3.36 \(\int \frac{x^3 (A+B x)}{\left (a+b x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=79 \[ -\frac{2 A+3 B x}{3 b^2 \sqrt{a+b x^2}}-\frac{x^2 (A+B x)}{3 b \left (a+b x^2\right )^{3/2}}+\frac{B \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{b^{5/2}} \]

[Out]

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

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Rubi [A]  time = 0.162611, antiderivative size = 79, 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 \[ -\frac{2 A+3 B x}{3 b^2 \sqrt{a+b x^2}}-\frac{x^2 (A+B x)}{3 b \left (a+b x^2\right )^{3/2}}+\frac{B \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{b^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[(x^3*(A + B*x))/(a + b*x^2)^(5/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 16.1205, size = 73, normalized size = 0.92 \[ \frac{B \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )}}{b^{\frac{5}{2}}} - \frac{x^{2} \left (2 A + 2 B x\right )}{6 b \left (a + b x^{2}\right )^{\frac{3}{2}}} - \frac{4 A + 6 B x}{6 b^{2} \sqrt{a + b x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*(B*x+A)/(b*x**2+a)**(5/2),x)

[Out]

B*atanh(sqrt(b)*x/sqrt(a + b*x**2))/b**(5/2) - x**2*(2*A + 2*B*x)/(6*b*(a + b*x*
*2)**(3/2)) - (4*A + 6*B*x)/(6*b**2*sqrt(a + b*x**2))

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Mathematica [A]  time = 0.152609, size = 73, normalized size = 0.92 \[ \frac{-\left (a+b x^2\right ) (3 A+4 B x)+a A+a B x}{3 b^2 \left (a+b x^2\right )^{3/2}}+\frac{B \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )}{b^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^3*(A + B*x))/(a + b*x^2)^(5/2),x]

[Out]

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

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Maple [A]  time = 0.012, size = 91, normalized size = 1.2 \[ -{\frac{A{x}^{2}}{b} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}-{\frac{2\,Aa}{3\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}-{\frac{{x}^{3}B}{3\,b} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}-{\frac{Bx}{{b}^{2}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}+{B\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(B*x+A)/(b*x^2+a)^(5/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.270001, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \,{\left (4 \, B b x^{3} + 3 \, A b x^{2} + 3 \, B a x + 2 \, A a\right )} \sqrt{b x^{2} + a} \sqrt{b} - 3 \,{\left (B b^{2} x^{4} + 2 \, B a b x^{2} + B a^{2}\right )} \log \left (-2 \, \sqrt{b x^{2} + a} b x -{\left (2 \, b x^{2} + a\right )} \sqrt{b}\right )}{6 \,{\left (b^{4} x^{4} + 2 \, a b^{3} x^{2} + a^{2} b^{2}\right )} \sqrt{b}}, -\frac{{\left (4 \, B b x^{3} + 3 \, A b x^{2} + 3 \, B a x + 2 \, A a\right )} \sqrt{b x^{2} + a} \sqrt{-b} - 3 \,{\left (B b^{2} x^{4} + 2 \, B a b x^{2} + B a^{2}\right )} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right )}{3 \,{\left (b^{4} x^{4} + 2 \, a b^{3} x^{2} + a^{2} b^{2}\right )} \sqrt{-b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*x^3/(b*x^2 + a)^(5/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 22.7279, size = 400, normalized size = 5.06 \[ A \left (\begin{cases} - \frac{2 a}{3 a b^{2} \sqrt{a + b x^{2}} + 3 b^{3} x^{2} \sqrt{a + b x^{2}}} - \frac{3 b x^{2}}{3 a b^{2} \sqrt{a + b x^{2}} + 3 b^{3} x^{2} \sqrt{a + b x^{2}}} & \text{for}\: b \neq 0 \\\frac{x^{4}}{4 a^{\frac{5}{2}}} & \text{otherwise} \end{cases}\right ) + B \left (\frac{3 a^{\frac{39}{2}} b^{11} \sqrt{1 + \frac{b x^{2}}{a}} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{3 a^{\frac{39}{2}} b^{\frac{27}{2}} \sqrt{1 + \frac{b x^{2}}{a}} + 3 a^{\frac{37}{2}} b^{\frac{29}{2}} x^{2} \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{3 a^{\frac{37}{2}} b^{12} x^{2} \sqrt{1 + \frac{b x^{2}}{a}} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{3 a^{\frac{39}{2}} b^{\frac{27}{2}} \sqrt{1 + \frac{b x^{2}}{a}} + 3 a^{\frac{37}{2}} b^{\frac{29}{2}} x^{2} \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{3 a^{19} b^{\frac{23}{2}} x}{3 a^{\frac{39}{2}} b^{\frac{27}{2}} \sqrt{1 + \frac{b x^{2}}{a}} + 3 a^{\frac{37}{2}} b^{\frac{29}{2}} x^{2} \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{4 a^{18} b^{\frac{25}{2}} x^{3}}{3 a^{\frac{39}{2}} b^{\frac{27}{2}} \sqrt{1 + \frac{b x^{2}}{a}} + 3 a^{\frac{37}{2}} b^{\frac{29}{2}} x^{2} \sqrt{1 + \frac{b x^{2}}{a}}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3*(B*x+A)/(b*x**2+a)**(5/2),x)

[Out]

A*Piecewise((-2*a/(3*a*b**2*sqrt(a + b*x**2) + 3*b**3*x**2*sqrt(a + b*x**2)) - 3
*b*x**2/(3*a*b**2*sqrt(a + b*x**2) + 3*b**3*x**2*sqrt(a + b*x**2)), Ne(b, 0)), (
x**4/(4*a**(5/2)), True)) + B*(3*a**(39/2)*b**11*sqrt(1 + b*x**2/a)*asinh(sqrt(b
)*x/sqrt(a))/(3*a**(39/2)*b**(27/2)*sqrt(1 + b*x**2/a) + 3*a**(37/2)*b**(29/2)*x
**2*sqrt(1 + b*x**2/a)) + 3*a**(37/2)*b**12*x**2*sqrt(1 + b*x**2/a)*asinh(sqrt(b
)*x/sqrt(a))/(3*a**(39/2)*b**(27/2)*sqrt(1 + b*x**2/a) + 3*a**(37/2)*b**(29/2)*x
**2*sqrt(1 + b*x**2/a)) - 3*a**19*b**(23/2)*x/(3*a**(39/2)*b**(27/2)*sqrt(1 + b*
x**2/a) + 3*a**(37/2)*b**(29/2)*x**2*sqrt(1 + b*x**2/a)) - 4*a**18*b**(25/2)*x**
3/(3*a**(39/2)*b**(27/2)*sqrt(1 + b*x**2/a) + 3*a**(37/2)*b**(29/2)*x**2*sqrt(1
+ b*x**2/a)))

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GIAC/XCAS [A]  time = 0.228951, size = 95, normalized size = 1.2 \[ -\frac{{\left ({\left (\frac{4 \, B x}{b} + \frac{3 \, A}{b}\right )} x + \frac{3 \, B a}{b^{2}}\right )} x + \frac{2 \, A a}{b^{2}}}{3 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}}} - \frac{B{\rm ln}\left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{b^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*x^3/(b*x^2 + a)^(5/2),x, algorithm="giac")

[Out]

-1/3*(((4*B*x/b + 3*A/b)*x + 3*B*a/b^2)*x + 2*A*a/b^2)/(b*x^2 + a)^(3/2) - B*ln(
abs(-sqrt(b)*x + sqrt(b*x^2 + a)))/b^(5/2)