3.169 \(\int \frac{x^{5/2} (A+B x)}{b x+c x^2} \, dx\)

Optimal. Leaf size=90 \[ -\frac{2 b^{3/2} (b B-A c) \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{c^{7/2}}+\frac{2 b \sqrt{x} (b B-A c)}{c^3}-\frac{2 x^{3/2} (b B-A c)}{3 c^2}+\frac{2 B x^{5/2}}{5 c} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.12507, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ -\frac{2 b^{3/2} (b B-A c) \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{c^{7/2}}+\frac{2 b \sqrt{x} (b B-A c)}{c^3}-\frac{2 x^{3/2} (b B-A c)}{3 c^2}+\frac{2 B x^{5/2}}{5 c} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 15.6919, size = 83, normalized size = 0.92 \[ \frac{2 B x^{\frac{5}{2}}}{5 c} + \frac{2 b^{\frac{3}{2}} \left (A c - B b\right ) \operatorname{atan}{\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}} \right )}}{c^{\frac{7}{2}}} - \frac{2 b \sqrt{x} \left (A c - B b\right )}{c^{3}} + \frac{2 x^{\frac{3}{2}} \left (A c - B b\right )}{3 c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.131296, size = 81, normalized size = 0.9 \[ \frac{2 \sqrt{x} \left (-5 b c (3 A+B x)+c^2 x (5 A+3 B x)+15 b^2 B\right )}{15 c^3}-\frac{2 b^{3/2} (b B-A c) \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{c^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.012, size = 102, normalized size = 1.1 \[{\frac{2\,B}{5\,c}{x}^{{\frac{5}{2}}}}+{\frac{2\,A}{3\,c}{x}^{{\frac{3}{2}}}}-{\frac{2\,Bb}{3\,{c}^{2}}{x}^{{\frac{3}{2}}}}-2\,{\frac{Ab\sqrt{x}}{{c}^{2}}}+2\,{\frac{{b}^{2}B\sqrt{x}}{{c}^{3}}}+2\,{\frac{{b}^{2}A}{{c}^{2}\sqrt{bc}}\arctan \left ({\frac{c\sqrt{x}}{\sqrt{bc}}} \right ) }-2\,{\frac{B{b}^{3}}{{c}^{3}\sqrt{bc}}\arctan \left ({\frac{c\sqrt{x}}{\sqrt{bc}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.299489, size = 1, normalized size = 0.01 \[ \left [-\frac{15 \,{\left (B b^{2} - A b c\right )} \sqrt{-\frac{b}{c}} \log \left (\frac{c x + 2 \, c \sqrt{x} \sqrt{-\frac{b}{c}} - b}{c x + b}\right ) - 2 \,{\left (3 \, B c^{2} x^{2} + 15 \, B b^{2} - 15 \, A b c - 5 \,{\left (B b c - A c^{2}\right )} x\right )} \sqrt{x}}{15 \, c^{3}}, -\frac{2 \,{\left (15 \,{\left (B b^{2} - A b c\right )} \sqrt{\frac{b}{c}} \arctan \left (\frac{\sqrt{x}}{\sqrt{\frac{b}{c}}}\right ) -{\left (3 \, B c^{2} x^{2} + 15 \, B b^{2} - 15 \, A b c - 5 \,{\left (B b c - A c^{2}\right )} x\right )} \sqrt{x}\right )}}{15 \, c^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/15*(15*(B*b^2 - A*b*c)*sqrt(-b/c)*log((c*x + 2*c*sqrt(x)*sqrt(-b/c) - b)/(c*
x + b)) - 2*(3*B*c^2*x^2 + 15*B*b^2 - 15*A*b*c - 5*(B*b*c - A*c^2)*x)*sqrt(x))/c
^3, -2/15*(15*(B*b^2 - A*b*c)*sqrt(b/c)*arctan(sqrt(x)/sqrt(b/c)) - (3*B*c^2*x^2
+ 15*B*b^2 - 15*A*b*c - 5*(B*b*c - A*c^2)*x)*sqrt(x))/c^3]

_______________________________________________________________________________________

Sympy [A]  time = 19.7478, size = 128, normalized size = 1.42 \[ \frac{2 A b^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}} \right )}}{c^{\frac{5}{2}}} - \frac{2 A b \sqrt{x}}{c^{2}} + \frac{2 A x^{\frac{3}{2}}}{3 c} - \frac{2 B b^{\frac{5}{2}} \operatorname{atan}{\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}} \right )}}{c^{\frac{7}{2}}} + \frac{2 B b^{2} \sqrt{x}}{c^{3}} - \frac{2 B b x^{\frac{3}{2}}}{3 c^{2}} + \frac{2 B x^{\frac{5}{2}}}{5 c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.270688, size = 123, normalized size = 1.37 \[ -\frac{2 \,{\left (B b^{3} - A b^{2} c\right )} \arctan \left (\frac{c \sqrt{x}}{\sqrt{b c}}\right )}{\sqrt{b c} c^{3}} + \frac{2 \,{\left (3 \, B c^{4} x^{\frac{5}{2}} - 5 \, B b c^{3} x^{\frac{3}{2}} + 5 \, A c^{4} x^{\frac{3}{2}} + 15 \, B b^{2} c^{2} \sqrt{x} - 15 \, A b c^{3} \sqrt{x}\right )}}{15 \, c^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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