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

Optimal. Leaf size=96 \[ \frac{4 b \sqrt{b x+c x^2} (4 b B-5 A c)}{15 c^3 \sqrt{x}}-\frac{2 \sqrt{x} \sqrt{b x+c x^2} (4 b B-5 A c)}{15 c^2}+\frac{2 B x^{3/2} \sqrt{b x+c x^2}}{5 c} \]

[Out]

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

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Rubi [A]  time = 0.201713, antiderivative size = 96, 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 \[ \frac{4 b \sqrt{b x+c x^2} (4 b B-5 A c)}{15 c^3 \sqrt{x}}-\frac{2 \sqrt{x} \sqrt{b x+c x^2} (4 b B-5 A c)}{15 c^2}+\frac{2 B x^{3/2} \sqrt{b x+c x^2}}{5 c} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 11.8392, size = 92, normalized size = 0.96 \[ \frac{2 B x^{\frac{3}{2}} \sqrt{b x + c x^{2}}}{5 c} - \frac{4 b \left (5 A c - 4 B b\right ) \sqrt{b x + c x^{2}}}{15 c^{3} \sqrt{x}} + \frac{2 \sqrt{x} \left (5 A c - 4 B b\right ) \sqrt{b x + c x^{2}}}{15 c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0668697, size = 55, normalized size = 0.57 \[ \frac{2 \sqrt{x (b+c x)} \left (-2 b c (5 A+2 B x)+c^2 x (5 A+3 B x)+8 b^2 B\right )}{15 c^3 \sqrt{x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.006, size = 59, normalized size = 0.6 \[ -{\frac{ \left ( 2\,cx+2\,b \right ) \left ( -3\,B{c}^{2}{x}^{2}-5\,Ax{c}^{2}+4\,Bxbc+10\,Abc-8\,{b}^{2}B \right ) }{15\,{c}^{3}}\sqrt{x}{\frac{1}{\sqrt{c{x}^{2}+bx}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/15*(c*x+b)*(-3*B*c^2*x^2-5*A*c^2*x+4*B*b*c*x+10*A*b*c-8*B*b^2)*x^(1/2)/c^3/(c
*x^2+b*x)^(1/2)

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Maxima [A]  time = 0.702023, size = 101, normalized size = 1.05 \[ \frac{2 \,{\left (c^{2} x^{2} - b c x - 2 \, b^{2}\right )} A}{3 \, \sqrt{c x + b} c^{2}} + \frac{2 \,{\left (3 \, c^{3} x^{3} - b c^{2} x^{2} + 4 \, b^{2} c x + 8 \, b^{3}\right )} B}{15 \, \sqrt{c x + b} c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*(c^2*x^2 - b*c*x - 2*b^2)*A/(sqrt(c*x + b)*c^2) + 2/15*(3*c^3*x^3 - b*c^2*x^
2 + 4*b^2*c*x + 8*b^3)*B/(sqrt(c*x + b)*c^3)

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Fricas [A]  time = 0.292595, size = 112, normalized size = 1.17 \[ \frac{2 \,{\left (3 \, B c^{3} x^{4} -{\left (B b c^{2} - 5 \, A c^{3}\right )} x^{3} +{\left (4 \, B b^{2} c - 5 \, A b c^{2}\right )} x^{2} + 2 \,{\left (4 \, B b^{3} - 5 \, A b^{2} c\right )} x\right )}}{15 \, \sqrt{c x^{2} + b x} c^{3} \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.272768, size = 112, normalized size = 1.17 \[ \frac{2 \,{\left (3 \,{\left (c x + b\right )}^{\frac{5}{2}} B - 10 \,{\left (c x + b\right )}^{\frac{3}{2}} B b + 15 \, \sqrt{c x + b} B b^{2} + 5 \,{\left (c x + b\right )}^{\frac{3}{2}} A c - 15 \, \sqrt{c x + b} A b c\right )}}{15 \, c^{3}} - \frac{4 \,{\left (4 \, B b^{\frac{5}{2}} - 5 \, A b^{\frac{3}{2}} c\right )}}{15 \, c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15*(3*(c*x + b)^(5/2)*B - 10*(c*x + b)^(3/2)*B*b + 15*sqrt(c*x + b)*B*b^2 + 5*
(c*x + b)^(3/2)*A*c - 15*sqrt(c*x + b)*A*b*c)/c^3 - 4/15*(4*B*b^(5/2) - 5*A*b^(3
/2)*c)/c^3