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

Optimal. Leaf size=81 \[ \frac{2 A \sqrt{b x+c x^2}}{\sqrt{x}}-2 A \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )+\frac{2 B \left (b x+c x^2\right )^{3/2}}{3 c x^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.164302, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{2 A \sqrt{b x+c x^2}}{\sqrt{x}}-2 A \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )+\frac{2 B \left (b x+c x^2\right )^{3/2}}{3 c x^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 12.0584, size = 75, normalized size = 0.93 \[ - 2 A \sqrt{b} \operatorname{atanh}{\left (\frac{\sqrt{b x + c x^{2}}}{\sqrt{b} \sqrt{x}} \right )} + \frac{2 A \sqrt{b x + c x^{2}}}{\sqrt{x}} + \frac{2 B \left (b x + c x^{2}\right )^{\frac{3}{2}}}{3 c x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.115209, size = 80, normalized size = 0.99 \[ \frac{2 \sqrt{x} \sqrt{b+c x} \left (\sqrt{b+c x} (3 A c+b B+B c x)-3 A \sqrt{b} c \tanh ^{-1}\left (\frac{\sqrt{b+c x}}{\sqrt{b}}\right )\right )}{3 c \sqrt{x (b+c x)}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.018, size = 79, normalized size = 1. \[ -{\frac{2}{3\,c}\sqrt{x \left ( cx+b \right ) } \left ( 3\,A\sqrt{b}c{\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \right ) -Bxc\sqrt{cx+b}-3\,A\sqrt{cx+b}c-Bb\sqrt{cx+b} \right ){\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{cx+b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3*(x*(c*x+b))^(1/2)*(3*A*b^(1/2)*c*arctanh((c*x+b)^(1/2)/b^(1/2))-B*x*c*(c*x+
b)^(1/2)-3*A*(c*x+b)^(1/2)*c-B*b*(c*x+b)^(1/2))/x^(1/2)/(c*x+b)^(1/2)/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(sqrt(c*x^2 + b*x)*(B*x + A)/x^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.291958, size = 1, normalized size = 0.01 \[ \left [\frac{2 \, B c^{2} x^{3} + 3 \, \sqrt{c x^{2} + b x} A \sqrt{b} c \sqrt{x} \log \left (-\frac{c x^{2} + 2 \, b x - 2 \, \sqrt{c x^{2} + b x} \sqrt{b} \sqrt{x}}{x^{2}}\right ) + 2 \,{\left (2 \, B b c + 3 \, A c^{2}\right )} x^{2} + 2 \,{\left (B b^{2} + 3 \, A b c\right )} x}{3 \, \sqrt{c x^{2} + b x} c \sqrt{x}}, \frac{2 \,{\left (B c^{2} x^{3} - 3 \, \sqrt{c x^{2} + b x} A \sqrt{-b} c \sqrt{x} \arctan \left (\frac{b \sqrt{x}}{\sqrt{c x^{2} + b x} \sqrt{-b}}\right ) +{\left (2 \, B b c + 3 \, A c^{2}\right )} x^{2} +{\left (B b^{2} + 3 \, A b c\right )} x\right )}}{3 \, \sqrt{c x^{2} + b x} c \sqrt{x}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.275502, size = 139, normalized size = 1.72 \[ \frac{2 \, A b \arctan \left (\frac{\sqrt{c x + b}}{\sqrt{-b}}\right )}{\sqrt{-b}} - \frac{2 \,{\left (3 \, A b c \arctan \left (\frac{\sqrt{b}}{\sqrt{-b}}\right ) + B \sqrt{-b} b^{\frac{3}{2}} + 3 \, A \sqrt{-b} \sqrt{b} c\right )}}{3 \, \sqrt{-b} c} + \frac{2 \,{\left ({\left (c x + b\right )}^{\frac{3}{2}} B c^{2} + 3 \, \sqrt{c x + b} A c^{3}\right )}}{3 \, c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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