3.1488 \(\int (A+B x) (d+e x)^m \left (a+c x^2\right )^2 \, dx\)
Optimal. Leaf size=234 \[ -\frac{\left (a e^2+c d^2\right )^2 (B d-A e) (d+e x)^{m+1}}{e^6 (m+1)}+\frac{\left (a e^2+c d^2\right ) (d+e x)^{m+2} \left (a B e^2-4 A c d e+5 B c d^2\right )}{e^6 (m+2)}+\frac{2 c (d+e x)^{m+4} \left (a B e^2-2 A c d e+5 B c d^2\right )}{e^6 (m+4)}-\frac{2 c (d+e x)^{m+3} \left (-a A e^3+3 a B d e^2-3 A c d^2 e+5 B c d^3\right )}{e^6 (m+3)}-\frac{c^2 (5 B d-A e) (d+e x)^{m+5}}{e^6 (m+5)}+\frac{B c^2 (d+e x)^{m+6}}{e^6 (m+6)} \]
[Out]
-(((B*d - A*e)*(c*d^2 + a*e^2)^2*(d + e*x)^(1 + m))/(e^6*(1 + m))) + ((c*d^2 + a
*e^2)*(5*B*c*d^2 - 4*A*c*d*e + a*B*e^2)*(d + e*x)^(2 + m))/(e^6*(2 + m)) - (2*c*
(5*B*c*d^3 - 3*A*c*d^2*e + 3*a*B*d*e^2 - a*A*e^3)*(d + e*x)^(3 + m))/(e^6*(3 + m
)) + (2*c*(5*B*c*d^2 - 2*A*c*d*e + a*B*e^2)*(d + e*x)^(4 + m))/(e^6*(4 + m)) - (
c^2*(5*B*d - A*e)*(d + e*x)^(5 + m))/(e^6*(5 + m)) + (B*c^2*(d + e*x)^(6 + m))/(
e^6*(6 + m))
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Rubi [A] time = 0.333203, antiderivative size = 234, 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
\[ -\frac{\left (a e^2+c d^2\right )^2 (B d-A e) (d+e x)^{m+1}}{e^6 (m+1)}+\frac{\left (a e^2+c d^2\right ) (d+e x)^{m+2} \left (a B e^2-4 A c d e+5 B c d^2\right )}{e^6 (m+2)}+\frac{2 c (d+e x)^{m+4} \left (a B e^2-2 A c d e+5 B c d^2\right )}{e^6 (m+4)}-\frac{2 c (d+e x)^{m+3} \left (-a A e^3+3 a B d e^2-3 A c d^2 e+5 B c d^3\right )}{e^6 (m+3)}-\frac{c^2 (5 B d-A e) (d+e x)^{m+5}}{e^6 (m+5)}+\frac{B c^2 (d+e x)^{m+6}}{e^6 (m+6)} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x)*(d + e*x)^m*(a + c*x^2)^2,x]
[Out]
-(((B*d - A*e)*(c*d^2 + a*e^2)^2*(d + e*x)^(1 + m))/(e^6*(1 + m))) + ((c*d^2 + a
*e^2)*(5*B*c*d^2 - 4*A*c*d*e + a*B*e^2)*(d + e*x)^(2 + m))/(e^6*(2 + m)) - (2*c*
(5*B*c*d^3 - 3*A*c*d^2*e + 3*a*B*d*e^2 - a*A*e^3)*(d + e*x)^(3 + m))/(e^6*(3 + m
)) + (2*c*(5*B*c*d^2 - 2*A*c*d*e + a*B*e^2)*(d + e*x)^(4 + m))/(e^6*(4 + m)) - (
c^2*(5*B*d - A*e)*(d + e*x)^(5 + m))/(e^6*(5 + m)) + (B*c^2*(d + e*x)^(6 + m))/(
e^6*(6 + m))
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Rubi in Sympy [A] time = 76.2719, size = 226, normalized size = 0.97 \[ \frac{B c^{2} \left (d + e x\right )^{m + 6}}{e^{6} \left (m + 6\right )} + \frac{c^{2} \left (d + e x\right )^{m + 5} \left (A e - 5 B d\right )}{e^{6} \left (m + 5\right )} + \frac{2 c \left (d + e x\right )^{m + 3} \left (A a e^{3} + 3 A c d^{2} e - 3 B a d e^{2} - 5 B c d^{3}\right )}{e^{6} \left (m + 3\right )} + \frac{2 c \left (d + e x\right )^{m + 4} \left (- 2 A c d e + B a e^{2} + 5 B c d^{2}\right )}{e^{6} \left (m + 4\right )} + \frac{\left (d + e x\right )^{m + 1} \left (A e - B d\right ) \left (a e^{2} + c d^{2}\right )^{2}}{e^{6} \left (m + 1\right )} + \frac{\left (d + e x\right )^{m + 2} \left (a e^{2} + c d^{2}\right ) \left (- 4 A c d e + B a e^{2} + 5 B c d^{2}\right )}{e^{6} \left (m + 2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(e*x+d)**m*(c*x**2+a)**2,x)
[Out]
B*c**2*(d + e*x)**(m + 6)/(e**6*(m + 6)) + c**2*(d + e*x)**(m + 5)*(A*e - 5*B*d)
/(e**6*(m + 5)) + 2*c*(d + e*x)**(m + 3)*(A*a*e**3 + 3*A*c*d**2*e - 3*B*a*d*e**2
- 5*B*c*d**3)/(e**6*(m + 3)) + 2*c*(d + e*x)**(m + 4)*(-2*A*c*d*e + B*a*e**2 +
5*B*c*d**2)/(e**6*(m + 4)) + (d + e*x)**(m + 1)*(A*e - B*d)*(a*e**2 + c*d**2)**2
/(e**6*(m + 1)) + (d + e*x)**(m + 2)*(a*e**2 + c*d**2)*(-4*A*c*d*e + B*a*e**2 +
5*B*c*d**2)/(e**6*(m + 2))
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Mathematica [A] time = 0.830991, size = 433, normalized size = 1.85 \[ \frac{(d+e x)^{m+1} \left (A e (m+6) \left (a^2 e^4 \left (m^4+14 m^3+71 m^2+154 m+120\right )+2 a c e^2 \left (m^2+9 m+20\right ) \left (2 d^2-2 d e (m+1) x+e^2 \left (m^2+3 m+2\right ) x^2\right )+c^2 \left (24 d^4-24 d^3 e (m+1) x+12 d^2 e^2 \left (m^2+3 m+2\right ) x^2-4 d e^3 \left (m^3+6 m^2+11 m+6\right ) x^3+e^4 \left (m^4+10 m^3+35 m^2+50 m+24\right ) x^4\right )\right )-B \left (a^2 e^4 \left (m^4+18 m^3+119 m^2+342 m+360\right ) (d-e (m+1) x)-2 a c e^2 \left (m^2+11 m+30\right ) \left (-6 d^3+6 d^2 e (m+1) x-3 d e^2 \left (m^2+3 m+2\right ) x^2+e^3 \left (m^3+6 m^2+11 m+6\right ) x^3\right )+c^2 \left (120 d^5-120 d^4 e (m+1) x+60 d^3 e^2 \left (m^2+3 m+2\right ) x^2-20 d^2 e^3 \left (m^3+6 m^2+11 m+6\right ) x^3+5 d e^4 \left (m^4+10 m^3+35 m^2+50 m+24\right ) x^4-e^5 \left (m^5+15 m^4+85 m^3+225 m^2+274 m+120\right ) x^5\right )\right )\right )}{e^6 (m+1) (m+2) (m+3) (m+4) (m+5) (m+6)} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x)*(d + e*x)^m*(a + c*x^2)^2,x]
[Out]
((d + e*x)^(1 + m)*(A*e*(6 + m)*(a^2*e^4*(120 + 154*m + 71*m^2 + 14*m^3 + m^4) +
2*a*c*e^2*(20 + 9*m + m^2)*(2*d^2 - 2*d*e*(1 + m)*x + e^2*(2 + 3*m + m^2)*x^2)
+ c^2*(24*d^4 - 24*d^3*e*(1 + m)*x + 12*d^2*e^2*(2 + 3*m + m^2)*x^2 - 4*d*e^3*(6
+ 11*m + 6*m^2 + m^3)*x^3 + e^4*(24 + 50*m + 35*m^2 + 10*m^3 + m^4)*x^4)) - B*(
a^2*e^4*(360 + 342*m + 119*m^2 + 18*m^3 + m^4)*(d - e*(1 + m)*x) - 2*a*c*e^2*(30
+ 11*m + m^2)*(-6*d^3 + 6*d^2*e*(1 + m)*x - 3*d*e^2*(2 + 3*m + m^2)*x^2 + e^3*(
6 + 11*m + 6*m^2 + m^3)*x^3) + c^2*(120*d^5 - 120*d^4*e*(1 + m)*x + 60*d^3*e^2*(
2 + 3*m + m^2)*x^2 - 20*d^2*e^3*(6 + 11*m + 6*m^2 + m^3)*x^3 + 5*d*e^4*(24 + 50*
m + 35*m^2 + 10*m^3 + m^4)*x^4 - e^5*(120 + 274*m + 225*m^2 + 85*m^3 + 15*m^4 +
m^5)*x^5))))/(e^6*(1 + m)*(2 + m)*(3 + m)*(4 + m)*(5 + m)*(6 + m))
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Maple [B] time = 0.019, size = 1271, normalized size = 5.4 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(e*x+d)^m*(c*x^2+a)^2,x)
[Out]
(e*x+d)^(1+m)*(B*c^2*e^5*m^5*x^5+A*c^2*e^5*m^5*x^4+15*B*c^2*e^5*m^4*x^5+16*A*c^2
*e^5*m^4*x^4+2*B*a*c*e^5*m^5*x^3-5*B*c^2*d*e^4*m^4*x^4+85*B*c^2*e^5*m^3*x^5+2*A*
a*c*e^5*m^5*x^2-4*A*c^2*d*e^4*m^4*x^3+95*A*c^2*e^5*m^3*x^4+34*B*a*c*e^5*m^4*x^3-
50*B*c^2*d*e^4*m^3*x^4+225*B*c^2*e^5*m^2*x^5+36*A*a*c*e^5*m^4*x^2-48*A*c^2*d*e^4
*m^3*x^3+260*A*c^2*e^5*m^2*x^4+B*a^2*e^5*m^5*x-6*B*a*c*d*e^4*m^4*x^2+214*B*a*c*e
^5*m^3*x^3+20*B*c^2*d^2*e^3*m^3*x^3-175*B*c^2*d*e^4*m^2*x^4+274*B*c^2*e^5*m*x^5+
A*a^2*e^5*m^5-4*A*a*c*d*e^4*m^4*x+242*A*a*c*e^5*m^3*x^2+12*A*c^2*d^2*e^3*m^3*x^2
-188*A*c^2*d*e^4*m^2*x^3+324*A*c^2*e^5*m*x^4+19*B*a^2*e^5*m^4*x-84*B*a*c*d*e^4*m
^3*x^2+614*B*a*c*e^5*m^2*x^3+120*B*c^2*d^2*e^3*m^2*x^3-250*B*c^2*d*e^4*m*x^4+120
*B*c^2*e^5*x^5+20*A*a^2*e^5*m^4-64*A*a*c*d*e^4*m^3*x+744*A*a*c*e^5*m^2*x^2+108*A
*c^2*d^2*e^3*m^2*x^2-288*A*c^2*d*e^4*m*x^3+144*A*c^2*e^5*x^4-B*a^2*d*e^4*m^4+137
*B*a^2*e^5*m^3*x+12*B*a*c*d^2*e^3*m^3*x-390*B*a*c*d*e^4*m^2*x^2+792*B*a*c*e^5*m*
x^3-60*B*c^2*d^3*e^2*m^2*x^2+220*B*c^2*d^2*e^3*m*x^3-120*B*c^2*d*e^4*x^4+155*A*a
^2*e^5*m^3+4*A*a*c*d^2*e^3*m^3-356*A*a*c*d*e^4*m^2*x+1016*A*a*c*e^5*m*x^2-24*A*c
^2*d^3*e^2*m^2*x+240*A*c^2*d^2*e^3*m*x^2-144*A*c^2*d*e^4*x^3-18*B*a^2*d*e^4*m^3+
461*B*a^2*e^5*m^2*x+144*B*a*c*d^2*e^3*m^2*x-672*B*a*c*d*e^4*m*x^2+360*B*a*c*e^5*
x^3-180*B*c^2*d^3*e^2*m*x^2+120*B*c^2*d^2*e^3*x^3+580*A*a^2*e^5*m^2+60*A*a*c*d^2
*e^3*m^2-776*A*a*c*d*e^4*m*x+480*A*a*c*e^5*x^2-168*A*c^2*d^3*e^2*m*x+144*A*c^2*d
^2*e^3*x^2-119*B*a^2*d*e^4*m^2+702*B*a^2*e^5*m*x-12*B*a*c*d^3*e^2*m^2+492*B*a*c*
d^2*e^3*m*x-360*B*a*c*d*e^4*x^2+120*B*c^2*d^4*e*m*x-120*B*c^2*d^3*e^2*x^2+1044*A
*a^2*e^5*m+296*A*a*c*d^2*e^3*m-480*A*a*c*d*e^4*x+24*A*c^2*d^4*e*m-144*A*c^2*d^3*
e^2*x-342*B*a^2*d*e^4*m+360*B*a^2*e^5*x-132*B*a*c*d^3*e^2*m+360*B*a*c*d^2*e^3*x+
120*B*c^2*d^4*e*x+720*A*a^2*e^5+480*A*a*c*d^2*e^3+144*A*c^2*d^4*e-360*B*a^2*d*e^
4-360*B*a*c*d^3*e^2-120*B*c^2*d^5)/e^6/(m^6+21*m^5+175*m^4+735*m^3+1624*m^2+1764
*m+720)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^2*(B*x + A)*(e*x + d)^m,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 0.290317, size = 1854, normalized size = 7.92 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^2*(B*x + A)*(e*x + d)^m,x, algorithm="fricas")
[Out]
(A*a^2*d*e^5*m^5 - 120*B*c^2*d^6 + 144*A*c^2*d^5*e - 360*B*a*c*d^4*e^2 + 480*A*a
*c*d^3*e^3 - 360*B*a^2*d^2*e^4 + 720*A*a^2*d*e^5 + (B*c^2*e^6*m^5 + 15*B*c^2*e^6
*m^4 + 85*B*c^2*e^6*m^3 + 225*B*c^2*e^6*m^2 + 274*B*c^2*e^6*m + 120*B*c^2*e^6)*x
^6 + (144*A*c^2*e^6 + (B*c^2*d*e^5 + A*c^2*e^6)*m^5 + 2*(5*B*c^2*d*e^5 + 8*A*c^2
*e^6)*m^4 + 5*(7*B*c^2*d*e^5 + 19*A*c^2*e^6)*m^3 + 10*(5*B*c^2*d*e^5 + 26*A*c^2*
e^6)*m^2 + 12*(2*B*c^2*d*e^5 + 27*A*c^2*e^6)*m)*x^5 - (B*a^2*d^2*e^4 - 20*A*a^2*
d*e^5)*m^4 + (360*B*a*c*e^6 + (A*c^2*d*e^5 + 2*B*a*c*e^6)*m^5 - (5*B*c^2*d^2*e^4
- 12*A*c^2*d*e^5 - 34*B*a*c*e^6)*m^4 - (30*B*c^2*d^2*e^4 - 47*A*c^2*d*e^5 - 214
*B*a*c*e^6)*m^3 - (55*B*c^2*d^2*e^4 - 72*A*c^2*d*e^5 - 614*B*a*c*e^6)*m^2 - 6*(5
*B*c^2*d^2*e^4 - 6*A*c^2*d*e^5 - 132*B*a*c*e^6)*m)*x^4 + (4*A*a*c*d^3*e^3 - 18*B
*a^2*d^2*e^4 + 155*A*a^2*d*e^5)*m^3 + 2*(240*A*a*c*e^6 + (B*a*c*d*e^5 + A*a*c*e^
6)*m^5 - 2*(A*c^2*d^2*e^4 - 7*B*a*c*d*e^5 - 9*A*a*c*e^6)*m^4 + (10*B*c^2*d^3*e^3
- 18*A*c^2*d^2*e^4 + 65*B*a*c*d*e^5 + 121*A*a*c*e^6)*m^3 + 2*(15*B*c^2*d^3*e^3
- 20*A*c^2*d^2*e^4 + 56*B*a*c*d*e^5 + 186*A*a*c*e^6)*m^2 + 4*(5*B*c^2*d^3*e^3 -
6*A*c^2*d^2*e^4 + 15*B*a*c*d*e^5 + 127*A*a*c*e^6)*m)*x^3 - (12*B*a*c*d^4*e^2 - 6
0*A*a*c*d^3*e^3 + 119*B*a^2*d^2*e^4 - 580*A*a^2*d*e^5)*m^2 + (360*B*a^2*e^6 + (2
*A*a*c*d*e^5 + B*a^2*e^6)*m^5 - (6*B*a*c*d^2*e^4 - 32*A*a*c*d*e^5 - 19*B*a^2*e^6
)*m^4 + (12*A*c^2*d^3*e^3 - 72*B*a*c*d^2*e^4 + 178*A*a*c*d*e^5 + 137*B*a^2*e^6)*
m^3 - (60*B*c^2*d^4*e^2 - 84*A*c^2*d^3*e^3 + 246*B*a*c*d^2*e^4 - 388*A*a*c*d*e^5
- 461*B*a^2*e^6)*m^2 - 6*(10*B*c^2*d^4*e^2 - 12*A*c^2*d^3*e^3 + 30*B*a*c*d^2*e^
4 - 40*A*a*c*d*e^5 - 117*B*a^2*e^6)*m)*x^2 + 2*(12*A*c^2*d^5*e - 66*B*a*c*d^4*e^
2 + 148*A*a*c*d^3*e^3 - 171*B*a^2*d^2*e^4 + 522*A*a^2*d*e^5)*m + (720*A*a^2*e^6
+ (B*a^2*d*e^5 + A*a^2*e^6)*m^5 - 2*(2*A*a*c*d^2*e^4 - 9*B*a^2*d*e^5 - 10*A*a^2*
e^6)*m^4 + (12*B*a*c*d^3*e^3 - 60*A*a*c*d^2*e^4 + 119*B*a^2*d*e^5 + 155*A*a^2*e^
6)*m^3 - 2*(12*A*c^2*d^4*e^2 - 66*B*a*c*d^3*e^3 + 148*A*a*c*d^2*e^4 - 171*B*a^2*
d*e^5 - 290*A*a^2*e^6)*m^2 + 12*(10*B*c^2*d^5*e - 12*A*c^2*d^4*e^2 + 30*B*a*c*d^
3*e^3 - 40*A*a*c*d^2*e^4 + 30*B*a^2*d*e^5 + 87*A*a^2*e^6)*m)*x)*(e*x + d)^m/(e^6
*m^6 + 21*e^6*m^5 + 175*e^6*m^4 + 735*e^6*m^3 + 1624*e^6*m^2 + 1764*e^6*m + 720*
e^6)
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*(e*x+d)**m*(c*x**2+a)**2,x)
[Out]
Timed out
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GIAC/XCAS [A] time = 0.275413, size = 1, normalized size = 0. \[ \mathit{Done} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^2*(B*x + A)*(e*x + d)^m,x, algorithm="giac")
[Out]
Done