3.571 \(\int \frac{x^4 \left (A+B x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=119 \[ -\frac{3 a (4 A b-5 a B) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{8 b^{7/2}}+\frac{3 x \sqrt{a+b x^2} (4 A b-5 a B)}{8 b^3}-\frac{x^3 (4 A b-5 a B)}{4 b^2 \sqrt{a+b x^2}}+\frac{B x^5}{4 b \sqrt{a+b x^2}} \]

[Out]

-((4*A*b - 5*a*B)*x^3)/(4*b^2*Sqrt[a + b*x^2]) + (B*x^5)/(4*b*Sqrt[a + b*x^2]) +
(3*(4*A*b - 5*a*B)*x*Sqrt[a + b*x^2])/(8*b^3) - (3*a*(4*A*b - 5*a*B)*ArcTanh[(S
qrt[b]*x)/Sqrt[a + b*x^2]])/(8*b^(7/2))

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Rubi [A]  time = 0.165322, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ -\frac{3 a (4 A b-5 a B) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{8 b^{7/2}}+\frac{3 x \sqrt{a+b x^2} (4 A b-5 a B)}{8 b^3}-\frac{x^3 (4 A b-5 a B)}{4 b^2 \sqrt{a+b x^2}}+\frac{B x^5}{4 b \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(x^4*(A + B*x^2))/(a + b*x^2)^(3/2),x]

[Out]

-((4*A*b - 5*a*B)*x^3)/(4*b^2*Sqrt[a + b*x^2]) + (B*x^5)/(4*b*Sqrt[a + b*x^2]) +
(3*(4*A*b - 5*a*B)*x*Sqrt[a + b*x^2])/(8*b^3) - (3*a*(4*A*b - 5*a*B)*ArcTanh[(S
qrt[b]*x)/Sqrt[a + b*x^2]])/(8*b^(7/2))

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Rubi in Sympy [A]  time = 18.0281, size = 114, normalized size = 0.96 \[ \frac{B x^{5}}{4 b \sqrt{a + b x^{2}}} - \frac{3 a \left (4 A b - 5 B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )}}{8 b^{\frac{7}{2}}} - \frac{x^{3} \left (4 A b - 5 B a\right )}{4 b^{2} \sqrt{a + b x^{2}}} + \frac{3 x \sqrt{a + b x^{2}} \left (4 A b - 5 B a\right )}{8 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4*(B*x**2+A)/(b*x**2+a)**(3/2),x)

[Out]

B*x**5/(4*b*sqrt(a + b*x**2)) - 3*a*(4*A*b - 5*B*a)*atanh(sqrt(b)*x/sqrt(a + b*x
**2))/(8*b**(7/2)) - x**3*(4*A*b - 5*B*a)/(4*b**2*sqrt(a + b*x**2)) + 3*x*sqrt(a
+ b*x**2)*(4*A*b - 5*B*a)/(8*b**3)

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Mathematica [A]  time = 0.16007, size = 99, normalized size = 0.83 \[ \frac{-15 a^2 B x+a b x \left (12 A-5 B x^2\right )+2 b^2 x^3 \left (2 A+B x^2\right )}{8 b^3 \sqrt{a+b x^2}}+\frac{3 a (5 a B-4 A b) \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )}{8 b^{7/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^4*(A + B*x^2))/(a + b*x^2)^(3/2),x]

[Out]

(-15*a^2*B*x + a*b*x*(12*A - 5*B*x^2) + 2*b^2*x^3*(2*A + B*x^2))/(8*b^3*Sqrt[a +
b*x^2]) + (3*a*(-4*A*b + 5*a*B)*Log[b*x + Sqrt[b]*Sqrt[a + b*x^2]])/(8*b^(7/2))

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Maple [A]  time = 0.01, size = 141, normalized size = 1.2 \[{\frac{A{x}^{3}}{2\,b}{\frac{1}{\sqrt{b{x}^{2}+a}}}}+{\frac{3\,aAx}{2\,{b}^{2}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}-{\frac{3\,Aa}{2}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{5}{2}}}}+{\frac{{x}^{5}B}{4\,b}{\frac{1}{\sqrt{b{x}^{2}+a}}}}-{\frac{5\,Ba{x}^{3}}{8\,{b}^{2}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}-{\frac{15\,Bx{a}^{2}}{8\,{b}^{3}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}+{\frac{15\,{a}^{2}B}{8}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4*(B*x^2+A)/(b*x^2+a)^(3/2),x)

[Out]

1/2*A*x^3/b/(b*x^2+a)^(1/2)+3/2*A*a/b^2*x/(b*x^2+a)^(1/2)-3/2*A*a/b^(5/2)*ln(x*b
^(1/2)+(b*x^2+a)^(1/2))+1/4*B*x^5/b/(b*x^2+a)^(1/2)-5/8*B*a/b^2*x^3/(b*x^2+a)^(1
/2)-15/8*B*a^2/b^3*x/(b*x^2+a)^(1/2)+15/8*B*a^2/b^(7/2)*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^2 + A)*x^4/(b*x^2 + a)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.232635, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (2 \, B b^{2} x^{5} -{\left (5 \, B a b - 4 \, A b^{2}\right )} x^{3} - 3 \,{\left (5 \, B a^{2} - 4 \, A a b\right )} x\right )} \sqrt{b x^{2} + a} \sqrt{b} - 3 \,{\left (5 \, B a^{3} - 4 \, A a^{2} b +{\left (5 \, B a^{2} b - 4 \, A a b^{2}\right )} x^{2}\right )} \log \left (2 \, \sqrt{b x^{2} + a} b x -{\left (2 \, b x^{2} + a\right )} \sqrt{b}\right )}{16 \,{\left (b^{4} x^{2} + a b^{3}\right )} \sqrt{b}}, \frac{{\left (2 \, B b^{2} x^{5} -{\left (5 \, B a b - 4 \, A b^{2}\right )} x^{3} - 3 \,{\left (5 \, B a^{2} - 4 \, A a b\right )} x\right )} \sqrt{b x^{2} + a} \sqrt{-b} + 3 \,{\left (5 \, B a^{3} - 4 \, A a^{2} b +{\left (5 \, B a^{2} b - 4 \, A a b^{2}\right )} x^{2}\right )} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right )}{8 \,{\left (b^{4} x^{2} + a b^{3}\right )} \sqrt{-b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*x^4/(b*x^2 + a)^(3/2),x, algorithm="fricas")

[Out]

[1/16*(2*(2*B*b^2*x^5 - (5*B*a*b - 4*A*b^2)*x^3 - 3*(5*B*a^2 - 4*A*a*b)*x)*sqrt(
b*x^2 + a)*sqrt(b) - 3*(5*B*a^3 - 4*A*a^2*b + (5*B*a^2*b - 4*A*a*b^2)*x^2)*log(2
*sqrt(b*x^2 + a)*b*x - (2*b*x^2 + a)*sqrt(b)))/((b^4*x^2 + a*b^3)*sqrt(b)), 1/8*
((2*B*b^2*x^5 - (5*B*a*b - 4*A*b^2)*x^3 - 3*(5*B*a^2 - 4*A*a*b)*x)*sqrt(b*x^2 +
a)*sqrt(-b) + 3*(5*B*a^3 - 4*A*a^2*b + (5*B*a^2*b - 4*A*a*b^2)*x^2)*arctan(sqrt(
-b)*x/sqrt(b*x^2 + a)))/((b^4*x^2 + a*b^3)*sqrt(-b))]

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Sympy [A]  time = 36.9472, size = 177, normalized size = 1.49 \[ A \left (\frac{3 \sqrt{a} x}{2 b^{2} \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{3 a \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2 b^{\frac{5}{2}}} + \frac{x^{3}}{2 \sqrt{a} b \sqrt{1 + \frac{b x^{2}}{a}}}\right ) + B \left (- \frac{15 a^{\frac{3}{2}} x}{8 b^{3} \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{5 \sqrt{a} x^{3}}{8 b^{2} \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{15 a^{2} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{8 b^{\frac{7}{2}}} + \frac{x^{5}}{4 \sqrt{a} b \sqrt{1 + \frac{b x^{2}}{a}}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4*(B*x**2+A)/(b*x**2+a)**(3/2),x)

[Out]

A*(3*sqrt(a)*x/(2*b**2*sqrt(1 + b*x**2/a)) - 3*a*asinh(sqrt(b)*x/sqrt(a))/(2*b**
(5/2)) + x**3/(2*sqrt(a)*b*sqrt(1 + b*x**2/a))) + B*(-15*a**(3/2)*x/(8*b**3*sqrt
(1 + b*x**2/a)) - 5*sqrt(a)*x**3/(8*b**2*sqrt(1 + b*x**2/a)) + 15*a**2*asinh(sqr
t(b)*x/sqrt(a))/(8*b**(7/2)) + x**5/(4*sqrt(a)*b*sqrt(1 + b*x**2/a)))

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GIAC/XCAS [A]  time = 0.230698, size = 140, normalized size = 1.18 \[ \frac{{\left ({\left (\frac{2 \, B x^{2}}{b} - \frac{5 \, B a b^{3} - 4 \, A b^{4}}{b^{5}}\right )} x^{2} - \frac{3 \,{\left (5 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )}}{b^{5}}\right )} x}{8 \, \sqrt{b x^{2} + a}} - \frac{3 \,{\left (5 \, B a^{2} - 4 \, A a b\right )}{\rm ln}\left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{8 \, b^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*x^4/(b*x^2 + a)^(3/2),x, algorithm="giac")

[Out]

1/8*((2*B*x^2/b - (5*B*a*b^3 - 4*A*b^4)/b^5)*x^2 - 3*(5*B*a^2*b^2 - 4*A*a*b^3)/b
^5)*x/sqrt(b*x^2 + a) - 3/8*(5*B*a^2 - 4*A*a*b)*ln(abs(-sqrt(b)*x + sqrt(b*x^2 +
a)))/b^(7/2)