3.1989 \(\int (a+b x) (d+e x)^8 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx\)

Optimal. Leaf size=362 \[ \frac{15 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{11} (b d-a e)^4}{11 e^7 (a+b x)}-\frac{3 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{10} (b d-a e)^5}{5 e^7 (a+b x)}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^9 (b d-a e)^6}{9 e^7 (a+b x)}+\frac{b^6 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{15}}{15 e^7 (a+b x)}-\frac{3 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{14} (b d-a e)}{7 e^7 (a+b x)}+\frac{15 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{13} (b d-a e)^2}{13 e^7 (a+b x)}-\frac{5 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{12} (b d-a e)^3}{3 e^7 (a+b x)} \]

[Out]

((b*d - a*e)^6*(d + e*x)^9*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*e^7*(a + b*x)) - (3
*b*(b*d - a*e)^5*(d + e*x)^10*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^7*(a + b*x)) +
(15*b^2*(b*d - a*e)^4*(d + e*x)^11*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*e^7*(a +
b*x)) - (5*b^3*(b*d - a*e)^3*(d + e*x)^12*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^7*
(a + b*x)) + (15*b^4*(b*d - a*e)^2*(d + e*x)^13*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(
13*e^7*(a + b*x)) - (3*b^5*(b*d - a*e)*(d + e*x)^14*Sqrt[a^2 + 2*a*b*x + b^2*x^2
])/(7*e^7*(a + b*x)) + (b^6*(d + e*x)^15*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(15*e^7*
(a + b*x))

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Rubi [A]  time = 1.34424, antiderivative size = 362, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{15 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{11} (b d-a e)^4}{11 e^7 (a+b x)}-\frac{3 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{10} (b d-a e)^5}{5 e^7 (a+b x)}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^9 (b d-a e)^6}{9 e^7 (a+b x)}+\frac{b^6 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{15}}{15 e^7 (a+b x)}-\frac{3 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{14} (b d-a e)}{7 e^7 (a+b x)}+\frac{15 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{13} (b d-a e)^2}{13 e^7 (a+b x)}-\frac{5 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{12} (b d-a e)^3}{3 e^7 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)*(d + e*x)^8*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

((b*d - a*e)^6*(d + e*x)^9*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*e^7*(a + b*x)) - (3
*b*(b*d - a*e)^5*(d + e*x)^10*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^7*(a + b*x)) +
(15*b^2*(b*d - a*e)^4*(d + e*x)^11*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*e^7*(a +
b*x)) - (5*b^3*(b*d - a*e)^3*(d + e*x)^12*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^7*
(a + b*x)) + (15*b^4*(b*d - a*e)^2*(d + e*x)^13*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(
13*e^7*(a + b*x)) - (3*b^5*(b*d - a*e)*(d + e*x)^14*Sqrt[a^2 + 2*a*b*x + b^2*x^2
])/(7*e^7*(a + b*x)) + (b^6*(d + e*x)^15*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(15*e^7*
(a + b*x))

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Rubi in Sympy [A]  time = 69.3526, size = 299, normalized size = 0.83 \[ \frac{\left (a + b x\right ) \left (d + e x\right )^{9} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{15 e} + \frac{\left (d + e x\right )^{9} \left (a e - b d\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{35 e^{2}} + \frac{\left (5 a + 5 b x\right ) \left (d + e x\right )^{9} \left (a e - b d\right )^{2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{455 e^{3}} + \frac{\left (d + e x\right )^{9} \left (a e - b d\right )^{3} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{273 e^{4}} + \frac{\left (3 a + 3 b x\right ) \left (d + e x\right )^{9} \left (a e - b d\right )^{4} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{3003 e^{5}} + \frac{\left (d + e x\right )^{9} \left (a e - b d\right )^{5} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{5005 e^{6}} + \frac{\left (d + e x\right )^{9} \left (a e - b d\right )^{6} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{45045 e^{7} \left (a + b x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)*(e*x+d)**8*(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

(a + b*x)*(d + e*x)**9*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)/(15*e) + (d + e*x)**9
*(a*e - b*d)*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)/(35*e**2) + (5*a + 5*b*x)*(d +
e*x)**9*(a*e - b*d)**2*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(455*e**3) + (d + e*x
)**9*(a*e - b*d)**3*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(273*e**4) + (3*a + 3*b*
x)*(d + e*x)**9*(a*e - b*d)**4*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(3003*e**5) + (d
+ e*x)**9*(a*e - b*d)**5*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(5005*e**6) + (d + e*
x)**9*(a*e - b*d)**6*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(45045*e**7*(a + b*x))

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Mathematica [A]  time = 0.521544, size = 679, normalized size = 1.88 \[ \frac{x \sqrt{(a+b x)^2} \left (5005 a^6 \left (9 d^8+36 d^7 e x+84 d^6 e^2 x^2+126 d^5 e^3 x^3+126 d^4 e^4 x^4+84 d^3 e^5 x^5+36 d^2 e^6 x^6+9 d e^7 x^7+e^8 x^8\right )+3003 a^5 b x \left (45 d^8+240 d^7 e x+630 d^6 e^2 x^2+1008 d^5 e^3 x^3+1050 d^4 e^4 x^4+720 d^3 e^5 x^5+315 d^2 e^6 x^6+80 d e^7 x^7+9 e^8 x^8\right )+1365 a^4 b^2 x^2 \left (165 d^8+990 d^7 e x+2772 d^6 e^2 x^2+4620 d^5 e^3 x^3+4950 d^4 e^4 x^4+3465 d^3 e^5 x^5+1540 d^2 e^6 x^6+396 d e^7 x^7+45 e^8 x^8\right )+455 a^3 b^3 x^3 \left (495 d^8+3168 d^7 e x+9240 d^6 e^2 x^2+15840 d^5 e^3 x^3+17325 d^4 e^4 x^4+12320 d^3 e^5 x^5+5544 d^2 e^6 x^6+1440 d e^7 x^7+165 e^8 x^8\right )+105 a^2 b^4 x^4 \left (1287 d^8+8580 d^7 e x+25740 d^6 e^2 x^2+45045 d^5 e^3 x^3+50050 d^4 e^4 x^4+36036 d^3 e^5 x^5+16380 d^2 e^6 x^6+4290 d e^7 x^7+495 e^8 x^8\right )+15 a b^5 x^5 \left (3003 d^8+20592 d^7 e x+63063 d^6 e^2 x^2+112112 d^5 e^3 x^3+126126 d^4 e^4 x^4+91728 d^3 e^5 x^5+42042 d^2 e^6 x^6+11088 d e^7 x^7+1287 e^8 x^8\right )+b^6 x^6 \left (6435 d^8+45045 d^7 e x+140140 d^6 e^2 x^2+252252 d^5 e^3 x^3+286650 d^4 e^4 x^4+210210 d^3 e^5 x^5+97020 d^2 e^6 x^6+25740 d e^7 x^7+3003 e^8 x^8\right )\right )}{45045 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)*(d + e*x)^8*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

(x*Sqrt[(a + b*x)^2]*(5005*a^6*(9*d^8 + 36*d^7*e*x + 84*d^6*e^2*x^2 + 126*d^5*e^
3*x^3 + 126*d^4*e^4*x^4 + 84*d^3*e^5*x^5 + 36*d^2*e^6*x^6 + 9*d*e^7*x^7 + e^8*x^
8) + 3003*a^5*b*x*(45*d^8 + 240*d^7*e*x + 630*d^6*e^2*x^2 + 1008*d^5*e^3*x^3 + 1
050*d^4*e^4*x^4 + 720*d^3*e^5*x^5 + 315*d^2*e^6*x^6 + 80*d*e^7*x^7 + 9*e^8*x^8)
+ 1365*a^4*b^2*x^2*(165*d^8 + 990*d^7*e*x + 2772*d^6*e^2*x^2 + 4620*d^5*e^3*x^3
+ 4950*d^4*e^4*x^4 + 3465*d^3*e^5*x^5 + 1540*d^2*e^6*x^6 + 396*d*e^7*x^7 + 45*e^
8*x^8) + 455*a^3*b^3*x^3*(495*d^8 + 3168*d^7*e*x + 9240*d^6*e^2*x^2 + 15840*d^5*
e^3*x^3 + 17325*d^4*e^4*x^4 + 12320*d^3*e^5*x^5 + 5544*d^2*e^6*x^6 + 1440*d*e^7*
x^7 + 165*e^8*x^8) + 105*a^2*b^4*x^4*(1287*d^8 + 8580*d^7*e*x + 25740*d^6*e^2*x^
2 + 45045*d^5*e^3*x^3 + 50050*d^4*e^4*x^4 + 36036*d^3*e^5*x^5 + 16380*d^2*e^6*x^
6 + 4290*d*e^7*x^7 + 495*e^8*x^8) + 15*a*b^5*x^5*(3003*d^8 + 20592*d^7*e*x + 630
63*d^6*e^2*x^2 + 112112*d^5*e^3*x^3 + 126126*d^4*e^4*x^4 + 91728*d^3*e^5*x^5 + 4
2042*d^2*e^6*x^6 + 11088*d*e^7*x^7 + 1287*e^8*x^8) + b^6*x^6*(6435*d^8 + 45045*d
^7*e*x + 140140*d^6*e^2*x^2 + 252252*d^5*e^3*x^3 + 286650*d^4*e^4*x^4 + 210210*d
^3*e^5*x^5 + 97020*d^2*e^6*x^6 + 25740*d*e^7*x^7 + 3003*e^8*x^8)))/(45045*(a + b
*x))

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Maple [B]  time = 0.014, size = 925, normalized size = 2.6 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)*(e*x+d)^8*(b^2*x^2+2*a*b*x+a^2)^(5/2),x)

[Out]

1/45045*x*(3003*b^6*e^8*x^14+19305*a*b^5*e^8*x^13+25740*b^6*d*e^7*x^13+51975*a^2
*b^4*e^8*x^12+166320*a*b^5*d*e^7*x^12+97020*b^6*d^2*e^6*x^12+75075*a^3*b^3*e^8*x
^11+450450*a^2*b^4*d*e^7*x^11+630630*a*b^5*d^2*e^6*x^11+210210*b^6*d^3*e^5*x^11+
61425*a^4*b^2*e^8*x^10+655200*a^3*b^3*d*e^7*x^10+1719900*a^2*b^4*d^2*e^6*x^10+13
75920*a*b^5*d^3*e^5*x^10+286650*b^6*d^4*e^4*x^10+27027*a^5*b*e^8*x^9+540540*a^4*
b^2*d*e^7*x^9+2522520*a^3*b^3*d^2*e^6*x^9+3783780*a^2*b^4*d^3*e^5*x^9+1891890*a*
b^5*d^4*e^4*x^9+252252*b^6*d^5*e^3*x^9+5005*a^6*e^8*x^8+240240*a^5*b*d*e^7*x^8+2
102100*a^4*b^2*d^2*e^6*x^8+5605600*a^3*b^3*d^3*e^5*x^8+5255250*a^2*b^4*d^4*e^4*x
^8+1681680*a*b^5*d^5*e^3*x^8+140140*b^6*d^6*e^2*x^8+45045*a^6*d*e^7*x^7+945945*a
^5*b*d^2*e^6*x^7+4729725*a^4*b^2*d^3*e^5*x^7+7882875*a^3*b^3*d^4*e^4*x^7+4729725
*a^2*b^4*d^5*e^3*x^7+945945*a*b^5*d^6*e^2*x^7+45045*b^6*d^7*e*x^7+180180*a^6*d^2
*e^6*x^6+2162160*a^5*b*d^3*e^5*x^6+6756750*a^4*b^2*d^4*e^4*x^6+7207200*a^3*b^3*d
^5*e^3*x^6+2702700*a^2*b^4*d^6*e^2*x^6+308880*a*b^5*d^7*e*x^6+6435*b^6*d^8*x^6+4
20420*a^6*d^3*e^5*x^5+3153150*a^5*b*d^4*e^4*x^5+6306300*a^4*b^2*d^5*e^3*x^5+4204
200*a^3*b^3*d^6*e^2*x^5+900900*a^2*b^4*d^7*e*x^5+45045*a*b^5*d^8*x^5+630630*a^6*
d^4*e^4*x^4+3027024*a^5*b*d^5*e^3*x^4+3783780*a^4*b^2*d^6*e^2*x^4+1441440*a^3*b^
3*d^7*e*x^4+135135*a^2*b^4*d^8*x^4+630630*a^6*d^5*e^3*x^3+1891890*a^5*b*d^6*e^2*
x^3+1351350*a^4*b^2*d^7*e*x^3+225225*a^3*b^3*d^8*x^3+420420*a^6*d^6*e^2*x^2+7207
20*a^5*b*d^7*e*x^2+225225*a^4*b^2*d^8*x^2+180180*a^6*d^7*e*x+135135*a^5*b*d^8*x+
45045*a^6*d^8)*((b*x+a)^2)^(5/2)/(b*x+a)^5

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(b*x + a)*(e*x + d)^8,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.283017, size = 1076, normalized size = 2.97 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(b*x + a)*(e*x + d)^8,x, algorithm="fricas")

[Out]

1/15*b^6*e^8*x^15 + a^6*d^8*x + 1/7*(4*b^6*d*e^7 + 3*a*b^5*e^8)*x^14 + 1/13*(28*
b^6*d^2*e^6 + 48*a*b^5*d*e^7 + 15*a^2*b^4*e^8)*x^13 + 1/3*(14*b^6*d^3*e^5 + 42*a
*b^5*d^2*e^6 + 30*a^2*b^4*d*e^7 + 5*a^3*b^3*e^8)*x^12 + 1/11*(70*b^6*d^4*e^4 + 3
36*a*b^5*d^3*e^5 + 420*a^2*b^4*d^2*e^6 + 160*a^3*b^3*d*e^7 + 15*a^4*b^2*e^8)*x^1
1 + 1/5*(28*b^6*d^5*e^3 + 210*a*b^5*d^4*e^4 + 420*a^2*b^4*d^3*e^5 + 280*a^3*b^3*
d^2*e^6 + 60*a^4*b^2*d*e^7 + 3*a^5*b*e^8)*x^10 + 1/9*(28*b^6*d^6*e^2 + 336*a*b^5
*d^5*e^3 + 1050*a^2*b^4*d^4*e^4 + 1120*a^3*b^3*d^3*e^5 + 420*a^4*b^2*d^2*e^6 + 4
8*a^5*b*d*e^7 + a^6*e^8)*x^9 + (b^6*d^7*e + 21*a*b^5*d^6*e^2 + 105*a^2*b^4*d^5*e
^3 + 175*a^3*b^3*d^4*e^4 + 105*a^4*b^2*d^3*e^5 + 21*a^5*b*d^2*e^6 + a^6*d*e^7)*x
^8 + 1/7*(b^6*d^8 + 48*a*b^5*d^7*e + 420*a^2*b^4*d^6*e^2 + 1120*a^3*b^3*d^5*e^3
+ 1050*a^4*b^2*d^4*e^4 + 336*a^5*b*d^3*e^5 + 28*a^6*d^2*e^6)*x^7 + 1/3*(3*a*b^5*
d^8 + 60*a^2*b^4*d^7*e + 280*a^3*b^3*d^6*e^2 + 420*a^4*b^2*d^5*e^3 + 210*a^5*b*d
^4*e^4 + 28*a^6*d^3*e^5)*x^6 + 1/5*(15*a^2*b^4*d^8 + 160*a^3*b^3*d^7*e + 420*a^4
*b^2*d^6*e^2 + 336*a^5*b*d^5*e^3 + 70*a^6*d^4*e^4)*x^5 + (5*a^3*b^3*d^8 + 30*a^4
*b^2*d^7*e + 42*a^5*b*d^6*e^2 + 14*a^6*d^5*e^3)*x^4 + 1/3*(15*a^4*b^2*d^8 + 48*a
^5*b*d^7*e + 28*a^6*d^6*e^2)*x^3 + (3*a^5*b*d^8 + 4*a^6*d^7*e)*x^2

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)*(e*x+d)**8*(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.296975, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(b*x + a)*(e*x + d)^8,x, algorithm="giac")

[Out]

Done