3.1190 \(\int \frac{(A+B x) \left (b x+c x^2\right )^{5/2}}{(d+e x)^5} \, dx\)

Optimal. Leaf size=633 \[ -\frac{5 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right ) \left (4 A c e (2 c d-b e)-B \left (3 b^2 e^2-20 b c d e+24 c^2 d^2\right )\right )}{4 e^7}-\frac{5 \left (b x+c x^2\right )^{3/2} \left (3 e x \left (A e \left (b^2 e^2-8 b c d e+8 c^2 d^2\right )-B d \left (9 b^2 e^2-32 b c d e+24 c^2 d^2\right )\right )+d \left (A e \left (-b^2 e^2-12 b c d e+16 c^2 d^2\right )-B d \left (7 b^2 e^2-52 b c d e+48 c^2 d^2\right )\right )\right )}{96 d e^4 (d+e x)^3 (c d-b e)}-\frac{5 \sqrt{b x+c x^2} \left (-2 c e x \left (A e \left (b^2 e^2-16 b c d e+16 c^2 d^2\right )-B d \left (17 b^2 e^2-64 b c d e+48 c^2 d^2\right )\right )-A e \left (b^3 e^3+16 b^2 c d e^2-80 b c^2 d^2 e+64 c^3 d^3\right )+B d \left (-7 b^3 e^3+120 b^2 c d e^2-304 b c^2 d^2 e+192 c^3 d^3\right )\right )}{64 d e^6 (d+e x) (c d-b e)}+\frac{5 \left (A e \left (-b^4 e^4-16 b^3 c d e^3+144 b^2 c^2 d^2 e^2-256 b c^3 d^3 e+128 c^4 d^4\right )-B d \left (7 b^4 e^4-168 b^3 c d e^3+672 b^2 c^2 d^2 e^2-896 b c^3 d^3 e+384 c^4 d^4\right )\right ) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{128 d^{3/2} e^7 (c d-b e)^{3/2}}+\frac{\left (b x+c x^2\right )^{5/2} (-A e+3 B d+2 B e x)}{4 e^2 (d+e x)^4} \]

[Out]

(-5*(B*d*(192*c^3*d^3 - 304*b*c^2*d^2*e + 120*b^2*c*d*e^2 - 7*b^3*e^3) - A*e*(64
*c^3*d^3 - 80*b*c^2*d^2*e + 16*b^2*c*d*e^2 + b^3*e^3) - 2*c*e*(A*e*(16*c^2*d^2 -
16*b*c*d*e + b^2*e^2) - B*d*(48*c^2*d^2 - 64*b*c*d*e + 17*b^2*e^2))*x)*Sqrt[b*x
+ c*x^2])/(64*d*e^6*(c*d - b*e)*(d + e*x)) - (5*(d*(A*e*(16*c^2*d^2 - 12*b*c*d*
e - b^2*e^2) - B*d*(48*c^2*d^2 - 52*b*c*d*e + 7*b^2*e^2)) + 3*e*(A*e*(8*c^2*d^2
- 8*b*c*d*e + b^2*e^2) - B*d*(24*c^2*d^2 - 32*b*c*d*e + 9*b^2*e^2))*x)*(b*x + c*
x^2)^(3/2))/(96*d*e^4*(c*d - b*e)*(d + e*x)^3) + ((3*B*d - A*e + 2*B*e*x)*(b*x +
c*x^2)^(5/2))/(4*e^2*(d + e*x)^4) - (5*Sqrt[c]*(4*A*c*e*(2*c*d - b*e) - B*(24*c
^2*d^2 - 20*b*c*d*e + 3*b^2*e^2))*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(4*e^7
) + (5*(A*e*(128*c^4*d^4 - 256*b*c^3*d^3*e + 144*b^2*c^2*d^2*e^2 - 16*b^3*c*d*e^
3 - b^4*e^4) - B*d*(384*c^4*d^4 - 896*b*c^3*d^3*e + 672*b^2*c^2*d^2*e^2 - 168*b^
3*c*d*e^3 + 7*b^4*e^4))*ArcTanh[(b*d + (2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*
e]*Sqrt[b*x + c*x^2])])/(128*d^(3/2)*e^7*(c*d - b*e)^(3/2))

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Rubi [A]  time = 2.09394, antiderivative size = 633, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{5 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right ) \left (4 A c e (2 c d-b e)-B \left (3 b^2 e^2-20 b c d e+24 c^2 d^2\right )\right )}{4 e^7}-\frac{5 \left (b x+c x^2\right )^{3/2} \left (3 e x \left (A e \left (b^2 e^2-8 b c d e+8 c^2 d^2\right )-B d \left (9 b^2 e^2-32 b c d e+24 c^2 d^2\right )\right )+d \left (A e \left (-b^2 e^2-12 b c d e+16 c^2 d^2\right )-B d \left (7 b^2 e^2-52 b c d e+48 c^2 d^2\right )\right )\right )}{96 d e^4 (d+e x)^3 (c d-b e)}-\frac{5 \sqrt{b x+c x^2} \left (-2 c e x \left (A e \left (b^2 e^2-16 b c d e+16 c^2 d^2\right )-B d \left (17 b^2 e^2-64 b c d e+48 c^2 d^2\right )\right )-A e \left (b^3 e^3+16 b^2 c d e^2-80 b c^2 d^2 e+64 c^3 d^3\right )+B d \left (-7 b^3 e^3+120 b^2 c d e^2-304 b c^2 d^2 e+192 c^3 d^3\right )\right )}{64 d e^6 (d+e x) (c d-b e)}+\frac{5 \left (A e \left (-b^4 e^4-16 b^3 c d e^3+144 b^2 c^2 d^2 e^2-256 b c^3 d^3 e+128 c^4 d^4\right )-B d \left (7 b^4 e^4-168 b^3 c d e^3+672 b^2 c^2 d^2 e^2-896 b c^3 d^3 e+384 c^4 d^4\right )\right ) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{128 d^{3/2} e^7 (c d-b e)^{3/2}}+\frac{\left (b x+c x^2\right )^{5/2} (-A e+3 B d+2 B e x)}{4 e^2 (d+e x)^4} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(b*x + c*x^2)^(5/2))/(d + e*x)^5,x]

[Out]

(-5*(B*d*(192*c^3*d^3 - 304*b*c^2*d^2*e + 120*b^2*c*d*e^2 - 7*b^3*e^3) - A*e*(64
*c^3*d^3 - 80*b*c^2*d^2*e + 16*b^2*c*d*e^2 + b^3*e^3) - 2*c*e*(A*e*(16*c^2*d^2 -
16*b*c*d*e + b^2*e^2) - B*d*(48*c^2*d^2 - 64*b*c*d*e + 17*b^2*e^2))*x)*Sqrt[b*x
+ c*x^2])/(64*d*e^6*(c*d - b*e)*(d + e*x)) - (5*(d*(A*e*(16*c^2*d^2 - 12*b*c*d*
e - b^2*e^2) - B*d*(48*c^2*d^2 - 52*b*c*d*e + 7*b^2*e^2)) + 3*e*(A*e*(8*c^2*d^2
- 8*b*c*d*e + b^2*e^2) - B*d*(24*c^2*d^2 - 32*b*c*d*e + 9*b^2*e^2))*x)*(b*x + c*
x^2)^(3/2))/(96*d*e^4*(c*d - b*e)*(d + e*x)^3) + ((3*B*d - A*e + 2*B*e*x)*(b*x +
c*x^2)^(5/2))/(4*e^2*(d + e*x)^4) - (5*Sqrt[c]*(4*A*c*e*(2*c*d - b*e) - B*(24*c
^2*d^2 - 20*b*c*d*e + 3*b^2*e^2))*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(4*e^7
) + (5*(A*e*(128*c^4*d^4 - 256*b*c^3*d^3*e + 144*b^2*c^2*d^2*e^2 - 16*b^3*c*d*e^
3 - b^4*e^4) - B*d*(384*c^4*d^4 - 896*b*c^3*d^3*e + 672*b^2*c^2*d^2*e^2 - 168*b^
3*c*d*e^3 + 7*b^4*e^4))*ArcTanh[(b*d + (2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*
e]*Sqrt[b*x + c*x^2])])/(128*d^(3/2)*e^7*(c*d - b*e)^(3/2))

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(c*x**2+b*x)**(5/2)/(e*x+d)**5,x)

[Out]

Timed out

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Mathematica [A]  time = 5.43294, size = 551, normalized size = 0.87 \[ \frac{(x (b+c x))^{5/2} \left (\frac{240 \sqrt{c} \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right ) \left (4 A c e (b e-2 c d)+B \left (3 b^2 e^2-20 b c d e+24 c^2 d^2\right )\right )}{(b+c x)^{5/2}}+\frac{e \sqrt{x} \left (\frac{2 \left (A e \left (-59 b^2 e^2+344 b c d e-344 c^2 d^2\right )+B d \left (163 b^2 e^2-656 b c d e+552 c^2 d^2\right )\right )}{(d+e x)^2}+\frac{A e \left (-15 b^3 e^3+646 b^2 c d e^2-1848 b c^2 d^2 e+1232 c^3 d^3\right )+B d \left (279 b^3 e^3-2414 b^2 c d e^2+4856 b c^2 d^2 e-2736 c^3 d^3\right )}{d (d+e x) (c d-b e)}+\frac{48 d^2 (B d-A e) (c d-b e)^2}{(d+e x)^4}-\frac{8 d (c d-b e) (17 A e (b e-2 c d)+B d (42 c d-25 b e))}{(d+e x)^3}+48 c (4 A c e+9 b B e-20 B c d)+96 B c^2 e x\right )}{(b+c x)^2}+\frac{15 \left (A e \left (b^4 e^4+16 b^3 c d e^3-144 b^2 c^2 d^2 e^2+256 b c^3 d^3 e-128 c^4 d^4\right )+B d \left (7 b^4 e^4-168 b^3 c d e^3+672 b^2 c^2 d^2 e^2-896 b c^3 d^3 e+384 c^4 d^4\right )\right ) \tan ^{-1}\left (\frac{\sqrt{x} \sqrt{b e-c d}}{\sqrt{d} \sqrt{b+c x}}\right )}{d^{3/2} (b+c x)^{5/2} (b e-c d)^{3/2}}\right )}{192 e^7 x^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(b*x + c*x^2)^(5/2))/(d + e*x)^5,x]

[Out]

((x*(b + c*x))^(5/2)*((e*Sqrt[x]*(48*c*(-20*B*c*d + 9*b*B*e + 4*A*c*e) + 96*B*c^
2*e*x + (48*d^2*(B*d - A*e)*(c*d - b*e)^2)/(d + e*x)^4 - (8*d*(c*d - b*e)*(B*d*(
42*c*d - 25*b*e) + 17*A*e*(-2*c*d + b*e)))/(d + e*x)^3 + (2*(A*e*(-344*c^2*d^2 +
344*b*c*d*e - 59*b^2*e^2) + B*d*(552*c^2*d^2 - 656*b*c*d*e + 163*b^2*e^2)))/(d
+ e*x)^2 + (A*e*(1232*c^3*d^3 - 1848*b*c^2*d^2*e + 646*b^2*c*d*e^2 - 15*b^3*e^3)
+ B*d*(-2736*c^3*d^3 + 4856*b*c^2*d^2*e - 2414*b^2*c*d*e^2 + 279*b^3*e^3))/(d*(
c*d - b*e)*(d + e*x))))/(b + c*x)^2 + (15*(A*e*(-128*c^4*d^4 + 256*b*c^3*d^3*e -
144*b^2*c^2*d^2*e^2 + 16*b^3*c*d*e^3 + b^4*e^4) + B*d*(384*c^4*d^4 - 896*b*c^3*
d^3*e + 672*b^2*c^2*d^2*e^2 - 168*b^3*c*d*e^3 + 7*b^4*e^4))*ArcTan[(Sqrt[-(c*d)
+ b*e]*Sqrt[x])/(Sqrt[d]*Sqrt[b + c*x])])/(d^(3/2)*(-(c*d) + b*e)^(3/2)*(b + c*x
)^(5/2)) + (240*Sqrt[c]*(4*A*c*e*(-2*c*d + b*e) + B*(24*c^2*d^2 - 20*b*c*d*e + 3
*b^2*e^2))*Log[c*Sqrt[x] + Sqrt[c]*Sqrt[b + c*x]])/(b + c*x)^(5/2)))/(192*e^7*x^
(5/2))

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Maple [B]  time = 0.055, size = 23819, normalized size = 37.6 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+b*x)^(5/2)/(e*x+d)^5,x)

[Out]

result too large to display

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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*x^2 + b*x)^(5/2)*(B*x + A)/(e*x + d)^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 60.6373, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(5/2)*(B*x + A)/(e*x + d)^5,x, algorithm="fricas")

[Out]

[1/384*(240*(24*B*c^3*d^8 - 4*(11*B*b*c^2 + 2*A*c^3)*d^7*e + (23*B*b^2*c + 12*A*
b*c^2)*d^6*e^2 - (3*B*b^3 + 4*A*b^2*c)*d^5*e^3 + (24*B*c^3*d^4*e^4 - 4*(11*B*b*c
^2 + 2*A*c^3)*d^3*e^5 + (23*B*b^2*c + 12*A*b*c^2)*d^2*e^6 - (3*B*b^3 + 4*A*b^2*c
)*d*e^7)*x^4 + 4*(24*B*c^3*d^5*e^3 - 4*(11*B*b*c^2 + 2*A*c^3)*d^4*e^4 + (23*B*b^
2*c + 12*A*b*c^2)*d^3*e^5 - (3*B*b^3 + 4*A*b^2*c)*d^2*e^6)*x^3 + 6*(24*B*c^3*d^6
*e^2 - 4*(11*B*b*c^2 + 2*A*c^3)*d^5*e^3 + (23*B*b^2*c + 12*A*b*c^2)*d^4*e^4 - (3
*B*b^3 + 4*A*b^2*c)*d^3*e^5)*x^2 + 4*(24*B*c^3*d^7*e - 4*(11*B*b*c^2 + 2*A*c^3)*
d^6*e^2 + (23*B*b^2*c + 12*A*b*c^2)*d^5*e^3 - (3*B*b^3 + 4*A*b^2*c)*d^4*e^4)*x)*
sqrt(c*d^2 - b*d*e)*sqrt(c)*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c)) - 2*(28
80*B*c^3*d^7*e - 15*A*b^3*d^3*e^5 - 240*(19*B*b*c^2 + 4*A*c^3)*d^6*e^2 + 600*(3*
B*b^2*c + 2*A*b*c^2)*d^5*e^3 - 15*(7*B*b^3 + 16*A*b^2*c)*d^4*e^4 - 96*(B*c^3*d^2
*e^6 - B*b*c^2*d*e^7)*x^5 + 48*(12*B*c^3*d^3*e^5 - (21*B*b*c^2 + 4*A*c^3)*d^2*e^
6 + (9*B*b^2*c + 4*A*b*c^2)*d*e^7)*x^4 + (6000*B*c^3*d^4*e^4 + 15*A*b^3*e^8 - 8*
(1231*B*b*c^2 + 250*A*c^3)*d^3*e^5 + 218*(19*B*b^2*c + 12*A*b*c^2)*d^2*e^6 - (27
9*B*b^3 + 646*A*b^2*c)*d*e^7)*x^3 + (12480*B*c^3*d^5*e^3 - 73*A*b^3*d*e^7 - 40*(
503*B*b*c^2 + 104*A*c^3)*d^4*e^4 + 4*(2049*B*b^2*c + 1330*A*b*c^2)*d^3*e^5 - (51
1*B*b^3 + 1132*A*b^2*c)*d^2*e^6)*x^2 + 5*(2016*B*c^3*d^6*e^2 - 11*A*b^3*d^2*e^6
- 48*(67*B*b*c^2 + 14*A*c^3)*d^5*e^3 + 2*(643*B*b^2*c + 424*A*b*c^2)*d^4*e^4 - (
77*B*b^3 + 174*A*b^2*c)*d^3*e^5)*x)*sqrt(c*d^2 - b*d*e)*sqrt(c*x^2 + b*x) - 15*(
384*B*c^4*d^9 + A*b^4*d^4*e^5 - 128*(7*B*b*c^3 + A*c^4)*d^8*e + 32*(21*B*b^2*c^2
+ 8*A*b*c^3)*d^7*e^2 - 24*(7*B*b^3*c + 6*A*b^2*c^2)*d^6*e^3 + (7*B*b^4 + 16*A*b
^3*c)*d^5*e^4 + (384*B*c^4*d^5*e^4 + A*b^4*e^9 - 128*(7*B*b*c^3 + A*c^4)*d^4*e^5
+ 32*(21*B*b^2*c^2 + 8*A*b*c^3)*d^3*e^6 - 24*(7*B*b^3*c + 6*A*b^2*c^2)*d^2*e^7
+ (7*B*b^4 + 16*A*b^3*c)*d*e^8)*x^4 + 4*(384*B*c^4*d^6*e^3 + A*b^4*d*e^8 - 128*(
7*B*b*c^3 + A*c^4)*d^5*e^4 + 32*(21*B*b^2*c^2 + 8*A*b*c^3)*d^4*e^5 - 24*(7*B*b^3
*c + 6*A*b^2*c^2)*d^3*e^6 + (7*B*b^4 + 16*A*b^3*c)*d^2*e^7)*x^3 + 6*(384*B*c^4*d
^7*e^2 + A*b^4*d^2*e^7 - 128*(7*B*b*c^3 + A*c^4)*d^6*e^3 + 32*(21*B*b^2*c^2 + 8*
A*b*c^3)*d^5*e^4 - 24*(7*B*b^3*c + 6*A*b^2*c^2)*d^4*e^5 + (7*B*b^4 + 16*A*b^3*c)
*d^3*e^6)*x^2 + 4*(384*B*c^4*d^8*e + A*b^4*d^3*e^6 - 128*(7*B*b*c^3 + A*c^4)*d^7
*e^2 + 32*(21*B*b^2*c^2 + 8*A*b*c^3)*d^6*e^3 - 24*(7*B*b^3*c + 6*A*b^2*c^2)*d^5*
e^4 + (7*B*b^4 + 16*A*b^3*c)*d^4*e^5)*x)*log((2*(c*d^2 - b*d*e)*sqrt(c*x^2 + b*x
) + sqrt(c*d^2 - b*d*e)*(b*d + (2*c*d - b*e)*x))/(e*x + d)))/((c*d^6*e^7 - b*d^5
*e^8 + (c*d^2*e^11 - b*d*e^12)*x^4 + 4*(c*d^3*e^10 - b*d^2*e^11)*x^3 + 6*(c*d^4*
e^9 - b*d^3*e^10)*x^2 + 4*(c*d^5*e^8 - b*d^4*e^9)*x)*sqrt(c*d^2 - b*d*e)), 1/192
*(120*(24*B*c^3*d^8 - 4*(11*B*b*c^2 + 2*A*c^3)*d^7*e + (23*B*b^2*c + 12*A*b*c^2)
*d^6*e^2 - (3*B*b^3 + 4*A*b^2*c)*d^5*e^3 + (24*B*c^3*d^4*e^4 - 4*(11*B*b*c^2 + 2
*A*c^3)*d^3*e^5 + (23*B*b^2*c + 12*A*b*c^2)*d^2*e^6 - (3*B*b^3 + 4*A*b^2*c)*d*e^
7)*x^4 + 4*(24*B*c^3*d^5*e^3 - 4*(11*B*b*c^2 + 2*A*c^3)*d^4*e^4 + (23*B*b^2*c +
12*A*b*c^2)*d^3*e^5 - (3*B*b^3 + 4*A*b^2*c)*d^2*e^6)*x^3 + 6*(24*B*c^3*d^6*e^2 -
4*(11*B*b*c^2 + 2*A*c^3)*d^5*e^3 + (23*B*b^2*c + 12*A*b*c^2)*d^4*e^4 - (3*B*b^3
+ 4*A*b^2*c)*d^3*e^5)*x^2 + 4*(24*B*c^3*d^7*e - 4*(11*B*b*c^2 + 2*A*c^3)*d^6*e^
2 + (23*B*b^2*c + 12*A*b*c^2)*d^5*e^3 - (3*B*b^3 + 4*A*b^2*c)*d^4*e^4)*x)*sqrt(-
c*d^2 + b*d*e)*sqrt(c)*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c)) - (2880*B*c^
3*d^7*e - 15*A*b^3*d^3*e^5 - 240*(19*B*b*c^2 + 4*A*c^3)*d^6*e^2 + 600*(3*B*b^2*c
+ 2*A*b*c^2)*d^5*e^3 - 15*(7*B*b^3 + 16*A*b^2*c)*d^4*e^4 - 96*(B*c^3*d^2*e^6 -
B*b*c^2*d*e^7)*x^5 + 48*(12*B*c^3*d^3*e^5 - (21*B*b*c^2 + 4*A*c^3)*d^2*e^6 + (9*
B*b^2*c + 4*A*b*c^2)*d*e^7)*x^4 + (6000*B*c^3*d^4*e^4 + 15*A*b^3*e^8 - 8*(1231*B
*b*c^2 + 250*A*c^3)*d^3*e^5 + 218*(19*B*b^2*c + 12*A*b*c^2)*d^2*e^6 - (279*B*b^3
+ 646*A*b^2*c)*d*e^7)*x^3 + (12480*B*c^3*d^5*e^3 - 73*A*b^3*d*e^7 - 40*(503*B*b
*c^2 + 104*A*c^3)*d^4*e^4 + 4*(2049*B*b^2*c + 1330*A*b*c^2)*d^3*e^5 - (511*B*b^3
+ 1132*A*b^2*c)*d^2*e^6)*x^2 + 5*(2016*B*c^3*d^6*e^2 - 11*A*b^3*d^2*e^6 - 48*(6
7*B*b*c^2 + 14*A*c^3)*d^5*e^3 + 2*(643*B*b^2*c + 424*A*b*c^2)*d^4*e^4 - (77*B*b^
3 + 174*A*b^2*c)*d^3*e^5)*x)*sqrt(-c*d^2 + b*d*e)*sqrt(c*x^2 + b*x) + 15*(384*B*
c^4*d^9 + A*b^4*d^4*e^5 - 128*(7*B*b*c^3 + A*c^4)*d^8*e + 32*(21*B*b^2*c^2 + 8*A
*b*c^3)*d^7*e^2 - 24*(7*B*b^3*c + 6*A*b^2*c^2)*d^6*e^3 + (7*B*b^4 + 16*A*b^3*c)*
d^5*e^4 + (384*B*c^4*d^5*e^4 + A*b^4*e^9 - 128*(7*B*b*c^3 + A*c^4)*d^4*e^5 + 32*
(21*B*b^2*c^2 + 8*A*b*c^3)*d^3*e^6 - 24*(7*B*b^3*c + 6*A*b^2*c^2)*d^2*e^7 + (7*B
*b^4 + 16*A*b^3*c)*d*e^8)*x^4 + 4*(384*B*c^4*d^6*e^3 + A*b^4*d*e^8 - 128*(7*B*b*
c^3 + A*c^4)*d^5*e^4 + 32*(21*B*b^2*c^2 + 8*A*b*c^3)*d^4*e^5 - 24*(7*B*b^3*c + 6
*A*b^2*c^2)*d^3*e^6 + (7*B*b^4 + 16*A*b^3*c)*d^2*e^7)*x^3 + 6*(384*B*c^4*d^7*e^2
+ A*b^4*d^2*e^7 - 128*(7*B*b*c^3 + A*c^4)*d^6*e^3 + 32*(21*B*b^2*c^2 + 8*A*b*c^
3)*d^5*e^4 - 24*(7*B*b^3*c + 6*A*b^2*c^2)*d^4*e^5 + (7*B*b^4 + 16*A*b^3*c)*d^3*e
^6)*x^2 + 4*(384*B*c^4*d^8*e + A*b^4*d^3*e^6 - 128*(7*B*b*c^3 + A*c^4)*d^7*e^2 +
32*(21*B*b^2*c^2 + 8*A*b*c^3)*d^6*e^3 - 24*(7*B*b^3*c + 6*A*b^2*c^2)*d^5*e^4 +
(7*B*b^4 + 16*A*b^3*c)*d^4*e^5)*x)*arctan(-sqrt(-c*d^2 + b*d*e)*sqrt(c*x^2 + b*x
)/((c*d - b*e)*x)))/((c*d^6*e^7 - b*d^5*e^8 + (c*d^2*e^11 - b*d*e^12)*x^4 + 4*(c
*d^3*e^10 - b*d^2*e^11)*x^3 + 6*(c*d^4*e^9 - b*d^3*e^10)*x^2 + 4*(c*d^5*e^8 - b*
d^4*e^9)*x)*sqrt(-c*d^2 + b*d*e)), 1/384*(480*(24*B*c^3*d^8 - 4*(11*B*b*c^2 + 2*
A*c^3)*d^7*e + (23*B*b^2*c + 12*A*b*c^2)*d^6*e^2 - (3*B*b^3 + 4*A*b^2*c)*d^5*e^3
+ (24*B*c^3*d^4*e^4 - 4*(11*B*b*c^2 + 2*A*c^3)*d^3*e^5 + (23*B*b^2*c + 12*A*b*c
^2)*d^2*e^6 - (3*B*b^3 + 4*A*b^2*c)*d*e^7)*x^4 + 4*(24*B*c^3*d^5*e^3 - 4*(11*B*b
*c^2 + 2*A*c^3)*d^4*e^4 + (23*B*b^2*c + 12*A*b*c^2)*d^3*e^5 - (3*B*b^3 + 4*A*b^2
*c)*d^2*e^6)*x^3 + 6*(24*B*c^3*d^6*e^2 - 4*(11*B*b*c^2 + 2*A*c^3)*d^5*e^3 + (23*
B*b^2*c + 12*A*b*c^2)*d^4*e^4 - (3*B*b^3 + 4*A*b^2*c)*d^3*e^5)*x^2 + 4*(24*B*c^3
*d^7*e - 4*(11*B*b*c^2 + 2*A*c^3)*d^6*e^2 + (23*B*b^2*c + 12*A*b*c^2)*d^5*e^3 -
(3*B*b^3 + 4*A*b^2*c)*d^4*e^4)*x)*sqrt(c*d^2 - b*d*e)*sqrt(-c)*arctan(sqrt(c*x^2
+ b*x)/(sqrt(-c)*x)) - 2*(2880*B*c^3*d^7*e - 15*A*b^3*d^3*e^5 - 240*(19*B*b*c^2
+ 4*A*c^3)*d^6*e^2 + 600*(3*B*b^2*c + 2*A*b*c^2)*d^5*e^3 - 15*(7*B*b^3 + 16*A*b
^2*c)*d^4*e^4 - 96*(B*c^3*d^2*e^6 - B*b*c^2*d*e^7)*x^5 + 48*(12*B*c^3*d^3*e^5 -
(21*B*b*c^2 + 4*A*c^3)*d^2*e^6 + (9*B*b^2*c + 4*A*b*c^2)*d*e^7)*x^4 + (6000*B*c^
3*d^4*e^4 + 15*A*b^3*e^8 - 8*(1231*B*b*c^2 + 250*A*c^3)*d^3*e^5 + 218*(19*B*b^2*
c + 12*A*b*c^2)*d^2*e^6 - (279*B*b^3 + 646*A*b^2*c)*d*e^7)*x^3 + (12480*B*c^3*d^
5*e^3 - 73*A*b^3*d*e^7 - 40*(503*B*b*c^2 + 104*A*c^3)*d^4*e^4 + 4*(2049*B*b^2*c
+ 1330*A*b*c^2)*d^3*e^5 - (511*B*b^3 + 1132*A*b^2*c)*d^2*e^6)*x^2 + 5*(2016*B*c^
3*d^6*e^2 - 11*A*b^3*d^2*e^6 - 48*(67*B*b*c^2 + 14*A*c^3)*d^5*e^3 + 2*(643*B*b^2
*c + 424*A*b*c^2)*d^4*e^4 - (77*B*b^3 + 174*A*b^2*c)*d^3*e^5)*x)*sqrt(c*d^2 - b*
d*e)*sqrt(c*x^2 + b*x) - 15*(384*B*c^4*d^9 + A*b^4*d^4*e^5 - 128*(7*B*b*c^3 + A*
c^4)*d^8*e + 32*(21*B*b^2*c^2 + 8*A*b*c^3)*d^7*e^2 - 24*(7*B*b^3*c + 6*A*b^2*c^2
)*d^6*e^3 + (7*B*b^4 + 16*A*b^3*c)*d^5*e^4 + (384*B*c^4*d^5*e^4 + A*b^4*e^9 - 12
8*(7*B*b*c^3 + A*c^4)*d^4*e^5 + 32*(21*B*b^2*c^2 + 8*A*b*c^3)*d^3*e^6 - 24*(7*B*
b^3*c + 6*A*b^2*c^2)*d^2*e^7 + (7*B*b^4 + 16*A*b^3*c)*d*e^8)*x^4 + 4*(384*B*c^4*
d^6*e^3 + A*b^4*d*e^8 - 128*(7*B*b*c^3 + A*c^4)*d^5*e^4 + 32*(21*B*b^2*c^2 + 8*A
*b*c^3)*d^4*e^5 - 24*(7*B*b^3*c + 6*A*b^2*c^2)*d^3*e^6 + (7*B*b^4 + 16*A*b^3*c)*
d^2*e^7)*x^3 + 6*(384*B*c^4*d^7*e^2 + A*b^4*d^2*e^7 - 128*(7*B*b*c^3 + A*c^4)*d^
6*e^3 + 32*(21*B*b^2*c^2 + 8*A*b*c^3)*d^5*e^4 - 24*(7*B*b^3*c + 6*A*b^2*c^2)*d^4
*e^5 + (7*B*b^4 + 16*A*b^3*c)*d^3*e^6)*x^2 + 4*(384*B*c^4*d^8*e + A*b^4*d^3*e^6
- 128*(7*B*b*c^3 + A*c^4)*d^7*e^2 + 32*(21*B*b^2*c^2 + 8*A*b*c^3)*d^6*e^3 - 24*(
7*B*b^3*c + 6*A*b^2*c^2)*d^5*e^4 + (7*B*b^4 + 16*A*b^3*c)*d^4*e^5)*x)*log((2*(c*
d^2 - b*d*e)*sqrt(c*x^2 + b*x) + sqrt(c*d^2 - b*d*e)*(b*d + (2*c*d - b*e)*x))/(e
*x + d)))/((c*d^6*e^7 - b*d^5*e^8 + (c*d^2*e^11 - b*d*e^12)*x^4 + 4*(c*d^3*e^10
- b*d^2*e^11)*x^3 + 6*(c*d^4*e^9 - b*d^3*e^10)*x^2 + 4*(c*d^5*e^8 - b*d^4*e^9)*x
)*sqrt(c*d^2 - b*d*e)), 1/192*(240*(24*B*c^3*d^8 - 4*(11*B*b*c^2 + 2*A*c^3)*d^7*
e + (23*B*b^2*c + 12*A*b*c^2)*d^6*e^2 - (3*B*b^3 + 4*A*b^2*c)*d^5*e^3 + (24*B*c^
3*d^4*e^4 - 4*(11*B*b*c^2 + 2*A*c^3)*d^3*e^5 + (23*B*b^2*c + 12*A*b*c^2)*d^2*e^6
- (3*B*b^3 + 4*A*b^2*c)*d*e^7)*x^4 + 4*(24*B*c^3*d^5*e^3 - 4*(11*B*b*c^2 + 2*A*
c^3)*d^4*e^4 + (23*B*b^2*c + 12*A*b*c^2)*d^3*e^5 - (3*B*b^3 + 4*A*b^2*c)*d^2*e^6
)*x^3 + 6*(24*B*c^3*d^6*e^2 - 4*(11*B*b*c^2 + 2*A*c^3)*d^5*e^3 + (23*B*b^2*c + 1
2*A*b*c^2)*d^4*e^4 - (3*B*b^3 + 4*A*b^2*c)*d^3*e^5)*x^2 + 4*(24*B*c^3*d^7*e - 4*
(11*B*b*c^2 + 2*A*c^3)*d^6*e^2 + (23*B*b^2*c + 12*A*b*c^2)*d^5*e^3 - (3*B*b^3 +
4*A*b^2*c)*d^4*e^4)*x)*sqrt(-c*d^2 + b*d*e)*sqrt(-c)*arctan(sqrt(c*x^2 + b*x)/(s
qrt(-c)*x)) - (2880*B*c^3*d^7*e - 15*A*b^3*d^3*e^5 - 240*(19*B*b*c^2 + 4*A*c^3)*
d^6*e^2 + 600*(3*B*b^2*c + 2*A*b*c^2)*d^5*e^3 - 15*(7*B*b^3 + 16*A*b^2*c)*d^4*e^
4 - 96*(B*c^3*d^2*e^6 - B*b*c^2*d*e^7)*x^5 + 48*(12*B*c^3*d^3*e^5 - (21*B*b*c^2
+ 4*A*c^3)*d^2*e^6 + (9*B*b^2*c + 4*A*b*c^2)*d*e^7)*x^4 + (6000*B*c^3*d^4*e^4 +
15*A*b^3*e^8 - 8*(1231*B*b*c^2 + 250*A*c^3)*d^3*e^5 + 218*(19*B*b^2*c + 12*A*b*c
^2)*d^2*e^6 - (279*B*b^3 + 646*A*b^2*c)*d*e^7)*x^3 + (12480*B*c^3*d^5*e^3 - 73*A
*b^3*d*e^7 - 40*(503*B*b*c^2 + 104*A*c^3)*d^4*e^4 + 4*(2049*B*b^2*c + 1330*A*b*c
^2)*d^3*e^5 - (511*B*b^3 + 1132*A*b^2*c)*d^2*e^6)*x^2 + 5*(2016*B*c^3*d^6*e^2 -
11*A*b^3*d^2*e^6 - 48*(67*B*b*c^2 + 14*A*c^3)*d^5*e^3 + 2*(643*B*b^2*c + 424*A*b
*c^2)*d^4*e^4 - (77*B*b^3 + 174*A*b^2*c)*d^3*e^5)*x)*sqrt(-c*d^2 + b*d*e)*sqrt(c
*x^2 + b*x) + 15*(384*B*c^4*d^9 + A*b^4*d^4*e^5 - 128*(7*B*b*c^3 + A*c^4)*d^8*e
+ 32*(21*B*b^2*c^2 + 8*A*b*c^3)*d^7*e^2 - 24*(7*B*b^3*c + 6*A*b^2*c^2)*d^6*e^3 +
(7*B*b^4 + 16*A*b^3*c)*d^5*e^4 + (384*B*c^4*d^5*e^4 + A*b^4*e^9 - 128*(7*B*b*c^
3 + A*c^4)*d^4*e^5 + 32*(21*B*b^2*c^2 + 8*A*b*c^3)*d^3*e^6 - 24*(7*B*b^3*c + 6*A
*b^2*c^2)*d^2*e^7 + (7*B*b^4 + 16*A*b^3*c)*d*e^8)*x^4 + 4*(384*B*c^4*d^6*e^3 + A
*b^4*d*e^8 - 128*(7*B*b*c^3 + A*c^4)*d^5*e^4 + 32*(21*B*b^2*c^2 + 8*A*b*c^3)*d^4
*e^5 - 24*(7*B*b^3*c + 6*A*b^2*c^2)*d^3*e^6 + (7*B*b^4 + 16*A*b^3*c)*d^2*e^7)*x^
3 + 6*(384*B*c^4*d^7*e^2 + A*b^4*d^2*e^7 - 128*(7*B*b*c^3 + A*c^4)*d^6*e^3 + 32*
(21*B*b^2*c^2 + 8*A*b*c^3)*d^5*e^4 - 24*(7*B*b^3*c + 6*A*b^2*c^2)*d^4*e^5 + (7*B
*b^4 + 16*A*b^3*c)*d^3*e^6)*x^2 + 4*(384*B*c^4*d^8*e + A*b^4*d^3*e^6 - 128*(7*B*
b*c^3 + A*c^4)*d^7*e^2 + 32*(21*B*b^2*c^2 + 8*A*b*c^3)*d^6*e^3 - 24*(7*B*b^3*c +
6*A*b^2*c^2)*d^5*e^4 + (7*B*b^4 + 16*A*b^3*c)*d^4*e^5)*x)*arctan(-sqrt(-c*d^2 +
b*d*e)*sqrt(c*x^2 + b*x)/((c*d - b*e)*x)))/((c*d^6*e^7 - b*d^5*e^8 + (c*d^2*e^1
1 - b*d*e^12)*x^4 + 4*(c*d^3*e^10 - b*d^2*e^11)*x^3 + 6*(c*d^4*e^9 - b*d^3*e^10)
*x^2 + 4*(c*d^5*e^8 - b*d^4*e^9)*x)*sqrt(-c*d^2 + b*d*e))]

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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((B*x+A)*(c*x**2+b*x)**(5/2)/(e*x+d)**5,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*x^2 + b*x)^(5/2)*(B*x + A)/(e*x + d)^5,x, algorithm="giac")

[Out]

Timed out