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

Optimal. Leaf size=95 \[ \frac{3 A b \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{2 a^{5/2}}-\frac{3 A \sqrt{a+b x^2}}{2 a^2 x^2}-\frac{2 B \sqrt{a+b x^2}}{a^2 x}+\frac{A+B x}{a x^2 \sqrt{a+b x^2}} \]

[Out]

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

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Rubi [A]  time = 0.303504, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3 \[ \frac{3 A b \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{2 a^{5/2}}-\frac{3 A \sqrt{a+b x^2}}{2 a^2 x^2}-\frac{2 B \sqrt{a+b x^2}}{a^2 x}+\frac{A+B x}{a x^2 \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 31.5056, size = 88, normalized size = 0.93 \[ - \frac{3 A \sqrt{a + b x^{2}}}{2 a^{2} x^{2}} + \frac{3 A b \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{2}}}{\sqrt{a}} \right )}}{2 a^{\frac{5}{2}}} - \frac{2 B \sqrt{a + b x^{2}}}{a^{2} x} + \frac{A + B x}{a x^{2} \sqrt{a + b x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.266282, size = 83, normalized size = 0.87 \[ \frac{-\frac{\sqrt{a} \left (a (A+2 B x)+b x^2 (3 A+4 B x)\right )}{x^2 \sqrt{a+b x^2}}+3 A b \log \left (\sqrt{a} \sqrt{a+b x^2}+a\right )-3 A b \log (x)}{2 a^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-((Sqrt[a]*(a*(A + 2*B*x) + b*x^2*(3*A + 4*B*x)))/(x^2*Sqrt[a + b*x^2])) - 3*A*
b*Log[x] + 3*A*b*Log[a + Sqrt[a]*Sqrt[a + b*x^2]])/(2*a^(5/2))

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Maple [A]  time = 0.011, size = 101, normalized size = 1.1 \[ -{\frac{A}{2\,a{x}^{2}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}-{\frac{3\,Ab}{2\,{a}^{2}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}+{\frac{3\,Ab}{2}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{5}{2}}}}-{\frac{B}{ax}{\frac{1}{\sqrt{b{x}^{2}+a}}}}-2\,{\frac{bBx}{{a}^{2}\sqrt{b{x}^{2}+a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.268861, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \,{\left (4 \, B b x^{3} + 3 \, A b x^{2} + 2 \, B a x + A a\right )} \sqrt{b x^{2} + a} \sqrt{a} - 3 \,{\left (A b^{2} x^{4} + A a b x^{2}\right )} \log \left (-\frac{{\left (b x^{2} + 2 \, a\right )} \sqrt{a} + 2 \, \sqrt{b x^{2} + a} a}{x^{2}}\right )}{4 \,{\left (a^{2} b x^{4} + a^{3} x^{2}\right )} \sqrt{a}}, -\frac{{\left (4 \, B b x^{3} + 3 \, A b x^{2} + 2 \, B a x + A a\right )} \sqrt{b x^{2} + a} \sqrt{-a} - 3 \,{\left (A b^{2} x^{4} + A a b x^{2}\right )} \arctan \left (\frac{\sqrt{-a}}{\sqrt{b x^{2} + a}}\right )}{2 \,{\left (a^{2} b x^{4} + a^{3} x^{2}\right )} \sqrt{-a}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 13.0872, size = 124, normalized size = 1.31 \[ A \left (- \frac{1}{2 a \sqrt{b} x^{3} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{3 \sqrt{b}}{2 a^{2} x \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{3 b \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{2 a^{\frac{5}{2}}}\right ) + B \left (- \frac{1}{a \sqrt{b} x^{2} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{2 \sqrt{b}}{a^{2} \sqrt{\frac{a}{b x^{2}} + 1}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)/x**3/(b*x**2+a)**(3/2),x)

[Out]

A*(-1/(2*a*sqrt(b)*x**3*sqrt(a/(b*x**2) + 1)) - 3*sqrt(b)/(2*a**2*x*sqrt(a/(b*x*
*2) + 1)) + 3*b*asinh(sqrt(a)/(sqrt(b)*x))/(2*a**(5/2))) + B*(-1/(a*sqrt(b)*x**2
*sqrt(a/(b*x**2) + 1)) - 2*sqrt(b)/(a**2*sqrt(a/(b*x**2) + 1)))

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GIAC/XCAS [A]  time = 0.220413, size = 231, normalized size = 2.43 \[ -\frac{\frac{B b x}{a^{2}} + \frac{A b}{a^{2}}}{\sqrt{b x^{2} + a}} - \frac{3 \, A b \arctan \left (-\frac{\sqrt{b} x - \sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{2}} + \frac{{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{3} A b + 2 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} B a \sqrt{b} +{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )} A a b - 2 \, B a^{2} \sqrt{b}}{{\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} - a\right )}^{2} a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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