3.237 \(\int \frac{\sqrt{x} (A+B x)}{\left (b x+c x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=68 \[ -\frac{2 A \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{b^{3/2}}-\frac{2 \sqrt{x} (b B-A c)}{b c \sqrt{b x+c x^2}} \]

[Out]

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

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Rubi [A]  time = 0.137286, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{2 A \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{b^{3/2}}-\frac{2 \sqrt{x} (b B-A c)}{b c \sqrt{b x+c x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(Sqrt[x]*(A + B*x))/(b*x + c*x^2)^(3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 10.2637, size = 60, normalized size = 0.88 \[ - \frac{2 A \operatorname{atanh}{\left (\frac{\sqrt{b x + c x^{2}}}{\sqrt{b} \sqrt{x}} \right )}}{b^{\frac{3}{2}}} + \frac{2 \sqrt{x} \left (A c - B b\right )}{b c \sqrt{b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*x**(1/2)/(c*x**2+b*x)**(3/2),x)

[Out]

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

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Mathematica [A]  time = 0.0846726, size = 69, normalized size = 1.01 \[ -\frac{2 \sqrt{x} \left (\sqrt{b} (b B-A c)+A c \sqrt{b+c x} \tanh ^{-1}\left (\frac{\sqrt{b+c x}}{\sqrt{b}}\right )\right )}{b^{3/2} c \sqrt{x (b+c x)}} \]

Antiderivative was successfully verified.

[In]  Integrate[(Sqrt[x]*(A + B*x))/(b*x + c*x^2)^(3/2),x]

[Out]

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

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Maple [A]  time = 0.02, size = 63, normalized size = 0.9 \[ -2\,{\frac{\sqrt{x \left ( cx+b \right ) }}{{b}^{3/2}\sqrt{x} \left ( cx+b \right ) c} \left ( A{\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \right ) c\sqrt{cx+b}-Ac\sqrt{b}+B{b}^{3/2} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*x^(1/2)/(c*x^2+b*x)^(3/2),x)

[Out]

-2*(x*(c*x+b))^(1/2)/b^(3/2)*(A*arctanh((c*x+b)^(1/2)/b^(1/2))*c*(c*x+b)^(1/2)-A
*c*b^(1/2)+B*b^(3/2))/x^(1/2)/(c*x+b)/c

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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(x)/(c*x^2 + b*x)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.301568, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \, \sqrt{c x^{2} + b x}{\left (B b - A c\right )} \sqrt{b} \sqrt{x} -{\left (A c^{2} x^{2} + A b c x\right )} \log \left (\frac{2 \, \sqrt{c x^{2} + b x} b \sqrt{x} -{\left (c x^{2} + 2 \, b x\right )} \sqrt{b}}{x^{2}}\right )}{{\left (b c^{2} x^{2} + b^{2} c x\right )} \sqrt{b}}, -\frac{2 \,{\left (\sqrt{c x^{2} + b x}{\left (B b - A c\right )} \sqrt{-b} \sqrt{x} +{\left (A c^{2} x^{2} + A b c x\right )} \arctan \left (\frac{\sqrt{-b} \sqrt{x}}{\sqrt{c x^{2} + b x}}\right )\right )}}{{\left (b c^{2} x^{2} + b^{2} c x\right )} \sqrt{-b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*sqrt(x)/(c*x^2 + b*x)^(3/2),x, algorithm="fricas")

[Out]

[-(2*sqrt(c*x^2 + b*x)*(B*b - A*c)*sqrt(b)*sqrt(x) - (A*c^2*x^2 + A*b*c*x)*log((
2*sqrt(c*x^2 + b*x)*b*sqrt(x) - (c*x^2 + 2*b*x)*sqrt(b))/x^2))/((b*c^2*x^2 + b^2
*c*x)*sqrt(b)), -2*(sqrt(c*x^2 + b*x)*(B*b - A*c)*sqrt(-b)*sqrt(x) + (A*c^2*x^2
+ A*b*c*x)*arctan(sqrt(-b)*sqrt(x)/sqrt(c*x^2 + b*x)))/((b*c^2*x^2 + b^2*c*x)*sq
rt(-b))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{x} \left (A + B x\right )}{\left (x \left (b + c x\right )\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*x**(1/2)/(c*x**2+b*x)**(3/2),x)

[Out]

Integral(sqrt(x)*(A + B*x)/(x*(b + c*x))**(3/2), x)

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GIAC/XCAS [A]  time = 0.27696, size = 130, normalized size = 1.91 \[ \frac{2 \, A \arctan \left (\frac{\sqrt{c x + b}}{\sqrt{-b}}\right )}{\sqrt{-b} b} - \frac{2 \,{\left (B b - A c\right )}}{\sqrt{c x + b} b c} - \frac{2 \,{\left (A \sqrt{b} c \arctan \left (\frac{\sqrt{b}}{\sqrt{-b}}\right ) - B \sqrt{-b} b + A \sqrt{-b} c\right )}}{\sqrt{-b} b^{\frac{3}{2}} c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*sqrt(x)/(c*x^2 + b*x)^(3/2),x, algorithm="giac")

[Out]

2*A*arctan(sqrt(c*x + b)/sqrt(-b))/(sqrt(-b)*b) - 2*(B*b - A*c)/(sqrt(c*x + b)*b
*c) - 2*(A*sqrt(b)*c*arctan(sqrt(b)/sqrt(-b)) - B*sqrt(-b)*b + A*sqrt(-b)*c)/(sq
rt(-b)*b^(3/2)*c)