3.2196 \(\int (d+e x)^2 (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2} \, dx\)

Optimal. Leaf size=487 \[ \frac{5 (2 c d-b e)^8 (-11 b e g+4 c d g+18 c e f) \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{65536 c^{13/2} e^2}+\frac{5 (b+2 c x) (2 c d-b e)^6 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-11 b e g+4 c d g+18 c e f)}{32768 c^6 e}+\frac{5 (b+2 c x) (2 c d-b e)^4 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-11 b e g+4 c d g+18 c e f)}{12288 c^5 e}+\frac{(b+2 c x) (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-11 b e g+4 c d g+18 c e f)}{768 c^4 e}-\frac{(2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-11 b e g+4 c d g+18 c e f)}{224 c^3 e^2}-\frac{(d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-11 b e g+4 c d g+18 c e f)}{144 c^2 e^2}-\frac{g (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{9 c e^2} \]

[Out]

(5*(2*c*d - b*e)^6*(18*c*e*f + 4*c*d*g - 11*b*e*g)*(b + 2*c*x)*Sqrt[d*(c*d - b*e
) - b*e^2*x - c*e^2*x^2])/(32768*c^6*e) + (5*(2*c*d - b*e)^4*(18*c*e*f + 4*c*d*g
- 11*b*e*g)*(b + 2*c*x)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(12288*c^5
*e) + ((2*c*d - b*e)^2*(18*c*e*f + 4*c*d*g - 11*b*e*g)*(b + 2*c*x)*(d*(c*d - b*e
) - b*e^2*x - c*e^2*x^2)^(5/2))/(768*c^4*e) - ((2*c*d - b*e)*(18*c*e*f + 4*c*d*g
- 11*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(224*c^3*e^2) - ((18*c
*e*f + 4*c*d*g - 11*b*e*g)*(d + e*x)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2)
)/(144*c^2*e^2) - (g*(d + e*x)^2*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(9
*c*e^2) + (5*(2*c*d - b*e)^8*(18*c*e*f + 4*c*d*g - 11*b*e*g)*ArcTan[(e*(b + 2*c*
x))/(2*Sqrt[c]*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])])/(65536*c^(13/2)*e^2)

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Rubi [A]  time = 1.36789, antiderivative size = 487, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 44, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.159 \[ \frac{5 (2 c d-b e)^8 (-11 b e g+4 c d g+18 c e f) \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{65536 c^{13/2} e^2}+\frac{5 (b+2 c x) (2 c d-b e)^6 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-11 b e g+4 c d g+18 c e f)}{32768 c^6 e}+\frac{5 (b+2 c x) (2 c d-b e)^4 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-11 b e g+4 c d g+18 c e f)}{12288 c^5 e}+\frac{(b+2 c x) (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-11 b e g+4 c d g+18 c e f)}{768 c^4 e}-\frac{(2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-11 b e g+4 c d g+18 c e f)}{224 c^3 e^2}-\frac{(d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-11 b e g+4 c d g+18 c e f)}{144 c^2 e^2}-\frac{g (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{9 c e^2} \]

Antiderivative was successfully verified.

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

[Out]

(5*(2*c*d - b*e)^6*(18*c*e*f + 4*c*d*g - 11*b*e*g)*(b + 2*c*x)*Sqrt[d*(c*d - b*e
) - b*e^2*x - c*e^2*x^2])/(32768*c^6*e) + (5*(2*c*d - b*e)^4*(18*c*e*f + 4*c*d*g
- 11*b*e*g)*(b + 2*c*x)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(12288*c^5
*e) + ((2*c*d - b*e)^2*(18*c*e*f + 4*c*d*g - 11*b*e*g)*(b + 2*c*x)*(d*(c*d - b*e
) - b*e^2*x - c*e^2*x^2)^(5/2))/(768*c^4*e) - ((2*c*d - b*e)*(18*c*e*f + 4*c*d*g
- 11*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(224*c^3*e^2) - ((18*c
*e*f + 4*c*d*g - 11*b*e*g)*(d + e*x)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2)
)/(144*c^2*e^2) - (g*(d + e*x)^2*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(9
*c*e^2) + (5*(2*c*d - b*e)^8*(18*c*e*f + 4*c*d*g - 11*b*e*g)*ArcTan[(e*(b + 2*c*
x))/(2*Sqrt[c]*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])])/(65536*c^(13/2)*e^2)

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Rubi in Sympy [A]  time = 153.905, size = 474, normalized size = 0.97 \[ - \frac{g \left (d + e x\right )^{2} \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{7}{2}}}{9 c e^{2}} + \frac{\left (d + e x\right ) \left (11 b e g - 4 c d g - 18 c e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{7}{2}}}{144 c^{2} e^{2}} - \frac{\left (b e - 2 c d\right ) \left (11 b e g - 4 c d g - 18 c e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{7}{2}}}{224 c^{3} e^{2}} - \frac{\left (b + 2 c x\right ) \left (b e - 2 c d\right )^{2} \left (11 b e g - 4 c d g - 18 c e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{5}{2}}}{768 c^{4} e} - \frac{5 \left (b + 2 c x\right ) \left (b e - 2 c d\right )^{4} \left (11 b e g - 4 c d g - 18 c e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{3}{2}}}{12288 c^{5} e} - \frac{5 \left (b + 2 c x\right ) \left (b e - 2 c d\right )^{6} \left (11 b e g - 4 c d g - 18 c e f\right ) \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}}{32768 c^{6} e} - \frac{5 \left (b e - 2 c d\right )^{8} \left (11 b e g - 4 c d g - 18 c e f\right ) \operatorname{atan}{\left (- \frac{e \left (- b - 2 c x\right )}{2 \sqrt{c} \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}} \right )}}{65536 c^{\frac{13}{2}} e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-g*(d + e*x)**2*(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))**(7/2)/(9*c*e**2) + (
d + e*x)*(11*b*e*g - 4*c*d*g - 18*c*e*f)*(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*
d))**(7/2)/(144*c**2*e**2) - (b*e - 2*c*d)*(11*b*e*g - 4*c*d*g - 18*c*e*f)*(-b*e
**2*x - c*e**2*x**2 + d*(-b*e + c*d))**(7/2)/(224*c**3*e**2) - (b + 2*c*x)*(b*e
- 2*c*d)**2*(11*b*e*g - 4*c*d*g - 18*c*e*f)*(-b*e**2*x - c*e**2*x**2 + d*(-b*e +
c*d))**(5/2)/(768*c**4*e) - 5*(b + 2*c*x)*(b*e - 2*c*d)**4*(11*b*e*g - 4*c*d*g
- 18*c*e*f)*(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))**(3/2)/(12288*c**5*e) - 5
*(b + 2*c*x)*(b*e - 2*c*d)**6*(11*b*e*g - 4*c*d*g - 18*c*e*f)*sqrt(-b*e**2*x - c
*e**2*x**2 + d*(-b*e + c*d))/(32768*c**6*e) - 5*(b*e - 2*c*d)**8*(11*b*e*g - 4*c
*d*g - 18*c*e*f)*atan(-e*(-b - 2*c*x)/(2*sqrt(c)*sqrt(-b*e**2*x - c*e**2*x**2 +
d*(-b*e + c*d))))/(65536*c**(13/2)*e**2)

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Mathematica [C]  time = 6.54096, size = 1226, normalized size = 2.52 \[ \frac{5 i (18 c e f+4 c d g-11 b e g) ((d+e x) (c (d-e x)-b e))^{5/2} \log \left (2 \sqrt{d+e x} \sqrt{c d-b e-c e x}-\frac{i e (b+2 c x)}{\sqrt{c}}\right ) (b e-2 c d)^8}{65536 c^{13/2} e^2 (d+e x)^{5/2} (c d-b e-c e x)^{5/2}}+\frac{\left (\frac{1}{9} c^2 e^6 g x^8+\frac{1}{144} c e^5 (18 c e f+36 c d g+37 b e g) x^7+\frac{e^4 \left (576 d e f c^2-320 d^2 g c^2+594 b e^2 f c+1796 b d e g c+309 b^2 e^2 g\right ) x^6}{2016}+\frac{e^3 \left (-1512 d^2 e f c^3-5712 d^3 g c^3+8424 b d e^2 f c^2+5484 b d^2 e g c^2+1458 b^2 e^3 f c+5928 b^2 d e^2 g c+5 b^3 e^3 g\right ) x^5}{8064 c}+\frac{e^2 \left (-13824 d^3 e f c^4-3072 d^4 g c^4+13176 b d^2 e^2 f c^3-15504 b d^3 e g c^3+14472 b^2 d e^3 f c^2+22044 b^2 d^2 e^2 g c^2+18 b^3 e^4 f c+120 b^3 d e^3 g c-11 b^4 e^4 g\right ) x^4}{16128 c^2}-\frac{e \left (30240 d^4 e f c^5-79296 d^5 g c^5+160704 b d^3 e^2 f c^4+232272 b d^4 e g c^4-225936 b^2 d^2 e^3 f c^3-148416 b^2 d^3 e^2 g c^3-1872 b^3 d e^4 f c^2-5704 b^3 d^2 e^3 g c^2+162 b^4 e^5 f c+1180 b^4 d e^4 g c-99 b^5 e^5 g\right ) x^3}{129024 c^3}+\frac{\left (221184 d^5 e f c^6+106496 d^6 g c^6-643680 b d^4 e^2 f c^5-192192 b d^5 e g c^5+402624 b^2 d^3 e^3 f c^4+52752 b^2 d^4 e^2 g c^4+24624 b^3 d^2 e^4 f c^3+46144 b^3 d^3 e^3 g c^3-4752 b^4 d e^5 f c^2-16104 b^4 d^2 e^4 g c^2+378 b^5 e^6 f c+2988 b^5 d e^5 g c-231 b^6 e^6 g\right ) x^2}{258048 c^4}-\frac{\left (-669312 d^6 e f c^7+80640 d^7 g c^7+1123200 b d^5 e^2 f c^6-373568 b d^6 e g c^6-116640 b^2 d^4 e^3 f c^5+663936 b^2 d^5 e^2 g c^5-459072 b^3 d^3 e^4 f c^4-604176 b^3 d^4 e^3 g c^4+147528 b^4 d^2 e^5 f c^3+313328 b^4 d^3 e^4 g c^3-25704 b^5 d e^6 f c^2-95868 b^5 d^2 e^5 g c^2+1890 b^6 e^7 f c+16128 b^6 d e^6 g c-1155 b^7 e^7 g\right ) x}{1032192 c^5 e}+\frac{-589824 d^7 e f c^8-360448 d^8 g c^8+2733696 b d^6 e^2 f c^7+1656064 b d^7 e g c^7-4662144 b^2 d^5 e^3 f c^6-3394752 b^2 d^6 e^2 g c^6+3924000 b^3 d^4 e^4 f c^5+3950464 b^3 d^5 e^3 g c^5-1847232 b^4 d^3 e^5 f c^4-2808496 b^4 d^4 e^4 g c^4+524664 b^5 d^2 e^6 f c^3+1245456 b^5 d^3 e^5 g c^3-83160 b^6 d e^7 f c^2-339108 b^6 d^2 e^6 g c^2+5670 b^7 e^8 f c+52080 b^7 d e^7 g c-3465 b^8 e^8 g}{2064384 c^6 e^2}\right ) ((d+e x) (c (d-e x)-b e))^{5/2}}{(d+e x)^2 (c d-b e-c e x)^2} \]

Antiderivative was successfully verified.

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

[Out]

(((-589824*c^8*d^7*e*f + 2733696*b*c^7*d^6*e^2*f - 4662144*b^2*c^6*d^5*e^3*f + 3
924000*b^3*c^5*d^4*e^4*f - 1847232*b^4*c^4*d^3*e^5*f + 524664*b^5*c^3*d^2*e^6*f
- 83160*b^6*c^2*d*e^7*f + 5670*b^7*c*e^8*f - 360448*c^8*d^8*g + 1656064*b*c^7*d^
7*e*g - 3394752*b^2*c^6*d^6*e^2*g + 3950464*b^3*c^5*d^5*e^3*g - 2808496*b^4*c^4*
d^4*e^4*g + 1245456*b^5*c^3*d^3*e^5*g - 339108*b^6*c^2*d^2*e^6*g + 52080*b^7*c*d
*e^7*g - 3465*b^8*e^8*g)/(2064384*c^6*e^2) - ((-669312*c^7*d^6*e*f + 1123200*b*c
^6*d^5*e^2*f - 116640*b^2*c^5*d^4*e^3*f - 459072*b^3*c^4*d^3*e^4*f + 147528*b^4*
c^3*d^2*e^5*f - 25704*b^5*c^2*d*e^6*f + 1890*b^6*c*e^7*f + 80640*c^7*d^7*g - 373
568*b*c^6*d^6*e*g + 663936*b^2*c^5*d^5*e^2*g - 604176*b^3*c^4*d^4*e^3*g + 313328
*b^4*c^3*d^3*e^4*g - 95868*b^5*c^2*d^2*e^5*g + 16128*b^6*c*d*e^6*g - 1155*b^7*e^
7*g)*x)/(1032192*c^5*e) + ((221184*c^6*d^5*e*f - 643680*b*c^5*d^4*e^2*f + 402624
*b^2*c^4*d^3*e^3*f + 24624*b^3*c^3*d^2*e^4*f - 4752*b^4*c^2*d*e^5*f + 378*b^5*c*
e^6*f + 106496*c^6*d^6*g - 192192*b*c^5*d^5*e*g + 52752*b^2*c^4*d^4*e^2*g + 4614
4*b^3*c^3*d^3*e^3*g - 16104*b^4*c^2*d^2*e^4*g + 2988*b^5*c*d*e^5*g - 231*b^6*e^6
*g)*x^2)/(258048*c^4) - (e*(30240*c^5*d^4*e*f + 160704*b*c^4*d^3*e^2*f - 225936*
b^2*c^3*d^2*e^3*f - 1872*b^3*c^2*d*e^4*f + 162*b^4*c*e^5*f - 79296*c^5*d^5*g + 2
32272*b*c^4*d^4*e*g - 148416*b^2*c^3*d^3*e^2*g - 5704*b^3*c^2*d^2*e^3*g + 1180*b
^4*c*d*e^4*g - 99*b^5*e^5*g)*x^3)/(129024*c^3) + (e^2*(-13824*c^4*d^3*e*f + 1317
6*b*c^3*d^2*e^2*f + 14472*b^2*c^2*d*e^3*f + 18*b^3*c*e^4*f - 3072*c^4*d^4*g - 15
504*b*c^3*d^3*e*g + 22044*b^2*c^2*d^2*e^2*g + 120*b^3*c*d*e^3*g - 11*b^4*e^4*g)*
x^4)/(16128*c^2) + (e^3*(-1512*c^3*d^2*e*f + 8424*b*c^2*d*e^2*f + 1458*b^2*c*e^3
*f - 5712*c^3*d^3*g + 5484*b*c^2*d^2*e*g + 5928*b^2*c*d*e^2*g + 5*b^3*e^3*g)*x^5
)/(8064*c) + (e^4*(576*c^2*d*e*f + 594*b*c*e^2*f - 320*c^2*d^2*g + 1796*b*c*d*e*
g + 309*b^2*e^2*g)*x^6)/2016 + (c*e^5*(18*c*e*f + 36*c*d*g + 37*b*e*g)*x^7)/144
+ (c^2*e^6*g*x^8)/9)*((d + e*x)*(-(b*e) + c*(d - e*x)))^(5/2))/((d + e*x)^2*(c*d
- b*e - c*e*x)^2) + (((5*I)/65536)*(-2*c*d + b*e)^8*(18*c*e*f + 4*c*d*g - 11*b*
e*g)*((d + e*x)*(-(b*e) + c*(d - e*x)))^(5/2)*Log[((-I)*e*(b + 2*c*x))/Sqrt[c] +
2*Sqrt[d + e*x]*Sqrt[c*d - b*e - c*e*x]])/(c^(13/2)*e^2*(d + e*x)^(5/2)*(c*d -
b*e - c*e*x)^(5/2))

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Maple [B]  time = 0.025, size = 4332, normalized size = 8.9 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-55/12288*e^4*g*b^6/c^5*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)-55/32768*e^6*g*b^
8/c^6*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)+11/144*g*b/c^2*x*(-c*e^2*x^2-b*e^2*
x-b*d*e+c*d^2)^(7/2)+5/192/e*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*b*d^5*g+45/1
28*d^8*f*c^3/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-
b*d*e+c*d^2)^(1/2))+45/128*d^6*f*c^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x+1/
24/e*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)*x*d^3*g-95/768/c*(-c*e^2*x^2-b*e^2*x
-b*d*e+c*d^2)^(3/2)*b^2*d^4*g-95/384*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*x*b*
d^4*g+3/128*b^3/c^3*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)*e^2*f-1/9*g*x^2*(-c*e
^2*x^2-b*e^2*x-b*d*e+c*d^2)^(7/2)/c-11/224*g*b^2/c^3*(-c*e^2*x^2-b*e^2*x-b*d*e+c
*d^2)^(7/2)+15/128*d^4*f*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*b+3/16*d^2*f*(-c
*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)*x+9/112*b/c^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^
2)^(7/2)*f-1/8*x*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(7/2)/c*f-115/512*(-c*e^2*x^2-
b*e^2*x-b*d*e+c*d^2)^(1/2)*b^2*d^6*g-225/256*d^3*f/c*e^3*(-c*e^2*x^2-b*e^2*x-b*d
*e+c*d^2)^(1/2)*x*b^3-185/768*b^3/c^2*e^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)
*x*d^2*g-15/128*b^3/c^2*e^3*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*x*d*f-885/409
6*b^5/c^3*e^4*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*d^2*g-135/2048*b^5/c^3*e^
5*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*d*f-1025/1024*b^3/c*e^2*(-c*e^2*x^2-b
*e^2*x-b*d*e+c*d^2)^(1/2)*x*d^4*g+35/96*b^2/c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(
3/2)*x*d^3*e*g+625/1024*b^4/c^2*e^3*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*d^3
*g-2205/2048*b^5/c^2*e^4/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*
x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))*d^4*g+945/512*b^4/c*e^3/(c*e^2)^(1/2)*arctan((c*
e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))*d^5*g+45/32*b^2*c
/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)
^(1/2))*d^7*e*g+85/2048*b^6/c^4*e^5*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*d*g
+115/1536*b^4/c^3*e^3*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*x*d*g+1/8*b^2/c^2*(
-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)*x*d*e*g-405/4096*b^7/c^4*e^6/(c*e^2)^(1/2)
*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))*d^2*g-
45/2048*b^7/c^4*e^7/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b
*e^2*x-b*d*e+c*d^2)^(1/2))*d*f+105/256*b^6/c^3*e^5/(c*e^2)^(1/2)*arctan((c*e^2)^
(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))*d^3*g+225/16384*b^8/c^
5*e^7/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c
*d^2)^(1/2))*d*g-3/16/c*e*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)*x*b*d*f-2/7*(-c
*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(7/2)/c/e*d*f-55/16384*e^6*g*b^7/c^5*(-c*e^2*x^2-b
*e^2*x-b*d*e+c*d^2)^(1/2)*x-55/6144*e^4*g*b^5/c^4*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^
2)^(3/2)*x-55/65536*e^8*g*b^9/c^6/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)
/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))-11/384*e^2*g*b^3/c^3*(-c*e^2*x^2-b*e^2*
x-b*d*e+c*d^2)^(5/2)*x-225/512*d^3*f/c^2*e^3*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1
/2)*b^4-15/64*d^3*f/c*e*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*b^2+675/512*d^4*f
*e^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*b^2-315/128*d^5*f*e^3/(c*e^2)^(1/2
)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))*b^3+6
75/4096*d^2*f/c^3*e^4*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*b^5-15/32*d^3*f*e*(
-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*x*b+45/256*d^2*f/c^2*e^2*(-c*e^2*x^2-b*e^2
*x-b*d*e+c*d^2)^(3/2)*b^3+675/1024*d^4*f/c*e^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^
(1/2)*b^3-1025/2048*b^4/c^2*e^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*d^4*g-185
/1536*b^4/c^3*e^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*d^2*g-15/256*b^4/c^3*e^
3*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*d*f+45/8192*b^6/c^4*e^6*(-c*e^2*x^2-b*e
^2*x-b*d*e+c*d^2)^(1/2)*x*f+15/1024*b^4/c^3*e^4*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)
^(3/2)*x*f+45/32768*b^8/c^5*e^8/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(
-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))*f+35/192*b^3/c^2*(-c*e^2*x^2-b*e^2*x-b*d*
e+c*d^2)^(3/2)*d^3*e*g+5/64*c^2/e*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*d^7*g
+1/48/c/e*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)*b*d^3*g-3/32/c^2*e*(-c*e^2*x^2-
b*e^2*x-b*d*e+c*d^2)^(5/2)*b^2*d*f+5/128*c/e*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1
/2)*b*d^7*g-1/4*x*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(7/2)/c/e*d*g+1/16*b^3/c^3*(-
c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)*d*e*g-525/256*b^3*e^2/(c*e^2)^(1/2)*arctan(
(c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))*d^6*g+15/16*b^
2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*d^5*e*g-885/8192*b^6/c^4*e^4*(-c*e^2*
x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*d^2*g-135/4096*b^6/c^4*e^5*(-c*e^2*x^2-b*e^2*x-b*
d*e+c*d^2)^(1/2)*d*f+5/96*c/e*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*x*d^5*g+625
/2048*b^5/c^3*e^3*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*d^3*g+97/504*b/c^2*(-c*
e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(7/2)/e*d*g+15/32*b^3/c*(-c*e^2*x^2-b*e^2*x-b*d*e+c
*d^2)^(1/2)*d^5*e*g+5/64*c^3/e/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-
c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))*d^9*g+3/64*b^2/c^2*(-c*e^2*x^2-b*e^2*x-b*d
*e+c*d^2)^(5/2)*x*e^2*f-115/256*c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*d^6*b
*g-5/32/c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)*x*b*d^2*g-135/256*c^2/(c*e^2)^(
1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))*d^
8*b*g+115/3072*b^5/c^4*e^3*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*d*g+85/4096*b^
7/c^5*e^5*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*d*g-135/128*d^5*f*c*e*(-c*e^2*x
^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*b+675/2048*d^2*f/c^2*e^4*(-c*e^2*x^2-b*e^2*x-b*d
*e+c*d^2)^(1/2)*x*b^4-45/32*d^7*f*c^2*e/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/
2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))*b-315/512*d^3*f/c^2*e^5/(c*e^2)^(
1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))*b^
5+45/128*d^2*f/c*e^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*x*b^2+315/2048*d^2*f
/c^3*e^6/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*
e+c*d^2)^(1/2))*b^6+315/128*d^6*f*c*e^2/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/
2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))*b^2+1575/1024*d^4*f/c*e^4/(c*e^2)
^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))*
b^4+15/2048*b^5/c^4*e^4*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*f+45/16384*b^7/c^
5*e^6*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*f-5/64/c^2*(-c*e^2*x^2-b*e^2*x-b*d*
e+c*d^2)^(5/2)*b^2*d^2*g-135/256*d^5*f*e*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*
b^2+45/256*d^6*f*c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*b+15/64*d^4*f*c*(-c*e^
2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*x+3/32*d^2*f/c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)
^(5/2)*b-11/768*e^2*g*b^4/c^4*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)-11/63*(-c*e
^2*x^2-b*e^2*x-b*d*e+c*d^2)^(7/2)/c/e^2*d^2*g

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

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)^2*(g*x + f),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 22.2891, size = 1, normalized size = 0. \[ \text{result too large to display} \]

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)^2*(g*x + f),x, algorithm="fricas")

[Out]

[1/8257536*(4*(229376*c^8*e^8*g*x^8 + 14336*(18*c^8*e^8*f + (36*c^8*d*e^7 + 37*b
*c^7*e^8)*g)*x^7 + 1024*(18*(32*c^8*d*e^7 + 33*b*c^7*e^8)*f - (320*c^8*d^2*e^6 -
1796*b*c^7*d*e^7 - 309*b^2*c^6*e^8)*g)*x^6 - 256*(54*(28*c^8*d^2*e^6 - 156*b*c^
7*d*e^7 - 27*b^2*c^6*e^8)*f + (5712*c^8*d^3*e^5 - 5484*b*c^7*d^2*e^6 - 5928*b^2*
c^6*d*e^7 - 5*b^3*c^5*e^8)*g)*x^5 - 128*(18*(768*c^8*d^3*e^5 - 732*b*c^7*d^2*e^6
- 804*b^2*c^6*d*e^7 - b^3*c^5*e^8)*f + (3072*c^8*d^4*e^4 + 15504*b*c^7*d^3*e^5
- 22044*b^2*c^6*d^2*e^6 - 120*b^3*c^5*d*e^7 + 11*b^4*c^4*e^8)*g)*x^4 - 16*(18*(1
680*c^8*d^4*e^4 + 8928*b*c^7*d^3*e^5 - 12552*b^2*c^6*d^2*e^6 - 104*b^3*c^5*d*e^7
+ 9*b^4*c^4*e^8)*f - (79296*c^8*d^5*e^3 - 232272*b*c^7*d^4*e^4 + 148416*b^2*c^6
*d^3*e^5 + 5704*b^3*c^5*d^2*e^6 - 1180*b^4*c^4*d*e^7 + 99*b^5*c^3*e^8)*g)*x^3 +
8*(54*(4096*c^8*d^5*e^3 - 11920*b*c^7*d^4*e^4 + 7456*b^2*c^6*d^3*e^5 + 456*b^3*c
^5*d^2*e^6 - 88*b^4*c^4*d*e^7 + 7*b^5*c^3*e^8)*f + (106496*c^8*d^6*e^2 - 192192*
b*c^7*d^5*e^3 + 52752*b^2*c^6*d^4*e^4 + 46144*b^3*c^5*d^3*e^5 - 16104*b^4*c^4*d^
2*e^6 + 2988*b^5*c^3*d*e^7 - 231*b^6*c^2*e^8)*g)*x^2 - 18*(32768*c^8*d^7*e - 151
872*b*c^7*d^6*e^2 + 259008*b^2*c^6*d^5*e^3 - 218000*b^3*c^5*d^4*e^4 + 102624*b^4
*c^4*d^3*e^5 - 29148*b^5*c^3*d^2*e^6 + 4620*b^6*c^2*d*e^7 - 315*b^7*c*e^8)*f - (
360448*c^8*d^8 - 1656064*b*c^7*d^7*e + 3394752*b^2*c^6*d^6*e^2 - 3950464*b^3*c^5
*d^5*e^3 + 2808496*b^4*c^4*d^4*e^4 - 1245456*b^5*c^3*d^3*e^5 + 339108*b^6*c^2*d^
2*e^6 - 52080*b^7*c*d*e^7 + 3465*b^8*e^8)*g + 2*(18*(37184*c^8*d^6*e^2 - 62400*b
*c^7*d^5*e^3 + 6480*b^2*c^6*d^4*e^4 + 25504*b^3*c^5*d^3*e^5 - 8196*b^4*c^4*d^2*e
^6 + 1428*b^5*c^3*d*e^7 - 105*b^6*c^2*e^8)*f - (80640*c^8*d^7*e - 373568*b*c^7*d
^6*e^2 + 663936*b^2*c^6*d^5*e^3 - 604176*b^3*c^5*d^4*e^4 + 313328*b^4*c^4*d^3*e^
5 - 95868*b^5*c^3*d^2*e^6 + 16128*b^6*c^2*d*e^7 - 1155*b^7*c*e^8)*g)*x)*sqrt(-c*
e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*sqrt(-c) - 315*(18*(256*c^9*d^8*e - 1024*b*c^
8*d^7*e^2 + 1792*b^2*c^7*d^6*e^3 - 1792*b^3*c^6*d^5*e^4 + 1120*b^4*c^5*d^4*e^5 -
448*b^5*c^4*d^3*e^6 + 112*b^6*c^3*d^2*e^7 - 16*b^7*c^2*d*e^8 + b^8*c*e^9)*f + (
1024*c^9*d^9 - 6912*b*c^8*d^8*e + 18432*b^2*c^7*d^7*e^2 - 26880*b^3*c^6*d^6*e^3
+ 24192*b^4*c^5*d^5*e^4 - 14112*b^5*c^4*d^4*e^5 + 5376*b^6*c^3*d^3*e^6 - 1296*b^
7*c^2*d^2*e^7 + 180*b^8*c*d*e^8 - 11*b^9*e^9)*g)*log(-4*sqrt(-c*e^2*x^2 - b*e^2*
x + c*d^2 - b*d*e)*(2*c^2*e*x + b*c*e) + (8*c^2*e^2*x^2 + 8*b*c*e^2*x - 4*c^2*d^
2 + 4*b*c*d*e + b^2*e^2)*sqrt(-c)))/(sqrt(-c)*c^6*e^2), 1/4128768*(2*(229376*c^8
*e^8*g*x^8 + 14336*(18*c^8*e^8*f + (36*c^8*d*e^7 + 37*b*c^7*e^8)*g)*x^7 + 1024*(
18*(32*c^8*d*e^7 + 33*b*c^7*e^8)*f - (320*c^8*d^2*e^6 - 1796*b*c^7*d*e^7 - 309*b
^2*c^6*e^8)*g)*x^6 - 256*(54*(28*c^8*d^2*e^6 - 156*b*c^7*d*e^7 - 27*b^2*c^6*e^8)
*f + (5712*c^8*d^3*e^5 - 5484*b*c^7*d^2*e^6 - 5928*b^2*c^6*d*e^7 - 5*b^3*c^5*e^8
)*g)*x^5 - 128*(18*(768*c^8*d^3*e^5 - 732*b*c^7*d^2*e^6 - 804*b^2*c^6*d*e^7 - b^
3*c^5*e^8)*f + (3072*c^8*d^4*e^4 + 15504*b*c^7*d^3*e^5 - 22044*b^2*c^6*d^2*e^6 -
120*b^3*c^5*d*e^7 + 11*b^4*c^4*e^8)*g)*x^4 - 16*(18*(1680*c^8*d^4*e^4 + 8928*b*
c^7*d^3*e^5 - 12552*b^2*c^6*d^2*e^6 - 104*b^3*c^5*d*e^7 + 9*b^4*c^4*e^8)*f - (79
296*c^8*d^5*e^3 - 232272*b*c^7*d^4*e^4 + 148416*b^2*c^6*d^3*e^5 + 5704*b^3*c^5*d
^2*e^6 - 1180*b^4*c^4*d*e^7 + 99*b^5*c^3*e^8)*g)*x^3 + 8*(54*(4096*c^8*d^5*e^3 -
11920*b*c^7*d^4*e^4 + 7456*b^2*c^6*d^3*e^5 + 456*b^3*c^5*d^2*e^6 - 88*b^4*c^4*d
*e^7 + 7*b^5*c^3*e^8)*f + (106496*c^8*d^6*e^2 - 192192*b*c^7*d^5*e^3 + 52752*b^2
*c^6*d^4*e^4 + 46144*b^3*c^5*d^3*e^5 - 16104*b^4*c^4*d^2*e^6 + 2988*b^5*c^3*d*e^
7 - 231*b^6*c^2*e^8)*g)*x^2 - 18*(32768*c^8*d^7*e - 151872*b*c^7*d^6*e^2 + 25900
8*b^2*c^6*d^5*e^3 - 218000*b^3*c^5*d^4*e^4 + 102624*b^4*c^4*d^3*e^5 - 29148*b^5*
c^3*d^2*e^6 + 4620*b^6*c^2*d*e^7 - 315*b^7*c*e^8)*f - (360448*c^8*d^8 - 1656064*
b*c^7*d^7*e + 3394752*b^2*c^6*d^6*e^2 - 3950464*b^3*c^5*d^5*e^3 + 2808496*b^4*c^
4*d^4*e^4 - 1245456*b^5*c^3*d^3*e^5 + 339108*b^6*c^2*d^2*e^6 - 52080*b^7*c*d*e^7
+ 3465*b^8*e^8)*g + 2*(18*(37184*c^8*d^6*e^2 - 62400*b*c^7*d^5*e^3 + 6480*b^2*c
^6*d^4*e^4 + 25504*b^3*c^5*d^3*e^5 - 8196*b^4*c^4*d^2*e^6 + 1428*b^5*c^3*d*e^7 -
105*b^6*c^2*e^8)*f - (80640*c^8*d^7*e - 373568*b*c^7*d^6*e^2 + 663936*b^2*c^6*d
^5*e^3 - 604176*b^3*c^5*d^4*e^4 + 313328*b^4*c^4*d^3*e^5 - 95868*b^5*c^3*d^2*e^6
+ 16128*b^6*c^2*d*e^7 - 1155*b^7*c*e^8)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2
- b*d*e)*sqrt(c) + 315*(18*(256*c^9*d^8*e - 1024*b*c^8*d^7*e^2 + 1792*b^2*c^7*d
^6*e^3 - 1792*b^3*c^6*d^5*e^4 + 1120*b^4*c^5*d^4*e^5 - 448*b^5*c^4*d^3*e^6 + 112
*b^6*c^3*d^2*e^7 - 16*b^7*c^2*d*e^8 + b^8*c*e^9)*f + (1024*c^9*d^9 - 6912*b*c^8*
d^8*e + 18432*b^2*c^7*d^7*e^2 - 26880*b^3*c^6*d^6*e^3 + 24192*b^4*c^5*d^5*e^4 -
14112*b^5*c^4*d^4*e^5 + 5376*b^6*c^3*d^3*e^6 - 1296*b^7*c^2*d^2*e^7 + 180*b^8*c*
d*e^8 - 11*b^9*e^9)*g)*arctan(1/2*(2*c*e*x + b*e)/(sqrt(-c*e^2*x^2 - b*e^2*x + c
*d^2 - b*d*e)*sqrt(c))))/(c^(13/2)*e^2)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.332417, size = 1, normalized size = 0. \[ \mathit{Done} \]

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)^2*(g*x + f),x, algorithm="giac")

[Out]

Done