3.2270 \(\int \frac{\left (a+b x+c x^2\right )^2}{(d+e x)^{7/2}} \, dx\)

Optimal. Leaf size=162 \[ -\frac{2 \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{e^5 \sqrt{d+e x}}+\frac{4 (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{3 e^5 (d+e x)^{3/2}}-\frac{2 \left (a e^2-b d e+c d^2\right )^2}{5 e^5 (d+e x)^{5/2}}-\frac{4 c \sqrt{d+e x} (2 c d-b e)}{e^5}+\frac{2 c^2 (d+e x)^{3/2}}{3 e^5} \]

[Out]

(-2*(c*d^2 - b*d*e + a*e^2)^2)/(5*e^5*(d + e*x)^(5/2)) + (4*(2*c*d - b*e)*(c*d^2
- b*d*e + a*e^2))/(3*e^5*(d + e*x)^(3/2)) - (2*(6*c^2*d^2 + b^2*e^2 - 2*c*e*(3*
b*d - a*e)))/(e^5*Sqrt[d + e*x]) - (4*c*(2*c*d - b*e)*Sqrt[d + e*x])/e^5 + (2*c^
2*(d + e*x)^(3/2))/(3*e^5)

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Rubi [A]  time = 0.19535, antiderivative size = 162, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045 \[ -\frac{2 \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{e^5 \sqrt{d+e x}}+\frac{4 (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{3 e^5 (d+e x)^{3/2}}-\frac{2 \left (a e^2-b d e+c d^2\right )^2}{5 e^5 (d+e x)^{5/2}}-\frac{4 c \sqrt{d+e x} (2 c d-b e)}{e^5}+\frac{2 c^2 (d+e x)^{3/2}}{3 e^5} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(c*d^2 - b*d*e + a*e^2)^2)/(5*e^5*(d + e*x)^(5/2)) + (4*(2*c*d - b*e)*(c*d^2
- b*d*e + a*e^2))/(3*e^5*(d + e*x)^(3/2)) - (2*(6*c^2*d^2 + b^2*e^2 - 2*c*e*(3*
b*d - a*e)))/(e^5*Sqrt[d + e*x]) - (4*c*(2*c*d - b*e)*Sqrt[d + e*x])/e^5 + (2*c^
2*(d + e*x)^(3/2))/(3*e^5)

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Rubi in Sympy [A]  time = 40.6452, size = 158, normalized size = 0.98 \[ \frac{2 c^{2} \left (d + e x\right )^{\frac{3}{2}}}{3 e^{5}} + \frac{4 c \sqrt{d + e x} \left (b e - 2 c d\right )}{e^{5}} - \frac{2 \left (2 a c e^{2} + b^{2} e^{2} - 6 b c d e + 6 c^{2} d^{2}\right )}{e^{5} \sqrt{d + e x}} - \frac{4 \left (b e - 2 c d\right ) \left (a e^{2} - b d e + c d^{2}\right )}{3 e^{5} \left (d + e x\right )^{\frac{3}{2}}} - \frac{2 \left (a e^{2} - b d e + c d^{2}\right )^{2}}{5 e^{5} \left (d + e x\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*c**2*(d + e*x)**(3/2)/(3*e**5) + 4*c*sqrt(d + e*x)*(b*e - 2*c*d)/e**5 - 2*(2*a
*c*e**2 + b**2*e**2 - 6*b*c*d*e + 6*c**2*d**2)/(e**5*sqrt(d + e*x)) - 4*(b*e - 2
*c*d)*(a*e**2 - b*d*e + c*d**2)/(3*e**5*(d + e*x)**(3/2)) - 2*(a*e**2 - b*d*e +
c*d**2)**2/(5*e**5*(d + e*x)**(5/2))

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Mathematica [A]  time = 0.385446, size = 134, normalized size = 0.83 \[ \frac{2 \sqrt{d+e x} \left (-\frac{15 \left (2 c e (a e-3 b d)+b^2 e^2+6 c^2 d^2\right )}{d+e x}+\frac{10 (2 c d-b e) \left (e (a e-b d)+c d^2\right )}{(d+e x)^2}-\frac{3 \left (e (a e-b d)+c d^2\right )^2}{(d+e x)^3}+5 c (6 b e-11 c d)+5 c^2 e x\right )}{15 e^5} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[d + e*x]*(5*c*(-11*c*d + 6*b*e) + 5*c^2*e*x - (3*(c*d^2 + e*(-(b*d) + a*
e))^2)/(d + e*x)^3 + (10*(2*c*d - b*e)*(c*d^2 + e*(-(b*d) + a*e)))/(d + e*x)^2 -
(15*(6*c^2*d^2 + b^2*e^2 + 2*c*e*(-3*b*d + a*e)))/(d + e*x)))/(15*e^5)

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Maple [A]  time = 0.01, size = 194, normalized size = 1.2 \[ -{\frac{-10\,{x}^{4}{c}^{2}{e}^{4}-60\,bc{e}^{4}{x}^{3}+80\,{x}^{3}{c}^{2}d{e}^{3}+60\,{x}^{2}ac{e}^{4}+30\,{b}^{2}{e}^{4}{x}^{2}-360\,bcd{e}^{3}{x}^{2}+480\,{x}^{2}{c}^{2}{d}^{2}{e}^{2}+20\,ab{e}^{4}x+80\,xacd{e}^{3}+40\,{b}^{2}d{e}^{3}x-480\,bc{d}^{2}{e}^{2}x+640\,x{c}^{2}{d}^{3}e+6\,{a}^{2}{e}^{4}+8\,abd{e}^{3}+32\,ac{d}^{2}{e}^{2}+16\,{b}^{2}{d}^{2}{e}^{2}-192\,bc{d}^{3}e+256\,{c}^{2}{d}^{4}}{15\,{e}^{5}} \left ( ex+d \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/15/(e*x+d)^(5/2)*(-5*c^2*e^4*x^4-30*b*c*e^4*x^3+40*c^2*d*e^3*x^3+30*a*c*e^4*x
^2+15*b^2*e^4*x^2-180*b*c*d*e^3*x^2+240*c^2*d^2*e^2*x^2+10*a*b*e^4*x+40*a*c*d*e^
3*x+20*b^2*d*e^3*x-240*b*c*d^2*e^2*x+320*c^2*d^3*e*x+3*a^2*e^4+4*a*b*d*e^3+16*a*
c*d^2*e^2+8*b^2*d^2*e^2-96*b*c*d^3*e+128*c^2*d^4)/e^5

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Maxima [A]  time = 0.70525, size = 250, normalized size = 1.54 \[ \frac{2 \,{\left (\frac{5 \,{\left ({\left (e x + d\right )}^{\frac{3}{2}} c^{2} - 6 \,{\left (2 \, c^{2} d - b c e\right )} \sqrt{e x + d}\right )}}{e^{4}} - \frac{3 \, c^{2} d^{4} - 6 \, b c d^{3} e - 6 \, a b d e^{3} + 3 \, a^{2} e^{4} + 3 \,{\left (b^{2} + 2 \, a c\right )} d^{2} e^{2} + 15 \,{\left (6 \, c^{2} d^{2} - 6 \, b c d e +{\left (b^{2} + 2 \, a c\right )} e^{2}\right )}{\left (e x + d\right )}^{2} - 10 \,{\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e - a b e^{3} +{\left (b^{2} + 2 \, a c\right )} d e^{2}\right )}{\left (e x + d\right )}}{{\left (e x + d\right )}^{\frac{5}{2}} e^{4}}\right )}}{15 \, e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^2/(e*x + d)^(7/2),x, algorithm="maxima")

[Out]

2/15*(5*((e*x + d)^(3/2)*c^2 - 6*(2*c^2*d - b*c*e)*sqrt(e*x + d))/e^4 - (3*c^2*d
^4 - 6*b*c*d^3*e - 6*a*b*d*e^3 + 3*a^2*e^4 + 3*(b^2 + 2*a*c)*d^2*e^2 + 15*(6*c^2
*d^2 - 6*b*c*d*e + (b^2 + 2*a*c)*e^2)*(e*x + d)^2 - 10*(2*c^2*d^3 - 3*b*c*d^2*e
- a*b*e^3 + (b^2 + 2*a*c)*d*e^2)*(e*x + d))/((e*x + d)^(5/2)*e^4))/e

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Fricas [A]  time = 0.207911, size = 263, normalized size = 1.62 \[ \frac{2 \,{\left (5 \, c^{2} e^{4} x^{4} - 128 \, c^{2} d^{4} + 96 \, b c d^{3} e - 4 \, a b d e^{3} - 3 \, a^{2} e^{4} - 8 \,{\left (b^{2} + 2 \, a c\right )} d^{2} e^{2} - 10 \,{\left (4 \, c^{2} d e^{3} - 3 \, b c e^{4}\right )} x^{3} - 15 \,{\left (16 \, c^{2} d^{2} e^{2} - 12 \, b c d e^{3} +{\left (b^{2} + 2 \, a c\right )} e^{4}\right )} x^{2} - 10 \,{\left (32 \, c^{2} d^{3} e - 24 \, b c d^{2} e^{2} + a b e^{4} + 2 \,{\left (b^{2} + 2 \, a c\right )} d e^{3}\right )} x\right )}}{15 \,{\left (e^{7} x^{2} + 2 \, d e^{6} x + d^{2} e^{5}\right )} \sqrt{e x + d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^2/(e*x + d)^(7/2),x, algorithm="fricas")

[Out]

2/15*(5*c^2*e^4*x^4 - 128*c^2*d^4 + 96*b*c*d^3*e - 4*a*b*d*e^3 - 3*a^2*e^4 - 8*(
b^2 + 2*a*c)*d^2*e^2 - 10*(4*c^2*d*e^3 - 3*b*c*e^4)*x^3 - 15*(16*c^2*d^2*e^2 - 1
2*b*c*d*e^3 + (b^2 + 2*a*c)*e^4)*x^2 - 10*(32*c^2*d^3*e - 24*b*c*d^2*e^2 + a*b*e
^4 + 2*(b^2 + 2*a*c)*d*e^3)*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 = 10.8464, size = 1180, normalized size = 7.28 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-6*a**2*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)) - 8*a*b*d*e**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)) - 20*a*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)) - 32*a*
c*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)) - 80*a*c*d*e**3*x/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sq
rt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) - 60*a*c*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*sqrt(d + e*x)) - 16*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*sq
rt(d + e*x)) - 40*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)) - 30*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*sqrt(d + e*x)) + 192*b*c*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)) + 480*b*c*d**2*e**2*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)) + 360*b*c*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)) + 60*b*c*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*c**2*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*c**2*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*c**2*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*c**2*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*c**2*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**2
*x + a*b*x**2 + 2*a*c*x**3/3 + b**2*x**3/3 + b*c*x**4/2 + c**2*x**5/5)/d**(7/2),
True))

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GIAC/XCAS [A]  time = 0.208848, size = 325, normalized size = 2.01 \[ \frac{2}{3} \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} c^{2} e^{10} - 12 \, \sqrt{x e + d} c^{2} d e^{10} + 6 \, \sqrt{x e + d} b c e^{11}\right )} e^{\left (-15\right )} - \frac{2 \,{\left (90 \,{\left (x e + d\right )}^{2} c^{2} d^{2} - 20 \,{\left (x e + d\right )} c^{2} d^{3} + 3 \, c^{2} d^{4} - 90 \,{\left (x e + d\right )}^{2} b c d e + 30 \,{\left (x e + d\right )} b c d^{2} e - 6 \, b c d^{3} e + 15 \,{\left (x e + d\right )}^{2} b^{2} e^{2} + 30 \,{\left (x e + d\right )}^{2} a c e^{2} - 10 \,{\left (x e + d\right )} b^{2} d e^{2} - 20 \,{\left (x e + d\right )} a c d e^{2} + 3 \, b^{2} d^{2} e^{2} + 6 \, a c d^{2} e^{2} + 10 \,{\left (x e + d\right )} a b e^{3} - 6 \, a b d e^{3} + 3 \, a^{2} e^{4}\right )} e^{\left (-5\right )}}{15 \,{\left (x e + d\right )}^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^2/(e*x + d)^(7/2),x, algorithm="giac")

[Out]

2/3*((x*e + d)^(3/2)*c^2*e^10 - 12*sqrt(x*e + d)*c^2*d*e^10 + 6*sqrt(x*e + d)*b*
c*e^11)*e^(-15) - 2/15*(90*(x*e + d)^2*c^2*d^2 - 20*(x*e + d)*c^2*d^3 + 3*c^2*d^
4 - 90*(x*e + d)^2*b*c*d*e + 30*(x*e + d)*b*c*d^2*e - 6*b*c*d^3*e + 15*(x*e + d)
^2*b^2*e^2 + 30*(x*e + d)^2*a*c*e^2 - 10*(x*e + d)*b^2*d*e^2 - 20*(x*e + d)*a*c*
d*e^2 + 3*b^2*d^2*e^2 + 6*a*c*d^2*e^2 + 10*(x*e + d)*a*b*e^3 - 6*a*b*d*e^3 + 3*a
^2*e^4)*e^(-5)/(x*e + d)^(5/2)