3.1625 \(\int \frac{\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{7/2}} \, dx\)

Optimal. Leaf size=125 \[ -\frac{8 b^3 \sqrt{d+e x} (b d-a e)}{e^5}-\frac{12 b^2 (b d-a e)^2}{e^5 \sqrt{d+e x}}+\frac{8 b (b d-a e)^3}{3 e^5 (d+e x)^{3/2}}-\frac{2 (b d-a e)^4}{5 e^5 (d+e x)^{5/2}}+\frac{2 b^4 (d+e x)^{3/2}}{3 e^5} \]

[Out]

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

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Rubi [A]  time = 0.130972, antiderivative size = 125, 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 \[ -\frac{8 b^3 \sqrt{d+e x} (b d-a e)}{e^5}-\frac{12 b^2 (b d-a e)^2}{e^5 \sqrt{d+e x}}+\frac{8 b (b d-a e)^3}{3 e^5 (d+e x)^{3/2}}-\frac{2 (b d-a e)^4}{5 e^5 (d+e x)^{5/2}}+\frac{2 b^4 (d+e x)^{3/2}}{3 e^5} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x + b^2*x^2)^2/(d + e*x)^(7/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 53.4297, size = 116, normalized size = 0.93 \[ \frac{2 b^{4} \left (d + e x\right )^{\frac{3}{2}}}{3 e^{5}} + \frac{8 b^{3} \sqrt{d + e x} \left (a e - b d\right )}{e^{5}} - \frac{12 b^{2} \left (a e - b d\right )^{2}}{e^{5} \sqrt{d + e x}} - \frac{8 b \left (a e - b d\right )^{3}}{3 e^{5} \left (d + e x\right )^{\frac{3}{2}}} - \frac{2 \left (a e - b d\right )^{4}}{5 e^{5} \left (d + e x\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b**2*x**2+2*a*b*x+a**2)**2/(e*x+d)**(7/2),x)

[Out]

2*b**4*(d + e*x)**(3/2)/(3*e**5) + 8*b**3*sqrt(d + e*x)*(a*e - b*d)/e**5 - 12*b*
*2*(a*e - b*d)**2/(e**5*sqrt(d + e*x)) - 8*b*(a*e - b*d)**3/(3*e**5*(d + e*x)**(
3/2)) - 2*(a*e - b*d)**4/(5*e**5*(d + e*x)**(5/2))

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Mathematica [A]  time = 0.230343, size = 98, normalized size = 0.78 \[ \frac{2 \sqrt{d+e x} \left (60 a b^3 e-\frac{90 b^2 (b d-a e)^2}{d+e x}+\frac{20 b (b d-a e)^3}{(d+e x)^2}-\frac{3 (b d-a e)^4}{(d+e x)^3}-55 b^4 d+5 b^4 e x\right )}{15 e^5} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x + b^2*x^2)^2/(d + e*x)^(7/2),x]

[Out]

(2*Sqrt[d + e*x]*(-55*b^4*d + 60*a*b^3*e + 5*b^4*e*x - (3*(b*d - a*e)^4)/(d + e*
x)^3 + (20*b*(b*d - a*e)^3)/(d + e*x)^2 - (90*b^2*(b*d - a*e)^2)/(d + e*x)))/(15
*e^5)

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Maple [A]  time = 0.012, size = 186, normalized size = 1.5 \[ -{\frac{-10\,{x}^{4}{b}^{4}{e}^{4}-120\,{x}^{3}a{b}^{3}{e}^{4}+80\,{x}^{3}{b}^{4}d{e}^{3}+180\,{x}^{2}{a}^{2}{b}^{2}{e}^{4}-720\,{x}^{2}a{b}^{3}d{e}^{3}+480\,{x}^{2}{b}^{4}{d}^{2}{e}^{2}+40\,x{a}^{3}b{e}^{4}+240\,x{a}^{2}{b}^{2}d{e}^{3}-960\,xa{b}^{3}{d}^{2}{e}^{2}+640\,x{b}^{4}{d}^{3}e+6\,{a}^{4}{e}^{4}+16\,{a}^{3}bd{e}^{3}+96\,{a}^{2}{b}^{2}{d}^{2}{e}^{2}-384\,a{b}^{3}{d}^{3}e+256\,{b}^{4}{d}^{4}}{15\,{e}^{5}} \left ( ex+d \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b^2*x^2+2*a*b*x+a^2)^2/(e*x+d)^(7/2),x)

[Out]

-2/15*(-5*b^4*e^4*x^4-60*a*b^3*e^4*x^3+40*b^4*d*e^3*x^3+90*a^2*b^2*e^4*x^2-360*a
*b^3*d*e^3*x^2+240*b^4*d^2*e^2*x^2+20*a^3*b*e^4*x+120*a^2*b^2*d*e^3*x-480*a*b^3*
d^2*e^2*x+320*b^4*d^3*e*x+3*a^4*e^4+8*a^3*b*d*e^3+48*a^2*b^2*d^2*e^2-192*a*b^3*d
^3*e+128*b^4*d^4)/(e*x+d)^(5/2)/e^5

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Maxima [A]  time = 0.744081, size = 255, normalized size = 2.04 \[ \frac{2 \,{\left (\frac{5 \,{\left ({\left (e x + d\right )}^{\frac{3}{2}} b^{4} - 12 \,{\left (b^{4} d - a b^{3} e\right )} \sqrt{e x + d}\right )}}{e^{4}} - \frac{3 \, b^{4} d^{4} - 12 \, a b^{3} d^{3} e + 18 \, a^{2} b^{2} d^{2} e^{2} - 12 \, a^{3} b d e^{3} + 3 \, a^{4} e^{4} + 90 \,{\left (b^{4} d^{2} - 2 \, a b^{3} d e + a^{2} b^{2} e^{2}\right )}{\left (e x + d\right )}^{2} - 20 \,{\left (b^{4} d^{3} - 3 \, a b^{3} d^{2} e + 3 \, a^{2} b^{2} d e^{2} - a^{3} b e^{3}\right )}{\left (e x + d\right )}}{{\left (e x + d\right )}^{\frac{5}{2}} e^{4}}\right )}}{15 \, e} \]

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)^(7/2),x, algorithm="maxima")

[Out]

2/15*(5*((e*x + d)^(3/2)*b^4 - 12*(b^4*d - a*b^3*e)*sqrt(e*x + d))/e^4 - (3*b^4*
d^4 - 12*a*b^3*d^3*e + 18*a^2*b^2*d^2*e^2 - 12*a^3*b*d*e^3 + 3*a^4*e^4 + 90*(b^4
*d^2 - 2*a*b^3*d*e + a^2*b^2*e^2)*(e*x + d)^2 - 20*(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))/((e*x + d)^(5/2)*e^4))/e

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Fricas [A]  time = 0.206054, size = 273, normalized size = 2.18 \[ \frac{2 \,{\left (5 \, b^{4} e^{4} x^{4} - 128 \, b^{4} d^{4} + 192 \, a b^{3} d^{3} e - 48 \, a^{2} b^{2} d^{2} e^{2} - 8 \, a^{3} b d e^{3} - 3 \, a^{4} e^{4} - 20 \,{\left (2 \, b^{4} d e^{3} - 3 \, a b^{3} e^{4}\right )} x^{3} - 30 \,{\left (8 \, b^{4} d^{2} e^{2} - 12 \, a b^{3} d e^{3} + 3 \, a^{2} b^{2} e^{4}\right )} x^{2} - 20 \,{\left (16 \, b^{4} d^{3} e - 24 \, a b^{3} d^{2} e^{2} + 6 \, a^{2} b^{2} d e^{3} + a^{3} b e^{4}\right )} x\right )}}{15 \,{\left (e^{7} x^{2} + 2 \, d e^{6} x + d^{2} e^{5}\right )} \sqrt{e x + d}} \]

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)^(7/2),x, algorithm="fricas")

[Out]

2/15*(5*b^4*e^4*x^4 - 128*b^4*d^4 + 192*a*b^3*d^3*e - 48*a^2*b^2*d^2*e^2 - 8*a^3
*b*d*e^3 - 3*a^4*e^4 - 20*(2*b^4*d*e^3 - 3*a*b^3*e^4)*x^3 - 30*(8*b^4*d^2*e^2 -
12*a*b^3*d*e^3 + 3*a^2*b^2*e^4)*x^2 - 20*(16*b^4*d^3*e - 24*a*b^3*d^2*e^2 + 6*a^
2*b^2*d*e^3 + a^3*b*e^4)*x)/((e^7*x^2 + 2*d*e^6*x + d^2*e^5)*sqrt(e*x + d))

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Sympy [A]  time = 10.4266, size = 1008, normalized size = 8.06 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b**2*x**2+2*a*b*x+a**2)**2/(e*x+d)**(7/2),x)

[Out]

Piecewise((-6*a**4*e**4/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x)
+ 15*e**7*x**2*sqrt(d + e*x)) - 16*a**3*b*d*e**3/(15*d**2*e**5*sqrt(d + e*x) + 3
0*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) - 40*a**3*b*e**4*x/(15*d*
*2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x))
- 96*a**2*b**2*d**2*e**2/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x)
+ 15*e**7*x**2*sqrt(d + e*x)) - 240*a**2*b**2*d*e**3*x/(15*d**2*e**5*sqrt(d + e
*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) - 180*a**2*b**2*e*
*4*x**2/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*s
qrt(d + e*x)) + 384*a*b**3*d**3*e/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt
(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) + 960*a*b**3*d**2*e**2*x/(15*d**2*e**5*s
qrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) + 720*a*b
**3*d*e**3*x**2/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**
7*x**2*sqrt(d + e*x)) + 120*a*b**3*e**4*x**3/(15*d**2*e**5*sqrt(d + e*x) + 30*d*
e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) - 256*b**4*d**4/(15*d**2*e**5
*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) - 640*b
**4*d**3*e*x/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x
**2*sqrt(d + e*x)) - 480*b**4*d**2*e**2*x**2/(15*d**2*e**5*sqrt(d + e*x) + 30*d*
e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) - 80*b**4*d*e**3*x**3/(15*d**
2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) +
10*b**4*e**4*x**4/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*
e**7*x**2*sqrt(d + e*x)), Ne(e, 0)), ((a**4*x + 2*a**3*b*x**2 + 2*a**2*b**2*x**3
+ a*b**3*x**4 + b**4*x**5/5)/d**(7/2), True))

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GIAC/XCAS [A]  time = 0.213718, size = 305, normalized size = 2.44 \[ \frac{2}{3} \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} b^{4} e^{10} - 12 \, \sqrt{x e + d} b^{4} d e^{10} + 12 \, \sqrt{x e + d} a b^{3} e^{11}\right )} e^{\left (-15\right )} - \frac{2 \,{\left (90 \,{\left (x e + d\right )}^{2} b^{4} d^{2} - 20 \,{\left (x e + d\right )} b^{4} d^{3} + 3 \, b^{4} d^{4} - 180 \,{\left (x e + d\right )}^{2} a b^{3} d e + 60 \,{\left (x e + d\right )} a b^{3} d^{2} e - 12 \, a b^{3} d^{3} e + 90 \,{\left (x e + d\right )}^{2} a^{2} b^{2} e^{2} - 60 \,{\left (x e + d\right )} a^{2} b^{2} d e^{2} + 18 \, a^{2} b^{2} d^{2} e^{2} + 20 \,{\left (x e + d\right )} a^{3} b e^{3} - 12 \, a^{3} b d e^{3} + 3 \, a^{4} e^{4}\right )} e^{\left (-5\right )}}{15 \,{\left (x e + d\right )}^{\frac{5}{2}}} \]

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)^(7/2),x, algorithm="giac")

[Out]

2/3*((x*e + d)^(3/2)*b^4*e^10 - 12*sqrt(x*e + d)*b^4*d*e^10 + 12*sqrt(x*e + d)*a
*b^3*e^11)*e^(-15) - 2/15*(90*(x*e + d)^2*b^4*d^2 - 20*(x*e + d)*b^4*d^3 + 3*b^4
*d^4 - 180*(x*e + d)^2*a*b^3*d*e + 60*(x*e + d)*a*b^3*d^2*e - 12*a*b^3*d^3*e + 9
0*(x*e + d)^2*a^2*b^2*e^2 - 60*(x*e + d)*a^2*b^2*d*e^2 + 18*a^2*b^2*d^2*e^2 + 20
*(x*e + d)*a^3*b*e^3 - 12*a^3*b*d*e^3 + 3*a^4*e^4)*e^(-5)/(x*e + d)^(5/2)