3.1195 \(\int \frac{A+B x}{(d+e x) \sqrt{b x+c x^2}} \, dx\)

Optimal. Leaf size=113 \[ \frac{2 B \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{\sqrt{c} e}-\frac{(B d-A e) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{\sqrt{d} e \sqrt{c d-b e}} \]

[Out]

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

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Rubi [A]  time = 0.233531, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{2 B \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{\sqrt{c} e}-\frac{(B d-A e) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{\sqrt{d} e \sqrt{c d-b e}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 27.6356, size = 97, normalized size = 0.86 \[ \frac{2 B \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )}}{\sqrt{c} e} + \frac{\left (A e - B d\right ) \operatorname{atan}{\left (\frac{- b d + x \left (b e - 2 c d\right )}{2 \sqrt{d} \sqrt{b e - c d} \sqrt{b x + c x^{2}}} \right )}}{\sqrt{d} e \sqrt{b e - c d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.164296, size = 144, normalized size = 1.27 \[ \frac{2 \sqrt{x} \sqrt{b+c x} \left (B \sqrt{d} \sqrt{b e-c d} \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right )-\sqrt{c} (B d-A e) \tan ^{-1}\left (\frac{\sqrt{x} \sqrt{b e-c d}}{\sqrt{d} \sqrt{b+c x}}\right )\right )}{\sqrt{c} \sqrt{d} e \sqrt{x (b+c x)} \sqrt{b e-c d}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[x]*Sqrt[b + c*x]*(-(Sqrt[c]*(B*d - A*e)*ArcTan[(Sqrt[-(c*d) + b*e]*Sqrt[
x])/(Sqrt[d]*Sqrt[b + c*x])]) + B*Sqrt[d]*Sqrt[-(c*d) + b*e]*Log[c*Sqrt[x] + Sqr
t[c]*Sqrt[b + c*x]]))/(Sqrt[c]*Sqrt[d]*e*Sqrt[-(c*d) + b*e]*Sqrt[x*(b + c*x)])

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Maple [B]  time = 0.01, size = 298, normalized size = 2.6 \[{\frac{B}{e}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){\frac{1}{\sqrt{c}}}}-{\frac{A}{e}\ln \left ({1 \left ( -2\,{\frac{d \left ( be-cd \right ) }{{e}^{2}}}+{\frac{be-2\,cd}{e} \left ({\frac{d}{e}}+x \right ) }+2\,\sqrt{-{\frac{d \left ( be-cd \right ) }{{e}^{2}}}}\sqrt{c \left ({\frac{d}{e}}+x \right ) ^{2}+{\frac{be-2\,cd}{e} \left ({\frac{d}{e}}+x \right ) }-{\frac{d \left ( be-cd \right ) }{{e}^{2}}}} \right ) \left ({\frac{d}{e}}+x \right ) ^{-1}} \right ){\frac{1}{\sqrt{-{\frac{d \left ( be-cd \right ) }{{e}^{2}}}}}}}+{\frac{Bd}{{e}^{2}}\ln \left ({1 \left ( -2\,{\frac{d \left ( be-cd \right ) }{{e}^{2}}}+{\frac{be-2\,cd}{e} \left ({\frac{d}{e}}+x \right ) }+2\,\sqrt{-{\frac{d \left ( be-cd \right ) }{{e}^{2}}}}\sqrt{c \left ({\frac{d}{e}}+x \right ) ^{2}+{\frac{be-2\,cd}{e} \left ({\frac{d}{e}}+x \right ) }-{\frac{d \left ( be-cd \right ) }{{e}^{2}}}} \right ) \left ({\frac{d}{e}}+x \right ) ^{-1}} \right ){\frac{1}{\sqrt{-{\frac{d \left ( be-cd \right ) }{{e}^{2}}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B/e*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2))/c^(1/2)-1/e/(-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+1/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))*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)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.358037, size = 1, normalized size = 0.01 \[ \left [\frac{\sqrt{c d^{2} - b d e} B \log \left ({\left (2 \, c x + b\right )} \sqrt{c} + 2 \, \sqrt{c x^{2} + b x} c\right ) -{\left (B d - A e\right )} \sqrt{c} \log \left (\frac{2 \,{\left (c d^{2} - b d e\right )} \sqrt{c x^{2} + b x} + \sqrt{c d^{2} - b d e}{\left (b d +{\left (2 \, c d - b e\right )} x\right )}}{e x + d}\right )}{\sqrt{c d^{2} - b d e} \sqrt{c} e}, \frac{2 \,{\left (B d - A e\right )} \sqrt{c} \arctan \left (-\frac{\sqrt{-c d^{2} + b d e} \sqrt{c x^{2} + b x}}{{\left (c d - b e\right )} x}\right ) + \sqrt{-c d^{2} + b d e} B \log \left ({\left (2 \, c x + b\right )} \sqrt{c} + 2 \, \sqrt{c x^{2} + b x} c\right )}{\sqrt{-c d^{2} + b d e} \sqrt{c} e}, \frac{2 \, \sqrt{c d^{2} - b d e} B \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right ) -{\left (B d - A e\right )} \sqrt{-c} \log \left (\frac{2 \,{\left (c d^{2} - b d e\right )} \sqrt{c x^{2} + b x} + \sqrt{c d^{2} - b d e}{\left (b d +{\left (2 \, c d - b e\right )} x\right )}}{e x + d}\right )}{\sqrt{c d^{2} - b d e} \sqrt{-c} e}, \frac{2 \,{\left ({\left (B d - A e\right )} \sqrt{-c} \arctan \left (-\frac{\sqrt{-c d^{2} + b d e} \sqrt{c x^{2} + b x}}{{\left (c d - b e\right )} x}\right ) + \sqrt{-c d^{2} + b d e} B \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right )\right )}}{\sqrt{-c d^{2} + b d e} \sqrt{-c} e}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError