3.292 \(\int \frac{\sqrt{b x+c x^2}}{(d+e x)^4} \, dx\)
Optimal. Leaf size=183 \[ -\frac{b^2 (2 c d-b e) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{16 d^{5/2} (c d-b e)^{5/2}}+\frac{\sqrt{b x+c x^2} (2 c d-b e) (x (2 c d-b e)+b d)}{8 d^2 (d+e x)^2 (c d-b e)^2}-\frac{e \left (b x+c x^2\right )^{3/2}}{3 d (d+e x)^3 (c d-b e)} \]
[Out]
((2*c*d - b*e)*(b*d + (2*c*d - b*e)*x)*Sqrt[b*x + c*x^2])/(8*d^2*(c*d - b*e)^2*(
d + e*x)^2) - (e*(b*x + c*x^2)^(3/2))/(3*d*(c*d - b*e)*(d + e*x)^3) - (b^2*(2*c*
d - b*e)*ArcTanh[(b*d + (2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x + c
*x^2])])/(16*d^(5/2)*(c*d - b*e)^(5/2))
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Rubi [A] time = 0.37386, antiderivative size = 183, normalized size of antiderivative = 1.,
number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19
\[ -\frac{b^2 (2 c d-b e) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{16 d^{5/2} (c d-b e)^{5/2}}+\frac{\sqrt{b x+c x^2} (2 c d-b e) (x (2 c d-b e)+b d)}{8 d^2 (d+e x)^2 (c d-b e)^2}-\frac{e \left (b x+c x^2\right )^{3/2}}{3 d (d+e x)^3 (c d-b e)} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[b*x + c*x^2]/(d + e*x)^4,x]
[Out]
((2*c*d - b*e)*(b*d + (2*c*d - b*e)*x)*Sqrt[b*x + c*x^2])/(8*d^2*(c*d - b*e)^2*(
d + e*x)^2) - (e*(b*x + c*x^2)^(3/2))/(3*d*(c*d - b*e)*(d + e*x)^3) - (b^2*(2*c*
d - b*e)*ArcTanh[(b*d + (2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x + c
*x^2])])/(16*d^(5/2)*(c*d - b*e)^(5/2))
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Rubi in Sympy [A] time = 40.5197, size = 156, normalized size = 0.85 \[ \frac{b^{2} \left (\frac{b e}{2} - c d\right ) \operatorname{atan}{\left (\frac{- b d + x \left (b e - 2 c d\right )}{2 \sqrt{d} \sqrt{b e - c d} \sqrt{b x + c x^{2}}} \right )}}{8 d^{\frac{5}{2}} \left (b e - c d\right )^{\frac{5}{2}}} + \frac{e \left (b x + c x^{2}\right )^{\frac{3}{2}}}{3 d \left (d + e x\right )^{3} \left (b e - c d\right )} - \frac{\left (b d - x \left (b e - 2 c d\right )\right ) \left (\frac{b e}{2} - c d\right ) \sqrt{b x + c x^{2}}}{4 d^{2} \left (d + e x\right )^{2} \left (b e - c d\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**2+b*x)**(1/2)/(e*x+d)**4,x)
[Out]
b**2*(b*e/2 - c*d)*atan((-b*d + x*(b*e - 2*c*d))/(2*sqrt(d)*sqrt(b*e - c*d)*sqrt
(b*x + c*x**2)))/(8*d**(5/2)*(b*e - c*d)**(5/2)) + e*(b*x + c*x**2)**(3/2)/(3*d*
(d + e*x)**3*(b*e - c*d)) - (b*d - x*(b*e - 2*c*d))*(b*e/2 - c*d)*sqrt(b*x + c*x
**2)/(4*d**2*(d + e*x)**2*(b*e - c*d)**2)
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Mathematica [A] time = 0.730887, size = 186, normalized size = 1.02 \[ \frac{\sqrt{x (b+c x)} \left (\frac{\sqrt{x} \left (b^2 e \left (-3 d^2+8 d e x+3 e^2 x^2\right )+2 b c d \left (3 d^2-7 d e x-2 e^2 x^2\right )+4 c^2 d^2 x (3 d+e x)\right )}{3 d^2 (d+e x)^3 (c d-b e)^2}+\frac{b^2 (b e-2 c d) \tan ^{-1}\left (\frac{\sqrt{x} \sqrt{b e-c d}}{\sqrt{d} \sqrt{b+c x}}\right )}{d^{5/2} \sqrt{b+c x} (b e-c d)^{5/2}}\right )}{8 \sqrt{x}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[b*x + c*x^2]/(d + e*x)^4,x]
[Out]
(Sqrt[x*(b + c*x)]*((Sqrt[x]*(4*c^2*d^2*x*(3*d + e*x) + 2*b*c*d*(3*d^2 - 7*d*e*x
- 2*e^2*x^2) + b^2*e*(-3*d^2 + 8*d*e*x + 3*e^2*x^2)))/(3*d^2*(c*d - b*e)^2*(d +
e*x)^3) + (b^2*(-2*c*d + b*e)*ArcTan[(Sqrt[-(c*d) + b*e]*Sqrt[x])/(Sqrt[d]*Sqrt
[b + c*x])])/(d^(5/2)*(-(c*d) + b*e)^(5/2)*Sqrt[b + c*x])))/(8*Sqrt[x])
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Maple [B] time = 0.016, size = 2891, normalized size = 15.8 \[ \text{output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+b*x)^(1/2)/(e*x+d)^4,x)
[Out]
1/2/e/d/(b*e-c*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+7/16/d/(b*e-c*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*c-
1/16*e/d^2/(b*e-c*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^4+1/8*e^2/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^2-1/2/e/d/(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)*c+1/8*e/d^2/(b*e-c*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+3/4/e^2/(b*e-c*d)^2*c^2/(-d*(b*e-c*d)/e^2)^(1/2)*l
n((-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+1/2*e/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)*x*b-1/2*e/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)*b*c
-1/4/e/d/(b*e-c*d)^2*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^2+5/4/e^2*d/(b*e-c*d)^3/(-d*(b*e-c*d)/e^2)^(1/2)*l
n((-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^3-1/8*e^2/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)*x*b^2+5/8*e
/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)*b^2*c
-1/2/e^3*d^2/(b*e-c*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))*c^4-1/2/e^3*d/(b*e-c*d)^2*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))-9/8/e/(b*e-c*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^2*c^2-1/2/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)*x-1/4/d^2/(b*e-c*d)^2*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-1/8/d^2/(b*e-c*d)^2*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^2+1/2/d/(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)*c^2+
1/3/e^2/d/(b*e-c*d)/(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)-1/2/e^2*d/(b*e-c*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^(7/2)-1/8*e^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)*b^3+1/2/e/d/(b*e-c*d
)^2*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)+1/4/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)*b-1/d/
(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)*b*c^2-5/8/
d/(b*e-c*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^(3/2)*b^2+1/e/(b*e-c*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^(5/2)*b+1/2/e/(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)*c^3-1/2/e^2/(b*e-c*d)^2*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))
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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)^4,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 0.251355, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^2 + b*x)/(e*x + d)^4,x, algorithm="fricas")
[Out]
[1/48*(2*(6*b*c*d^3 - 3*b^2*d^2*e + (4*c^2*d^2*e - 4*b*c*d*e^2 + 3*b^2*e^3)*x^2
+ 2*(6*c^2*d^3 - 7*b*c*d^2*e + 4*b^2*d*e^2)*x)*sqrt(c*d^2 - b*d*e)*sqrt(c*x^2 +
b*x) - 3*(2*b^2*c*d^4 - b^3*d^3*e + (2*b^2*c*d*e^3 - b^3*e^4)*x^3 + 3*(2*b^2*c*d
^2*e^2 - b^3*d*e^3)*x^2 + 3*(2*b^2*c*d^3*e - b^3*d^2*e^2)*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^2*d^7 - 2*b*c*d^6*e + b^2*d^5*e^2 + (c^2*d^4*e^3 - 2*b*c*d^3*e^4 + b^2*d^2*
e^5)*x^3 + 3*(c^2*d^5*e^2 - 2*b*c*d^4*e^3 + b^2*d^3*e^4)*x^2 + 3*(c^2*d^6*e - 2*
b*c*d^5*e^2 + b^2*d^4*e^3)*x)*sqrt(c*d^2 - b*d*e)), 1/24*((6*b*c*d^3 - 3*b^2*d^2
*e + (4*c^2*d^2*e - 4*b*c*d*e^2 + 3*b^2*e^3)*x^2 + 2*(6*c^2*d^3 - 7*b*c*d^2*e +
4*b^2*d*e^2)*x)*sqrt(-c*d^2 + b*d*e)*sqrt(c*x^2 + b*x) + 3*(2*b^2*c*d^4 - b^3*d^
3*e + (2*b^2*c*d*e^3 - b^3*e^4)*x^3 + 3*(2*b^2*c*d^2*e^2 - b^3*d*e^3)*x^2 + 3*(2
*b^2*c*d^3*e - b^3*d^2*e^2)*x)*arctan(-sqrt(-c*d^2 + b*d*e)*sqrt(c*x^2 + b*x)/((
c*d - b*e)*x)))/((c^2*d^7 - 2*b*c*d^6*e + b^2*d^5*e^2 + (c^2*d^4*e^3 - 2*b*c*d^3
*e^4 + b^2*d^2*e^5)*x^3 + 3*(c^2*d^5*e^2 - 2*b*c*d^4*e^3 + b^2*d^3*e^4)*x^2 + 3*
(c^2*d^6*e - 2*b*c*d^5*e^2 + b^2*d^4*e^3)*x)*sqrt(-c*d^2 + b*d*e))]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{x \left (b + c x\right )}}{\left (d + e x\right )^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**2+b*x)**(1/2)/(e*x+d)**4,x)
[Out]
Integral(sqrt(x*(b + c*x))/(d + e*x)**4, x)
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GIAC/XCAS [A] time = 0.58698, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^2 + b*x)/(e*x + d)^4,x, algorithm="giac")
[Out]
sage0*x