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

Optimal. Leaf size=66 \[ -\frac{(2 b B-A c) \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{b^{3/2}}-\frac{A \sqrt{b x+c x^2}}{b x^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.142382, antiderivative size = 66, 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 \[ -\frac{(2 b B-A c) \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{b^{3/2}}-\frac{A \sqrt{b x+c x^2}}{b x^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 9.57728, size = 58, normalized size = 0.88 \[ - \frac{A \sqrt{b x + c x^{2}}}{b x^{\frac{3}{2}}} + \frac{2 \left (\frac{A c}{2} - B b\right ) \operatorname{atanh}{\left (\frac{\sqrt{b x + c x^{2}}}{\sqrt{b} \sqrt{x}} \right )}}{b^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0893354, size = 73, normalized size = 1.11 \[ \frac{-x \sqrt{b+c x} (2 b B-A c) \tanh ^{-1}\left (\frac{\sqrt{b+c x}}{\sqrt{b}}\right )-A \sqrt{b} (b+c x)}{b^{3/2} \sqrt{x} \sqrt{x (b+c x)}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.025, size = 71, normalized size = 1.1 \[{1\sqrt{x \left ( cx+b \right ) } \left ( A{\it Artanh} \left ({1\sqrt{cx+b}{\frac{1}{\sqrt{b}}}} \right ) xc-2\,B{\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \right ) xb-A\sqrt{cx+b}\sqrt{b} \right ){b}^{-{\frac{3}{2}}}{x}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{cx+b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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)*x^(3/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.296445, size = 1, normalized size = 0.02 \[ \left [-\frac{{\left (2 \, B b - A c\right )} x^{2} \log \left (-\frac{2 \, \sqrt{c x^{2} + b x} b \sqrt{x} +{\left (c x^{2} + 2 \, b x\right )} \sqrt{b}}{x^{2}}\right ) + 2 \, \sqrt{c x^{2} + b x} A \sqrt{b} \sqrt{x}}{2 \, b^{\frac{3}{2}} x^{2}}, -\frac{{\left (2 \, B b - A c\right )} x^{2} \arctan \left (\frac{\sqrt{-b} \sqrt{x}}{\sqrt{c x^{2} + b x}}\right ) + \sqrt{c x^{2} + b x} A \sqrt{-b} \sqrt{x}}{\sqrt{-b} b x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.295632, size = 78, normalized size = 1.18 \[ -\frac{\frac{\sqrt{c x + b} A c}{b x} - \frac{{\left (2 \, B b c - A c^{2}\right )} \arctan \left (\frac{\sqrt{c x + b}}{\sqrt{-b}}\right )}{\sqrt{-b} b}}{c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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