3.1605 \(\int \frac{(b+2 c x) \left (a+b x+c x^2\right )^2}{(d+e x)^{3/2}} \, dx\)

Optimal. Leaf size=248 \[ \frac{8 c (d+e x)^{5/2} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{5 e^6}-\frac{2 (d+e x)^{3/2} (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{3 e^6}+\frac{4 \sqrt{d+e x} \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^6}+\frac{2 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{e^6 \sqrt{d+e x}}-\frac{10 c^2 (d+e x)^{7/2} (2 c d-b e)}{7 e^6}+\frac{4 c^3 (d+e x)^{9/2}}{9 e^6} \]

[Out]

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

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Rubi [A]  time = 0.338705, antiderivative size = 248, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.036 \[ \frac{8 c (d+e x)^{5/2} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{5 e^6}-\frac{2 (d+e x)^{3/2} (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{3 e^6}+\frac{4 \sqrt{d+e x} \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^6}+\frac{2 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{e^6 \sqrt{d+e x}}-\frac{10 c^2 (d+e x)^{7/2} (2 c d-b e)}{7 e^6}+\frac{4 c^3 (d+e x)^{9/2}}{9 e^6} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 68.4362, size = 246, normalized size = 0.99 \[ \frac{4 c^{3} \left (d + e x\right )^{\frac{9}{2}}}{9 e^{6}} + \frac{10 c^{2} \left (d + e x\right )^{\frac{7}{2}} \left (b e - 2 c d\right )}{7 e^{6}} + \frac{8 c \left (d + e x\right )^{\frac{5}{2}} \left (a c e^{2} + b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right )}{5 e^{6}} + \frac{2 \left (d + e x\right )^{\frac{3}{2}} \left (b e - 2 c d\right ) \left (6 a c e^{2} + b^{2} e^{2} - 10 b c d e + 10 c^{2} d^{2}\right )}{3 e^{6}} + \frac{4 \sqrt{d + e x} \left (a e^{2} - b d e + c d^{2}\right ) \left (a c e^{2} + b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right )}{e^{6}} - \frac{2 \left (b e - 2 c d\right ) \left (a e^{2} - b d e + c d^{2}\right )^{2}}{e^{6} \sqrt{d + e x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4*c**3*(d + e*x)**(9/2)/(9*e**6) + 10*c**2*(d + e*x)**(7/2)*(b*e - 2*c*d)/(7*e**
6) + 8*c*(d + e*x)**(5/2)*(a*c*e**2 + b**2*e**2 - 5*b*c*d*e + 5*c**2*d**2)/(5*e*
*6) + 2*(d + e*x)**(3/2)*(b*e - 2*c*d)*(6*a*c*e**2 + b**2*e**2 - 10*b*c*d*e + 10
*c**2*d**2)/(3*e**6) + 4*sqrt(d + e*x)*(a*e**2 - b*d*e + c*d**2)*(a*c*e**2 + b**
2*e**2 - 5*b*c*d*e + 5*c**2*d**2)/e**6 - 2*(b*e - 2*c*d)*(a*e**2 - b*d*e + c*d**
2)**2/(e**6*sqrt(d + e*x))

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Mathematica [A]  time = 0.307767, size = 287, normalized size = 1.16 \[ \frac{252 c e^2 \left (5 a^2 e^2 (2 d+e x)+5 a b e \left (-8 d^2-4 d e x+e^2 x^2\right )+2 b^2 \left (16 d^3+8 d^2 e x-2 d e^2 x^2+e^3 x^3\right )\right )-210 b e^3 \left (3 a^2 e^2-6 a b e (2 d+e x)+b^2 \left (8 d^2+4 d e x-e^2 x^2\right )\right )-18 c^2 e \left (5 b \left (128 d^4+64 d^3 e x-16 d^2 e^2 x^2+8 d e^3 x^3-5 e^4 x^4\right )-28 a e \left (16 d^3+8 d^2 e x-2 d e^2 x^2+e^3 x^3\right )\right )+20 c^3 \left (256 d^5+128 d^4 e x-32 d^3 e^2 x^2+16 d^2 e^3 x^3-10 d e^4 x^4+7 e^5 x^5\right )}{315 e^6 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

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

[Out]

(20*c^3*(256*d^5 + 128*d^4*e*x - 32*d^3*e^2*x^2 + 16*d^2*e^3*x^3 - 10*d*e^4*x^4
+ 7*e^5*x^5) - 210*b*e^3*(3*a^2*e^2 - 6*a*b*e*(2*d + e*x) + b^2*(8*d^2 + 4*d*e*x
- e^2*x^2)) + 252*c*e^2*(5*a^2*e^2*(2*d + e*x) + 5*a*b*e*(-8*d^2 - 4*d*e*x + e^
2*x^2) + 2*b^2*(16*d^3 + 8*d^2*e*x - 2*d*e^2*x^2 + e^3*x^3)) - 18*c^2*e*(-28*a*e
*(16*d^3 + 8*d^2*e*x - 2*d*e^2*x^2 + e^3*x^3) + 5*b*(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)))/(315*e^6*Sqrt[d + e*x])

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Maple [A]  time = 0.012, size = 359, normalized size = 1.5 \[ -{\frac{-140\,{c}^{3}{x}^{5}{e}^{5}-450\,b{c}^{2}{e}^{5}{x}^{4}+200\,{c}^{3}d{e}^{4}{x}^{4}-504\,a{c}^{2}{e}^{5}{x}^{3}-504\,{b}^{2}c{e}^{5}{x}^{3}+720\,b{c}^{2}d{e}^{4}{x}^{3}-320\,{c}^{3}{d}^{2}{e}^{3}{x}^{3}-1260\,abc{e}^{5}{x}^{2}+1008\,a{c}^{2}d{e}^{4}{x}^{2}-210\,{b}^{3}{e}^{5}{x}^{2}+1008\,{b}^{2}cd{e}^{4}{x}^{2}-1440\,b{c}^{2}{d}^{2}{e}^{3}{x}^{2}+640\,{c}^{3}{d}^{3}{e}^{2}{x}^{2}-1260\,{a}^{2}c{e}^{5}x-1260\,a{b}^{2}{e}^{5}x+5040\,abcd{e}^{4}x-4032\,a{c}^{2}{d}^{2}{e}^{3}x+840\,{b}^{3}d{e}^{4}x-4032\,{b}^{2}c{d}^{2}{e}^{3}x+5760\,b{c}^{2}{d}^{3}{e}^{2}x-2560\,{c}^{3}{d}^{4}ex+630\,{a}^{2}b{e}^{5}-2520\,{a}^{2}cd{e}^{4}-2520\,a{b}^{2}d{e}^{4}+10080\,abc{d}^{2}{e}^{3}-8064\,a{c}^{2}{d}^{3}{e}^{2}+1680\,{b}^{3}{d}^{2}{e}^{3}-8064\,{b}^{2}c{d}^{3}{e}^{2}+11520\,b{c}^{2}{d}^{4}e-5120\,{c}^{3}{d}^{5}}{315\,{e}^{6}}{\frac{1}{\sqrt{ex+d}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/315/(e*x+d)^(1/2)*(-70*c^3*e^5*x^5-225*b*c^2*e^5*x^4+100*c^3*d*e^4*x^4-252*a*
c^2*e^5*x^3-252*b^2*c*e^5*x^3+360*b*c^2*d*e^4*x^3-160*c^3*d^2*e^3*x^3-630*a*b*c*
e^5*x^2+504*a*c^2*d*e^4*x^2-105*b^3*e^5*x^2+504*b^2*c*d*e^4*x^2-720*b*c^2*d^2*e^
3*x^2+320*c^3*d^3*e^2*x^2-630*a^2*c*e^5*x-630*a*b^2*e^5*x+2520*a*b*c*d*e^4*x-201
6*a*c^2*d^2*e^3*x+420*b^3*d*e^4*x-2016*b^2*c*d^2*e^3*x+2880*b*c^2*d^3*e^2*x-1280
*c^3*d^4*e*x+315*a^2*b*e^5-1260*a^2*c*d*e^4-1260*a*b^2*d*e^4+5040*a*b*c*d^2*e^3-
4032*a*c^2*d^3*e^2+840*b^3*d^2*e^3-4032*b^2*c*d^3*e^2+5760*b*c^2*d^4*e-2560*c^3*
d^5)/e^6

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Maxima [A]  time = 0.722163, size = 427, normalized size = 1.72 \[ \frac{2 \,{\left (\frac{70 \,{\left (e x + d\right )}^{\frac{9}{2}} c^{3} - 225 \,{\left (2 \, c^{3} d - b c^{2} e\right )}{\left (e x + d\right )}^{\frac{7}{2}} + 252 \,{\left (5 \, c^{3} d^{2} - 5 \, b c^{2} d e +{\left (b^{2} c + a c^{2}\right )} e^{2}\right )}{\left (e x + d\right )}^{\frac{5}{2}} - 105 \,{\left (20 \, c^{3} d^{3} - 30 \, b c^{2} d^{2} e + 12 \,{\left (b^{2} c + a c^{2}\right )} d e^{2} -{\left (b^{3} + 6 \, a b c\right )} e^{3}\right )}{\left (e x + d\right )}^{\frac{3}{2}} + 630 \,{\left (5 \, c^{3} d^{4} - 10 \, b c^{2} d^{3} e + 6 \,{\left (b^{2} c + a c^{2}\right )} d^{2} e^{2} -{\left (b^{3} + 6 \, a b c\right )} d e^{3} +{\left (a b^{2} + a^{2} c\right )} e^{4}\right )} \sqrt{e x + d}}{e^{5}} + \frac{315 \,{\left (2 \, c^{3} d^{5} - 5 \, b c^{2} d^{4} e - a^{2} b e^{5} + 4 \,{\left (b^{2} c + a c^{2}\right )} d^{3} e^{2} -{\left (b^{3} + 6 \, a b c\right )} d^{2} e^{3} + 2 \,{\left (a b^{2} + a^{2} c\right )} d e^{4}\right )}}{\sqrt{e x + d} e^{5}}\right )}}{315 \, e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^2*(2*c*x + b)/(e*x + d)^(3/2),x, algorithm="maxima")

[Out]

2/315*((70*(e*x + d)^(9/2)*c^3 - 225*(2*c^3*d - b*c^2*e)*(e*x + d)^(7/2) + 252*(
5*c^3*d^2 - 5*b*c^2*d*e + (b^2*c + a*c^2)*e^2)*(e*x + d)^(5/2) - 105*(20*c^3*d^3
- 30*b*c^2*d^2*e + 12*(b^2*c + a*c^2)*d*e^2 - (b^3 + 6*a*b*c)*e^3)*(e*x + d)^(3
/2) + 630*(5*c^3*d^4 - 10*b*c^2*d^3*e + 6*(b^2*c + a*c^2)*d^2*e^2 - (b^3 + 6*a*b
*c)*d*e^3 + (a*b^2 + a^2*c)*e^4)*sqrt(e*x + d))/e^5 + 315*(2*c^3*d^5 - 5*b*c^2*d
^4*e - a^2*b*e^5 + 4*(b^2*c + a*c^2)*d^3*e^2 - (b^3 + 6*a*b*c)*d^2*e^3 + 2*(a*b^
2 + a^2*c)*d*e^4)/(sqrt(e*x + d)*e^5))/e

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Fricas [A]  time = 0.282133, size = 413, normalized size = 1.67 \[ \frac{2 \,{\left (70 \, c^{3} e^{5} x^{5} + 2560 \, c^{3} d^{5} - 5760 \, b c^{2} d^{4} e - 315 \, a^{2} b e^{5} + 4032 \,{\left (b^{2} c + a c^{2}\right )} d^{3} e^{2} - 840 \,{\left (b^{3} + 6 \, a b c\right )} d^{2} e^{3} + 1260 \,{\left (a b^{2} + a^{2} c\right )} d e^{4} - 25 \,{\left (4 \, c^{3} d e^{4} - 9 \, b c^{2} e^{5}\right )} x^{4} + 4 \,{\left (40 \, c^{3} d^{2} e^{3} - 90 \, b c^{2} d e^{4} + 63 \,{\left (b^{2} c + a c^{2}\right )} e^{5}\right )} x^{3} -{\left (320 \, c^{3} d^{3} e^{2} - 720 \, b c^{2} d^{2} e^{3} + 504 \,{\left (b^{2} c + a c^{2}\right )} d e^{4} - 105 \,{\left (b^{3} + 6 \, a b c\right )} e^{5}\right )} x^{2} + 2 \,{\left (640 \, c^{3} d^{4} e - 1440 \, b c^{2} d^{3} e^{2} + 1008 \,{\left (b^{2} c + a c^{2}\right )} d^{2} e^{3} - 210 \,{\left (b^{3} + 6 \, a b c\right )} d e^{4} + 315 \,{\left (a b^{2} + a^{2} c\right )} e^{5}\right )} x\right )}}{315 \, \sqrt{e x + d} e^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^2*(2*c*x + b)/(e*x + d)^(3/2),x, algorithm="fricas")

[Out]

2/315*(70*c^3*e^5*x^5 + 2560*c^3*d^5 - 5760*b*c^2*d^4*e - 315*a^2*b*e^5 + 4032*(
b^2*c + a*c^2)*d^3*e^2 - 840*(b^3 + 6*a*b*c)*d^2*e^3 + 1260*(a*b^2 + a^2*c)*d*e^
4 - 25*(4*c^3*d*e^4 - 9*b*c^2*e^5)*x^4 + 4*(40*c^3*d^2*e^3 - 90*b*c^2*d*e^4 + 63
*(b^2*c + a*c^2)*e^5)*x^3 - (320*c^3*d^3*e^2 - 720*b*c^2*d^2*e^3 + 504*(b^2*c +
a*c^2)*d*e^4 - 105*(b^3 + 6*a*b*c)*e^5)*x^2 + 2*(640*c^3*d^4*e - 1440*b*c^2*d^3*
e^2 + 1008*(b^2*c + a*c^2)*d^2*e^3 - 210*(b^3 + 6*a*b*c)*d*e^4 + 315*(a*b^2 + a^
2*c)*e^5)*x)/(sqrt(e*x + d)*e^6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.278405, size = 621, normalized size = 2.5 \[ \frac{2}{315} \,{\left (70 \,{\left (x e + d\right )}^{\frac{9}{2}} c^{3} e^{48} - 450 \,{\left (x e + d\right )}^{\frac{7}{2}} c^{3} d e^{48} + 1260 \,{\left (x e + d\right )}^{\frac{5}{2}} c^{3} d^{2} e^{48} - 2100 \,{\left (x e + d\right )}^{\frac{3}{2}} c^{3} d^{3} e^{48} + 3150 \, \sqrt{x e + d} c^{3} d^{4} e^{48} + 225 \,{\left (x e + d\right )}^{\frac{7}{2}} b c^{2} e^{49} - 1260 \,{\left (x e + d\right )}^{\frac{5}{2}} b c^{2} d e^{49} + 3150 \,{\left (x e + d\right )}^{\frac{3}{2}} b c^{2} d^{2} e^{49} - 6300 \, \sqrt{x e + d} b c^{2} d^{3} e^{49} + 252 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{2} c e^{50} + 252 \,{\left (x e + d\right )}^{\frac{5}{2}} a c^{2} e^{50} - 1260 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{2} c d e^{50} - 1260 \,{\left (x e + d\right )}^{\frac{3}{2}} a c^{2} d e^{50} + 3780 \, \sqrt{x e + d} b^{2} c d^{2} e^{50} + 3780 \, \sqrt{x e + d} a c^{2} d^{2} e^{50} + 105 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{3} e^{51} + 630 \,{\left (x e + d\right )}^{\frac{3}{2}} a b c e^{51} - 630 \, \sqrt{x e + d} b^{3} d e^{51} - 3780 \, \sqrt{x e + d} a b c d e^{51} + 630 \, \sqrt{x e + d} a b^{2} e^{52} + 630 \, \sqrt{x e + d} a^{2} c e^{52}\right )} e^{\left (-54\right )} + \frac{2 \,{\left (2 \, c^{3} d^{5} - 5 \, b c^{2} d^{4} e + 4 \, b^{2} c d^{3} e^{2} + 4 \, a c^{2} d^{3} e^{2} - b^{3} d^{2} e^{3} - 6 \, a b c d^{2} e^{3} + 2 \, a b^{2} d e^{4} + 2 \, a^{2} c d e^{4} - a^{2} b e^{5}\right )} e^{\left (-6\right )}}{\sqrt{x e + d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^2*(2*c*x + b)/(e*x + d)^(3/2),x, algorithm="giac")

[Out]

2/315*(70*(x*e + d)^(9/2)*c^3*e^48 - 450*(x*e + d)^(7/2)*c^3*d*e^48 + 1260*(x*e
+ d)^(5/2)*c^3*d^2*e^48 - 2100*(x*e + d)^(3/2)*c^3*d^3*e^48 + 3150*sqrt(x*e + d)
*c^3*d^4*e^48 + 225*(x*e + d)^(7/2)*b*c^2*e^49 - 1260*(x*e + d)^(5/2)*b*c^2*d*e^
49 + 3150*(x*e + d)^(3/2)*b*c^2*d^2*e^49 - 6300*sqrt(x*e + d)*b*c^2*d^3*e^49 + 2
52*(x*e + d)^(5/2)*b^2*c*e^50 + 252*(x*e + d)^(5/2)*a*c^2*e^50 - 1260*(x*e + d)^
(3/2)*b^2*c*d*e^50 - 1260*(x*e + d)^(3/2)*a*c^2*d*e^50 + 3780*sqrt(x*e + d)*b^2*
c*d^2*e^50 + 3780*sqrt(x*e + d)*a*c^2*d^2*e^50 + 105*(x*e + d)^(3/2)*b^3*e^51 +
630*(x*e + d)^(3/2)*a*b*c*e^51 - 630*sqrt(x*e + d)*b^3*d*e^51 - 3780*sqrt(x*e +
d)*a*b*c*d*e^51 + 630*sqrt(x*e + d)*a*b^2*e^52 + 630*sqrt(x*e + d)*a^2*c*e^52)*e
^(-54) + 2*(2*c^3*d^5 - 5*b*c^2*d^4*e + 4*b^2*c*d^3*e^2 + 4*a*c^2*d^3*e^2 - b^3*
d^2*e^3 - 6*a*b*c*d^2*e^3 + 2*a*b^2*d*e^4 + 2*a^2*c*d*e^4 - a^2*b*e^5)*e^(-6)/sq
rt(x*e + d)