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

Optimal. Leaf size=66 \[ \frac{A \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{b^{3/2}}-\frac{x (A+B x)}{b \sqrt{a+b x^2}}+\frac{2 B \sqrt{a+b x^2}}{b^2} \]

[Out]

-((x*(A + B*x))/(b*Sqrt[a + b*x^2])) + (2*B*Sqrt[a + b*x^2])/b^2 + (A*ArcTanh[(S
qrt[b]*x)/Sqrt[a + b*x^2]])/b^(3/2)

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Rubi [A]  time = 0.13668, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{A \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{b^{3/2}}-\frac{x (A+B x)}{b \sqrt{a+b x^2}}+\frac{2 B \sqrt{a+b x^2}}{b^2} \]

Antiderivative was successfully verified.

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

[Out]

-((x*(A + B*x))/(b*Sqrt[a + b*x^2])) + (2*B*Sqrt[a + b*x^2])/b^2 + (A*ArcTanh[(S
qrt[b]*x)/Sqrt[a + b*x^2]])/b^(3/2)

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Rubi in Sympy [A]  time = 14.7082, size = 65, normalized size = 0.98 \[ \frac{A \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )}}{b^{\frac{3}{2}}} + \frac{2 B \sqrt{a + b x^{2}}}{b^{2}} - \frac{x \left (2 A + 2 B x\right )}{2 b \sqrt{a + b x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.105024, size = 60, normalized size = 0.91 \[ \frac{A \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )}{b^{3/2}}+\frac{2 a B+b x (B x-A)}{b^2 \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.009, size = 72, normalized size = 1.1 \[ -{\frac{Ax}{b}{\frac{1}{\sqrt{b{x}^{2}+a}}}}+{A\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{3}{2}}}}+{\frac{B{x}^{2}}{b}{\frac{1}{\sqrt{b{x}^{2}+a}}}}+2\,{\frac{Ba}{{b}^{2}\sqrt{b{x}^{2}+a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.26142, size = 1, normalized size = 0.02 \[ \left [\frac{2 \,{\left (B b x^{2} - A b x + 2 \, B a\right )} \sqrt{b x^{2} + a} \sqrt{b} +{\left (A b^{2} x^{2} + A a b\right )} \log \left (-2 \, \sqrt{b x^{2} + a} b x -{\left (2 \, b x^{2} + a\right )} \sqrt{b}\right )}{2 \,{\left (b^{3} x^{2} + a b^{2}\right )} \sqrt{b}}, \frac{{\left (B b x^{2} - A b x + 2 \, B a\right )} \sqrt{b x^{2} + a} \sqrt{-b} +{\left (A b^{2} x^{2} + A a b\right )} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right )}{{\left (b^{3} x^{2} + a b^{2}\right )} \sqrt{-b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 6.06385, size = 83, normalized size = 1.26 \[ A \left (\frac{\operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{b^{\frac{3}{2}}} - \frac{x}{\sqrt{a} b \sqrt{1 + \frac{b x^{2}}{a}}}\right ) + B \left (\begin{cases} \frac{2 a}{b^{2} \sqrt{a + b x^{2}}} + \frac{x^{2}}{b \sqrt{a + b x^{2}}} & \text{for}\: b \neq 0 \\\frac{x^{4}}{4 a^{\frac{3}{2}}} & \text{otherwise} \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*(asinh(sqrt(b)*x/sqrt(a))/b**(3/2) - x/(sqrt(a)*b*sqrt(1 + b*x**2/a))) + B*Pie
cewise((2*a/(b**2*sqrt(a + b*x**2)) + x**2/(b*sqrt(a + b*x**2)), Ne(b, 0)), (x**
4/(4*a**(3/2)), True))

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GIAC/XCAS [A]  time = 0.220276, size = 78, normalized size = 1.18 \[ \frac{{\left (\frac{B x}{b} - \frac{A}{b}\right )} x + \frac{2 \, B a}{b^{2}}}{\sqrt{b x^{2} + a}} - \frac{A{\rm ln}\left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{b^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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