3.1604 \(\int \frac{(b+2 c x) \left (a+b x+c x^2\right )^2}{\sqrt{d+e x}} \, dx\)

Optimal. Leaf size=250 \[ \frac{8 c (d+e x)^{7/2} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{7 e^6}-\frac{2 (d+e x)^{5/2} (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{5 e^6}+\frac{4 (d+e x)^{3/2} \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{3 e^6}-\frac{2 \sqrt{d+e x} (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{e^6}-\frac{10 c^2 (d+e x)^{9/2} (2 c d-b e)}{9 e^6}+\frac{4 c^3 (d+e x)^{11/2}}{11 e^6} \]

[Out]

(-2*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^2*Sqrt[d + e*x])/e^6 + (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))*(d + e*x)^(3/2))/(3*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)^(5/2)
)/(5*e^6) + (8*c*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e))*(d + e*x)^(7/2))/(7*e
^6) - (10*c^2*(2*c*d - b*e)*(d + e*x)^(9/2))/(9*e^6) + (4*c^3*(d + e*x)^(11/2))/
(11*e^6)

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Rubi [A]  time = 0.334924, antiderivative size = 250, 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 \[ \frac{8 c (d+e x)^{7/2} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{7 e^6}-\frac{2 (d+e x)^{5/2} (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{5 e^6}+\frac{4 (d+e x)^{3/2} \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{3 e^6}-\frac{2 \sqrt{d+e x} (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{e^6}-\frac{10 c^2 (d+e x)^{9/2} (2 c d-b e)}{9 e^6}+\frac{4 c^3 (d+e x)^{11/2}}{11 e^6} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^2*Sqrt[d + e*x])/e^6 + (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))*(d + e*x)^(3/2))/(3*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)^(5/2)
)/(5*e^6) + (8*c*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e))*(d + e*x)^(7/2))/(7*e
^6) - (10*c^2*(2*c*d - b*e)*(d + e*x)^(9/2))/(9*e^6) + (4*c^3*(d + e*x)^(11/2))/
(11*e^6)

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Rubi in Sympy [A]  time = 68.2161, size = 248, normalized size = 0.99 \[ \frac{4 c^{3} \left (d + e x\right )^{\frac{11}{2}}}{11 e^{6}} + \frac{10 c^{2} \left (d + e x\right )^{\frac{9}{2}} \left (b e - 2 c d\right )}{9 e^{6}} + \frac{8 c \left (d + e x\right )^{\frac{7}{2}} \left (a c e^{2} + b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right )}{7 e^{6}} + \frac{2 \left (d + e x\right )^{\frac{5}{2}} \left (b e - 2 c d\right ) \left (6 a c e^{2} + b^{2} e^{2} - 10 b c d e + 10 c^{2} d^{2}\right )}{5 e^{6}} + \frac{4 \left (d + e x\right )^{\frac{3}{2}} \left (a e^{2} - b d e + c d^{2}\right ) \left (a c e^{2} + b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right )}{3 e^{6}} + \frac{2 \sqrt{d + e x} \left (b e - 2 c d\right ) \left (a e^{2} - b d e + c d^{2}\right )^{2}}{e^{6}} \]

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)**(1/2),x)

[Out]

4*c**3*(d + e*x)**(11/2)/(11*e**6) + 10*c**2*(d + e*x)**(9/2)*(b*e - 2*c*d)/(9*e
**6) + 8*c*(d + e*x)**(7/2)*(a*c*e**2 + b**2*e**2 - 5*b*c*d*e + 5*c**2*d**2)/(7*
e**6) + 2*(d + e*x)**(5/2)*(b*e - 2*c*d)*(6*a*c*e**2 + b**2*e**2 - 10*b*c*d*e +
10*c**2*d**2)/(5*e**6) + 4*(d + e*x)**(3/2)*(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)/(3*e**6) + 2*sqrt(d + e*x)*(b*e - 2*c*d)*
(a*e**2 - b*d*e + c*d**2)**2/e**6

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Mathematica [A]  time = 0.399369, size = 290, normalized size = 1.16 \[ \frac{2 \sqrt{d+e x} \left (-66 c e^2 \left (-35 a^2 e^2 (e x-2 d)-21 a b e \left (8 d^2-4 d e x+3 e^2 x^2\right )+6 b^2 \left (16 d^3-8 d^2 e x+6 d e^2 x^2-5 e^3 x^3\right )\right )+231 b e^3 \left (15 a^2 e^2+10 a b e (e x-2 d)+b^2 \left (8 d^2-4 d e x+3 e^2 x^2\right )\right )+11 c^2 e \left (36 a e \left (-16 d^3+8 d^2 e x-6 d e^2 x^2+5 e^3 x^3\right )+5 b \left (128 d^4-64 d^3 e x+48 d^2 e^2 x^2-40 d e^3 x^3+35 e^4 x^4\right )\right )-10 c^3 \left (256 d^5-128 d^4 e x+96 d^3 e^2 x^2-80 d^2 e^3 x^3+70 d e^4 x^4-63 e^5 x^5\right )\right )}{3465 e^6} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[d + e*x]*(-10*c^3*(256*d^5 - 128*d^4*e*x + 96*d^3*e^2*x^2 - 80*d^2*e^3*x
^3 + 70*d*e^4*x^4 - 63*e^5*x^5) + 231*b*e^3*(15*a^2*e^2 + 10*a*b*e*(-2*d + e*x)
+ b^2*(8*d^2 - 4*d*e*x + 3*e^2*x^2)) - 66*c*e^2*(-35*a^2*e^2*(-2*d + e*x) - 21*a
*b*e*(8*d^2 - 4*d*e*x + 3*e^2*x^2) + 6*b^2*(16*d^3 - 8*d^2*e*x + 6*d*e^2*x^2 - 5
*e^3*x^3)) + 11*c^2*e*(36*a*e*(-16*d^3 + 8*d^2*e*x - 6*d*e^2*x^2 + 5*e^3*x^3) +
5*b*(128*d^4 - 64*d^3*e*x + 48*d^2*e^2*x^2 - 40*d*e^3*x^3 + 35*e^4*x^4))))/(3465
*e^6)

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Maple [A]  time = 0.012, size = 359, normalized size = 1.4 \[{\frac{1260\,{c}^{3}{x}^{5}{e}^{5}+3850\,b{c}^{2}{e}^{5}{x}^{4}-1400\,{c}^{3}d{e}^{4}{x}^{4}+3960\,a{c}^{2}{e}^{5}{x}^{3}+3960\,{b}^{2}c{e}^{5}{x}^{3}-4400\,b{c}^{2}d{e}^{4}{x}^{3}+1600\,{c}^{3}{d}^{2}{e}^{3}{x}^{3}+8316\,abc{e}^{5}{x}^{2}-4752\,a{c}^{2}d{e}^{4}{x}^{2}+1386\,{b}^{3}{e}^{5}{x}^{2}-4752\,{b}^{2}cd{e}^{4}{x}^{2}+5280\,b{c}^{2}{d}^{2}{e}^{3}{x}^{2}-1920\,{c}^{3}{d}^{3}{e}^{2}{x}^{2}+4620\,{a}^{2}c{e}^{5}x+4620\,a{b}^{2}{e}^{5}x-11088\,abcd{e}^{4}x+6336\,a{c}^{2}{d}^{2}{e}^{3}x-1848\,{b}^{3}d{e}^{4}x+6336\,{b}^{2}c{d}^{2}{e}^{3}x-7040\,b{c}^{2}{d}^{3}{e}^{2}x+2560\,{c}^{3}{d}^{4}ex+6930\,{a}^{2}b{e}^{5}-9240\,{a}^{2}cd{e}^{4}-9240\,a{b}^{2}d{e}^{4}+22176\,abc{d}^{2}{e}^{3}-12672\,a{c}^{2}{d}^{3}{e}^{2}+3696\,{b}^{3}{d}^{2}{e}^{3}-12672\,{b}^{2}c{d}^{3}{e}^{2}+14080\,b{c}^{2}{d}^{4}e-5120\,{c}^{3}{d}^{5}}{3465\,{e}^{6}}\sqrt{ex+d}} \]

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)^(1/2),x)

[Out]

2/3465*(e*x+d)^(1/2)*(630*c^3*e^5*x^5+1925*b*c^2*e^5*x^4-700*c^3*d*e^4*x^4+1980*
a*c^2*e^5*x^3+1980*b^2*c*e^5*x^3-2200*b*c^2*d*e^4*x^3+800*c^3*d^2*e^3*x^3+4158*a
*b*c*e^5*x^2-2376*a*c^2*d*e^4*x^2+693*b^3*e^5*x^2-2376*b^2*c*d*e^4*x^2+2640*b*c^
2*d^2*e^3*x^2-960*c^3*d^3*e^2*x^2+2310*a^2*c*e^5*x+2310*a*b^2*e^5*x-5544*a*b*c*d
*e^4*x+3168*a*c^2*d^2*e^3*x-924*b^3*d*e^4*x+3168*b^2*c*d^2*e^3*x-3520*b*c^2*d^3*
e^2*x+1280*c^3*d^4*e*x+3465*a^2*b*e^5-4620*a^2*c*d*e^4-4620*a*b^2*d*e^4+11088*a*
b*c*d^2*e^3-6336*a*c^2*d^3*e^2+1848*b^3*d^2*e^3-6336*b^2*c*d^3*e^2+7040*b*c^2*d^
4*e-2560*c^3*d^5)/e^6

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Maxima [A]  time = 0.708319, size = 416, normalized size = 1.66 \[ \frac{2 \,{\left (630 \,{\left (e x + d\right )}^{\frac{11}{2}} c^{3} - 1925 \,{\left (2 \, c^{3} d - b c^{2} e\right )}{\left (e x + d\right )}^{\frac{9}{2}} + 1980 \,{\left (5 \, c^{3} d^{2} - 5 \, b c^{2} d e +{\left (b^{2} c + a c^{2}\right )} e^{2}\right )}{\left (e x + d\right )}^{\frac{7}{2}} - 693 \,{\left (20 \, c^{3} d^{3} - 30 \, b c^{2} d^{2} e + 12 \,{\left (b^{2} c + a c^{2}\right )} d e^{2} -{\left (b^{3} + 6 \, a b c\right )} e^{3}\right )}{\left (e x + d\right )}^{\frac{5}{2}} + 2310 \,{\left (5 \, c^{3} d^{4} - 10 \, b c^{2} d^{3} e + 6 \,{\left (b^{2} c + a c^{2}\right )} d^{2} e^{2} -{\left (b^{3} + 6 \, a b c\right )} d e^{3} +{\left (a b^{2} + a^{2} c\right )} e^{4}\right )}{\left (e x + d\right )}^{\frac{3}{2}} - 3465 \,{\left (2 \, c^{3} d^{5} - 5 \, b c^{2} d^{4} e - a^{2} b e^{5} + 4 \,{\left (b^{2} c + a c^{2}\right )} d^{3} e^{2} -{\left (b^{3} + 6 \, a b c\right )} d^{2} e^{3} + 2 \,{\left (a b^{2} + a^{2} c\right )} d e^{4}\right )} \sqrt{e x + d}\right )}}{3465 \, e^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3465*(630*(e*x + d)^(11/2)*c^3 - 1925*(2*c^3*d - b*c^2*e)*(e*x + d)^(9/2) + 19
80*(5*c^3*d^2 - 5*b*c^2*d*e + (b^2*c + a*c^2)*e^2)*(e*x + d)^(7/2) - 693*(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
)^(5/2) + 2310*(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)*(e*x + d)^(3/2) - 3465*(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^6

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Fricas [A]  time = 0.272535, size = 413, normalized size = 1.65 \[ \frac{2 \,{\left (630 \, c^{3} e^{5} x^{5} - 2560 \, c^{3} d^{5} + 7040 \, b c^{2} d^{4} e + 3465 \, a^{2} b e^{5} - 6336 \,{\left (b^{2} c + a c^{2}\right )} d^{3} e^{2} + 1848 \,{\left (b^{3} + 6 \, a b c\right )} d^{2} e^{3} - 4620 \,{\left (a b^{2} + a^{2} c\right )} d e^{4} - 175 \,{\left (4 \, c^{3} d e^{4} - 11 \, b c^{2} e^{5}\right )} x^{4} + 20 \,{\left (40 \, c^{3} d^{2} e^{3} - 110 \, b c^{2} d e^{4} + 99 \,{\left (b^{2} c + a c^{2}\right )} e^{5}\right )} x^{3} - 3 \,{\left (320 \, c^{3} d^{3} e^{2} - 880 \, b c^{2} d^{2} e^{3} + 792 \,{\left (b^{2} c + a c^{2}\right )} d e^{4} - 231 \,{\left (b^{3} + 6 \, a b c\right )} e^{5}\right )} x^{2} + 2 \,{\left (640 \, c^{3} d^{4} e - 1760 \, b c^{2} d^{3} e^{2} + 1584 \,{\left (b^{2} c + a c^{2}\right )} d^{2} e^{3} - 462 \,{\left (b^{3} + 6 \, a b c\right )} d e^{4} + 1155 \,{\left (a b^{2} + a^{2} c\right )} e^{5}\right )} x\right )} \sqrt{e x + d}}{3465 \, e^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3465*(630*c^3*e^5*x^5 - 2560*c^3*d^5 + 7040*b*c^2*d^4*e + 3465*a^2*b*e^5 - 633
6*(b^2*c + a*c^2)*d^3*e^2 + 1848*(b^3 + 6*a*b*c)*d^2*e^3 - 4620*(a*b^2 + a^2*c)*
d*e^4 - 175*(4*c^3*d*e^4 - 11*b*c^2*e^5)*x^4 + 20*(40*c^3*d^2*e^3 - 110*b*c^2*d*
e^4 + 99*(b^2*c + a*c^2)*e^5)*x^3 - 3*(320*c^3*d^3*e^2 - 880*b*c^2*d^2*e^3 + 792
*(b^2*c + a*c^2)*d*e^4 - 231*(b^3 + 6*a*b*c)*e^5)*x^2 + 2*(640*c^3*d^4*e - 1760*
b*c^2*d^3*e^2 + 1584*(b^2*c + a*c^2)*d^2*e^3 - 462*(b^3 + 6*a*b*c)*d*e^4 + 1155*
(a*b^2 + a^2*c)*e^5)*x)*sqrt(e*x + d)/e^6

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Sympy [A]  time = 74.4187, size = 1025, normalized size = 4.1 \[ \text{result too large to display} \]

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)**(1/2),x)

[Out]

Piecewise((-(2*a**2*b*d/sqrt(d + e*x) + 2*a**2*b*(-d/sqrt(d + e*x) - sqrt(d + e*
x)) + 4*a**2*c*d*(-d/sqrt(d + e*x) - sqrt(d + e*x))/e + 4*a**2*c*(d**2/sqrt(d +
e*x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3)/e + 4*a*b**2*d*(-d/sqrt(d + e*x)
- sqrt(d + e*x))/e + 4*a*b**2*(d**2/sqrt(d + e*x) + 2*d*sqrt(d + e*x) - (d + e*x
)**(3/2)/3)/e + 12*a*b*c*d*(d**2/sqrt(d + e*x) + 2*d*sqrt(d + e*x) - (d + e*x)**
(3/2)/3)/e**2 + 12*a*b*c*(-d**3/sqrt(d + e*x) - 3*d**2*sqrt(d + e*x) + d*(d + e*
x)**(3/2) - (d + e*x)**(5/2)/5)/e**2 + 8*a*c**2*d*(-d**3/sqrt(d + e*x) - 3*d**2*
sqrt(d + e*x) + d*(d + e*x)**(3/2) - (d + e*x)**(5/2)/5)/e**3 + 8*a*c**2*(d**4/s
qrt(d + e*x) + 4*d**3*sqrt(d + e*x) - 2*d**2*(d + e*x)**(3/2) + 4*d*(d + e*x)**(
5/2)/5 - (d + e*x)**(7/2)/7)/e**3 + 2*b**3*d*(d**2/sqrt(d + e*x) + 2*d*sqrt(d +
e*x) - (d + e*x)**(3/2)/3)/e**2 + 2*b**3*(-d**3/sqrt(d + e*x) - 3*d**2*sqrt(d +
e*x) + d*(d + e*x)**(3/2) - (d + e*x)**(5/2)/5)/e**2 + 8*b**2*c*d*(-d**3/sqrt(d
+ e*x) - 3*d**2*sqrt(d + e*x) + d*(d + e*x)**(3/2) - (d + e*x)**(5/2)/5)/e**3 +
8*b**2*c*(d**4/sqrt(d + e*x) + 4*d**3*sqrt(d + e*x) - 2*d**2*(d + e*x)**(3/2) +
4*d*(d + e*x)**(5/2)/5 - (d + e*x)**(7/2)/7)/e**3 + 10*b*c**2*d*(d**4/sqrt(d + e
*x) + 4*d**3*sqrt(d + e*x) - 2*d**2*(d + e*x)**(3/2) + 4*d*(d + e*x)**(5/2)/5 -
(d + e*x)**(7/2)/7)/e**4 + 10*b*c**2*(-d**5/sqrt(d + e*x) - 5*d**4*sqrt(d + e*x)
+ 10*d**3*(d + e*x)**(3/2)/3 - 2*d**2*(d + e*x)**(5/2) + 5*d*(d + e*x)**(7/2)/7
- (d + e*x)**(9/2)/9)/e**4 + 4*c**3*d*(-d**5/sqrt(d + e*x) - 5*d**4*sqrt(d + e*
x) + 10*d**3*(d + e*x)**(3/2)/3 - 2*d**2*(d + e*x)**(5/2) + 5*d*(d + e*x)**(7/2)
/7 - (d + e*x)**(9/2)/9)/e**5 + 4*c**3*(d**6/sqrt(d + e*x) + 6*d**5*sqrt(d + e*x
) - 5*d**4*(d + e*x)**(3/2) + 4*d**3*(d + e*x)**(5/2) - 15*d**2*(d + e*x)**(7/2)
/7 + 2*d*(d + e*x)**(9/2)/3 - (d + e*x)**(11/2)/11)/e**5)/e, Ne(e, 0)), ((a + b*
x + c*x**2)**3/(3*sqrt(d)), True))

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GIAC/XCAS [A]  time = 0.271218, size = 636, normalized size = 2.54 \[ \frac{2}{3465} \,{\left (2310 \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} - 3 \, \sqrt{x e + d} d\right )} a b^{2} e^{\left (-1\right )} + 2310 \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} - 3 \, \sqrt{x e + d} d\right )} a^{2} c e^{\left (-1\right )} + 231 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} e^{8} - 10 \,{\left (x e + d\right )}^{\frac{3}{2}} d e^{8} + 15 \, \sqrt{x e + d} d^{2} e^{8}\right )} b^{3} e^{\left (-10\right )} + 1386 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} e^{8} - 10 \,{\left (x e + d\right )}^{\frac{3}{2}} d e^{8} + 15 \, \sqrt{x e + d} d^{2} e^{8}\right )} a b c e^{\left (-10\right )} + 396 \,{\left (5 \,{\left (x e + d\right )}^{\frac{7}{2}} e^{18} - 21 \,{\left (x e + d\right )}^{\frac{5}{2}} d e^{18} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2} e^{18} - 35 \, \sqrt{x e + d} d^{3} e^{18}\right )} b^{2} c e^{\left (-21\right )} + 396 \,{\left (5 \,{\left (x e + d\right )}^{\frac{7}{2}} e^{18} - 21 \,{\left (x e + d\right )}^{\frac{5}{2}} d e^{18} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2} e^{18} - 35 \, \sqrt{x e + d} d^{3} e^{18}\right )} a c^{2} e^{\left (-21\right )} + 55 \,{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} e^{32} - 180 \,{\left (x e + d\right )}^{\frac{7}{2}} d e^{32} + 378 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} e^{32} - 420 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3} e^{32} + 315 \, \sqrt{x e + d} d^{4} e^{32}\right )} b c^{2} e^{\left (-36\right )} + 10 \,{\left (63 \,{\left (x e + d\right )}^{\frac{11}{2}} e^{50} - 385 \,{\left (x e + d\right )}^{\frac{9}{2}} d e^{50} + 990 \,{\left (x e + d\right )}^{\frac{7}{2}} d^{2} e^{50} - 1386 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{3} e^{50} + 1155 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{4} e^{50} - 693 \, \sqrt{x e + d} d^{5} e^{50}\right )} c^{3} e^{\left (-55\right )} + 3465 \, \sqrt{x e + d} a^{2} b\right )} e^{\left (-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3465*(2310*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a*b^2*e^(-1) + 2310*((x*e + d
)^(3/2) - 3*sqrt(x*e + d)*d)*a^2*c*e^(-1) + 231*(3*(x*e + d)^(5/2)*e^8 - 10*(x*e
+ d)^(3/2)*d*e^8 + 15*sqrt(x*e + d)*d^2*e^8)*b^3*e^(-10) + 1386*(3*(x*e + d)^(5
/2)*e^8 - 10*(x*e + d)^(3/2)*d*e^8 + 15*sqrt(x*e + d)*d^2*e^8)*a*b*c*e^(-10) + 3
96*(5*(x*e + d)^(7/2)*e^18 - 21*(x*e + d)^(5/2)*d*e^18 + 35*(x*e + d)^(3/2)*d^2*
e^18 - 35*sqrt(x*e + d)*d^3*e^18)*b^2*c*e^(-21) + 396*(5*(x*e + d)^(7/2)*e^18 -
21*(x*e + d)^(5/2)*d*e^18 + 35*(x*e + d)^(3/2)*d^2*e^18 - 35*sqrt(x*e + d)*d^3*e
^18)*a*c^2*e^(-21) + 55*(35*(x*e + d)^(9/2)*e^32 - 180*(x*e + d)^(7/2)*d*e^32 +
378*(x*e + d)^(5/2)*d^2*e^32 - 420*(x*e + d)^(3/2)*d^3*e^32 + 315*sqrt(x*e + d)*
d^4*e^32)*b*c^2*e^(-36) + 10*(63*(x*e + d)^(11/2)*e^50 - 385*(x*e + d)^(9/2)*d*e
^50 + 990*(x*e + d)^(7/2)*d^2*e^50 - 1386*(x*e + d)^(5/2)*d^3*e^50 + 1155*(x*e +
d)^(3/2)*d^4*e^50 - 693*sqrt(x*e + d)*d^5*e^50)*c^3*e^(-55) + 3465*sqrt(x*e + d
)*a^2*b)*e^(-1)