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

Optimal. Leaf size=90 \[ \frac{4 c \sqrt{b x+c x^2} (5 b B-4 A c)}{15 b^3 x}-\frac{2 \sqrt{b x+c x^2} (5 b B-4 A c)}{15 b^2 x^2}-\frac{2 A \sqrt{b x+c x^2}}{5 b x^3} \]

[Out]

(-2*A*Sqrt[b*x + c*x^2])/(5*b*x^3) - (2*(5*b*B - 4*A*c)*Sqrt[b*x + c*x^2])/(15*b
^2*x^2) + (4*c*(5*b*B - 4*A*c)*Sqrt[b*x + c*x^2])/(15*b^3*x)

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Rubi [A]  time = 0.202773, antiderivative size = 90, 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 \[ \frac{4 c \sqrt{b x+c x^2} (5 b B-4 A c)}{15 b^3 x}-\frac{2 \sqrt{b x+c x^2} (5 b B-4 A c)}{15 b^2 x^2}-\frac{2 A \sqrt{b x+c x^2}}{5 b x^3} \]

Antiderivative was successfully verified.

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

[Out]

(-2*A*Sqrt[b*x + c*x^2])/(5*b*x^3) - (2*(5*b*B - 4*A*c)*Sqrt[b*x + c*x^2])/(15*b
^2*x^2) + (4*c*(5*b*B - 4*A*c)*Sqrt[b*x + c*x^2])/(15*b^3*x)

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Rubi in Sympy [A]  time = 12.5121, size = 85, normalized size = 0.94 \[ - \frac{2 A \sqrt{b x + c x^{2}}}{5 b x^{3}} + \frac{2 \left (4 A c - 5 B b\right ) \sqrt{b x + c x^{2}}}{15 b^{2} x^{2}} - \frac{4 c \left (4 A c - 5 B b\right ) \sqrt{b x + c x^{2}}}{15 b^{3} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*A*sqrt(b*x + c*x**2)/(5*b*x**3) + 2*(4*A*c - 5*B*b)*sqrt(b*x + c*x**2)/(15*b*
*2*x**2) - 4*c*(4*A*c - 5*B*b)*sqrt(b*x + c*x**2)/(15*b**3*x)

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Mathematica [A]  time = 0.0811509, size = 54, normalized size = 0.6 \[ -\frac{2 \sqrt{x (b+c x)} \left (A \left (3 b^2-4 b c x+8 c^2 x^2\right )+5 b B x (b-2 c x)\right )}{15 b^3 x^3} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[x*(b + c*x)]*(5*b*B*x*(b - 2*c*x) + A*(3*b^2 - 4*b*c*x + 8*c^2*x^2)))/(
15*b^3*x^3)

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Maple [A]  time = 0.008, size = 62, normalized size = 0.7 \[ -{\frac{ \left ( 2\,cx+2\,b \right ) \left ( 8\,A{c}^{2}{x}^{2}-10\,B{x}^{2}bc-4\,Abcx+5\,{b}^{2}Bx+3\,{b}^{2}A \right ) }{15\,{x}^{2}{b}^{3}}{\frac{1}{\sqrt{c{x}^{2}+bx}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/15*(c*x+b)*(8*A*c^2*x^2-10*B*b*c*x^2-4*A*b*c*x+5*B*b^2*x+3*A*b^2)/x^2/b^3/(c*
x^2+b*x)^(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),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.272067, size = 77, normalized size = 0.86 \[ -\frac{2 \,{\left (3 \, A b^{2} - 2 \,{\left (5 \, B b c - 4 \, A c^{2}\right )} x^{2} +{\left (5 \, B b^{2} - 4 \, A b c\right )} x\right )} \sqrt{c x^{2} + b x}}{15 \, b^{3} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/15*(3*A*b^2 - 2*(5*B*b*c - 4*A*c^2)*x^2 + (5*B*b^2 - 4*A*b*c)*x)*sqrt(c*x^2 +
b*x)/(b^3*x^3)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.280254, size = 180, normalized size = 2. \[ \frac{2 \,{\left (15 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{3} B \sqrt{c} + 5 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{2} B b + 20 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{2} A c + 15 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} A b \sqrt{c} + 3 \, A b^{2}\right )}}{15 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15*(15*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*B*sqrt(c) + 5*(sqrt(c)*x - sqrt(c*x^2
+ b*x))^2*B*b + 20*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*A*c + 15*(sqrt(c)*x - sqrt
(c*x^2 + b*x))*A*b*sqrt(c) + 3*A*b^2)/(sqrt(c)*x - sqrt(c*x^2 + b*x))^5