3.1623 \(\int \frac{\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{3/2}} \, dx\)

Optimal. Leaf size=123 \[ -\frac{8 b^3 (d+e x)^{5/2} (b d-a e)}{5 e^5}+\frac{4 b^2 (d+e x)^{3/2} (b d-a e)^2}{e^5}-\frac{8 b \sqrt{d+e x} (b d-a e)^3}{e^5}-\frac{2 (b d-a e)^4}{e^5 \sqrt{d+e x}}+\frac{2 b^4 (d+e x)^{7/2}}{7 e^5} \]

[Out]

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

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Rubi [A]  time = 0.130095, antiderivative size = 123, 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 \[ -\frac{8 b^3 (d+e x)^{5/2} (b d-a e)}{5 e^5}+\frac{4 b^2 (d+e x)^{3/2} (b d-a e)^2}{e^5}-\frac{8 b \sqrt{d+e x} (b d-a e)^3}{e^5}-\frac{2 (b d-a e)^4}{e^5 \sqrt{d+e x}}+\frac{2 b^4 (d+e x)^{7/2}}{7 e^5} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 54.6062, size = 114, normalized size = 0.93 \[ \frac{2 b^{4} \left (d + e x\right )^{\frac{7}{2}}}{7 e^{5}} + \frac{8 b^{3} \left (d + e x\right )^{\frac{5}{2}} \left (a e - b d\right )}{5 e^{5}} + \frac{4 b^{2} \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )^{2}}{e^{5}} + \frac{8 b \sqrt{d + e x} \left (a e - b d\right )^{3}}{e^{5}} - \frac{2 \left (a e - b d\right )^{4}}{e^{5} \sqrt{d + e x}} \]

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)**(3/2),x)

[Out]

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

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Mathematica [A]  time = 0.167363, size = 151, normalized size = 1.23 \[ \frac{2 \left (-35 a^4 e^4+140 a^3 b e^3 (2 d+e x)+70 a^2 b^2 e^2 \left (-8 d^2-4 d e x+e^2 x^2\right )+28 a b^3 e \left (16 d^3+8 d^2 e x-2 d e^2 x^2+e^3 x^3\right )+b^4 \left (-128 d^4-64 d^3 e x+16 d^2 e^2 x^2-8 d e^3 x^3+5 e^4 x^4\right )\right )}{35 e^5 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(-35*a^4*e^4 + 140*a^3*b*e^3*(2*d + e*x) + 70*a^2*b^2*e^2*(-8*d^2 - 4*d*e*x +
e^2*x^2) + 28*a*b^3*e*(16*d^3 + 8*d^2*e*x - 2*d*e^2*x^2 + e^3*x^3) + b^4*(-128*
d^4 - 64*d^3*e*x + 16*d^2*e^2*x^2 - 8*d*e^3*x^3 + 5*e^4*x^4)))/(35*e^5*Sqrt[d +
e*x])

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Maple [A]  time = 0.011, size = 186, normalized size = 1.5 \[ -{\frac{-10\,{x}^{4}{b}^{4}{e}^{4}-56\,{x}^{3}a{b}^{3}{e}^{4}+16\,{x}^{3}{b}^{4}d{e}^{3}-140\,{x}^{2}{a}^{2}{b}^{2}{e}^{4}+112\,{x}^{2}a{b}^{3}d{e}^{3}-32\,{x}^{2}{b}^{4}{d}^{2}{e}^{2}-280\,x{a}^{3}b{e}^{4}+560\,x{a}^{2}{b}^{2}d{e}^{3}-448\,xa{b}^{3}{d}^{2}{e}^{2}+128\,x{b}^{4}{d}^{3}e+70\,{a}^{4}{e}^{4}-560\,{a}^{3}bd{e}^{3}+1120\,{a}^{2}{b}^{2}{d}^{2}{e}^{2}-896\,a{b}^{3}{d}^{3}e+256\,{b}^{4}{d}^{4}}{35\,{e}^{5}}{\frac{1}{\sqrt{ex+d}}}} \]

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)^(3/2),x)

[Out]

-2/35*(-5*b^4*e^4*x^4-28*a*b^3*e^4*x^3+8*b^4*d*e^3*x^3-70*a^2*b^2*e^4*x^2+56*a*b
^3*d*e^3*x^2-16*b^4*d^2*e^2*x^2-140*a^3*b*e^4*x+280*a^2*b^2*d*e^3*x-224*a*b^3*d^
2*e^2*x+64*b^4*d^3*e*x+35*a^4*e^4-280*a^3*b*d*e^3+560*a^2*b^2*d^2*e^2-448*a*b^3*
d^3*e+128*b^4*d^4)/(e*x+d)^(1/2)/e^5

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Maxima [A]  time = 0.738014, size = 255, normalized size = 2.07 \[ \frac{2 \,{\left (\frac{5 \,{\left (e x + d\right )}^{\frac{7}{2}} b^{4} - 28 \,{\left (b^{4} d - a b^{3} e\right )}{\left (e x + d\right )}^{\frac{5}{2}} + 70 \,{\left (b^{4} d^{2} - 2 \, a b^{3} d e + a^{2} b^{2} e^{2}\right )}{\left (e x + d\right )}^{\frac{3}{2}} - 140 \,{\left (b^{4} d^{3} - 3 \, a b^{3} d^{2} e + 3 \, a^{2} b^{2} d e^{2} - a^{3} b e^{3}\right )} \sqrt{e x + d}}{e^{4}} - \frac{35 \,{\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )}}{\sqrt{e x + d} e^{4}}\right )}}{35 \, e} \]

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)^(3/2),x, algorithm="maxima")

[Out]

2/35*((5*(e*x + d)^(7/2)*b^4 - 28*(b^4*d - a*b^3*e)*(e*x + d)^(5/2) + 70*(b^4*d^
2 - 2*a*b^3*d*e + a^2*b^2*e^2)*(e*x + d)^(3/2) - 140*(b^4*d^3 - 3*a*b^3*d^2*e +
3*a^2*b^2*d*e^2 - a^3*b*e^3)*sqrt(e*x + d))/e^4 - 35*(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)/(sqrt(e*x + d)*e^4))/e

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Fricas [A]  time = 0.20546, size = 246, normalized size = 2. \[ \frac{2 \,{\left (5 \, b^{4} e^{4} x^{4} - 128 \, b^{4} d^{4} + 448 \, a b^{3} d^{3} e - 560 \, a^{2} b^{2} d^{2} e^{2} + 280 \, a^{3} b d e^{3} - 35 \, a^{4} e^{4} - 4 \,{\left (2 \, b^{4} d e^{3} - 7 \, a b^{3} e^{4}\right )} x^{3} + 2 \,{\left (8 \, b^{4} d^{2} e^{2} - 28 \, a b^{3} d e^{3} + 35 \, a^{2} b^{2} e^{4}\right )} x^{2} - 4 \,{\left (16 \, b^{4} d^{3} e - 56 \, a b^{3} d^{2} e^{2} + 70 \, a^{2} b^{2} d e^{3} - 35 \, a^{3} b e^{4}\right )} x\right )}}{35 \, \sqrt{e x + d} e^{5}} \]

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]

2/35*(5*b^4*e^4*x^4 - 128*b^4*d^4 + 448*a*b^3*d^3*e - 560*a^2*b^2*d^2*e^2 + 280*
a^3*b*d*e^3 - 35*a^4*e^4 - 4*(2*b^4*d*e^3 - 7*a*b^3*e^4)*x^3 + 2*(8*b^4*d^2*e^2
- 28*a*b^3*d*e^3 + 35*a^2*b^2*e^4)*x^2 - 4*(16*b^4*d^3*e - 56*a*b^3*d^2*e^2 + 70
*a^2*b^2*d*e^3 - 35*a^3*b*e^4)*x)/(sqrt(e*x + d)*e^5)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + b x\right )^{4}}{\left (d + e x\right )^{\frac{3}{2}}}\, dx \]

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)

[Out]

Integral((a + b*x)**4/(d + e*x)**(3/2), x)

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GIAC/XCAS [A]  time = 0.218859, size = 320, normalized size = 2.6 \[ \frac{2}{35} \,{\left (5 \,{\left (x e + d\right )}^{\frac{7}{2}} b^{4} e^{30} - 28 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{4} d e^{30} + 70 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{4} d^{2} e^{30} - 140 \, \sqrt{x e + d} b^{4} d^{3} e^{30} + 28 \,{\left (x e + d\right )}^{\frac{5}{2}} a b^{3} e^{31} - 140 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{3} d e^{31} + 420 \, \sqrt{x e + d} a b^{3} d^{2} e^{31} + 70 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{2} b^{2} e^{32} - 420 \, \sqrt{x e + d} a^{2} b^{2} d e^{32} + 140 \, \sqrt{x e + d} a^{3} b e^{33}\right )} e^{\left (-35\right )} - \frac{2 \,{\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )} e^{\left (-5\right )}}{\sqrt{x e + d}} \]

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)^(3/2),x, algorithm="giac")

[Out]

2/35*(5*(x*e + d)^(7/2)*b^4*e^30 - 28*(x*e + d)^(5/2)*b^4*d*e^30 + 70*(x*e + d)^
(3/2)*b^4*d^2*e^30 - 140*sqrt(x*e + d)*b^4*d^3*e^30 + 28*(x*e + d)^(5/2)*a*b^3*e
^31 - 140*(x*e + d)^(3/2)*a*b^3*d*e^31 + 420*sqrt(x*e + d)*a*b^3*d^2*e^31 + 70*(
x*e + d)^(3/2)*a^2*b^2*e^32 - 420*sqrt(x*e + d)*a^2*b^2*d*e^32 + 140*sqrt(x*e +
d)*a^3*b*e^33)*e^(-35) - 2*(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^(-5)/sqrt(x*e + d)