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

Optimal. Leaf size=130 \[ \frac{c^{3/2} (5 b B-7 A c) \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{b^{9/2}}+\frac{c (5 b B-7 A c)}{b^4 \sqrt{x}}-\frac{5 b B-7 A c}{3 b^3 x^{3/2}}+\frac{5 b B-7 A c}{5 b^2 c x^{5/2}}-\frac{b B-A c}{b c x^{5/2} (b+c x)} \]

[Out]

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

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Rubi [A]  time = 0.167846, antiderivative size = 130, 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 \[ \frac{c^{3/2} (5 b B-7 A c) \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{b^{9/2}}+\frac{c (5 b B-7 A c)}{b^4 \sqrt{x}}-\frac{5 b B-7 A c}{3 b^3 x^{3/2}}+\frac{5 b B-7 A c}{5 b^2 c x^{5/2}}-\frac{b B-A c}{b c x^{5/2} (b+c x)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 22.0989, size = 119, normalized size = 0.92 \[ \frac{A c - B b}{b c x^{\frac{5}{2}} \left (b + c x\right )} - \frac{7 A c - 5 B b}{5 b^{2} c x^{\frac{5}{2}}} + \frac{7 A c - 5 B b}{3 b^{3} x^{\frac{3}{2}}} - \frac{c \left (7 A c - 5 B b\right )}{b^{4} \sqrt{x}} - \frac{c^{\frac{3}{2}} \left (7 A c - 5 B b\right ) \operatorname{atan}{\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}} \right )}}{b^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.199423, size = 115, normalized size = 0.88 \[ \frac{c^{3/2} (5 b B-7 A c) \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{b^{9/2}}+\frac{5 b B x \left (-2 b^2+10 b c x+15 c^2 x^2\right )-A \left (6 b^3-14 b^2 c x+70 b c^2 x^2+105 c^3 x^3\right )}{15 b^4 x^{5/2} (b+c x)} \]

Antiderivative was successfully verified.

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

[Out]

(5*b*B*x*(-2*b^2 + 10*b*c*x + 15*c^2*x^2) - A*(6*b^3 - 14*b^2*c*x + 70*b*c^2*x^2
+ 105*c^3*x^3))/(15*b^4*x^(5/2)*(b + c*x)) + (c^(3/2)*(5*b*B - 7*A*c)*ArcTan[(S
qrt[c]*Sqrt[x])/Sqrt[b]])/b^(9/2)

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Maple [A]  time = 0.031, size = 139, normalized size = 1.1 \[ -{\frac{2\,A}{5\,{b}^{2}}{x}^{-{\frac{5}{2}}}}+{\frac{4\,Ac}{3\,{b}^{3}}{x}^{-{\frac{3}{2}}}}-{\frac{2\,B}{3\,{b}^{2}}{x}^{-{\frac{3}{2}}}}-6\,{\frac{A{c}^{2}}{{b}^{4}\sqrt{x}}}+4\,{\frac{Bc}{{b}^{3}\sqrt{x}}}-{\frac{A{c}^{3}}{{b}^{4} \left ( cx+b \right ) }\sqrt{x}}+{\frac{B{c}^{2}}{{b}^{3} \left ( cx+b \right ) }\sqrt{x}}-7\,{\frac{A{c}^{3}}{{b}^{4}\sqrt{bc}}\arctan \left ({\frac{c\sqrt{x}}{\sqrt{bc}}} \right ) }+5\,{\frac{B{c}^{2}}{{b}^{3}\sqrt{bc}}\arctan \left ({\frac{c\sqrt{x}}{\sqrt{bc}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.300995, size = 1, normalized size = 0.01 \[ \left [-\frac{12 \, A b^{3} - 30 \,{\left (5 \, B b c^{2} - 7 \, A c^{3}\right )} x^{3} - 20 \,{\left (5 \, B b^{2} c - 7 \, A b c^{2}\right )} x^{2} + 15 \,{\left ({\left (5 \, B b c^{2} - 7 \, A c^{3}\right )} x^{3} +{\left (5 \, B b^{2} c - 7 \, A b c^{2}\right )} x^{2}\right )} \sqrt{x} \sqrt{-\frac{c}{b}} \log \left (\frac{c x - 2 \, b \sqrt{x} \sqrt{-\frac{c}{b}} - b}{c x + b}\right ) + 4 \,{\left (5 \, B b^{3} - 7 \, A b^{2} c\right )} x}{30 \,{\left (b^{4} c x^{3} + b^{5} x^{2}\right )} \sqrt{x}}, -\frac{6 \, A b^{3} - 15 \,{\left (5 \, B b c^{2} - 7 \, A c^{3}\right )} x^{3} - 10 \,{\left (5 \, B b^{2} c - 7 \, A b c^{2}\right )} x^{2} + 15 \,{\left ({\left (5 \, B b c^{2} - 7 \, A c^{3}\right )} x^{3} +{\left (5 \, B b^{2} c - 7 \, A b c^{2}\right )} x^{2}\right )} \sqrt{x} \sqrt{\frac{c}{b}} \arctan \left (\frac{b \sqrt{\frac{c}{b}}}{c \sqrt{x}}\right ) + 2 \,{\left (5 \, B b^{3} - 7 \, A b^{2} c\right )} x}{15 \,{\left (b^{4} c x^{3} + b^{5} x^{2}\right )} \sqrt{x}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/30*(12*A*b^3 - 30*(5*B*b*c^2 - 7*A*c^3)*x^3 - 20*(5*B*b^2*c - 7*A*b*c^2)*x^2
+ 15*((5*B*b*c^2 - 7*A*c^3)*x^3 + (5*B*b^2*c - 7*A*b*c^2)*x^2)*sqrt(x)*sqrt(-c/
b)*log((c*x - 2*b*sqrt(x)*sqrt(-c/b) - b)/(c*x + b)) + 4*(5*B*b^3 - 7*A*b^2*c)*x
)/((b^4*c*x^3 + b^5*x^2)*sqrt(x)), -1/15*(6*A*b^3 - 15*(5*B*b*c^2 - 7*A*c^3)*x^3
- 10*(5*B*b^2*c - 7*A*b*c^2)*x^2 + 15*((5*B*b*c^2 - 7*A*c^3)*x^3 + (5*B*b^2*c -
7*A*b*c^2)*x^2)*sqrt(x)*sqrt(c/b)*arctan(b*sqrt(c/b)/(c*sqrt(x))) + 2*(5*B*b^3
- 7*A*b^2*c)*x)/((b^4*c*x^3 + b^5*x^2)*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((B*x+A)/x**(3/2)/(c*x**2+b*x)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.272072, size = 149, normalized size = 1.15 \[ \frac{{\left (5 \, B b c^{2} - 7 \, A c^{3}\right )} \arctan \left (\frac{c \sqrt{x}}{\sqrt{b c}}\right )}{\sqrt{b c} b^{4}} + \frac{B b c^{2} \sqrt{x} - A c^{3} \sqrt{x}}{{\left (c x + b\right )} b^{4}} + \frac{2 \,{\left (30 \, B b c x^{2} - 45 \, A c^{2} x^{2} - 5 \, B b^{2} x + 10 \, A b c x - 3 \, A b^{2}\right )}}{15 \, b^{4} x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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