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

Optimal. Leaf size=67 \[ -\frac{2 A \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{a^{5/2}}+\frac{2 A}{a^2 \sqrt{a+b x}}+\frac{2 (A b-a B)}{3 a b (a+b x)^{3/2}} \]

[Out]

(2*(A*b - a*B))/(3*a*b*(a + b*x)^(3/2)) + (2*A)/(a^2*Sqrt[a + b*x]) - (2*A*ArcTa
nh[Sqrt[a + b*x]/Sqrt[a]])/a^(5/2)

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Rubi [A]  time = 0.0813486, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ -\frac{2 A \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{a^{5/2}}+\frac{2 A}{a^2 \sqrt{a+b x}}+\frac{2 (A b-a B)}{3 a b (a+b x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(A*b - a*B))/(3*a*b*(a + b*x)^(3/2)) + (2*A)/(a^2*Sqrt[a + b*x]) - (2*A*ArcTa
nh[Sqrt[a + b*x]/Sqrt[a]])/a^(5/2)

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Rubi in Sympy [A]  time = 9.37235, size = 61, normalized size = 0.91 \[ \frac{2 A}{a^{2} \sqrt{a + b x}} - \frac{2 A \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{a^{\frac{5}{2}}} + \frac{2 \left (A b - B a\right )}{3 a b \left (a + b x\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.173855, size = 63, normalized size = 0.94 \[ \frac{-2 a^2 B+8 a A b+6 A b^2 x}{3 a^2 b (a+b x)^{3/2}}-\frac{2 A \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{a^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(8*a*A*b - 2*a^2*B + 6*A*b^2*x)/(3*a^2*b*(a + b*x)^(3/2)) - (2*A*ArcTanh[Sqrt[a
+ b*x]/Sqrt[a]])/a^(5/2)

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Maple [A]  time = 0.014, size = 59, normalized size = 0.9 \[ 2\,{\frac{1}{b} \left ( -1/3\,{\frac{-Ab+Ba}{a \left ( bx+a \right ) ^{3/2}}}+{\frac{Ab}{{a}^{2}\sqrt{bx+a}}}-{\frac{Ab}{{a}^{5/2}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) } \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/x/(b*x+a)^(5/2),x)

[Out]

2/b*(-1/3*(-A*b+B*a)/a/(b*x+a)^(3/2)+1/a^2/(b*x+a)^(1/2)*A*b-A*b/a^(5/2)*arctanh
((b*x+a)^(1/2)/a^(1/2)))

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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)^(5/2)*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.235753, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (A b^{2} x + A a b\right )} \sqrt{b x + a} \log \left (\frac{{\left (b x + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{b x + a} a}{x}\right ) + 2 \,{\left (3 \, A b^{2} x - B a^{2} + 4 \, A a b\right )} \sqrt{a}}{3 \,{\left (a^{2} b^{2} x + a^{3} b\right )} \sqrt{b x + a} \sqrt{a}}, \frac{2 \,{\left (3 \,{\left (A b^{2} x + A a b\right )} \sqrt{b x + a} \arctan \left (\frac{a}{\sqrt{b x + a} \sqrt{-a}}\right ) +{\left (3 \, A b^{2} x - B a^{2} + 4 \, A a b\right )} \sqrt{-a}\right )}}{3 \,{\left (a^{2} b^{2} x + a^{3} b\right )} \sqrt{b x + a} \sqrt{-a}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 19.2874, size = 714, normalized size = 10.66 \[ A \left (\frac{8 a^{7} \sqrt{1 + \frac{b x}{a}}}{3 a^{\frac{19}{2}} + 9 a^{\frac{17}{2}} b x + 9 a^{\frac{15}{2}} b^{2} x^{2} + 3 a^{\frac{13}{2}} b^{3} x^{3}} + \frac{3 a^{7} \log{\left (\frac{b x}{a} \right )}}{3 a^{\frac{19}{2}} + 9 a^{\frac{17}{2}} b x + 9 a^{\frac{15}{2}} b^{2} x^{2} + 3 a^{\frac{13}{2}} b^{3} x^{3}} - \frac{6 a^{7} \log{\left (\sqrt{1 + \frac{b x}{a}} + 1 \right )}}{3 a^{\frac{19}{2}} + 9 a^{\frac{17}{2}} b x + 9 a^{\frac{15}{2}} b^{2} x^{2} + 3 a^{\frac{13}{2}} b^{3} x^{3}} + \frac{14 a^{6} b x \sqrt{1 + \frac{b x}{a}}}{3 a^{\frac{19}{2}} + 9 a^{\frac{17}{2}} b x + 9 a^{\frac{15}{2}} b^{2} x^{2} + 3 a^{\frac{13}{2}} b^{3} x^{3}} + \frac{9 a^{6} b x \log{\left (\frac{b x}{a} \right )}}{3 a^{\frac{19}{2}} + 9 a^{\frac{17}{2}} b x + 9 a^{\frac{15}{2}} b^{2} x^{2} + 3 a^{\frac{13}{2}} b^{3} x^{3}} - \frac{18 a^{6} b x \log{\left (\sqrt{1 + \frac{b x}{a}} + 1 \right )}}{3 a^{\frac{19}{2}} + 9 a^{\frac{17}{2}} b x + 9 a^{\frac{15}{2}} b^{2} x^{2} + 3 a^{\frac{13}{2}} b^{3} x^{3}} + \frac{6 a^{5} b^{2} x^{2} \sqrt{1 + \frac{b x}{a}}}{3 a^{\frac{19}{2}} + 9 a^{\frac{17}{2}} b x + 9 a^{\frac{15}{2}} b^{2} x^{2} + 3 a^{\frac{13}{2}} b^{3} x^{3}} + \frac{9 a^{5} b^{2} x^{2} \log{\left (\frac{b x}{a} \right )}}{3 a^{\frac{19}{2}} + 9 a^{\frac{17}{2}} b x + 9 a^{\frac{15}{2}} b^{2} x^{2} + 3 a^{\frac{13}{2}} b^{3} x^{3}} - \frac{18 a^{5} b^{2} x^{2} \log{\left (\sqrt{1 + \frac{b x}{a}} + 1 \right )}}{3 a^{\frac{19}{2}} + 9 a^{\frac{17}{2}} b x + 9 a^{\frac{15}{2}} b^{2} x^{2} + 3 a^{\frac{13}{2}} b^{3} x^{3}} + \frac{3 a^{4} b^{3} x^{3} \log{\left (\frac{b x}{a} \right )}}{3 a^{\frac{19}{2}} + 9 a^{\frac{17}{2}} b x + 9 a^{\frac{15}{2}} b^{2} x^{2} + 3 a^{\frac{13}{2}} b^{3} x^{3}} - \frac{6 a^{4} b^{3} x^{3} \log{\left (\sqrt{1 + \frac{b x}{a}} + 1 \right )}}{3 a^{\frac{19}{2}} + 9 a^{\frac{17}{2}} b x + 9 a^{\frac{15}{2}} b^{2} x^{2} + 3 a^{\frac{13}{2}} b^{3} x^{3}}\right ) - \frac{2 B}{3 b \left (a + b x\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*(8*a**7*sqrt(1 + b*x/a)/(3*a**(19/2) + 9*a**(17/2)*b*x + 9*a**(15/2)*b**2*x**2
 + 3*a**(13/2)*b**3*x**3) + 3*a**7*log(b*x/a)/(3*a**(19/2) + 9*a**(17/2)*b*x + 9
*a**(15/2)*b**2*x**2 + 3*a**(13/2)*b**3*x**3) - 6*a**7*log(sqrt(1 + b*x/a) + 1)/
(3*a**(19/2) + 9*a**(17/2)*b*x + 9*a**(15/2)*b**2*x**2 + 3*a**(13/2)*b**3*x**3)
+ 14*a**6*b*x*sqrt(1 + b*x/a)/(3*a**(19/2) + 9*a**(17/2)*b*x + 9*a**(15/2)*b**2*
x**2 + 3*a**(13/2)*b**3*x**3) + 9*a**6*b*x*log(b*x/a)/(3*a**(19/2) + 9*a**(17/2)
*b*x + 9*a**(15/2)*b**2*x**2 + 3*a**(13/2)*b**3*x**3) - 18*a**6*b*x*log(sqrt(1 +
 b*x/a) + 1)/(3*a**(19/2) + 9*a**(17/2)*b*x + 9*a**(15/2)*b**2*x**2 + 3*a**(13/2
)*b**3*x**3) + 6*a**5*b**2*x**2*sqrt(1 + b*x/a)/(3*a**(19/2) + 9*a**(17/2)*b*x +
 9*a**(15/2)*b**2*x**2 + 3*a**(13/2)*b**3*x**3) + 9*a**5*b**2*x**2*log(b*x/a)/(3
*a**(19/2) + 9*a**(17/2)*b*x + 9*a**(15/2)*b**2*x**2 + 3*a**(13/2)*b**3*x**3) -
18*a**5*b**2*x**2*log(sqrt(1 + b*x/a) + 1)/(3*a**(19/2) + 9*a**(17/2)*b*x + 9*a*
*(15/2)*b**2*x**2 + 3*a**(13/2)*b**3*x**3) + 3*a**4*b**3*x**3*log(b*x/a)/(3*a**(
19/2) + 9*a**(17/2)*b*x + 9*a**(15/2)*b**2*x**2 + 3*a**(13/2)*b**3*x**3) - 6*a**
4*b**3*x**3*log(sqrt(1 + b*x/a) + 1)/(3*a**(19/2) + 9*a**(17/2)*b*x + 9*a**(15/2
)*b**2*x**2 + 3*a**(13/2)*b**3*x**3)) - 2*B/(3*b*(a + b*x)**(3/2))

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GIAC/XCAS [A]  time = 0.210275, size = 82, normalized size = 1.22 \[ \frac{2 \, A \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{2}} - \frac{2 \,{\left (B a^{2} - 3 \,{\left (b x + a\right )} A b - A a b\right )}}{3 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{2} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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