3.4 \(\int (e x)^m (A+B x^2) (c+d x^2) \, dx\)

Optimal. Leaf size=60 \[ \frac {(e x)^{m+3} (A d+B c)}{e^3 (m+3)}+\frac {A c (e x)^{m+1}}{e (m+1)}+\frac {B d (e x)^{m+5}}{e^5 (m+5)} \]

[Out]

A*c*(e*x)^(1+m)/e/(1+m)+(A*d+B*c)*(e*x)^(3+m)/e^3/(3+m)+B*d*(e*x)^(5+m)/e^5/(5+m)

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Rubi [A]  time = 0.03, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {448} \[ \frac {(e x)^{m+3} (A d+B c)}{e^3 (m+3)}+\frac {A c (e x)^{m+1}}{e (m+1)}+\frac {B d (e x)^{m+5}}{e^5 (m+5)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(A*c*(e*x)^(1 + m))/(e*(1 + m)) + ((B*c + A*d)*(e*x)^(3 + m))/(e^3*(3 + m)) + (B*d*(e*x)^(5 + m))/(e^5*(5 + m)
)

Rule 448

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[ExpandI
ntegrand[(e*x)^m*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] &
& IGtQ[p, 0] && IGtQ[q, 0]

Rubi steps

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

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Mathematica [A]  time = 0.04, size = 43, normalized size = 0.72 \[ x (e x)^m \left (\frac {x^2 (A d+B c)}{m+3}+\frac {A c}{m+1}+\frac {B d x^4}{m+5}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 1.04, size = 94, normalized size = 1.57 \[ \frac {{\left ({\left (B d m^{2} + 4 \, B d m + 3 \, B d\right )} x^{5} + {\left ({\left (B c + A d\right )} m^{2} + 5 \, B c + 5 \, A d + 6 \, {\left (B c + A d\right )} m\right )} x^{3} + {\left (A c m^{2} + 8 \, A c m + 15 \, A c\right )} x\right )} \left (e x\right )^{m}}{m^{3} + 9 \, m^{2} + 23 \, m + 15} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((B*d*m^2 + 4*B*d*m + 3*B*d)*x^5 + ((B*c + A*d)*m^2 + 5*B*c + 5*A*d + 6*(B*c + A*d)*m)*x^3 + (A*c*m^2 + 8*A*c*
m + 15*A*c)*x)*(e*x)^m/(m^3 + 9*m^2 + 23*m + 15)

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giac [B]  time = 0.50, size = 167, normalized size = 2.78 \[ \frac {B d m^{2} x^{5} x^{m} e^{m} + 4 \, B d m x^{5} x^{m} e^{m} + B c m^{2} x^{3} x^{m} e^{m} + A d m^{2} x^{3} x^{m} e^{m} + 3 \, B d x^{5} x^{m} e^{m} + 6 \, B c m x^{3} x^{m} e^{m} + 6 \, A d m x^{3} x^{m} e^{m} + A c m^{2} x x^{m} e^{m} + 5 \, B c x^{3} x^{m} e^{m} + 5 \, A d x^{3} x^{m} e^{m} + 8 \, A c m x x^{m} e^{m} + 15 \, A c x x^{m} e^{m}}{m^{3} + 9 \, m^{2} + 23 \, m + 15} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(B*d*m^2*x^5*x^m*e^m + 4*B*d*m*x^5*x^m*e^m + B*c*m^2*x^3*x^m*e^m + A*d*m^2*x^3*x^m*e^m + 3*B*d*x^5*x^m*e^m + 6
*B*c*m*x^3*x^m*e^m + 6*A*d*m*x^3*x^m*e^m + A*c*m^2*x*x^m*e^m + 5*B*c*x^3*x^m*e^m + 5*A*d*x^3*x^m*e^m + 8*A*c*m
*x*x^m*e^m + 15*A*c*x*x^m*e^m)/(m^3 + 9*m^2 + 23*m + 15)

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maple [A]  time = 0.00, size = 111, normalized size = 1.85 \[ \frac {\left (B d \,m^{2} x^{4}+4 B d m \,x^{4}+A d \,m^{2} x^{2}+B c \,m^{2} x^{2}+3 B d \,x^{4}+6 A d m \,x^{2}+6 B c m \,x^{2}+A c \,m^{2}+5 A d \,x^{2}+5 B c \,x^{2}+8 A c m +15 A c \right ) x \left (e x \right )^{m}}{\left (m +5\right ) \left (m +3\right ) \left (m +1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*(B*d*m^2*x^4+4*B*d*m*x^4+A*d*m^2*x^2+B*c*m^2*x^2+3*B*d*x^4+6*A*d*m*x^2+6*B*c*m*x^2+A*c*m^2+5*A*d*x^2+5*B*c*x
^2+8*A*c*m+15*A*c)*(e*x)^m/(m+5)/(m+3)/(m+1)

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maxima [A]  time = 1.45, size = 70, normalized size = 1.17 \[ \frac {B d e^{m} x^{5} x^{m}}{m + 5} + \frac {B c e^{m} x^{3} x^{m}}{m + 3} + \frac {A d e^{m} x^{3} x^{m}}{m + 3} + \frac {\left (e x\right )^{m + 1} A c}{e {\left (m + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

B*d*e^m*x^5*x^m/(m + 5) + B*c*e^m*x^3*x^m/(m + 3) + A*d*e^m*x^3*x^m/(m + 3) + (e*x)^(m + 1)*A*c/(e*(m + 1))

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mupad [B]  time = 0.96, size = 97, normalized size = 1.62 \[ {\left (e\,x\right )}^m\,\left (\frac {x^3\,\left (A\,d+B\,c\right )\,\left (m^2+6\,m+5\right )}{m^3+9\,m^2+23\,m+15}+\frac {B\,d\,x^5\,\left (m^2+4\,m+3\right )}{m^3+9\,m^2+23\,m+15}+\frac {A\,c\,x\,\left (m^2+8\,m+15\right )}{m^3+9\,m^2+23\,m+15}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(e*x)^m*((x^3*(A*d + B*c)*(6*m + m^2 + 5))/(23*m + 9*m^2 + m^3 + 15) + (B*d*x^5*(4*m + m^2 + 3))/(23*m + 9*m^2
+ m^3 + 15) + (A*c*x*(8*m + m^2 + 15))/(23*m + 9*m^2 + m^3 + 15))

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sympy [A]  time = 1.93, size = 459, normalized size = 7.65 \[ \begin {cases} \frac {- \frac {A c}{4 x^{4}} - \frac {A d}{2 x^{2}} - \frac {B c}{2 x^{2}} + B d \log {\relax (x )}}{e^{5}} & \text {for}\: m = -5 \\\frac {- \frac {A c}{2 x^{2}} + A d \log {\relax (x )} + B c \log {\relax (x )} + \frac {B d x^{2}}{2}}{e^{3}} & \text {for}\: m = -3 \\\frac {A c \log {\relax (x )} + \frac {A d x^{2}}{2} + \frac {B c x^{2}}{2} + \frac {B d x^{4}}{4}}{e} & \text {for}\: m = -1 \\\frac {A c e^{m} m^{2} x x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac {8 A c e^{m} m x x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac {15 A c e^{m} x x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac {A d e^{m} m^{2} x^{3} x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac {6 A d e^{m} m x^{3} x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac {5 A d e^{m} x^{3} x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac {B c e^{m} m^{2} x^{3} x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac {6 B c e^{m} m x^{3} x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac {5 B c e^{m} x^{3} x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac {B d e^{m} m^{2} x^{5} x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac {4 B d e^{m} m x^{5} x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac {3 B d e^{m} x^{5} x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise(((-A*c/(4*x**4) - A*d/(2*x**2) - B*c/(2*x**2) + B*d*log(x))/e**5, Eq(m, -5)), ((-A*c/(2*x**2) + A*d*
log(x) + B*c*log(x) + B*d*x**2/2)/e**3, Eq(m, -3)), ((A*c*log(x) + A*d*x**2/2 + B*c*x**2/2 + B*d*x**4/4)/e, Eq
(m, -1)), (A*c*e**m*m**2*x*x**m/(m**3 + 9*m**2 + 23*m + 15) + 8*A*c*e**m*m*x*x**m/(m**3 + 9*m**2 + 23*m + 15)
+ 15*A*c*e**m*x*x**m/(m**3 + 9*m**2 + 23*m + 15) + A*d*e**m*m**2*x**3*x**m/(m**3 + 9*m**2 + 23*m + 15) + 6*A*d
*e**m*m*x**3*x**m/(m**3 + 9*m**2 + 23*m + 15) + 5*A*d*e**m*x**3*x**m/(m**3 + 9*m**2 + 23*m + 15) + B*c*e**m*m*
*2*x**3*x**m/(m**3 + 9*m**2 + 23*m + 15) + 6*B*c*e**m*m*x**3*x**m/(m**3 + 9*m**2 + 23*m + 15) + 5*B*c*e**m*x**
3*x**m/(m**3 + 9*m**2 + 23*m + 15) + B*d*e**m*m**2*x**5*x**m/(m**3 + 9*m**2 + 23*m + 15) + 4*B*d*e**m*m*x**5*x
**m/(m**3 + 9*m**2 + 23*m + 15) + 3*B*d*e**m*x**5*x**m/(m**3 + 9*m**2 + 23*m + 15), True))

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