3.1992 \(\int (a+b x) (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx\)

Optimal. Leaf size=263 \[ \frac{5 e^4 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^{10} (b d-a e)}{11 b^6}+\frac{e^3 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^9 (b d-a e)^2}{b^6}+\frac{10 e^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^8 (b d-a e)^3}{9 b^6}+\frac{5 e \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^7 (b d-a e)^4}{8 b^6}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^6 (b d-a e)^5}{7 b^6}+\frac{e^5 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^{11}}{12 b^6} \]

[Out]

((b*d - a*e)^5*(a + b*x)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*b^6) + (5*e*(b*d -
a*e)^4*(a + b*x)^7*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(8*b^6) + (10*e^2*(b*d - a*e)^
3*(a + b*x)^8*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*b^6) + (e^3*(b*d - a*e)^2*(a + b
*x)^9*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/b^6 + (5*e^4*(b*d - a*e)*(a + b*x)^10*Sqrt[
a^2 + 2*a*b*x + b^2*x^2])/(11*b^6) + (e^5*(a + b*x)^11*Sqrt[a^2 + 2*a*b*x + b^2*
x^2])/(12*b^6)

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Rubi [A]  time = 0.832427, antiderivative size = 263, 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 \[ \frac{5 e^4 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^{10} (b d-a e)}{11 b^6}+\frac{e^3 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^9 (b d-a e)^2}{b^6}+\frac{10 e^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^8 (b d-a e)^3}{9 b^6}+\frac{5 e \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^7 (b d-a e)^4}{8 b^6}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^6 (b d-a e)^5}{7 b^6}+\frac{e^5 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^{11}}{12 b^6} \]

Antiderivative was successfully verified.

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

[Out]

((b*d - a*e)^5*(a + b*x)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*b^6) + (5*e*(b*d -
a*e)^4*(a + b*x)^7*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(8*b^6) + (10*e^2*(b*d - a*e)^
3*(a + b*x)^8*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*b^6) + (e^3*(b*d - a*e)^2*(a + b
*x)^9*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/b^6 + (5*e^4*(b*d - a*e)*(a + b*x)^10*Sqrt[
a^2 + 2*a*b*x + b^2*x^2])/(11*b^6) + (e^5*(a + b*x)^11*Sqrt[a^2 + 2*a*b*x + b^2*
x^2])/(12*b^6)

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Rubi in Sympy [A]  time = 80.3723, size = 224, normalized size = 0.85 \[ \frac{\left (d + e x\right )^{5} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{7}{2}}}{12 b} - \frac{5 \left (d + e x\right )^{4} \left (a e - b d\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{7}{2}}}{132 b^{2}} + \frac{\left (d + e x\right )^{3} \left (a e - b d\right )^{2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{7}{2}}}{66 b^{3}} - \frac{\left (d + e x\right )^{2} \left (a e - b d\right )^{3} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{7}{2}}}{198 b^{4}} + \frac{\left (d + e x\right ) \left (a e - b d\right )^{4} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{7}{2}}}{792 b^{5}} - \frac{\left (a e - b d\right )^{5} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{7}{2}}}{5544 b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(d + e*x)**5*(a**2 + 2*a*b*x + b**2*x**2)**(7/2)/(12*b) - 5*(d + e*x)**4*(a*e -
b*d)*(a**2 + 2*a*b*x + b**2*x**2)**(7/2)/(132*b**2) + (d + e*x)**3*(a*e - b*d)**
2*(a**2 + 2*a*b*x + b**2*x**2)**(7/2)/(66*b**3) - (d + e*x)**2*(a*e - b*d)**3*(a
**2 + 2*a*b*x + b**2*x**2)**(7/2)/(198*b**4) + (d + e*x)*(a*e - b*d)**4*(a**2 +
2*a*b*x + b**2*x**2)**(7/2)/(792*b**5) - (a*e - b*d)**5*(a**2 + 2*a*b*x + b**2*x
**2)**(7/2)/(5544*b**6)

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Mathematica [A]  time = 0.334226, size = 448, normalized size = 1.7 \[ \frac{x \sqrt{(a+b x)^2} \left (924 a^6 \left (6 d^5+15 d^4 e x+20 d^3 e^2 x^2+15 d^2 e^3 x^3+6 d e^4 x^4+e^5 x^5\right )+792 a^5 b x \left (21 d^5+70 d^4 e x+105 d^3 e^2 x^2+84 d^2 e^3 x^3+35 d e^4 x^4+6 e^5 x^5\right )+495 a^4 b^2 x^2 \left (56 d^5+210 d^4 e x+336 d^3 e^2 x^2+280 d^2 e^3 x^3+120 d e^4 x^4+21 e^5 x^5\right )+220 a^3 b^3 x^3 \left (126 d^5+504 d^4 e x+840 d^3 e^2 x^2+720 d^2 e^3 x^3+315 d e^4 x^4+56 e^5 x^5\right )+66 a^2 b^4 x^4 \left (252 d^5+1050 d^4 e x+1800 d^3 e^2 x^2+1575 d^2 e^3 x^3+700 d e^4 x^4+126 e^5 x^5\right )+12 a b^5 x^5 \left (462 d^5+1980 d^4 e x+3465 d^3 e^2 x^2+3080 d^2 e^3 x^3+1386 d e^4 x^4+252 e^5 x^5\right )+b^6 x^6 \left (792 d^5+3465 d^4 e x+6160 d^3 e^2 x^2+5544 d^2 e^3 x^3+2520 d e^4 x^4+462 e^5 x^5\right )\right )}{5544 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

(x*Sqrt[(a + b*x)^2]*(924*a^6*(6*d^5 + 15*d^4*e*x + 20*d^3*e^2*x^2 + 15*d^2*e^3*
x^3 + 6*d*e^4*x^4 + e^5*x^5) + 792*a^5*b*x*(21*d^5 + 70*d^4*e*x + 105*d^3*e^2*x^
2 + 84*d^2*e^3*x^3 + 35*d*e^4*x^4 + 6*e^5*x^5) + 495*a^4*b^2*x^2*(56*d^5 + 210*d
^4*e*x + 336*d^3*e^2*x^2 + 280*d^2*e^3*x^3 + 120*d*e^4*x^4 + 21*e^5*x^5) + 220*a
^3*b^3*x^3*(126*d^5 + 504*d^4*e*x + 840*d^3*e^2*x^2 + 720*d^2*e^3*x^3 + 315*d*e^
4*x^4 + 56*e^5*x^5) + 66*a^2*b^4*x^4*(252*d^5 + 1050*d^4*e*x + 1800*d^3*e^2*x^2
+ 1575*d^2*e^3*x^3 + 700*d*e^4*x^4 + 126*e^5*x^5) + 12*a*b^5*x^5*(462*d^5 + 1980
*d^4*e*x + 3465*d^3*e^2*x^2 + 3080*d^2*e^3*x^3 + 1386*d*e^4*x^4 + 252*e^5*x^5) +
b^6*x^6*(792*d^5 + 3465*d^4*e*x + 6160*d^3*e^2*x^2 + 5544*d^2*e^3*x^3 + 2520*d*
e^4*x^4 + 462*e^5*x^5)))/(5544*(a + b*x))

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Maple [B]  time = 0.013, size = 598, normalized size = 2.3 \[{\frac{x \left ( 462\,{e}^{5}{b}^{6}{x}^{11}+3024\,{x}^{10}{e}^{5}a{b}^{5}+2520\,{x}^{10}d{e}^{4}{b}^{6}+8316\,{x}^{9}{e}^{5}{a}^{2}{b}^{4}+16632\,{x}^{9}d{e}^{4}a{b}^{5}+5544\,{x}^{9}{d}^{2}{e}^{3}{b}^{6}+12320\,{x}^{8}{e}^{5}{a}^{3}{b}^{3}+46200\,{x}^{8}d{e}^{4}{a}^{2}{b}^{4}+36960\,{x}^{8}{d}^{2}{e}^{3}a{b}^{5}+6160\,{x}^{8}{d}^{3}{e}^{2}{b}^{6}+10395\,{x}^{7}{e}^{5}{b}^{2}{a}^{4}+69300\,{x}^{7}d{e}^{4}{a}^{3}{b}^{3}+103950\,{x}^{7}{d}^{2}{e}^{3}{a}^{2}{b}^{4}+41580\,{x}^{7}{d}^{3}{e}^{2}a{b}^{5}+3465\,{x}^{7}{d}^{4}e{b}^{6}+4752\,{x}^{6}{a}^{5}b{e}^{5}+59400\,{x}^{6}{a}^{4}{b}^{2}d{e}^{4}+158400\,{x}^{6}{a}^{3}{b}^{3}{d}^{2}{e}^{3}+118800\,{x}^{6}{a}^{2}{b}^{4}{d}^{3}{e}^{2}+23760\,{x}^{6}a{b}^{5}{d}^{4}e+792\,{x}^{6}{d}^{5}{b}^{6}+924\,{x}^{5}{e}^{5}{a}^{6}+27720\,{x}^{5}d{e}^{4}{a}^{5}b+138600\,{x}^{5}{d}^{2}{e}^{3}{b}^{2}{a}^{4}+184800\,{x}^{5}{d}^{3}{e}^{2}{a}^{3}{b}^{3}+69300\,{x}^{5}{d}^{4}e{a}^{2}{b}^{4}+5544\,{x}^{5}{d}^{5}a{b}^{5}+5544\,{a}^{6}d{e}^{4}{x}^{4}+66528\,{a}^{5}b{d}^{2}{e}^{3}{x}^{4}+166320\,{a}^{4}{b}^{2}{d}^{3}{e}^{2}{x}^{4}+110880\,{a}^{3}{b}^{3}{d}^{4}e{x}^{4}+16632\,{a}^{2}{b}^{4}{d}^{5}{x}^{4}+13860\,{x}^{3}{d}^{2}{e}^{3}{a}^{6}+83160\,{x}^{3}{d}^{3}{e}^{2}{a}^{5}b+103950\,{x}^{3}{d}^{4}e{b}^{2}{a}^{4}+27720\,{x}^{3}{d}^{5}{a}^{3}{b}^{3}+18480\,{x}^{2}{d}^{3}{e}^{2}{a}^{6}+55440\,{x}^{2}{d}^{4}e{a}^{5}b+27720\,{x}^{2}{d}^{5}{b}^{2}{a}^{4}+13860\,x{d}^{4}e{a}^{6}+16632\,x{d}^{5}{a}^{5}b+5544\,{d}^{5}{a}^{6} \right ) }{5544\, \left ( bx+a \right ) ^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5544*x*(462*b^6*e^5*x^11+3024*a*b^5*e^5*x^10+2520*b^6*d*e^4*x^10+8316*a^2*b^4*
e^5*x^9+16632*a*b^5*d*e^4*x^9+5544*b^6*d^2*e^3*x^9+12320*a^3*b^3*e^5*x^8+46200*a
^2*b^4*d*e^4*x^8+36960*a*b^5*d^2*e^3*x^8+6160*b^6*d^3*e^2*x^8+10395*a^4*b^2*e^5*
x^7+69300*a^3*b^3*d*e^4*x^7+103950*a^2*b^4*d^2*e^3*x^7+41580*a*b^5*d^3*e^2*x^7+3
465*b^6*d^4*e*x^7+4752*a^5*b*e^5*x^6+59400*a^4*b^2*d*e^4*x^6+158400*a^3*b^3*d^2*
e^3*x^6+118800*a^2*b^4*d^3*e^2*x^6+23760*a*b^5*d^4*e*x^6+792*b^6*d^5*x^6+924*a^6
*e^5*x^5+27720*a^5*b*d*e^4*x^5+138600*a^4*b^2*d^2*e^3*x^5+184800*a^3*b^3*d^3*e^2
*x^5+69300*a^2*b^4*d^4*e*x^5+5544*a*b^5*d^5*x^5+5544*a^6*d*e^4*x^4+66528*a^5*b*d
^2*e^3*x^4+166320*a^4*b^2*d^3*e^2*x^4+110880*a^3*b^3*d^4*e*x^4+16632*a^2*b^4*d^5
*x^4+13860*a^6*d^2*e^3*x^3+83160*a^5*b*d^3*e^2*x^3+103950*a^4*b^2*d^4*e*x^3+2772
0*a^3*b^3*d^5*x^3+18480*a^6*d^3*e^2*x^2+55440*a^5*b*d^4*e*x^2+27720*a^4*b^2*d^5*
x^2+13860*a^6*d^4*e*x+16632*a^5*b*d^5*x+5544*a^6*d^5)*((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)^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.280683, size = 698, normalized size = 2.65 \[ \frac{1}{12} \, b^{6} e^{5} x^{12} + a^{6} d^{5} x + \frac{1}{11} \,{\left (5 \, b^{6} d e^{4} + 6 \, a b^{5} e^{5}\right )} x^{11} + \frac{1}{2} \,{\left (2 \, b^{6} d^{2} e^{3} + 6 \, a b^{5} d e^{4} + 3 \, a^{2} b^{4} e^{5}\right )} x^{10} + \frac{5}{9} \,{\left (2 \, b^{6} d^{3} e^{2} + 12 \, a b^{5} d^{2} e^{3} + 15 \, a^{2} b^{4} d e^{4} + 4 \, a^{3} b^{3} e^{5}\right )} x^{9} + \frac{5}{8} \,{\left (b^{6} d^{4} e + 12 \, a b^{5} d^{3} e^{2} + 30 \, a^{2} b^{4} d^{2} e^{3} + 20 \, a^{3} b^{3} d e^{4} + 3 \, a^{4} b^{2} e^{5}\right )} x^{8} + \frac{1}{7} \,{\left (b^{6} d^{5} + 30 \, a b^{5} d^{4} e + 150 \, a^{2} b^{4} d^{3} e^{2} + 200 \, a^{3} b^{3} d^{2} e^{3} + 75 \, a^{4} b^{2} d e^{4} + 6 \, a^{5} b e^{5}\right )} x^{7} + \frac{1}{6} \,{\left (6 \, a b^{5} d^{5} + 75 \, a^{2} b^{4} d^{4} e + 200 \, a^{3} b^{3} d^{3} e^{2} + 150 \, a^{4} b^{2} d^{2} e^{3} + 30 \, a^{5} b d e^{4} + a^{6} e^{5}\right )} x^{6} +{\left (3 \, a^{2} b^{4} d^{5} + 20 \, a^{3} b^{3} d^{4} e + 30 \, a^{4} b^{2} d^{3} e^{2} + 12 \, a^{5} b d^{2} e^{3} + a^{6} d e^{4}\right )} x^{5} + \frac{5}{4} \,{\left (4 \, a^{3} b^{3} d^{5} + 15 \, a^{4} b^{2} d^{4} e + 12 \, a^{5} b d^{3} e^{2} + 2 \, a^{6} d^{2} e^{3}\right )} x^{4} + \frac{5}{3} \,{\left (3 \, a^{4} b^{2} d^{5} + 6 \, a^{5} b d^{4} e + 2 \, a^{6} d^{3} e^{2}\right )} x^{3} + \frac{1}{2} \,{\left (6 \, a^{5} b d^{5} + 5 \, a^{6} d^{4} e\right )} x^{2} \]

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)^5,x, algorithm="fricas")

[Out]

1/12*b^6*e^5*x^12 + a^6*d^5*x + 1/11*(5*b^6*d*e^4 + 6*a*b^5*e^5)*x^11 + 1/2*(2*b
^6*d^2*e^3 + 6*a*b^5*d*e^4 + 3*a^2*b^4*e^5)*x^10 + 5/9*(2*b^6*d^3*e^2 + 12*a*b^5
*d^2*e^3 + 15*a^2*b^4*d*e^4 + 4*a^3*b^3*e^5)*x^9 + 5/8*(b^6*d^4*e + 12*a*b^5*d^3
*e^2 + 30*a^2*b^4*d^2*e^3 + 20*a^3*b^3*d*e^4 + 3*a^4*b^2*e^5)*x^8 + 1/7*(b^6*d^5
+ 30*a*b^5*d^4*e + 150*a^2*b^4*d^3*e^2 + 200*a^3*b^3*d^2*e^3 + 75*a^4*b^2*d*e^4
+ 6*a^5*b*e^5)*x^7 + 1/6*(6*a*b^5*d^5 + 75*a^2*b^4*d^4*e + 200*a^3*b^3*d^3*e^2
+ 150*a^4*b^2*d^2*e^3 + 30*a^5*b*d*e^4 + a^6*e^5)*x^6 + (3*a^2*b^4*d^5 + 20*a^3*
b^3*d^4*e + 30*a^4*b^2*d^3*e^2 + 12*a^5*b*d^2*e^3 + a^6*d*e^4)*x^5 + 5/4*(4*a^3*
b^3*d^5 + 15*a^4*b^2*d^4*e + 12*a^5*b*d^3*e^2 + 2*a^6*d^2*e^3)*x^4 + 5/3*(3*a^4*
b^2*d^5 + 6*a^5*b*d^4*e + 2*a^6*d^3*e^2)*x^3 + 1/2*(6*a^5*b*d^5 + 5*a^6*d^4*e)*x
^2

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \left (a + b x\right ) \left (d + e x\right )^{5} \left (\left (a + b x\right )^{2}\right )^{\frac{5}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((a + b*x)*(d + e*x)**5*((a + b*x)**2)**(5/2), x)

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GIAC/XCAS [A]  time = 0.304878, size = 1094, normalized size = 4.16 \[ \text{result too large to display} \]

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)^5,x, algorithm="giac")

[Out]

1/12*b^6*x^12*e^5*sign(b*x + a) + 5/11*b^6*d*x^11*e^4*sign(b*x + a) + b^6*d^2*x^
10*e^3*sign(b*x + a) + 10/9*b^6*d^3*x^9*e^2*sign(b*x + a) + 5/8*b^6*d^4*x^8*e*si
gn(b*x + a) + 1/7*b^6*d^5*x^7*sign(b*x + a) + 6/11*a*b^5*x^11*e^5*sign(b*x + a)
+ 3*a*b^5*d*x^10*e^4*sign(b*x + a) + 20/3*a*b^5*d^2*x^9*e^3*sign(b*x + a) + 15/2
*a*b^5*d^3*x^8*e^2*sign(b*x + a) + 30/7*a*b^5*d^4*x^7*e*sign(b*x + a) + a*b^5*d^
5*x^6*sign(b*x + a) + 3/2*a^2*b^4*x^10*e^5*sign(b*x + a) + 25/3*a^2*b^4*d*x^9*e^
4*sign(b*x + a) + 75/4*a^2*b^4*d^2*x^8*e^3*sign(b*x + a) + 150/7*a^2*b^4*d^3*x^7
*e^2*sign(b*x + a) + 25/2*a^2*b^4*d^4*x^6*e*sign(b*x + a) + 3*a^2*b^4*d^5*x^5*si
gn(b*x + a) + 20/9*a^3*b^3*x^9*e^5*sign(b*x + a) + 25/2*a^3*b^3*d*x^8*e^4*sign(b
*x + a) + 200/7*a^3*b^3*d^2*x^7*e^3*sign(b*x + a) + 100/3*a^3*b^3*d^3*x^6*e^2*si
gn(b*x + a) + 20*a^3*b^3*d^4*x^5*e*sign(b*x + a) + 5*a^3*b^3*d^5*x^4*sign(b*x +
a) + 15/8*a^4*b^2*x^8*e^5*sign(b*x + a) + 75/7*a^4*b^2*d*x^7*e^4*sign(b*x + a) +
25*a^4*b^2*d^2*x^6*e^3*sign(b*x + a) + 30*a^4*b^2*d^3*x^5*e^2*sign(b*x + a) + 7
5/4*a^4*b^2*d^4*x^4*e*sign(b*x + a) + 5*a^4*b^2*d^5*x^3*sign(b*x + a) + 6/7*a^5*
b*x^7*e^5*sign(b*x + a) + 5*a^5*b*d*x^6*e^4*sign(b*x + a) + 12*a^5*b*d^2*x^5*e^3
*sign(b*x + a) + 15*a^5*b*d^3*x^4*e^2*sign(b*x + a) + 10*a^5*b*d^4*x^3*e*sign(b*
x + a) + 3*a^5*b*d^5*x^2*sign(b*x + a) + 1/6*a^6*x^6*e^5*sign(b*x + a) + a^6*d*x
^5*e^4*sign(b*x + a) + 5/2*a^6*d^2*x^4*e^3*sign(b*x + a) + 10/3*a^6*d^3*x^3*e^2*
sign(b*x + a) + 5/2*a^6*d^4*x^2*e*sign(b*x + a) + a^6*d^5*x*sign(b*x + a)