3.1729 \(\int \frac{A+B x}{(a+b x) \sqrt{d+e x}} \, dx\)

Optimal. Leaf size=74 \[ \frac{2 B \sqrt{d+e x}}{b e}-\frac{2 (A b-a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{3/2} \sqrt{b d-a e}} \]

[Out]

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

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Rubi [A]  time = 0.114113, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{2 B \sqrt{d+e x}}{b e}-\frac{2 (A b-a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{3/2} \sqrt{b d-a e}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 12.8449, size = 63, normalized size = 0.85 \[ \frac{2 B \sqrt{d + e x}}{b e} + \frac{2 \left (A b - B a\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{a e - b d}} \right )}}{b^{\frac{3}{2}} \sqrt{a e - b d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.116443, size = 74, normalized size = 1. \[ \frac{2 B \sqrt{d+e x}}{b e}-\frac{2 (A b-a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{3/2} \sqrt{b d-a e}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.013, size = 96, normalized size = 1.3 \[ 2\,{\frac{B\sqrt{ex+d}}{be}}+2\,{\frac{A}{\sqrt{ \left ( ae-bd \right ) b}}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) }-2\,{\frac{Ba}{b\sqrt{ \left ( ae-bd \right ) b}}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*B*(e*x+d)^(1/2)/b/e+2/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)
^(1/2))*A-2/b/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*B*
a

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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)/((b*x + a)*sqrt(e*x + d)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.221832, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (B a - A b\right )} e \log \left (\frac{\sqrt{b^{2} d - a b e}{\left (b e x + 2 \, b d - a e\right )} - 2 \,{\left (b^{2} d - a b e\right )} \sqrt{e x + d}}{b x + a}\right ) - 2 \, \sqrt{b^{2} d - a b e} \sqrt{e x + d} B}{\sqrt{b^{2} d - a b e} b e}, \frac{2 \,{\left ({\left (B a - A b\right )} e \arctan \left (-\frac{b d - a e}{\sqrt{-b^{2} d + a b e} \sqrt{e x + d}}\right ) + \sqrt{-b^{2} d + a b e} \sqrt{e x + d} B\right )}}{\sqrt{-b^{2} d + a b e} b e}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 13.1921, size = 211, normalized size = 2.85 \[ \frac{2 B \sqrt{d + e x}}{b e} + \frac{2 \left (- A b + B a\right ) \left (\begin{cases} \frac{\operatorname{atan}{\left (\frac{1}{\sqrt{\frac{b}{a e - b d}} \sqrt{d + e x}} \right )}}{\sqrt{\frac{b}{a e - b d}} \left (a e - b d\right )} & \text{for}\: \frac{b}{a e - b d} > 0 \\- \frac{\operatorname{acoth}{\left (\frac{1}{\sqrt{- \frac{b}{a e - b d}} \sqrt{d + e x}} \right )}}{\sqrt{- \frac{b}{a e - b d}} \left (a e - b d\right )} & \text{for}\: \frac{1}{d + e x} > - \frac{b}{a e - b d} \wedge \frac{b}{a e - b d} < 0 \\- \frac{\operatorname{atanh}{\left (\frac{1}{\sqrt{- \frac{b}{a e - b d}} \sqrt{d + e x}} \right )}}{\sqrt{- \frac{b}{a e - b d}} \left (a e - b d\right )} & \text{for}\: \frac{b}{a e - b d} < 0 \wedge \frac{1}{d + e x} < - \frac{b}{a e - b d} \end{cases}\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*B*sqrt(d + e*x)/(b*e) + 2*(-A*b + B*a)*Piecewise((atan(1/(sqrt(b/(a*e - b*d))*
sqrt(d + e*x)))/(sqrt(b/(a*e - b*d))*(a*e - b*d)), b/(a*e - b*d) > 0), (-acoth(1
/(sqrt(-b/(a*e - b*d))*sqrt(d + e*x)))/(sqrt(-b/(a*e - b*d))*(a*e - b*d)), (b/(a
*e - b*d) < 0) & (1/(d + e*x) > -b/(a*e - b*d))), (-atanh(1/(sqrt(-b/(a*e - b*d)
)*sqrt(d + e*x)))/(sqrt(-b/(a*e - b*d))*(a*e - b*d)), (b/(a*e - b*d) < 0) & (1/(
d + e*x) < -b/(a*e - b*d))))/b

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GIAC/XCAS [A]  time = 0.220934, size = 93, normalized size = 1.26 \[ \frac{2 \, \sqrt{x e + d} B e^{\left (-1\right )}}{b} - \frac{2 \,{\left (B a - A b\right )} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{\sqrt{-b^{2} d + a b e} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(x*e + d)*B*e^(-1)/b - 2*(B*a - A*b)*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d +
a*b*e))/(sqrt(-b^2*d + a*b*e)*b)