3.1612 \(\int \frac{(b+2 c x) \left (a+b x+c x^2\right )^3}{(d+e x)^{5/2}} \, dx\)

Optimal. Leaf size=421 \[ \frac{2 (d+e x)^{3/2} \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{3 e^8}+\frac{6 c^2 (d+e x)^{7/2} \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{7 e^8}-\frac{2 c (d+e x)^{5/2} (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^8}-\frac{6 \sqrt{d+e x} (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^8}-\frac{2 \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^8 \sqrt{d+e x}}+\frac{2 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{3 e^8 (d+e x)^{3/2}}-\frac{14 c^3 (d+e x)^{9/2} (2 c d-b e)}{9 e^8}+\frac{4 c^4 (d+e x)^{11/2}}{11 e^8} \]

[Out]

(2*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^3)/(3*e^8*(d + e*x)^(3/2)) - (2*(c*d^2
- b*d*e + a*e^2)^2*(14*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(7*b*d - a*e)))/(e^8*Sqrt[d +
e*x]) - (6*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)*(7*c^2*d^2 + b^2*e^2 - c*e*(7*
b*d - 3*a*e))*Sqrt[d + e*x])/e^8 + (2*(70*c^4*d^4 + b^4*e^4 - 4*b^2*c*e^3*(5*b*d
- 3*a*e) - 20*c^3*d^2*e*(7*b*d - 3*a*e) + 6*c^2*e^2*(15*b^2*d^2 - 10*a*b*d*e +
a^2*e^2))*(d + e*x)^(3/2))/(3*e^8) - (2*c*(2*c*d - b*e)*(7*c^2*d^2 + b^2*e^2 - c
*e*(7*b*d - 3*a*e))*(d + e*x)^(5/2))/e^8 + (6*c^2*(14*c^2*d^2 + 3*b^2*e^2 - 2*c*
e*(7*b*d - a*e))*(d + e*x)^(7/2))/(7*e^8) - (14*c^3*(2*c*d - b*e)*(d + e*x)^(9/2
))/(9*e^8) + (4*c^4*(d + e*x)^(11/2))/(11*e^8)

_______________________________________________________________________________________

Rubi [A]  time = 0.640624, antiderivative size = 421, 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 \[ \frac{2 (d+e x)^{3/2} \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{3 e^8}+\frac{6 c^2 (d+e x)^{7/2} \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{7 e^8}-\frac{2 c (d+e x)^{5/2} (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^8}-\frac{6 \sqrt{d+e x} (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^8}-\frac{2 \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^8 \sqrt{d+e x}}+\frac{2 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{3 e^8 (d+e x)^{3/2}}-\frac{14 c^3 (d+e x)^{9/2} (2 c d-b e)}{9 e^8}+\frac{4 c^4 (d+e x)^{11/2}}{11 e^8} \]

Antiderivative was successfully verified.

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

[Out]

(2*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^3)/(3*e^8*(d + e*x)^(3/2)) - (2*(c*d^2
- b*d*e + a*e^2)^2*(14*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(7*b*d - a*e)))/(e^8*Sqrt[d +
e*x]) - (6*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)*(7*c^2*d^2 + b^2*e^2 - c*e*(7*
b*d - 3*a*e))*Sqrt[d + e*x])/e^8 + (2*(70*c^4*d^4 + b^4*e^4 - 4*b^2*c*e^3*(5*b*d
- 3*a*e) - 20*c^3*d^2*e*(7*b*d - 3*a*e) + 6*c^2*e^2*(15*b^2*d^2 - 10*a*b*d*e +
a^2*e^2))*(d + e*x)^(3/2))/(3*e^8) - (2*c*(2*c*d - b*e)*(7*c^2*d^2 + b^2*e^2 - c
*e*(7*b*d - 3*a*e))*(d + e*x)^(5/2))/e^8 + (6*c^2*(14*c^2*d^2 + 3*b^2*e^2 - 2*c*
e*(7*b*d - a*e))*(d + e*x)^(7/2))/(7*e^8) - (14*c^3*(2*c*d - b*e)*(d + e*x)^(9/2
))/(9*e^8) + (4*c^4*(d + e*x)^(11/2))/(11*e^8)

_______________________________________________________________________________________

Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

_______________________________________________________________________________________

Mathematica [A]  time = 0.988713, size = 598, normalized size = 1.42 \[ \frac{-198 c^2 e^2 \left (14 a^2 e^2 \left (16 d^3+24 d^2 e x+6 d e^2 x^2-e^3 x^3\right )-7 a b e \left (128 d^4+192 d^3 e x+48 d^2 e^2 x^2-8 d e^3 x^3+3 e^4 x^4\right )+3 b^2 \left (256 d^5+384 d^4 e x+96 d^3 e^2 x^2-16 d^2 e^3 x^3+6 d e^4 x^4-3 e^5 x^5\right )\right )+462 c e^3 \left (-2 a^3 e^3 (2 d+3 e x)+9 a^2 b e^2 \left (8 d^2+12 d e x+3 e^2 x^2\right )+12 a b^2 e \left (-16 d^3-24 d^2 e x-6 d e^2 x^2+e^3 x^3\right )+b^3 \left (128 d^4+192 d^3 e x+48 d^2 e^2 x^2-8 d e^3 x^3+3 e^4 x^4\right )\right )-462 b e^4 \left (a^3 e^3+3 a^2 b e^2 (2 d+3 e x)-3 a b^2 e \left (8 d^2+12 d e x+3 e^2 x^2\right )+b^3 \left (16 d^3+24 d^2 e x+6 d e^2 x^2-e^3 x^3\right )\right )+22 c^3 e \left (7 b \left (1024 d^6+1536 d^5 e x+384 d^4 e^2 x^2-64 d^3 e^3 x^3+24 d^2 e^4 x^4-12 d e^5 x^5+7 e^6 x^6\right )-18 a e \left (256 d^5+384 d^4 e x+96 d^3 e^2 x^2-16 d^2 e^3 x^3+6 d e^4 x^4-3 e^5 x^5\right )\right )-28 c^4 \left (2048 d^7+3072 d^6 e x+768 d^5 e^2 x^2-128 d^4 e^3 x^3+48 d^3 e^4 x^4-24 d^2 e^5 x^5+14 d e^6 x^6-9 e^7 x^7\right )}{693 e^8 (d+e x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-28*c^4*(2048*d^7 + 3072*d^6*e*x + 768*d^5*e^2*x^2 - 128*d^4*e^3*x^3 + 48*d^3*e
^4*x^4 - 24*d^2*e^5*x^5 + 14*d*e^6*x^6 - 9*e^7*x^7) - 462*b*e^4*(a^3*e^3 + 3*a^2
*b*e^2*(2*d + 3*e*x) - 3*a*b^2*e*(8*d^2 + 12*d*e*x + 3*e^2*x^2) + b^3*(16*d^3 +
24*d^2*e*x + 6*d*e^2*x^2 - e^3*x^3)) + 462*c*e^3*(-2*a^3*e^3*(2*d + 3*e*x) + 9*a
^2*b*e^2*(8*d^2 + 12*d*e*x + 3*e^2*x^2) + 12*a*b^2*e*(-16*d^3 - 24*d^2*e*x - 6*d
*e^2*x^2 + e^3*x^3) + b^3*(128*d^4 + 192*d^3*e*x + 48*d^2*e^2*x^2 - 8*d*e^3*x^3
+ 3*e^4*x^4)) - 198*c^2*e^2*(14*a^2*e^2*(16*d^3 + 24*d^2*e*x + 6*d*e^2*x^2 - e^3
*x^3) - 7*a*b*e*(128*d^4 + 192*d^3*e*x + 48*d^2*e^2*x^2 - 8*d*e^3*x^3 + 3*e^4*x^
4) + 3*b^2*(256*d^5 + 384*d^4*e*x + 96*d^3*e^2*x^2 - 16*d^2*e^3*x^3 + 6*d*e^4*x^
4 - 3*e^5*x^5)) + 22*c^3*e*(-18*a*e*(256*d^5 + 384*d^4*e*x + 96*d^3*e^2*x^2 - 16
*d^2*e^3*x^3 + 6*d*e^4*x^4 - 3*e^5*x^5) + 7*b*(1024*d^6 + 1536*d^5*e*x + 384*d^4
*e^2*x^2 - 64*d^3*e^3*x^3 + 24*d^2*e^4*x^4 - 12*d*e^5*x^5 + 7*e^6*x^6)))/(693*e^
8*(d + e*x)^(3/2))

_______________________________________________________________________________________

Maple [B]  time = 0.013, size = 795, normalized size = 1.9 \[ -{\frac{-252\,{c}^{4}{x}^{7}{e}^{7}-1078\,b{c}^{3}{e}^{7}{x}^{6}+392\,{c}^{4}d{e}^{6}{x}^{6}-1188\,a{c}^{3}{e}^{7}{x}^{5}-1782\,{b}^{2}{c}^{2}{e}^{7}{x}^{5}+1848\,b{c}^{3}d{e}^{6}{x}^{5}-672\,{c}^{4}{d}^{2}{e}^{5}{x}^{5}-4158\,ab{c}^{2}{e}^{7}{x}^{4}+2376\,a{c}^{3}d{e}^{6}{x}^{4}-1386\,{b}^{3}c{e}^{7}{x}^{4}+3564\,{b}^{2}{c}^{2}d{e}^{6}{x}^{4}-3696\,b{c}^{3}{d}^{2}{e}^{5}{x}^{4}+1344\,{c}^{4}{d}^{3}{e}^{4}{x}^{4}-2772\,{a}^{2}{c}^{2}{e}^{7}{x}^{3}-5544\,a{b}^{2}c{e}^{7}{x}^{3}+11088\,ab{c}^{2}d{e}^{6}{x}^{3}-6336\,a{c}^{3}{d}^{2}{e}^{5}{x}^{3}-462\,{b}^{4}{e}^{7}{x}^{3}+3696\,{b}^{3}cd{e}^{6}{x}^{3}-9504\,{b}^{2}{c}^{2}{d}^{2}{e}^{5}{x}^{3}+9856\,b{c}^{3}{d}^{3}{e}^{4}{x}^{3}-3584\,{c}^{4}{d}^{4}{e}^{3}{x}^{3}-12474\,{a}^{2}bc{e}^{7}{x}^{2}+16632\,{a}^{2}{c}^{2}d{e}^{6}{x}^{2}-4158\,a{b}^{3}{e}^{7}{x}^{2}+33264\,a{b}^{2}cd{e}^{6}{x}^{2}-66528\,ab{c}^{2}{d}^{2}{e}^{5}{x}^{2}+38016\,a{c}^{3}{d}^{3}{e}^{4}{x}^{2}+2772\,{b}^{4}d{e}^{6}{x}^{2}-22176\,{b}^{3}c{d}^{2}{e}^{5}{x}^{2}+57024\,{b}^{2}{c}^{2}{d}^{3}{e}^{4}{x}^{2}-59136\,b{c}^{3}{d}^{4}{e}^{3}{x}^{2}+21504\,{c}^{4}{d}^{5}{e}^{2}{x}^{2}+2772\,{a}^{3}c{e}^{7}x+4158\,{a}^{2}{b}^{2}{e}^{7}x-49896\,{a}^{2}bcd{e}^{6}x+66528\,{a}^{2}{c}^{2}{d}^{2}{e}^{5}x-16632\,a{b}^{3}d{e}^{6}x+133056\,a{b}^{2}c{d}^{2}{e}^{5}x-266112\,ab{c}^{2}{d}^{3}{e}^{4}x+152064\,a{c}^{3}{d}^{4}{e}^{3}x+11088\,{b}^{4}{d}^{2}{e}^{5}x-88704\,{b}^{3}c{d}^{3}{e}^{4}x+228096\,{b}^{2}{c}^{2}{d}^{4}{e}^{3}x-236544\,b{c}^{3}{d}^{5}{e}^{2}x+86016\,{c}^{4}{d}^{6}ex+462\,{a}^{3}b{e}^{7}+1848\,{a}^{3}cd{e}^{6}+2772\,{a}^{2}{b}^{2}d{e}^{6}-33264\,{a}^{2}bc{d}^{2}{e}^{5}+44352\,{a}^{2}{c}^{2}{d}^{3}{e}^{4}-11088\,a{b}^{3}{d}^{2}{e}^{5}+88704\,a{b}^{2}c{d}^{3}{e}^{4}-177408\,ab{c}^{2}{d}^{4}{e}^{3}+101376\,a{c}^{3}{d}^{5}{e}^{2}+7392\,{b}^{4}{d}^{3}{e}^{4}-59136\,{b}^{3}c{d}^{4}{e}^{3}+152064\,{b}^{2}{c}^{2}{d}^{5}{e}^{2}-157696\,b{c}^{3}{d}^{6}e+57344\,{c}^{4}{d}^{7}}{693\,{e}^{8}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/693/(e*x+d)^(3/2)*(-126*c^4*e^7*x^7-539*b*c^3*e^7*x^6+196*c^4*d*e^6*x^6-594*a
*c^3*e^7*x^5-891*b^2*c^2*e^7*x^5+924*b*c^3*d*e^6*x^5-336*c^4*d^2*e^5*x^5-2079*a*
b*c^2*e^7*x^4+1188*a*c^3*d*e^6*x^4-693*b^3*c*e^7*x^4+1782*b^2*c^2*d*e^6*x^4-1848
*b*c^3*d^2*e^5*x^4+672*c^4*d^3*e^4*x^4-1386*a^2*c^2*e^7*x^3-2772*a*b^2*c*e^7*x^3
+5544*a*b*c^2*d*e^6*x^3-3168*a*c^3*d^2*e^5*x^3-231*b^4*e^7*x^3+1848*b^3*c*d*e^6*
x^3-4752*b^2*c^2*d^2*e^5*x^3+4928*b*c^3*d^3*e^4*x^3-1792*c^4*d^4*e^3*x^3-6237*a^
2*b*c*e^7*x^2+8316*a^2*c^2*d*e^6*x^2-2079*a*b^3*e^7*x^2+16632*a*b^2*c*d*e^6*x^2-
33264*a*b*c^2*d^2*e^5*x^2+19008*a*c^3*d^3*e^4*x^2+1386*b^4*d*e^6*x^2-11088*b^3*c
*d^2*e^5*x^2+28512*b^2*c^2*d^3*e^4*x^2-29568*b*c^3*d^4*e^3*x^2+10752*c^4*d^5*e^2
*x^2+1386*a^3*c*e^7*x+2079*a^2*b^2*e^7*x-24948*a^2*b*c*d*e^6*x+33264*a^2*c^2*d^2
*e^5*x-8316*a*b^3*d*e^6*x+66528*a*b^2*c*d^2*e^5*x-133056*a*b*c^2*d^3*e^4*x+76032
*a*c^3*d^4*e^3*x+5544*b^4*d^2*e^5*x-44352*b^3*c*d^3*e^4*x+114048*b^2*c^2*d^4*e^3
*x-118272*b*c^3*d^5*e^2*x+43008*c^4*d^6*e*x+231*a^3*b*e^7+924*a^3*c*d*e^6+1386*a
^2*b^2*d*e^6-16632*a^2*b*c*d^2*e^5+22176*a^2*c^2*d^3*e^4-5544*a*b^3*d^2*e^5+4435
2*a*b^2*c*d^3*e^4-88704*a*b*c^2*d^4*e^3+50688*a*c^3*d^5*e^2+3696*b^4*d^3*e^4-295
68*b^3*c*d^4*e^3+76032*b^2*c^2*d^5*e^2-78848*b*c^3*d^6*e+28672*c^4*d^7)/e^8

_______________________________________________________________________________________

Maxima [A]  time = 0.715746, size = 879, normalized size = 2.09 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/693*((126*(e*x + d)^(11/2)*c^4 - 539*(2*c^4*d - b*c^3*e)*(e*x + d)^(9/2) + 297
*(14*c^4*d^2 - 14*b*c^3*d*e + (3*b^2*c^2 + 2*a*c^3)*e^2)*(e*x + d)^(7/2) - 693*(
14*c^4*d^3 - 21*b*c^3*d^2*e + 3*(3*b^2*c^2 + 2*a*c^3)*d*e^2 - (b^3*c + 3*a*b*c^2
)*e^3)*(e*x + d)^(5/2) + 231*(70*c^4*d^4 - 140*b*c^3*d^3*e + 30*(3*b^2*c^2 + 2*a
*c^3)*d^2*e^2 - 20*(b^3*c + 3*a*b*c^2)*d*e^3 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*e^
4)*(e*x + d)^(3/2) - 2079*(14*c^4*d^5 - 35*b*c^3*d^4*e + 10*(3*b^2*c^2 + 2*a*c^3
)*d^3*e^2 - 10*(b^3*c + 3*a*b*c^2)*d^2*e^3 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d*e^
4 - (a*b^3 + 3*a^2*b*c)*e^5)*sqrt(e*x + d))/e^7 + 231*(2*c^4*d^7 - 7*b*c^3*d^6*e
- a^3*b*e^7 + 3*(3*b^2*c^2 + 2*a*c^3)*d^5*e^2 - 5*(b^3*c + 3*a*b*c^2)*d^4*e^3 +
(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^3*e^4 - 3*(a*b^3 + 3*a^2*b*c)*d^2*e^5 + (3*a^2
*b^2 + 2*a^3*c)*d*e^6 - 3*(14*c^4*d^6 - 42*b*c^3*d^5*e + 15*(3*b^2*c^2 + 2*a*c^3
)*d^4*e^2 - 20*(b^3*c + 3*a*b*c^2)*d^3*e^3 + 3*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^
2*e^4 - 6*(a*b^3 + 3*a^2*b*c)*d*e^5 + (3*a^2*b^2 + 2*a^3*c)*e^6)*(e*x + d))/((e*
x + d)^(3/2)*e^7))/e

_______________________________________________________________________________________

Fricas [A]  time = 0.279656, size = 888, normalized size = 2.11 \[ \frac{2 \,{\left (126 \, c^{4} e^{7} x^{7} - 28672 \, c^{4} d^{7} + 78848 \, b c^{3} d^{6} e - 231 \, a^{3} b e^{7} - 25344 \,{\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{5} e^{2} + 29568 \,{\left (b^{3} c + 3 \, a b c^{2}\right )} d^{4} e^{3} - 3696 \,{\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{3} e^{4} + 5544 \,{\left (a b^{3} + 3 \, a^{2} b c\right )} d^{2} e^{5} - 462 \,{\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d e^{6} - 49 \,{\left (4 \, c^{4} d e^{6} - 11 \, b c^{3} e^{7}\right )} x^{6} + 3 \,{\left (112 \, c^{4} d^{2} e^{5} - 308 \, b c^{3} d e^{6} + 99 \,{\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} e^{7}\right )} x^{5} - 3 \,{\left (224 \, c^{4} d^{3} e^{4} - 616 \, b c^{3} d^{2} e^{5} + 198 \,{\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d e^{6} - 231 \,{\left (b^{3} c + 3 \, a b c^{2}\right )} e^{7}\right )} x^{4} +{\left (1792 \, c^{4} d^{4} e^{3} - 4928 \, b c^{3} d^{3} e^{4} + 1584 \,{\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{2} e^{5} - 1848 \,{\left (b^{3} c + 3 \, a b c^{2}\right )} d e^{6} + 231 \,{\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} e^{7}\right )} x^{3} - 3 \,{\left (3584 \, c^{4} d^{5} e^{2} - 9856 \, b c^{3} d^{4} e^{3} + 3168 \,{\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{3} e^{4} - 3696 \,{\left (b^{3} c + 3 \, a b c^{2}\right )} d^{2} e^{5} + 462 \,{\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d e^{6} - 693 \,{\left (a b^{3} + 3 \, a^{2} b c\right )} e^{7}\right )} x^{2} - 3 \,{\left (14336 \, c^{4} d^{6} e - 39424 \, b c^{3} d^{5} e^{2} + 12672 \,{\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{4} e^{3} - 14784 \,{\left (b^{3} c + 3 \, a b c^{2}\right )} d^{3} e^{4} + 1848 \,{\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{2} e^{5} - 2772 \,{\left (a b^{3} + 3 \, a^{2} b c\right )} d e^{6} + 231 \,{\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} e^{7}\right )} x\right )}}{693 \,{\left (e^{9} x + d e^{8}\right )} \sqrt{e x + d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/693*(126*c^4*e^7*x^7 - 28672*c^4*d^7 + 78848*b*c^3*d^6*e - 231*a^3*b*e^7 - 253
44*(3*b^2*c^2 + 2*a*c^3)*d^5*e^2 + 29568*(b^3*c + 3*a*b*c^2)*d^4*e^3 - 3696*(b^4
+ 12*a*b^2*c + 6*a^2*c^2)*d^3*e^4 + 5544*(a*b^3 + 3*a^2*b*c)*d^2*e^5 - 462*(3*a
^2*b^2 + 2*a^3*c)*d*e^6 - 49*(4*c^4*d*e^6 - 11*b*c^3*e^7)*x^6 + 3*(112*c^4*d^2*e
^5 - 308*b*c^3*d*e^6 + 99*(3*b^2*c^2 + 2*a*c^3)*e^7)*x^5 - 3*(224*c^4*d^3*e^4 -
616*b*c^3*d^2*e^5 + 198*(3*b^2*c^2 + 2*a*c^3)*d*e^6 - 231*(b^3*c + 3*a*b*c^2)*e^
7)*x^4 + (1792*c^4*d^4*e^3 - 4928*b*c^3*d^3*e^4 + 1584*(3*b^2*c^2 + 2*a*c^3)*d^2
*e^5 - 1848*(b^3*c + 3*a*b*c^2)*d*e^6 + 231*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*e^7)*
x^3 - 3*(3584*c^4*d^5*e^2 - 9856*b*c^3*d^4*e^3 + 3168*(3*b^2*c^2 + 2*a*c^3)*d^3*
e^4 - 3696*(b^3*c + 3*a*b*c^2)*d^2*e^5 + 462*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d*e^
6 - 693*(a*b^3 + 3*a^2*b*c)*e^7)*x^2 - 3*(14336*c^4*d^6*e - 39424*b*c^3*d^5*e^2
+ 12672*(3*b^2*c^2 + 2*a*c^3)*d^4*e^3 - 14784*(b^3*c + 3*a*b*c^2)*d^3*e^4 + 1848
*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^2*e^5 - 2772*(a*b^3 + 3*a^2*b*c)*d*e^6 + 231*(
3*a^2*b^2 + 2*a^3*c)*e^7)*x)/((e^9*x + d*e^8)*sqrt(e*x + d))

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (b + 2 c x\right ) \left (a + b x + c x^{2}\right )^{3}}{\left (d + e x\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((b + 2*c*x)*(a + b*x + c*x**2)**3/(d + e*x)**(5/2), x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.287178, 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*(2*c*x + b)/(e*x + d)^(5/2),x, algorithm="giac")

[Out]

Done