∖int(f+gx)∖left(cd2−bde−be2x−ce2x2∖right)5∕2 ____________________________________________________________________(d+ex)4∖,dx

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3.2201 \(\int \frac{(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^4} \, dx\)

Optimal. Leaf size=342 \[ -\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{e^2 (d+e x)^4 (2 c d-b e)}-\frac{2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (b e g-8 c d g+6 c e f)}{e^2 (d+e x)^2 (2 c d-b e)}-\frac{5 c \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (b e g-8 c d g+6 c e f)}{3 e^2 (2 c d-b e)}-\frac{5 (b+2 c x) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (b e g-8 c d g+6 c e f)}{8 e}-\frac{5 (2 c d-b e)^2 (b e g-8 c d g+6 c e f) \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{16 \sqrt{c} e^2} \]

[Out]

(-5*(6*c*e*f - 8*c*d*g + b*e*g)*(b + 2*c*x)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2

*x^2])/(8*e) - (5*c*(6*c*e*f - 8*c*d*g + b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2

*x^2)^(3/2))/(3*e^2*(2*c*d - b*e)) - (2*(6*c*e*f - 8*c*d*g + b*e*g)*(d*(c*d - b*

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

)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(e^2*(2*c*d - b*e)*(d + e*x)^4) -

(5*(2*c*d - b*e)^2*(6*c*e*f - 8*c*d*g + 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])])/(16*Sqrt[c]*e^2)

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Rubi [A] time = 1.21323, antiderivative size = 342, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 44, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ -\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{e^2 (d+e x)^4 (2 c d-b e)}-\frac{2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (b e g-8 c d g+6 c e f)}{e^2 (d+e x)^2 (2 c d-b e)}-\frac{5 c \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (b e g-8 c d g+6 c e f)}{3 e^2 (2 c d-b e)}-\frac{5 (b+2 c x) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (b e g-8 c d g+6 c e f)}{8 e}-\frac{5 (2 c d-b e)^2 (b e g-8 c d g+6 c e f) \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{16 \sqrt{c} e^2} \]

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))/(d + e*x)^4,x]

[Out]

(-5*(6*c*e*f - 8*c*d*g + b*e*g)*(b + 2*c*x)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2

*x^2])/(8*e) - (5*c*(6*c*e*f - 8*c*d*g + b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2

*x^2)^(3/2))/(3*e^2*(2*c*d - b*e)) - (2*(6*c*e*f - 8*c*d*g + b*e*g)*(d*(c*d - b*

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

)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(e^2*(2*c*d - b*e)*(d + e*x)^4) -

(5*(2*c*d - b*e)^2*(6*c*e*f - 8*c*d*g + 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])])/(16*Sqrt[c]*e^2)

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Rubi in Sympy [A] time = 117.959, size = 332, normalized size = 0.97 \[ \frac{5 c \left (b e g - 8 c d g + 6 c e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{3}{2}}}{3 e^{2} \left (b e - 2 c d\right )} - \frac{5 \left (b + 2 c x\right ) \left (\frac{b e g}{2} - 4 c d g + 3 c e f\right ) \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}}{4 e} + \frac{4 \left (\frac{b e g}{2} - 4 c d g + 3 c e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{5}{2}}}{e^{2} \left (d + e x\right )^{2} \left (b e - 2 c d\right )} - \frac{2 \left (d g - e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{7}{2}}}{e^{2} \left (d + e x\right )^{4} \left (b e - 2 c d\right )} - \frac{5 \left (b e - 2 c d\right )^{2} \left (b e g - 8 c d g + 6 c e f\right ) \operatorname{atan}{\left (- \frac{e \left (- b - 2 c x\right )}{2 \sqrt{c} \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}} \right )}}{16 \sqrt{c} e^{2}} \]

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)**4,x)

[Out]

5*c*(b*e*g - 8*c*d*g + 6*c*e*f)*(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))**(3/2

)/(3*e**2*(b*e - 2*c*d)) - 5*(b + 2*c*x)*(b*e*g/2 - 4*c*d*g + 3*c*e*f)*sqrt(-b*e

**2*x - c*e**2*x**2 + d*(-b*e + c*d))/(4*e) + 4*(b*e*g/2 - 4*c*d*g + 3*c*e*f)*(-

b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))**(5/2)/(e**2*(d + e*x)**2*(b*e - 2*c*d)

) - 2*(d*g - e*f)*(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))**(7/2)/(e**2*(d + e

*x)**4*(b*e - 2*c*d)) - 5*(b*e - 2*c*d)**2*(b*e*g - 8*c*d*g + 6*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))))/(16*sqrt

(c)*e**2)

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Mathematica [C] time = 1.80804, size = 273, normalized size = 0.8 \[ \frac{((d+e x) (c (d-e x)-b e))^{5/2} \left (\frac{6 b^2 e^2 (27 d g-16 e f+11 e g x)+4 b c e \left (-176 d^2 g+d e (123 f-67 g x)+e^2 x (27 f+13 g x)\right )+8 c^2 \left (94 d^3 g+d^2 e (34 g x-72 f)-d e^2 x (21 f+10 g x)+e^3 x^2 (3 f+2 g x)\right )}{(d+e x)^3 (b e-c d+c e x)^2}-\frac{15 i (b e-2 c d)^2 (b e g-8 c d g+6 c e f) \log \left (2 \sqrt{d+e x} \sqrt{c (d-e x)-b e}-\frac{i e (b+2 c x)}{\sqrt{c}}\right )}{\sqrt{c} (d+e x)^{5/2} (c (d-e x)-b e)^{5/2}}\right )}{48 e^2} \]

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))/(d + e*x)^4,x]

[Out]

(((d + e*x)*(-(b*e) + c*(d - e*x)))^(5/2)*((6*b^2*e^2*(-16*e*f + 27*d*g + 11*e*g

*x) + 4*b*c*e*(-176*d^2*g + d*e*(123*f - 67*g*x) + e^2*x*(27*f + 13*g*x)) + 8*c^

2*(94*d^3*g + e^3*x^2*(3*f + 2*g*x) - d*e^2*x*(21*f + 10*g*x) + d^2*e*(-72*f + 3

4*g*x)))/((d + e*x)^3*(-(c*d) + b*e + c*e*x)^2) - ((15*I)*(-2*c*d + b*e)^2*(6*c*

e*f - 8*c*d*g + b*e*g)*Log[((-I)*e*(b + 2*c*x))/Sqrt[c] + 2*Sqrt[d + e*x]*Sqrt[-

(b*e) + c*(d - e*x)]])/(Sqrt[c]*(d + e*x)^(5/2)*(-(b*e) + c*(d - e*x))^(5/2))))/

(48*e^2)

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Maple [B] time = 0.03, size = 4987, normalized size = 14.6 \[ \text{output too large to display} \]

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)^4,x)

[Out]

-5/8*g*e^4/(-b*e^2+2*c*d*e)^2*b^4*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(1

/2)+32/e*c^2/(-b*e^2+2*c*d*e)^3/(d/e+x)^2*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/

e+x))^(7/2)*d*g-5/16*g*e^6/(-b*e^2+2*c*d*e)^2*b^5/(c*e^2)^(1/2)*arctan((c*e^2)^(

1/2)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+

x))^(1/2))-5/3*g*e^2*c/(-b*e^2+2*c*d*e)^2*b^2*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)

*(d/e+x))^(3/2)+16/3*g/e^2*c/(-b*e^2+2*c*d*e)^2/(d/e+x)^2*(-c*(d/e+x)^2*e^2+(-b*

e^2+2*c*d*e)*(d/e+x))^(7/2)+32*e*c^3/(-b*e^2+2*c*d*e)^3*(-c*(d/e+x)^2*e^2+(-b*e^

2+2*c*d*e)*(d/e+x))^(5/2)*d*g+10*e^4*c^2/(-b*e^2+2*c*d*e)^3*b^2*(-c*(d/e+x)^2*e^

2+(-b*e^2+2*c*d*e)*(d/e+x))^(3/2)*f+15/4*e^6*c/(-b*e^2+2*c*d*e)^3*b^4*(-c*(d/e+x

)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(1/2)*f-12/e^2*c/(-b*e^2+2*c*d*e)^2/(d/e+x)^3*

(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(7/2)*f+2/e^5/(-b*e^2+2*c*d*e)/(d/e+

x)^4*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(7/2)*d*g+45*e^4*c^3/(-b*e^2+2*

c*d*e)^3*b^2*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(1/2)*d^2*f-45*e^3*c^3/

(-b*e^2+2*c*d*e)^3*b^2*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(1/2)*d^3*g+1

5/2*e^6*c^2/(-b*e^2+2*c*d*e)^3*b^3*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(

1/2)*x*f+12/e^3*c/(-b*e^2+2*c*d*e)^2/(d/e+x)^3*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e

)*(d/e+x))^(7/2)*d*g+45/2*e^4*c^2/(-b*e^2+2*c*d*e)^3*b^3*(-c*(d/e+x)^2*e^2+(-b*e

^2+2*c*d*e)*(d/e+x))^(1/2)*d^2*g-45/2*e^5*c^2/(-b*e^2+2*c*d*e)^3*b^3*(-c*(d/e+x)

^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(1/2)*d*f+15/8*e^8*c/(-b*e^2+2*c*d*e)^3*b^5/(c*

e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-c*(d/e+x)^2

*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(1/2))*f-10*e^3*c^2/(-b*e^2+2*c*d*e)^3*b^2*(-c*(d

/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(3/2)*d*g+10*g*e*c^5/(-b*e^2+2*c*d*e)^2*d^

5/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-c*(d/e

+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(1/2))-5/4*g*e^4*c/(-b*e^2+2*c*d*e)^2*b^3*(-

c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(1/2)*x-15/4*e^5*c/(-b*e^2+2*c*d*e)^3*

b^4*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(1/2)*d*g+20*e^4*c^3/(-b*e^2+2*c

*d*e)^3*b*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(3/2)*x*f+40*e^2*c^4/(-b*e

^2+2*c*d*e)^3*d^2*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(3/2)*x*g-40*e^3*c

^4/(-b*e^2+2*c*d*e)^3*d*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(3/2)*x*f+20

*e^2*c^3/(-b*e^2+2*c*d*e)^3*d^2*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(3/2

)*b*g-20*e^3*c^3/(-b*e^2+2*c*d*e)^3*d*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x)

)^(3/2)*b*f+60*e^2*c^5/(-b*e^2+2*c*d*e)^3*d^4*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)

*(d/e+x))^(1/2)*x*g-60*e^3*c^5/(-b*e^2+2*c*d*e)^3*d^3*(-c*(d/e+x)^2*e^2+(-b*e^2+

2*c*d*e)*(d/e+x))^(1/2)*x*f+30*e^2*c^4/(-b*e^2+2*c*d*e)^3*d^4*(-c*(d/e+x)^2*e^2+

(-b*e^2+2*c*d*e)*(d/e+x))^(1/2)*b*g-30*e^3*c^4/(-b*e^2+2*c*d*e)^3*d^3*(-c*(d/e+x

)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(1/2)*b*f+60*e^2*c^6/(-b*e^2+2*c*d*e)^3*d^6/(c

*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-c*(d/e+x)^

2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(1/2))*g-60*e^3*c^6/(-b*e^2+2*c*d*e)^3*d^5/(c*e^

2)^(1/2)*arctan((c*e^2)^(1/2)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-c*(d/e+x)^2*e

^2+(-b*e^2+2*c*d*e)*(d/e+x))^(1/2))*f-150*e^3*c^5/(-b*e^2+2*c*d*e)^3*b/(c*e^2)^(

1/2)*arctan((c*e^2)^(1/2)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-c*(d/e+x)^2*e^2+(

-b*e^2+2*c*d*e)*(d/e+x))^(1/2))*d^5*g-15/2*e^5*c^2/(-b*e^2+2*c*d*e)^3*b^3*(-c*(d

/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(1/2)*x*d*g+150*e^4*c^4/(-b*e^2+2*c*d*e)^3

*b^2/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-c*(

d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(1/2))*d^4*g+75*e^6*c^3/(-b*e^2+2*c*d*e)^

3*b^3/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-c*

(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(1/2))*d^2*f-25/2*g*e^4*c^2/(-b*e^2+2*c*

d*e)^2*b^3/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)

/(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(1/2))*d^2-25*g*e^2*c^4/(-b*e^2+2*c

*d*e)^2*b/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/

(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(1/2))*d^4+15/2*g*e^3*c^2/(-b*e^2+2*

c*d*e)^2*b^2*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(1/2)*x*d-15*g*e^2*c^3/

(-b*e^2+2*c*d*e)^2*b*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(1/2)*x*d^2-15/

8*e^7*c/(-b*e^2+2*c*d*e)^3*b^5/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+d/e-1/2*(-b

*e^2+2*c*d*e)/c/e^2)/(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(1/2))*d*g-20*e

^3*c^3/(-b*e^2+2*c*d*e)^3*b*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(3/2)*x*

d*g-75*e^5*c^3/(-b*e^2+2*c*d*e)^3*b^3/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+d/e-

1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(1/2))*d

^3*g+45*e^4*c^3/(-b*e^2+2*c*d*e)^3*b^2*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x

))^(1/2)*x*d^2*g+150*e^4*c^5/(-b*e^2+2*c*d*e)^3*b/(c*e^2)^(1/2)*arctan((c*e^2)^(

1/2)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+

x))^(1/2))*d^4*f-75/4*e^7*c^2/(-b*e^2+2*c*d*e)^3*b^4/(c*e^2)^(1/2)*arctan((c*e^2

)^(1/2)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d

/e+x))^(1/2))*d*f+90*e^4*c^4/(-b*e^2+2*c*d*e)^3*b*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*

d*e)*(d/e+x))^(1/2)*x*d^2*f+75/4*e^6*c^2/(-b*e^2+2*c*d*e)^3*b^4/(c*e^2)^(1/2)*ar

ctan((c*e^2)^(1/2)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-c*(d/e+x)^2*e^2+(-b*e^2+

2*c*d*e)*(d/e+x))^(1/2))*d^2*g+15/4*g*e^3*c/(-b*e^2+2*c*d*e)^2*b^3*(-c*(d/e+x)^2

*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(1/2)*d-15/2*g*e^2*c^2/(-b*e^2+2*c*d*e)^2*b^2*(-c

*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(1/2)*d^2-10/3*g*e^2*c^2/(-b*e^2+2*c*d*

e)^2*b*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(3/2)*x+20/3*g*e*c^3/(-b*e^2+

2*c*d*e)^2*d*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(3/2)*x+10/3*g*e*c^2/(-

b*e^2+2*c*d*e)^2*d*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(3/2)*b+10*g*e*c^

4/(-b*e^2+2*c*d*e)^2*d^3*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(1/2)*x+5*g

*e*c^3/(-b*e^2+2*c*d*e)^2*d^3*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(1/2)*

b+25*g*e^3*c^3/(-b*e^2+2*c*d*e)^2*b^2/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+d/e-

1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(1/2))*d

^3-150*e^5*c^4/(-b*e^2+2*c*d*e)^3*b^2/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+d/e-

1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(1/2))*d

^3*f-45*e^5*c^3/(-b*e^2+2*c*d*e)^3*b^2*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x

))^(1/2)*x*d*f-90*e^3*c^4/(-b*e^2+2*c*d*e)^3*b*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e

)*(d/e+x))^(1/2)*x*d^3*g+2*g/e^4/(-b*e^2+2*c*d*e)/(d/e+x)^3*(-c*(d/e+x)^2*e^2+(-

b*e^2+2*c*d*e)*(d/e+x))^(7/2)+16/3*g*c^2/(-b*e^2+2*c*d*e)^2*(-c*(d/e+x)^2*e^2+(-

b*e^2+2*c*d*e)*(d/e+x))^(5/2)-32*c^2/(-b*e^2+2*c*d*e)^3/(d/e+x)^2*(-c*(d/e+x)^2*

e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(7/2)*f-2/e^4/(-b*e^2+2*c*d*e)/(d/e+x)^4*(-c*(d/e+

x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(7/2)*f-32*e^2*c^3/(-b*e^2+2*c*d*e)^3*(-c*(d/

e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(5/2)*f+25/8*g*e^5*c/(-b*e^2+2*c*d*e)^2*b^4

/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-c*(d/e+

x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(1/2))*d

_______________________________________________________________________________________

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)*(g*x + f)/(e*x + d)^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A] time = 1.63587, 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)*(g*x + f)/(e*x + d)^4,x, algorithm="fricas")

[Out]

[1/96*(4*(8*c^2*e^3*g*x^3 + 2*(6*c^2*e^3*f - (20*c^2*d*e^2 - 13*b*c*e^3)*g)*x^2

- 6*(48*c^2*d^2*e - 41*b*c*d*e^2 + 8*b^2*e^3)*f + (376*c^2*d^3 - 352*b*c*d^2*e +

81*b^2*d*e^2)*g - (6*(14*c^2*d*e^2 - 9*b*c*e^3)*f - (136*c^2*d^2*e - 134*b*c*d*

e^2 + 33*b^2*e^3)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*sqrt(-c) + 15

*(6*(4*c^3*d^3*e - 4*b*c^2*d^2*e^2 + b^2*c*d*e^3)*f - (32*c^3*d^4 - 36*b*c^2*d^3

*e + 12*b^2*c*d^2*e^2 - b^3*d*e^3)*g + (6*(4*c^3*d^2*e^2 - 4*b*c^2*d*e^3 + b^2*c

*e^4)*f - (32*c^3*d^3*e - 36*b*c^2*d^2*e^2 + 12*b^2*c*d*e^3 - b^3*e^4)*g)*x)*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)))/((e^3*x + d*e^2)

*sqrt(-c)), 1/48*(2*(8*c^2*e^3*g*x^3 + 2*(6*c^2*e^3*f - (20*c^2*d*e^2 - 13*b*c*e

^3)*g)*x^2 - 6*(48*c^2*d^2*e - 41*b*c*d*e^2 + 8*b^2*e^3)*f + (376*c^2*d^3 - 352*

b*c*d^2*e + 81*b^2*d*e^2)*g - (6*(14*c^2*d*e^2 - 9*b*c*e^3)*f - (136*c^2*d^2*e -

134*b*c*d*e^2 + 33*b^2*e^3)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*sq

rt(c) - 15*(6*(4*c^3*d^3*e - 4*b*c^2*d^2*e^2 + b^2*c*d*e^3)*f - (32*c^3*d^4 - 36

*b*c^2*d^3*e + 12*b^2*c*d^2*e^2 - b^3*d*e^3)*g + (6*(4*c^3*d^2*e^2 - 4*b*c^2*d*e

^3 + b^2*c*e^4)*f - (32*c^3*d^3*e - 36*b*c^2*d^2*e^2 + 12*b^2*c*d*e^3 - b^3*e^4)

*g)*x)*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)*sq

rt(c))))/((e^3*x + d*e^2)*sqrt(c))]

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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)**4,x)

[Out]

Timed out

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GIAC/XCAS [A] time = 1.03355, size = 4, normalized size = 0.01 \[ \mathit{sage}_{0} x \]

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)/(e*x + d)^4,x, algorithm="giac")

[Out]

sage0*x

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