3.2052 \(\int (a+b x) (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx\)

Optimal. Leaf size=158 \[ -\frac{10 b^4 (d+e x)^{13/2} (b d-a e)}{13 e^6}+\frac{20 b^3 (d+e x)^{11/2} (b d-a e)^2}{11 e^6}-\frac{20 b^2 (d+e x)^{9/2} (b d-a e)^3}{9 e^6}+\frac{10 b (d+e x)^{7/2} (b d-a e)^4}{7 e^6}-\frac{2 (d+e x)^{5/2} (b d-a e)^5}{5 e^6}+\frac{2 b^5 (d+e x)^{15/2}}{15 e^6} \]

[Out]

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

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Rubi [A]  time = 0.140308, antiderivative size = 158, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.061 \[ -\frac{10 b^4 (d+e x)^{13/2} (b d-a e)}{13 e^6}+\frac{20 b^3 (d+e x)^{11/2} (b d-a e)^2}{11 e^6}-\frac{20 b^2 (d+e x)^{9/2} (b d-a e)^3}{9 e^6}+\frac{10 b (d+e x)^{7/2} (b d-a e)^4}{7 e^6}-\frac{2 (d+e x)^{5/2} (b d-a e)^5}{5 e^6}+\frac{2 b^5 (d+e x)^{15/2}}{15 e^6} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 72.9121, size = 146, normalized size = 0.92 \[ \frac{2 b^{5} \left (d + e x\right )^{\frac{15}{2}}}{15 e^{6}} + \frac{10 b^{4} \left (d + e x\right )^{\frac{13}{2}} \left (a e - b d\right )}{13 e^{6}} + \frac{20 b^{3} \left (d + e x\right )^{\frac{11}{2}} \left (a e - b d\right )^{2}}{11 e^{6}} + \frac{20 b^{2} \left (d + e x\right )^{\frac{9}{2}} \left (a e - b d\right )^{3}}{9 e^{6}} + \frac{10 b \left (d + e x\right )^{\frac{7}{2}} \left (a e - b d\right )^{4}}{7 e^{6}} + \frac{2 \left (d + e x\right )^{\frac{5}{2}} \left (a e - b d\right )^{5}}{5 e^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.297916, size = 217, normalized size = 1.37 \[ \frac{2 (d+e x)^{5/2} \left (9009 a^5 e^5+6435 a^4 b e^4 (5 e x-2 d)+1430 a^3 b^2 e^3 \left (8 d^2-20 d e x+35 e^2 x^2\right )+390 a^2 b^3 e^2 \left (-16 d^3+40 d^2 e x-70 d e^2 x^2+105 e^3 x^3\right )+15 a b^4 e \left (128 d^4-320 d^3 e x+560 d^2 e^2 x^2-840 d e^3 x^3+1155 e^4 x^4\right )+b^5 \left (-256 d^5+640 d^4 e x-1120 d^3 e^2 x^2+1680 d^2 e^3 x^3-2310 d e^4 x^4+3003 e^5 x^5\right )\right )}{45045 e^6} \]

Antiderivative was successfully verified.

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

[Out]

(2*(d + e*x)^(5/2)*(9009*a^5*e^5 + 6435*a^4*b*e^4*(-2*d + 5*e*x) + 1430*a^3*b^2*
e^3*(8*d^2 - 20*d*e*x + 35*e^2*x^2) + 390*a^2*b^3*e^2*(-16*d^3 + 40*d^2*e*x - 70
*d*e^2*x^2 + 105*e^3*x^3) + 15*a*b^4*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) + b^5*(-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)))/(45045*e^6)

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Maple [B]  time = 0.011, size = 273, normalized size = 1.7 \[{\frac{6006\,{x}^{5}{b}^{5}{e}^{5}+34650\,{x}^{4}a{b}^{4}{e}^{5}-4620\,{x}^{4}{b}^{5}d{e}^{4}+81900\,{x}^{3}{a}^{2}{b}^{3}{e}^{5}-25200\,{x}^{3}a{b}^{4}d{e}^{4}+3360\,{x}^{3}{b}^{5}{d}^{2}{e}^{3}+100100\,{x}^{2}{a}^{3}{b}^{2}{e}^{5}-54600\,{x}^{2}{a}^{2}{b}^{3}d{e}^{4}+16800\,{x}^{2}a{b}^{4}{d}^{2}{e}^{3}-2240\,{x}^{2}{b}^{5}{d}^{3}{e}^{2}+64350\,x{a}^{4}b{e}^{5}-57200\,x{a}^{3}{b}^{2}d{e}^{4}+31200\,x{a}^{2}{b}^{3}{d}^{2}{e}^{3}-9600\,xa{b}^{4}{d}^{3}{e}^{2}+1280\,x{b}^{5}{d}^{4}e+18018\,{a}^{5}{e}^{5}-25740\,{a}^{4}bd{e}^{4}+22880\,{a}^{3}{b}^{2}{d}^{2}{e}^{3}-12480\,{a}^{2}{b}^{3}{d}^{3}{e}^{2}+3840\,a{b}^{4}{d}^{4}e-512\,{b}^{5}{d}^{5}}{45045\,{e}^{6}} \left ( ex+d \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/45045*(e*x+d)^(5/2)*(3003*b^5*e^5*x^5+17325*a*b^4*e^5*x^4-2310*b^5*d*e^4*x^4+4
0950*a^2*b^3*e^5*x^3-12600*a*b^4*d*e^4*x^3+1680*b^5*d^2*e^3*x^3+50050*a^3*b^2*e^
5*x^2-27300*a^2*b^3*d*e^4*x^2+8400*a*b^4*d^2*e^3*x^2-1120*b^5*d^3*e^2*x^2+32175*
a^4*b*e^5*x-28600*a^3*b^2*d*e^4*x+15600*a^2*b^3*d^2*e^3*x-4800*a*b^4*d^3*e^2*x+6
40*b^5*d^4*e*x+9009*a^5*e^5-12870*a^4*b*d*e^4+11440*a^3*b^2*d^2*e^3-6240*a^2*b^3
*d^3*e^2+1920*a*b^4*d^4*e-256*b^5*d^5)/e^6

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Maxima [A]  time = 0.72226, size = 350, normalized size = 2.22 \[ \frac{2 \,{\left (3003 \,{\left (e x + d\right )}^{\frac{15}{2}} b^{5} - 17325 \,{\left (b^{5} d - a b^{4} e\right )}{\left (e x + d\right )}^{\frac{13}{2}} + 40950 \,{\left (b^{5} d^{2} - 2 \, a b^{4} d e + a^{2} b^{3} e^{2}\right )}{\left (e x + d\right )}^{\frac{11}{2}} - 50050 \,{\left (b^{5} d^{3} - 3 \, a b^{4} d^{2} e + 3 \, a^{2} b^{3} d e^{2} - a^{3} b^{2} e^{3}\right )}{\left (e x + d\right )}^{\frac{9}{2}} + 32175 \,{\left (b^{5} d^{4} - 4 \, a b^{4} d^{3} e + 6 \, a^{2} b^{3} d^{2} e^{2} - 4 \, a^{3} b^{2} d e^{3} + a^{4} b e^{4}\right )}{\left (e x + d\right )}^{\frac{7}{2}} - 9009 \,{\left (b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}\right )}{\left (e x + d\right )}^{\frac{5}{2}}\right )}}{45045 \, e^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/45045*(3003*(e*x + d)^(15/2)*b^5 - 17325*(b^5*d - a*b^4*e)*(e*x + d)^(13/2) +
40950*(b^5*d^2 - 2*a*b^4*d*e + a^2*b^3*e^2)*(e*x + d)^(11/2) - 50050*(b^5*d^3 -
3*a*b^4*d^2*e + 3*a^2*b^3*d*e^2 - a^3*b^2*e^3)*(e*x + d)^(9/2) + 32175*(b^5*d^4
- 4*a*b^4*d^3*e + 6*a^2*b^3*d^2*e^2 - 4*a^3*b^2*d*e^3 + a^4*b*e^4)*(e*x + d)^(7/
2) - 9009*(b^5*d^5 - 5*a*b^4*d^4*e + 10*a^2*b^3*d^3*e^2 - 10*a^3*b^2*d^2*e^3 + 5
*a^4*b*d*e^4 - a^5*e^5)*(e*x + d)^(5/2))/e^6

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Fricas [A]  time = 0.282003, size = 564, normalized size = 3.57 \[ \frac{2 \,{\left (3003 \, b^{5} e^{7} x^{7} - 256 \, b^{5} d^{7} + 1920 \, a b^{4} d^{6} e - 6240 \, a^{2} b^{3} d^{5} e^{2} + 11440 \, a^{3} b^{2} d^{4} e^{3} - 12870 \, a^{4} b d^{3} e^{4} + 9009 \, a^{5} d^{2} e^{5} + 231 \,{\left (16 \, b^{5} d e^{6} + 75 \, a b^{4} e^{7}\right )} x^{6} + 63 \,{\left (b^{5} d^{2} e^{5} + 350 \, a b^{4} d e^{6} + 650 \, a^{2} b^{3} e^{7}\right )} x^{5} - 35 \,{\left (2 \, b^{5} d^{3} e^{4} - 15 \, a b^{4} d^{2} e^{5} - 1560 \, a^{2} b^{3} d e^{6} - 1430 \, a^{3} b^{2} e^{7}\right )} x^{4} + 5 \,{\left (16 \, b^{5} d^{4} e^{3} - 120 \, a b^{4} d^{3} e^{4} + 390 \, a^{2} b^{3} d^{2} e^{5} + 14300 \, a^{3} b^{2} d e^{6} + 6435 \, a^{4} b e^{7}\right )} x^{3} - 3 \,{\left (32 \, b^{5} d^{5} e^{2} - 240 \, a b^{4} d^{4} e^{3} + 780 \, a^{2} b^{3} d^{3} e^{4} - 1430 \, a^{3} b^{2} d^{2} e^{5} - 17160 \, a^{4} b d e^{6} - 3003 \, a^{5} e^{7}\right )} x^{2} +{\left (128 \, b^{5} d^{6} e - 960 \, a b^{4} d^{5} e^{2} + 3120 \, a^{2} b^{3} d^{4} e^{3} - 5720 \, a^{3} b^{2} d^{3} e^{4} + 6435 \, a^{4} b d^{2} e^{5} + 18018 \, a^{5} d e^{6}\right )} x\right )} \sqrt{e x + d}}{45045 \, e^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/45045*(3003*b^5*e^7*x^7 - 256*b^5*d^7 + 1920*a*b^4*d^6*e - 6240*a^2*b^3*d^5*e^
2 + 11440*a^3*b^2*d^4*e^3 - 12870*a^4*b*d^3*e^4 + 9009*a^5*d^2*e^5 + 231*(16*b^5
*d*e^6 + 75*a*b^4*e^7)*x^6 + 63*(b^5*d^2*e^5 + 350*a*b^4*d*e^6 + 650*a^2*b^3*e^7
)*x^5 - 35*(2*b^5*d^3*e^4 - 15*a*b^4*d^2*e^5 - 1560*a^2*b^3*d*e^6 - 1430*a^3*b^2
*e^7)*x^4 + 5*(16*b^5*d^4*e^3 - 120*a*b^4*d^3*e^4 + 390*a^2*b^3*d^2*e^5 + 14300*
a^3*b^2*d*e^6 + 6435*a^4*b*e^7)*x^3 - 3*(32*b^5*d^5*e^2 - 240*a*b^4*d^4*e^3 + 78
0*a^2*b^3*d^3*e^4 - 1430*a^3*b^2*d^2*e^5 - 17160*a^4*b*d*e^6 - 3003*a^5*e^7)*x^2
+ (128*b^5*d^6*e - 960*a*b^4*d^5*e^2 + 3120*a^2*b^3*d^4*e^3 - 5720*a^3*b^2*d^3*
e^4 + 6435*a^4*b*d^2*e^5 + 18018*a^5*d*e^6)*x)*sqrt(e*x + d)/e^6

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Sympy [A]  time = 12.178, size = 763, normalized size = 4.83 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**5*d*Piecewise((sqrt(d)*x, Eq(e, 0)), (2*(d + e*x)**(3/2)/(3*e), True)) + 2*a*
*5*(-d*(d + e*x)**(3/2)/3 + (d + e*x)**(5/2)/5)/e + 10*a**4*b*d*(-d*(d + e*x)**(
3/2)/3 + (d + e*x)**(5/2)/5)/e**2 + 10*a**4*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 + 20*a**3*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 + 20*a**3*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 + 20*a**2*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 + 20*a**2*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 + 10*a*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 + 10*a*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 + 2*b**5*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**5*
(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

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GIAC/XCAS [A]  time = 0.289321, 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*(b*x + a)*(e*x + d)^(3/2),x, algorithm="giac")

[Out]

Done