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

Optimal. Leaf size=131 \[ -\frac{b^{3/2} (7 b B-5 A c) \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{c^{9/2}}+\frac{b \sqrt{x} (7 b B-5 A c)}{c^4}-\frac{x^{3/2} (7 b B-5 A c)}{3 c^3}+\frac{x^{5/2} (7 b B-5 A c)}{5 b c^2}-\frac{x^{7/2} (b B-A c)}{b c (b+c x)} \]

[Out]

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

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Rubi [A]  time = 0.180302, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ -\frac{b^{3/2} (7 b B-5 A c) \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{c^{9/2}}+\frac{b \sqrt{x} (7 b B-5 A c)}{c^4}-\frac{x^{3/2} (7 b B-5 A c)}{3 c^3}+\frac{x^{5/2} (7 b B-5 A c)}{5 b c^2}-\frac{x^{7/2} (b B-A c)}{b c (b+c x)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 22.0272, size = 119, normalized size = 0.91 \[ \frac{b^{\frac{3}{2}} \left (5 A c - 7 B b\right ) \operatorname{atan}{\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}} \right )}}{c^{\frac{9}{2}}} - \frac{b \sqrt{x} \left (5 A c - 7 B b\right )}{c^{4}} + \frac{x^{\frac{3}{2}} \left (5 A c - 7 B b\right )}{3 c^{3}} + \frac{x^{\frac{7}{2}} \left (A c - B b\right )}{b c \left (b + c x\right )} - \frac{x^{\frac{5}{2}} \left (5 A c - 7 B b\right )}{5 b c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.169204, size = 110, normalized size = 0.84 \[ \frac{\sqrt{x} \left (b^2 (70 B c x-75 A c)-2 b c^2 x (25 A+7 B x)+2 c^3 x^2 (5 A+3 B x)+105 b^3 B\right )}{15 c^4 (b+c x)}-\frac{b^{3/2} (7 b B-5 A c) \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{c^{9/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[x]*(105*b^3*B + 2*c^3*x^2*(5*A + 3*B*x) - 2*b*c^2*x*(25*A + 7*B*x) + b^2*(
-75*A*c + 70*B*c*x)))/(15*c^4*(b + c*x)) - (b^(3/2)*(7*b*B - 5*A*c)*ArcTan[(Sqrt
[c]*Sqrt[x])/Sqrt[b]])/c^(9/2)

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Maple [A]  time = 0.019, size = 139, normalized size = 1.1 \[{\frac{2\,B}{5\,{c}^{2}}{x}^{{\frac{5}{2}}}}+{\frac{2\,A}{3\,{c}^{2}}{x}^{{\frac{3}{2}}}}-{\frac{4\,Bb}{3\,{c}^{3}}{x}^{{\frac{3}{2}}}}-4\,{\frac{Ab\sqrt{x}}{{c}^{3}}}+6\,{\frac{{b}^{2}B\sqrt{x}}{{c}^{4}}}-{\frac{{b}^{2}A}{{c}^{3} \left ( cx+b \right ) }\sqrt{x}}+{\frac{B{b}^{3}}{{c}^{4} \left ( cx+b \right ) }\sqrt{x}}+5\,{\frac{{b}^{2}A}{{c}^{3}\sqrt{bc}}\arctan \left ({\frac{c\sqrt{x}}{\sqrt{bc}}} \right ) }-7\,{\frac{B{b}^{3}}{{c}^{4}\sqrt{bc}}\arctan \left ({\frac{c\sqrt{x}}{\sqrt{bc}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/5/c^2*B*x^(5/2)+2/3/c^2*A*x^(3/2)-4/3/c^3*B*x^(3/2)*b-4/c^3*A*x^(1/2)*b+6/c^4*
x^(1/2)*B*b^2-b^2/c^3*x^(1/2)/(c*x+b)*A+b^3/c^4*x^(1/2)/(c*x+b)*B+5*b^2/c^3/(b*c
)^(1/2)*arctan(c*x^(1/2)/(b*c)^(1/2))*A-7*b^3/c^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)*x^(9/2)/(c*x^2 + b*x)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.296948, size = 1, normalized size = 0.01 \[ \left [-\frac{15 \,{\left (7 \, B b^{3} - 5 \, A b^{2} c +{\left (7 \, B b^{2} c - 5 \, A b c^{2}\right )} x\right )} \sqrt{-\frac{b}{c}} \log \left (\frac{c x + 2 \, c \sqrt{x} \sqrt{-\frac{b}{c}} - b}{c x + b}\right ) - 2 \,{\left (6 \, B c^{3} x^{3} + 105 \, B b^{3} - 75 \, A b^{2} c - 2 \,{\left (7 \, B b c^{2} - 5 \, A c^{3}\right )} x^{2} + 10 \,{\left (7 \, B b^{2} c - 5 \, A b c^{2}\right )} x\right )} \sqrt{x}}{30 \,{\left (c^{5} x + b c^{4}\right )}}, -\frac{15 \,{\left (7 \, B b^{3} - 5 \, A b^{2} c +{\left (7 \, B b^{2} c - 5 \, A b c^{2}\right )} x\right )} \sqrt{\frac{b}{c}} \arctan \left (\frac{\sqrt{x}}{\sqrt{\frac{b}{c}}}\right ) -{\left (6 \, B c^{3} x^{3} + 105 \, B b^{3} - 75 \, A b^{2} c - 2 \,{\left (7 \, B b c^{2} - 5 \, A c^{3}\right )} x^{2} + 10 \,{\left (7 \, B b^{2} c - 5 \, A b c^{2}\right )} x\right )} \sqrt{x}}{15 \,{\left (c^{5} x + b c^{4}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/30*(15*(7*B*b^3 - 5*A*b^2*c + (7*B*b^2*c - 5*A*b*c^2)*x)*sqrt(-b/c)*log((c*x
+ 2*c*sqrt(x)*sqrt(-b/c) - b)/(c*x + b)) - 2*(6*B*c^3*x^3 + 105*B*b^3 - 75*A*b^
2*c - 2*(7*B*b*c^2 - 5*A*c^3)*x^2 + 10*(7*B*b^2*c - 5*A*b*c^2)*x)*sqrt(x))/(c^5*
x + b*c^4), -1/15*(15*(7*B*b^3 - 5*A*b^2*c + (7*B*b^2*c - 5*A*b*c^2)*x)*sqrt(b/c
)*arctan(sqrt(x)/sqrt(b/c)) - (6*B*c^3*x^3 + 105*B*b^3 - 75*A*b^2*c - 2*(7*B*b*c
^2 - 5*A*c^3)*x^2 + 10*(7*B*b^2*c - 5*A*b*c^2)*x)*sqrt(x))/(c^5*x + b*c^4)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{\frac{5}{2}} \left (A + B x\right )}{\left (b + c x\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.270383, size = 165, normalized size = 1.26 \[ -\frac{{\left (7 \, B b^{3} - 5 \, A b^{2} c\right )} \arctan \left (\frac{c \sqrt{x}}{\sqrt{b c}}\right )}{\sqrt{b c} c^{4}} + \frac{B b^{3} \sqrt{x} - A b^{2} c \sqrt{x}}{{\left (c x + b\right )} c^{4}} + \frac{2 \,{\left (3 \, B c^{8} x^{\frac{5}{2}} - 10 \, B b c^{7} x^{\frac{3}{2}} + 5 \, A c^{8} x^{\frac{3}{2}} + 45 \, B b^{2} c^{6} \sqrt{x} - 30 \, A b c^{7} \sqrt{x}\right )}}{15 \, c^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(7*B*b^3 - 5*A*b^2*c)*arctan(c*sqrt(x)/sqrt(b*c))/(sqrt(b*c)*c^4) + (B*b^3*sqrt
(x) - A*b^2*c*sqrt(x))/((c*x + b)*c^4) + 2/15*(3*B*c^8*x^(5/2) - 10*B*b*c^7*x^(3
/2) + 5*A*c^8*x^(3/2) + 45*B*b^2*c^6*sqrt(x) - 30*A*b*c^7*sqrt(x))/c^10