3.2274 \(\int \frac{\left (a+b x+c x^2\right )^3}{\sqrt{d+e x}} \, dx\)

Optimal. Leaf size=282 \[ \frac{2 c (d+e x)^{9/2} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{3 e^7}-\frac{2 (d+e x)^{7/2} (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{7 e^7}+\frac{6 (d+e x)^{5/2} \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{5 e^7}-\frac{2 (d+e x)^{3/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{e^7}+\frac{2 \sqrt{d+e x} \left (a e^2-b d e+c d^2\right )^3}{e^7}-\frac{6 c^2 (d+e x)^{11/2} (2 c d-b e)}{11 e^7}+\frac{2 c^3 (d+e x)^{13/2}}{13 e^7} \]

[Out]

(2*(c*d^2 - b*d*e + a*e^2)^3*Sqrt[d + e*x])/e^7 - (2*(2*c*d - b*e)*(c*d^2 - b*d*
e + a*e^2)^2*(d + e*x)^(3/2))/e^7 + (6*(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)^(5/2))/(5*e^7) - (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)^(7/2))/(7*e^7) + (2*c*(5*c^2*d^2 +
b^2*e^2 - c*e*(5*b*d - a*e))*(d + e*x)^(9/2))/(3*e^7) - (6*c^2*(2*c*d - b*e)*(d
+ e*x)^(11/2))/(11*e^7) + (2*c^3*(d + e*x)^(13/2))/(13*e^7)

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Rubi [A]  time = 0.390044, antiderivative size = 282, 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 \[ \frac{2 c (d+e x)^{9/2} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{3 e^7}-\frac{2 (d+e x)^{7/2} (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{7 e^7}+\frac{6 (d+e x)^{5/2} \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{5 e^7}-\frac{2 (d+e x)^{3/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{e^7}+\frac{2 \sqrt{d+e x} \left (a e^2-b d e+c d^2\right )^3}{e^7}-\frac{6 c^2 (d+e x)^{11/2} (2 c d-b e)}{11 e^7}+\frac{2 c^3 (d+e x)^{13/2}}{13 e^7} \]

Antiderivative was successfully verified.

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

[Out]

(2*(c*d^2 - b*d*e + a*e^2)^3*Sqrt[d + e*x])/e^7 - (2*(2*c*d - b*e)*(c*d^2 - b*d*
e + a*e^2)^2*(d + e*x)^(3/2))/e^7 + (6*(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)^(5/2))/(5*e^7) - (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)^(7/2))/(7*e^7) + (2*c*(5*c^2*d^2 +
b^2*e^2 - c*e*(5*b*d - a*e))*(d + e*x)^(9/2))/(3*e^7) - (6*c^2*(2*c*d - b*e)*(d
+ e*x)^(11/2))/(11*e^7) + (2*c^3*(d + e*x)^(13/2))/(13*e^7)

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Rubi in Sympy [A]  time = 72.5509, size = 279, normalized size = 0.99 \[ \frac{2 c^{3} \left (d + e x\right )^{\frac{13}{2}}}{13 e^{7}} + \frac{6 c^{2} \left (d + e x\right )^{\frac{11}{2}} \left (b e - 2 c d\right )}{11 e^{7}} + \frac{2 c \left (d + e x\right )^{\frac{9}{2}} \left (a c e^{2} + b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right )}{3 e^{7}} + \frac{2 \left (d + e x\right )^{\frac{7}{2}} \left (b e - 2 c d\right ) \left (6 a c e^{2} + b^{2} e^{2} - 10 b c d e + 10 c^{2} d^{2}\right )}{7 e^{7}} + \frac{6 \left (d + e x\right )^{\frac{5}{2}} \left (a e^{2} - b d e + c d^{2}\right ) \left (a c e^{2} + b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right )}{5 e^{7}} + \frac{2 \left (d + e x\right )^{\frac{3}{2}} \left (b e - 2 c d\right ) \left (a e^{2} - b d e + c d^{2}\right )^{2}}{e^{7}} + \frac{2 \sqrt{d + e x} \left (a e^{2} - b d e + c d^{2}\right )^{3}}{e^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+b*x+a)**3/(e*x+d)**(1/2),x)

[Out]

2*c**3*(d + e*x)**(13/2)/(13*e**7) + 6*c**2*(d + e*x)**(11/2)*(b*e - 2*c*d)/(11*
e**7) + 2*c*(d + e*x)**(9/2)*(a*c*e**2 + b**2*e**2 - 5*b*c*d*e + 5*c**2*d**2)/(3
*e**7) + 2*(d + e*x)**(7/2)*(b*e - 2*c*d)*(6*a*c*e**2 + b**2*e**2 - 10*b*c*d*e +
10*c**2*d**2)/(7*e**7) + 6*(d + e*x)**(5/2)*(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)/(5*e**7) + 2*(d + e*x)**(3/2)*(b*e - 2*c
*d)*(a*e**2 - b*d*e + c*d**2)**2/e**7 + 2*sqrt(d + e*x)*(a*e**2 - b*d*e + c*d**2
)**3/e**7

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Mathematica [A]  time = 0.417069, size = 396, normalized size = 1.4 \[ \frac{2 \sqrt{d+e x} \left (143 c e^2 \left (21 a^2 e^2 \left (8 d^2-4 d e x+3 e^2 x^2\right )+18 a b e \left (-16 d^3+8 d^2 e x-6 d e^2 x^2+5 e^3 x^3\right )+b^2 \left (128 d^4-64 d^3 e x+48 d^2 e^2 x^2-40 d e^3 x^3+35 e^4 x^4\right )\right )+429 e^3 \left (35 a^3 e^3+35 a^2 b e^2 (e x-2 d)+7 a b^2 e \left (8 d^2-4 d e x+3 e^2 x^2\right )+b^3 \left (-16 d^3+8 d^2 e x-6 d e^2 x^2+5 e^3 x^3\right )\right )-13 c^2 e \left (5 b \left (256 d^5-128 d^4 e x+96 d^3 e^2 x^2-80 d^2 e^3 x^3+70 d e^4 x^4-63 e^5 x^5\right )-11 a e \left (128 d^4-64 d^3 e x+48 d^2 e^2 x^2-40 d e^3 x^3+35 e^4 x^4\right )\right )+5 c^3 \left (1024 d^6-512 d^5 e x+384 d^4 e^2 x^2-320 d^3 e^3 x^3+280 d^2 e^4 x^4-252 d e^5 x^5+231 e^6 x^6\right )\right )}{15015 e^7} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[d + e*x]*(5*c^3*(1024*d^6 - 512*d^5*e*x + 384*d^4*e^2*x^2 - 320*d^3*e^3*
x^3 + 280*d^2*e^4*x^4 - 252*d*e^5*x^5 + 231*e^6*x^6) + 429*e^3*(35*a^3*e^3 + 35*
a^2*b*e^2*(-2*d + e*x) + 7*a*b^2*e*(8*d^2 - 4*d*e*x + 3*e^2*x^2) + b^3*(-16*d^3
+ 8*d^2*e*x - 6*d*e^2*x^2 + 5*e^3*x^3)) + 143*c*e^2*(21*a^2*e^2*(8*d^2 - 4*d*e*x
+ 3*e^2*x^2) + 18*a*b*e*(-16*d^3 + 8*d^2*e*x - 6*d*e^2*x^2 + 5*e^3*x^3) + b^2*(
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)) - 13*c^2*e*(
-11*a*e*(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) + 5*
b*(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))))/(15015*e^7)

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Maple [A]  time = 0.012, size = 495, normalized size = 1.8 \[{\frac{2310\,{c}^{3}{x}^{6}{e}^{6}+8190\,b{c}^{2}{e}^{6}{x}^{5}-2520\,{c}^{3}d{e}^{5}{x}^{5}+10010\,{x}^{4}a{c}^{2}{e}^{6}+10010\,{b}^{2}c{e}^{6}{x}^{4}-9100\,b{c}^{2}d{e}^{5}{x}^{4}+2800\,{x}^{4}{c}^{3}{d}^{2}{e}^{4}+25740\,abc{e}^{6}{x}^{3}-11440\,{x}^{3}a{c}^{2}d{e}^{5}+4290\,{b}^{3}{e}^{6}{x}^{3}-11440\,{b}^{2}cd{e}^{5}{x}^{3}+10400\,b{c}^{2}{d}^{2}{e}^{4}{x}^{3}-3200\,{x}^{3}{c}^{3}{d}^{3}{e}^{3}+18018\,{x}^{2}{a}^{2}c{e}^{6}+18018\,a{b}^{2}{e}^{6}{x}^{2}-30888\,abcd{e}^{5}{x}^{2}+13728\,{x}^{2}a{c}^{2}{d}^{2}{e}^{4}-5148\,{b}^{3}d{e}^{5}{x}^{2}+13728\,{b}^{2}c{d}^{2}{e}^{4}{x}^{2}-12480\,b{c}^{2}{d}^{3}{e}^{3}{x}^{2}+3840\,{x}^{2}{c}^{3}{d}^{4}{e}^{2}+30030\,{a}^{2}b{e}^{6}x-24024\,x{a}^{2}cd{e}^{5}-24024\,a{b}^{2}d{e}^{5}x+41184\,abc{d}^{2}{e}^{4}x-18304\,xa{c}^{2}{d}^{3}{e}^{3}+6864\,{b}^{3}{d}^{2}{e}^{4}x-18304\,{b}^{2}c{d}^{3}{e}^{3}x+16640\,b{c}^{2}{d}^{4}{e}^{2}x-5120\,{c}^{3}{d}^{5}ex+30030\,{a}^{3}{e}^{6}-60060\,{a}^{2}bd{e}^{5}+48048\,{a}^{2}c{d}^{2}{e}^{4}+48048\,a{b}^{2}{d}^{2}{e}^{4}-82368\,abc{d}^{3}{e}^{3}+36608\,{c}^{2}{d}^{4}a{e}^{2}-13728\,{b}^{3}{d}^{3}{e}^{3}+36608\,{b}^{2}c{d}^{4}{e}^{2}-33280\,b{c}^{2}{d}^{5}e+10240\,{c}^{3}{d}^{6}}{15015\,{e}^{7}}\sqrt{ex+d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x+a)^3/(e*x+d)^(1/2),x)

[Out]

2/15015*(e*x+d)^(1/2)*(1155*c^3*e^6*x^6+4095*b*c^2*e^6*x^5-1260*c^3*d*e^5*x^5+50
05*a*c^2*e^6*x^4+5005*b^2*c*e^6*x^4-4550*b*c^2*d*e^5*x^4+1400*c^3*d^2*e^4*x^4+12
870*a*b*c*e^6*x^3-5720*a*c^2*d*e^5*x^3+2145*b^3*e^6*x^3-5720*b^2*c*d*e^5*x^3+520
0*b*c^2*d^2*e^4*x^3-1600*c^3*d^3*e^3*x^3+9009*a^2*c*e^6*x^2+9009*a*b^2*e^6*x^2-1
5444*a*b*c*d*e^5*x^2+6864*a*c^2*d^2*e^4*x^2-2574*b^3*d*e^5*x^2+6864*b^2*c*d^2*e^
4*x^2-6240*b*c^2*d^3*e^3*x^2+1920*c^3*d^4*e^2*x^2+15015*a^2*b*e^6*x-12012*a^2*c*
d*e^5*x-12012*a*b^2*d*e^5*x+20592*a*b*c*d^2*e^4*x-9152*a*c^2*d^3*e^3*x+3432*b^3*
d^2*e^4*x-9152*b^2*c*d^3*e^3*x+8320*b*c^2*d^4*e^2*x-2560*c^3*d^5*e*x+15015*a^3*e
^6-30030*a^2*b*d*e^5+24024*a^2*c*d^2*e^4+24024*a*b^2*d^2*e^4-41184*a*b*c*d^3*e^3
+18304*a*c^2*d^4*e^2-6864*b^3*d^3*e^3+18304*b^2*c*d^4*e^2-16640*b*c^2*d^5*e+5120
*c^3*d^6)/e^7

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Maxima [A]  time = 0.713478, size = 709, normalized size = 2.51 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15015*(15015*sqrt(e*x + d)*a^3 + 3003*a^2*(5*((e*x + d)^(3/2) - 3*sqrt(e*x + d
)*d)*b/e + (3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*c/e
^2) + 143*a*(21*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2
)*b^2/e^2 + 18*(5*(e*x + d)^(7/2) - 21*(e*x + d)^(5/2)*d + 35*(e*x + d)^(3/2)*d^
2 - 35*sqrt(e*x + d)*d^3)*b*c/e^3 + (35*(e*x + d)^(9/2) - 180*(e*x + d)^(7/2)*d
+ 378*(e*x + d)^(5/2)*d^2 - 420*(e*x + d)^(3/2)*d^3 + 315*sqrt(e*x + d)*d^4)*c^2
/e^4) + 429*(5*(e*x + d)^(7/2) - 21*(e*x + d)^(5/2)*d + 35*(e*x + d)^(3/2)*d^2 -
35*sqrt(e*x + d)*d^3)*b^3/e^3 + 143*(35*(e*x + d)^(9/2) - 180*(e*x + d)^(7/2)*d
+ 378*(e*x + d)^(5/2)*d^2 - 420*(e*x + d)^(3/2)*d^3 + 315*sqrt(e*x + d)*d^4)*b^
2*c/e^4 + 65*(63*(e*x + d)^(11/2) - 385*(e*x + d)^(9/2)*d + 990*(e*x + d)^(7/2)*
d^2 - 1386*(e*x + d)^(5/2)*d^3 + 1155*(e*x + d)^(3/2)*d^4 - 693*sqrt(e*x + d)*d^
5)*b*c^2/e^5 + 5*(231*(e*x + d)^(13/2) - 1638*(e*x + d)^(11/2)*d + 5005*(e*x + d
)^(9/2)*d^2 - 8580*(e*x + d)^(7/2)*d^3 + 9009*(e*x + d)^(5/2)*d^4 - 6006*(e*x +
d)^(3/2)*d^5 + 3003*sqrt(e*x + d)*d^6)*c^3/e^6)/e

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Fricas [A]  time = 0.20835, size = 551, normalized size = 1.95 \[ \frac{2 \,{\left (1155 \, c^{3} e^{6} x^{6} + 5120 \, c^{3} d^{6} - 16640 \, b c^{2} d^{5} e - 30030 \, a^{2} b d e^{5} + 15015 \, a^{3} e^{6} + 18304 \,{\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} - 6864 \,{\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} + 24024 \,{\left (a b^{2} + a^{2} c\right )} d^{2} e^{4} - 315 \,{\left (4 \, c^{3} d e^{5} - 13 \, b c^{2} e^{6}\right )} x^{5} + 35 \,{\left (40 \, c^{3} d^{2} e^{4} - 130 \, b c^{2} d e^{5} + 143 \,{\left (b^{2} c + a c^{2}\right )} e^{6}\right )} x^{4} - 5 \,{\left (320 \, c^{3} d^{3} e^{3} - 1040 \, b c^{2} d^{2} e^{4} + 1144 \,{\left (b^{2} c + a c^{2}\right )} d e^{5} - 429 \,{\left (b^{3} + 6 \, a b c\right )} e^{6}\right )} x^{3} + 3 \,{\left (640 \, c^{3} d^{4} e^{2} - 2080 \, b c^{2} d^{3} e^{3} + 2288 \,{\left (b^{2} c + a c^{2}\right )} d^{2} e^{4} - 858 \,{\left (b^{3} + 6 \, a b c\right )} d e^{5} + 3003 \,{\left (a b^{2} + a^{2} c\right )} e^{6}\right )} x^{2} -{\left (2560 \, c^{3} d^{5} e - 8320 \, b c^{2} d^{4} e^{2} - 15015 \, a^{2} b e^{6} + 9152 \,{\left (b^{2} c + a c^{2}\right )} d^{3} e^{3} - 3432 \,{\left (b^{3} + 6 \, a b c\right )} d^{2} e^{4} + 12012 \,{\left (a b^{2} + a^{2} c\right )} d e^{5}\right )} x\right )} \sqrt{e x + d}}{15015 \, e^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15015*(1155*c^3*e^6*x^6 + 5120*c^3*d^6 - 16640*b*c^2*d^5*e - 30030*a^2*b*d*e^5
+ 15015*a^3*e^6 + 18304*(b^2*c + a*c^2)*d^4*e^2 - 6864*(b^3 + 6*a*b*c)*d^3*e^3
+ 24024*(a*b^2 + a^2*c)*d^2*e^4 - 315*(4*c^3*d*e^5 - 13*b*c^2*e^6)*x^5 + 35*(40*
c^3*d^2*e^4 - 130*b*c^2*d*e^5 + 143*(b^2*c + a*c^2)*e^6)*x^4 - 5*(320*c^3*d^3*e^
3 - 1040*b*c^2*d^2*e^4 + 1144*(b^2*c + a*c^2)*d*e^5 - 429*(b^3 + 6*a*b*c)*e^6)*x
^3 + 3*(640*c^3*d^4*e^2 - 2080*b*c^2*d^3*e^3 + 2288*(b^2*c + a*c^2)*d^2*e^4 - 85
8*(b^3 + 6*a*b*c)*d*e^5 + 3003*(a*b^2 + a^2*c)*e^6)*x^2 - (2560*c^3*d^5*e - 8320
*b*c^2*d^4*e^2 - 15015*a^2*b*e^6 + 9152*(b^2*c + a*c^2)*d^3*e^3 - 3432*(b^3 + 6*
a*b*c)*d^2*e^4 + 12012*(a*b^2 + a^2*c)*d*e^5)*x)*sqrt(e*x + d)/e^7

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Sympy [A]  time = 105.516, size = 1406, normalized size = 4.99 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+b*x+a)**3/(e*x+d)**(1/2),x)

[Out]

Piecewise((-(2*a**3*d/sqrt(d + e*x) + 2*a**3*(-d/sqrt(d + e*x) - sqrt(d + e*x))
+ 6*a**2*b*d*(-d/sqrt(d + e*x) - sqrt(d + e*x))/e + 6*a**2*b*(d**2/sqrt(d + e*x)
+ 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3)/e + 6*a**2*c*d*(d**2/sqrt(d + e*x) +
2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3)/e**2 + 6*a**2*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 + 6*a*b**2*d
*(d**2/sqrt(d + e*x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3)/e**2 + 6*a*b**2*(
-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 + 12*a*b*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 + 12*a*b*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 + 6*a*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 + 6*a*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 + 2*b**3*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 + 2*b**3*(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 + 6*b**2*c*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 + 6*b**2*c*(-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 + 6*b*c**2*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 + 6*b*c**2*(d**6/sqrt(d + e*x) + 6*d**5*sqr
t(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 + 2*c**3*d*(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**6 + 2*c**3*(-d**7/sqrt(d + e*x) - 7*d**6*sqrt(d + e*x) + 7*d**5*(d
+ e*x)**(3/2) - 7*d**4*(d + e*x)**(5/2) + 5*d**3*(d + e*x)**(7/2) - 7*d**2*(d +
e*x)**(9/2)/3 + 7*d*(d + e*x)**(11/2)/11 - (d + e*x)**(13/2)/13)/e**6)/e, Ne(e,
0)), ((a**3*x + 3*a**2*b*x**2/2 + b*c**2*x**6/2 + c**3*x**7/7 + x**5*(3*a*c**2 +
3*b**2*c)/5 + x**4*(6*a*b*c + b**3)/4 + x**3*(3*a**2*c + 3*a*b**2)/3)/sqrt(d),
True))

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GIAC/XCAS [A]  time = 0.209837, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done