3.1167 \(\int \frac{(A+B x) \sqrt{b x+c x^2}}{(d+e x)^2} \, dx\)
Optimal. Leaf size=185 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right ) (-2 A c e-b B e+4 B c d)}{\sqrt{c} e^3}+\frac{(B d (4 c d-3 b e)-A e (2 c d-b e)) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{2 \sqrt{d} e^3 \sqrt{c d-b e}}+\frac{\sqrt{b x+c x^2} (-A e+2 B d+B e x)}{e^2 (d+e x)} \]
[Out]
((2*B*d - A*e + B*e*x)*Sqrt[b*x + c*x^2])/(e^2*(d + e*x)) - ((4*B*c*d - b*B*e -
2*A*c*e)*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(Sqrt[c]*e^3) + ((B*d*(4*c*d -
3*b*e) - A*e*(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])])/(2*Sqrt[d]*e^3*Sqrt[c*d - b*e])
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Rubi [A] time = 0.472826, antiderivative size = 185, 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
\[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right ) (-2 A c e-b B e+4 B c d)}{\sqrt{c} e^3}+\frac{(B d (4 c d-3 b e)-A e (2 c d-b e)) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{2 \sqrt{d} e^3 \sqrt{c d-b e}}+\frac{\sqrt{b x+c x^2} (-A e+2 B d+B e x)}{e^2 (d+e x)} \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*Sqrt[b*x + c*x^2])/(d + e*x)^2,x]
[Out]
((2*B*d - A*e + B*e*x)*Sqrt[b*x + c*x^2])/(e^2*(d + e*x)) - ((4*B*c*d - b*B*e -
2*A*c*e)*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(Sqrt[c]*e^3) + ((B*d*(4*c*d -
3*b*e) - A*e*(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])])/(2*Sqrt[d]*e^3*Sqrt[c*d - b*e])
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Rubi in Sympy [A] time = 61.3986, size = 177, normalized size = 0.96 \[ - \frac{\sqrt{b x + c x^{2}} \left (A e - 2 B d - B e x\right )}{e^{2} \left (d + e x\right )} + \frac{\left (A b e^{2} - 2 A c d e - 3 B b d e + 4 B c d^{2}\right ) \operatorname{atan}{\left (\frac{- b d + x \left (b e - 2 c d\right )}{2 \sqrt{d} \sqrt{b e - c d} \sqrt{b x + c x^{2}}} \right )}}{2 \sqrt{d} e^{3} \sqrt{b e - c d}} + \frac{\left (2 A c e + B b e - 4 B c d\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )}}{\sqrt{c} e^{3}} \]
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)**2,x)
[Out]
-sqrt(b*x + c*x**2)*(A*e - 2*B*d - B*e*x)/(e**2*(d + e*x)) + (A*b*e**2 - 2*A*c*d
*e - 3*B*b*d*e + 4*B*c*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)))/(2*sqrt(d)*e**3*sqrt(b*e - c*d)) + (2*A*c*e + B*b*e -
4*B*c*d)*atanh(sqrt(c)*x/sqrt(b*x + c*x**2))/(sqrt(c)*e**3)
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Mathematica [A] time = 0.562531, size = 192, normalized size = 1.04 \[ \frac{\sqrt{x (b+c x)} \left (-\frac{\log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right ) (-2 A c e-b B e+4 B c d)}{\sqrt{c} \sqrt{b+c x}}+\frac{(A e (b e-2 c d)+B d (4 c d-3 b e)) \tan ^{-1}\left (\frac{\sqrt{x} \sqrt{b e-c d}}{\sqrt{d} \sqrt{b+c x}}\right )}{\sqrt{d} \sqrt{b+c x} \sqrt{b e-c d}}-\frac{e \sqrt{x} (A e-B d)}{d+e x}+B e \sqrt{x}\right )}{e^3 \sqrt{x}} \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*Sqrt[b*x + c*x^2])/(d + e*x)^2,x]
[Out]
(Sqrt[x*(b + c*x)]*(B*e*Sqrt[x] - (e*(-(B*d) + A*e)*Sqrt[x])/(d + e*x) + ((B*d*(
4*c*d - 3*b*e) + A*e*(-2*c*d + b*e))*ArcTan[(Sqrt[-(c*d) + b*e]*Sqrt[x])/(Sqrt[d
]*Sqrt[b + c*x])])/(Sqrt[d]*Sqrt[-(c*d) + b*e]*Sqrt[b + c*x]) - ((4*B*c*d - b*B*
e - 2*A*c*e)*Log[c*Sqrt[x] + Sqrt[c]*Sqrt[b + c*x]])/(Sqrt[c]*Sqrt[b + c*x])))/(
e^3*Sqrt[x])
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Maple [B] time = 0.015, size = 2285, normalized size = 12.4 \[ \text{result too large to display} \]
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)^2,x)
[Out]
B/e^2*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)+1/2*B/e^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-B/e^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)*d+B/e^3*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-B/e^4
*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+1/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)*A-1/e/(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-1/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*A+1/e/(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*B+1/e/(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*A-1/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*
B*d+1/e/(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*A-1/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*B*d-1/e^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))*c^(3/2)*A+1/e^3*d^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^(3/2)*B-1/2/e/(b*e-c*d)/(-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^2*A+1/2/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*B*d+3/2/e^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))*b*c*A-3/2/e^3*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))*b*c*B-
1/e^3*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*A+1/e^4*d^3/(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-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*A+1/e*c/(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*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)^2,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 0.87517, 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)*(B*x + A)/(e*x + d)^2,x, algorithm="fricas")
[Out]
[1/2*(2*(B*e^2*x + 2*B*d*e - A*e^2)*sqrt(c*d^2 - b*d*e)*sqrt(c*x^2 + b*x)*sqrt(c
) - (4*B*c*d^2 - (B*b + 2*A*c)*d*e + (4*B*c*d*e - (B*b + 2*A*c)*e^2)*x)*sqrt(c*d
^2 - b*d*e)*log((2*c*x + b)*sqrt(c) + 2*sqrt(c*x^2 + b*x)*c) + (4*B*c*d^3 + A*b*
d*e^2 - (3*B*b + 2*A*c)*d^2*e + (4*B*c*d^2*e + A*b*e^3 - (3*B*b + 2*A*c)*d*e^2)*
x)*sqrt(c)*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)))/((e^4*x + d*e^3)*sqrt(c*d^2 - b*d*e)*sqrt(c)), 1/
2*(2*(B*e^2*x + 2*B*d*e - A*e^2)*sqrt(-c*d^2 + b*d*e)*sqrt(c*x^2 + b*x)*sqrt(c)
- 2*(4*B*c*d^3 + A*b*d*e^2 - (3*B*b + 2*A*c)*d^2*e + (4*B*c*d^2*e + A*b*e^3 - (3
*B*b + 2*A*c)*d*e^2)*x)*sqrt(c)*arctan(-sqrt(-c*d^2 + b*d*e)*sqrt(c*x^2 + b*x)/(
(c*d - b*e)*x)) - (4*B*c*d^2 - (B*b + 2*A*c)*d*e + (4*B*c*d*e - (B*b + 2*A*c)*e^
2)*x)*sqrt(-c*d^2 + b*d*e)*log((2*c*x + b)*sqrt(c) + 2*sqrt(c*x^2 + b*x)*c))/((e
^4*x + d*e^3)*sqrt(-c*d^2 + b*d*e)*sqrt(c)), 1/2*(2*(B*e^2*x + 2*B*d*e - A*e^2)*
sqrt(c*d^2 - b*d*e)*sqrt(c*x^2 + b*x)*sqrt(-c) - 2*(4*B*c*d^2 - (B*b + 2*A*c)*d*
e + (4*B*c*d*e - (B*b + 2*A*c)*e^2)*x)*sqrt(c*d^2 - b*d*e)*arctan(sqrt(c*x^2 + b
*x)*sqrt(-c)/(c*x)) + (4*B*c*d^3 + A*b*d*e^2 - (3*B*b + 2*A*c)*d^2*e + (4*B*c*d^
2*e + A*b*e^3 - (3*B*b + 2*A*c)*d*e^2)*x)*sqrt(-c)*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)))/((e^4*x +
d*e^3)*sqrt(c*d^2 - b*d*e)*sqrt(-c)), ((B*e^2*x + 2*B*d*e - A*e^2)*sqrt(-c*d^2
+ b*d*e)*sqrt(c*x^2 + b*x)*sqrt(-c) - (4*B*c*d^3 + A*b*d*e^2 - (3*B*b + 2*A*c)*d
^2*e + (4*B*c*d^2*e + A*b*e^3 - (3*B*b + 2*A*c)*d*e^2)*x)*sqrt(-c)*arctan(-sqrt(
-c*d^2 + b*d*e)*sqrt(c*x^2 + b*x)/((c*d - b*e)*x)) - (4*B*c*d^2 - (B*b + 2*A*c)*
d*e + (4*B*c*d*e - (B*b + 2*A*c)*e^2)*x)*sqrt(-c*d^2 + b*d*e)*arctan(sqrt(c*x^2
+ b*x)*sqrt(-c)/(c*x)))/((e^4*x + d*e^3)*sqrt(-c*d^2 + b*d*e)*sqrt(-c))]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{x \left (b + c x\right )} \left (A + B x\right )}{\left (d + e x\right )^{2}}\, dx \]
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)**2,x)
[Out]
Integral(sqrt(x*(b + c*x))*(A + B*x)/(d + e*x)**2, x)
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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(sqrt(c*x^2 + b*x)*(B*x + A)/(e*x + d)^2,x, algorithm="giac")
[Out]
Timed out