*m**2 + 1689*m + 945) + 260*B*a*b*c*m**2*x**5*(e*x)**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689
*m + 945) + 600*B*a*b*c*m*x**5*(e*x)**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 378*B*a*b*c*x*
*5*(e*x)**m/(m**5 + 25*m**4 + 230*m**3 + 950*m*...

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1009 vs. \(2 (144) = 288\).
time = 1.18, size = 1009, normalized size = 7.01 \begin {gather*} \frac {B b^{2} d m^{4} x^{9} x^{m} e^{m} + 16 \, B b^{2} d m^{3} x^{9} x^{m} e^{m} + B b^{2} c m^{4} x^{7} x^{m} e^{m} + 2 \, B a b d m^{4} x^{7} x^{m} e^{m} + A b^{2} d m^{4} x^{7} x^{m} e^{m} + 86 \, B b^{2} d m^{2} x^{9} x^{m} e^{m} + 18 \, B b^{2} c m^{3} x^{7} x^{m} e^{m} + 36 \, B a b d m^{3} x^{7} x^{m} e^{m} + 18 \, A b^{2} d m^{3} x^{7} x^{m} e^{m} + 176 \, B b^{2} d m x^{9} x^{m} e^{m} + 2 \, B a b c m^{4} x^{5} x^{m} e^{m} + A b^{2} c m^{4} x^{5} x^{m} e^{m} + B a^{2} d m^{4} x^{5} x^{m} e^{m} + 2 \, A a b d m^{4} x^{5} x^{m} e^{m} + 104 \, B b^{2} c m^{2} x^{7} x^{m} e^{m} + 208 \, B a b d m^{2} x^{7} x^{m} e^{m} + 104 \, A b^{2} d m^{2} x^{7} x^{m} e^{m} + 105 \, B b^{2} d x^{9} x^{m} e^{m} + 40 \, B a b c m^{3} x^{5} x^{m} e^{m} + 20 \, A b^{2} c m^{3} x^{5} x^{m} e^{m} + 20 \, B a^{2} d m^{3} x^{5} x^{m} e^{m} + 40 \, A a b d m^{3} x^{5} x^{m} e^{m} + 222 \, B b^{2} c m x^{7} x^{m} e^{m} + 444 \, B a b d m x^{7} x^{m} e^{m} + 222 \, A b^{2} d m x^{7} x^{m} e^{m} + B a^{2} c m^{4} x^{3} x^{m} e^{m} + 2 \, A a b c m^{4} x^{3} x^{m} e^{m} + A a^{2} d m^{4} x^{3} x^{m} e^{m} + 260 \, B a b c m^{2} x^{5} x^{m} e^{m} + 130 \, A b^{2} c m^{2} x^{5} x^{m} e^{m} + 130 \, B a^{2} d m^{2} x^{5} x^{m} e^{m} + 260 \, A a b d m^{2} x^{5} x^{m} e^{m} + 135 \, B b^{2} c x^{7} x^{m} e^{m} + 270 \, B a b d x^{7} x^{m} e^{m} + 135 \, A b^{2} d x^{7} x^{m} e^{m} + 22 \, B a^{2} c m^{3} x^{3} x^{m} e^{m} + 44 \, A a b c m^{3} x^{3} x^{m} e^{m} + 22 \, A a^{2} d m^{3} x^{3} x^{m} e^{m} + 600 \, B a b c m x^{5} x^{m} e^{m} + 300 \, A b^{2} c m x^{5} x^{m} e^{m} + 300 \, B a^{2} d m x^{5} x^{m} e^{m} + 600 \, A a b d m x^{5} x^{m} e^{m} + A a^{2} c m^{4} x x^{m} e^{m} + 164 \, B a^{2} c m^{2} x^{3} x^{m} e^{m} + 328 \, A a b c m^{2} x^{3} x^{m} e^{m} + 164 \, A a^{2} d m^{2} x^{3} x^{m} e^{m} + 378 \, B a b c x^{5} x^{m} e^{m} + 189 \, A b^{2} c x^{5} x^{m} e^{m} + 189 \, B a^{2} d x^{5} x^{m} e^{m} + 378 \, A a b d x^{5} x^{m} e^{m} + 24 \, A a^{2} c m^{3} x x^{m} e^{m} + 458 \, B a^{2} c m x^{3} x^{m} e^{m} + 916 \, A a b c m x^{3} x^{m} e^{m} + 458 \, A a^{2} d m x^{3} x^{m} e^{m} + 206 \, A a^{2} c m^{2} x x^{m} e^{m} + 315 \, B a^{2} c x^{3} x^{m} e^{m} + 630 \, A a b c x^{3} x^{m} e^{m} + 315 \, A a^{2} d x^{3} x^{m} e^{m} + 744 \, A a^{2} c m x x^{m} e^{m} + 945 \, A a^{2} c x x^{m} e^{m}}{m^{5} + 25 \, m^{4} + 230 \, m^{3} + 950 \, m^{2} + 1689 \, m + 945} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(B*b^2*d*m^4*x^9*x^m*e^m + 16*B*b^2*d*m^3*x^9*x^m*e^m + B*b^2*c*m^4*x^7*x^m*e^m + 2*B*a*b*d*m^4*x^7*x^m*e^m +
A*b^2*d*m^4*x^7*x^m*e^m + 86*B*b^2*d*m^2*x^9*x^m*e^m + 18*B*b^2*c*m^3*x^7*x^m*e^m + 36*B*a*b*d*m^3*x^7*x^m*e^m
+ 18*A*b^2*d*m^3*x^7*x^m*e^m + 176*B*b^2*d*m*x^9*x^m*e^m + 2*B*a*b*c*m^4*x^5*x^m*e^m + A*b^2*c*m^4*x^5*x^m*e^
m + B*a^2*d*m^4*x^5*x^m*e^m + 2*A*a*b*d*m^4*x^5*x^m*e^m + 104*B*b^2*c*m^2*x^7*x^m*e^m + 208*B*a*b*d*m^2*x^7*x^
m*e^m + 104*A*b^2*d*m^2*x^7*x^m*e^m + 105*B*b^2*d*x^9*x^m*e^m + 40*B*a*b*c*m^3*x^5*x^m*e^m + 20*A*b^2*c*m^3*x^
5*x^m*e^m + 20*B*a^2*d*m^3*x^5*x^m*e^m + 40*A*a*b*d*m^3*x^5*x^m*e^m + 222*B*b^2*c*m*x^7*x^m*e^m + 444*B*a*b*d*
m*x^7*x^m*e^m + 222*A*b^2*d*m*x^7*x^m*e^m + B*a^2*c*m^4*x^3*x^m*e^m + 2*A*a*b*c*m^4*x^3*x^m*e^m + A*a^2*d*m^4*
x^3*x^m*e^m + 260*B*a*b*c*m^2*x^5*x^m*e^m + 130*A*b^2*c*m^2*x^5*x^m*e^m + 130*B*a^2*d*m^2*x^5*x^m*e^m + 260*A*
a*b*d*m^2*x^5*x^m*e^m + 135*B*b^2*c*x^7*x^m*e^m + 270*B*a*b*d*x^7*x^m*e^m + 135*A*b^2*d*x^7*x^m*e^m + 22*B*a^2
*c*m^3*x^3*x^m*e^m + 44*A*a*b*c*m^3*x^3*x^m*e^m + 22*A*a^2*d*m^3*x^3*x^m*e^m + 600*B*a*b*c*m*x^5*x^m*e^m + 300
*A*b^2*c*m*x^5*x^m*e^m + 300*B*a^2*d*m*x^5*x^m*e^m + 600*A*a*b*d*m*x^5*x^m*e^m + A*a^2*c*m^4*x*x^m*e^m + 164*B
*a^2*c*m^2*x^3*x^m*e^m + 328*A*a*b*c*m^2*x^3*x^m*e^m + 164*A*a^2*d*m^2*x^3*x^m*e^m + 378*B*a*b*c*x^5*x^m*e^m +
189*A*b^2*c*x^5*x^m*e^m + 189*B*a^2*d*x^5*x^m*e^m + 378*A*a*b*d*x^5*x^m*e^m + 24*A*a^2*c*m^3*x*x^m*e^m + 458*
B*a^2*c*m*x^3*x^m*e^m + 916*A*a*b*c*m*x^3*x^m*e^m + 458*A*a^2*d*m*x^3*x^m*e^m + 206*A*a^2*c*m^2*x*x^m*e^m + 31
5*B*a^2*c*x^3*x^m*e^m + 630*A*a*b*c*x^3*x^m*e^m + 315*A*a^2*d*x^3*x^m*e^m + 744*A*a^2*c*m*x*x^m*e^m + 945*A*a^
2*c*x*x^m*e^m)/(m^5 + 25*m^4 + 230*m^3 + 950*m^2 + 1689*m + 945)

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Mupad [B]
time = 1.20, size = 305, normalized size = 2.12 \begin {gather*} {\left (e\,x\right )}^m\,\left (\frac {x^5\,\left (A\,b^2\,c+B\,a^2\,d+2\,A\,a\,b\,d+2\,B\,a\,b\,c\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 {a\,x^3\,\left (A\,a\,d+2\,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 {b\,x^7\,\left (A\,b\,d+2\,B\,a\,d+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^2\,c\,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^2\,d\,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 ) \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)^2*(c + d*x^2),x)

[Out]

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