3.189 \(\int \frac{x^{5/2} (A+B x)}{\left (b x+c x^2\right )^3} \, dx\)

Optimal. Leaf size=100 \[ \frac{(3 A c+b B) \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{4 b^{5/2} c^{3/2}}+\frac{\sqrt{x} (3 A c+b B)}{4 b^2 c (b+c x)}-\frac{\sqrt{x} (b B-A c)}{2 b c (b+c x)^2} \]

[Out]

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

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Rubi [A]  time = 0.118334, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ \frac{(3 A c+b B) \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{4 b^{5/2} c^{3/2}}+\frac{\sqrt{x} (3 A c+b B)}{4 b^2 c (b+c x)}-\frac{\sqrt{x} (b B-A c)}{2 b c (b+c x)^2} \]

Antiderivative was successfully verified.

[In]  Int[(x^(5/2)*(A + B*x))/(b*x + c*x^2)^3,x]

[Out]

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

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Rubi in Sympy [A]  time = 15.3699, size = 85, normalized size = 0.85 \[ \frac{\sqrt{x} \left (A c - B b\right )}{2 b c \left (b + c x\right )^{2}} + \frac{\sqrt{x} \left (3 A c + B b\right )}{4 b^{2} c \left (b + c x\right )} + \frac{\left (3 A c + B b\right ) \operatorname{atan}{\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}} \right )}}{4 b^{\frac{5}{2}} c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(5/2)*(B*x+A)/(c*x**2+b*x)**3,x)

[Out]

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

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Mathematica [A]  time = 0.140577, size = 86, normalized size = 0.86 \[ \frac{(3 A c+b B) \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{4 b^{5/2} c^{3/2}}+\frac{\sqrt{x} \left (b c (5 A+B x)+3 A c^2 x+b^2 (-B)\right )}{4 b^2 c (b+c x)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^(5/2)*(A + B*x))/(b*x + c*x^2)^3,x]

[Out]

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

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Maple [A]  time = 0.019, size = 95, normalized size = 1. \[ 2\,{\frac{1}{ \left ( cx+b \right ) ^{2}} \left ( 1/8\,{\frac{ \left ( 3\,Ac+Bb \right ){x}^{3/2}}{{b}^{2}}}+1/8\,{\frac{ \left ( 5\,Ac-Bb \right ) \sqrt{x}}{bc}} \right ) }+{\frac{3\,A}{4\,{b}^{2}}\arctan \left ({c\sqrt{x}{\frac{1}{\sqrt{bc}}}} \right ){\frac{1}{\sqrt{bc}}}}+{\frac{B}{4\,bc}\arctan \left ({c\sqrt{x}{\frac{1}{\sqrt{bc}}}} \right ){\frac{1}{\sqrt{bc}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(5/2)*(B*x+A)/(c*x^2+b*x)^3,x)

[Out]

2*(1/8*(3*A*c+B*b)/b^2*x^(3/2)+1/8*(5*A*c-B*b)/b/c*x^(1/2))/(c*x+b)^2+3/4/b^2/(b
*c)^(1/2)*arctan(c*x^(1/2)/(b*c)^(1/2))*A+1/4/b/c/(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^(5/2)/(c*x^2 + b*x)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.283056, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \,{\left (B b^{2} - 5 \, A b c -{\left (B b c + 3 \, A c^{2}\right )} x\right )} \sqrt{-b c} \sqrt{x} -{\left (B b^{3} + 3 \, A b^{2} c +{\left (B b c^{2} + 3 \, A c^{3}\right )} x^{2} + 2 \,{\left (B b^{2} c + 3 \, A b c^{2}\right )} x\right )} \log \left (\frac{2 \, b c \sqrt{x} + \sqrt{-b c}{\left (c x - b\right )}}{c x + b}\right )}{8 \,{\left (b^{2} c^{3} x^{2} + 2 \, b^{3} c^{2} x + b^{4} c\right )} \sqrt{-b c}}, -\frac{{\left (B b^{2} - 5 \, A b c -{\left (B b c + 3 \, A c^{2}\right )} x\right )} \sqrt{b c} \sqrt{x} +{\left (B b^{3} + 3 \, A b^{2} c +{\left (B b c^{2} + 3 \, A c^{3}\right )} x^{2} + 2 \,{\left (B b^{2} c + 3 \, A b c^{2}\right )} x\right )} \arctan \left (\frac{b}{\sqrt{b c} \sqrt{x}}\right )}{4 \,{\left (b^{2} c^{3} x^{2} + 2 \, b^{3} c^{2} x + b^{4} c\right )} \sqrt{b c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*x^(5/2)/(c*x^2 + b*x)^3,x, algorithm="fricas")

[Out]

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

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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**(5/2)*(B*x+A)/(c*x**2+b*x)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.27143, size = 111, normalized size = 1.11 \[ \frac{{\left (B b + 3 \, A c\right )} \arctan \left (\frac{c \sqrt{x}}{\sqrt{b c}}\right )}{4 \, \sqrt{b c} b^{2} c} + \frac{B b c x^{\frac{3}{2}} + 3 \, A c^{2} x^{\frac{3}{2}} - B b^{2} \sqrt{x} + 5 \, A b c \sqrt{x}}{4 \,{\left (c x + b\right )}^{2} b^{2} c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*x^(5/2)/(c*x^2 + b*x)^3,x, algorithm="giac")

[Out]

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