3.176 \(\int \frac{A+B x}{x^{9/2} \left (b x+c x^2\right )} \, dx\)

Optimal. Leaf size=136 \[ \frac{2 c^{7/2} (b B-A c) \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{b^{11/2}}+\frac{2 c^3 (b B-A c)}{b^5 \sqrt{x}}-\frac{2 c^2 (b B-A c)}{3 b^4 x^{3/2}}+\frac{2 c (b B-A c)}{5 b^3 x^{5/2}}-\frac{2 (b B-A c)}{7 b^2 x^{7/2}}-\frac{2 A}{9 b x^{9/2}} \]

[Out]

(-2*A)/(9*b*x^(9/2)) - (2*(b*B - A*c))/(7*b^2*x^(7/2)) + (2*c*(b*B - A*c))/(5*b^
3*x^(5/2)) - (2*c^2*(b*B - A*c))/(3*b^4*x^(3/2)) + (2*c^3*(b*B - A*c))/(b^5*Sqrt
[x]) + (2*c^(7/2)*(b*B - A*c)*ArcTan[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/b^(11/2)

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Rubi [A]  time = 0.191533, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ \frac{2 c^{7/2} (b B-A c) \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{b^{11/2}}+\frac{2 c^3 (b B-A c)}{b^5 \sqrt{x}}-\frac{2 c^2 (b B-A c)}{3 b^4 x^{3/2}}+\frac{2 c (b B-A c)}{5 b^3 x^{5/2}}-\frac{2 (b B-A c)}{7 b^2 x^{7/2}}-\frac{2 A}{9 b x^{9/2}} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)/(x^(9/2)*(b*x + c*x^2)),x]

[Out]

(-2*A)/(9*b*x^(9/2)) - (2*(b*B - A*c))/(7*b^2*x^(7/2)) + (2*c*(b*B - A*c))/(5*b^
3*x^(5/2)) - (2*c^2*(b*B - A*c))/(3*b^4*x^(3/2)) + (2*c^3*(b*B - A*c))/(b^5*Sqrt
[x]) + (2*c^(7/2)*(b*B - A*c)*ArcTan[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/b^(11/2)

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Rubi in Sympy [A]  time = 25.005, size = 129, normalized size = 0.95 \[ - \frac{2 A}{9 b x^{\frac{9}{2}}} + \frac{2 \left (A c - B b\right )}{7 b^{2} x^{\frac{7}{2}}} - \frac{2 c \left (A c - B b\right )}{5 b^{3} x^{\frac{5}{2}}} + \frac{2 c^{2} \left (A c - B b\right )}{3 b^{4} x^{\frac{3}{2}}} - \frac{2 c^{3} \left (A c - B b\right )}{b^{5} \sqrt{x}} - \frac{2 c^{\frac{7}{2}} \left (A c - B b\right ) \operatorname{atan}{\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}} \right )}}{b^{\frac{11}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)/x**(9/2)/(c*x**2+b*x),x)

[Out]

-2*A/(9*b*x**(9/2)) + 2*(A*c - B*b)/(7*b**2*x**(7/2)) - 2*c*(A*c - B*b)/(5*b**3*
x**(5/2)) + 2*c**2*(A*c - B*b)/(3*b**4*x**(3/2)) - 2*c**3*(A*c - B*b)/(b**5*sqrt
(x)) - 2*c**(7/2)*(A*c - B*b)*atan(sqrt(c)*sqrt(x)/sqrt(b))/b**(11/2)

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Mathematica [A]  time = 0.304365, size = 129, normalized size = 0.95 \[ \frac{2 c^{7/2} (b B-A c) \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{b^{11/2}}-\frac{2 \left (A \left (35 b^4-45 b^3 c x+63 b^2 c^2 x^2-105 b c^3 x^3+315 c^4 x^4\right )+3 b B x \left (15 b^3-21 b^2 c x+35 b c^2 x^2-105 c^3 x^3\right )\right )}{315 b^5 x^{9/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)/(x^(9/2)*(b*x + c*x^2)),x]

[Out]

(-2*(3*b*B*x*(15*b^3 - 21*b^2*c*x + 35*b*c^2*x^2 - 105*c^3*x^3) + A*(35*b^4 - 45
*b^3*c*x + 63*b^2*c^2*x^2 - 105*b*c^3*x^3 + 315*c^4*x^4)))/(315*b^5*x^(9/2)) + (
2*c^(7/2)*(b*B - A*c)*ArcTan[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/b^(11/2)

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Maple [A]  time = 0.02, size = 150, normalized size = 1.1 \[ -{\frac{2\,A}{9\,b}{x}^{-{\frac{9}{2}}}}+{\frac{2\,Ac}{7\,{b}^{2}}{x}^{-{\frac{7}{2}}}}-{\frac{2\,B}{7\,b}{x}^{-{\frac{7}{2}}}}-2\,{\frac{{c}^{4}A}{{b}^{5}\sqrt{x}}}+2\,{\frac{B{c}^{3}}{{b}^{4}\sqrt{x}}}-{\frac{2\,A{c}^{2}}{5\,{b}^{3}}{x}^{-{\frac{5}{2}}}}+{\frac{2\,Bc}{5\,{b}^{2}}{x}^{-{\frac{5}{2}}}}+{\frac{2\,A{c}^{3}}{3\,{b}^{4}}{x}^{-{\frac{3}{2}}}}-{\frac{2\,B{c}^{2}}{3\,{b}^{3}}{x}^{-{\frac{3}{2}}}}-2\,{\frac{A{c}^{5}}{{b}^{5}\sqrt{bc}}\arctan \left ({\frac{c\sqrt{x}}{\sqrt{bc}}} \right ) }+2\,{\frac{{c}^{4}B}{{b}^{4}\sqrt{bc}}\arctan \left ({\frac{c\sqrt{x}}{\sqrt{bc}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/x^(9/2)/(c*x^2+b*x),x)

[Out]

-2/9*A/b/x^(9/2)+2/7/b^2/x^(7/2)*A*c-2/7/b/x^(7/2)*B-2/b^5*c^4/x^(1/2)*A+2/b^4*c
^3/x^(1/2)*B-2/5/b^3*c^2/x^(5/2)*A+2/5/b^2*c/x^(5/2)*B+2/3/b^4*c^3/x^(3/2)*A-2/3
/b^3*c^2/x^(3/2)*B-2*c^5/b^5/(b*c)^(1/2)*arctan(c*x^(1/2)/(b*c)^(1/2))*A+2*c^4/b
^4/(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)/((c*x^2 + b*x)*x^(9/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.302253, size = 1, normalized size = 0.01 \[ \left [-\frac{315 \,{\left (B b c^{3} - A c^{4}\right )} x^{\frac{9}{2}} \sqrt{-\frac{c}{b}} \log \left (\frac{c x - 2 \, b \sqrt{x} \sqrt{-\frac{c}{b}} - b}{c x + b}\right ) + 70 \, A b^{4} - 630 \,{\left (B b c^{3} - A c^{4}\right )} x^{4} + 210 \,{\left (B b^{2} c^{2} - A b c^{3}\right )} x^{3} - 126 \,{\left (B b^{3} c - A b^{2} c^{2}\right )} x^{2} + 90 \,{\left (B b^{4} - A b^{3} c\right )} x}{315 \, b^{5} x^{\frac{9}{2}}}, -\frac{2 \,{\left (315 \,{\left (B b c^{3} - A c^{4}\right )} x^{\frac{9}{2}} \sqrt{\frac{c}{b}} \arctan \left (\frac{b \sqrt{\frac{c}{b}}}{c \sqrt{x}}\right ) + 35 \, A b^{4} - 315 \,{\left (B b c^{3} - A c^{4}\right )} x^{4} + 105 \,{\left (B b^{2} c^{2} - A b c^{3}\right )} x^{3} - 63 \,{\left (B b^{3} c - A b^{2} c^{2}\right )} x^{2} + 45 \,{\left (B b^{4} - A b^{3} c\right )} x\right )}}{315 \, b^{5} x^{\frac{9}{2}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((c*x^2 + b*x)*x^(9/2)),x, algorithm="fricas")

[Out]

[-1/315*(315*(B*b*c^3 - A*c^4)*x^(9/2)*sqrt(-c/b)*log((c*x - 2*b*sqrt(x)*sqrt(-c
/b) - b)/(c*x + b)) + 70*A*b^4 - 630*(B*b*c^3 - A*c^4)*x^4 + 210*(B*b^2*c^2 - A*
b*c^3)*x^3 - 126*(B*b^3*c - A*b^2*c^2)*x^2 + 90*(B*b^4 - A*b^3*c)*x)/(b^5*x^(9/2
)), -2/315*(315*(B*b*c^3 - A*c^4)*x^(9/2)*sqrt(c/b)*arctan(b*sqrt(c/b)/(c*sqrt(x
))) + 35*A*b^4 - 315*(B*b*c^3 - A*c^4)*x^4 + 105*(B*b^2*c^2 - A*b*c^3)*x^3 - 63*
(B*b^3*c - A*b^2*c^2)*x^2 + 45*(B*b^4 - A*b^3*c)*x)/(b^5*x^(9/2))]

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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((B*x+A)/x**(9/2)/(c*x**2+b*x),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.270923, size = 173, normalized size = 1.27 \[ \frac{2 \,{\left (B b c^{4} - A c^{5}\right )} \arctan \left (\frac{c \sqrt{x}}{\sqrt{b c}}\right )}{\sqrt{b c} b^{5}} + \frac{2 \,{\left (315 \, B b c^{3} x^{4} - 315 \, A c^{4} x^{4} - 105 \, B b^{2} c^{2} x^{3} + 105 \, A b c^{3} x^{3} + 63 \, B b^{3} c x^{2} - 63 \, A b^{2} c^{2} x^{2} - 45 \, B b^{4} x + 45 \, A b^{3} c x - 35 \, A b^{4}\right )}}{315 \, b^{5} x^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((c*x^2 + b*x)*x^(9/2)),x, algorithm="giac")

[Out]

2*(B*b*c^4 - A*c^5)*arctan(c*sqrt(x)/sqrt(b*c))/(sqrt(b*c)*b^5) + 2/315*(315*B*b
*c^3*x^4 - 315*A*c^4*x^4 - 105*B*b^2*c^2*x^3 + 105*A*b*c^3*x^3 + 63*B*b^3*c*x^2
- 63*A*b^2*c^2*x^2 - 45*B*b^4*x + 45*A*b^3*c*x - 35*A*b^4)/(b^5*x^(9/2))