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

Optimal. Leaf size=123 \[ \frac{3 (b B-5 A c) \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{4 b^{7/2} \sqrt{c}}+\frac{3 (b B-5 A c)}{4 b^3 c \sqrt{x}}-\frac{b B-5 A c}{4 b^2 c \sqrt{x} (b+c x)}-\frac{b B-A c}{2 b c \sqrt{x} (b+c x)^2} \]

[Out]

(3*(b*B - 5*A*c))/(4*b^3*c*Sqrt[x]) - (b*B - A*c)/(2*b*c*Sqrt[x]*(b + c*x)^2) -
(b*B - 5*A*c)/(4*b^2*c*Sqrt[x]*(b + c*x)) + (3*(b*B - 5*A*c)*ArcTan[(Sqrt[c]*Sqr
t[x])/Sqrt[b]])/(4*b^(7/2)*Sqrt[c])

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Rubi [A]  time = 0.146427, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ \frac{3 (b B-5 A c) \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{4 b^{7/2} \sqrt{c}}+\frac{3 (b B-5 A c)}{4 b^3 c \sqrt{x}}-\frac{b B-5 A c}{4 b^2 c \sqrt{x} (b+c x)}-\frac{b B-A c}{2 b c \sqrt{x} (b+c x)^2} \]

Antiderivative was successfully verified.

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

[Out]

(3*(b*B - 5*A*c))/(4*b^3*c*Sqrt[x]) - (b*B - A*c)/(2*b*c*Sqrt[x]*(b + c*x)^2) -
(b*B - 5*A*c)/(4*b^2*c*Sqrt[x]*(b + c*x)) + (3*(b*B - 5*A*c)*ArcTan[(Sqrt[c]*Sqr
t[x])/Sqrt[b]])/(4*b^(7/2)*Sqrt[c])

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Rubi in Sympy [A]  time = 18.5203, size = 109, normalized size = 0.89 \[ \frac{A c - B b}{2 b c \sqrt{x} \left (b + c x\right )^{2}} + \frac{5 A c - B b}{4 b^{2} c \sqrt{x} \left (b + c x\right )} - \frac{3 \left (5 A c - B b\right )}{4 b^{3} c \sqrt{x}} - \frac{3 \left (5 A c - B b\right ) \operatorname{atan}{\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}} \right )}}{4 b^{\frac{7}{2}} \sqrt{c}} \]

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,x)

[Out]

(A*c - B*b)/(2*b*c*sqrt(x)*(b + c*x)**2) + (5*A*c - B*b)/(4*b**2*c*sqrt(x)*(b +
c*x)) - 3*(5*A*c - B*b)/(4*b**3*c*sqrt(x)) - 3*(5*A*c - B*b)*atan(sqrt(c)*sqrt(x
)/sqrt(b))/(4*b**(7/2)*sqrt(c))

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Mathematica [A]  time = 0.141738, size = 94, normalized size = 0.76 \[ \frac{3 (b B-5 A c) \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{4 b^{7/2} \sqrt{c}}+\frac{b B x (5 b+3 c x)-A \left (8 b^2+25 b c x+15 c^2 x^2\right )}{4 b^3 \sqrt{x} (b+c x)^2} \]

Antiderivative was successfully verified.

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

[Out]

(b*B*x*(5*b + 3*c*x) - A*(8*b^2 + 25*b*c*x + 15*c^2*x^2))/(4*b^3*Sqrt[x]*(b + c*
x)^2) + (3*(b*B - 5*A*c)*ArcTan[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(4*b^(7/2)*Sqrt[c])

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Maple [A]  time = 0.025, size = 125, normalized size = 1. \[ -2\,{\frac{A}{{b}^{3}\sqrt{x}}}-{\frac{7\,A{c}^{2}}{4\,{b}^{3} \left ( cx+b \right ) ^{2}}{x}^{{\frac{3}{2}}}}+{\frac{3\,Bc}{4\,{b}^{2} \left ( cx+b \right ) ^{2}}{x}^{{\frac{3}{2}}}}-{\frac{9\,Ac}{4\,{b}^{2} \left ( cx+b \right ) ^{2}}\sqrt{x}}+{\frac{5\,B}{4\,b \left ( cx+b \right ) ^{2}}\sqrt{x}}-{\frac{15\,Ac}{4\,{b}^{3}}\arctan \left ({c\sqrt{x}{\frac{1}{\sqrt{bc}}}} \right ){\frac{1}{\sqrt{bc}}}}+{\frac{3\,B}{4\,{b}^{2}}\arctan \left ({c\sqrt{x}{\frac{1}{\sqrt{bc}}}} \right ){\frac{1}{\sqrt{bc}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*A/b^3/x^(1/2)-7/4/b^3/(c*x+b)^2*x^(3/2)*A*c^2+3/4/b^2/(c*x+b)^2*x^(3/2)*B*c-9
/4/b^2/(c*x+b)^2*A*x^(1/2)*c+5/4/b/(c*x+b)^2*B*x^(1/2)-15/4/b^3/(b*c)^(1/2)*arct
an(c*x^(1/2)/(b*c)^(1/2))*A*c+3/4/b^2/(b*c)^(1/2)*arctan(c*x^(1/2)/(b*c)^(1/2))*
B

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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)*x^(3/2)/(c*x^2 + b*x)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.291311, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \,{\left (B b^{3} - 5 \, A b^{2} c +{\left (B b c^{2} - 5 \, A c^{3}\right )} x^{2} + 2 \,{\left (B b^{2} c - 5 \, A b c^{2}\right )} x\right )} \sqrt{x} \log \left (-\frac{2 \, b c \sqrt{x} - \sqrt{-b c}{\left (c x - b\right )}}{c x + b}\right ) + 2 \,{\left (8 \, A b^{2} - 3 \,{\left (B b c - 5 \, A c^{2}\right )} x^{2} - 5 \,{\left (B b^{2} - 5 \, A b c\right )} x\right )} \sqrt{-b c}}{8 \,{\left (b^{3} c^{2} x^{2} + 2 \, b^{4} c x + b^{5}\right )} \sqrt{-b c} \sqrt{x}}, -\frac{3 \,{\left (B b^{3} - 5 \, A b^{2} c +{\left (B b c^{2} - 5 \, A c^{3}\right )} x^{2} + 2 \,{\left (B b^{2} c - 5 \, A b c^{2}\right )} x\right )} \sqrt{x} \arctan \left (\frac{b}{\sqrt{b c} \sqrt{x}}\right ) +{\left (8 \, A b^{2} - 3 \,{\left (B b c - 5 \, A c^{2}\right )} x^{2} - 5 \,{\left (B b^{2} - 5 \, A b c\right )} x\right )} \sqrt{b c}}{4 \,{\left (b^{3} c^{2} x^{2} + 2 \, b^{4} c x + b^{5}\right )} \sqrt{b c} \sqrt{x}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/8*(3*(B*b^3 - 5*A*b^2*c + (B*b*c^2 - 5*A*c^3)*x^2 + 2*(B*b^2*c - 5*A*b*c^2)*
x)*sqrt(x)*log(-(2*b*c*sqrt(x) - sqrt(-b*c)*(c*x - b))/(c*x + b)) + 2*(8*A*b^2 -
3*(B*b*c - 5*A*c^2)*x^2 - 5*(B*b^2 - 5*A*b*c)*x)*sqrt(-b*c))/((b^3*c^2*x^2 + 2*
b^4*c*x + b^5)*sqrt(-b*c)*sqrt(x)), -1/4*(3*(B*b^3 - 5*A*b^2*c + (B*b*c^2 - 5*A*
c^3)*x^2 + 2*(B*b^2*c - 5*A*b*c^2)*x)*sqrt(x)*arctan(b/(sqrt(b*c)*sqrt(x))) + (8
*A*b^2 - 3*(B*b*c - 5*A*c^2)*x^2 - 5*(B*b^2 - 5*A*b*c)*x)*sqrt(b*c))/((b^3*c^2*x
^2 + 2*b^4*c*x + b^5)*sqrt(b*c)*sqrt(x))]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.275074, size = 116, normalized size = 0.94 \[ \frac{3 \,{\left (B b - 5 \, A c\right )} \arctan \left (\frac{c \sqrt{x}}{\sqrt{b c}}\right )}{4 \, \sqrt{b c} b^{3}} - \frac{2 \, A}{b^{3} \sqrt{x}} + \frac{3 \, B b c x^{\frac{3}{2}} - 7 \, A c^{2} x^{\frac{3}{2}} + 5 \, B b^{2} \sqrt{x} - 9 \, A b c \sqrt{x}}{4 \,{\left (c x + b\right )}^{2} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/4*(B*b - 5*A*c)*arctan(c*sqrt(x)/sqrt(b*c))/(sqrt(b*c)*b^3) - 2*A/(b^3*sqrt(x)
) + 1/4*(3*B*b*c*x^(3/2) - 7*A*c^2*x^(3/2) + 5*B*b^2*sqrt(x) - 9*A*b*c*sqrt(x))/
((c*x + b)^2*b^3)