3.1624 \(\int \frac{\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{5/2}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.128688, 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 \[ -\frac{8 b^3 (d+e x)^{3/2} (b d-a e)}{3 e^5}+\frac{12 b^2 \sqrt{d+e x} (b d-a e)^2}{e^5}+\frac{8 b (b d-a e)^3}{e^5 \sqrt{d+e x}}-\frac{2 (b d-a e)^4}{3 e^5 (d+e x)^{3/2}}+\frac{2 b^4 (d+e x)^{5/2}}{5 e^5} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 53.615, size = 116, normalized size = 0.93 \[ \frac{2 b^{4} \left (d + e x\right )^{\frac{5}{2}}}{5 e^{5}} + \frac{8 b^{3} \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )}{3 e^{5}} + \frac{12 b^{2} \sqrt{d + e x} \left (a e - b d\right )^{2}}{e^{5}} - \frac{8 b \left (a e - b d\right )^{3}}{e^{5} \sqrt{d + e x}} - \frac{2 \left (a e - b d\right )^{4}}{3 e^{5} \left (d + e x\right )^{\frac{3}{2}}} \]

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)**(5/2),x)

[Out]

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

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Mathematica [A]  time = 0.20862, size = 110, normalized size = 0.88 \[ \frac{2 \sqrt{d+e x} \left (b^2 \left (90 a^2 e^2-160 a b d e+73 b^2 d^2\right )-2 b^3 e x (7 b d-10 a e)+\frac{60 b (b d-a e)^3}{d+e x}-\frac{5 (b d-a e)^4}{(d+e x)^2}+3 b^4 e^2 x^2\right )}{15 e^5} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[d + e*x]*(b^2*(73*b^2*d^2 - 160*a*b*d*e + 90*a^2*e^2) - 2*b^3*e*(7*b*d -
10*a*e)*x + 3*b^4*e^2*x^2 - (5*(b*d - a*e)^4)/(d + e*x)^2 + (60*b*(b*d - a*e)^3
)/(d + e*x)))/(15*e^5)

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Maple [A]  time = 0.011, size = 186, normalized size = 1.5 \[ -{\frac{-6\,{x}^{4}{b}^{4}{e}^{4}-40\,{x}^{3}a{b}^{3}{e}^{4}+16\,{x}^{3}{b}^{4}d{e}^{3}-180\,{x}^{2}{a}^{2}{b}^{2}{e}^{4}+240\,{x}^{2}a{b}^{3}d{e}^{3}-96\,{x}^{2}{b}^{4}{d}^{2}{e}^{2}+120\,x{a}^{3}b{e}^{4}-720\,x{a}^{2}{b}^{2}d{e}^{3}+960\,xa{b}^{3}{d}^{2}{e}^{2}-384\,x{b}^{4}{d}^{3}e+10\,{a}^{4}{e}^{4}+80\,{a}^{3}bd{e}^{3}-480\,{a}^{2}{b}^{2}{d}^{2}{e}^{2}+640\,a{b}^{3}{d}^{3}e-256\,{b}^{4}{d}^{4}}{15\,{e}^{5}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}} \]

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)^(5/2),x)

[Out]

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

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Maxima [A]  time = 0.739769, size = 252, normalized size = 2.02 \[ \frac{2 \,{\left (\frac{3 \,{\left (e x + d\right )}^{\frac{5}{2}} b^{4} - 20 \,{\left (b^{4} d - a b^{3} e\right )}{\left (e x + d\right )}^{\frac{3}{2}} + 90 \,{\left (b^{4} d^{2} - 2 \, a b^{3} d e + a^{2} b^{2} e^{2}\right )} \sqrt{e x + d}}{e^{4}} - \frac{5 \,{\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4} - 12 \,{\left (b^{4} d^{3} - 3 \, a b^{3} d^{2} e + 3 \, a^{2} b^{2} d e^{2} - a^{3} b e^{3}\right )}{\left (e x + d\right )}\right )}}{{\left (e x + d\right )}^{\frac{3}{2}} e^{4}}\right )}}{15 \, e} \]

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)^(5/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.205044, size = 259, normalized size = 2.07 \[ \frac{2 \,{\left (3 \, b^{4} e^{4} x^{4} + 128 \, b^{4} d^{4} - 320 \, a b^{3} d^{3} e + 240 \, a^{2} b^{2} d^{2} e^{2} - 40 \, a^{3} b d e^{3} - 5 \, a^{4} e^{4} - 4 \,{\left (2 \, b^{4} d e^{3} - 5 \, a b^{3} e^{4}\right )} x^{3} + 6 \,{\left (8 \, b^{4} d^{2} e^{2} - 20 \, a b^{3} d e^{3} + 15 \, a^{2} b^{2} e^{4}\right )} x^{2} + 12 \,{\left (16 \, b^{4} d^{3} e - 40 \, a b^{3} d^{2} e^{2} + 30 \, a^{2} b^{2} d e^{3} - 5 \, a^{3} b e^{4}\right )} x\right )}}{15 \,{\left (e^{6} x + d e^{5}\right )} \sqrt{e x + d}} \]

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)^(5/2),x, algorithm="fricas")

[Out]

2/15*(3*b^4*e^4*x^4 + 128*b^4*d^4 - 320*a*b^3*d^3*e + 240*a^2*b^2*d^2*e^2 - 40*a
^3*b*d*e^3 - 5*a^4*e^4 - 4*(2*b^4*d*e^3 - 5*a*b^3*e^4)*x^3 + 6*(8*b^4*d^2*e^2 -
20*a*b^3*d*e^3 + 15*a^2*b^2*e^4)*x^2 + 12*(16*b^4*d^3*e - 40*a*b^3*d^2*e^2 + 30*
a^2*b^2*d*e^3 - 5*a^3*b*e^4)*x)/((e^6*x + d*e^5)*sqrt(e*x + d))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + b x\right )^{4}}{\left (d + e x\right )^{\frac{5}{2}}}\, dx \]

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)**(5/2),x)

[Out]

Integral((a + b*x)**4/(d + e*x)**(5/2), x)

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GIAC/XCAS [A]  time = 0.213063, size = 309, normalized size = 2.47 \[ \frac{2}{15} \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{4} e^{20} - 20 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{4} d e^{20} + 90 \, \sqrt{x e + d} b^{4} d^{2} e^{20} + 20 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{3} e^{21} - 180 \, \sqrt{x e + d} a b^{3} d e^{21} + 90 \, \sqrt{x e + d} a^{2} b^{2} e^{22}\right )} e^{\left (-25\right )} + \frac{2 \,{\left (12 \,{\left (x e + d\right )} b^{4} d^{3} - b^{4} d^{4} - 36 \,{\left (x e + d\right )} a b^{3} d^{2} e + 4 \, a b^{3} d^{3} e + 36 \,{\left (x e + d\right )} a^{2} b^{2} d e^{2} - 6 \, a^{2} b^{2} d^{2} e^{2} - 12 \,{\left (x e + d\right )} a^{3} b e^{3} + 4 \, a^{3} b d e^{3} - a^{4} e^{4}\right )} e^{\left (-5\right )}}{3 \,{\left (x e + d\right )}^{\frac{3}{2}}} \]

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)^(5/2),x, algorithm="giac")

[Out]

2/15*(3*(x*e + d)^(5/2)*b^4*e^20 - 20*(x*e + d)^(3/2)*b^4*d*e^20 + 90*sqrt(x*e +
d)*b^4*d^2*e^20 + 20*(x*e + d)^(3/2)*a*b^3*e^21 - 180*sqrt(x*e + d)*a*b^3*d*e^2
1 + 90*sqrt(x*e + d)*a^2*b^2*e^22)*e^(-25) + 2/3*(12*(x*e + d)*b^4*d^3 - b^4*d^4
- 36*(x*e + d)*a*b^3*d^2*e + 4*a*b^3*d^3*e + 36*(x*e + d)*a^2*b^2*d*e^2 - 6*a^2
*b^2*d^2*e^2 - 12*(x*e + d)*a^3*b*e^3 + 4*a^3*b*d*e^3 - a^4*e^4)*e^(-5)/(x*e + d
)^(3/2)