3.1720 \(\int (a+b x)^3 (A+B x) \sqrt{d+e x} \, dx\)

Optimal. Leaf size=173 \[ -\frac{2 b^2 (d+e x)^{9/2} (-3 a B e-A b e+4 b B d)}{9 e^5}+\frac{6 b (d+e x)^{7/2} (b d-a e) (-a B e-A b e+2 b B d)}{7 e^5}-\frac{2 (d+e x)^{5/2} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{5 e^5}+\frac{2 (d+e x)^{3/2} (b d-a e)^3 (B d-A e)}{3 e^5}+\frac{2 b^3 B (d+e x)^{11/2}}{11 e^5} \]

[Out]

(2*(b*d - a*e)^3*(B*d - A*e)*(d + e*x)^(3/2))/(3*e^5) - (2*(b*d - a*e)^2*(4*b*B*
d - 3*A*b*e - a*B*e)*(d + e*x)^(5/2))/(5*e^5) + (6*b*(b*d - a*e)*(2*b*B*d - A*b*
e - a*B*e)*(d + e*x)^(7/2))/(7*e^5) - (2*b^2*(4*b*B*d - A*b*e - 3*a*B*e)*(d + e*
x)^(9/2))/(9*e^5) + (2*b^3*B*(d + e*x)^(11/2))/(11*e^5)

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Rubi [A]  time = 0.211043, antiderivative size = 173, 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 \[ -\frac{2 b^2 (d+e x)^{9/2} (-3 a B e-A b e+4 b B d)}{9 e^5}+\frac{6 b (d+e x)^{7/2} (b d-a e) (-a B e-A b e+2 b B d)}{7 e^5}-\frac{2 (d+e x)^{5/2} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{5 e^5}+\frac{2 (d+e x)^{3/2} (b d-a e)^3 (B d-A e)}{3 e^5}+\frac{2 b^3 B (d+e x)^{11/2}}{11 e^5} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^3*(A + B*x)*Sqrt[d + e*x],x]

[Out]

(2*(b*d - a*e)^3*(B*d - A*e)*(d + e*x)^(3/2))/(3*e^5) - (2*(b*d - a*e)^2*(4*b*B*
d - 3*A*b*e - a*B*e)*(d + e*x)^(5/2))/(5*e^5) + (6*b*(b*d - a*e)*(2*b*B*d - A*b*
e - a*B*e)*(d + e*x)^(7/2))/(7*e^5) - (2*b^2*(4*b*B*d - A*b*e - 3*a*B*e)*(d + e*
x)^(9/2))/(9*e^5) + (2*b^3*B*(d + e*x)^(11/2))/(11*e^5)

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Rubi in Sympy [A]  time = 44.356, size = 170, normalized size = 0.98 \[ \frac{2 B b^{3} \left (d + e x\right )^{\frac{11}{2}}}{11 e^{5}} + \frac{2 b^{2} \left (d + e x\right )^{\frac{9}{2}} \left (A b e + 3 B a e - 4 B b d\right )}{9 e^{5}} + \frac{6 b \left (d + e x\right )^{\frac{7}{2}} \left (a e - b d\right ) \left (A b e + B a e - 2 B b d\right )}{7 e^{5}} + \frac{2 \left (d + e x\right )^{\frac{5}{2}} \left (a e - b d\right )^{2} \left (3 A b e + B a e - 4 B b d\right )}{5 e^{5}} + \frac{2 \left (d + e x\right )^{\frac{3}{2}} \left (A e - B d\right ) \left (a e - b d\right )^{3}}{3 e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**3*(B*x+A)*(e*x+d)**(1/2),x)

[Out]

2*B*b**3*(d + e*x)**(11/2)/(11*e**5) + 2*b**2*(d + e*x)**(9/2)*(A*b*e + 3*B*a*e
- 4*B*b*d)/(9*e**5) + 6*b*(d + e*x)**(7/2)*(a*e - b*d)*(A*b*e + B*a*e - 2*B*b*d)
/(7*e**5) + 2*(d + e*x)**(5/2)*(a*e - b*d)**2*(3*A*b*e + B*a*e - 4*B*b*d)/(5*e**
5) + 2*(d + e*x)**(3/2)*(A*e - B*d)*(a*e - b*d)**3/(3*e**5)

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Mathematica [A]  time = 0.235004, size = 227, normalized size = 1.31 \[ \frac{2 (d+e x)^{3/2} \left (231 a^3 e^3 (5 A e-2 B d+3 B e x)+99 a^2 b e^2 \left (7 A e (3 e x-2 d)+B \left (8 d^2-12 d e x+15 e^2 x^2\right )\right )-33 a b^2 e \left (B \left (16 d^3-24 d^2 e x+30 d e^2 x^2-35 e^3 x^3\right )-3 A e \left (8 d^2-12 d e x+15 e^2 x^2\right )\right )+b^3 \left (11 A e \left (-16 d^3+24 d^2 e x-30 d e^2 x^2+35 e^3 x^3\right )+B \left (128 d^4-192 d^3 e x+240 d^2 e^2 x^2-280 d e^3 x^3+315 e^4 x^4\right )\right )\right )}{3465 e^5} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^3*(A + B*x)*Sqrt[d + e*x],x]

[Out]

(2*(d + e*x)^(3/2)*(231*a^3*e^3*(-2*B*d + 5*A*e + 3*B*e*x) + 99*a^2*b*e^2*(7*A*e
*(-2*d + 3*e*x) + B*(8*d^2 - 12*d*e*x + 15*e^2*x^2)) - 33*a*b^2*e*(-3*A*e*(8*d^2
- 12*d*e*x + 15*e^2*x^2) + B*(16*d^3 - 24*d^2*e*x + 30*d*e^2*x^2 - 35*e^3*x^3))
+ b^3*(11*A*e*(-16*d^3 + 24*d^2*e*x - 30*d*e^2*x^2 + 35*e^3*x^3) + B*(128*d^4 -
192*d^3*e*x + 240*d^2*e^2*x^2 - 280*d*e^3*x^3 + 315*e^4*x^4))))/(3465*e^5)

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Maple [A]  time = 0.01, size = 301, normalized size = 1.7 \[{\frac{630\,B{b}^{3}{x}^{4}{e}^{4}+770\,A{b}^{3}{e}^{4}{x}^{3}+2310\,Ba{b}^{2}{e}^{4}{x}^{3}-560\,B{b}^{3}d{e}^{3}{x}^{3}+2970\,Aa{b}^{2}{e}^{4}{x}^{2}-660\,A{b}^{3}d{e}^{3}{x}^{2}+2970\,B{a}^{2}b{e}^{4}{x}^{2}-1980\,Ba{b}^{2}d{e}^{3}{x}^{2}+480\,B{b}^{3}{d}^{2}{e}^{2}{x}^{2}+4158\,A{a}^{2}b{e}^{4}x-2376\,Aa{b}^{2}d{e}^{3}x+528\,A{b}^{3}{d}^{2}{e}^{2}x+1386\,B{a}^{3}{e}^{4}x-2376\,B{a}^{2}bd{e}^{3}x+1584\,Ba{b}^{2}{d}^{2}{e}^{2}x-384\,B{b}^{3}{d}^{3}ex+2310\,{a}^{3}A{e}^{4}-2772\,A{a}^{2}bd{e}^{3}+1584\,Aa{b}^{2}{d}^{2}{e}^{2}-352\,A{b}^{3}{d}^{3}e-924\,B{a}^{3}d{e}^{3}+1584\,B{a}^{2}b{d}^{2}{e}^{2}-1056\,Ba{b}^{2}{d}^{3}e+256\,B{b}^{3}{d}^{4}}{3465\,{e}^{5}} \left ( ex+d \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^3*(B*x+A)*(e*x+d)^(1/2),x)

[Out]

2/3465*(e*x+d)^(3/2)*(315*B*b^3*e^4*x^4+385*A*b^3*e^4*x^3+1155*B*a*b^2*e^4*x^3-2
80*B*b^3*d*e^3*x^3+1485*A*a*b^2*e^4*x^2-330*A*b^3*d*e^3*x^2+1485*B*a^2*b*e^4*x^2
-990*B*a*b^2*d*e^3*x^2+240*B*b^3*d^2*e^2*x^2+2079*A*a^2*b*e^4*x-1188*A*a*b^2*d*e
^3*x+264*A*b^3*d^2*e^2*x+693*B*a^3*e^4*x-1188*B*a^2*b*d*e^3*x+792*B*a*b^2*d^2*e^
2*x-192*B*b^3*d^3*e*x+1155*A*a^3*e^4-1386*A*a^2*b*d*e^3+792*A*a*b^2*d^2*e^2-176*
A*b^3*d^3*e-462*B*a^3*d*e^3+792*B*a^2*b*d^2*e^2-528*B*a*b^2*d^3*e+128*B*b^3*d^4)
/e^5

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Maxima [A]  time = 1.35613, size = 358, normalized size = 2.07 \[ \frac{2 \,{\left (315 \,{\left (e x + d\right )}^{\frac{11}{2}} B b^{3} - 385 \,{\left (4 \, B b^{3} d -{\left (3 \, B a b^{2} + A b^{3}\right )} e\right )}{\left (e x + d\right )}^{\frac{9}{2}} + 1485 \,{\left (2 \, B b^{3} d^{2} -{\left (3 \, B a b^{2} + A b^{3}\right )} d e +{\left (B a^{2} b + A a b^{2}\right )} e^{2}\right )}{\left (e x + d\right )}^{\frac{7}{2}} - 693 \,{\left (4 \, B b^{3} d^{3} - 3 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e + 6 \,{\left (B a^{2} b + A a b^{2}\right )} d e^{2} -{\left (B a^{3} + 3 \, A a^{2} b\right )} e^{3}\right )}{\left (e x + d\right )}^{\frac{5}{2}} + 1155 \,{\left (B b^{3} d^{4} + A a^{3} e^{4} -{\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 3 \,{\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} -{\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3}\right )}{\left (e x + d\right )}^{\frac{3}{2}}\right )}}{3465 \, e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3465*(315*(e*x + d)^(11/2)*B*b^3 - 385*(4*B*b^3*d - (3*B*a*b^2 + A*b^3)*e)*(e*
x + d)^(9/2) + 1485*(2*B*b^3*d^2 - (3*B*a*b^2 + A*b^3)*d*e + (B*a^2*b + A*a*b^2)
*e^2)*(e*x + d)^(7/2) - 693*(4*B*b^3*d^3 - 3*(3*B*a*b^2 + A*b^3)*d^2*e + 6*(B*a^
2*b + A*a*b^2)*d*e^2 - (B*a^3 + 3*A*a^2*b)*e^3)*(e*x + d)^(5/2) + 1155*(B*b^3*d^
4 + A*a^3*e^4 - (3*B*a*b^2 + A*b^3)*d^3*e + 3*(B*a^2*b + A*a*b^2)*d^2*e^2 - (B*a
^3 + 3*A*a^2*b)*d*e^3)*(e*x + d)^(3/2))/e^5

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Fricas [A]  time = 0.223963, size = 477, normalized size = 2.76 \[ \frac{2 \,{\left (315 \, B b^{3} e^{5} x^{5} + 128 \, B b^{3} d^{5} + 1155 \, A a^{3} d e^{4} - 176 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{4} e + 792 \,{\left (B a^{2} b + A a b^{2}\right )} d^{3} e^{2} - 462 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d^{2} e^{3} + 35 \,{\left (B b^{3} d e^{4} + 11 \,{\left (3 \, B a b^{2} + A b^{3}\right )} e^{5}\right )} x^{4} - 5 \,{\left (8 \, B b^{3} d^{2} e^{3} - 11 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d e^{4} - 297 \,{\left (B a^{2} b + A a b^{2}\right )} e^{5}\right )} x^{3} + 3 \,{\left (16 \, B b^{3} d^{3} e^{2} - 22 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{3} + 99 \,{\left (B a^{2} b + A a b^{2}\right )} d e^{4} + 231 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} e^{5}\right )} x^{2} -{\left (64 \, B b^{3} d^{4} e - 1155 \, A a^{3} e^{5} - 88 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e^{2} + 396 \,{\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{3} - 231 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{4}\right )} x\right )} \sqrt{e x + d}}{3465 \, e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3465*(315*B*b^3*e^5*x^5 + 128*B*b^3*d^5 + 1155*A*a^3*d*e^4 - 176*(3*B*a*b^2 +
A*b^3)*d^4*e + 792*(B*a^2*b + A*a*b^2)*d^3*e^2 - 462*(B*a^3 + 3*A*a^2*b)*d^2*e^3
+ 35*(B*b^3*d*e^4 + 11*(3*B*a*b^2 + A*b^3)*e^5)*x^4 - 5*(8*B*b^3*d^2*e^3 - 11*(
3*B*a*b^2 + A*b^3)*d*e^4 - 297*(B*a^2*b + A*a*b^2)*e^5)*x^3 + 3*(16*B*b^3*d^3*e^
2 - 22*(3*B*a*b^2 + A*b^3)*d^2*e^3 + 99*(B*a^2*b + A*a*b^2)*d*e^4 + 231*(B*a^3 +
3*A*a^2*b)*e^5)*x^2 - (64*B*b^3*d^4*e - 1155*A*a^3*e^5 - 88*(3*B*a*b^2 + A*b^3)
*d^3*e^2 + 396*(B*a^2*b + A*a*b^2)*d^2*e^3 - 231*(B*a^3 + 3*A*a^2*b)*d*e^4)*x)*s
qrt(e*x + d)/e^5

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Sympy [A]  time = 6.65397, size = 342, normalized size = 1.98 \[ \frac{2 \left (\frac{B b^{3} \left (d + e x\right )^{\frac{11}{2}}}{11 e^{4}} + \frac{\left (d + e x\right )^{\frac{9}{2}} \left (A b^{3} e + 3 B a b^{2} e - 4 B b^{3} d\right )}{9 e^{4}} + \frac{\left (d + e x\right )^{\frac{7}{2}} \left (3 A a b^{2} e^{2} - 3 A b^{3} d e + 3 B a^{2} b e^{2} - 9 B a b^{2} d e + 6 B b^{3} d^{2}\right )}{7 e^{4}} + \frac{\left (d + e x\right )^{\frac{5}{2}} \left (3 A a^{2} b e^{3} - 6 A a b^{2} d e^{2} + 3 A b^{3} d^{2} e + B a^{3} e^{3} - 6 B a^{2} b d e^{2} + 9 B a b^{2} d^{2} e - 4 B b^{3} d^{3}\right )}{5 e^{4}} + \frac{\left (d + e x\right )^{\frac{3}{2}} \left (A a^{3} e^{4} - 3 A a^{2} b d e^{3} + 3 A a b^{2} d^{2} e^{2} - A b^{3} d^{3} e - B a^{3} d e^{3} + 3 B a^{2} b d^{2} e^{2} - 3 B a b^{2} d^{3} e + B b^{3} d^{4}\right )}{3 e^{4}}\right )}{e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**3*(B*x+A)*(e*x+d)**(1/2),x)

[Out]

2*(B*b**3*(d + e*x)**(11/2)/(11*e**4) + (d + e*x)**(9/2)*(A*b**3*e + 3*B*a*b**2*
e - 4*B*b**3*d)/(9*e**4) + (d + e*x)**(7/2)*(3*A*a*b**2*e**2 - 3*A*b**3*d*e + 3*
B*a**2*b*e**2 - 9*B*a*b**2*d*e + 6*B*b**3*d**2)/(7*e**4) + (d + e*x)**(5/2)*(3*A
*a**2*b*e**3 - 6*A*a*b**2*d*e**2 + 3*A*b**3*d**2*e + B*a**3*e**3 - 6*B*a**2*b*d*
e**2 + 9*B*a*b**2*d**2*e - 4*B*b**3*d**3)/(5*e**4) + (d + e*x)**(3/2)*(A*a**3*e*
*4 - 3*A*a**2*b*d*e**3 + 3*A*a*b**2*d**2*e**2 - A*b**3*d**3*e - B*a**3*d*e**3 +
3*B*a**2*b*d**2*e**2 - 3*B*a*b**2*d**3*e + B*b**3*d**4)/(3*e**4))/e

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GIAC/XCAS [A]  time = 0.215393, size = 522, normalized size = 3.02 \[ \frac{2}{3465} \,{\left (231 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} d\right )} B a^{3} e^{\left (-1\right )} + 693 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} d\right )} A a^{2} b e^{\left (-1\right )} + 99 \,{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} e^{12} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} d e^{12} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2} e^{12}\right )} B a^{2} b e^{\left (-14\right )} + 99 \,{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} e^{12} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} d e^{12} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2} e^{12}\right )} A a b^{2} e^{\left (-14\right )} + 33 \,{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} e^{24} - 135 \,{\left (x e + d\right )}^{\frac{7}{2}} d e^{24} + 189 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} e^{24} - 105 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3} e^{24}\right )} B a b^{2} e^{\left (-27\right )} + 11 \,{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} e^{24} - 135 \,{\left (x e + d\right )}^{\frac{7}{2}} d e^{24} + 189 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} e^{24} - 105 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3} e^{24}\right )} A b^{3} e^{\left (-27\right )} +{\left (315 \,{\left (x e + d\right )}^{\frac{11}{2}} e^{40} - 1540 \,{\left (x e + d\right )}^{\frac{9}{2}} d e^{40} + 2970 \,{\left (x e + d\right )}^{\frac{7}{2}} d^{2} e^{40} - 2772 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{3} e^{40} + 1155 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{4} e^{40}\right )} B b^{3} e^{\left (-44\right )} + 1155 \,{\left (x e + d\right )}^{\frac{3}{2}} A a^{3}\right )} e^{\left (-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3465*(231*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*B*a^3*e^(-1) + 693*(3*(x*e
+ d)^(5/2) - 5*(x*e + d)^(3/2)*d)*A*a^2*b*e^(-1) + 99*(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*a^2*b*e^(-14) + 99*
(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*a*b^2*e^(-14) + 33*(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*a*b^2*e^(-27) +
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*b^3*e^(-27) + (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*b^3*e^(-44) + 1155*(x*e + d)
^(3/2)*A*a^3)*e^(-1)