3.2276 \(\int \frac{(d+e x)^{13/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=448 \[ -\frac{256 \sqrt{d+e x} (2 c d-b e)^3 (-10 b e g+13 c d g+7 c e f)}{105 c^6 e^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{128 (d+e x)^{3/2} (2 c d-b e)^2 (-10 b e g+13 c d g+7 c e f)}{105 c^5 e^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{32 (d+e x)^{5/2} (2 c d-b e) (-10 b e g+13 c d g+7 c e f)}{105 c^4 e^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{16 (d+e x)^{7/2} (-10 b e g+13 c d g+7 c e f)}{105 c^3 e^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{2 (d+e x)^{9/2} (-10 b e g+13 c d g+7 c e f)}{21 c^2 e^2 (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{2 (d+e x)^{13/2} (-b e g+c d g+c e f)}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}} \]

[Out]

(2*(c*e*f + c*d*g - b*e*g)*(d + e*x)^(13/2))/(3*c*e^2*(2*c*d - b*e)*(d*(c*d - b*
e) - b*e^2*x - c*e^2*x^2)^(3/2)) - (256*(2*c*d - b*e)^3*(7*c*e*f + 13*c*d*g - 10
*b*e*g)*Sqrt[d + e*x])/(105*c^6*e^2*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]) +
(128*(2*c*d - b*e)^2*(7*c*e*f + 13*c*d*g - 10*b*e*g)*(d + e*x)^(3/2))/(105*c^5*
e^2*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]) + (32*(2*c*d - b*e)*(7*c*e*f + 13
*c*d*g - 10*b*e*g)*(d + e*x)^(5/2))/(105*c^4*e^2*Sqrt[d*(c*d - b*e) - b*e^2*x -
c*e^2*x^2]) + (16*(7*c*e*f + 13*c*d*g - 10*b*e*g)*(d + e*x)^(7/2))/(105*c^3*e^2*
Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]) + (2*(7*c*e*f + 13*c*d*g - 10*b*e*g)*
(d + e*x)^(9/2))/(21*c^2*e^2*(2*c*d - b*e)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*
x^2])

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Rubi [A]  time = 1.53977, antiderivative size = 448, 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 \[ -\frac{256 \sqrt{d+e x} (2 c d-b e)^3 (-10 b e g+13 c d g+7 c e f)}{105 c^6 e^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{128 (d+e x)^{3/2} (2 c d-b e)^2 (-10 b e g+13 c d g+7 c e f)}{105 c^5 e^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{32 (d+e x)^{5/2} (2 c d-b e) (-10 b e g+13 c d g+7 c e f)}{105 c^4 e^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{16 (d+e x)^{7/2} (-10 b e g+13 c d g+7 c e f)}{105 c^3 e^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{2 (d+e x)^{9/2} (-10 b e g+13 c d g+7 c e f)}{21 c^2 e^2 (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{2 (d+e x)^{13/2} (-b e g+c d g+c e f)}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[((d + e*x)^(13/2)*(f + g*x))/(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2),x]

[Out]

(2*(c*e*f + c*d*g - b*e*g)*(d + e*x)^(13/2))/(3*c*e^2*(2*c*d - b*e)*(d*(c*d - b*
e) - b*e^2*x - c*e^2*x^2)^(3/2)) - (256*(2*c*d - b*e)^3*(7*c*e*f + 13*c*d*g - 10
*b*e*g)*Sqrt[d + e*x])/(105*c^6*e^2*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]) +
(128*(2*c*d - b*e)^2*(7*c*e*f + 13*c*d*g - 10*b*e*g)*(d + e*x)^(3/2))/(105*c^5*
e^2*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]) + (32*(2*c*d - b*e)*(7*c*e*f + 13
*c*d*g - 10*b*e*g)*(d + e*x)^(5/2))/(105*c^4*e^2*Sqrt[d*(c*d - b*e) - b*e^2*x -
c*e^2*x^2]) + (16*(7*c*e*f + 13*c*d*g - 10*b*e*g)*(d + e*x)^(7/2))/(105*c^3*e^2*
Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]) + (2*(7*c*e*f + 13*c*d*g - 10*b*e*g)*
(d + e*x)^(9/2))/(21*c^2*e^2*(2*c*d - b*e)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*
x^2])

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Rubi in Sympy [A]  time = 171.106, size = 435, normalized size = 0.97 \[ \frac{2 \left (d + e x\right )^{\frac{13}{2}} \left (b e g - c d g - c e f\right )}{3 c e^{2} \left (b e - 2 c d\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{3}{2}}} + \frac{2 \left (d + e x\right )^{\frac{9}{2}} \left (10 b e g - 13 c d g - 7 c e f\right )}{21 c^{2} e^{2} \left (b e - 2 c d\right ) \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}} - \frac{16 \left (d + e x\right )^{\frac{7}{2}} \left (10 b e g - 13 c d g - 7 c e f\right )}{105 c^{3} e^{2} \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}} + \frac{32 \left (d + e x\right )^{\frac{5}{2}} \left (b e - 2 c d\right ) \left (10 b e g - 13 c d g - 7 c e f\right )}{105 c^{4} e^{2} \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}} - \frac{128 \left (d + e x\right )^{\frac{3}{2}} \left (b e - 2 c d\right )^{2} \left (10 b e g - 13 c d g - 7 c e f\right )}{105 c^{5} e^{2} \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}} - \frac{256 \sqrt{d + e x} \left (b e - 2 c d\right )^{3} \left (10 b e g - 13 c d g - 7 c e f\right )}{105 c^{6} e^{2} \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**(13/2)*(g*x+f)/(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(5/2),x)

[Out]

2*(d + e*x)**(13/2)*(b*e*g - c*d*g - c*e*f)/(3*c*e**2*(b*e - 2*c*d)*(-b*e**2*x -
c*e**2*x**2 + d*(-b*e + c*d))**(3/2)) + 2*(d + e*x)**(9/2)*(10*b*e*g - 13*c*d*g
- 7*c*e*f)/(21*c**2*e**2*(b*e - 2*c*d)*sqrt(-b*e**2*x - c*e**2*x**2 + d*(-b*e +
c*d))) - 16*(d + e*x)**(7/2)*(10*b*e*g - 13*c*d*g - 7*c*e*f)/(105*c**3*e**2*sqr
t(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))) + 32*(d + e*x)**(5/2)*(b*e - 2*c*d)
*(10*b*e*g - 13*c*d*g - 7*c*e*f)/(105*c**4*e**2*sqrt(-b*e**2*x - c*e**2*x**2 + d
*(-b*e + c*d))) - 128*(d + e*x)**(3/2)*(b*e - 2*c*d)**2*(10*b*e*g - 13*c*d*g - 7
*c*e*f)/(105*c**5*e**2*sqrt(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))) - 256*sqr
t(d + e*x)*(b*e - 2*c*d)**3*(10*b*e*g - 13*c*d*g - 7*c*e*f)/(105*c**6*e**2*sqrt(
-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d)))

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Mathematica [A]  time = 0.803494, size = 366, normalized size = 0.82 \[ \frac{2 \sqrt{d+e x} \left (-1280 b^5 e^5 g+128 b^4 c e^4 (78 d g+7 e f-15 e g x)-32 b^3 c^2 e^3 \left (953 d^2 g+2 d e (91 f-204 g x)+3 e^2 x (5 g x-14 f)\right )+16 b^2 c^3 e^2 \left (2844 d^3 g+3 d^2 e (287 f-681 g x)+6 d e^2 x (29 g x-77 f)+e^3 x^2 (21 f+5 g x)\right )-2 b c^4 e \left (16563 d^4 g+12 d^3 e (581 f-1482 g x)+6 d^2 e^2 x (449 g x-1106 f)+12 d e^3 x^2 (63 f+16 g x)+e^4 x^3 (28 f+15 g x)\right )+c^5 \left (9414 d^5 g+3 d^4 e (1687 f-4707 g x)+12 d^3 e^2 x (292 g x-637 f)+2 d^2 e^3 x^2 (903 f+257 g x)+2 d e^4 x^3 (98 f+57 g x)+3 e^5 x^4 (7 f+5 g x)\right )\right )}{105 c^6 e^2 (b e-c d+c e x) \sqrt{(d+e x) (c (d-e x)-b e)}} \]

Antiderivative was successfully verified.

[In]  Integrate[((d + e*x)^(13/2)*(f + g*x))/(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2),x]

[Out]

(2*Sqrt[d + e*x]*(-1280*b^5*e^5*g + 128*b^4*c*e^4*(7*e*f + 78*d*g - 15*e*g*x) -
32*b^3*c^2*e^3*(953*d^2*g + 2*d*e*(91*f - 204*g*x) + 3*e^2*x*(-14*f + 5*g*x)) +
16*b^2*c^3*e^2*(2844*d^3*g + 3*d^2*e*(287*f - 681*g*x) + e^3*x^2*(21*f + 5*g*x)
+ 6*d*e^2*x*(-77*f + 29*g*x)) + c^5*(9414*d^5*g + 3*d^4*e*(1687*f - 4707*g*x) +
3*e^5*x^4*(7*f + 5*g*x) + 2*d*e^4*x^3*(98*f + 57*g*x) + 2*d^2*e^3*x^2*(903*f + 2
57*g*x) + 12*d^3*e^2*x*(-637*f + 292*g*x)) - 2*b*c^4*e*(16563*d^4*g + 12*d^3*e*(
581*f - 1482*g*x) + e^4*x^3*(28*f + 15*g*x) + 12*d*e^3*x^2*(63*f + 16*g*x) + 6*d
^2*e^2*x*(-1106*f + 449*g*x))))/(105*c^6*e^2*(-(c*d) + b*e + c*e*x)*Sqrt[(d + e*
x)*(-(b*e) + c*(d - e*x))])

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Maple [A]  time = 0.012, size = 535, normalized size = 1.2 \[ -{\frac{ \left ( 2\,cex+2\,be-2\,cd \right ) \left ( -15\,g{e}^{5}{x}^{5}{c}^{5}+30\,b{c}^{4}{e}^{5}g{x}^{4}-114\,{c}^{5}d{e}^{4}g{x}^{4}-21\,{c}^{5}{e}^{5}f{x}^{4}-80\,{b}^{2}{c}^{3}{e}^{5}g{x}^{3}+384\,b{c}^{4}d{e}^{4}g{x}^{3}+56\,b{c}^{4}{e}^{5}f{x}^{3}-514\,{c}^{5}{d}^{2}{e}^{3}g{x}^{3}-196\,{c}^{5}d{e}^{4}f{x}^{3}+480\,{b}^{3}{c}^{2}{e}^{5}g{x}^{2}-2784\,{b}^{2}{c}^{3}d{e}^{4}g{x}^{2}-336\,{b}^{2}{c}^{3}{e}^{5}f{x}^{2}+5388\,b{c}^{4}{d}^{2}{e}^{3}g{x}^{2}+1512\,b{c}^{4}d{e}^{4}f{x}^{2}-3504\,{c}^{5}{d}^{3}{e}^{2}g{x}^{2}-1806\,{c}^{5}{d}^{2}{e}^{3}f{x}^{2}+1920\,{b}^{4}c{e}^{5}gx-13056\,{b}^{3}{c}^{2}d{e}^{4}gx-1344\,{b}^{3}{c}^{2}{e}^{5}fx+32688\,{b}^{2}{c}^{3}{d}^{2}{e}^{3}gx+7392\,{b}^{2}{c}^{3}d{e}^{4}fx-35568\,b{c}^{4}{d}^{3}{e}^{2}gx-13272\,b{c}^{4}{d}^{2}{e}^{3}fx+14121\,{c}^{5}{d}^{4}egx+7644\,{c}^{5}{d}^{3}{e}^{2}fx+1280\,{b}^{5}{e}^{5}g-9984\,{b}^{4}cd{e}^{4}g-896\,{b}^{4}c{e}^{5}f+30496\,{b}^{3}{c}^{2}{d}^{2}{e}^{3}g+5824\,{b}^{3}{c}^{2}d{e}^{4}f-45504\,{b}^{2}{c}^{3}{d}^{3}{e}^{2}g-13776\,{b}^{2}{c}^{3}{d}^{2}{e}^{3}f+33126\,b{c}^{4}{d}^{4}eg+13944\,b{c}^{4}{d}^{3}{e}^{2}f-9414\,{c}^{5}{d}^{5}g-5061\,f{d}^{4}{c}^{5}e \right ) }{105\,{c}^{6}{e}^{2}} \left ( ex+d \right ) ^{{\frac{5}{2}}} \left ( -c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2} \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^(13/2)*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x)

[Out]

-2/105*(c*e*x+b*e-c*d)*(-15*c^5*e^5*g*x^5+30*b*c^4*e^5*g*x^4-114*c^5*d*e^4*g*x^4
-21*c^5*e^5*f*x^4-80*b^2*c^3*e^5*g*x^3+384*b*c^4*d*e^4*g*x^3+56*b*c^4*e^5*f*x^3-
514*c^5*d^2*e^3*g*x^3-196*c^5*d*e^4*f*x^3+480*b^3*c^2*e^5*g*x^2-2784*b^2*c^3*d*e
^4*g*x^2-336*b^2*c^3*e^5*f*x^2+5388*b*c^4*d^2*e^3*g*x^2+1512*b*c^4*d*e^4*f*x^2-3
504*c^5*d^3*e^2*g*x^2-1806*c^5*d^2*e^3*f*x^2+1920*b^4*c*e^5*g*x-13056*b^3*c^2*d*
e^4*g*x-1344*b^3*c^2*e^5*f*x+32688*b^2*c^3*d^2*e^3*g*x+7392*b^2*c^3*d*e^4*f*x-35
568*b*c^4*d^3*e^2*g*x-13272*b*c^4*d^2*e^3*f*x+14121*c^5*d^4*e*g*x+7644*c^5*d^3*e
^2*f*x+1280*b^5*e^5*g-9984*b^4*c*d*e^4*g-896*b^4*c*e^5*f+30496*b^3*c^2*d^2*e^3*g
+5824*b^3*c^2*d*e^4*f-45504*b^2*c^3*d^3*e^2*g-13776*b^2*c^3*d^2*e^3*f+33126*b*c^
4*d^4*e*g+13944*b*c^4*d^3*e^2*f-9414*c^5*d^5*g-5061*c^5*d^4*e*f)*(e*x+d)^(5/2)/c
^6/e^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)

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Maxima [A]  time = 0.756779, size = 690, normalized size = 1.54 \[ \frac{2 \,{\left (3 \, c^{4} e^{4} x^{4} + 723 \, c^{4} d^{4} - 1992 \, b c^{3} d^{3} e + 1968 \, b^{2} c^{2} d^{2} e^{2} - 832 \, b^{3} c d e^{3} + 128 \, b^{4} e^{4} + 4 \,{\left (7 \, c^{4} d e^{3} - 2 \, b c^{3} e^{4}\right )} x^{3} + 6 \,{\left (43 \, c^{4} d^{2} e^{2} - 36 \, b c^{3} d e^{3} + 8 \, b^{2} c^{2} e^{4}\right )} x^{2} - 12 \,{\left (91 \, c^{4} d^{3} e - 158 \, b c^{3} d^{2} e^{2} + 88 \, b^{2} c^{2} d e^{3} - 16 \, b^{3} c e^{4}\right )} x\right )} f}{15 \,{\left (c^{6} e^{2} x - c^{6} d e + b c^{5} e^{2}\right )} \sqrt{-c e x + c d - b e}} + \frac{2 \,{\left (15 \, c^{5} e^{5} x^{5} + 9414 \, c^{5} d^{5} - 33126 \, b c^{4} d^{4} e + 45504 \, b^{2} c^{3} d^{3} e^{2} - 30496 \, b^{3} c^{2} d^{2} e^{3} + 9984 \, b^{4} c d e^{4} - 1280 \, b^{5} e^{5} + 6 \,{\left (19 \, c^{5} d e^{4} - 5 \, b c^{4} e^{5}\right )} x^{4} + 2 \,{\left (257 \, c^{5} d^{2} e^{3} - 192 \, b c^{4} d e^{4} + 40 \, b^{2} c^{3} e^{5}\right )} x^{3} + 12 \,{\left (292 \, c^{5} d^{3} e^{2} - 449 \, b c^{4} d^{2} e^{3} + 232 \, b^{2} c^{3} d e^{4} - 40 \, b^{3} c^{2} e^{5}\right )} x^{2} - 3 \,{\left (4707 \, c^{5} d^{4} e - 11856 \, b c^{4} d^{3} e^{2} + 10896 \, b^{2} c^{3} d^{2} e^{3} - 4352 \, b^{3} c^{2} d e^{4} + 640 \, b^{4} c e^{5}\right )} x\right )} g}{105 \,{\left (c^{7} e^{3} x - c^{7} d e^{2} + b c^{6} e^{3}\right )} \sqrt{-c e x + c d - b e}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(13/2)*(g*x + f)/(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)^(5/2),x, algorithm="maxima")

[Out]

2/15*(3*c^4*e^4*x^4 + 723*c^4*d^4 - 1992*b*c^3*d^3*e + 1968*b^2*c^2*d^2*e^2 - 83
2*b^3*c*d*e^3 + 128*b^4*e^4 + 4*(7*c^4*d*e^3 - 2*b*c^3*e^4)*x^3 + 6*(43*c^4*d^2*
e^2 - 36*b*c^3*d*e^3 + 8*b^2*c^2*e^4)*x^2 - 12*(91*c^4*d^3*e - 158*b*c^3*d^2*e^2
+ 88*b^2*c^2*d*e^3 - 16*b^3*c*e^4)*x)*f/((c^6*e^2*x - c^6*d*e + b*c^5*e^2)*sqrt
(-c*e*x + c*d - b*e)) + 2/105*(15*c^5*e^5*x^5 + 9414*c^5*d^5 - 33126*b*c^4*d^4*e
+ 45504*b^2*c^3*d^3*e^2 - 30496*b^3*c^2*d^2*e^3 + 9984*b^4*c*d*e^4 - 1280*b^5*e
^5 + 6*(19*c^5*d*e^4 - 5*b*c^4*e^5)*x^4 + 2*(257*c^5*d^2*e^3 - 192*b*c^4*d*e^4 +
40*b^2*c^3*e^5)*x^3 + 12*(292*c^5*d^3*e^2 - 449*b*c^4*d^2*e^3 + 232*b^2*c^3*d*e
^4 - 40*b^3*c^2*e^5)*x^2 - 3*(4707*c^5*d^4*e - 11856*b*c^4*d^3*e^2 + 10896*b^2*c
^3*d^2*e^3 - 4352*b^3*c^2*d*e^4 + 640*b^4*c*e^5)*x)*g/((c^7*e^3*x - c^7*d*e^2 +
b*c^6*e^3)*sqrt(-c*e*x + c*d - b*e))

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Fricas [A]  time = 0.294625, size = 886, normalized size = 1.98 \[ \frac{2 \,{\left (15 \, c^{5} e^{6} g x^{6} + 3 \,{\left (7 \, c^{5} e^{6} f +{\left (43 \, c^{5} d e^{5} - 10 \, b c^{4} e^{6}\right )} g\right )} x^{5} +{\left (7 \,{\left (31 \, c^{5} d e^{5} - 8 \, b c^{4} e^{6}\right )} f + 2 \,{\left (314 \, c^{5} d^{2} e^{4} - 207 \, b c^{4} d e^{5} + 40 \, b^{2} c^{3} e^{6}\right )} g\right )} x^{4} + 2 \,{\left (7 \,{\left (143 \, c^{5} d^{2} e^{4} - 112 \, b c^{4} d e^{5} + 24 \, b^{2} c^{3} e^{6}\right )} f +{\left (2009 \, c^{5} d^{3} e^{3} - 2886 \, b c^{4} d^{2} e^{4} + 1432 \, b^{2} c^{3} d e^{5} - 240 \, b^{3} c^{2} e^{6}\right )} g\right )} x^{3} - 3 \,{\left (14 \,{\left (139 \, c^{5} d^{3} e^{3} - 280 \, b c^{4} d^{2} e^{4} + 168 \, b^{2} c^{3} d e^{5} - 32 \, b^{3} c^{2} e^{6}\right )} f +{\left (3539 \, c^{5} d^{4} e^{2} - 10060 \, b c^{4} d^{3} e^{3} + 9968 \, b^{2} c^{3} d^{2} e^{4} - 4192 \, b^{3} c^{2} d e^{5} + 640 \, b^{4} c e^{6}\right )} g\right )} x^{2} + 7 \,{\left (723 \, c^{5} d^{5} e - 1992 \, b c^{4} d^{4} e^{2} + 1968 \, b^{2} c^{3} d^{3} e^{3} - 832 \, b^{3} c^{2} d^{2} e^{4} + 128 \, b^{4} c d e^{5}\right )} f + 2 \,{\left (4707 \, c^{5} d^{6} - 16563 \, b c^{4} d^{5} e + 22752 \, b^{2} c^{3} d^{4} e^{2} - 15248 \, b^{3} c^{2} d^{3} e^{3} + 4992 \, b^{4} c d^{2} e^{4} - 640 \, b^{5} d e^{5}\right )} g -{\left (7 \,{\left (369 \, c^{5} d^{4} e^{2} + 96 \, b c^{4} d^{3} e^{3} - 912 \, b^{2} c^{3} d^{2} e^{4} + 640 \, b^{3} c^{2} d e^{5} - 128 \, b^{4} c e^{6}\right )} f +{\left (4707 \, c^{5} d^{5} e - 2442 \, b c^{4} d^{4} e^{2} - 12816 \, b^{2} c^{3} d^{3} e^{3} + 17440 \, b^{3} c^{2} d^{2} e^{4} - 8064 \, b^{4} c d e^{5} + 1280 \, b^{5} e^{6}\right )} g\right )} x\right )}}{105 \,{\left (c^{7} e^{3} x - c^{7} d e^{2} + b c^{6} e^{3}\right )} \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt{e x + d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(13/2)*(g*x + f)/(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)^(5/2),x, algorithm="fricas")

[Out]

2/105*(15*c^5*e^6*g*x^6 + 3*(7*c^5*e^6*f + (43*c^5*d*e^5 - 10*b*c^4*e^6)*g)*x^5
+ (7*(31*c^5*d*e^5 - 8*b*c^4*e^6)*f + 2*(314*c^5*d^2*e^4 - 207*b*c^4*d*e^5 + 40*
b^2*c^3*e^6)*g)*x^4 + 2*(7*(143*c^5*d^2*e^4 - 112*b*c^4*d*e^5 + 24*b^2*c^3*e^6)*
f + (2009*c^5*d^3*e^3 - 2886*b*c^4*d^2*e^4 + 1432*b^2*c^3*d*e^5 - 240*b^3*c^2*e^
6)*g)*x^3 - 3*(14*(139*c^5*d^3*e^3 - 280*b*c^4*d^2*e^4 + 168*b^2*c^3*d*e^5 - 32*
b^3*c^2*e^6)*f + (3539*c^5*d^4*e^2 - 10060*b*c^4*d^3*e^3 + 9968*b^2*c^3*d^2*e^4
- 4192*b^3*c^2*d*e^5 + 640*b^4*c*e^6)*g)*x^2 + 7*(723*c^5*d^5*e - 1992*b*c^4*d^4
*e^2 + 1968*b^2*c^3*d^3*e^3 - 832*b^3*c^2*d^2*e^4 + 128*b^4*c*d*e^5)*f + 2*(4707
*c^5*d^6 - 16563*b*c^4*d^5*e + 22752*b^2*c^3*d^4*e^2 - 15248*b^3*c^2*d^3*e^3 + 4
992*b^4*c*d^2*e^4 - 640*b^5*d*e^5)*g - (7*(369*c^5*d^4*e^2 + 96*b*c^4*d^3*e^3 -
912*b^2*c^3*d^2*e^4 + 640*b^3*c^2*d*e^5 - 128*b^4*c*e^6)*f + (4707*c^5*d^5*e - 2
442*b*c^4*d^4*e^2 - 12816*b^2*c^3*d^3*e^3 + 17440*b^3*c^2*d^2*e^4 - 8064*b^4*c*d
*e^5 + 1280*b^5*e^6)*g)*x)/((c^7*e^3*x - c^7*d*e^2 + b*c^6*e^3)*sqrt(-c*e^2*x^2
- b*e^2*x + c*d^2 - b*d*e)*sqrt(e*x + d))

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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)**(13/2)*(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 [A]  time = 0.905095, size = 4, normalized size = 0.01 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(13/2)*(g*x + f)/(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)^(5/2),x, algorithm="giac")

[Out]

sage0*x