3.182 \(\int \frac{A+B x}{\sqrt{x} \left (b x+c x^2\right )^2} \, dx\)

Optimal. Leaf size=110 \[ -\frac{\sqrt{c} (3 b B-5 A c) \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{b^{7/2}}-\frac{3 b B-5 A c}{b^3 \sqrt{x}}+\frac{3 b B-5 A c}{3 b^2 c x^{3/2}}-\frac{b B-A c}{b c x^{3/2} (b+c x)} \]

[Out]

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

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Rubi [A]  time = 0.137408, antiderivative size = 110, 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 \[ -\frac{\sqrt{c} (3 b B-5 A c) \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{b^{7/2}}-\frac{3 b B-5 A c}{b^3 \sqrt{x}}+\frac{3 b B-5 A c}{3 b^2 c x^{3/2}}-\frac{b B-A c}{b c x^{3/2} (b+c x)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 17.9075, size = 97, normalized size = 0.88 \[ \frac{A c - B b}{b c x^{\frac{3}{2}} \left (b + c x\right )} - \frac{5 A c - 3 B b}{3 b^{2} c x^{\frac{3}{2}}} + \frac{5 A c - 3 B b}{b^{3} \sqrt{x}} + \frac{\sqrt{c} \left (5 A c - 3 B b\right ) \operatorname{atan}{\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}} \right )}}{b^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)/(c*x**2+b*x)**2/x**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.184391, size = 92, normalized size = 0.84 \[ \frac{\sqrt{c} (5 A c-3 b B) \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{b^{7/2}}+\frac{A \left (-2 b^2+10 b c x+15 c^2 x^2\right )-3 b B x (2 b+3 c x)}{3 b^3 x^{3/2} (b+c x)} \]

Antiderivative was successfully verified.

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

[Out]

(-3*b*B*x*(2*b + 3*c*x) + A*(-2*b^2 + 10*b*c*x + 15*c^2*x^2))/(3*b^3*x^(3/2)*(b
+ c*x)) + (Sqrt[c]*(-3*b*B + 5*A*c)*ArcTan[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/b^(7/2)

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Maple [A]  time = 0.026, size = 113, normalized size = 1. \[ -{\frac{2\,A}{3\,{b}^{2}}{x}^{-{\frac{3}{2}}}}+4\,{\frac{Ac}{\sqrt{x}{b}^{3}}}-2\,{\frac{B}{\sqrt{x}{b}^{2}}}+{\frac{A{c}^{2}}{{b}^{3} \left ( cx+b \right ) }\sqrt{x}}-{\frac{Bc}{{b}^{2} \left ( cx+b \right ) }\sqrt{x}}+5\,{\frac{A{c}^{2}}{{b}^{3}\sqrt{bc}}\arctan \left ({\frac{c\sqrt{x}}{\sqrt{bc}}} \right ) }-3\,{\frac{Bc}{{b}^{2}\sqrt{bc}}\arctan \left ({\frac{c\sqrt{x}}{\sqrt{bc}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/(c*x^2+b*x)^2/x^(1/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.313601, size = 1, normalized size = 0.01 \[ \left [-\frac{4 \, A b^{2} + 6 \,{\left (3 \, B b c - 5 \, A c^{2}\right )} x^{2} + 3 \,{\left ({\left (3 \, B b c - 5 \, A c^{2}\right )} x^{2} +{\left (3 \, B b^{2} - 5 \, A b c\right )} x\right )} \sqrt{x} \sqrt{-\frac{c}{b}} \log \left (\frac{c x + 2 \, b \sqrt{x} \sqrt{-\frac{c}{b}} - b}{c x + b}\right ) + 4 \,{\left (3 \, B b^{2} - 5 \, A b c\right )} x}{6 \,{\left (b^{3} c x^{2} + b^{4} x\right )} \sqrt{x}}, -\frac{2 \, A b^{2} + 3 \,{\left (3 \, B b c - 5 \, A c^{2}\right )} x^{2} - 3 \,{\left ({\left (3 \, B b c - 5 \, A c^{2}\right )} x^{2} +{\left (3 \, B b^{2} - 5 \, A b c\right )} x\right )} \sqrt{x} \sqrt{\frac{c}{b}} \arctan \left (\frac{b \sqrt{\frac{c}{b}}}{c \sqrt{x}}\right ) + 2 \,{\left (3 \, B b^{2} - 5 \, A b c\right )} x}{3 \,{\left (b^{3} c x^{2} + b^{4} x\right )} \sqrt{x}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/6*(4*A*b^2 + 6*(3*B*b*c - 5*A*c^2)*x^2 + 3*((3*B*b*c - 5*A*c^2)*x^2 + (3*B*b
^2 - 5*A*b*c)*x)*sqrt(x)*sqrt(-c/b)*log((c*x + 2*b*sqrt(x)*sqrt(-c/b) - b)/(c*x
+ b)) + 4*(3*B*b^2 - 5*A*b*c)*x)/((b^3*c*x^2 + b^4*x)*sqrt(x)), -1/3*(2*A*b^2 +
3*(3*B*b*c - 5*A*c^2)*x^2 - 3*((3*B*b*c - 5*A*c^2)*x^2 + (3*B*b^2 - 5*A*b*c)*x)*
sqrt(x)*sqrt(c/b)*arctan(b*sqrt(c/b)/(c*sqrt(x))) + 2*(3*B*b^2 - 5*A*b*c)*x)/((b
^3*c*x^2 + b^4*x)*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)/(c*x**2+b*x)**2/x**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.269698, size = 115, normalized size = 1.05 \[ -\frac{{\left (3 \, B b c - 5 \, A c^{2}\right )} \arctan \left (\frac{c \sqrt{x}}{\sqrt{b c}}\right )}{\sqrt{b c} b^{3}} - \frac{B b c \sqrt{x} - A c^{2} \sqrt{x}}{{\left (c x + b\right )} b^{3}} - \frac{2 \,{\left (3 \, B b x - 6 \, A c x + A b\right )}}{3 \, b^{3} x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(3*B*b*c - 5*A*c^2)*arctan(c*sqrt(x)/sqrt(b*c))/(sqrt(b*c)*b^3) - (B*b*c*sqrt(x
) - A*c^2*sqrt(x))/((c*x + b)*b^3) - 2/3*(3*B*b*x - 6*A*c*x + A*b)/(b^3*x^(3/2))