3.1620 \(\int (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx\)

Optimal. Leaf size=129 \[ -\frac{8 b^3 (d+e x)^{11/2} (b d-a e)}{11 e^5}+\frac{4 b^2 (d+e x)^{9/2} (b d-a e)^2}{3 e^5}-\frac{8 b (d+e x)^{7/2} (b d-a e)^3}{7 e^5}+\frac{2 (d+e x)^{5/2} (b d-a e)^4}{5 e^5}+\frac{2 b^4 (d+e x)^{13/2}}{13 e^5} \]

[Out]

(2*(b*d - a*e)^4*(d + e*x)^(5/2))/(5*e^5) - (8*b*(b*d - a*e)^3*(d + e*x)^(7/2))/
(7*e^5) + (4*b^2*(b*d - a*e)^2*(d + e*x)^(9/2))/(3*e^5) - (8*b^3*(b*d - a*e)*(d
+ e*x)^(11/2))/(11*e^5) + (2*b^4*(d + e*x)^(13/2))/(13*e^5)

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Rubi [A]  time = 0.136755, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071 \[ -\frac{8 b^3 (d+e x)^{11/2} (b d-a e)}{11 e^5}+\frac{4 b^2 (d+e x)^{9/2} (b d-a e)^2}{3 e^5}-\frac{8 b (d+e x)^{7/2} (b d-a e)^3}{7 e^5}+\frac{2 (d+e x)^{5/2} (b d-a e)^4}{5 e^5}+\frac{2 b^4 (d+e x)^{13/2}}{13 e^5} \]

Antiderivative was successfully verified.

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

[Out]

(2*(b*d - a*e)^4*(d + e*x)^(5/2))/(5*e^5) - (8*b*(b*d - a*e)^3*(d + e*x)^(7/2))/
(7*e^5) + (4*b^2*(b*d - a*e)^2*(d + e*x)^(9/2))/(3*e^5) - (8*b^3*(b*d - a*e)*(d
+ e*x)^(11/2))/(11*e^5) + (2*b^4*(d + e*x)^(13/2))/(13*e^5)

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Rubi in Sympy [A]  time = 56.0825, size = 119, normalized size = 0.92 \[ \frac{2 b^{4} \left (d + e x\right )^{\frac{13}{2}}}{13 e^{5}} + \frac{8 b^{3} \left (d + e x\right )^{\frac{11}{2}} \left (a e - b d\right )}{11 e^{5}} + \frac{4 b^{2} \left (d + e x\right )^{\frac{9}{2}} \left (a e - b d\right )^{2}}{3 e^{5}} + \frac{8 b \left (d + e x\right )^{\frac{7}{2}} \left (a e - b d\right )^{3}}{7 e^{5}} + \frac{2 \left (d + e x\right )^{\frac{5}{2}} \left (a e - b d\right )^{4}}{5 e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*b**4*(d + e*x)**(13/2)/(13*e**5) + 8*b**3*(d + e*x)**(11/2)*(a*e - b*d)/(11*e*
*5) + 4*b**2*(d + e*x)**(9/2)*(a*e - b*d)**2/(3*e**5) + 8*b*(d + e*x)**(7/2)*(a*
e - b*d)**3/(7*e**5) + 2*(d + e*x)**(5/2)*(a*e - b*d)**4/(5*e**5)

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Mathematica [A]  time = 0.159305, size = 154, normalized size = 1.19 \[ \frac{2 (d+e x)^{5/2} \left (3003 a^4 e^4+1716 a^3 b e^3 (5 e x-2 d)+286 a^2 b^2 e^2 \left (8 d^2-20 d e x+35 e^2 x^2\right )+52 a b^3 e \left (-16 d^3+40 d^2 e x-70 d e^2 x^2+105 e^3 x^3\right )+b^4 \left (128 d^4-320 d^3 e x+560 d^2 e^2 x^2-840 d e^3 x^3+1155 e^4 x^4\right )\right )}{15015 e^5} \]

Antiderivative was successfully verified.

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

[Out]

(2*(d + e*x)^(5/2)*(3003*a^4*e^4 + 1716*a^3*b*e^3*(-2*d + 5*e*x) + 286*a^2*b^2*e
^2*(8*d^2 - 20*d*e*x + 35*e^2*x^2) + 52*a*b^3*e*(-16*d^3 + 40*d^2*e*x - 70*d*e^2
*x^2 + 105*e^3*x^3) + b^4*(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)))/(15015*e^5)

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Maple [A]  time = 0.012, size = 186, normalized size = 1.4 \[{\frac{2310\,{x}^{4}{b}^{4}{e}^{4}+10920\,{x}^{3}a{b}^{3}{e}^{4}-1680\,{x}^{3}{b}^{4}d{e}^{3}+20020\,{x}^{2}{a}^{2}{b}^{2}{e}^{4}-7280\,{x}^{2}a{b}^{3}d{e}^{3}+1120\,{x}^{2}{b}^{4}{d}^{2}{e}^{2}+17160\,x{a}^{3}b{e}^{4}-11440\,x{a}^{2}{b}^{2}d{e}^{3}+4160\,xa{b}^{3}{d}^{2}{e}^{2}-640\,x{b}^{4}{d}^{3}e+6006\,{a}^{4}{e}^{4}-6864\,{a}^{3}bd{e}^{3}+4576\,{a}^{2}{b}^{2}{d}^{2}{e}^{2}-1664\,a{b}^{3}{d}^{3}e+256\,{b}^{4}{d}^{4}}{15015\,{e}^{5}} \left ( ex+d \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15015*(e*x+d)^(5/2)*(1155*b^4*e^4*x^4+5460*a*b^3*e^4*x^3-840*b^4*d*e^3*x^3+100
10*a^2*b^2*e^4*x^2-3640*a*b^3*d*e^3*x^2+560*b^4*d^2*e^2*x^2+8580*a^3*b*e^4*x-572
0*a^2*b^2*d*e^3*x+2080*a*b^3*d^2*e^2*x-320*b^4*d^3*e*x+3003*a^4*e^4-3432*a^3*b*d
*e^3+2288*a^2*b^2*d^2*e^2-832*a*b^3*d^3*e+128*b^4*d^4)/e^5

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Maxima [A]  time = 0.734359, size = 244, normalized size = 1.89 \[ \frac{2 \,{\left (1155 \,{\left (e x + d\right )}^{\frac{13}{2}} b^{4} - 5460 \,{\left (b^{4} d - a b^{3} e\right )}{\left (e x + d\right )}^{\frac{11}{2}} + 10010 \,{\left (b^{4} d^{2} - 2 \, a b^{3} d e + a^{2} b^{2} e^{2}\right )}{\left (e x + d\right )}^{\frac{9}{2}} - 8580 \,{\left (b^{4} d^{3} - 3 \, a b^{3} d^{2} e + 3 \, a^{2} b^{2} d e^{2} - a^{3} b e^{3}\right )}{\left (e x + d\right )}^{\frac{7}{2}} + 3003 \,{\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )}{\left (e x + d\right )}^{\frac{5}{2}}\right )}}{15015 \, e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15015*(1155*(e*x + d)^(13/2)*b^4 - 5460*(b^4*d - a*b^3*e)*(e*x + d)^(11/2) + 1
0010*(b^4*d^2 - 2*a*b^3*d*e + a^2*b^2*e^2)*(e*x + d)^(9/2) - 8580*(b^4*d^3 - 3*a
*b^3*d^2*e + 3*a^2*b^2*d*e^2 - a^3*b*e^3)*(e*x + d)^(7/2) + 3003*(b^4*d^4 - 4*a*
b^3*d^3*e + 6*a^2*b^2*d^2*e^2 - 4*a^3*b*d*e^3 + a^4*e^4)*(e*x + d)^(5/2))/e^5

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Fricas [A]  time = 0.206734, size = 420, normalized size = 3.26 \[ \frac{2 \,{\left (1155 \, b^{4} e^{6} x^{6} + 128 \, b^{4} d^{6} - 832 \, a b^{3} d^{5} e + 2288 \, a^{2} b^{2} d^{4} e^{2} - 3432 \, a^{3} b d^{3} e^{3} + 3003 \, a^{4} d^{2} e^{4} + 210 \,{\left (7 \, b^{4} d e^{5} + 26 \, a b^{3} e^{6}\right )} x^{5} + 35 \,{\left (b^{4} d^{2} e^{4} + 208 \, a b^{3} d e^{5} + 286 \, a^{2} b^{2} e^{6}\right )} x^{4} - 20 \,{\left (2 \, b^{4} d^{3} e^{3} - 13 \, a b^{3} d^{2} e^{4} - 715 \, a^{2} b^{2} d e^{5} - 429 \, a^{3} b e^{6}\right )} x^{3} + 3 \,{\left (16 \, b^{4} d^{4} e^{2} - 104 \, a b^{3} d^{3} e^{3} + 286 \, a^{2} b^{2} d^{2} e^{4} + 4576 \, a^{3} b d e^{5} + 1001 \, a^{4} e^{6}\right )} x^{2} - 2 \,{\left (32 \, b^{4} d^{5} e - 208 \, a b^{3} d^{4} e^{2} + 572 \, a^{2} b^{2} d^{3} e^{3} - 858 \, a^{3} b d^{2} e^{4} - 3003 \, a^{4} d e^{5}\right )} x\right )} \sqrt{e x + d}}{15015 \, e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15015*(1155*b^4*e^6*x^6 + 128*b^4*d^6 - 832*a*b^3*d^5*e + 2288*a^2*b^2*d^4*e^2
- 3432*a^3*b*d^3*e^3 + 3003*a^4*d^2*e^4 + 210*(7*b^4*d*e^5 + 26*a*b^3*e^6)*x^5
+ 35*(b^4*d^2*e^4 + 208*a*b^3*d*e^5 + 286*a^2*b^2*e^6)*x^4 - 20*(2*b^4*d^3*e^3 -
13*a*b^3*d^2*e^4 - 715*a^2*b^2*d*e^5 - 429*a^3*b*e^6)*x^3 + 3*(16*b^4*d^4*e^2 -
104*a*b^3*d^3*e^3 + 286*a^2*b^2*d^2*e^4 + 4576*a^3*b*d*e^5 + 1001*a^4*e^6)*x^2
- 2*(32*b^4*d^5*e - 208*a*b^3*d^4*e^2 + 572*a^2*b^2*d^3*e^3 - 858*a^3*b*d^2*e^4
- 3003*a^4*d*e^5)*x)*sqrt(e*x + d)/e^5

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Sympy [A]  time = 9.06034, size = 559, normalized size = 4.33 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**4*d*Piecewise((sqrt(d)*x, Eq(e, 0)), (2*(d + e*x)**(3/2)/(3*e), True)) + 2*a*
*4*(-d*(d + e*x)**(3/2)/3 + (d + e*x)**(5/2)/5)/e + 8*a**3*b*d*(-d*(d + e*x)**(3
/2)/3 + (d + e*x)**(5/2)/5)/e**2 + 8*a**3*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 + 12*a**2*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 + 12*a**2*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 + 8*a*b**3*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 + 8*a*b**3*(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*b**4*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*b**4*(-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

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GIAC/XCAS [A]  time = 0.224812, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done