3.1186 \(\int \frac{(A+B x) \left (b x+c x^2\right )^{5/2}}{d+e x} \, dx\)

Optimal. Leaf size=703 \[ \frac{\left (b x+c x^2\right )^{3/2} \left (-2 c e x \left (12 A c e (2 c d-b e)-B \left (-5 b^2 e^2-12 b c d e+24 c^2 d^2\right )\right )+4 A c e \left (3 b^2 e^2-22 b c d e+16 c^2 d^2\right )-B \left (5 b^3 e^3+12 b^2 c d e^2-88 b c^2 d^2 e+64 c^3 d^3\right )\right )}{192 c^2 e^4}+\frac{\sqrt{b x+c x^2} \left (3 \left (4 A c e \left (-3 b^4 e^4-10 b^3 c d e^3+176 b^2 c^2 d^2 e^2-288 b c^3 d^3 e+128 c^4 d^4\right )-B \left (-5 b^5 e^5-12 b^4 c d e^4-40 b^3 c^2 d^2 e^3+704 b^2 c^3 d^3 e^2-1152 b c^4 d^4 e+512 c^5 d^5\right )\right )-2 c e x \left (\left (-3 b^2 e^2-8 b c d e+16 c^2 d^2\right ) \left (12 A c e (2 c d-b e)-B \left (-5 b^2 e^2-12 b c d e+24 c^2 d^2\right )\right )+8 b c d e (2 c d-b e) (-12 A c e-5 b B e+12 B c d)\right )\right )}{1536 c^3 e^6}-\frac{\tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right ) \left (4 A c e \left (-3 b^5 e^5-10 b^4 c d e^4-80 b^3 c^2 d^2 e^3+480 b^2 c^3 d^3 e^2-640 b c^4 d^4 e+256 c^5 d^5\right )-B \left (-5 b^6 e^6-12 b^5 c d e^5-40 b^4 c^2 d^2 e^4-320 b^3 c^3 d^3 e^3+1920 b^2 c^4 d^4 e^2-2560 b c^5 d^5 e+1024 c^6 d^6\right )\right )}{512 c^{7/2} e^7}-\frac{d^{5/2} (B d-A e) (c d-b e)^{5/2} \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{e^7}-\frac{\left (b x+c x^2\right )^{5/2} (-12 A c e-5 b B e+12 B c d-10 B c e x)}{60 c e^2} \]

[Out]

((3*(4*A*c*e*(128*c^4*d^4 - 288*b*c^3*d^3*e + 176*b^2*c^2*d^2*e^2 - 10*b^3*c*d*e
^3 - 3*b^4*e^4) - B*(512*c^5*d^5 - 1152*b*c^4*d^4*e + 704*b^2*c^3*d^3*e^2 - 40*b
^3*c^2*d^2*e^3 - 12*b^4*c*d*e^4 - 5*b^5*e^5)) - 2*c*e*(8*b*c*d*e*(2*c*d - b*e)*(
12*B*c*d - 5*b*B*e - 12*A*c*e) + (16*c^2*d^2 - 8*b*c*d*e - 3*b^2*e^2)*(12*A*c*e*
(2*c*d - b*e) - B*(24*c^2*d^2 - 12*b*c*d*e - 5*b^2*e^2)))*x)*Sqrt[b*x + c*x^2])/
(1536*c^3*e^6) + ((4*A*c*e*(16*c^2*d^2 - 22*b*c*d*e + 3*b^2*e^2) - B*(64*c^3*d^3
- 88*b*c^2*d^2*e + 12*b^2*c*d*e^2 + 5*b^3*e^3) - 2*c*e*(12*A*c*e*(2*c*d - b*e)
- B*(24*c^2*d^2 - 12*b*c*d*e - 5*b^2*e^2))*x)*(b*x + c*x^2)^(3/2))/(192*c^2*e^4)
- ((12*B*c*d - 5*b*B*e - 12*A*c*e - 10*B*c*e*x)*(b*x + c*x^2)^(5/2))/(60*c*e^2)
- ((4*A*c*e*(256*c^5*d^5 - 640*b*c^4*d^4*e + 480*b^2*c^3*d^3*e^2 - 80*b^3*c^2*d
^2*e^3 - 10*b^4*c*d*e^4 - 3*b^5*e^5) - B*(1024*c^6*d^6 - 2560*b*c^5*d^5*e + 1920
*b^2*c^4*d^4*e^2 - 320*b^3*c^3*d^3*e^3 - 40*b^4*c^2*d^2*e^4 - 12*b^5*c*d*e^5 - 5
*b^6*e^6))*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(512*c^(7/2)*e^7) - (d^(5/2)*
(B*d - A*e)*(c*d - b*e)^(5/2)*ArcTanh[(b*d + (2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*
d - b*e]*Sqrt[b*x + c*x^2])])/e^7

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Rubi [A]  time = 2.62361, antiderivative size = 703, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ \frac{\left (b x+c x^2\right )^{3/2} \left (-2 c e x \left (12 A c e (2 c d-b e)-B \left (-5 b^2 e^2-12 b c d e+24 c^2 d^2\right )\right )+4 A c e \left (3 b^2 e^2-22 b c d e+16 c^2 d^2\right )-B \left (5 b^3 e^3+12 b^2 c d e^2-88 b c^2 d^2 e+64 c^3 d^3\right )\right )}{192 c^2 e^4}+\frac{\sqrt{b x+c x^2} \left (3 \left (4 A c e \left (-3 b^4 e^4-10 b^3 c d e^3+176 b^2 c^2 d^2 e^2-288 b c^3 d^3 e+128 c^4 d^4\right )-B \left (-5 b^5 e^5-12 b^4 c d e^4-40 b^3 c^2 d^2 e^3+704 b^2 c^3 d^3 e^2-1152 b c^4 d^4 e+512 c^5 d^5\right )\right )-2 c e x \left (\left (-3 b^2 e^2-8 b c d e+16 c^2 d^2\right ) \left (12 A c e (2 c d-b e)-B \left (-5 b^2 e^2-12 b c d e+24 c^2 d^2\right )\right )+8 b c d e (2 c d-b e) (-12 A c e-5 b B e+12 B c d)\right )\right )}{1536 c^3 e^6}-\frac{\tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right ) \left (4 A c e \left (-3 b^5 e^5-10 b^4 c d e^4-80 b^3 c^2 d^2 e^3+480 b^2 c^3 d^3 e^2-640 b c^4 d^4 e+256 c^5 d^5\right )-B \left (-5 b^6 e^6-12 b^5 c d e^5-40 b^4 c^2 d^2 e^4-320 b^3 c^3 d^3 e^3+1920 b^2 c^4 d^4 e^2-2560 b c^5 d^5 e+1024 c^6 d^6\right )\right )}{512 c^{7/2} e^7}-\frac{d^{5/2} (B d-A e) (c d-b e)^{5/2} \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{e^7}-\frac{\left (b x+c x^2\right )^{5/2} (-12 A c e-5 b B e+12 B c d-10 B c e x)}{60 c e^2} \]

Antiderivative was successfully verified.

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

[Out]

((3*(4*A*c*e*(128*c^4*d^4 - 288*b*c^3*d^3*e + 176*b^2*c^2*d^2*e^2 - 10*b^3*c*d*e
^3 - 3*b^4*e^4) - B*(512*c^5*d^5 - 1152*b*c^4*d^4*e + 704*b^2*c^3*d^3*e^2 - 40*b
^3*c^2*d^2*e^3 - 12*b^4*c*d*e^4 - 5*b^5*e^5)) - 2*c*e*(8*b*c*d*e*(2*c*d - b*e)*(
12*B*c*d - 5*b*B*e - 12*A*c*e) + (16*c^2*d^2 - 8*b*c*d*e - 3*b^2*e^2)*(12*A*c*e*
(2*c*d - b*e) - B*(24*c^2*d^2 - 12*b*c*d*e - 5*b^2*e^2)))*x)*Sqrt[b*x + c*x^2])/
(1536*c^3*e^6) + ((4*A*c*e*(16*c^2*d^2 - 22*b*c*d*e + 3*b^2*e^2) - B*(64*c^3*d^3
- 88*b*c^2*d^2*e + 12*b^2*c*d*e^2 + 5*b^3*e^3) - 2*c*e*(12*A*c*e*(2*c*d - b*e)
- B*(24*c^2*d^2 - 12*b*c*d*e - 5*b^2*e^2))*x)*(b*x + c*x^2)^(3/2))/(192*c^2*e^4)
- ((12*B*c*d - 5*b*B*e - 12*A*c*e - 10*B*c*e*x)*(b*x + c*x^2)^(5/2))/(60*c*e^2)
- ((4*A*c*e*(256*c^5*d^5 - 640*b*c^4*d^4*e + 480*b^2*c^3*d^3*e^2 - 80*b^3*c^2*d
^2*e^3 - 10*b^4*c*d*e^4 - 3*b^5*e^5) - B*(1024*c^6*d^6 - 2560*b*c^5*d^5*e + 1920
*b^2*c^4*d^4*e^2 - 320*b^3*c^3*d^3*e^3 - 40*b^4*c^2*d^2*e^4 - 12*b^5*c*d*e^5 - 5
*b^6*e^6))*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(512*c^(7/2)*e^7) - (d^(5/2)*
(B*d - A*e)*(c*d - b*e)^(5/2)*ArcTanh[(b*d + (2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*
d - b*e]*Sqrt[b*x + c*x^2])])/e^7

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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),x)

[Out]

Timed out

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Mathematica [A]  time = 2.55575, size = 649, normalized size = 0.92 \[ \frac{(x (b+c x))^{5/2} \left (\frac{e \sqrt{x} \left (B \left (75 b^5 e^5+10 b^4 c e^4 (18 d-5 e x)+40 b^3 c^2 e^3 \left (15 d^2-3 d e x+e^2 x^2\right )+16 b^2 c^3 e^2 \left (-660 d^3+295 d^2 e x-186 d e^2 x^2+135 e^3 x^3\right )+64 b c^4 e \left (270 d^4-130 d^3 e x+85 d^2 e^2 x^2-63 d e^3 x^3+50 e^4 x^4\right )-128 c^5 \left (60 d^5-30 d^4 e x+20 d^3 e^2 x^2-15 d^2 e^3 x^3+12 d e^4 x^4-10 e^5 x^5\right )\right )-4 A c e \left (45 b^4 e^4-30 b^3 c e^3 (e x-5 d)-4 b^2 c^2 e^2 \left (660 d^2-295 d e x+186 e^2 x^2\right )+16 b c^3 e \left (270 d^3-130 d^2 e x+85 d e^2 x^2-63 e^3 x^3\right )-32 c^4 \left (60 d^4-30 d^3 e x+20 d^2 e^2 x^2-15 d e^3 x^3+12 e^4 x^4\right )\right )\right )}{c^3 (b+c x)^2}+\frac{15 \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right ) \left (4 A c e \left (3 b^5 e^5+10 b^4 c d e^4+80 b^3 c^2 d^2 e^3-480 b^2 c^3 d^3 e^2+640 b c^4 d^4 e-256 c^5 d^5\right )+B \left (-5 b^6 e^6-12 b^5 c d e^5-40 b^4 c^2 d^2 e^4-320 b^3 c^3 d^3 e^3+1920 b^2 c^4 d^4 e^2-2560 b c^5 d^5 e+1024 c^6 d^6\right )\right )}{c^{7/2} (b+c x)^{5/2}}+\frac{15360 d^{5/2} (B d-A e) (b e-c d)^{5/2} \tan ^{-1}\left (\frac{\sqrt{x} \sqrt{b e-c d}}{\sqrt{d} \sqrt{b+c x}}\right )}{(b+c x)^{5/2}}\right )}{7680 e^7 x^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

((x*(b + c*x))^(5/2)*((e*Sqrt[x]*(-4*A*c*e*(45*b^4*e^4 - 30*b^3*c*e^3*(-5*d + e*
x) - 4*b^2*c^2*e^2*(660*d^2 - 295*d*e*x + 186*e^2*x^2) + 16*b*c^3*e*(270*d^3 - 1
30*d^2*e*x + 85*d*e^2*x^2 - 63*e^3*x^3) - 32*c^4*(60*d^4 - 30*d^3*e*x + 20*d^2*e
^2*x^2 - 15*d*e^3*x^3 + 12*e^4*x^4)) + B*(75*b^5*e^5 + 10*b^4*c*e^4*(18*d - 5*e*
x) + 40*b^3*c^2*e^3*(15*d^2 - 3*d*e*x + e^2*x^2) + 16*b^2*c^3*e^2*(-660*d^3 + 29
5*d^2*e*x - 186*d*e^2*x^2 + 135*e^3*x^3) + 64*b*c^4*e*(270*d^4 - 130*d^3*e*x + 8
5*d^2*e^2*x^2 - 63*d*e^3*x^3 + 50*e^4*x^4) - 128*c^5*(60*d^5 - 30*d^4*e*x + 20*d
^3*e^2*x^2 - 15*d^2*e^3*x^3 + 12*d*e^4*x^4 - 10*e^5*x^5))))/(c^3*(b + c*x)^2) +
(15360*d^(5/2)*(B*d - A*e)*(-(c*d) + b*e)^(5/2)*ArcTan[(Sqrt[-(c*d) + b*e]*Sqrt[
x])/(Sqrt[d]*Sqrt[b + c*x])])/(b + c*x)^(5/2) + (15*(4*A*c*e*(-256*c^5*d^5 + 640
*b*c^4*d^4*e - 480*b^2*c^3*d^3*e^2 + 80*b^3*c^2*d^2*e^3 + 10*b^4*c*d*e^4 + 3*b^5
*e^5) + B*(1024*c^6*d^6 - 2560*b*c^5*d^5*e + 1920*b^2*c^4*d^4*e^2 - 320*b^3*c^3*
d^3*e^3 - 40*b^4*c^2*d^2*e^4 - 12*b^5*c*d*e^5 - 5*b^6*e^6))*Log[c*Sqrt[x] + Sqrt
[c]*Sqrt[b + c*x]])/(c^(7/2)*(b + c*x)^(5/2))))/(7680*e^7*x^(5/2))

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Maple [B]  time = 0.016, size = 4097, normalized size = 5.8 \[ \text{output too large to display} \]

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),x)

[Out]

1/12*B/e/c*(c*x^2+b*x)^(5/2)*b+1/3/e^3*d^2*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*
(b*e-c*d)/e^2)^(3/2)*c*A-3/e^5*d^4/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e
^2+(b*e-2*c*d)/e*(d/e+x)+2*(-d*(b*e-c*d)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(
d/e+x)-d*(b*e-c*d)/e^2)^(1/2))/(d/e+x))*b^2*c*A+3/e^6*d^5/(-d*(b*e-c*d)/e^2)^(1/
2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*(d/e+x)+2*(-d*(b*e-c*d)/e^2)^(1/2)*(c*(d
/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))/(d/e+x))*b^2*c*B+5/32/e^3*
b^2*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*x*d^2*B-5/64/e^2/c
*b^3*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*d*A+5/64/e^3/c*b^
3*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*d^2*B+1/e^4*d^3/(-d*
(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*(d/e+x)+2*(-d*(b*e-c*d
)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))/(d/e+x))
*b^3*A+11/24/e^3*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(3/2)*b*d^2
*B-3/128/e/c^2*b^4*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*A+3
/256/e/c^(5/2)*b^5*ln((1/2*(b*e-2*c*d)/e+c*(d/e+x))/c^(1/2)+(c*(d/e+x)^2+(b*e-2*
c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))*A-1/e^6*d^5*ln((1/2*(b*e-2*c*d)/e+c*(d/e+
x))/c^(1/2)+(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))*c^(5/2)*A
-5/192*B/e*b^3/c^2*(c*x^2+b*x)^(3/2)+5/512*B/e*b^5/c^3*(c*x^2+b*x)^(1/2)-5/1024*
B/e*b^6/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2))+1/e^7*d^6*ln((1/2*(b*e
-2*c*d)/e+c*(d/e+x))/c^(1/2)+(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)
^(1/2))*c^(5/2)*B+11/8/e^3*d^2*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^
2)^(1/2)*b^2*A-11/8/e^4*d^3*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^
(1/2)*b^2*B+1/e^5*d^4*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*
c^2*A-1/e^6*d^5*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*c^2*B+
1/5/e*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(5/2)*A+1/16/e/c*(c*(d
/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(3/2)*b^2*A-11/24/e^2*(c*(d/e+x)^
2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(3/2)*b*d*A-1/3/e^4*d^3*(c*(d/e+x)^2+(b
*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(3/2)*c*B+1/8/e*(c*(d/e+x)^2+(b*e-2*c*d)/e*
(d/e+x)-d*(b*e-c*d)/e^2)^(3/2)*x*b*A+1/6*B/e*(c*x^2+b*x)^(5/2)*x-1/5/e^2*(c*(d/e
+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(5/2)*B*d-5/96*B/e*b^2/c*(c*x^2+b*x
)^(3/2)*x+5/256*B/e*b^4/c^2*(c*x^2+b*x)^(1/2)*x-1/2/e^4*d^3*(c*(d/e+x)^2+(b*e-2*
c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*x*c^2*A+1/2/e^5*d^4*(c*(d/e+x)^2+(b*e-2*c*
d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*x*c^2*B-9/4/e^4*d^3*(c*(d/e+x)^2+(b*e-2*c*d)
/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*b*c*A+9/4/e^5*d^4*(c*(d/e+x)^2+(b*e-2*c*d)/e*(
d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*b*c*B+5/16/e^3*d^2*ln((1/2*(b*e-2*c*d)/e+c*(d/e+x)
)/c^(1/2)+(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))/c^(1/2)*b^3
*A-5/16/e^4*d^3*ln((1/2*(b*e-2*c*d)/e+c*(d/e+x))/c^(1/2)+(c*(d/e+x)^2+(b*e-2*c*d
)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))/c^(1/2)*b^3*B-15/8/e^4*d^3*ln((1/2*(b*e-2*c*
d)/e+c*(d/e+x))/c^(1/2)+(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2
))*c^(1/2)*b^2*A+15/8/e^5*d^4*ln((1/2*(b*e-2*c*d)/e+c*(d/e+x))/c^(1/2)+(c*(d/e+x
)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))*c^(1/2)*b^2*B+5/2/e^5*d^4*ln((
1/2*(b*e-2*c*d)/e+c*(d/e+x))/c^(1/2)+(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c
*d)/e^2)^(1/2))*c^(3/2)*b*A-5/2/e^6*d^5*ln((1/2*(b*e-2*c*d)/e+c*(d/e+x))/c^(1/2)
+(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))*c^(3/2)*b*B-1/4/e^2*
(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(3/2)*x*c*d*A+1/4/e^3*(c*(d/
e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(3/2)*x*c*d^2*B-3/64/e/c*b^3*(c*(d
/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*x*A-1/e^5*d^4/(-d*(b*e-c*d)
/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*(d/e+x)+2*(-d*(b*e-c*d)/e^2)^(1
/2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))/(d/e+x))*b^3*B-1/
e^7*d^6/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*(d/e+x)+2*
(-d*(b*e-c*d)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/
2))/(d/e+x))*c^3*A+1/e^8*d^7/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*
e-2*c*d)/e*(d/e+x)+2*(-d*(b*e-c*d)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)
-d*(b*e-c*d)/e^2)^(1/2))/(d/e+x))*c^3*B+5/128/e^2*d/c^(3/2)*ln((1/2*(b*e-2*c*d)/
e+c*(d/e+x))/c^(1/2)+(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))*
b^4*A-5/128/e^3*d^2/c^(3/2)*ln((1/2*(b*e-2*c*d)/e+c*(d/e+x))/c^(1/2)+(c*(d/e+x)^
2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))*b^4*B-1/8/e^2*(c*(d/e+x)^2+(b*e-
2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(3/2)*x*b*B*d-1/16/e^2/c*(c*(d/e+x)^2+(b*e-2*c
*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(3/2)*b^2*B*d+3/128/e^2/c^2*b^4*(c*(d/e+x)^2+(b*e
-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*B*d-3/256/e^2/c^(5/2)*b^5*ln((1/2*(b*e-
2*c*d)/e+c*(d/e+x))/c^(1/2)+(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^
(1/2))*B*d-5/32/e^2*b^2*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2
)*x*d*A-3/e^7*d^6/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*
(d/e+x)+2*(-d*(b*e-c*d)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d
)/e^2)^(1/2))/(d/e+x))*b*c^2*B+3/4/e^3*d^2*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*
(b*e-c*d)/e^2)^(1/2)*x*b*c*A-3/4/e^4*d^3*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b
*e-c*d)/e^2)^(1/2)*x*b*c*B+3/64/e^2/c*b^3*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(
b*e-c*d)/e^2)^(1/2)*x*B*d+3/e^6*d^5/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/
e^2+(b*e-2*c*d)/e*(d/e+x)+2*(-d*(b*e-c*d)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*
(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))/(d/e+x))*b*c^2*A

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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),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [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),x, algorithm="fricas")

[Out]

Timed out

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

[Out]

Exception raised: TypeError