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

Optimal. Leaf size=70 \[ \frac{2 \sqrt{b x+c x^2} (2 b B-A c)}{b c^2 \sqrt{x}}-\frac{2 x^{3/2} (b B-A c)}{b c \sqrt{b x+c x^2}} \]

[Out]

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

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Rubi [A]  time = 0.138337, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{2 \sqrt{b x+c x^2} (2 b B-A c)}{b c^2 \sqrt{x}}-\frac{2 x^{3/2} (b B-A c)}{b c \sqrt{b x+c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 9.83566, size = 60, normalized size = 0.86 \[ \frac{2 x^{\frac{3}{2}} \left (A c - B b\right )}{b c \sqrt{b x + c x^{2}}} - \frac{4 \left (\frac{A c}{2} - B b\right ) \sqrt{b x + c x^{2}}}{b c^{2} \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0362186, size = 34, normalized size = 0.49 \[ \frac{2 \sqrt{x} (-A c+2 b B+B c x)}{c^2 \sqrt{x (b+c x)}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.005, size = 38, normalized size = 0.5 \[ -2\,{\frac{ \left ( cx+b \right ) \left ( -Bcx+Ac-2\,Bb \right ){x}^{3/2}}{{c}^{2} \left ( c{x}^{2}+bx \right ) ^{3/2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.760242, size = 69, normalized size = 0.99 \[ 2 \, B{\left (\frac{\sqrt{c x + b} x}{c^{2} x + b c} + \frac{2 \, b}{\sqrt{c x + b} c^{2}}\right )} - \frac{2 \, A}{\sqrt{c x + b} c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.286583, size = 50, normalized size = 0.71 \[ \frac{2 \,{\left (B c x^{2} +{\left (2 \, B b - A c\right )} x\right )}}{\sqrt{c x^{2} + b x} c^{2} \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.273702, size = 66, normalized size = 0.94 \[ \frac{2 \,{\left (\sqrt{c x + b} B + \frac{B b - A c}{\sqrt{c x + b}}\right )}}{c^{2}} - \frac{2 \,{\left (2 \, B b - A c\right )}}{\sqrt{b} c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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