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

Optimal. Leaf size=107 \[ \frac{\sqrt{b} (5 A b-3 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{7/2}}+\frac{5 A b-3 a B}{a^3 \sqrt{x}}-\frac{5 A b-3 a B}{3 a^2 b x^{3/2}}+\frac{A b-a B}{a b x^{3/2} (a+b x)} \]

[Out]

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

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Rubi [A]  time = 0.138668, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{\sqrt{b} (5 A b-3 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{7/2}}+\frac{5 A b-3 a B}{a^3 \sqrt{x}}-\frac{5 A b-3 a B}{3 a^2 b x^{3/2}}+\frac{A b-a B}{a b x^{3/2} (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 16.8235, size = 97, normalized size = 0.91 \[ \frac{A b - B a}{a b x^{\frac{3}{2}} \left (a + b x\right )} - \frac{5 A b - 3 B a}{3 a^{2} b x^{\frac{3}{2}}} + \frac{5 A b - 3 B a}{a^{3} \sqrt{x}} + \frac{\sqrt{b} \left (5 A b - 3 B a\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{a^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.15924, size = 90, normalized size = 0.84 \[ \frac{\sqrt{b} (5 A b-3 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{7/2}}+\frac{-2 a^2 (A+3 B x)+a b x (10 A-9 B x)+15 A b^2 x^2}{3 a^3 x^{3/2} (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

(15*A*b^2*x^2 + a*b*x*(10*A - 9*B*x) - 2*a^2*(A + 3*B*x))/(3*a^3*x^(3/2)*(a + b*
x)) + (Sqrt[b]*(5*A*b - 3*a*B)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/a^(7/2)

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Maple [A]  time = 0.024, size = 113, normalized size = 1.1 \[ -{\frac{2\,A}{3\,{a}^{2}}{x}^{-{\frac{3}{2}}}}+4\,{\frac{Ab}{\sqrt{x}{a}^{3}}}-2\,{\frac{B}{\sqrt{x}{a}^{2}}}+{\frac{{b}^{2}A}{{a}^{3} \left ( bx+a \right ) }\sqrt{x}}-{\frac{Bb}{{a}^{2} \left ( bx+a \right ) }\sqrt{x}}+5\,{\frac{{b}^{2}A}{{a}^{3}\sqrt{ab}}\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ) }-3\,{\frac{Bb}{{a}^{2}\sqrt{ab}}\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.235961, size = 1, normalized size = 0.01 \[ \left [-\frac{4 \, A a^{2} + 6 \,{\left (3 \, B a b - 5 \, A b^{2}\right )} x^{2} + 3 \,{\left ({\left (3 \, B a b - 5 \, A b^{2}\right )} x^{2} +{\left (3 \, B a^{2} - 5 \, A a b\right )} x\right )} \sqrt{x} \sqrt{-\frac{b}{a}} \log \left (\frac{b x + 2 \, a \sqrt{x} \sqrt{-\frac{b}{a}} - a}{b x + a}\right ) + 4 \,{\left (3 \, B a^{2} - 5 \, A a b\right )} x}{6 \,{\left (a^{3} b x^{2} + a^{4} x\right )} \sqrt{x}}, -\frac{2 \, A a^{2} + 3 \,{\left (3 \, B a b - 5 \, A b^{2}\right )} x^{2} - 3 \,{\left ({\left (3 \, B a b - 5 \, A b^{2}\right )} x^{2} +{\left (3 \, B a^{2} - 5 \, A a b\right )} x\right )} \sqrt{x} \sqrt{\frac{b}{a}} \arctan \left (\frac{a \sqrt{\frac{b}{a}}}{b \sqrt{x}}\right ) + 2 \,{\left (3 \, B a^{2} - 5 \, A a b\right )} x}{3 \,{\left (a^{3} b x^{2} + a^{4} x\right )} \sqrt{x}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.23504, size = 115, normalized size = 1.07 \[ -\frac{{\left (3 \, B a b - 5 \, A b^{2}\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} a^{3}} - \frac{B a b \sqrt{x} - A b^{2} \sqrt{x}}{{\left (b x + a\right )} a^{3}} - \frac{2 \,{\left (3 \, B a x - 6 \, A b x + A a\right )}}{3 \, a^{3} x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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