3.1223 \(\int (A+B x) (d+e x)^{3/2} \left (b x+c x^2\right )^2 \, dx\)

Optimal. Leaf size=267 \[ -\frac{2 (d+e x)^{11/2} \left (2 A c e (2 c d-b e)-B \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )\right )}{11 e^6}+\frac{2 (d+e x)^{9/2} \left (A e \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )-B d \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )\right )}{9 e^6}-\frac{2 d^2 (d+e x)^{5/2} (B d-A e) (c d-b e)^2}{5 e^6}-\frac{2 c (d+e x)^{13/2} (-A c e-2 b B e+5 B c d)}{13 e^6}+\frac{2 d (d+e x)^{7/2} (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{7 e^6}+\frac{2 B c^2 (d+e x)^{15/2}}{15 e^6} \]

[Out]

(-2*d^2*(B*d - A*e)*(c*d - b*e)^2*(d + e*x)^(5/2))/(5*e^6) + (2*d*(c*d - b*e)*(B
*d*(5*c*d - 3*b*e) - 2*A*e*(2*c*d - b*e))*(d + e*x)^(7/2))/(7*e^6) + (2*(A*e*(6*
c^2*d^2 - 6*b*c*d*e + b^2*e^2) - B*d*(10*c^2*d^2 - 12*b*c*d*e + 3*b^2*e^2))*(d +
e*x)^(9/2))/(9*e^6) - (2*(2*A*c*e*(2*c*d - b*e) - B*(10*c^2*d^2 - 8*b*c*d*e + b
^2*e^2))*(d + e*x)^(11/2))/(11*e^6) - (2*c*(5*B*c*d - 2*b*B*e - A*c*e)*(d + e*x)
^(13/2))/(13*e^6) + (2*B*c^2*(d + e*x)^(15/2))/(15*e^6)

_______________________________________________________________________________________

Rubi [A]  time = 0.469419, antiderivative size = 267, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ -\frac{2 (d+e x)^{11/2} \left (2 A c e (2 c d-b e)-B \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )\right )}{11 e^6}+\frac{2 (d+e x)^{9/2} \left (A e \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )-B d \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )\right )}{9 e^6}-\frac{2 d^2 (d+e x)^{5/2} (B d-A e) (c d-b e)^2}{5 e^6}-\frac{2 c (d+e x)^{13/2} (-A c e-2 b B e+5 B c d)}{13 e^6}+\frac{2 d (d+e x)^{7/2} (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{7 e^6}+\frac{2 B c^2 (d+e x)^{15/2}}{15 e^6} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)*(d + e*x)^(3/2)*(b*x + c*x^2)^2,x]

[Out]

(-2*d^2*(B*d - A*e)*(c*d - b*e)^2*(d + e*x)^(5/2))/(5*e^6) + (2*d*(c*d - b*e)*(B
*d*(5*c*d - 3*b*e) - 2*A*e*(2*c*d - b*e))*(d + e*x)^(7/2))/(7*e^6) + (2*(A*e*(6*
c^2*d^2 - 6*b*c*d*e + b^2*e^2) - B*d*(10*c^2*d^2 - 12*b*c*d*e + 3*b^2*e^2))*(d +
e*x)^(9/2))/(9*e^6) - (2*(2*A*c*e*(2*c*d - b*e) - B*(10*c^2*d^2 - 8*b*c*d*e + b
^2*e^2))*(d + e*x)^(11/2))/(11*e^6) - (2*c*(5*B*c*d - 2*b*B*e - A*c*e)*(d + e*x)
^(13/2))/(13*e^6) + (2*B*c^2*(d + e*x)^(15/2))/(15*e^6)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 103.842, size = 292, normalized size = 1.09 \[ \frac{2 B c^{2} \left (d + e x\right )^{\frac{15}{2}}}{15 e^{6}} + \frac{2 c \left (d + e x\right )^{\frac{13}{2}} \left (A c e + 2 B b e - 5 B c d\right )}{13 e^{6}} + \frac{2 d^{2} \left (d + e x\right )^{\frac{5}{2}} \left (A e - B d\right ) \left (b e - c d\right )^{2}}{5 e^{6}} - \frac{2 d \left (d + e x\right )^{\frac{7}{2}} \left (b e - c d\right ) \left (2 A b e^{2} - 4 A c d e - 3 B b d e + 5 B c d^{2}\right )}{7 e^{6}} + \frac{2 \left (d + e x\right )^{\frac{11}{2}} \left (2 A b c e^{2} - 4 A c^{2} d e + B b^{2} e^{2} - 8 B b c d e + 10 B c^{2} d^{2}\right )}{11 e^{6}} + \frac{2 \left (d + e x\right )^{\frac{9}{2}} \left (A b^{2} e^{3} - 6 A b c d e^{2} + 6 A c^{2} d^{2} e - 3 B b^{2} d e^{2} + 12 B b c d^{2} e - 10 B c^{2} d^{3}\right )}{9 e^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(e*x+d)**(3/2)*(c*x**2+b*x)**2,x)

[Out]

2*B*c**2*(d + e*x)**(15/2)/(15*e**6) + 2*c*(d + e*x)**(13/2)*(A*c*e + 2*B*b*e -
5*B*c*d)/(13*e**6) + 2*d**2*(d + e*x)**(5/2)*(A*e - B*d)*(b*e - c*d)**2/(5*e**6)
- 2*d*(d + e*x)**(7/2)*(b*e - c*d)*(2*A*b*e**2 - 4*A*c*d*e - 3*B*b*d*e + 5*B*c*
d**2)/(7*e**6) + 2*(d + e*x)**(11/2)*(2*A*b*c*e**2 - 4*A*c**2*d*e + B*b**2*e**2
- 8*B*b*c*d*e + 10*B*c**2*d**2)/(11*e**6) + 2*(d + e*x)**(9/2)*(A*b**2*e**3 - 6*
A*b*c*d*e**2 + 6*A*c**2*d**2*e - 3*B*b**2*d*e**2 + 12*B*b*c*d**2*e - 10*B*c**2*d
**3)/(9*e**6)

_______________________________________________________________________________________

Mathematica [A]  time = 0.587316, size = 272, normalized size = 1.02 \[ \frac{2 (d+e x)^{5/2} \left (A e \left (143 b^2 e^2 \left (8 d^2-20 d e x+35 e^2 x^2\right )+78 b c e \left (-16 d^3+40 d^2 e x-70 d e^2 x^2+105 e^3 x^3\right )+3 c^2 \left (128 d^4-320 d^3 e x+560 d^2 e^2 x^2-840 d e^3 x^3+1155 e^4 x^4\right )\right )+B \left (39 b^2 e^2 \left (-16 d^3+40 d^2 e x-70 d e^2 x^2+105 e^3 x^3\right )+6 b c e \left (128 d^4-320 d^3 e x+560 d^2 e^2 x^2-840 d e^3 x^3+1155 e^4 x^4\right )+c^2 \left (-256 d^5+640 d^4 e x-1120 d^3 e^2 x^2+1680 d^2 e^3 x^3-2310 d e^4 x^4+3003 e^5 x^5\right )\right )\right )}{45045 e^6} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)*(d + e*x)^(3/2)*(b*x + c*x^2)^2,x]

[Out]

(2*(d + e*x)^(5/2)*(A*e*(143*b^2*e^2*(8*d^2 - 20*d*e*x + 35*e^2*x^2) + 78*b*c*e*
(-16*d^3 + 40*d^2*e*x - 70*d*e^2*x^2 + 105*e^3*x^3) + 3*c^2*(128*d^4 - 320*d^3*e
*x + 560*d^2*e^2*x^2 - 840*d*e^3*x^3 + 1155*e^4*x^4)) + B*(39*b^2*e^2*(-16*d^3 +
40*d^2*e*x - 70*d*e^2*x^2 + 105*e^3*x^3) + 6*b*c*e*(128*d^4 - 320*d^3*e*x + 560
*d^2*e^2*x^2 - 840*d*e^3*x^3 + 1155*e^4*x^4) + c^2*(-256*d^5 + 640*d^4*e*x - 112
0*d^3*e^2*x^2 + 1680*d^2*e^3*x^3 - 2310*d*e^4*x^4 + 3003*e^5*x^5))))/(45045*e^6)

_______________________________________________________________________________________

Maple [A]  time = 0.01, size = 341, normalized size = 1.3 \[{\frac{6006\,B{c}^{2}{x}^{5}{e}^{5}+6930\,A{c}^{2}{e}^{5}{x}^{4}+13860\,Bbc{e}^{5}{x}^{4}-4620\,B{c}^{2}d{e}^{4}{x}^{4}+16380\,Abc{e}^{5}{x}^{3}-5040\,A{c}^{2}d{e}^{4}{x}^{3}+8190\,B{b}^{2}{e}^{5}{x}^{3}-10080\,Bbcd{e}^{4}{x}^{3}+3360\,B{c}^{2}{d}^{2}{e}^{3}{x}^{3}+10010\,A{b}^{2}{e}^{5}{x}^{2}-10920\,Abcd{e}^{4}{x}^{2}+3360\,A{c}^{2}{d}^{2}{e}^{3}{x}^{2}-5460\,B{b}^{2}d{e}^{4}{x}^{2}+6720\,Bbc{d}^{2}{e}^{3}{x}^{2}-2240\,B{c}^{2}{d}^{3}{e}^{2}{x}^{2}-5720\,A{b}^{2}d{e}^{4}x+6240\,Abc{d}^{2}{e}^{3}x-1920\,A{c}^{2}{d}^{3}{e}^{2}x+3120\,B{b}^{2}{d}^{2}{e}^{3}x-3840\,Bbc{d}^{3}{e}^{2}x+1280\,B{c}^{2}{d}^{4}ex+2288\,A{b}^{2}{d}^{2}{e}^{3}-2496\,Abc{d}^{3}{e}^{2}+768\,A{c}^{2}{d}^{4}e-1248\,B{b}^{2}{d}^{3}{e}^{2}+1536\,Bbc{d}^{4}e-512\,B{c}^{2}{d}^{5}}{45045\,{e}^{6}} \left ( ex+d \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(e*x+d)^(3/2)*(c*x^2+b*x)^2,x)

[Out]

2/45045*(e*x+d)^(5/2)*(3003*B*c^2*e^5*x^5+3465*A*c^2*e^5*x^4+6930*B*b*c*e^5*x^4-
2310*B*c^2*d*e^4*x^4+8190*A*b*c*e^5*x^3-2520*A*c^2*d*e^4*x^3+4095*B*b^2*e^5*x^3-
5040*B*b*c*d*e^4*x^3+1680*B*c^2*d^2*e^3*x^3+5005*A*b^2*e^5*x^2-5460*A*b*c*d*e^4*
x^2+1680*A*c^2*d^2*e^3*x^2-2730*B*b^2*d*e^4*x^2+3360*B*b*c*d^2*e^3*x^2-1120*B*c^
2*d^3*e^2*x^2-2860*A*b^2*d*e^4*x+3120*A*b*c*d^2*e^3*x-960*A*c^2*d^3*e^2*x+1560*B
*b^2*d^2*e^3*x-1920*B*b*c*d^3*e^2*x+640*B*c^2*d^4*e*x+1144*A*b^2*d^2*e^3-1248*A*
b*c*d^3*e^2+384*A*c^2*d^4*e-624*B*b^2*d^3*e^2+768*B*b*c*d^4*e-256*B*c^2*d^5)/e^6

_______________________________________________________________________________________

Maxima [A]  time = 0.704221, size = 393, normalized size = 1.47 \[ \frac{2 \,{\left (3003 \,{\left (e x + d\right )}^{\frac{15}{2}} B c^{2} - 3465 \,{\left (5 \, B c^{2} d -{\left (2 \, B b c + A c^{2}\right )} e\right )}{\left (e x + d\right )}^{\frac{13}{2}} + 4095 \,{\left (10 \, B c^{2} d^{2} - 4 \,{\left (2 \, B b c + A c^{2}\right )} d e +{\left (B b^{2} + 2 \, A b c\right )} e^{2}\right )}{\left (e x + d\right )}^{\frac{11}{2}} - 5005 \,{\left (10 \, B c^{2} d^{3} - A b^{2} e^{3} - 6 \,{\left (2 \, B b c + A c^{2}\right )} d^{2} e + 3 \,{\left (B b^{2} + 2 \, A b c\right )} d e^{2}\right )}{\left (e x + d\right )}^{\frac{9}{2}} + 6435 \,{\left (5 \, B c^{2} d^{4} - 2 \, A b^{2} d e^{3} - 4 \,{\left (2 \, B b c + A c^{2}\right )} d^{3} e + 3 \,{\left (B b^{2} + 2 \, A b c\right )} d^{2} e^{2}\right )}{\left (e x + d\right )}^{\frac{7}{2}} - 9009 \,{\left (B c^{2} d^{5} - A b^{2} d^{2} e^{3} -{\left (2 \, B b c + A c^{2}\right )} d^{4} e +{\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{2}\right )}{\left (e x + d\right )}^{\frac{5}{2}}\right )}}{45045 \, e^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^2*(B*x + A)*(e*x + d)^(3/2),x, algorithm="maxima")

[Out]

2/45045*(3003*(e*x + d)^(15/2)*B*c^2 - 3465*(5*B*c^2*d - (2*B*b*c + A*c^2)*e)*(e
*x + d)^(13/2) + 4095*(10*B*c^2*d^2 - 4*(2*B*b*c + A*c^2)*d*e + (B*b^2 + 2*A*b*c
)*e^2)*(e*x + d)^(11/2) - 5005*(10*B*c^2*d^3 - A*b^2*e^3 - 6*(2*B*b*c + A*c^2)*d
^2*e + 3*(B*b^2 + 2*A*b*c)*d*e^2)*(e*x + d)^(9/2) + 6435*(5*B*c^2*d^4 - 2*A*b^2*
d*e^3 - 4*(2*B*b*c + A*c^2)*d^3*e + 3*(B*b^2 + 2*A*b*c)*d^2*e^2)*(e*x + d)^(7/2)
- 9009*(B*c^2*d^5 - A*b^2*d^2*e^3 - (2*B*b*c + A*c^2)*d^4*e + (B*b^2 + 2*A*b*c)
*d^3*e^2)*(e*x + d)^(5/2))/e^6

_______________________________________________________________________________________

Fricas [A]  time = 0.30274, size = 574, normalized size = 2.15 \[ \frac{2 \,{\left (3003 \, B c^{2} e^{7} x^{7} - 256 \, B c^{2} d^{7} + 1144 \, A b^{2} d^{4} e^{3} + 384 \,{\left (2 \, B b c + A c^{2}\right )} d^{6} e - 624 \,{\left (B b^{2} + 2 \, A b c\right )} d^{5} e^{2} + 231 \,{\left (16 \, B c^{2} d e^{6} + 15 \,{\left (2 \, B b c + A c^{2}\right )} e^{7}\right )} x^{6} + 63 \,{\left (B c^{2} d^{2} e^{5} + 70 \,{\left (2 \, B b c + A c^{2}\right )} d e^{6} + 65 \,{\left (B b^{2} + 2 \, A b c\right )} e^{7}\right )} x^{5} - 35 \,{\left (2 \, B c^{2} d^{3} e^{4} - 143 \, A b^{2} e^{7} - 3 \,{\left (2 \, B b c + A c^{2}\right )} d^{2} e^{5} - 156 \,{\left (B b^{2} + 2 \, A b c\right )} d e^{6}\right )} x^{4} + 5 \,{\left (16 \, B c^{2} d^{4} e^{3} + 1430 \, A b^{2} d e^{6} - 24 \,{\left (2 \, B b c + A c^{2}\right )} d^{3} e^{4} + 39 \,{\left (B b^{2} + 2 \, A b c\right )} d^{2} e^{5}\right )} x^{3} - 3 \,{\left (32 \, B c^{2} d^{5} e^{2} - 143 \, A b^{2} d^{2} e^{5} - 48 \,{\left (2 \, B b c + A c^{2}\right )} d^{4} e^{3} + 78 \,{\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{4}\right )} x^{2} + 4 \,{\left (32 \, B c^{2} d^{6} e - 143 \, A b^{2} d^{3} e^{4} - 48 \,{\left (2 \, B b c + A c^{2}\right )} d^{5} e^{2} + 78 \,{\left (B b^{2} + 2 \, A b c\right )} d^{4} e^{3}\right )} x\right )} \sqrt{e x + d}}{45045 \, e^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^2*(B*x + A)*(e*x + d)^(3/2),x, algorithm="fricas")

[Out]

2/45045*(3003*B*c^2*e^7*x^7 - 256*B*c^2*d^7 + 1144*A*b^2*d^4*e^3 + 384*(2*B*b*c
+ A*c^2)*d^6*e - 624*(B*b^2 + 2*A*b*c)*d^5*e^2 + 231*(16*B*c^2*d*e^6 + 15*(2*B*b
*c + A*c^2)*e^7)*x^6 + 63*(B*c^2*d^2*e^5 + 70*(2*B*b*c + A*c^2)*d*e^6 + 65*(B*b^
2 + 2*A*b*c)*e^7)*x^5 - 35*(2*B*c^2*d^3*e^4 - 143*A*b^2*e^7 - 3*(2*B*b*c + A*c^2
)*d^2*e^5 - 156*(B*b^2 + 2*A*b*c)*d*e^6)*x^4 + 5*(16*B*c^2*d^4*e^3 + 1430*A*b^2*
d*e^6 - 24*(2*B*b*c + A*c^2)*d^3*e^4 + 39*(B*b^2 + 2*A*b*c)*d^2*e^5)*x^3 - 3*(32
*B*c^2*d^5*e^2 - 143*A*b^2*d^2*e^5 - 48*(2*B*b*c + A*c^2)*d^4*e^3 + 78*(B*b^2 +
2*A*b*c)*d^3*e^4)*x^2 + 4*(32*B*c^2*d^6*e - 143*A*b^2*d^3*e^4 - 48*(2*B*b*c + A*
c^2)*d^5*e^2 + 78*(B*b^2 + 2*A*b*c)*d^4*e^3)*x)*sqrt(e*x + d)/e^6

_______________________________________________________________________________________

Sympy [A]  time = 11.2614, size = 937, normalized size = 3.51 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(e*x+d)**(3/2)*(c*x**2+b*x)**2,x)

[Out]

2*A*b**2*d*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/
7)/e**3 + 2*A*b**2*(-d**3*(d + e*x)**(3/2)/3 + 3*d**2*(d + e*x)**(5/2)/5 - 3*d*(
d + e*x)**(7/2)/7 + (d + e*x)**(9/2)/9)/e**3 + 4*A*b*c*d*(-d**3*(d + e*x)**(3/2)
/3 + 3*d**2*(d + e*x)**(5/2)/5 - 3*d*(d + e*x)**(7/2)/7 + (d + e*x)**(9/2)/9)/e*
*4 + 4*A*b*c*(d**4*(d + e*x)**(3/2)/3 - 4*d**3*(d + e*x)**(5/2)/5 + 6*d**2*(d +
e*x)**(7/2)/7 - 4*d*(d + e*x)**(9/2)/9 + (d + e*x)**(11/2)/11)/e**4 + 2*A*c**2*d
*(d**4*(d + e*x)**(3/2)/3 - 4*d**3*(d + e*x)**(5/2)/5 + 6*d**2*(d + e*x)**(7/2)/
7 - 4*d*(d + e*x)**(9/2)/9 + (d + e*x)**(11/2)/11)/e**5 + 2*A*c**2*(-d**5*(d + e
*x)**(3/2)/3 + d**4*(d + e*x)**(5/2) - 10*d**3*(d + e*x)**(7/2)/7 + 10*d**2*(d +
e*x)**(9/2)/9 - 5*d*(d + e*x)**(11/2)/11 + (d + e*x)**(13/2)/13)/e**5 + 2*B*b**
2*d*(-d**3*(d + e*x)**(3/2)/3 + 3*d**2*(d + e*x)**(5/2)/5 - 3*d*(d + e*x)**(7/2)
/7 + (d + e*x)**(9/2)/9)/e**4 + 2*B*b**2*(d**4*(d + e*x)**(3/2)/3 - 4*d**3*(d +
e*x)**(5/2)/5 + 6*d**2*(d + e*x)**(7/2)/7 - 4*d*(d + e*x)**(9/2)/9 + (d + e*x)**
(11/2)/11)/e**4 + 4*B*b*c*d*(d**4*(d + e*x)**(3/2)/3 - 4*d**3*(d + e*x)**(5/2)/5
+ 6*d**2*(d + e*x)**(7/2)/7 - 4*d*(d + e*x)**(9/2)/9 + (d + e*x)**(11/2)/11)/e*
*5 + 4*B*b*c*(-d**5*(d + e*x)**(3/2)/3 + d**4*(d + e*x)**(5/2) - 10*d**3*(d + e*
x)**(7/2)/7 + 10*d**2*(d + e*x)**(9/2)/9 - 5*d*(d + e*x)**(11/2)/11 + (d + e*x)*
*(13/2)/13)/e**5 + 2*B*c**2*d*(-d**5*(d + e*x)**(3/2)/3 + d**4*(d + e*x)**(5/2)
- 10*d**3*(d + e*x)**(7/2)/7 + 10*d**2*(d + e*x)**(9/2)/9 - 5*d*(d + e*x)**(11/2
)/11 + (d + e*x)**(13/2)/13)/e**6 + 2*B*c**2*(d**6*(d + e*x)**(3/2)/3 - 6*d**5*(
d + e*x)**(5/2)/5 + 15*d**4*(d + e*x)**(7/2)/7 - 20*d**3*(d + e*x)**(9/2)/9 + 15
*d**2*(d + e*x)**(11/2)/11 - 6*d*(d + e*x)**(13/2)/13 + (d + e*x)**(15/2)/15)/e*
*6

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.297255, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^2*(B*x + A)*(e*x + d)^(3/2),x, algorithm="giac")

[Out]

Done