3.2240 \(\int (d+e x)^{5/2} (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2} \, dx\)

Optimal. Leaf size=419 \[ -\frac{256 (2 c d-b e)^4 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-2 b e g+c d g+3 c e f)}{45045 c^6 e^2 (d+e x)^{5/2}}-\frac{128 (2 c d-b e)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-2 b e g+c d g+3 c e f)}{9009 c^5 e^2 (d+e x)^{3/2}}-\frac{32 (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-2 b e g+c d g+3 c e f)}{1287 c^4 e^2 \sqrt{d+e x}}-\frac{16 \sqrt{d+e x} (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-2 b e g+c d g+3 c e f)}{429 c^3 e^2}-\frac{2 (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-2 b e g+c d g+3 c e f)}{39 c^2 e^2}-\frac{2 g (d+e x)^{5/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{15 c e^2} \]

[Out]

(-256*(2*c*d - b*e)^4*(3*c*e*f + c*d*g - 2*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e
^2*x^2)^(5/2))/(45045*c^6*e^2*(d + e*x)^(5/2)) - (128*(2*c*d - b*e)^3*(3*c*e*f +
c*d*g - 2*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(9009*c^5*e^2*(d
+ e*x)^(3/2)) - (32*(2*c*d - b*e)^2*(3*c*e*f + c*d*g - 2*b*e*g)*(d*(c*d - b*e) -
b*e^2*x - c*e^2*x^2)^(5/2))/(1287*c^4*e^2*Sqrt[d + e*x]) - (16*(2*c*d - b*e)*(3
*c*e*f + c*d*g - 2*b*e*g)*Sqrt[d + e*x]*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5
/2))/(429*c^3*e^2) - (2*(3*c*e*f + c*d*g - 2*b*e*g)*(d + e*x)^(3/2)*(d*(c*d - b*
e) - b*e^2*x - c*e^2*x^2)^(5/2))/(39*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)^(5/2))/(15*c*e^2)

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Rubi [A]  time = 1.49235, antiderivative size = 419, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065 \[ -\frac{256 (2 c d-b e)^4 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-2 b e g+c d g+3 c e f)}{45045 c^6 e^2 (d+e x)^{5/2}}-\frac{128 (2 c d-b e)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-2 b e g+c d g+3 c e f)}{9009 c^5 e^2 (d+e x)^{3/2}}-\frac{32 (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-2 b e g+c d g+3 c e f)}{1287 c^4 e^2 \sqrt{d+e x}}-\frac{16 \sqrt{d+e x} (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-2 b e g+c d g+3 c e f)}{429 c^3 e^2}-\frac{2 (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-2 b e g+c d g+3 c e f)}{39 c^2 e^2}-\frac{2 g (d+e x)^{5/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{15 c e^2} \]

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)^(3/2),x]

[Out]

(-256*(2*c*d - b*e)^4*(3*c*e*f + c*d*g - 2*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e
^2*x^2)^(5/2))/(45045*c^6*e^2*(d + e*x)^(5/2)) - (128*(2*c*d - b*e)^3*(3*c*e*f +
c*d*g - 2*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(9009*c^5*e^2*(d
+ e*x)^(3/2)) - (32*(2*c*d - b*e)^2*(3*c*e*f + c*d*g - 2*b*e*g)*(d*(c*d - b*e) -
b*e^2*x - c*e^2*x^2)^(5/2))/(1287*c^4*e^2*Sqrt[d + e*x]) - (16*(2*c*d - b*e)*(3
*c*e*f + c*d*g - 2*b*e*g)*Sqrt[d + e*x]*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5
/2))/(429*c^3*e^2) - (2*(3*c*e*f + c*d*g - 2*b*e*g)*(d + e*x)^(3/2)*(d*(c*d - b*
e) - b*e^2*x - c*e^2*x^2)^(5/2))/(39*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)^(5/2))/(15*c*e^2)

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Rubi in Sympy [A]  time = 152.43, size = 406, normalized size = 0.97 \[ - \frac{2 g \left (d + e x\right )^{\frac{5}{2}} \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{5}{2}}}{15 c e^{2}} + \frac{2 \left (d + e x\right )^{\frac{3}{2}} \left (2 b e g - c d g - 3 c e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{5}{2}}}{39 c^{2} e^{2}} - \frac{16 \sqrt{d + e x} \left (b e - 2 c d\right ) \left (2 b e g - c d g - 3 c e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{5}{2}}}{429 c^{3} e^{2}} + \frac{32 \left (b e - 2 c d\right )^{2} \left (2 b e g - c d g - 3 c e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{5}{2}}}{1287 c^{4} e^{2} \sqrt{d + e x}} - \frac{128 \left (b e - 2 c d\right )^{3} \left (2 b e g - c d g - 3 c e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{5}{2}}}{9009 c^{5} e^{2} \left (d + e x\right )^{\frac{3}{2}}} + \frac{256 \left (b e - 2 c d\right )^{4} \left (2 b e g - c d g - 3 c e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{5}{2}}}{45045 c^{6} e^{2} \left (d + e x\right )^{\frac{5}{2}}} \]

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)**(3/2),x)

[Out]

-2*g*(d + e*x)**(5/2)*(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))**(5/2)/(15*c*e*
*2) + 2*(d + e*x)**(3/2)*(2*b*e*g - c*d*g - 3*c*e*f)*(-b*e**2*x - c*e**2*x**2 +
d*(-b*e + c*d))**(5/2)/(39*c**2*e**2) - 16*sqrt(d + e*x)*(b*e - 2*c*d)*(2*b*e*g
- c*d*g - 3*c*e*f)*(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))**(5/2)/(429*c**3*e
**2) + 32*(b*e - 2*c*d)**2*(2*b*e*g - c*d*g - 3*c*e*f)*(-b*e**2*x - c*e**2*x**2
+ d*(-b*e + c*d))**(5/2)/(1287*c**4*e**2*sqrt(d + e*x)) - 128*(b*e - 2*c*d)**3*(
2*b*e*g - c*d*g - 3*c*e*f)*(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))**(5/2)/(90
09*c**5*e**2*(d + e*x)**(3/2)) + 256*(b*e - 2*c*d)**4*(2*b*e*g - c*d*g - 3*c*e*f
)*(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))**(5/2)/(45045*c**6*e**2*(d + e*x)**
(5/2))

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Mathematica [A]  time = 0.7552, size = 364, normalized size = 0.87 \[ -\frac{2 (b e-c d+c e x)^2 \sqrt{(d+e x) (c (d-e x)-b e)} \left (-256 b^5 e^5 g+128 b^4 c e^4 (22 d g+3 e f+5 e g x)-32 b^3 c^2 e^3 \left (389 d^2 g+2 d e (63 f+100 g x)+5 e^2 x (6 f+7 g x)\right )+16 b^2 c^3 e^2 \left (1724 d^3 g+3 d^2 e (347 f+515 g x)+30 d e^2 x (19 f+21 g x)+105 e^3 x^2 (f+g x)\right )-2 b c^4 e \left (15191 d^4 g+4 d^3 e (4131 f+5530 g x)+30 d^2 e^2 x (542 f+553 g x)+420 d e^3 x^2 (17 f+16 g x)+105 e^4 x^3 (12 f+11 g x)\right )+c^5 \left (12686 d^5 g+d^4 e (29049 f+31715 g x)+20 d^3 e^2 x (2505 f+2212 g x)+210 d^2 e^3 x^2 (203 f+173 g x)+210 d e^4 x^3 (90 f+77 g x)+231 e^5 x^4 (15 f+13 g x)\right )\right )}{45045 c^6 e^2 \sqrt{d+e x}} \]

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)^(3/2),x]

[Out]

(-2*(-(c*d) + b*e + c*e*x)^2*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*(-256*b^5*e^
5*g + 128*b^4*c*e^4*(3*e*f + 22*d*g + 5*e*g*x) - 32*b^3*c^2*e^3*(389*d^2*g + 5*e
^2*x*(6*f + 7*g*x) + 2*d*e*(63*f + 100*g*x)) + 16*b^2*c^3*e^2*(1724*d^3*g + 105*
e^3*x^2*(f + g*x) + 30*d*e^2*x*(19*f + 21*g*x) + 3*d^2*e*(347*f + 515*g*x)) - 2*
b*c^4*e*(15191*d^4*g + 105*e^4*x^3*(12*f + 11*g*x) + 420*d*e^3*x^2*(17*f + 16*g*
x) + 30*d^2*e^2*x*(542*f + 553*g*x) + 4*d^3*e*(4131*f + 5530*g*x)) + c^5*(12686*
d^5*g + 231*e^5*x^4*(15*f + 13*g*x) + 210*d*e^4*x^3*(90*f + 77*g*x) + 210*d^2*e^
3*x^2*(203*f + 173*g*x) + 20*d^3*e^2*x*(2505*f + 2212*g*x) + d^4*e*(29049*f + 31
715*g*x))))/(45045*c^6*e^2*Sqrt[d + e*x])

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Maple [A]  time = 0.012, size = 535, normalized size = 1.3 \[ -{\frac{ \left ( 2\,cex+2\,be-2\,cd \right ) \left ( -3003\,g{e}^{5}{x}^{5}{c}^{5}+2310\,b{c}^{4}{e}^{5}g{x}^{4}-16170\,{c}^{5}d{e}^{4}g{x}^{4}-3465\,{c}^{5}{e}^{5}f{x}^{4}-1680\,{b}^{2}{c}^{3}{e}^{5}g{x}^{3}+13440\,b{c}^{4}d{e}^{4}g{x}^{3}+2520\,b{c}^{4}{e}^{5}f{x}^{3}-36330\,{c}^{5}{d}^{2}{e}^{3}g{x}^{3}-18900\,{c}^{5}d{e}^{4}f{x}^{3}+1120\,{b}^{3}{c}^{2}{e}^{5}g{x}^{2}-10080\,{b}^{2}{c}^{3}d{e}^{4}g{x}^{2}-1680\,{b}^{2}{c}^{3}{e}^{5}f{x}^{2}+33180\,b{c}^{4}{d}^{2}{e}^{3}g{x}^{2}+14280\,b{c}^{4}d{e}^{4}f{x}^{2}-44240\,{c}^{5}{d}^{3}{e}^{2}g{x}^{2}-42630\,{c}^{5}{d}^{2}{e}^{3}f{x}^{2}-640\,{b}^{4}c{e}^{5}gx+6400\,{b}^{3}{c}^{2}d{e}^{4}gx+960\,{b}^{3}{c}^{2}{e}^{5}fx-24720\,{b}^{2}{c}^{3}{d}^{2}{e}^{3}gx-9120\,{b}^{2}{c}^{3}d{e}^{4}fx+44240\,b{c}^{4}{d}^{3}{e}^{2}gx+32520\,b{c}^{4}{d}^{2}{e}^{3}fx-31715\,{c}^{5}{d}^{4}egx-50100\,{c}^{5}{d}^{3}{e}^{2}fx+256\,{b}^{5}{e}^{5}g-2816\,{b}^{4}cd{e}^{4}g-384\,{b}^{4}c{e}^{5}f+12448\,{b}^{3}{c}^{2}{d}^{2}{e}^{3}g+4032\,{b}^{3}{c}^{2}d{e}^{4}f-27584\,{b}^{2}{c}^{3}{d}^{3}{e}^{2}g-16656\,{b}^{2}{c}^{3}{d}^{2}{e}^{3}f+30382\,b{c}^{4}{d}^{4}eg+33048\,b{c}^{4}{d}^{3}{e}^{2}f-12686\,{c}^{5}{d}^{5}g-29049\,f{d}^{4}{c}^{5}e \right ) }{45045\,{c}^{6}{e}^{2}} \left ( -c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2} \right ) ^{{\frac{3}{2}}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}} \]

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)^(3/2),x)

[Out]

-2/45045*(c*e*x+b*e-c*d)*(-3003*c^5*e^5*g*x^5+2310*b*c^4*e^5*g*x^4-16170*c^5*d*e
^4*g*x^4-3465*c^5*e^5*f*x^4-1680*b^2*c^3*e^5*g*x^3+13440*b*c^4*d*e^4*g*x^3+2520*
b*c^4*e^5*f*x^3-36330*c^5*d^2*e^3*g*x^3-18900*c^5*d*e^4*f*x^3+1120*b^3*c^2*e^5*g
*x^2-10080*b^2*c^3*d*e^4*g*x^2-1680*b^2*c^3*e^5*f*x^2+33180*b*c^4*d^2*e^3*g*x^2+
14280*b*c^4*d*e^4*f*x^2-44240*c^5*d^3*e^2*g*x^2-42630*c^5*d^2*e^3*f*x^2-640*b^4*
c*e^5*g*x+6400*b^3*c^2*d*e^4*g*x+960*b^3*c^2*e^5*f*x-24720*b^2*c^3*d^2*e^3*g*x-9
120*b^2*c^3*d*e^4*f*x+44240*b*c^4*d^3*e^2*g*x+32520*b*c^4*d^2*e^3*f*x-31715*c^5*
d^4*e*g*x-50100*c^5*d^3*e^2*f*x+256*b^5*e^5*g-2816*b^4*c*d*e^4*g-384*b^4*c*e^5*f
+12448*b^3*c^2*d^2*e^3*g+4032*b^3*c^2*d*e^4*f-27584*b^2*c^3*d^3*e^2*g-16656*b^2*
c^3*d^2*e^3*f+30382*b*c^4*d^4*e*g+33048*b*c^4*d^3*e^2*f-12686*c^5*d^5*g-29049*c^
5*d^4*e*f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)/c^6/e^2/(e*x+d)^(3/2)

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Maxima [A]  time = 0.776206, size = 1181, normalized size = 2.82 \[ \text{result too large to display} \]

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)^(3/2)*(e*x + d)^(5/2)*(g*x + f),x, algorithm="maxima")

[Out]

-2/15015*(1155*c^6*e^6*x^6 + 9683*c^6*d^6 - 30382*b*c^5*d^5*e + 37267*b^2*c^4*d^
4*e^2 - 23464*b^3*c^3*d^3*e^3 + 8368*b^4*c^2*d^2*e^4 - 1600*b^5*c*d*e^5 + 128*b^
6*e^6 + 210*(19*c^6*d*e^5 + 7*b*c^5*e^6)*x^5 + 35*(79*c^6*d^2*e^4 + 206*b*c^5*d*
e^5 + b^2*c^4*e^6)*x^4 - 20*(271*c^6*d^3*e^3 - 683*b*c^5*d^2*e^4 - 19*b^2*c^4*d*
e^5 + 2*b^3*c^3*e^6)*x^3 - 3*(3169*c^6*d^4*e^2 - 3628*b*c^5*d^3*e^3 - 694*b^2*c^
4*d^2*e^4 + 168*b^3*c^3*d*e^5 - 16*b^4*c^2*e^6)*x^2 - 2*(1333*c^6*d^5*e + 1421*b
*c^5*d^4*e^2 - 4142*b^2*c^4*d^3*e^3 + 1724*b^3*c^3*d^2*e^4 - 368*b^4*c^2*d*e^5 +
32*b^5*c*e^6)*x)*sqrt(-c*e*x + c*d - b*e)*(e*x + d)*f/(c^5*e^2*x + c^5*d*e) - 2
/45045*(3003*c^7*e^7*x^7 + 12686*c^7*d^7 - 55754*b*c^6*d^6*e + 101034*b^2*c^5*d^
5*e^2 - 97998*b^3*c^4*d^4*e^3 + 55296*b^4*c^3*d^3*e^4 - 18336*b^5*c^2*d^2*e^5 +
3328*b^6*c*d*e^6 - 256*b^7*e^7 + 924*(11*c^7*d*e^6 + 4*b*c^6*e^7)*x^6 + 63*(111*
c^7*d^2*e^5 + 278*b*c^6*d*e^6 + b^2*c^5*e^7)*x^5 - 70*(175*c^7*d^3*e^4 - 453*b*c
^6*d^2*e^5 - 9*b^2*c^5*d*e^6 + b^3*c^4*e^7)*x^4 - 5*(4087*c^7*d^4*e^3 - 4900*b*c
^6*d^3*e^4 - 618*b^2*c^5*d^2*e^5 + 160*b^3*c^4*d*e^6 - 16*b^4*c^3*e^7)*x^3 - 12*
(542*c^7*d^5*e^2 + 11*b*c^6*d^4*e^3 - 862*b^2*c^5*d^3*e^4 + 389*b^3*c^4*d^2*e^5
- 88*b^4*c^3*d*e^6 + 8*b^5*c^2*e^7)*x^2 + (6343*c^7*d^6*e - 21534*b*c^6*d^5*e^2
+ 28983*b^2*c^5*d^4*e^3 - 20016*b^3*c^4*d^3*e^4 + 7632*b^4*c^3*d^2*e^5 - 1536*b^
5*c^2*d*e^6 + 128*b^6*c*e^7)*x)*sqrt(-c*e*x + c*d - b*e)*(e*x + d)*g/(c^6*e^3*x
+ c^6*d*e^2)

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Fricas [A]  time = 0.322105, size = 1789, normalized size = 4.27 \[ \text{result too large to display} \]

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)^(3/2)*(e*x + d)^(5/2)*(g*x + f),x, algorithm="fricas")

[Out]

2/45045*(3003*c^8*e^9*g*x^9 + 231*(15*c^8*e^9*f + (44*c^8*d*e^8 + 29*b*c^7*e^9)*
g)*x^8 + 21*(15*(38*c^8*d*e^8 + 25*b*c^7*e^9)*f + (190*c^8*d^2*e^7 + 1461*b*c^7*
d*e^8 + 179*b^2*c^6*e^9)*g)*x^7 + 7*(15*(46*c^8*d^2*e^7 + 353*b*c^7*d*e^8 + 43*b
^2*c^6*e^9)*f - (3202*c^8*d^3*e^6 - 6453*b*c^7*d^2*e^7 - 3120*b^2*c^6*d*e^8 + b^
3*c^5*e^9)*g)*x^6 - (15*(1882*c^8*d^3*e^6 - 3789*b*c^7*d^2*e^7 - 1812*b^2*c^6*d*
e^8 + b^3*c^5*e^9)*f + (27428*c^8*d^4*e^5 - 1729*b*c^7*d^3*e^6 - 52251*b^2*c^6*d
^2*e^7 + 107*b^3*c^5*d*e^8 - 10*b^4*c^4*e^9)*g)*x^5 - (3*(12272*c^8*d^4*e^5 - 10
19*b*c^7*d^3*e^6 - 22917*b^2*c^6*d^2*e^7 + 89*b^3*c^5*d*e^8 - 8*b^4*c^4*e^9)*f -
(5746*c^8*d^5*e^4 - 64527*b*c^7*d^4*e^5 + 65924*b^2*c^6*d^3*e^6 - 878*b^3*c^5*d
^2*e^7 + 186*b^4*c^4*d*e^8 - 16*b^5*c^3*e^9)*g)*x^4 + (3*(2754*c^8*d^5*e^4 - 314
29*b*c^7*d^4*e^5 + 32448*b^2*c^6*d^3*e^6 - 946*b^3*c^5*d^2*e^7 + 192*b^4*c^4*d*e
^8 - 16*b^5*c^3*e^9)*f + (26778*c^8*d^6*e^3 - 72973*b*c^7*d^5*e^4 + 50261*b^2*c^
6*d^4*e^5 - 5782*b^3*c^5*d^3*e^6 + 2084*b^4*c^4*d^2*e^7 - 400*b^5*c^3*d*e^8 + 32
*b^6*c^2*e^9)*g)*x^3 + (3*(19190*c^8*d^6*e^3 - 53439*b*c^7*d^5*e^4 + 43227*b^2*c
^6*d^4*e^5 - 12594*b^3*c^5*d^3*e^6 + 4368*b^4*c^4*d^2*e^7 - 816*b^5*c^3*d*e^8 +
64*b^6*c^2*e^9)*f + (19190*c^8*d^7*e^2 - 55783*b*c^7*d^6*e^3 + 69024*b^2*c^6*d^5
*e^4 - 54003*b^3*c^5*d^4*e^5 + 29556*b^4*c^4*d^3*e^6 - 9552*b^5*c^3*d^2*e^7 + 16
96*b^6*c^2*d*e^8 - 128*b^7*c*e^9)*g)*x^2 - 3*(9683*c^8*d^8*e - 40065*b*c^7*d^7*e
^2 + 67649*b^2*c^6*d^6*e^3 - 60731*b^3*c^5*d^5*e^4 + 31832*b^4*c^4*d^4*e^5 - 996
8*b^5*c^3*d^3*e^6 + 1728*b^6*c^2*d^2*e^7 - 128*b^7*c*d*e^8)*f - 2*(6343*c^8*d^9
- 34220*b*c^7*d^8*e + 78394*b^2*c^6*d^7*e^2 - 99516*b^3*c^5*d^6*e^3 + 76647*b^4*
c^4*d^5*e^4 - 36816*b^5*c^3*d^4*e^5 + 10832*b^6*c^2*d^3*e^6 - 1792*b^7*c*d^2*e^7
+ 128*b^8*d*e^8)*g + (3*(2666*c^8*d^7*e^2 + 9859*b*c^7*d^6*e^3 - 41508*b^2*c^6*
d^5*e^4 + 48999*b^3*c^5*d^4*e^5 - 27648*b^4*c^4*d^3*e^6 + 9168*b^5*c^3*d^2*e^7 -
1664*b^6*c^2*d*e^8 + 128*b^7*c*e^9)*f - (6343*c^8*d^8*e - 40563*b*c^7*d^7*e^2 +
106271*b^2*c^6*d^6*e^3 - 150033*b^3*c^5*d^5*e^4 + 125646*b^4*c^4*d^4*e^5 - 6446
4*b^5*c^3*d^3*e^6 + 20000*b^6*c^2*d^2*e^7 - 3456*b^7*c*d*e^8 + 256*b^8*e^9)*g)*x
)/(sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*sqrt(e*x + d)*c^6*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)**(3/2),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)^(3/2)*(e*x + d)^(5/2)*(g*x + f),x, algorithm="giac")

[Out]

Timed out