3.1215 \(\int (A+B x) (d+e x)^{3/2} \left (b x+c x^2\right ) \, dx\)

Optimal. Leaf size=126 \[ -\frac{2 (d+e x)^{9/2} (-A c e-b B e+3 B c d)}{9 e^4}+\frac{2 (d+e x)^{7/2} (B d (3 c d-2 b e)-A e (2 c d-b e))}{7 e^4}-\frac{2 d (d+e x)^{5/2} (B d-A e) (c d-b e)}{5 e^4}+\frac{2 B c (d+e x)^{11/2}}{11 e^4} \]

[Out]

(-2*d*(B*d - A*e)*(c*d - b*e)*(d + e*x)^(5/2))/(5*e^4) + (2*(B*d*(3*c*d - 2*b*e)
- A*e*(2*c*d - b*e))*(d + e*x)^(7/2))/(7*e^4) - (2*(3*B*c*d - b*B*e - A*c*e)*(d
+ e*x)^(9/2))/(9*e^4) + (2*B*c*(d + e*x)^(11/2))/(11*e^4)

_______________________________________________________________________________________

Rubi [A]  time = 0.209876, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ -\frac{2 (d+e x)^{9/2} (-A c e-b B e+3 B c d)}{9 e^4}+\frac{2 (d+e x)^{7/2} (B d (3 c d-2 b e)-A e (2 c d-b e))}{7 e^4}-\frac{2 d (d+e x)^{5/2} (B d-A e) (c d-b e)}{5 e^4}+\frac{2 B c (d+e x)^{11/2}}{11 e^4} \]

Antiderivative was successfully verified.

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

[Out]

(-2*d*(B*d - A*e)*(c*d - b*e)*(d + e*x)^(5/2))/(5*e^4) + (2*(B*d*(3*c*d - 2*b*e)
- A*e*(2*c*d - b*e))*(d + e*x)^(7/2))/(7*e^4) - (2*(3*B*c*d - b*B*e - A*c*e)*(d
+ e*x)^(9/2))/(9*e^4) + (2*B*c*(d + e*x)^(11/2))/(11*e^4)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 33.6772, size = 128, normalized size = 1.02 \[ \frac{2 B c \left (d + e x\right )^{\frac{11}{2}}}{11 e^{4}} - \frac{2 d \left (d + e x\right )^{\frac{5}{2}} \left (A e - B d\right ) \left (b e - c d\right )}{5 e^{4}} + \frac{2 \left (d + e x\right )^{\frac{9}{2}} \left (A c e + B b e - 3 B c d\right )}{9 e^{4}} + \frac{2 \left (d + e x\right )^{\frac{7}{2}} \left (A b e^{2} - 2 A c d e - 2 B b d e + 3 B c d^{2}\right )}{7 e^{4}} \]

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),x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.189072, size = 114, normalized size = 0.9 \[ \frac{2 (d+e x)^{5/2} \left (11 A e \left (9 b e (5 e x-2 d)+c \left (8 d^2-20 d e x+35 e^2 x^2\right )\right )+B \left (11 b e \left (8 d^2-20 d e x+35 e^2 x^2\right )-3 c \left (16 d^3-40 d^2 e x+70 d e^2 x^2-105 e^3 x^3\right )\right )\right )}{3465 e^4} \]

Antiderivative was successfully verified.

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

[Out]

(2*(d + e*x)^(5/2)*(11*A*e*(9*b*e*(-2*d + 5*e*x) + c*(8*d^2 - 20*d*e*x + 35*e^2*
x^2)) + B*(11*b*e*(8*d^2 - 20*d*e*x + 35*e^2*x^2) - 3*c*(16*d^3 - 40*d^2*e*x + 7
0*d*e^2*x^2 - 105*e^3*x^3))))/(3465*e^4)

_______________________________________________________________________________________

Maple [A]  time = 0.007, size = 121, normalized size = 1. \[ -{\frac{-630\,Bc{x}^{3}{e}^{3}-770\,Ac{e}^{3}{x}^{2}-770\,Bb{e}^{3}{x}^{2}+420\,Bcd{e}^{2}{x}^{2}-990\,Ab{e}^{3}x+440\,Acd{e}^{2}x+440\,Bbd{e}^{2}x-240\,Bc{d}^{2}ex+396\,Abd{e}^{2}-176\,Ac{d}^{2}e-176\,Bb{d}^{2}e+96\,Bc{d}^{3}}{3465\,{e}^{4}} \left ( ex+d \right ) ^{{\frac{5}{2}}}} \]

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),x)

[Out]

-2/3465*(e*x+d)^(5/2)*(-315*B*c*e^3*x^3-385*A*c*e^3*x^2-385*B*b*e^3*x^2+210*B*c*
d*e^2*x^2-495*A*b*e^3*x+220*A*c*d*e^2*x+220*B*b*d*e^2*x-120*B*c*d^2*e*x+198*A*b*
d*e^2-88*A*c*d^2*e-88*B*b*d^2*e+48*B*c*d^3)/e^4

_______________________________________________________________________________________

Maxima [A]  time = 0.698205, size = 151, normalized size = 1.2 \[ \frac{2 \,{\left (315 \,{\left (e x + d\right )}^{\frac{11}{2}} B c - 385 \,{\left (3 \, B c d -{\left (B b + A c\right )} e\right )}{\left (e x + d\right )}^{\frac{9}{2}} + 495 \,{\left (3 \, B c d^{2} + A b e^{2} - 2 \,{\left (B b + A c\right )} d e\right )}{\left (e x + d\right )}^{\frac{7}{2}} - 693 \,{\left (B c d^{3} + A b d e^{2} -{\left (B b + A c\right )} d^{2} e\right )}{\left (e x + d\right )}^{\frac{5}{2}}\right )}}{3465 \, e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3465*(315*(e*x + d)^(11/2)*B*c - 385*(3*B*c*d - (B*b + A*c)*e)*(e*x + d)^(9/2)
+ 495*(3*B*c*d^2 + A*b*e^2 - 2*(B*b + A*c)*d*e)*(e*x + d)^(7/2) - 693*(B*c*d^3
+ A*b*d*e^2 - (B*b + A*c)*d^2*e)*(e*x + d)^(5/2))/e^4

_______________________________________________________________________________________

Fricas [A]  time = 0.319343, size = 257, normalized size = 2.04 \[ \frac{2 \,{\left (315 \, B c e^{5} x^{5} - 48 \, B c d^{5} - 198 \, A b d^{3} e^{2} + 88 \,{\left (B b + A c\right )} d^{4} e + 35 \,{\left (12 \, B c d e^{4} + 11 \,{\left (B b + A c\right )} e^{5}\right )} x^{4} + 5 \,{\left (3 \, B c d^{2} e^{3} + 99 \, A b e^{5} + 110 \,{\left (B b + A c\right )} d e^{4}\right )} x^{3} - 3 \,{\left (6 \, B c d^{3} e^{2} - 264 \, A b d e^{4} - 11 \,{\left (B b + A c\right )} d^{2} e^{3}\right )} x^{2} +{\left (24 \, B c d^{4} e + 99 \, A b d^{2} e^{3} - 44 \,{\left (B b + A c\right )} d^{3} e^{2}\right )} x\right )} \sqrt{e x + d}}{3465 \, e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3465*(315*B*c*e^5*x^5 - 48*B*c*d^5 - 198*A*b*d^3*e^2 + 88*(B*b + A*c)*d^4*e +
35*(12*B*c*d*e^4 + 11*(B*b + A*c)*e^5)*x^4 + 5*(3*B*c*d^2*e^3 + 99*A*b*e^5 + 110
*(B*b + A*c)*d*e^4)*x^3 - 3*(6*B*c*d^3*e^2 - 264*A*b*d*e^4 - 11*(B*b + A*c)*d^2*
e^3)*x^2 + (24*B*c*d^4*e + 99*A*b*d^2*e^3 - 44*(B*b + A*c)*d^3*e^2)*x)*sqrt(e*x
+ d)/e^4

_______________________________________________________________________________________

Sympy [A]  time = 7.90511, size = 434, normalized size = 3.44 \[ \frac{2 A b d \left (- \frac{d \left (d + e x\right )^{\frac{3}{2}}}{3} + \frac{\left (d + e x\right )^{\frac{5}{2}}}{5}\right )}{e^{2}} + \frac{2 A b \left (\frac{d^{2} \left (d + e x\right )^{\frac{3}{2}}}{3} - \frac{2 d \left (d + e x\right )^{\frac{5}{2}}}{5} + \frac{\left (d + e x\right )^{\frac{7}{2}}}{7}\right )}{e^{2}} + \frac{2 A c d \left (\frac{d^{2} \left (d + e x\right )^{\frac{3}{2}}}{3} - \frac{2 d \left (d + e x\right )^{\frac{5}{2}}}{5} + \frac{\left (d + e x\right )^{\frac{7}{2}}}{7}\right )}{e^{3}} + \frac{2 A c \left (- \frac{d^{3} \left (d + e x\right )^{\frac{3}{2}}}{3} + \frac{3 d^{2} \left (d + e x\right )^{\frac{5}{2}}}{5} - \frac{3 d \left (d + e x\right )^{\frac{7}{2}}}{7} + \frac{\left (d + e x\right )^{\frac{9}{2}}}{9}\right )}{e^{3}} + \frac{2 B b d \left (\frac{d^{2} \left (d + e x\right )^{\frac{3}{2}}}{3} - \frac{2 d \left (d + e x\right )^{\frac{5}{2}}}{5} + \frac{\left (d + e x\right )^{\frac{7}{2}}}{7}\right )}{e^{3}} + \frac{2 B b \left (- \frac{d^{3} \left (d + e x\right )^{\frac{3}{2}}}{3} + \frac{3 d^{2} \left (d + e x\right )^{\frac{5}{2}}}{5} - \frac{3 d \left (d + e x\right )^{\frac{7}{2}}}{7} + \frac{\left (d + e x\right )^{\frac{9}{2}}}{9}\right )}{e^{3}} + \frac{2 B c d \left (- \frac{d^{3} \left (d + e x\right )^{\frac{3}{2}}}{3} + \frac{3 d^{2} \left (d + e x\right )^{\frac{5}{2}}}{5} - \frac{3 d \left (d + e x\right )^{\frac{7}{2}}}{7} + \frac{\left (d + e x\right )^{\frac{9}{2}}}{9}\right )}{e^{4}} + \frac{2 B c \left (\frac{d^{4} \left (d + e x\right )^{\frac{3}{2}}}{3} - \frac{4 d^{3} \left (d + e x\right )^{\frac{5}{2}}}{5} + \frac{6 d^{2} \left (d + e x\right )^{\frac{7}{2}}}{7} - \frac{4 d \left (d + e x\right )^{\frac{9}{2}}}{9} + \frac{\left (d + e x\right )^{\frac{11}{2}}}{11}\right )}{e^{4}} \]

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),x)

[Out]

2*A*b*d*(-d*(d + e*x)**(3/2)/3 + (d + e*x)**(5/2)/5)/e**2 + 2*A*b*(d**2*(d + e*x
)**(3/2)/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**2 + 2*A*c*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*c*(-
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 + 2*B*b*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*B*b*(-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 + 2*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 + 2*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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.287779, size = 594, normalized size = 4.71 \[ \frac{2}{3465} \,{\left (231 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} d\right )} A b d e^{\left (-1\right )} + 33 \,{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} e^{12} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} d e^{12} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2} e^{12}\right )} B b d e^{\left (-14\right )} + 33 \,{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} e^{12} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} d e^{12} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2} e^{12}\right )} A c d e^{\left (-14\right )} + 11 \,{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} e^{24} - 135 \,{\left (x e + d\right )}^{\frac{7}{2}} d e^{24} + 189 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} e^{24} - 105 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3} e^{24}\right )} B c d e^{\left (-27\right )} + 33 \,{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} e^{12} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} d e^{12} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2} e^{12}\right )} A b e^{\left (-13\right )} + 11 \,{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} e^{24} - 135 \,{\left (x e + d\right )}^{\frac{7}{2}} d e^{24} + 189 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} e^{24} - 105 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3} e^{24}\right )} B b e^{\left (-26\right )} + 11 \,{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} e^{24} - 135 \,{\left (x e + d\right )}^{\frac{7}{2}} d e^{24} + 189 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} e^{24} - 105 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3} e^{24}\right )} A c e^{\left (-26\right )} +{\left (315 \,{\left (x e + d\right )}^{\frac{11}{2}} e^{40} - 1540 \,{\left (x e + d\right )}^{\frac{9}{2}} d e^{40} + 2970 \,{\left (x e + d\right )}^{\frac{7}{2}} d^{2} e^{40} - 2772 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{3} e^{40} + 1155 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{4} e^{40}\right )} B c e^{\left (-43\right )}\right )} e^{\left (-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3465*(231*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*A*b*d*e^(-1) + 33*(15*(x*e
+ d)^(7/2)*e^12 - 42*(x*e + d)^(5/2)*d*e^12 + 35*(x*e + d)^(3/2)*d^2*e^12)*B*b*
d*e^(-14) + 33*(15*(x*e + d)^(7/2)*e^12 - 42*(x*e + d)^(5/2)*d*e^12 + 35*(x*e +
d)^(3/2)*d^2*e^12)*A*c*d*e^(-14) + 11*(35*(x*e + d)^(9/2)*e^24 - 135*(x*e + d)^(
7/2)*d*e^24 + 189*(x*e + d)^(5/2)*d^2*e^24 - 105*(x*e + d)^(3/2)*d^3*e^24)*B*c*d
*e^(-27) + 33*(15*(x*e + d)^(7/2)*e^12 - 42*(x*e + d)^(5/2)*d*e^12 + 35*(x*e + d
)^(3/2)*d^2*e^12)*A*b*e^(-13) + 11*(35*(x*e + d)^(9/2)*e^24 - 135*(x*e + d)^(7/2
)*d*e^24 + 189*(x*e + d)^(5/2)*d^2*e^24 - 105*(x*e + d)^(3/2)*d^3*e^24)*B*b*e^(-
26) + 11*(35*(x*e + d)^(9/2)*e^24 - 135*(x*e + d)^(7/2)*d*e^24 + 189*(x*e + d)^(
5/2)*d^2*e^24 - 105*(x*e + d)^(3/2)*d^3*e^24)*A*c*e^(-26) + (315*(x*e + d)^(11/2
)*e^40 - 1540*(x*e + d)^(9/2)*d*e^40 + 2970*(x*e + d)^(7/2)*d^2*e^40 - 2772*(x*e
+ d)^(5/2)*d^3*e^40 + 1155*(x*e + d)^(3/2)*d^4*e^40)*B*c*e^(-43))*e^(-1)