3.2253 \(\int \frac{(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{\sqrt{d+e x}} \, dx\)

Optimal. Leaf size=270 \[ -\frac{16 (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-6 b e g-c d g+13 c e f)}{9009 c^4 e^2 (d+e x)^{7/2}}-\frac{8 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-6 b e g-c d g+13 c e f)}{1287 c^3 e^2 (d+e x)^{5/2}}-\frac{2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-6 b e g-c d g+13 c e f)}{143 c^2 e^2 (d+e x)^{3/2}}-\frac{2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{13 c e^2 \sqrt{d+e x}} \]

[Out]

(-16*(2*c*d - b*e)^2*(13*c*e*f - c*d*g - 6*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e
^2*x^2)^(7/2))/(9009*c^4*e^2*(d + e*x)^(7/2)) - (8*(2*c*d - b*e)*(13*c*e*f - c*d
*g - 6*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(1287*c^3*e^2*(d + e*
x)^(5/2)) - (2*(13*c*e*f - c*d*g - 6*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2
)^(7/2))/(143*c^2*e^2*(d + e*x)^(3/2)) - (2*g*(d*(c*d - b*e) - b*e^2*x - c*e^2*x
^2)^(7/2))/(13*c*e^2*Sqrt[d + e*x])

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Rubi [A]  time = 0.950962, antiderivative size = 270, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065 \[ -\frac{16 (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-6 b e g-c d g+13 c e f)}{9009 c^4 e^2 (d+e x)^{7/2}}-\frac{8 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-6 b e g-c d g+13 c e f)}{1287 c^3 e^2 (d+e x)^{5/2}}-\frac{2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-6 b e g-c d g+13 c e f)}{143 c^2 e^2 (d+e x)^{3/2}}-\frac{2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{13 c e^2 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

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

[Out]

(-16*(2*c*d - b*e)^2*(13*c*e*f - c*d*g - 6*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e
^2*x^2)^(7/2))/(9009*c^4*e^2*(d + e*x)^(7/2)) - (8*(2*c*d - b*e)*(13*c*e*f - c*d
*g - 6*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(1287*c^3*e^2*(d + e*
x)^(5/2)) - (2*(13*c*e*f - c*d*g - 6*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2
)^(7/2))/(143*c^2*e^2*(d + e*x)^(3/2)) - (2*g*(d*(c*d - b*e) - b*e^2*x - c*e^2*x
^2)^(7/2))/(13*c*e^2*Sqrt[d + e*x])

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Rubi in Sympy [A]  time = 91.4135, size = 257, normalized size = 0.95 \[ - \frac{2 g \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{7}{2}}}{13 c e^{2} \sqrt{d + e x}} + \frac{2 \left (6 b e g + c d g - 13 c e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{7}{2}}}{143 c^{2} e^{2} \left (d + e x\right )^{\frac{3}{2}}} - \frac{8 \left (b e - 2 c d\right ) \left (6 b e g + c d g - 13 c e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{7}{2}}}{1287 c^{3} e^{2} \left (d + e x\right )^{\frac{5}{2}}} + \frac{16 \left (b e - 2 c d\right )^{2} \left (6 b e g + c d g - 13 c e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{7}{2}}}{9009 c^{4} e^{2} \left (d + e x\right )^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((g*x+f)*(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(5/2)/(e*x+d)**(1/2),x)

[Out]

-2*g*(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))**(7/2)/(13*c*e**2*sqrt(d + e*x))
+ 2*(6*b*e*g + c*d*g - 13*c*e*f)*(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))**(7
/2)/(143*c**2*e**2*(d + e*x)**(3/2)) - 8*(b*e - 2*c*d)*(6*b*e*g + c*d*g - 13*c*e
*f)*(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))**(7/2)/(1287*c**3*e**2*(d + e*x)*
*(5/2)) + 16*(b*e - 2*c*d)**2*(6*b*e*g + c*d*g - 13*c*e*f)*(-b*e**2*x - c*e**2*x
**2 + d*(-b*e + c*d))**(7/2)/(9009*c**4*e**2*(d + e*x)**(7/2))

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Mathematica [A]  time = 0.46863, size = 183, normalized size = 0.68 \[ \frac{2 (b e-c d+c e x)^3 \sqrt{(d+e x) (c (d-e x)-b e)} \left (-48 b^3 e^3 g+8 b^2 c e^2 (44 d g+13 e f+21 e g x)-2 b c^2 e \left (423 d^2 g+d e (390 f+532 g x)+7 e^2 x (26 f+27 g x)\right )+c^3 \left (542 d^3 g+d^2 e (1963 f+1897 g x)+14 d e^2 x (169 f+144 g x)+63 e^3 x^2 (13 f+11 g x)\right )\right )}{9009 c^4 e^2 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(-(c*d) + b*e + c*e*x)^3*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*(-48*b^3*e^3*
g + 8*b^2*c*e^2*(13*e*f + 44*d*g + 21*e*g*x) - 2*b*c^2*e*(423*d^2*g + 7*e^2*x*(2
6*f + 27*g*x) + d*e*(390*f + 532*g*x)) + c^3*(542*d^3*g + 63*e^3*x^2*(13*f + 11*
g*x) + 14*d*e^2*x*(169*f + 144*g*x) + d^2*e*(1963*f + 1897*g*x))))/(9009*c^4*e^2
*Sqrt[d + e*x])

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Maple [A]  time = 0.013, size = 235, normalized size = 0.9 \[ -{\frac{ \left ( 2\,cex+2\,be-2\,cd \right ) \left ( -693\,{e}^{3}g{x}^{3}{c}^{3}+378\,b{c}^{2}{e}^{3}g{x}^{2}-2016\,{c}^{3}d{e}^{2}g{x}^{2}-819\,{c}^{3}{e}^{3}f{x}^{2}-168\,{b}^{2}c{e}^{3}gx+1064\,b{c}^{2}d{e}^{2}gx+364\,b{c}^{2}{e}^{3}fx-1897\,{c}^{3}{d}^{2}egx-2366\,{c}^{3}d{e}^{2}fx+48\,{b}^{3}{e}^{3}g-352\,{b}^{2}cd{e}^{2}g-104\,{b}^{2}c{e}^{3}f+846\,b{c}^{2}{d}^{2}eg+780\,b{c}^{2}d{e}^{2}f-542\,{c}^{3}{d}^{3}g-1963\,f{d}^{2}{c}^{3}e \right ) }{9009\,{c}^{4}{e}^{2}} \left ( -c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2} \right ) ^{{\frac{5}{2}}} \left ( ex+d \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/(e*x+d)^(1/2),x)

[Out]

-2/9009*(c*e*x+b*e-c*d)*(-693*c^3*e^3*g*x^3+378*b*c^2*e^3*g*x^2-2016*c^3*d*e^2*g
*x^2-819*c^3*e^3*f*x^2-168*b^2*c*e^3*g*x+1064*b*c^2*d*e^2*g*x+364*b*c^2*e^3*f*x-
1897*c^3*d^2*e*g*x-2366*c^3*d*e^2*f*x+48*b^3*e^3*g-352*b^2*c*d*e^2*g-104*b^2*c*e
^3*f+846*b*c^2*d^2*e*g+780*b*c^2*d*e^2*f-542*c^3*d^3*g-1963*c^3*d^2*e*f)*(-c*e^2
*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/c^4/e^2/(e*x+d)^(5/2)

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Maxima [A]  time = 0.739747, size = 861, normalized size = 3.19 \[ \frac{2 \,{\left (63 \, c^{5} e^{5} x^{5} - 151 \, c^{5} d^{5} + 513 \, b c^{4} d^{4} e - 641 \, b^{2} c^{3} d^{3} e^{2} + 355 \, b^{3} c^{2} d^{2} e^{3} - 84 \, b^{4} c d e^{4} + 8 \, b^{5} e^{5} - 7 \,{\left (c^{5} d e^{4} - 23 \, b c^{4} e^{5}\right )} x^{4} -{\left (206 \, c^{5} d^{2} e^{3} - 192 \, b c^{4} d e^{4} - 113 \, b^{2} c^{3} e^{5}\right )} x^{3} + 3 \,{\left (10 \, c^{5} d^{3} e^{2} - 118 \, b c^{4} d^{2} e^{3} + 107 \, b^{2} c^{3} d e^{4} + b^{3} c^{2} e^{5}\right )} x^{2} +{\left (271 \, c^{5} d^{4} e - 512 \, b c^{4} d^{3} e^{2} + 207 \, b^{2} c^{3} d^{2} e^{3} + 38 \, b^{3} c^{2} d e^{4} - 4 \, b^{4} c e^{5}\right )} x\right )} \sqrt{-c e x + c d - b e} f}{693 \, c^{3} e} + \frac{2 \,{\left (693 \, c^{6} e^{6} x^{6} - 542 \, c^{6} d^{6} + 2472 \, b c^{5} d^{5} e - 4516 \, b^{2} c^{4} d^{4} e^{2} + 4184 \, b^{3} c^{3} d^{3} e^{3} - 2046 \, b^{4} c^{2} d^{2} e^{4} + 496 \, b^{5} c d e^{5} - 48 \, b^{6} e^{6} - 63 \,{\left (c^{6} d e^{5} - 27 \, b c^{5} e^{6}\right )} x^{5} - 7 \,{\left (296 \, c^{6} d^{2} e^{4} - 280 \, b c^{5} d e^{5} - 159 \, b^{2} c^{4} e^{6}\right )} x^{4} +{\left (206 \, c^{6} d^{3} e^{3} - 3114 \, b c^{5} d^{2} e^{4} + 2893 \, b^{2} c^{4} d e^{5} + 15 \, b^{3} c^{3} e^{6}\right )} x^{3} + 3 \,{\left (683 \, c^{6} d^{4} e^{2} - 1328 \, b c^{5} d^{3} e^{3} + 601 \, b^{2} c^{4} d^{2} e^{4} + 50 \, b^{3} c^{3} d e^{5} - 6 \, b^{4} c^{2} e^{6}\right )} x^{2} -{\left (271 \, c^{6} d^{5} e - 965 \, b c^{5} d^{4} e^{2} + 1293 \, b^{2} c^{4} d^{3} e^{3} - 799 \, b^{3} c^{3} d^{2} e^{4} + 224 \, b^{4} c^{2} d e^{5} - 24 \, b^{5} c e^{6}\right )} x\right )} \sqrt{-c e x + c d - b e} g}{9009 \, c^{4} e^{2}} \]

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)*(g*x + f)/sqrt(e*x + d),x, algorithm="maxima")

[Out]

2/693*(63*c^5*e^5*x^5 - 151*c^5*d^5 + 513*b*c^4*d^4*e - 641*b^2*c^3*d^3*e^2 + 35
5*b^3*c^2*d^2*e^3 - 84*b^4*c*d*e^4 + 8*b^5*e^5 - 7*(c^5*d*e^4 - 23*b*c^4*e^5)*x^
4 - (206*c^5*d^2*e^3 - 192*b*c^4*d*e^4 - 113*b^2*c^3*e^5)*x^3 + 3*(10*c^5*d^3*e^
2 - 118*b*c^4*d^2*e^3 + 107*b^2*c^3*d*e^4 + b^3*c^2*e^5)*x^2 + (271*c^5*d^4*e -
512*b*c^4*d^3*e^2 + 207*b^2*c^3*d^2*e^3 + 38*b^3*c^2*d*e^4 - 4*b^4*c*e^5)*x)*sqr
t(-c*e*x + c*d - b*e)*f/(c^3*e) + 2/9009*(693*c^6*e^6*x^6 - 542*c^6*d^6 + 2472*b
*c^5*d^5*e - 4516*b^2*c^4*d^4*e^2 + 4184*b^3*c^3*d^3*e^3 - 2046*b^4*c^2*d^2*e^4
+ 496*b^5*c*d*e^5 - 48*b^6*e^6 - 63*(c^6*d*e^5 - 27*b*c^5*e^6)*x^5 - 7*(296*c^6*
d^2*e^4 - 280*b*c^5*d*e^5 - 159*b^2*c^4*e^6)*x^4 + (206*c^6*d^3*e^3 - 3114*b*c^5
*d^2*e^4 + 2893*b^2*c^4*d*e^5 + 15*b^3*c^3*e^6)*x^3 + 3*(683*c^6*d^4*e^2 - 1328*
b*c^5*d^3*e^3 + 601*b^2*c^4*d^2*e^4 + 50*b^3*c^3*d*e^5 - 6*b^4*c^2*e^6)*x^2 - (2
71*c^6*d^5*e - 965*b*c^5*d^4*e^2 + 1293*b^2*c^4*d^3*e^3 - 799*b^3*c^3*d^2*e^4 +
224*b^4*c^2*d*e^5 - 24*b^5*c*e^6)*x)*sqrt(-c*e*x + c*d - b*e)*g/(c^4*e^2)

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Fricas [A]  time = 0.30458, size = 1442, normalized size = 5.34 \[ \text{result too large to display} \]

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)*(g*x + f)/sqrt(e*x + d),x, algorithm="fricas")

[Out]

-2/9009*(693*c^7*e^8*g*x^8 + 63*(13*c^7*e^8*f - (c^7*d*e^7 - 38*b*c^6*e^8)*g)*x^
7 - 7*(13*(c^7*d*e^7 - 32*b*c^6*e^8)*f + (395*c^7*d^2*e^6 - 370*b*c^6*d*e^7 - 40
2*b^2*c^5*e^8)*g)*x^6 - (13*(269*c^7*d^2*e^6 - 248*b*c^6*d*e^7 - 274*b^2*c^5*e^8
)*f - (269*c^7*d^3*e^5 - 6950*b*c^6*d^2*e^6 + 6554*b^2*c^5*d*e^7 + 1128*b^3*c^4*
e^8)*g)*x^5 + (13*(37*c^7*d^3*e^5 - 728*b*c^6*d^2*e^6 + 674*b^2*c^5*d*e^7 + 116*
b^3*c^4*e^8)*f + (4121*c^7*d^4*e^4 - 7810*b*c^6*d^3*e^5 - 464*b^2*c^5*d^2*e^6 +
4156*b^3*c^4*d*e^7 - 3*b^4*c^3*e^8)*g)*x^4 + (13*(477*c^7*d^4*e^4 - 880*b*c^6*d^
3*e^5 - 68*b^2*c^5*d^2*e^6 + 472*b^3*c^4*d*e^7 - b^4*c^3*e^8)*f - (477*c^7*d^5*e
^3 - 6334*b*c^6*d^4*e^4 + 11284*b^2*c^5*d^3*e^5 - 5480*b^3*c^4*d^2*e^6 + 59*b^4*
c^3*d*e^7 - 6*b^5*c^2*e^8)*g)*x^3 - (13*(181*c^7*d^5*e^3 - 1168*b*c^6*d^4*e^4 +
1828*b^2*c^5*d^3*e^5 - 880*b^3*c^4*d^2*e^6 + 43*b^4*c^3*d*e^7 - 4*b^5*c^2*e^8)*f
+ (2591*c^7*d^6*e^2 - 8234*b*c^6*d^5*e^3 + 9338*b^2*c^5*d^4*e^4 - 4544*b^3*c^4*
d^3*e^5 + 1079*b^4*c^3*d^2*e^6 - 254*b^5*c^2*d*e^7 + 24*b^6*c*e^8)*g)*x^2 + 13*(
151*c^7*d^7*e - 664*b*c^6*d^6*e^2 + 1154*b^2*c^5*d^5*e^3 - 996*b^3*c^4*d^4*e^4 +
439*b^4*c^3*d^3*e^5 - 92*b^5*c^2*d^2*e^6 + 8*b^6*c*d*e^7)*f + 2*(271*c^7*d^8 -
1507*b*c^6*d^7*e + 3494*b^2*c^5*d^6*e^2 - 4350*b^3*c^4*d^5*e^3 + 3115*b^4*c^3*d^
4*e^4 - 1271*b^5*c^2*d^3*e^5 + 272*b^6*c*d^2*e^6 - 24*b^7*d*e^7)*g - (13*(271*c^
7*d^6*e^2 - 632*b*c^6*d^5*e^3 + 206*b^2*c^5*d^4*e^4 + 472*b^3*c^4*d^3*e^5 - 397*
b^4*c^3*d^2*e^6 + 88*b^5*c^2*d*e^7 - 8*b^6*c*e^8)*f - (271*c^7*d^7*e - 1778*b*c^
6*d^6*e^2 + 4730*b^2*c^5*d^5*e^3 - 6608*b^3*c^4*d^4*e^4 + 5207*b^4*c^3*d^3*e^5 -
2294*b^5*c^2*d^2*e^6 + 520*b^6*c*d*e^7 - 48*b^7*e^8)*g)*x)/(sqrt(-c*e^2*x^2 - b
*e^2*x + c*d^2 - b*d*e)*sqrt(e*x + d)*c^4*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((g*x+f)*(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(5/2)/(e*x+d)**(1/2),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)*(g*x + f)/sqrt(e*x + d),x, algorithm="giac")

[Out]

Timed out