3.293 \(\int \frac{\sqrt{b x+c x^2}}{(d+e x)^5} \, dx\)

Optimal. Leaf size=258 \[ -\frac{b^2 \left (5 b^2 e^2-16 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{128 d^{7/2} (c d-b e)^{7/2}}+\frac{\sqrt{b x+c x^2} \left (5 b^2 e^2-16 b c d e+16 c^2 d^2\right ) (x (2 c d-b e)+b d)}{64 d^3 (d+e x)^2 (c d-b e)^3}-\frac{5 e \left (b x+c x^2\right )^{3/2} (2 c d-b e)}{24 d^2 (d+e x)^3 (c d-b e)^2}-\frac{e \left (b x+c x^2\right )^{3/2}}{4 d (d+e x)^4 (c d-b e)} \]

[Out]

((16*c^2*d^2 - 16*b*c*d*e + 5*b^2*e^2)*(b*d + (2*c*d - b*e)*x)*Sqrt[b*x + c*x^2]
)/(64*d^3*(c*d - b*e)^3*(d + e*x)^2) - (e*(b*x + c*x^2)^(3/2))/(4*d*(c*d - b*e)*
(d + e*x)^4) - (5*e*(2*c*d - b*e)*(b*x + c*x^2)^(3/2))/(24*d^2*(c*d - b*e)^2*(d
+ e*x)^3) - (b^2*(16*c^2*d^2 - 16*b*c*d*e + 5*b^2*e^2)*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^(7/2)*(c*d - b*e)^
(7/2))

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Rubi [A]  time = 0.791436, antiderivative size = 258, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238 \[ -\frac{b^2 \left (5 b^2 e^2-16 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{128 d^{7/2} (c d-b e)^{7/2}}+\frac{\sqrt{b x+c x^2} \left (5 b^2 e^2-16 b c d e+16 c^2 d^2\right ) (x (2 c d-b e)+b d)}{64 d^3 (d+e x)^2 (c d-b e)^3}-\frac{5 e \left (b x+c x^2\right )^{3/2} (2 c d-b e)}{24 d^2 (d+e x)^3 (c d-b e)^2}-\frac{e \left (b x+c x^2\right )^{3/2}}{4 d (d+e x)^4 (c d-b e)} \]

Antiderivative was successfully verified.

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

[Out]

((16*c^2*d^2 - 16*b*c*d*e + 5*b^2*e^2)*(b*d + (2*c*d - b*e)*x)*Sqrt[b*x + c*x^2]
)/(64*d^3*(c*d - b*e)^3*(d + e*x)^2) - (e*(b*x + c*x^2)^(3/2))/(4*d*(c*d - b*e)*
(d + e*x)^4) - (5*e*(2*c*d - b*e)*(b*x + c*x^2)^(3/2))/(24*d^2*(c*d - b*e)^2*(d
+ e*x)^3) - (b^2*(16*c^2*d^2 - 16*b*c*d*e + 5*b^2*e^2)*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^(7/2)*(c*d - b*e)^
(7/2))

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Rubi in Sympy [A]  time = 78.8856, size = 235, normalized size = 0.91 \[ \frac{b^{2} \left (5 b^{2} e^{2} - 16 b c d e + 16 c^{2} d^{2}\right ) \operatorname{atan}{\left (\frac{- b d + x \left (b e - 2 c d\right )}{2 \sqrt{d} \sqrt{b e - c d} \sqrt{b x + c x^{2}}} \right )}}{128 d^{\frac{7}{2}} \left (b e - c d\right )^{\frac{7}{2}}} + \frac{e \left (b x + c x^{2}\right )^{\frac{3}{2}}}{4 d \left (d + e x\right )^{4} \left (b e - c d\right )} + \frac{5 e \left (b e - 2 c d\right ) \left (b x + c x^{2}\right )^{\frac{3}{2}}}{24 d^{2} \left (d + e x\right )^{3} \left (b e - c d\right )^{2}} - \frac{\left (b d - x \left (b e - 2 c d\right )\right ) \sqrt{b x + c x^{2}} \left (5 b^{2} e^{2} - 16 b c d e + 16 c^{2} d^{2}\right )}{64 d^{3} \left (d + e x\right )^{2} \left (b e - c d\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b**2*(5*b**2*e**2 - 16*b*c*d*e + 16*c**2*d**2)*atan((-b*d + x*(b*e - 2*c*d))/(2*
sqrt(d)*sqrt(b*e - c*d)*sqrt(b*x + c*x**2)))/(128*d**(7/2)*(b*e - c*d)**(7/2)) +
e*(b*x + c*x**2)**(3/2)/(4*d*(d + e*x)**4*(b*e - c*d)) + 5*e*(b*e - 2*c*d)*(b*x
+ c*x**2)**(3/2)/(24*d**2*(d + e*x)**3*(b*e - c*d)**2) - (b*d - x*(b*e - 2*c*d)
)*sqrt(b*x + c*x**2)*(5*b**2*e**2 - 16*b*c*d*e + 16*c**2*d**2)/(64*d**3*(d + e*x
)**2*(b*e - c*d)**3)

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Mathematica [A]  time = 0.625204, size = 272, normalized size = 1.05 \[ \frac{\sqrt{x (b+c x)} \left (-\frac{3 b^2 \left (5 b^2 e^2-16 b c d e+16 c^2 d^2\right ) \tan ^{-1}\left (\frac{\sqrt{x} \sqrt{b e-c d}}{\sqrt{d} \sqrt{b+c x}}\right )}{\sqrt{b+c x} \sqrt{b e-c d}}-\frac{\sqrt{d} \sqrt{x} \left (-2 d (d+e x)^2 \left (5 b^2 e^2-8 b c d e+8 c^2 d^2\right ) (c d-b e)-(d+e x)^3 \left (-15 b^3 e^3+38 b^2 c d e^2-24 b c^2 d^2 e+16 c^3 d^3\right )+48 d^3 (c d-b e)^3-8 d^2 (d+e x) (2 c d-b e) (c d-b e)^2\right )}{e (d+e x)^4}\right )}{192 d^{7/2} \sqrt{x} (c d-b e)^3} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[x*(b + c*x)]*(-((Sqrt[d]*Sqrt[x]*(48*d^3*(c*d - b*e)^3 - 8*d^2*(c*d - b*e)
^2*(2*c*d - b*e)*(d + e*x) - 2*d*(c*d - b*e)*(8*c^2*d^2 - 8*b*c*d*e + 5*b^2*e^2)
*(d + e*x)^2 - (16*c^3*d^3 - 24*b*c^2*d^2*e + 38*b^2*c*d*e^2 - 15*b^3*e^3)*(d +
e*x)^3))/(e*(d + e*x)^4)) - (3*b^2*(16*c^2*d^2 - 16*b*c*d*e + 5*b^2*e^2)*ArcTan[
(Sqrt[-(c*d) + b*e]*Sqrt[x])/(Sqrt[d]*Sqrt[b + c*x])])/(Sqrt[-(c*d) + b*e]*Sqrt[
b + c*x])))/(192*d^(7/2)*(c*d - b*e)^3*Sqrt[x])

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Maple [B]  time = 0.02, size = 4819, normalized size = 18.7 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-15/8/e^2*d/(b*e-c*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))*c^4*b-7/32*e/d^3/(b*e-c*d)^3*c*(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+3/4/e^3*d/(b*e-c*d)^3*c^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))+5/8/e/d
/(b*e-c*d)^3/(d/e+x)^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^2+1/8/e*c/d^2/(b*e-c*d)^2/(d/e+x)^2*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e
-c*d)/e^2)^(3/2)-3/2/e^2/(b*e-c*d)^3*c^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-1/16/e*c^(3/2)/d^2/(b*e-c*d)^2*l
n((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-5/64*e/d^3/(b*e-c*d)^3*c^(1/2)*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^3-35/6
4*e/d^2/(b*e-c*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^3-45/32*e/d^2/(b*e-c*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^2*c^2+5/64*e^2/d^3/(b*e-
c*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^4+5/32*e/d^3/(b*e-c*d)^3/(d/e+x)^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^2-5/128*e^2/d^3/(b*e-c*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^5-9/8/e/d/(b*e-c*d)^3*c^(5/2)*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+35/16/e/(b*e-c*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^3-3/16/d^2/(b*e-c*d)^3*c/(-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^3-5/4/d/(b*e-c*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*c^2-5/8/d
^2/(b*e-c*d)^3/(d/e+x)^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*b+5/8/e^3*d^2/(b*e-c*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))*c^5+35/64*e^2/d^3/(b*e-c*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^3*c+5/24/e/d^2/(b*e-c*d)^2/(d/e+x)
^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-5/12/e^2/d/(b*e-c
*d)^2/(d/e+x)^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+5/64
*e^3/d^4/(b*e-c*d)^4/(d/e+x)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)
^(3/2)*b^3+13/16/d^2/(b*e-c*d)^3*c^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+1/8/e^3*c^3/(b*e-c*d)^2/(-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))+1/4/e^3/d/(b*e-c*d)/(d/e+x)^4*(c*(
d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(3/2)+1/8/e^2*c^(5/2)/d/(b*e-c*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))-25/16/e/(b*e-c*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^(7/2)*b-1/8/e*
c^2/d^2/(b*e-c*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)+5/
8/e^2*d/(b*e-c*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^(9/2)+15/16/e/d/(b*e-c*d)^3*c^2/(-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
^2-15/32*e^2/d^3/(b*e-c*d)^4/(d/e+x)*(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*c-1/16*e*c^2/d^3/(b*e-c*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-1/8/e^2*c^2/d/(b*e-c*d)^2/(-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+15/16*e/d^2/(b*e-
c*d)^4/(d/e+x)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(3/2)*b*c^2+4
5/128*e/d^2/(b*e-c*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^4*c-5/8/d/(b*e-c*d)^4/(d/e+x)*(c*(d/e+x)^2+(b*e-2
*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(3/2)*c^3-3/4/e/d/(b*e-c*d)^3*c^3*(c*(d/e+x)^2+
(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)-5/64*e^3/d^4/(b*e-c*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^4+17/32/d^2/(b*e-c*d)^3*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^2-1/8*c^2/d^2/(b*e-c*d)^3/(d/e+x)*(c*(d/e+x)^2+(b*e-2*c*
d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(3/2)+25/16/d/(b*e-c*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^3*b+45/32/d/(b*e-c*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^(5/2)*b^2+1/8*c^3/d^2/(b*e-c*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+5/8/d/(b*e-c*d)^4*c^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+1/16*e*c/d^3/(b*e-c*d)^3/(d/e+x)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d
/e+x)-d*(b*e-c*d)/e^2)^(3/2)*b-15/16*e/d^2/(b*e-c*d)^4*c^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-5/64*e^3/d^4/(b*e-c*d)^4*c*(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^3+15/32*e^2/d^3/(b*e-c*d)^4*c^
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^2-5/8/e/(b*e-c*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^4+3/4/e^2/(b*e-c
*d)^3*c^(7/2)*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))

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.248709, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/384*(2*(48*b*c^2*d^5 - 48*b^2*c*d^4*e + 15*b^3*d^3*e^2 + (16*c^3*d^3*e^2 - 24
*b*c^2*d^2*e^3 + 38*b^2*c*d*e^4 - 15*b^3*e^5)*x^3 + (64*c^3*d^4*e - 104*b*c^2*d^
3*e^2 + 140*b^2*c*d^2*e^3 - 55*b^3*d*e^4)*x^2 + (96*c^3*d^5 - 176*b*c^2*d^4*e +
198*b^2*c*d^3*e^2 - 73*b^3*d^2*e^3)*x)*sqrt(c*d^2 - b*d*e)*sqrt(c*x^2 + b*x) - 3
*(16*b^2*c^2*d^6 - 16*b^3*c*d^5*e + 5*b^4*d^4*e^2 + (16*b^2*c^2*d^2*e^4 - 16*b^3
*c*d*e^5 + 5*b^4*e^6)*x^4 + 4*(16*b^2*c^2*d^3*e^3 - 16*b^3*c*d^2*e^4 + 5*b^4*d*e
^5)*x^3 + 6*(16*b^2*c^2*d^4*e^2 - 16*b^3*c*d^3*e^3 + 5*b^4*d^2*e^4)*x^2 + 4*(16*
b^2*c^2*d^5*e - 16*b^3*c*d^4*e^2 + 5*b^4*d^3*e^3)*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^3*d
^10 - 3*b*c^2*d^9*e + 3*b^2*c*d^8*e^2 - b^3*d^7*e^3 + (c^3*d^6*e^4 - 3*b*c^2*d^5
*e^5 + 3*b^2*c*d^4*e^6 - b^3*d^3*e^7)*x^4 + 4*(c^3*d^7*e^3 - 3*b*c^2*d^6*e^4 + 3
*b^2*c*d^5*e^5 - b^3*d^4*e^6)*x^3 + 6*(c^3*d^8*e^2 - 3*b*c^2*d^7*e^3 + 3*b^2*c*d
^6*e^4 - b^3*d^5*e^5)*x^2 + 4*(c^3*d^9*e - 3*b*c^2*d^8*e^2 + 3*b^2*c*d^7*e^3 - b
^3*d^6*e^4)*x)*sqrt(c*d^2 - b*d*e)), 1/192*((48*b*c^2*d^5 - 48*b^2*c*d^4*e + 15*
b^3*d^3*e^2 + (16*c^3*d^3*e^2 - 24*b*c^2*d^2*e^3 + 38*b^2*c*d*e^4 - 15*b^3*e^5)*
x^3 + (64*c^3*d^4*e - 104*b*c^2*d^3*e^2 + 140*b^2*c*d^2*e^3 - 55*b^3*d*e^4)*x^2
+ (96*c^3*d^5 - 176*b*c^2*d^4*e + 198*b^2*c*d^3*e^2 - 73*b^3*d^2*e^3)*x)*sqrt(-c
*d^2 + b*d*e)*sqrt(c*x^2 + b*x) + 3*(16*b^2*c^2*d^6 - 16*b^3*c*d^5*e + 5*b^4*d^4
*e^2 + (16*b^2*c^2*d^2*e^4 - 16*b^3*c*d*e^5 + 5*b^4*e^6)*x^4 + 4*(16*b^2*c^2*d^3
*e^3 - 16*b^3*c*d^2*e^4 + 5*b^4*d*e^5)*x^3 + 6*(16*b^2*c^2*d^4*e^2 - 16*b^3*c*d^
3*e^3 + 5*b^4*d^2*e^4)*x^2 + 4*(16*b^2*c^2*d^5*e - 16*b^3*c*d^4*e^2 + 5*b^4*d^3*
e^3)*x)*arctan(-sqrt(-c*d^2 + b*d*e)*sqrt(c*x^2 + b*x)/((c*d - b*e)*x)))/((c^3*d
^10 - 3*b*c^2*d^9*e + 3*b^2*c*d^8*e^2 - b^3*d^7*e^3 + (c^3*d^6*e^4 - 3*b*c^2*d^5
*e^5 + 3*b^2*c*d^4*e^6 - b^3*d^3*e^7)*x^4 + 4*(c^3*d^7*e^3 - 3*b*c^2*d^6*e^4 + 3
*b^2*c*d^5*e^5 - b^3*d^4*e^6)*x^3 + 6*(c^3*d^8*e^2 - 3*b*c^2*d^7*e^3 + 3*b^2*c*d
^6*e^4 - b^3*d^5*e^5)*x^2 + 4*(c^3*d^9*e - 3*b*c^2*d^8*e^2 + 3*b^2*c*d^7*e^3 - b
^3*d^6*e^4)*x)*sqrt(-c*d^2 + b*d*e))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{x \left (b + c x\right )}}{\left (d + e x\right )^{5}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+b*x)**(1/2)/(e*x+d)**5,x)

[Out]

Integral(sqrt(x*(b + c*x))/(d + e*x)**5, x)

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GIAC/XCAS [A]  time = 0.401369, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done