3.1196 \(\int \frac{A+B x}{(d+e x)^2 \sqrt{b x+c x^2}} \, dx\)

Optimal. Leaf size=128 \[ \frac{\sqrt{b x+c x^2} (B d-A e)}{d (d+e x) (c d-b e)}-\frac{(A b e-2 A c d+b B d) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{2 d^{3/2} (c d-b e)^{3/2}} \]

[Out]

((B*d - A*e)*Sqrt[b*x + c*x^2])/(d*(c*d - b*e)*(d + e*x)) - ((b*B*d - 2*A*c*d +
A*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*d^(3/2)*(c*d - b*e)^(3/2))

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Rubi [A]  time = 0.241654, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ \frac{\sqrt{b x+c x^2} (B d-A e)}{d (d+e x) (c d-b e)}-\frac{(A b e-2 A c d+b B d) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{2 d^{3/2} (c d-b e)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

((B*d - A*e)*Sqrt[b*x + c*x^2])/(d*(c*d - b*e)*(d + e*x)) - ((b*B*d - 2*A*c*d +
A*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*d^(3/2)*(c*d - b*e)^(3/2))

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Rubi in Sympy [A]  time = 27.1757, size = 105, normalized size = 0.82 \[ \frac{\left (A e - B d\right ) \sqrt{b x + c x^{2}}}{d \left (d + e x\right ) \left (b e - c d\right )} + \frac{\left (- A c d + \frac{b \left (A e + B d\right )}{2}\right ) \operatorname{atan}{\left (\frac{- b d + x \left (b e - 2 c d\right )}{2 \sqrt{d} \sqrt{b e - c d} \sqrt{b x + c x^{2}}} \right )}}{d^{\frac{3}{2}} \left (b e - c d\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(A*e - B*d)*sqrt(b*x + c*x**2)/(d*(d + e*x)*(b*e - c*d)) + (-A*c*d + b*(A*e + B*
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
)))/(d**(3/2)*(b*e - c*d)**(3/2))

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Mathematica [A]  time = 0.254098, size = 134, normalized size = 1.05 \[ \frac{\sqrt{x} \left (\frac{\sqrt{d} \sqrt{x} (b+c x) (B d-A e)}{d+e x}-\frac{\sqrt{b+c x} (A b e-2 A c d+b B d) \tan ^{-1}\left (\frac{\sqrt{x} \sqrt{b e-c d}}{\sqrt{d} \sqrt{b+c x}}\right )}{\sqrt{b e-c d}}\right )}{d^{3/2} \sqrt{x (b+c x)} (c d-b e)} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[x]*((Sqrt[d]*(B*d - A*e)*Sqrt[x]*(b + c*x))/(d + e*x) - ((b*B*d - 2*A*c*d
+ A*b*e)*Sqrt[b + c*x]*ArcTan[(Sqrt[-(c*d) + b*e]*Sqrt[x])/(Sqrt[d]*Sqrt[b + c*x
])])/Sqrt[-(c*d) + b*e]))/(d^(3/2)*(c*d - b*e)*Sqrt[x*(b + c*x)])

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Maple [B]  time = 0.015, size = 849, normalized size = 6.6 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-B/e^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/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)^(1/2)*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)^(1/2)*B-1/2/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*A+1/2/e/(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*B+1/e/(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*A-1/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))*c*B*d

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.319963, size = 1, normalized size = 0.01 \[ \left [\frac{2 \, \sqrt{c d^{2} - b d e} \sqrt{c x^{2} + b x}{\left (B d - A e\right )} -{\left (A b d e +{\left (B b - 2 \, A c\right )} d^{2} +{\left (A b e^{2} +{\left (B b - 2 \, A c\right )} d e\right )} x\right )} \log \left (\frac{2 \,{\left (c d^{2} - b d e\right )} \sqrt{c x^{2} + b x} + \sqrt{c d^{2} - b d e}{\left (b d +{\left (2 \, c d - b e\right )} x\right )}}{e x + d}\right )}{2 \,{\left (c d^{3} - b d^{2} e +{\left (c d^{2} e - b d e^{2}\right )} x\right )} \sqrt{c d^{2} - b d e}}, \frac{\sqrt{-c d^{2} + b d e} \sqrt{c x^{2} + b x}{\left (B d - A e\right )} +{\left (A b d e +{\left (B b - 2 \, A c\right )} d^{2} +{\left (A b e^{2} +{\left (B b - 2 \, A c\right )} d e\right )} x\right )} \arctan \left (-\frac{\sqrt{-c d^{2} + b d e} \sqrt{c x^{2} + b x}}{{\left (c d - b e\right )} x}\right )}{{\left (c d^{3} - b d^{2} e +{\left (c d^{2} e - b d e^{2}\right )} x\right )} \sqrt{-c d^{2} + b d e}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/2*(2*sqrt(c*d^2 - b*d*e)*sqrt(c*x^2 + b*x)*(B*d - A*e) - (A*b*d*e + (B*b - 2*
A*c)*d^2 + (A*b*e^2 + (B*b - 2*A*c)*d*e)*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*d^3 - b*d^2*
e + (c*d^2*e - b*d*e^2)*x)*sqrt(c*d^2 - b*d*e)), (sqrt(-c*d^2 + b*d*e)*sqrt(c*x^
2 + b*x)*(B*d - A*e) + (A*b*d*e + (B*b - 2*A*c)*d^2 + (A*b*e^2 + (B*b - 2*A*c)*d
*e)*x)*arctan(-sqrt(-c*d^2 + b*d*e)*sqrt(c*x^2 + b*x)/((c*d - b*e)*x)))/((c*d^3
- b*d^2*e + (c*d^2*e - b*d*e^2)*x)*sqrt(-c*d^2 + b*d*e))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{A + B x}{\sqrt{x \left (b + c x\right )} \left (d + e x\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((A + B*x)/(sqrt(x*(b + c*x))*(d + e*x)**2), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{B x + A}{\sqrt{c x^{2} + b x}{\left (e x + d\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((B*x + A)/(sqrt(c*x^2 + b*x)*(e*x + d)^2), x)