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

Optimal. Leaf size=91 \[ \frac{c (e x)^{m+3} (2 A d+B c)}{e^3 (m+3)}+\frac{d (e x)^{m+5} (A d+2 B c)}{e^5 (m+5)}+\frac{A c^2 (e x)^{m+1}}{e (m+1)}+\frac{B d^2 (e x)^{m+7}}{e^7 (m+7)} \]

[Out]

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

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Rubi [A]  time = 0.0686443, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {448} \[ \frac{c (e x)^{m+3} (2 A d+B c)}{e^3 (m+3)}+\frac{d (e x)^{m+5} (A d+2 B c)}{e^5 (m+5)}+\frac{A c^2 (e x)^{m+1}}{e (m+1)}+\frac{B d^2 (e x)^{m+7}}{e^7 (m+7)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0503973, size = 67, normalized size = 0.74 \[ x (e x)^m \left (\frac{d x^4 (A d+2 B c)}{m+5}+\frac{c x^2 (2 A d+B c)}{m+3}+\frac{A c^2}{m+1}+\frac{B d^2 x^6}{m+7}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.006, size = 263, normalized size = 2.9 \begin{align*}{\frac{ \left ( B{d}^{2}{m}^{3}{x}^{6}+9\,B{d}^{2}{m}^{2}{x}^{6}+A{d}^{2}{m}^{3}{x}^{4}+2\,Bcd{m}^{3}{x}^{4}+23\,B{d}^{2}m{x}^{6}+11\,A{d}^{2}{m}^{2}{x}^{4}+22\,Bcd{m}^{2}{x}^{4}+15\,B{d}^{2}{x}^{6}+2\,Acd{m}^{3}{x}^{2}+31\,A{d}^{2}m{x}^{4}+B{c}^{2}{m}^{3}{x}^{2}+62\,Bcdm{x}^{4}+26\,Acd{m}^{2}{x}^{2}+21\,A{d}^{2}{x}^{4}+13\,B{c}^{2}{m}^{2}{x}^{2}+42\,Bcd{x}^{4}+A{c}^{2}{m}^{3}+94\,Acdm{x}^{2}+47\,B{c}^{2}m{x}^{2}+15\,A{c}^{2}{m}^{2}+70\,Acd{x}^{2}+35\,B{c}^{2}{x}^{2}+71\,A{c}^{2}m+105\,A{c}^{2} \right ) x \left ( ex \right ) ^{m}}{ \left ( 7+m \right ) \left ( 5+m \right ) \left ( 3+m \right ) \left ( 1+m \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*(B*d^2*m^3*x^6+9*B*d^2*m^2*x^6+A*d^2*m^3*x^4+2*B*c*d*m^3*x^4+23*B*d^2*m*x^6+11*A*d^2*m^2*x^4+22*B*c*d*m^2*x^
4+15*B*d^2*x^6+2*A*c*d*m^3*x^2+31*A*d^2*m*x^4+B*c^2*m^3*x^2+62*B*c*d*m*x^4+26*A*c*d*m^2*x^2+21*A*d^2*x^4+13*B*
c^2*m^2*x^2+42*B*c*d*x^4+A*c^2*m^3+94*A*c*d*m*x^2+47*B*c^2*m*x^2+15*A*c^2*m^2+70*A*c*d*x^2+35*B*c^2*x^2+71*A*c
^2*m+105*A*c^2)*(e*x)^m/(7+m)/(5+m)/(3+m)/(1+m)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.58193, size = 495, normalized size = 5.44 \begin{align*} \frac{{\left ({\left (B d^{2} m^{3} + 9 \, B d^{2} m^{2} + 23 \, B d^{2} m + 15 \, B d^{2}\right )} x^{7} +{\left ({\left (2 \, B c d + A d^{2}\right )} m^{3} + 42 \, B c d + 21 \, A d^{2} + 11 \,{\left (2 \, B c d + A d^{2}\right )} m^{2} + 31 \,{\left (2 \, B c d + A d^{2}\right )} m\right )} x^{5} +{\left ({\left (B c^{2} + 2 \, A c d\right )} m^{3} + 35 \, B c^{2} + 70 \, A c d + 13 \,{\left (B c^{2} + 2 \, A c d\right )} m^{2} + 47 \,{\left (B c^{2} + 2 \, A c d\right )} m\right )} x^{3} +{\left (A c^{2} m^{3} + 15 \, A c^{2} m^{2} + 71 \, A c^{2} m + 105 \, A c^{2}\right )} x\right )} \left (e x\right )^{m}}{m^{4} + 16 \, m^{3} + 86 \, m^{2} + 176 \, m + 105} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((B*d^2*m^3 + 9*B*d^2*m^2 + 23*B*d^2*m + 15*B*d^2)*x^7 + ((2*B*c*d + A*d^2)*m^3 + 42*B*c*d + 21*A*d^2 + 11*(2*
B*c*d + A*d^2)*m^2 + 31*(2*B*c*d + A*d^2)*m)*x^5 + ((B*c^2 + 2*A*c*d)*m^3 + 35*B*c^2 + 70*A*c*d + 13*(B*c^2 +
2*A*c*d)*m^2 + 47*(B*c^2 + 2*A*c*d)*m)*x^3 + (A*c^2*m^3 + 15*A*c^2*m^2 + 71*A*c^2*m + 105*A*c^2)*x)*(e*x)^m/(m
^4 + 16*m^3 + 86*m^2 + 176*m + 105)

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Sympy [A]  time = 1.79833, size = 1137, normalized size = 12.49 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise(((-A*c**2/(6*x**6) - A*c*d/(2*x**4) - A*d**2/(2*x**2) - B*c**2/(4*x**4) - B*c*d/x**2 + B*d**2*log(x)
)/e**7, Eq(m, -7)), ((-A*c**2/(4*x**4) - A*c*d/x**2 + A*d**2*log(x) - B*c**2/(2*x**2) + 2*B*c*d*log(x) + B*d**
2*x**2/2)/e**5, Eq(m, -5)), ((-A*c**2/(2*x**2) + 2*A*c*d*log(x) + A*d**2*x**2/2 + B*c**2*log(x) + B*c*d*x**2 +
B*d**2*x**4/4)/e**3, Eq(m, -3)), ((A*c**2*log(x) + A*c*d*x**2 + A*d**2*x**4/4 + B*c**2*x**2/2 + B*c*d*x**4/2
+ B*d**2*x**6/6)/e, Eq(m, -1)), (A*c**2*e**m*m**3*x*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 15*A*c**2*
e**m*m**2*x*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 71*A*c**2*e**m*m*x*x**m/(m**4 + 16*m**3 + 86*m**2
+ 176*m + 105) + 105*A*c**2*e**m*x*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 2*A*c*d*e**m*m**3*x**3*x**m
/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 26*A*c*d*e**m*m**2*x**3*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 1
05) + 94*A*c*d*e**m*m*x**3*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 70*A*c*d*e**m*x**3*x**m/(m**4 + 16*
m**3 + 86*m**2 + 176*m + 105) + A*d**2*e**m*m**3*x**5*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 11*A*d**
2*e**m*m**2*x**5*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 31*A*d**2*e**m*m*x**5*x**m/(m**4 + 16*m**3 +
86*m**2 + 176*m + 105) + 21*A*d**2*e**m*x**5*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + B*c**2*e**m*m**3*
x**3*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 13*B*c**2*e**m*m**2*x**3*x**m/(m**4 + 16*m**3 + 86*m**2 +
176*m + 105) + 47*B*c**2*e**m*m*x**3*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 35*B*c**2*e**m*x**3*x**m
/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 2*B*c*d*e**m*m**3*x**5*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 10
5) + 22*B*c*d*e**m*m**2*x**5*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 62*B*c*d*e**m*m*x**5*x**m/(m**4 +
16*m**3 + 86*m**2 + 176*m + 105) + 42*B*c*d*e**m*x**5*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + B*d**2*
e**m*m**3*x**7*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 9*B*d**2*e**m*m**2*x**7*x**m/(m**4 + 16*m**3 +
86*m**2 + 176*m + 105) + 23*B*d**2*e**m*m*x**7*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 15*B*d**2*e**m*
x**7*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105), True))

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Giac [B]  time = 1.19729, size = 513, normalized size = 5.64 \begin{align*} \frac{B d^{2} m^{3} x^{7} x^{m} e^{m} + 9 \, B d^{2} m^{2} x^{7} x^{m} e^{m} + 2 \, B c d m^{3} x^{5} x^{m} e^{m} + A d^{2} m^{3} x^{5} x^{m} e^{m} + 23 \, B d^{2} m x^{7} x^{m} e^{m} + 22 \, B c d m^{2} x^{5} x^{m} e^{m} + 11 \, A d^{2} m^{2} x^{5} x^{m} e^{m} + 15 \, B d^{2} x^{7} x^{m} e^{m} + B c^{2} m^{3} x^{3} x^{m} e^{m} + 2 \, A c d m^{3} x^{3} x^{m} e^{m} + 62 \, B c d m x^{5} x^{m} e^{m} + 31 \, A d^{2} m x^{5} x^{m} e^{m} + 13 \, B c^{2} m^{2} x^{3} x^{m} e^{m} + 26 \, A c d m^{2} x^{3} x^{m} e^{m} + 42 \, B c d x^{5} x^{m} e^{m} + 21 \, A d^{2} x^{5} x^{m} e^{m} + A c^{2} m^{3} x x^{m} e^{m} + 47 \, B c^{2} m x^{3} x^{m} e^{m} + 94 \, A c d m x^{3} x^{m} e^{m} + 15 \, A c^{2} m^{2} x x^{m} e^{m} + 35 \, B c^{2} x^{3} x^{m} e^{m} + 70 \, A c d x^{3} x^{m} e^{m} + 71 \, A c^{2} m x x^{m} e^{m} + 105 \, A c^{2} x x^{m} e^{m}}{m^{4} + 16 \, m^{3} + 86 \, m^{2} + 176 \, m + 105} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(B*d^2*m^3*x^7*x^m*e^m + 9*B*d^2*m^2*x^7*x^m*e^m + 2*B*c*d*m^3*x^5*x^m*e^m + A*d^2*m^3*x^5*x^m*e^m + 23*B*d^2*
m*x^7*x^m*e^m + 22*B*c*d*m^2*x^5*x^m*e^m + 11*A*d^2*m^2*x^5*x^m*e^m + 15*B*d^2*x^7*x^m*e^m + B*c^2*m^3*x^3*x^m
*e^m + 2*A*c*d*m^3*x^3*x^m*e^m + 62*B*c*d*m*x^5*x^m*e^m + 31*A*d^2*m*x^5*x^m*e^m + 13*B*c^2*m^2*x^3*x^m*e^m +
26*A*c*d*m^2*x^3*x^m*e^m + 42*B*c*d*x^5*x^m*e^m + 21*A*d^2*x^5*x^m*e^m + A*c^2*m^3*x*x^m*e^m + 47*B*c^2*m*x^3*
x^m*e^m + 94*A*c*d*m*x^3*x^m*e^m + 15*A*c^2*m^2*x*x^m*e^m + 35*B*c^2*x^3*x^m*e^m + 70*A*c*d*x^3*x^m*e^m + 71*A
*c^2*m*x*x^m*e^m + 105*A*c^2*x*x^m*e^m)/(m^4 + 16*m^3 + 86*m^2 + 176*m + 105)