3.2271 \(\int (d+e x)^{5/2} \left (a+b x+c x^2\right )^3 \, dx\)

Optimal. Leaf size=286 \[ \frac{2 c (d+e x)^{15/2} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{5 e^7}-\frac{2 (d+e x)^{13/2} (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{13 e^7}+\frac{6 (d+e x)^{11/2} \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{11 e^7}-\frac{2 (d+e x)^{9/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{3 e^7}+\frac{2 (d+e x)^{7/2} \left (a e^2-b d e+c d^2\right )^3}{7 e^7}-\frac{6 c^2 (d+e x)^{17/2} (2 c d-b e)}{17 e^7}+\frac{2 c^3 (d+e x)^{19/2}}{19 e^7} \]

[Out]

(2*(c*d^2 - b*d*e + a*e^2)^3*(d + e*x)^(7/2))/(7*e^7) - (2*(2*c*d - b*e)*(c*d^2
- b*d*e + a*e^2)^2*(d + e*x)^(9/2))/(3*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)^(11/2))/(11*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)^(13/2))/(13*e^7) + (2*
c*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e))*(d + e*x)^(15/2))/(5*e^7) - (6*c^2*(
2*c*d - b*e)*(d + e*x)^(17/2))/(17*e^7) + (2*c^3*(d + e*x)^(19/2))/(19*e^7)

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Rubi [A]  time = 0.468484, antiderivative size = 286, 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 \[ \frac{2 c (d+e x)^{15/2} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{5 e^7}-\frac{2 (d+e x)^{13/2} (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{13 e^7}+\frac{6 (d+e x)^{11/2} \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{11 e^7}-\frac{2 (d+e x)^{9/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{3 e^7}+\frac{2 (d+e x)^{7/2} \left (a e^2-b d e+c d^2\right )^3}{7 e^7}-\frac{6 c^2 (d+e x)^{17/2} (2 c d-b e)}{17 e^7}+\frac{2 c^3 (d+e x)^{19/2}}{19 e^7} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^(5/2)*(a + b*x + c*x^2)^3,x]

[Out]

(2*(c*d^2 - b*d*e + a*e^2)^3*(d + e*x)^(7/2))/(7*e^7) - (2*(2*c*d - b*e)*(c*d^2
- b*d*e + a*e^2)^2*(d + e*x)^(9/2))/(3*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)^(11/2))/(11*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)^(13/2))/(13*e^7) + (2*
c*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e))*(d + e*x)^(15/2))/(5*e^7) - (6*c^2*(
2*c*d - b*e)*(d + e*x)^(17/2))/(17*e^7) + (2*c^3*(d + e*x)^(19/2))/(19*e^7)

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Rubi in Sympy [A]  time = 79.6031, size = 282, normalized size = 0.99 \[ \frac{2 c^{3} \left (d + e x\right )^{\frac{19}{2}}}{19 e^{7}} + \frac{6 c^{2} \left (d + e x\right )^{\frac{17}{2}} \left (b e - 2 c d\right )}{17 e^{7}} + \frac{2 c \left (d + e x\right )^{\frac{15}{2}} \left (a c e^{2} + b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right )}{5 e^{7}} + \frac{2 \left (d + e x\right )^{\frac{13}{2}} \left (b e - 2 c d\right ) \left (6 a c e^{2} + b^{2} e^{2} - 10 b c d e + 10 c^{2} d^{2}\right )}{13 e^{7}} + \frac{6 \left (d + e x\right )^{\frac{11}{2}} \left (a e^{2} - b d e + c d^{2}\right ) \left (a c e^{2} + b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right )}{11 e^{7}} + \frac{2 \left (d + e x\right )^{\frac{9}{2}} \left (b e - 2 c d\right ) \left (a e^{2} - b d e + c d^{2}\right )^{2}}{3 e^{7}} + \frac{2 \left (d + e x\right )^{\frac{7}{2}} \left (a e^{2} - b d e + c d^{2}\right )^{3}}{7 e^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**(5/2)*(c*x**2+b*x+a)**3,x)

[Out]

2*c**3*(d + e*x)**(19/2)/(19*e**7) + 6*c**2*(d + e*x)**(17/2)*(b*e - 2*c*d)/(17*
e**7) + 2*c*(d + e*x)**(15/2)*(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)**(13/2)*(b*e - 2*c*d)*(6*a*c*e**2 + b**2*e**2 - 10*b*c*d*e
+ 10*c**2*d**2)/(13*e**7) + 6*(d + e*x)**(11/2)*(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)/(11*e**7) + 2*(d + e*x)**(9/2)*(b*e
- 2*c*d)*(a*e**2 - b*d*e + c*d**2)**2/(3*e**7) + 2*(d + e*x)**(7/2)*(a*e**2 - b*
d*e + c*d**2)**3/(7*e**7)

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Mathematica [A]  time = 0.559089, size = 397, normalized size = 1.39 \[ \frac{2 (d+e x)^{7/2} \left (323 c e^2 \left (65 a^2 e^2 \left (8 d^2-28 d e x+63 e^2 x^2\right )+30 a b e \left (-16 d^3+56 d^2 e x-126 d e^2 x^2+231 e^3 x^3\right )+b^2 \left (128 d^4-448 d^3 e x+1008 d^2 e^2 x^2-1848 d e^3 x^3+3003 e^4 x^4\right )\right )+1615 e^3 \left (429 a^3 e^3+143 a^2 b e^2 (7 e x-2 d)+13 a b^2 e \left (8 d^2-28 d e x+63 e^2 x^2\right )+b^3 \left (-16 d^3+56 d^2 e x-126 d e^2 x^2+231 e^3 x^3\right )\right )-19 c^2 e \left (5 b \left (256 d^5-896 d^4 e x+2016 d^3 e^2 x^2-3696 d^2 e^3 x^3+6006 d e^4 x^4-9009 e^5 x^5\right )-17 a e \left (128 d^4-448 d^3 e x+1008 d^2 e^2 x^2-1848 d e^3 x^3+3003 e^4 x^4\right )\right )+5 c^3 \left (1024 d^6-3584 d^5 e x+8064 d^4 e^2 x^2-14784 d^3 e^3 x^3+24024 d^2 e^4 x^4-36036 d e^5 x^5+51051 e^6 x^6\right )\right )}{4849845 e^7} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^(5/2)*(a + b*x + c*x^2)^3,x]

[Out]

(2*(d + e*x)^(7/2)*(5*c^3*(1024*d^6 - 3584*d^5*e*x + 8064*d^4*e^2*x^2 - 14784*d^
3*e^3*x^3 + 24024*d^2*e^4*x^4 - 36036*d*e^5*x^5 + 51051*e^6*x^6) + 1615*e^3*(429
*a^3*e^3 + 143*a^2*b*e^2*(-2*d + 7*e*x) + 13*a*b^2*e*(8*d^2 - 28*d*e*x + 63*e^2*
x^2) + b^3*(-16*d^3 + 56*d^2*e*x - 126*d*e^2*x^2 + 231*e^3*x^3)) + 323*c*e^2*(65
*a^2*e^2*(8*d^2 - 28*d*e*x + 63*e^2*x^2) + 30*a*b*e*(-16*d^3 + 56*d^2*e*x - 126*
d*e^2*x^2 + 231*e^3*x^3) + b^2*(128*d^4 - 448*d^3*e*x + 1008*d^2*e^2*x^2 - 1848*
d*e^3*x^3 + 3003*e^4*x^4)) - 19*c^2*e*(-17*a*e*(128*d^4 - 448*d^3*e*x + 1008*d^2
*e^2*x^2 - 1848*d*e^3*x^3 + 3003*e^4*x^4) + 5*b*(256*d^5 - 896*d^4*e*x + 2016*d^
3*e^2*x^2 - 3696*d^2*e^3*x^3 + 6006*d*e^4*x^4 - 9009*e^5*x^5))))/(4849845*e^7)

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Maple [A]  time = 0.012, size = 495, normalized size = 1.7 \[{\frac{510510\,{c}^{3}{x}^{6}{e}^{6}+1711710\,b{c}^{2}{e}^{6}{x}^{5}-360360\,{c}^{3}d{e}^{5}{x}^{5}+1939938\,{x}^{4}a{c}^{2}{e}^{6}+1939938\,{b}^{2}c{e}^{6}{x}^{4}-1141140\,b{c}^{2}d{e}^{5}{x}^{4}+240240\,{x}^{4}{c}^{3}{d}^{2}{e}^{4}+4476780\,abc{e}^{6}{x}^{3}-1193808\,{x}^{3}a{c}^{2}d{e}^{5}+746130\,{b}^{3}{e}^{6}{x}^{3}-1193808\,{b}^{2}cd{e}^{5}{x}^{3}+702240\,b{c}^{2}{d}^{2}{e}^{4}{x}^{3}-147840\,{x}^{3}{c}^{3}{d}^{3}{e}^{3}+2645370\,{x}^{2}{a}^{2}c{e}^{6}+2645370\,a{b}^{2}{e}^{6}{x}^{2}-2441880\,abcd{e}^{5}{x}^{2}+651168\,{x}^{2}a{c}^{2}{d}^{2}{e}^{4}-406980\,{b}^{3}d{e}^{5}{x}^{2}+651168\,{b}^{2}c{d}^{2}{e}^{4}{x}^{2}-383040\,b{c}^{2}{d}^{3}{e}^{3}{x}^{2}+80640\,{x}^{2}{c}^{3}{d}^{4}{e}^{2}+3233230\,{a}^{2}b{e}^{6}x-1175720\,x{a}^{2}cd{e}^{5}-1175720\,a{b}^{2}d{e}^{5}x+1085280\,abc{d}^{2}{e}^{4}x-289408\,xa{c}^{2}{d}^{3}{e}^{3}+180880\,{b}^{3}{d}^{2}{e}^{4}x-289408\,{b}^{2}c{d}^{3}{e}^{3}x+170240\,b{c}^{2}{d}^{4}{e}^{2}x-35840\,{c}^{3}{d}^{5}ex+1385670\,{a}^{3}{e}^{6}-923780\,{a}^{2}bd{e}^{5}+335920\,{a}^{2}c{d}^{2}{e}^{4}+335920\,a{b}^{2}{d}^{2}{e}^{4}-310080\,abc{d}^{3}{e}^{3}+82688\,{c}^{2}{d}^{4}a{e}^{2}-51680\,{b}^{3}{d}^{3}{e}^{3}+82688\,{b}^{2}c{d}^{4}{e}^{2}-48640\,b{c}^{2}{d}^{5}e+10240\,{c}^{3}{d}^{6}}{4849845\,{e}^{7}} \left ( ex+d \right ) ^{{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^(5/2)*(c*x^2+b*x+a)^3,x)

[Out]

2/4849845*(e*x+d)^(7/2)*(255255*c^3*e^6*x^6+855855*b*c^2*e^6*x^5-180180*c^3*d*e^
5*x^5+969969*a*c^2*e^6*x^4+969969*b^2*c*e^6*x^4-570570*b*c^2*d*e^5*x^4+120120*c^
3*d^2*e^4*x^4+2238390*a*b*c*e^6*x^3-596904*a*c^2*d*e^5*x^3+373065*b^3*e^6*x^3-59
6904*b^2*c*d*e^5*x^3+351120*b*c^2*d^2*e^4*x^3-73920*c^3*d^3*e^3*x^3+1322685*a^2*
c*e^6*x^2+1322685*a*b^2*e^6*x^2-1220940*a*b*c*d*e^5*x^2+325584*a*c^2*d^2*e^4*x^2
-203490*b^3*d*e^5*x^2+325584*b^2*c*d^2*e^4*x^2-191520*b*c^2*d^3*e^3*x^2+40320*c^
3*d^4*e^2*x^2+1616615*a^2*b*e^6*x-587860*a^2*c*d*e^5*x-587860*a*b^2*d*e^5*x+5426
40*a*b*c*d^2*e^4*x-144704*a*c^2*d^3*e^3*x+90440*b^3*d^2*e^4*x-144704*b^2*c*d^3*e
^3*x+85120*b*c^2*d^4*e^2*x-17920*c^3*d^5*e*x+692835*a^3*e^6-461890*a^2*b*d*e^5+1
67960*a^2*c*d^2*e^4+167960*a*b^2*d^2*e^4-155040*a*b*c*d^3*e^3+41344*a*c^2*d^4*e^
2-25840*b^3*d^3*e^3+41344*b^2*c*d^4*e^2-24320*b*c^2*d^5*e+5120*c^3*d^6)/e^7

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Maxima [A]  time = 0.709144, size = 549, normalized size = 1.92 \[ \frac{2 \,{\left (255255 \,{\left (e x + d\right )}^{\frac{19}{2}} c^{3} - 855855 \,{\left (2 \, c^{3} d - b c^{2} e\right )}{\left (e x + d\right )}^{\frac{17}{2}} + 969969 \,{\left (5 \, c^{3} d^{2} - 5 \, b c^{2} d e +{\left (b^{2} c + a c^{2}\right )} e^{2}\right )}{\left (e x + d\right )}^{\frac{15}{2}} - 373065 \,{\left (20 \, c^{3} d^{3} - 30 \, b c^{2} d^{2} e + 12 \,{\left (b^{2} c + a c^{2}\right )} d e^{2} -{\left (b^{3} + 6 \, a b c\right )} e^{3}\right )}{\left (e x + d\right )}^{\frac{13}{2}} + 1322685 \,{\left (5 \, c^{3} d^{4} - 10 \, b c^{2} d^{3} e + 6 \,{\left (b^{2} c + a c^{2}\right )} d^{2} e^{2} -{\left (b^{3} + 6 \, a b c\right )} d e^{3} +{\left (a b^{2} + a^{2} c\right )} e^{4}\right )}{\left (e x + d\right )}^{\frac{11}{2}} - 1616615 \,{\left (2 \, c^{3} d^{5} - 5 \, b c^{2} d^{4} e - a^{2} b e^{5} + 4 \,{\left (b^{2} c + a c^{2}\right )} d^{3} e^{2} -{\left (b^{3} + 6 \, a b c\right )} d^{2} e^{3} + 2 \,{\left (a b^{2} + a^{2} c\right )} d e^{4}\right )}{\left (e x + d\right )}^{\frac{9}{2}} + 692835 \,{\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e - 3 \, a^{2} b d e^{5} + a^{3} e^{6} + 3 \,{\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} -{\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} + 3 \,{\left (a b^{2} + a^{2} c\right )} d^{2} e^{4}\right )}{\left (e x + d\right )}^{\frac{7}{2}}\right )}}{4849845 \, e^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^3*(e*x + d)^(5/2),x, algorithm="maxima")

[Out]

2/4849845*(255255*(e*x + d)^(19/2)*c^3 - 855855*(2*c^3*d - b*c^2*e)*(e*x + d)^(1
7/2) + 969969*(5*c^3*d^2 - 5*b*c^2*d*e + (b^2*c + a*c^2)*e^2)*(e*x + d)^(15/2) -
373065*(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)^(13/2) + 1322685*(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)^(11/2) - 1616
615*(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)*(e*x + d)^(9/2) + 692835*(c^3*d^6 -
3*b*c^2*d^5*e - 3*a^2*b*d*e^5 + a^3*e^6 + 3*(b^2*c + a*c^2)*d^4*e^2 - (b^3 + 6*a
*b*c)*d^3*e^3 + 3*(a*b^2 + a^2*c)*d^2*e^4)*(e*x + d)^(7/2))/e^7

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Fricas [A]  time = 0.210035, size = 981, normalized size = 3.43 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^3*(e*x + d)^(5/2),x, algorithm="fricas")

[Out]

2/4849845*(255255*c^3*e^9*x^9 + 5120*c^3*d^9 - 24320*b*c^2*d^8*e - 461890*a^2*b*
d^4*e^5 + 692835*a^3*d^3*e^6 + 41344*(b^2*c + a*c^2)*d^7*e^2 - 25840*(b^3 + 6*a*
b*c)*d^6*e^3 + 167960*(a*b^2 + a^2*c)*d^5*e^4 + 45045*(13*c^3*d*e^8 + 19*b*c^2*e
^9)*x^8 + 3003*(115*c^3*d^2*e^7 + 665*b*c^2*d*e^8 + 323*(b^2*c + a*c^2)*e^9)*x^7
+ 231*(5*c^3*d^3*e^6 + 5225*b*c^2*d^2*e^7 + 10013*(b^2*c + a*c^2)*d*e^8 + 1615*
(b^3 + 6*a*b*c)*e^9)*x^6 - 63*(20*c^3*d^4*e^5 - 95*b*c^2*d^3*e^6 - 22933*(b^2*c
+ a*c^2)*d^2*e^7 - 14535*(b^3 + 6*a*b*c)*d*e^8 - 20995*(a*b^2 + a^2*c)*e^9)*x^5
+ 35*(40*c^3*d^5*e^4 - 190*b*c^2*d^4*e^5 + 46189*a^2*b*e^9 + 323*(b^2*c + a*c^2)
*d^3*e^6 + 17119*(b^3 + 6*a*b*c)*d^2*e^7 + 96577*(a*b^2 + a^2*c)*d*e^8)*x^4 - 5*
(320*c^3*d^6*e^3 - 1520*b*c^2*d^5*e^4 - 877591*a^2*b*d*e^8 - 138567*a^3*e^9 + 25
84*(b^2*c + a*c^2)*d^4*e^5 - 1615*(b^3 + 6*a*b*c)*d^3*e^6 - 474487*(a*b^2 + a^2*
c)*d^2*e^7)*x^3 + 3*(640*c^3*d^7*e^2 - 3040*b*c^2*d^6*e^3 + 1154725*a^2*b*d^2*e^
7 + 692835*a^3*d*e^8 + 5168*(b^2*c + a*c^2)*d^5*e^4 - 3230*(b^3 + 6*a*b*c)*d^4*e
^5 + 20995*(a*b^2 + a^2*c)*d^3*e^6)*x^2 - (2560*c^3*d^8*e - 12160*b*c^2*d^7*e^2
- 230945*a^2*b*d^3*e^6 - 2078505*a^3*d^2*e^7 + 20672*(b^2*c + a*c^2)*d^6*e^3 - 1
2920*(b^3 + 6*a*b*c)*d^5*e^4 + 83980*(a*b^2 + a^2*c)*d^4*e^5)*x)*sqrt(e*x + d)/e
^7

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Sympy [A]  time = 22.5081, size = 2363, normalized size = 8.26 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**(5/2)*(c*x**2+b*x+a)**3,x)

[Out]

a**3*d**2*Piecewise((sqrt(d)*x, Eq(e, 0)), (2*(d + e*x)**(3/2)/(3*e), True)) + 4
*a**3*d*(-d*(d + e*x)**(3/2)/3 + (d + e*x)**(5/2)/5)/e + 2*a**3*(d**2*(d + e*x)*
*(3/2)/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e + 6*a**2*b*d**2*(-d*(d
+ e*x)**(3/2)/3 + (d + e*x)**(5/2)/5)/e**2 + 12*a**2*b*d*(d**2*(d + e*x)**(3/2)
/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**2 + 6*a**2*b*(-d**3*(d + e*
x)**(3/2)/3 + 3*d**2*(d + e*x)**(5/2)/5 - 3*d*(d + e*x)**(7/2)/7 + (d + e*x)**(9
/2)/9)/e**2 + 6*a**2*c*d**2*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d + e*x)**(5/2)/5 +
(d + e*x)**(7/2)/7)/e**3 + 12*a**2*c*d*(-d**3*(d + e*x)**(3/2)/3 + 3*d**2*(d + e
*x)**(5/2)/5 - 3*d*(d + e*x)**(7/2)/7 + (d + e*x)**(9/2)/9)/e**3 + 6*a**2*c*(d**
4*(d + e*x)**(3/2)/3 - 4*d**3*(d + e*x)**(5/2)/5 + 6*d**2*(d + e*x)**(7/2)/7 - 4
*d*(d + e*x)**(9/2)/9 + (d + e*x)**(11/2)/11)/e**3 + 6*a*b**2*d**2*(d**2*(d + e*
x)**(3/2)/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**3 + 12*a*b**2*d*(-
d**3*(d + e*x)**(3/2)/3 + 3*d**2*(d + e*x)**(5/2)/5 - 3*d*(d + e*x)**(7/2)/7 + (
d + e*x)**(9/2)/9)/e**3 + 6*a*b**2*(d**4*(d + e*x)**(3/2)/3 - 4*d**3*(d + e*x)**
(5/2)/5 + 6*d**2*(d + e*x)**(7/2)/7 - 4*d*(d + e*x)**(9/2)/9 + (d + e*x)**(11/2)
/11)/e**3 + 12*a*b*c*d**2*(-d**3*(d + e*x)**(3/2)/3 + 3*d**2*(d + e*x)**(5/2)/5
- 3*d*(d + e*x)**(7/2)/7 + (d + e*x)**(9/2)/9)/e**4 + 24*a*b*c*d*(d**4*(d + e*x)
**(3/2)/3 - 4*d**3*(d + e*x)**(5/2)/5 + 6*d**2*(d + e*x)**(7/2)/7 - 4*d*(d + e*x
)**(9/2)/9 + (d + e*x)**(11/2)/11)/e**4 + 12*a*b*c*(-d**5*(d + e*x)**(3/2)/3 + d
**4*(d + e*x)**(5/2) - 10*d**3*(d + e*x)**(7/2)/7 + 10*d**2*(d + e*x)**(9/2)/9 -
5*d*(d + e*x)**(11/2)/11 + (d + e*x)**(13/2)/13)/e**4 + 6*a*c**2*d**2*(d**4*(d
+ e*x)**(3/2)/3 - 4*d**3*(d + e*x)**(5/2)/5 + 6*d**2*(d + e*x)**(7/2)/7 - 4*d*(d
+ e*x)**(9/2)/9 + (d + e*x)**(11/2)/11)/e**5 + 12*a*c**2*d*(-d**5*(d + e*x)**(3
/2)/3 + d**4*(d + e*x)**(5/2) - 10*d**3*(d + e*x)**(7/2)/7 + 10*d**2*(d + e*x)**
(9/2)/9 - 5*d*(d + e*x)**(11/2)/11 + (d + e*x)**(13/2)/13)/e**5 + 6*a*c**2*(d**6
*(d + e*x)**(3/2)/3 - 6*d**5*(d + e*x)**(5/2)/5 + 15*d**4*(d + e*x)**(7/2)/7 - 2
0*d**3*(d + e*x)**(9/2)/9 + 15*d**2*(d + e*x)**(11/2)/11 - 6*d*(d + e*x)**(13/2)
/13 + (d + e*x)**(15/2)/15)/e**5 + 2*b**3*d**2*(-d**3*(d + e*x)**(3/2)/3 + 3*d**
2*(d + e*x)**(5/2)/5 - 3*d*(d + e*x)**(7/2)/7 + (d + e*x)**(9/2)/9)/e**4 + 4*b**
3*d*(d**4*(d + e*x)**(3/2)/3 - 4*d**3*(d + e*x)**(5/2)/5 + 6*d**2*(d + e*x)**(7/
2)/7 - 4*d*(d + e*x)**(9/2)/9 + (d + e*x)**(11/2)/11)/e**4 + 2*b**3*(-d**5*(d +
e*x)**(3/2)/3 + d**4*(d + e*x)**(5/2) - 10*d**3*(d + e*x)**(7/2)/7 + 10*d**2*(d
+ e*x)**(9/2)/9 - 5*d*(d + e*x)**(11/2)/11 + (d + e*x)**(13/2)/13)/e**4 + 6*b**2
*c*d**2*(d**4*(d + e*x)**(3/2)/3 - 4*d**3*(d + e*x)**(5/2)/5 + 6*d**2*(d + e*x)*
*(7/2)/7 - 4*d*(d + e*x)**(9/2)/9 + (d + e*x)**(11/2)/11)/e**5 + 12*b**2*c*d*(-d
**5*(d + e*x)**(3/2)/3 + d**4*(d + e*x)**(5/2) - 10*d**3*(d + e*x)**(7/2)/7 + 10
*d**2*(d + e*x)**(9/2)/9 - 5*d*(d + e*x)**(11/2)/11 + (d + e*x)**(13/2)/13)/e**5
+ 6*b**2*c*(d**6*(d + e*x)**(3/2)/3 - 6*d**5*(d + e*x)**(5/2)/5 + 15*d**4*(d +
e*x)**(7/2)/7 - 20*d**3*(d + e*x)**(9/2)/9 + 15*d**2*(d + e*x)**(11/2)/11 - 6*d*
(d + e*x)**(13/2)/13 + (d + e*x)**(15/2)/15)/e**5 + 6*b*c**2*d**2*(-d**5*(d + e*
x)**(3/2)/3 + d**4*(d + e*x)**(5/2) - 10*d**3*(d + e*x)**(7/2)/7 + 10*d**2*(d +
e*x)**(9/2)/9 - 5*d*(d + e*x)**(11/2)/11 + (d + e*x)**(13/2)/13)/e**6 + 12*b*c**
2*d*(d**6*(d + e*x)**(3/2)/3 - 6*d**5*(d + e*x)**(5/2)/5 + 15*d**4*(d + e*x)**(7
/2)/7 - 20*d**3*(d + e*x)**(9/2)/9 + 15*d**2*(d + e*x)**(11/2)/11 - 6*d*(d + e*x
)**(13/2)/13 + (d + e*x)**(15/2)/15)/e**6 + 6*b*c**2*(-d**7*(d + e*x)**(3/2)/3 +
7*d**6*(d + e*x)**(5/2)/5 - 3*d**5*(d + e*x)**(7/2) + 35*d**4*(d + e*x)**(9/2)/
9 - 35*d**3*(d + e*x)**(11/2)/11 + 21*d**2*(d + e*x)**(13/2)/13 - 7*d*(d + e*x)*
*(15/2)/15 + (d + e*x)**(17/2)/17)/e**6 + 2*c**3*d**2*(d**6*(d + e*x)**(3/2)/3 -
6*d**5*(d + e*x)**(5/2)/5 + 15*d**4*(d + e*x)**(7/2)/7 - 20*d**3*(d + e*x)**(9/
2)/9 + 15*d**2*(d + e*x)**(11/2)/11 - 6*d*(d + e*x)**(13/2)/13 + (d + e*x)**(15/
2)/15)/e**7 + 4*c**3*d*(-d**7*(d + e*x)**(3/2)/3 + 7*d**6*(d + e*x)**(5/2)/5 - 3
*d**5*(d + e*x)**(7/2) + 35*d**4*(d + e*x)**(9/2)/9 - 35*d**3*(d + e*x)**(11/2)/
11 + 21*d**2*(d + e*x)**(13/2)/13 - 7*d*(d + e*x)**(15/2)/15 + (d + e*x)**(17/2)
/17)/e**7 + 2*c**3*(d**8*(d + e*x)**(3/2)/3 - 8*d**7*(d + e*x)**(5/2)/5 + 4*d**6
*(d + e*x)**(7/2) - 56*d**5*(d + e*x)**(9/2)/9 + 70*d**4*(d + e*x)**(11/2)/11 -
56*d**3*(d + e*x)**(13/2)/13 + 28*d**2*(d + e*x)**(15/2)/15 - 8*d*(d + e*x)**(17
/2)/17 + (d + e*x)**(19/2)/19)/e**7

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.253272, 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*(e*x + d)^(5/2),x, algorithm="giac")

[Out]

Done