3.1168 \(\int \frac{(A+B x) \sqrt{b x+c x^2}}{(d+e x)^3} \, dx\)
Optimal. Leaf size=235 \[ -\frac{\left (A b^2 e^3+B d \left (3 b^2 e^2-12 b c d e+8 c^2 d^2\right )\right ) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{8 d^{3/2} e^3 (c d-b e)^{3/2}}+\frac{\sqrt{b x+c x^2} \left (d \left (A b e^2-B d (4 c d-3 b e)\right )-e x (B d (6 c d-5 b e)-A e (2 c d-b e))\right )}{4 d e^2 (d+e x)^2 (c d-b e)}+\frac{2 B \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{e^3} \]
[Out]
((d*(A*b*e^2 - B*d*(4*c*d - 3*b*e)) - e*(B*d*(6*c*d - 5*b*e) - A*e*(2*c*d - b*e)
)*x)*Sqrt[b*x + c*x^2])/(4*d*e^2*(c*d - b*e)*(d + e*x)^2) + (2*B*Sqrt[c]*ArcTanh
[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/e^3 - ((A*b^2*e^3 + B*d*(8*c^2*d^2 - 12*b*c*d*e
+ 3*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])])/(8*d^(3/2)*e^3*(c*d - b*e)^(3/2))
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Rubi [A] time = 0.797078, antiderivative size = 235, normalized size of antiderivative = 1.,
number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192
\[ -\frac{\left (A b^2 e^3+B d \left (3 b^2 e^2-12 b c d e+8 c^2 d^2\right )\right ) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{8 d^{3/2} e^3 (c d-b e)^{3/2}}+\frac{\sqrt{b x+c x^2} \left (d \left (A b e^2-B d (4 c d-3 b e)\right )-e x (B d (6 c d-5 b e)-A e (2 c d-b e))\right )}{4 d e^2 (d+e x)^2 (c d-b e)}+\frac{2 B \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{e^3} \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*Sqrt[b*x + c*x^2])/(d + e*x)^3,x]
[Out]
((d*(A*b*e^2 - B*d*(4*c*d - 3*b*e)) - e*(B*d*(6*c*d - 5*b*e) - A*e*(2*c*d - b*e)
)*x)*Sqrt[b*x + c*x^2])/(4*d*e^2*(c*d - b*e)*(d + e*x)^2) + (2*B*Sqrt[c]*ArcTanh
[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/e^3 - ((A*b^2*e^3 + B*d*(8*c^2*d^2 - 12*b*c*d*e
+ 3*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])])/(8*d^(3/2)*e^3*(c*d - b*e)^(3/2))
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Rubi in Sympy [A] time = 86.828, size = 233, normalized size = 0.99 \[ \frac{2 B \sqrt{c} \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )}}{e^{3}} - \frac{\sqrt{b x + c x^{2}} \left (\frac{d \left (A b e^{2} + 3 B b d e - 4 B c d^{2}\right )}{4} - \frac{e x \left (A b e^{2} - 2 A c d e - 5 B b d e + 6 B c d^{2}\right )}{4}\right )}{d e^{2} \left (d + e x\right )^{2} \left (b e - c d\right )} + \frac{\left (A b^{2} e^{3} + 3 B b^{2} d e^{2} - 12 B b c d^{2} e + 8 B c^{2} d^{3}\right ) \operatorname{atan}{\left (\frac{- b d + x \left (b e - 2 c d\right )}{2 \sqrt{d} \sqrt{b e - c d} \sqrt{b x + c x^{2}}} \right )}}{8 d^{\frac{3}{2}} e^{3} \left (b e - c d\right )^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(c*x**2+b*x)**(1/2)/(e*x+d)**3,x)
[Out]
2*B*sqrt(c)*atanh(sqrt(c)*x/sqrt(b*x + c*x**2))/e**3 - sqrt(b*x + c*x**2)*(d*(A*
b*e**2 + 3*B*b*d*e - 4*B*c*d**2)/4 - e*x*(A*b*e**2 - 2*A*c*d*e - 5*B*b*d*e + 6*B
*c*d**2)/4)/(d*e**2*(d + e*x)**2*(b*e - c*d)) + (A*b**2*e**3 + 3*B*b**2*d*e**2 -
12*B*b*c*d**2*e + 8*B*c**2*d**3)*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**(3/2)*e**3*(b*e - c*d)**(3/2))
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Mathematica [A] time = 0.914647, size = 235, normalized size = 1. \[ \frac{\sqrt{x (b+c x)} \left (\frac{\left (A b^2 e^3+B d \left (3 b^2 e^2-12 b c d e+8 c^2 d^2\right )\right ) \tan ^{-1}\left (\frac{\sqrt{x} \sqrt{b e-c d}}{\sqrt{d} \sqrt{b+c x}}\right )}{d^{3/2} \sqrt{b+c x} (b e-c d)^{3/2}}-\frac{e \sqrt{x} (A e (b e-2 c d)+B d (6 c d-5 b e))}{d (d+e x) (c d-b e)}-\frac{2 e \sqrt{x} (A e-B d)}{(d+e x)^2}+\frac{8 B \sqrt{c} \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right )}{\sqrt{b+c x}}\right )}{4 e^3 \sqrt{x}} \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*Sqrt[b*x + c*x^2])/(d + e*x)^3,x]
[Out]
(Sqrt[x*(b + c*x)]*((-2*e*(-(B*d) + A*e)*Sqrt[x])/(d + e*x)^2 - (e*(B*d*(6*c*d -
5*b*e) + A*e*(-2*c*d + b*e))*Sqrt[x])/(d*(c*d - b*e)*(d + e*x)) + ((A*b^2*e^3 +
B*d*(8*c^2*d^2 - 12*b*c*d*e + 3*b^2*e^2))*ArcTan[(Sqrt[-(c*d) + b*e]*Sqrt[x])/(
Sqrt[d]*Sqrt[b + c*x])])/(d^(3/2)*(-(c*d) + b*e)^(3/2)*Sqrt[b + c*x]) + (8*B*Sqr
t[c]*Log[c*Sqrt[x] + Sqrt[c]*Sqrt[b + c*x]])/Sqrt[b + c*x]))/(4*e^3*Sqrt[x])
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Maple [B] time = 0.018, size = 4316, normalized size = 18.4 \[ \text{output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(c*x^2+b*x)^(1/2)/(e*x+d)^3,x)
[Out]
1/2/e^2*c^(3/2)/(b*e-c*d)*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+1/2/e/d/(b*e-c*d)/(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)*A-1/2/e/(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)*x*B+3/4/d/(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)*b*c*A+1/2/e^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))*c^(5/2)*A-1/2/e^3*d^2/(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))*c^
(5/2)*B-1/4*e/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)*b^2*A+1/2/e^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)*c^2*B*d-3/4/e/(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))*c^(3/2)*b*A+B/e/d/(b*e
-c*d)/(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/e/d/(b
*e-c*d)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*b+5/4*B/e^2/(b
*e-c*d)*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-3/2*B/e^3*d/(b*e-c*d)*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)-1/2/e*c/d/(b*e-c*d)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^
(1/2)*A+1/4/d/(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)*x*b*B-1/2*B/e^2/(b*e-c*d)/(-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^2-1/4/d/(b*e-c*d)^2/(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*B-1/2/d/(b*e-c*d)^2/(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*A+1/4/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))*c^(1/2)*b^2*A+1/2/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)*x*A-1/8/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^3*A-1/4/e/(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))*c^(1/2)*b^2*B-3/2*B/e^4*d^2/(b*e-c*d)/(-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^2-B/e*c/d/(b*e-
c*d)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*x-1/2/e/(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)*c^2*A+1/4*e/d^2/(b*
e-c*d)^2/(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*A+1
/2/e^3*c^2*d/(b*e-c*d)/(-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))*A+2*B/e^3*d/(b*e-c*d)/(-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-5/8/e^2/(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^2*c*B*d-1/4*e/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)*x*b*A-1/e^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*c^2*A+5/8/e/(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^
2*c*A+1/2/e^3*d^2/(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))*c^3*A-1/4/e*c^(1/2)/d/(b*e-c*d)*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*A+3/4/e^2/(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))*c^(3/2)*b*B*d-1/2/e^2*c/(b*e-
c*d)/(-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*A-1/2/e^2/(b*e-c*d)/(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+1/4/d/(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)*b^2*B+3/2*B/e^2/(b*e-c*d)*(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/e/(b*e-c*d)^2/(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*B-3/4/e/(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)*b*c*B+1/8/e/(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^3*B+1/e^3*d^2/(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*c^2*B-1/2/e^4*d^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))*c^3*B
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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)*(B*x + A)/(e*x + d)^3,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 1.78854, 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)*(B*x + A)/(e*x + d)^3,x, algorithm="fricas")
[Out]
[1/8*(8*(B*c*d^4 - B*b*d^3*e + (B*c*d^2*e^2 - B*b*d*e^3)*x^2 + 2*(B*c*d^3*e - B*
b*d^2*e^2)*x)*sqrt(c*d^2 - b*d*e)*sqrt(c)*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sq
rt(c)) - 2*(4*B*c*d^3*e - 3*B*b*d^2*e^2 - A*b*d*e^3 + (6*B*c*d^2*e^2 + A*b*e^4 -
(5*B*b + 2*A*c)*d*e^3)*x)*sqrt(c*d^2 - b*d*e)*sqrt(c*x^2 + b*x) - (8*B*c^2*d^5
- 12*B*b*c*d^4*e + 3*B*b^2*d^3*e^2 + A*b^2*d^2*e^3 + (8*B*c^2*d^3*e^2 - 12*B*b*c
*d^2*e^3 + 3*B*b^2*d*e^4 + A*b^2*e^5)*x^2 + 2*(8*B*c^2*d^4*e - 12*B*b*c*d^3*e^2
+ 3*B*b^2*d^2*e^3 + A*b^2*d*e^4)*x)*log((2*(c*d^2 - b*d*e)*sqrt(c*x^2 + b*x) + s
qrt(c*d^2 - b*d*e)*(b*d + (2*c*d - b*e)*x))/(e*x + d)))/((c*d^4*e^3 - b*d^3*e^4
+ (c*d^2*e^5 - b*d*e^6)*x^2 + 2*(c*d^3*e^4 - b*d^2*e^5)*x)*sqrt(c*d^2 - b*d*e)),
1/4*(4*(B*c*d^4 - B*b*d^3*e + (B*c*d^2*e^2 - B*b*d*e^3)*x^2 + 2*(B*c*d^3*e - B*
b*d^2*e^2)*x)*sqrt(-c*d^2 + b*d*e)*sqrt(c)*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*s
qrt(c)) - (4*B*c*d^3*e - 3*B*b*d^2*e^2 - A*b*d*e^3 + (6*B*c*d^2*e^2 + A*b*e^4 -
(5*B*b + 2*A*c)*d*e^3)*x)*sqrt(-c*d^2 + b*d*e)*sqrt(c*x^2 + b*x) + (8*B*c^2*d^5
- 12*B*b*c*d^4*e + 3*B*b^2*d^3*e^2 + A*b^2*d^2*e^3 + (8*B*c^2*d^3*e^2 - 12*B*b*c
*d^2*e^3 + 3*B*b^2*d*e^4 + A*b^2*e^5)*x^2 + 2*(8*B*c^2*d^4*e - 12*B*b*c*d^3*e^2
+ 3*B*b^2*d^2*e^3 + A*b^2*d*e^4)*x)*arctan(-sqrt(-c*d^2 + b*d*e)*sqrt(c*x^2 + b*
x)/((c*d - b*e)*x)))/((c*d^4*e^3 - b*d^3*e^4 + (c*d^2*e^5 - b*d*e^6)*x^2 + 2*(c*
d^3*e^4 - b*d^2*e^5)*x)*sqrt(-c*d^2 + b*d*e)), 1/8*(16*(B*c*d^4 - B*b*d^3*e + (B
*c*d^2*e^2 - B*b*d*e^3)*x^2 + 2*(B*c*d^3*e - B*b*d^2*e^2)*x)*sqrt(c*d^2 - b*d*e)
*sqrt(-c)*arctan(sqrt(c*x^2 + b*x)/(sqrt(-c)*x)) - 2*(4*B*c*d^3*e - 3*B*b*d^2*e^
2 - A*b*d*e^3 + (6*B*c*d^2*e^2 + A*b*e^4 - (5*B*b + 2*A*c)*d*e^3)*x)*sqrt(c*d^2
- b*d*e)*sqrt(c*x^2 + b*x) - (8*B*c^2*d^5 - 12*B*b*c*d^4*e + 3*B*b^2*d^3*e^2 + A
*b^2*d^2*e^3 + (8*B*c^2*d^3*e^2 - 12*B*b*c*d^2*e^3 + 3*B*b^2*d*e^4 + A*b^2*e^5)*
x^2 + 2*(8*B*c^2*d^4*e - 12*B*b*c*d^3*e^2 + 3*B*b^2*d^2*e^3 + A*b^2*d*e^4)*x)*lo
g((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^4*e^3 - b*d^3*e^4 + (c*d^2*e^5 - b*d*e^6)*x^2 + 2*(c*d^3
*e^4 - b*d^2*e^5)*x)*sqrt(c*d^2 - b*d*e)), 1/4*(8*(B*c*d^4 - B*b*d^3*e + (B*c*d^
2*e^2 - B*b*d*e^3)*x^2 + 2*(B*c*d^3*e - B*b*d^2*e^2)*x)*sqrt(-c*d^2 + b*d*e)*sqr
t(-c)*arctan(sqrt(c*x^2 + b*x)/(sqrt(-c)*x)) - (4*B*c*d^3*e - 3*B*b*d^2*e^2 - A*
b*d*e^3 + (6*B*c*d^2*e^2 + A*b*e^4 - (5*B*b + 2*A*c)*d*e^3)*x)*sqrt(-c*d^2 + b*d
*e)*sqrt(c*x^2 + b*x) + (8*B*c^2*d^5 - 12*B*b*c*d^4*e + 3*B*b^2*d^3*e^2 + A*b^2*
d^2*e^3 + (8*B*c^2*d^3*e^2 - 12*B*b*c*d^2*e^3 + 3*B*b^2*d*e^4 + A*b^2*e^5)*x^2 +
2*(8*B*c^2*d^4*e - 12*B*b*c*d^3*e^2 + 3*B*b^2*d^2*e^3 + A*b^2*d*e^4)*x)*arctan(
-sqrt(-c*d^2 + b*d*e)*sqrt(c*x^2 + b*x)/((c*d - b*e)*x)))/((c*d^4*e^3 - b*d^3*e^
4 + (c*d^2*e^5 - b*d*e^6)*x^2 + 2*(c*d^3*e^4 - b*d^2*e^5)*x)*sqrt(-c*d^2 + b*d*e
))]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{x \left (b + c x\right )} \left (A + B x\right )}{\left (d + e x\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*(c*x**2+b*x)**(1/2)/(e*x+d)**3,x)
[Out]
Integral(sqrt(x*(b + c*x))*(A + B*x)/(d + e*x)**3, x)
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GIAC/XCAS [A] time = 0.638634, 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)*(B*x + A)/(e*x + d)^3,x, algorithm="giac")
[Out]
sage0*x