3.2192 \(\int \frac{(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^7} \, dx\)

Optimal. Leaf size=210 \[ -\frac{4 c \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-9 b e g+14 c d g+4 c e f)}{315 e^2 (d+e x)^5 (2 c d-b e)^3}-\frac{2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-9 b e g+14 c d g+4 c e f)}{63 e^2 (d+e x)^6 (2 c d-b e)^2}-\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{9 e^2 (d+e x)^7 (2 c d-b e)} \]

[Out]

(-2*(e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(9*e^2*(2*c*d - b*e
)*(d + e*x)^7) - (2*(4*c*e*f + 14*c*d*g - 9*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*
e^2*x^2)^(5/2))/(63*e^2*(2*c*d - b*e)^2*(d + e*x)^6) - (4*c*(4*c*e*f + 14*c*d*g
- 9*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(315*e^2*(2*c*d - b*e)^3
*(d + e*x)^5)

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Rubi [A]  time = 0.765177, antiderivative size = 210, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 44, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.068 \[ -\frac{4 c \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-9 b e g+14 c d g+4 c e f)}{315 e^2 (d+e x)^5 (2 c d-b e)^3}-\frac{2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-9 b e g+14 c d g+4 c e f)}{63 e^2 (d+e x)^6 (2 c d-b e)^2}-\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{9 e^2 (d+e x)^7 (2 c d-b e)} \]

Antiderivative was successfully verified.

[In]  Int[((f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(3/2))/(d + e*x)^7,x]

[Out]

(-2*(e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(9*e^2*(2*c*d - b*e
)*(d + e*x)^7) - (2*(4*c*e*f + 14*c*d*g - 9*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*
e^2*x^2)^(5/2))/(63*e^2*(2*c*d - b*e)^2*(d + e*x)^6) - (4*c*(4*c*e*f + 14*c*d*g
- 9*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(315*e^2*(2*c*d - b*e)^3
*(d + e*x)^5)

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Rubi in Sympy [A]  time = 78.6533, size = 199, normalized size = 0.95 \[ - \frac{4 c \left (9 b e g - 14 c d g - 4 c e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{5}{2}}}{315 e^{2} \left (d + e x\right )^{5} \left (b e - 2 c d\right )^{3}} + \frac{2 \left (9 b e g - 14 c d g - 4 c e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{5}{2}}}{63 e^{2} \left (d + e x\right )^{6} \left (b e - 2 c d\right )^{2}} - \frac{2 \left (d g - e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{5}{2}}}{9 e^{2} \left (d + e x\right )^{7} \left (b e - 2 c d\right )} \]

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)**(3/2)/(e*x+d)**7,x)

[Out]

-4*c*(9*b*e*g - 14*c*d*g - 4*c*e*f)*(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))**
(5/2)/(315*e**2*(d + e*x)**5*(b*e - 2*c*d)**3) + 2*(9*b*e*g - 14*c*d*g - 4*c*e*f
)*(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))**(5/2)/(63*e**2*(d + e*x)**6*(b*e -
2*c*d)**2) - 2*(d*g - e*f)*(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))**(5/2)/(9
*e**2*(d + e*x)**7*(b*e - 2*c*d))

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Mathematica [A]  time = 0.417496, size = 168, normalized size = 0.8 \[ \frac{2 (b e-c d+c e x)^2 \sqrt{(d+e x) (c (d-e x)-b e)} \left (5 b^2 e^2 (2 d g+7 e f+9 e g x)-2 b c e \left (19 d^2 g+d e (80 f+98 g x)+e^2 x (10 f+9 g x)\right )+4 c^2 \left (7 d^3 g+d^2 e (47 f+49 g x)+7 d e^2 x (2 f+g x)+2 e^3 f x^2\right )\right )}{315 e^2 (d+e x)^5 (b e-2 c d)^3} \]

Antiderivative was successfully verified.

[In]  Integrate[((f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(3/2))/(d + e*x)^7,x]

[Out]

(2*(-(c*d) + b*e + c*e*x)^2*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*(5*b^2*e^2*(7
*e*f + 2*d*g + 9*e*g*x) + 4*c^2*(7*d^3*g + 2*e^3*f*x^2 + 7*d*e^2*x*(2*f + g*x) +
d^2*e*(47*f + 49*g*x)) - 2*b*c*e*(19*d^2*g + e^2*x*(10*f + 9*g*x) + d*e*(80*f +
98*g*x))))/(315*e^2*(-2*c*d + b*e)^3*(d + e*x)^5)

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Maple [A]  time = 0.015, size = 236, normalized size = 1.1 \[ -{\frac{ \left ( 2\,cex+2\,be-2\,cd \right ) \left ( -18\,bc{e}^{3}g{x}^{2}+28\,{c}^{2}d{e}^{2}g{x}^{2}+8\,{c}^{2}{e}^{3}f{x}^{2}+45\,{b}^{2}{e}^{3}gx-196\,bcd{e}^{2}gx-20\,bc{e}^{3}fx+196\,{c}^{2}{d}^{2}egx+56\,{c}^{2}d{e}^{2}fx+10\,{b}^{2}d{e}^{2}g+35\,{b}^{2}{e}^{3}f-38\,bc{d}^{2}eg-160\,bcd{e}^{2}f+28\,{c}^{2}{d}^{3}g+188\,{c}^{2}{d}^{2}ef \right ) }{315\, \left ( ex+d \right ) ^{6} \left ({b}^{3}{e}^{3}-6\,{b}^{2}cd{e}^{2}+12\,b{c}^{2}{d}^{2}e-8\,{c}^{3}{d}^{3} \right ){e}^{2}} \left ( -c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2} \right ) ^{{\frac{3}{2}}}} \]

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)^(3/2)/(e*x+d)^7,x)

[Out]

-2/315*(c*e*x+b*e-c*d)*(-18*b*c*e^3*g*x^2+28*c^2*d*e^2*g*x^2+8*c^2*e^3*f*x^2+45*
b^2*e^3*g*x-196*b*c*d*e^2*g*x-20*b*c*e^3*f*x+196*c^2*d^2*e*g*x+56*c^2*d*e^2*f*x+
10*b^2*d*e^2*g+35*b^2*e^3*f-38*b*c*d^2*e*g-160*b*c*d*e^2*f+28*c^2*d^3*g+188*c^2*
d^2*e*f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)/(e*x+d)^6/(b^3*e^3-6*b^2*c*d*e^2
+12*b*c^2*d^2*e-8*c^3*d^3)/e^2

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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)^(3/2)*(g*x + f)/(e*x + d)^7,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 13.9758, size = 998, normalized size = 4.75 \[ -\frac{2 \, \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}{\left (2 \,{\left (4 \, c^{4} e^{5} f +{\left (14 \, c^{4} d e^{4} - 9 \, b c^{3} e^{5}\right )} g\right )} x^{4} +{\left (4 \,{\left (10 \, c^{4} d e^{4} - b c^{3} e^{5}\right )} f +{\left (140 \, c^{4} d^{2} e^{3} - 104 \, b c^{3} d e^{4} + 9 \, b^{2} c^{2} e^{5}\right )} g\right )} x^{3} + 3 \,{\left ({\left (28 \, c^{4} d^{2} e^{3} - 8 \, b c^{3} d e^{4} + b^{2} c^{2} e^{5}\right )} f - 8 \,{\left (14 \, c^{4} d^{3} e^{2} - 28 \, b c^{3} d^{2} e^{3} + 17 \, b^{2} c^{2} d e^{4} - 3 \, b^{3} c e^{5}\right )} g\right )} x^{2} +{\left (188 \, c^{4} d^{4} e - 536 \, b c^{3} d^{3} e^{2} + 543 \, b^{2} c^{2} d^{2} e^{3} - 230 \, b^{3} c d e^{4} + 35 \, b^{4} e^{5}\right )} f + 2 \,{\left (14 \, c^{4} d^{5} - 47 \, b c^{3} d^{4} e + 57 \, b^{2} c^{2} d^{3} e^{2} - 29 \, b^{3} c d^{2} e^{3} + 5 \, b^{4} d e^{4}\right )} g -{\left (2 \,{\left (160 \, c^{4} d^{3} e^{2} - 282 \, b c^{3} d^{2} e^{3} + 147 \, b^{2} c^{2} d e^{4} - 25 \, b^{3} c e^{5}\right )} f -{\left (140 \, c^{4} d^{4} e - 456 \, b c^{3} d^{3} e^{2} + 537 \, b^{2} c^{2} d^{2} e^{3} - 266 \, b^{3} c d e^{4} + 45 \, b^{4} e^{5}\right )} g\right )} x\right )}}{315 \,{\left (8 \, c^{3} d^{8} e^{2} - 12 \, b c^{2} d^{7} e^{3} + 6 \, b^{2} c d^{6} e^{4} - b^{3} d^{5} e^{5} +{\left (8 \, c^{3} d^{3} e^{7} - 12 \, b c^{2} d^{2} e^{8} + 6 \, b^{2} c d e^{9} - b^{3} e^{10}\right )} x^{5} + 5 \,{\left (8 \, c^{3} d^{4} e^{6} - 12 \, b c^{2} d^{3} e^{7} + 6 \, b^{2} c d^{2} e^{8} - b^{3} d e^{9}\right )} x^{4} + 10 \,{\left (8 \, c^{3} d^{5} e^{5} - 12 \, b c^{2} d^{4} e^{6} + 6 \, b^{2} c d^{3} e^{7} - b^{3} d^{2} e^{8}\right )} x^{3} + 10 \,{\left (8 \, c^{3} d^{6} e^{4} - 12 \, b c^{2} d^{5} e^{5} + 6 \, b^{2} c d^{4} e^{6} - b^{3} d^{3} e^{7}\right )} x^{2} + 5 \,{\left (8 \, c^{3} d^{7} e^{3} - 12 \, b c^{2} d^{6} e^{4} + 6 \, b^{2} c d^{5} e^{5} - b^{3} d^{4} e^{6}\right )} x\right )}} \]

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)*(g*x + f)/(e*x + d)^7,x, algorithm="fricas")

[Out]

-2/315*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(2*(4*c^4*e^5*f + (14*c^4*d*e^
4 - 9*b*c^3*e^5)*g)*x^4 + (4*(10*c^4*d*e^4 - b*c^3*e^5)*f + (140*c^4*d^2*e^3 - 1
04*b*c^3*d*e^4 + 9*b^2*c^2*e^5)*g)*x^3 + 3*((28*c^4*d^2*e^3 - 8*b*c^3*d*e^4 + b^
2*c^2*e^5)*f - 8*(14*c^4*d^3*e^2 - 28*b*c^3*d^2*e^3 + 17*b^2*c^2*d*e^4 - 3*b^3*c
*e^5)*g)*x^2 + (188*c^4*d^4*e - 536*b*c^3*d^3*e^2 + 543*b^2*c^2*d^2*e^3 - 230*b^
3*c*d*e^4 + 35*b^4*e^5)*f + 2*(14*c^4*d^5 - 47*b*c^3*d^4*e + 57*b^2*c^2*d^3*e^2
- 29*b^3*c*d^2*e^3 + 5*b^4*d*e^4)*g - (2*(160*c^4*d^3*e^2 - 282*b*c^3*d^2*e^3 +
147*b^2*c^2*d*e^4 - 25*b^3*c*e^5)*f - (140*c^4*d^4*e - 456*b*c^3*d^3*e^2 + 537*b
^2*c^2*d^2*e^3 - 266*b^3*c*d*e^4 + 45*b^4*e^5)*g)*x)/(8*c^3*d^8*e^2 - 12*b*c^2*d
^7*e^3 + 6*b^2*c*d^6*e^4 - b^3*d^5*e^5 + (8*c^3*d^3*e^7 - 12*b*c^2*d^2*e^8 + 6*b
^2*c*d*e^9 - b^3*e^10)*x^5 + 5*(8*c^3*d^4*e^6 - 12*b*c^2*d^3*e^7 + 6*b^2*c*d^2*e
^8 - b^3*d*e^9)*x^4 + 10*(8*c^3*d^5*e^5 - 12*b*c^2*d^4*e^6 + 6*b^2*c*d^3*e^7 - b
^3*d^2*e^8)*x^3 + 10*(8*c^3*d^6*e^4 - 12*b*c^2*d^5*e^5 + 6*b^2*c*d^4*e^6 - b^3*d
^3*e^7)*x^2 + 5*(8*c^3*d^7*e^3 - 12*b*c^2*d^6*e^4 + 6*b^2*c*d^5*e^5 - b^3*d^4*e^
6)*x)

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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)**(3/2)/(e*x+d)**7,x)

[Out]

Timed out

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

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)*(g*x + f)/(e*x + d)^7,x, algorithm="giac")

[Out]

Exception raised: TypeError