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

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

[Out]

A*c^2*(e*x)^(1+m)/e/(1+m)+c*(2*A*d+B*c)*(e*x)^(3+m)/e^3/(3+m)+d*(A*d+2*B*c)*(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 = 0.07, antiderivative size = 91, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.05, 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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fricas [B]  time = 1.11, size = 217, normalized size = 2.38 \[ \frac {{\left ({\left (B d^{2} m^{3} + 9 \, B d^{2} m^{2} + 23 \, B d^{2} m + 15 \, B d^{2}\right )} x^{7} + {\left ({\left (2 \, B c d + A d^{2}\right )} m^{3} + 42 \, B c d + 21 \, A d^{2} + 11 \, {\left (2 \, B c d + A d^{2}\right )} m^{2} + 31 \, {\left (2 \, B c d + A d^{2}\right )} m\right )} x^{5} + {\left ({\left (B c^{2} + 2 \, A c d\right )} m^{3} + 35 \, B c^{2} + 70 \, A c d + 13 \, {\left (B c^{2} + 2 \, A c d\right )} m^{2} + 47 \, {\left (B c^{2} + 2 \, A c d\right )} m\right )} x^{3} + {\left (A c^{2} m^{3} + 15 \, A c^{2} m^{2} + 71 \, A c^{2} m + 105 \, A c^{2}\right )} x\right )} \left (e x\right )^{m}}{m^{4} + 16 \, m^{3} + 86 \, m^{2} + 176 \, m + 105} \]

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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giac [B]  time = 0.45, size = 380, normalized size = 4.18 \[ \frac {B d^{2} m^{3} x^{7} x^{m} e^{m} + 9 \, B d^{2} m^{2} x^{7} x^{m} e^{m} + 2 \, B c d m^{3} x^{5} x^{m} e^{m} + A d^{2} m^{3} x^{5} x^{m} e^{m} + 23 \, B d^{2} m x^{7} x^{m} e^{m} + 22 \, B c d m^{2} x^{5} x^{m} e^{m} + 11 \, A d^{2} m^{2} x^{5} x^{m} e^{m} + 15 \, B d^{2} x^{7} x^{m} e^{m} + B c^{2} m^{3} x^{3} x^{m} e^{m} + 2 \, A c d m^{3} x^{3} x^{m} e^{m} + 62 \, B c d m x^{5} x^{m} e^{m} + 31 \, A d^{2} m x^{5} x^{m} e^{m} + 13 \, B c^{2} m^{2} x^{3} x^{m} e^{m} + 26 \, A c d m^{2} x^{3} x^{m} e^{m} + 42 \, B c d x^{5} x^{m} e^{m} + 21 \, A d^{2} x^{5} x^{m} e^{m} + A c^{2} m^{3} x x^{m} e^{m} + 47 \, B c^{2} m x^{3} x^{m} e^{m} + 94 \, A c d m x^{3} x^{m} e^{m} + 15 \, A c^{2} m^{2} x x^{m} e^{m} + 35 \, B c^{2} x^{3} x^{m} e^{m} + 70 \, A c d x^{3} x^{m} e^{m} + 71 \, A c^{2} m x x^{m} e^{m} + 105 \, A c^{2} x x^{m} e^{m}}{m^{4} + 16 \, m^{3} + 86 \, m^{2} + 176 \, m + 105} \]

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)

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maple [B]  time = 0.01, size = 263, normalized size = 2.89 \[ \frac {\left (B \,d^{2} m^{3} x^{6}+9 B \,d^{2} m^{2} x^{6}+A \,d^{2} m^{3} x^{4}+2 B c d \,m^{3} x^{4}+23 B \,d^{2} m \,x^{6}+11 A \,d^{2} m^{2} x^{4}+22 B c d \,m^{2} x^{4}+15 B \,d^{2} x^{6}+2 A c d \,m^{3} x^{2}+31 A \,d^{2} m \,x^{4}+B \,c^{2} m^{3} x^{2}+62 B c d m \,x^{4}+26 A c d \,m^{2} x^{2}+21 A \,d^{2} x^{4}+13 B \,c^{2} m^{2} x^{2}+42 B c d \,x^{4}+A \,c^{2} m^{3}+94 A c d m \,x^{2}+47 B \,c^{2} m \,x^{2}+15 A \,c^{2} m^{2}+70 A c d \,x^{2}+35 B \,c^{2} x^{2}+71 A \,c^{2} m +105 A \,c^{2}\right ) x \left (e x \right )^{m}}{\left (m +7\right ) \left (m +5\right ) \left (m +3\right ) \left (m +1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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maxima [A]  time = 1.51, size = 116, normalized size = 1.27 \[ \frac {B d^{2} e^{m} x^{7} x^{m}}{m + 7} + \frac {2 \, B c d e^{m} x^{5} x^{m}}{m + 5} + \frac {A d^{2} e^{m} x^{5} x^{m}}{m + 5} + \frac {B c^{2} e^{m} x^{3} x^{m}}{m + 3} + \frac {2 \, A c d e^{m} x^{3} x^{m}}{m + 3} + \frac {\left (e x\right )^{m + 1} A c^{2}}{e {\left (m + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 1.05, size = 179, normalized size = 1.97 \[ {\left (e\,x\right )}^m\,\left (\frac {B\,d^2\,x^7\,\left (m^3+9\,m^2+23\,m+15\right )}{m^4+16\,m^3+86\,m^2+176\,m+105}+\frac {A\,c^2\,x\,\left (m^3+15\,m^2+71\,m+105\right )}{m^4+16\,m^3+86\,m^2+176\,m+105}+\frac {c\,x^3\,\left (2\,A\,d+B\,c\right )\,\left (m^3+13\,m^2+47\,m+35\right )}{m^4+16\,m^3+86\,m^2+176\,m+105}+\frac {d\,x^5\,\left (A\,d+2\,B\,c\right )\,\left (m^3+11\,m^2+31\,m+21\right )}{m^4+16\,m^3+86\,m^2+176\,m+105}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 3.17, size = 1137, normalized size = 12.49 \[ \begin {cases} \frac {- \frac {A c^{2}}{6 x^{6}} - \frac {A c d}{2 x^{4}} - \frac {A d^{2}}{2 x^{2}} - \frac {B c^{2}}{4 x^{4}} - \frac {B c d}{x^{2}} + B d^{2} \log {\relax (x )}}{e^{7}} & \text {for}\: m = -7 \\\frac {- \frac {A c^{2}}{4 x^{4}} - \frac {A c d}{x^{2}} + A d^{2} \log {\relax (x )} - \frac {B c^{2}}{2 x^{2}} + 2 B c d \log {\relax (x )} + \frac {B d^{2} x^{2}}{2}}{e^{5}} & \text {for}\: m = -5 \\\frac {- \frac {A c^{2}}{2 x^{2}} + 2 A c d \log {\relax (x )} + \frac {A d^{2} x^{2}}{2} + B c^{2} \log {\relax (x )} + B c d x^{2} + \frac {B d^{2} x^{4}}{4}}{e^{3}} & \text {for}\: m = -3 \\\frac {A c^{2} \log {\relax (x )} + A c d x^{2} + \frac {A d^{2} x^{4}}{4} + \frac {B c^{2} x^{2}}{2} + \frac {B c d x^{4}}{2} + \frac {B d^{2} x^{6}}{6}}{e} & \text {for}\: m = -1 \\\frac {A c^{2} e^{m} m^{3} x x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {15 A c^{2} e^{m} m^{2} x x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {71 A c^{2} e^{m} m x x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {105 A c^{2} e^{m} x x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {2 A c d e^{m} m^{3} x^{3} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {26 A c d e^{m} m^{2} x^{3} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {94 A c d e^{m} m x^{3} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {70 A c d e^{m} x^{3} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {A d^{2} e^{m} m^{3} x^{5} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {11 A d^{2} e^{m} m^{2} x^{5} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {31 A d^{2} e^{m} m x^{5} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {21 A d^{2} e^{m} x^{5} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {B c^{2} e^{m} m^{3} x^{3} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {13 B c^{2} e^{m} m^{2} x^{3} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {47 B c^{2} e^{m} m x^{3} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {35 B c^{2} e^{m} x^{3} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {2 B c d e^{m} m^{3} x^{5} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {22 B c d e^{m} m^{2} x^{5} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {62 B c d e^{m} m x^{5} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {42 B c d e^{m} x^{5} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {B d^{2} e^{m} m^{3} x^{7} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {9 B d^{2} e^{m} m^{2} x^{7} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {23 B d^{2} e^{m} m x^{7} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {15 B d^{2} e^{m} x^{7} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} & \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)**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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