3.2250 \(\int (d+e x)^{5/2} (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2} \, dx\)

Optimal. Leaf size=501 \[ -\frac{512 (2 c d-b e)^5 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-12 b e g+5 c d g+19 c e f)}{2909907 c^7 e^2 (d+e x)^{7/2}}-\frac{256 (2 c d-b e)^4 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-12 b e g+5 c d g+19 c e f)}{415701 c^6 e^2 (d+e x)^{5/2}}-\frac{64 (2 c d-b e)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-12 b e g+5 c d g+19 c e f)}{46189 c^5 e^2 (d+e x)^{3/2}}-\frac{32 (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-12 b e g+5 c d g+19 c e f)}{12597 c^4 e^2 \sqrt{d+e x}}-\frac{4 \sqrt{d+e x} (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-12 b e g+5 c d g+19 c e f)}{969 c^3 e^2}-\frac{2 (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-12 b e g+5 c d g+19 c e f)}{323 c^2 e^2}-\frac{2 g (d+e x)^{5/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{19 c e^2} \]

[Out]

(-512*(2*c*d - b*e)^5*(19*c*e*f + 5*c*d*g - 12*b*e*g)*(d*(c*d - b*e) - b*e^2*x -
c*e^2*x^2)^(7/2))/(2909907*c^7*e^2*(d + e*x)^(7/2)) - (256*(2*c*d - b*e)^4*(19*
c*e*f + 5*c*d*g - 12*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(415701
*c^6*e^2*(d + e*x)^(5/2)) - (64*(2*c*d - b*e)^3*(19*c*e*f + 5*c*d*g - 12*b*e*g)*
(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(46189*c^5*e^2*(d + e*x)^(3/2)) - (
32*(2*c*d - b*e)^2*(19*c*e*f + 5*c*d*g - 12*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*
e^2*x^2)^(7/2))/(12597*c^4*e^2*Sqrt[d + e*x]) - (4*(2*c*d - b*e)*(19*c*e*f + 5*c
*d*g - 12*b*e*g)*Sqrt[d + e*x]*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(969
*c^3*e^2) - (2*(19*c*e*f + 5*c*d*g - 12*b*e*g)*(d + e*x)^(3/2)*(d*(c*d - b*e) -
b*e^2*x - c*e^2*x^2)^(7/2))/(323*c^2*e^2) - (2*g*(d + e*x)^(5/2)*(d*(c*d - b*e)
- b*e^2*x - c*e^2*x^2)^(7/2))/(19*c*e^2)

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Rubi [A]  time = 1.9061, antiderivative size = 501, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 3, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065 \[ -\frac{512 (2 c d-b e)^5 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-12 b e g+5 c d g+19 c e f)}{2909907 c^7 e^2 (d+e x)^{7/2}}-\frac{256 (2 c d-b e)^4 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-12 b e g+5 c d g+19 c e f)}{415701 c^6 e^2 (d+e x)^{5/2}}-\frac{64 (2 c d-b e)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-12 b e g+5 c d g+19 c e f)}{46189 c^5 e^2 (d+e x)^{3/2}}-\frac{32 (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-12 b e g+5 c d g+19 c e f)}{12597 c^4 e^2 \sqrt{d+e x}}-\frac{4 \sqrt{d+e x} (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-12 b e g+5 c d g+19 c e f)}{969 c^3 e^2}-\frac{2 (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-12 b e g+5 c d g+19 c e f)}{323 c^2 e^2}-\frac{2 g (d+e x)^{5/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{19 c e^2} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^(5/2)*(f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2),x]

[Out]

(-512*(2*c*d - b*e)^5*(19*c*e*f + 5*c*d*g - 12*b*e*g)*(d*(c*d - b*e) - b*e^2*x -
c*e^2*x^2)^(7/2))/(2909907*c^7*e^2*(d + e*x)^(7/2)) - (256*(2*c*d - b*e)^4*(19*
c*e*f + 5*c*d*g - 12*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(415701
*c^6*e^2*(d + e*x)^(5/2)) - (64*(2*c*d - b*e)^3*(19*c*e*f + 5*c*d*g - 12*b*e*g)*
(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(46189*c^5*e^2*(d + e*x)^(3/2)) - (
32*(2*c*d - b*e)^2*(19*c*e*f + 5*c*d*g - 12*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*
e^2*x^2)^(7/2))/(12597*c^4*e^2*Sqrt[d + e*x]) - (4*(2*c*d - b*e)*(19*c*e*f + 5*c
*d*g - 12*b*e*g)*Sqrt[d + e*x]*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(969
*c^3*e^2) - (2*(19*c*e*f + 5*c*d*g - 12*b*e*g)*(d + e*x)^(3/2)*(d*(c*d - b*e) -
b*e^2*x - c*e^2*x^2)^(7/2))/(323*c^2*e^2) - (2*g*(d + e*x)^(5/2)*(d*(c*d - b*e)
- b*e^2*x - c*e^2*x^2)^(7/2))/(19*c*e^2)

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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((e*x+d)**(5/2)*(g*x+f)*(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(5/2),x)

[Out]

Timed out

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Mathematica [A]  time = 1.40275, size = 484, normalized size = 0.97 \[ \frac{2 (b e-c d+c e x)^3 \sqrt{(d+e x) (c (d-e x)-b e)} \left (3072 b^6 e^6 g-256 b^5 c e^5 (167 d g+19 e f+42 e g x)+128 b^4 c^2 e^4 \left (1956 d^2 g+d e (513 f+1085 g x)+7 e^2 x (19 f+27 g x)\right )-32 b^3 c^3 e^3 \left (24701 d^3 g+d^2 e (11533 f+23044 g x)+7 d e^2 x (950 f+1287 g x)+63 e^3 x^2 (19 f+22 g x)\right )+8 b^2 c^4 e^2 \left (177311 d^4 g+2 d^3 e (68609 f+126819 g x)+42 d^2 e^2 x (3211 f+4080 g x)+42 d e^3 x^2 (1311 f+1441 g x)+231 e^4 x^3 (38 f+39 g x)\right )-2 b c^5 e \left (682101 d^5 g+d^4 e (894273 f+1467802 g x)+98 d^3 e^2 x (14098 f+16299 g x)+126 d^2 e^3 x^2 (7885 f+8052 g x)+1617 d e^4 x^3 (228 f+221 g x)+3003 e^5 x^4 (19 f+18 g x)\right )+c^6 \left (525458 d^6 g+d^5 e (1414759 f+1839103 g x)+7 d^4 e^2 x (499529 f+487215 g x)+42 d^3 e^3 x^2 (100719 f+91135 g x)+462 d^2 e^4 x^3 (6289 f+5590 g x)+3003 d e^5 x^4 (361 f+321 g x)+9009 e^6 x^5 (19 f+17 g x)\right )\right )}{2909907 c^7 e^2 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^(5/2)*(f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2),x]

[Out]

(2*(-(c*d) + b*e + c*e*x)^3*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*(3072*b^6*e^6
*g - 256*b^5*c*e^5*(19*e*f + 167*d*g + 42*e*g*x) + 128*b^4*c^2*e^4*(1956*d^2*g +
7*e^2*x*(19*f + 27*g*x) + d*e*(513*f + 1085*g*x)) - 32*b^3*c^3*e^3*(24701*d^3*g
+ 63*e^3*x^2*(19*f + 22*g*x) + 7*d*e^2*x*(950*f + 1287*g*x) + d^2*e*(11533*f +
23044*g*x)) + 8*b^2*c^4*e^2*(177311*d^4*g + 231*e^4*x^3*(38*f + 39*g*x) + 42*d*e
^3*x^2*(1311*f + 1441*g*x) + 42*d^2*e^2*x*(3211*f + 4080*g*x) + 2*d^3*e*(68609*f
+ 126819*g*x)) - 2*b*c^5*e*(682101*d^5*g + 3003*e^5*x^4*(19*f + 18*g*x) + 1617*
d*e^4*x^3*(228*f + 221*g*x) + 126*d^2*e^3*x^2*(7885*f + 8052*g*x) + 98*d^3*e^2*x
*(14098*f + 16299*g*x) + d^4*e*(894273*f + 1467802*g*x)) + c^6*(525458*d^6*g + 9
009*e^6*x^5*(19*f + 17*g*x) + 3003*d*e^5*x^4*(361*f + 321*g*x) + 462*d^2*e^4*x^3
*(6289*f + 5590*g*x) + 42*d^3*e^3*x^2*(100719*f + 91135*g*x) + 7*d^4*e^2*x*(4995
29*f + 487215*g*x) + d^5*e*(1414759*f + 1839103*g*x))))/(2909907*c^7*e^2*Sqrt[d
+ e*x])

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Maple [A]  time = 0.013, size = 739, normalized size = 1.5 \[{\frac{ \left ( 2\,cex+2\,be-2\,cd \right ) \left ( 153153\,g{e}^{6}{x}^{6}{c}^{6}-108108\,b{c}^{5}{e}^{6}g{x}^{5}+963963\,{c}^{6}d{e}^{5}g{x}^{5}+171171\,{c}^{6}{e}^{6}f{x}^{5}+72072\,{b}^{2}{c}^{4}{e}^{6}g{x}^{4}-714714\,b{c}^{5}d{e}^{5}g{x}^{4}-114114\,b{c}^{5}{e}^{6}f{x}^{4}+2582580\,{c}^{6}{d}^{2}{e}^{4}g{x}^{4}+1084083\,{c}^{6}d{e}^{5}f{x}^{4}-44352\,{b}^{3}{c}^{3}{e}^{6}g{x}^{3}+484176\,{b}^{2}{c}^{4}d{e}^{5}g{x}^{3}+70224\,{b}^{2}{c}^{4}{e}^{6}f{x}^{3}-2029104\,b{c}^{5}{d}^{2}{e}^{4}g{x}^{3}-737352\,b{c}^{5}d{e}^{5}f{x}^{3}+3827670\,{c}^{6}{d}^{3}{e}^{3}g{x}^{3}+2905518\,{c}^{6}{d}^{2}{e}^{4}f{x}^{3}+24192\,{b}^{4}{c}^{2}{e}^{6}g{x}^{2}-288288\,{b}^{3}{c}^{3}d{e}^{5}g{x}^{2}-38304\,{b}^{3}{c}^{3}{e}^{6}f{x}^{2}+1370880\,{b}^{2}{c}^{4}{d}^{2}{e}^{4}g{x}^{2}+440496\,{b}^{2}{c}^{4}d{e}^{5}f{x}^{2}-3194604\,b{c}^{5}{d}^{3}{e}^{3}g{x}^{2}-1987020\,b{c}^{5}{d}^{2}{e}^{4}f{x}^{2}+3410505\,{c}^{6}{d}^{4}{e}^{2}g{x}^{2}+4230198\,{c}^{6}{d}^{3}{e}^{3}f{x}^{2}-10752\,{b}^{5}c{e}^{6}gx+138880\,{b}^{4}{c}^{2}d{e}^{5}gx+17024\,{b}^{4}{c}^{2}{e}^{6}fx-737408\,{b}^{3}{c}^{3}{d}^{2}{e}^{4}gx-212800\,{b}^{3}{c}^{3}d{e}^{5}fx+2029104\,{b}^{2}{c}^{4}{d}^{3}{e}^{3}gx+1078896\,{b}^{2}{c}^{4}{d}^{2}{e}^{4}fx-2935604\,b{c}^{5}{d}^{4}{e}^{2}gx-2763208\,b{c}^{5}{d}^{3}{e}^{3}fx+1839103\,{c}^{6}{d}^{5}egx+3496703\,{c}^{6}{d}^{4}{e}^{2}fx+3072\,{b}^{6}{e}^{6}g-42752\,{b}^{5}cd{e}^{5}g-4864\,{b}^{5}c{e}^{6}f+250368\,{b}^{4}{c}^{2}{d}^{2}{e}^{4}g+65664\,{b}^{4}{c}^{2}d{e}^{5}f-790432\,{b}^{3}{c}^{3}{d}^{3}{e}^{3}g-369056\,{b}^{3}{c}^{3}{d}^{2}{e}^{4}f+1418488\,{b}^{2}{c}^{4}{d}^{4}{e}^{2}g+1097744\,{b}^{2}{c}^{4}{d}^{3}{e}^{3}f-1364202\,b{c}^{5}{d}^{5}eg-1788546\,b{c}^{5}{d}^{4}{e}^{2}f+525458\,{c}^{6}{d}^{6}g+1414759\,f{d}^{5}{c}^{6}e \right ) }{2909907\,{c}^{7}{e}^{2}} \left ( -c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2} \right ) ^{{\frac{5}{2}}} \left ( ex+d \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^(5/2)*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x)

[Out]

2/2909907*(c*e*x+b*e-c*d)*(153153*c^6*e^6*g*x^6-108108*b*c^5*e^6*g*x^5+963963*c^
6*d*e^5*g*x^5+171171*c^6*e^6*f*x^5+72072*b^2*c^4*e^6*g*x^4-714714*b*c^5*d*e^5*g*
x^4-114114*b*c^5*e^6*f*x^4+2582580*c^6*d^2*e^4*g*x^4+1084083*c^6*d*e^5*f*x^4-443
52*b^3*c^3*e^6*g*x^3+484176*b^2*c^4*d*e^5*g*x^3+70224*b^2*c^4*e^6*f*x^3-2029104*
b*c^5*d^2*e^4*g*x^3-737352*b*c^5*d*e^5*f*x^3+3827670*c^6*d^3*e^3*g*x^3+2905518*c
^6*d^2*e^4*f*x^3+24192*b^4*c^2*e^6*g*x^2-288288*b^3*c^3*d*e^5*g*x^2-38304*b^3*c^
3*e^6*f*x^2+1370880*b^2*c^4*d^2*e^4*g*x^2+440496*b^2*c^4*d*e^5*f*x^2-3194604*b*c
^5*d^3*e^3*g*x^2-1987020*b*c^5*d^2*e^4*f*x^2+3410505*c^6*d^4*e^2*g*x^2+4230198*c
^6*d^3*e^3*f*x^2-10752*b^5*c*e^6*g*x+138880*b^4*c^2*d*e^5*g*x+17024*b^4*c^2*e^6*
f*x-737408*b^3*c^3*d^2*e^4*g*x-212800*b^3*c^3*d*e^5*f*x+2029104*b^2*c^4*d^3*e^3*
g*x+1078896*b^2*c^4*d^2*e^4*f*x-2935604*b*c^5*d^4*e^2*g*x-2763208*b*c^5*d^3*e^3*
f*x+1839103*c^6*d^5*e*g*x+3496703*c^6*d^4*e^2*f*x+3072*b^6*e^6*g-42752*b^5*c*d*e
^5*g-4864*b^5*c*e^6*f+250368*b^4*c^2*d^2*e^4*g+65664*b^4*c^2*d*e^5*f-790432*b^3*
c^3*d^3*e^3*g-369056*b^3*c^3*d^2*e^4*f+1418488*b^2*c^4*d^4*e^2*g+1097744*b^2*c^4
*d^3*e^3*f-1364202*b*c^5*d^5*e*g-1788546*b*c^5*d^4*e^2*f+525458*c^6*d^6*g+141475
9*c^6*d^5*e*f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/c^7/e^2/(e*x+d)^(5/2)

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Maxima [A]  time = 0.78288, size = 1841, normalized size = 3.67 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/153153*(9009*c^8*e^8*x^8 - 74461*c^8*d^8 + 317517*b*c^7*d^7*e - 563561*b^2*c^6
*d^6*e^2 + 549615*b^3*c^5*d^5*e^3 - 329190*b^4*c^4*d^4*e^4 + 126672*b^5*c^3*d^3*
e^5 - 30560*b^6*c^2*d^2*e^6 + 4224*b^7*c*d*e^7 - 256*b^8*e^8 + 3003*(10*c^8*d*e^
7 + 7*b*c^7*e^8)*x^7 + 231*(38*c^8*d^2*e^6 + 417*b*c^7*d*e^7 + 55*b^2*c^6*e^8)*x
^6 - 63*(1174*c^8*d^3*e^5 - 2179*b*c^7*d^2*e^6 - 1204*b^2*c^6*d*e^7 - b^3*c^5*e^
8)*x^5 - 35*(2348*c^8*d^4*e^4 + 587*b*c^7*d^3*e^5 - 5343*b^2*c^6*d^2*e^6 - 25*b^
3*c^5*d*e^7 + 2*b^4*c^4*e^8)*x^4 + (37354*c^8*d^5*e^3 - 257745*b*c^7*d^4*e^4 + 2
37200*b^2*c^6*d^3*e^5 + 6070*b^3*c^5*d^2*e^6 - 1080*b^4*c^4*d*e^7 + 80*b^5*c^3*e
^8)*x^3 + 3*(35362*c^8*d^6*e^2 - 87409*b*c^7*d^5*e^3 + 44825*b^2*c^6*d^4*e^4 + 9
650*b^3*c^5*d^3*e^5 - 2860*b^4*c^4*d^2*e^6 + 464*b^5*c^3*d*e^7 - 32*b^6*c^2*e^8)
*x^2 + (39346*c^8*d^7*e - 31625*b*c^7*d^6*e^2 - 83676*b^2*c^6*d^5*e^3 + 114555*b
^3*c^5*d^4*e^4 - 50040*b^4*c^4*d^3*e^5 + 13296*b^5*c^3*d^2*e^6 - 1984*b^6*c^2*d*
e^7 + 128*b^7*c*e^8)*x)*sqrt(-c*e*x + c*d - b*e)*(e*x + d)*f/(c^6*e^2*x + c^6*d*
e) + 2/2909907*(153153*c^9*e^9*x^9 - 525458*c^9*d^9 + 2940576*b*c^8*d^8*e - 7087
468*b^2*c^7*d^7*e^2 + 9663960*b^3*c^6*d^6*e^3 - 8241330*b^4*c^5*d^5*e^4 + 458364
0*b^5*c^4*d^4*e^5 - 1672864*b^6*c^3*d^3*e^6 + 387840*b^7*c^2*d^2*e^7 - 51968*b^8
*c*d*e^8 + 3072*b^9*e^9 + 9009*(56*c^9*d*e^8 + 39*b*c^8*e^9)*x^8 + 3003*(50*c^9*
d^2*e^7 + 527*b*c^8*d*e^8 + 69*b^2*c^7*e^9)*x^7 - 231*(5114*c^9*d^3*e^6 - 9585*b
*c^8*d^2*e^7 - 5216*b^2*c^7*d*e^8 - 3*b^3*c^6*e^9)*x^6 - 63*(20456*c^9*d^4*e^5 +
4189*b*c^8*d^3*e^6 - 45509*b^2*c^7*d^2*e^7 - 143*b^3*c^6*d*e^8 + 12*b^4*c^5*e^9
)*x^5 + 7*(72574*c^9*d^5*e^4 - 530165*b*c^8*d^4*e^5 + 496980*b^2*c^7*d^3*e^6 + 8
230*b^3*c^6*d^2*e^7 - 1550*b^4*c^5*d*e^8 + 120*b^5*c^4*e^9)*x^4 + (1411994*c^9*d
^6*e^3 - 3574809*b*c^8*d^5*e^4 + 1981645*b^2*c^7*d^4*e^5 + 247010*b^3*c^6*d^3*e^
6 - 78240*b^4*c^5*d^2*e^7 + 13360*b^5*c^4*d*e^8 - 960*b^6*c^3*e^9)*x^3 + 3*(1768
10*c^9*d^7*e^2 - 248777*b*c^8*d^6*e^3 - 105344*b^2*c^7*d^5*e^4 + 276115*b^3*c^6*
d^4*e^5 - 130100*b^4*c^5*d^3*e^6 + 36640*b^5*c^4*d^2*e^7 - 5728*b^6*c^3*d*e^8 +
384*b^7*c^2*e^9)*x^2 - (262729*c^9*d^8*e - 1207559*b*c^8*d^7*e^2 + 2336175*b^2*c
^7*d^6*e^3 - 2495805*b^3*c^6*d^5*e^4 + 1624860*b^4*c^5*d^4*e^5 - 666960*b^5*c^4*
d^3*e^6 + 169472*b^6*c^3*d^2*e^7 - 24448*b^7*c^2*d*e^8 + 1536*b^8*c*e^9)*x)*sqrt
(-c*e*x + c*d - b*e)*(e*x + d)*g/(c^7*e^3*x + c^7*d*e^2)

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Fricas [A]  time = 0.388401, size = 2601, normalized size = 5.19 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/2909907*(153153*c^10*e^11*g*x^11 + 9009*(19*c^10*e^11*f + 56*(c^10*d*e^10 + b
*c^9*e^11)*g)*x^10 + 3003*(190*(c^10*d*e^10 + b*c^9*e^11)*f - (c^10*d^2*e^9 - 74
6*b*c^9*d*e^10 - 186*b^2*c^8*e^11)*g)*x^9 - 231*(19*(c^10*d^2*e^9 - 586*b*c^9*d*
e^10 - 146*b^2*c^8*e^11)*f + 2*(3649*c^10*d^3*e^8 - 5449*b*c^9*d^2*e^9 - 6794*b^
2*c^8*d*e^10 - 450*b^3*c^7*e^11)*g)*x^8 - 21*(152*(619*c^10*d^3*e^8 - 923*b*c^9*
d^2*e^9 - 1150*b^2*c^8*d*e^10 - 76*b^3*c^7*e^11)*f + (68518*c^10*d^4*e^7 + 13703
2*b*c^9*d^3*e^8 - 307456*b^2*c^8*d^2*e^9 - 67672*b^3*c^7*d*e^10 + 3*b^4*c^6*e^11
)*g)*x^7 - 7*(19*(12994*c^10*d^4*e^7 + 26008*b*c^9*d^3*e^8 - 58272*b^2*c^8*d^2*e
^9 - 12776*b^3*c^7*d*e^10 + b^4*c^6*e^11)*f - 4*(60334*c^10*d^5*e^6 - 299834*b*c
^9*d^4*e^7 + 150864*b^2*c^8*d^3*e^8 + 147460*b^3*c^7*d^2*e^9 - 41*b^4*c^6*d*e^10
+ 3*b^5*c^5*e^11)*g)*x^6 + (38*(55658*c^10*d^5*e^6 - 275582*b*c^9*d^4*e^7 + 139
040*b^2*c^8*d^3*e^8 + 134432*b^3*c^7*d^2*e^9 - 71*b^4*c^6*d*e^10 + 5*b^5*c^5*e^1
1)*f + (2700722*c^10*d^6*e^5 - 4091612*b*c^9*d^5*e^6 - 4860484*b^2*c^8*d^4*e^7 +
6583928*b^3*c^7*d^3*e^8 - 10865*b^4*c^6*d^2*e^9 + 1754*b^5*c^5*d*e^10 - 120*b^6
*c^4*e^11)*g)*x^5 + (19*(188266*c^10*d^6*e^5 - 286508*b*c^9*d^5*e^6 - 330820*b^2
*c^8*d^4*e^7 + 452280*b^3*c^7*d^3*e^8 - 1565*b^4*c^6*d^2*e^9 + 242*b^5*c^5*d*e^1
0 - 16*b^6*c^4*e^11)*f + 2*(11206*c^10*d^7*e^4 + 2442418*b*c^9*d^6*e^5 - 5540428
*b^2*c^8*d^5*e^6 + 3115620*b^3*c^7*d^4*e^7 - 37415*b^4*c^6*d^3*e^8 + 9995*b^5*c^
5*d^2*e^9 - 1492*b^6*c^4*d*e^10 + 96*b^7*c^3*e^11)*g)*x^4 + (76*(498*c^10*d^7*e^
4 + 92390*b*c^9*d^6*e^5 - 210212*b^2*c^8*d^5*e^6 + 120040*b^3*c^7*d^4*e^7 - 3485
*b^4*c^6*d^3*e^8 + 889*b^5*c^5*d^2*e^9 - 128*b^6*c^4*d*e^10 + 8*b^7*c^3*e^11)*f
- (1674723*c^10*d^8*e^3 - 6724792*b*c^9*d^7*e^4 + 8638960*b^2*c^8*d^6*e^5 - 3914
408*b^3*c^7*d^5*e^6 + 471265*b^4*c^6*d^4*e^7 - 185060*b^5*c^5*d^3*e^8 + 45232*b^
6*c^4*d^2*e^9 - 6304*b^7*c^3*d*e^10 + 384*b^8*c^2*e^11)*g)*x^3 - (19*(180547*c^1
0*d^8*e^3 - 725176*b*c^9*d^7*e^4 + 991888*b^2*c^8*d^6*e^5 - 571464*b^3*c^7*d^5*e
^6 + 177105*b^4*c^6*d^4*e^7 - 66660*b^5*c^5*d^3*e^8 + 15776*b^6*c^4*d^2*e^9 - 21
44*b^7*c^3*d*e^10 + 128*b^8*c^2*e^11)*f + 16*(65993*c^10*d^9*e^2 - 247163*b*c^9*
d^8*e^3 + 394388*b^2*c^8*d^7*e^4 - 386463*b^3*c^7*d^6*e^5 + 282930*b^4*c^6*d^5*e
^6 - 153660*b^5*c^5*d^4*e^7 + 54925*b^6*c^4*d^3*e^8 - 12502*b^7*c^3*d^2*e^9 + 16
48*b^8*c^2*d*e^10 - 96*b^9*c*e^11)*g)*x^2 + 19*(74461*c^10*d^10*e - 391978*b*c^9
*d^9*e^2 + 881078*b^2*c^8*d^8*e^3 - 1113176*b^3*c^7*d^7*e^4 + 878805*b^4*c^6*d^6
*e^5 - 455862*b^5*c^5*d^5*e^6 + 157232*b^6*c^4*d^4*e^7 - 34784*b^7*c^3*d^3*e^8 +
4480*b^8*c^2*d^2*e^9 - 256*b^9*c*d*e^10)*f + 2*(262729*c^10*d^11 - 1733017*b*c^
9*d^10*e + 5014022*b^2*c^8*d^9*e^2 - 8375714*b^3*c^7*d^8*e^3 + 8952645*b^4*c^6*d
^7*e^4 - 6412485*b^5*c^5*d^6*e^5 + 3128252*b^6*c^4*d^5*e^6 - 1030352*b^7*c^3*d^4
*e^7 + 219904*b^8*c^2*d^3*e^8 - 27520*b^9*c*d^2*e^9 + 1536*b^10*d*e^10)*g - (38*
(19673*c^10*d^9*e^2 + 1745*b*c^9*d^8*e^3 - 184784*b^2*c^8*d^7*e^4 + 380896*b^3*c
^7*d^6*e^5 - 357105*b^4*c^6*d^5*e^6 + 196263*b^5*c^5*d^4*e^7 - 70976*b^6*c^4*d^3
*e^8 + 16336*b^7*c^3*d^2*e^9 - 2176*b^8*c^2*d*e^10 + 128*b^9*c*e^11)*f - (262729
*c^10*d^10*e - 1995746*b*c^9*d^9*e^2 + 6484310*b^2*c^8*d^8*e^3 - 11919448*b^3*c^
7*d^7*e^4 + 13784625*b^4*c^6*d^6*e^5 - 10533150*b^5*c^5*d^5*e^6 + 5420072*b^6*c^
4*d^4*e^7 - 1866784*b^7*c^3*d^3*e^8 + 413824*b^8*c^2*d^2*e^9 - 53504*b^9*c*d*e^1
0 + 3072*b^10*e^11)*g)*x)/(sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*sqrt(e*x +
d)*c^7*e^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**(5/2)*(g*x+f)*(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)^(5/2)*(e*x + d)^(5/2)*(g*x + f),x, algorithm="giac")

[Out]

Timed out