3.74 \(\int \frac{(A+B x) \sqrt{b x+c x^2}}{x^4} \, dx\)

Optimal. Leaf size=57 \[ -\frac{2 \left (b x+c x^2\right )^{3/2} (5 b B-2 A c)}{15 b^2 x^3}-\frac{2 A \left (b x+c x^2\right )^{3/2}}{5 b x^4} \]

[Out]

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

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Rubi [A]  time = 0.13228, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ -\frac{2 \left (b x+c x^2\right )^{3/2} (5 b B-2 A c)}{15 b^2 x^3}-\frac{2 A \left (b x+c x^2\right )^{3/2}}{5 b x^4} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*Sqrt[b*x + c*x^2])/x^4,x]

[Out]

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

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Rubi in Sympy [A]  time = 8.28516, size = 53, normalized size = 0.93 \[ - \frac{2 A \left (b x + c x^{2}\right )^{\frac{3}{2}}}{5 b x^{4}} + \frac{4 \left (A c - \frac{5 B b}{2}\right ) \left (b x + c x^{2}\right )^{\frac{3}{2}}}{15 b^{2} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(c*x**2+b*x)**(1/2)/x**4,x)

[Out]

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

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Mathematica [A]  time = 0.056062, size = 36, normalized size = 0.63 \[ -\frac{2 (x (b+c x))^{3/2} (3 A b-2 A c x+5 b B x)}{15 b^2 x^4} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*Sqrt[b*x + c*x^2])/x^4,x]

[Out]

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

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Maple [A]  time = 0.008, size = 40, normalized size = 0.7 \[ -{\frac{ \left ( 2\,cx+2\,b \right ) \left ( -2\,Acx+5\,xBb+3\,Ab \right ) }{15\,{b}^{2}{x}^{3}}\sqrt{c{x}^{2}+bx}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+b*x)^(1/2)/x^4,x)

[Out]

-2/15*(c*x+b)*(-2*A*c*x+5*B*b*x+3*A*b)*(c*x^2+b*x)^(1/2)/b^2/x^3

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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^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.267802, size = 74, normalized size = 1.3 \[ -\frac{2 \,{\left (3 \, A b^{2} +{\left (5 \, B b c - 2 \, A c^{2}\right )} x^{2} +{\left (5 \, B b^{2} + A b c\right )} x\right )} \sqrt{c x^{2} + b x}}{15 \, b^{2} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + b*x)*(B*x + A)/x^4,x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{x \left (b + c x\right )} \left (A + B x\right )}{x^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(c*x**2+b*x)**(1/2)/x**4,x)

[Out]

Integral(sqrt(x*(b + c*x))*(A + B*x)/x**4, x)

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GIAC/XCAS [A]  time = 0.275469, size = 258, normalized size = 4.53 \[ \frac{2 \,{\left (15 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{4} B c + 15 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{3} B b \sqrt{c} + 15 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{3} A c^{\frac{3}{2}} + 5 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{2} B b^{2} + 25 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{2} A b c + 15 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} A b^{2} \sqrt{c} + 3 \, A b^{3}\right )}}{15 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + b*x)*(B*x + A)/x^4,x, algorithm="giac")

[Out]

2/15*(15*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*B*c + 15*(sqrt(c)*x - sqrt(c*x^2 + b*
x))^3*B*b*sqrt(c) + 15*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*A*c^(3/2) + 5*(sqrt(c)*
x - sqrt(c*x^2 + b*x))^2*B*b^2 + 25*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*A*b*c + 15
*(sqrt(c)*x - sqrt(c*x^2 + b*x))*A*b^2*sqrt(c) + 3*A*b^3)/(sqrt(c)*x - sqrt(c*x^
2 + b*x))^5