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

Optimal. Leaf size=122 \[ \frac{a^2 (6 A b-5 a B) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{16 b^{7/2}}-\frac{a x \sqrt{a+b x^2} (6 A b-5 a B)}{16 b^3}+\frac{x^3 \sqrt{a+b x^2} (6 A b-5 a B)}{24 b^2}+\frac{B x^5 \sqrt{a+b x^2}}{6 b} \]

[Out]

-(a*(6*A*b - 5*a*B)*x*Sqrt[a + b*x^2])/(16*b^3) + ((6*A*b - 5*a*B)*x^3*Sqrt[a +
b*x^2])/(24*b^2) + (B*x^5*Sqrt[a + b*x^2])/(6*b) + (a^2*(6*A*b - 5*a*B)*ArcTanh[
(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(16*b^(7/2))

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Rubi [A]  time = 0.171011, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{a^2 (6 A b-5 a B) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{16 b^{7/2}}-\frac{a x \sqrt{a+b x^2} (6 A b-5 a B)}{16 b^3}+\frac{x^3 \sqrt{a+b x^2} (6 A b-5 a B)}{24 b^2}+\frac{B x^5 \sqrt{a+b x^2}}{6 b} \]

Antiderivative was successfully verified.

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

[Out]

-(a*(6*A*b - 5*a*B)*x*Sqrt[a + b*x^2])/(16*b^3) + ((6*A*b - 5*a*B)*x^3*Sqrt[a +
b*x^2])/(24*b^2) + (B*x^5*Sqrt[a + b*x^2])/(6*b) + (a^2*(6*A*b - 5*a*B)*ArcTanh[
(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(16*b^(7/2))

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Rubi in Sympy [A]  time = 18.1425, size = 114, normalized size = 0.93 \[ \frac{B x^{5} \sqrt{a + b x^{2}}}{6 b} + \frac{a^{2} \left (6 A b - 5 B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )}}{16 b^{\frac{7}{2}}} - \frac{a x \sqrt{a + b x^{2}} \left (6 A b - 5 B a\right )}{16 b^{3}} + \frac{x^{3} \sqrt{a + b x^{2}} \left (6 A b - 5 B a\right )}{24 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4*(B*x**2+A)/(b*x**2+a)**(1/2),x)

[Out]

B*x**5*sqrt(a + b*x**2)/(6*b) + a**2*(6*A*b - 5*B*a)*atanh(sqrt(b)*x/sqrt(a + b*
x**2))/(16*b**(7/2)) - a*x*sqrt(a + b*x**2)*(6*A*b - 5*B*a)/(16*b**3) + x**3*sqr
t(a + b*x**2)*(6*A*b - 5*B*a)/(24*b**2)

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Mathematica [A]  time = 0.114659, size = 105, normalized size = 0.86 \[ \sqrt{a+b x^2} \left (\frac{a x (5 a B-6 A b)}{16 b^3}+\frac{x^3 (6 A b-5 a B)}{24 b^2}+\frac{B x^5}{6 b}\right )-\frac{a^2 (5 a B-6 A b) \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )}{16 b^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[a + b*x^2]*((a*(-6*A*b + 5*a*B)*x)/(16*b^3) + ((6*A*b - 5*a*B)*x^3)/(24*b^2
) + (B*x^5)/(6*b)) - (a^2*(-6*A*b + 5*a*B)*Log[b*x + Sqrt[b]*Sqrt[a + b*x^2]])/(
16*b^(7/2))

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Maple [A]  time = 0.011, size = 143, normalized size = 1.2 \[{\frac{A{x}^{3}}{4\,b}\sqrt{b{x}^{2}+a}}-{\frac{3\,aAx}{8\,{b}^{2}}\sqrt{b{x}^{2}+a}}+{\frac{3\,A{a}^{2}}{8}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{5}{2}}}}+{\frac{{x}^{5}B}{6\,b}\sqrt{b{x}^{2}+a}}-{\frac{5\,Ba{x}^{3}}{24\,{b}^{2}}\sqrt{b{x}^{2}+a}}+{\frac{5\,Bx{a}^{2}}{16\,{b}^{3}}\sqrt{b{x}^{2}+a}}-{\frac{5\,B{a}^{3}}{16}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4*(B*x^2+A)/(b*x^2+a)^(1/2),x)

[Out]

1/4*A*x^3/b*(b*x^2+a)^(1/2)-3/8*A*a/b^2*x*(b*x^2+a)^(1/2)+3/8*A*a^2/b^(5/2)*ln(x
*b^(1/2)+(b*x^2+a)^(1/2))+1/6*B*x^5*(b*x^2+a)^(1/2)/b-5/24*B*a/b^2*x^3*(b*x^2+a)
^(1/2)+5/16*B*a^2/b^3*x*(b*x^2+a)^(1/2)-5/16*B*a^3/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/sqrt(b*x^2 + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.258052, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (8 \, B b^{2} x^{5} - 2 \,{\left (5 \, B a b - 6 \, A b^{2}\right )} x^{3} + 3 \,{\left (5 \, B a^{2} - 6 \, A a b\right )} x\right )} \sqrt{b x^{2} + a} \sqrt{b} - 3 \,{\left (5 \, B a^{3} - 6 \, A a^{2} b\right )} \log \left (-2 \, \sqrt{b x^{2} + a} b x -{\left (2 \, b x^{2} + a\right )} \sqrt{b}\right )}{96 \, b^{\frac{7}{2}}}, \frac{{\left (8 \, B b^{2} x^{5} - 2 \,{\left (5 \, B a b - 6 \, A b^{2}\right )} x^{3} + 3 \,{\left (5 \, B a^{2} - 6 \, A a b\right )} x\right )} \sqrt{b x^{2} + a} \sqrt{-b} - 3 \,{\left (5 \, B a^{3} - 6 \, A a^{2} b\right )} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right )}{48 \, \sqrt{-b} b^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/96*(2*(8*B*b^2*x^5 - 2*(5*B*a*b - 6*A*b^2)*x^3 + 3*(5*B*a^2 - 6*A*a*b)*x)*sqr
t(b*x^2 + a)*sqrt(b) - 3*(5*B*a^3 - 6*A*a^2*b)*log(-2*sqrt(b*x^2 + a)*b*x - (2*b
*x^2 + a)*sqrt(b)))/b^(7/2), 1/48*((8*B*b^2*x^5 - 2*(5*B*a*b - 6*A*b^2)*x^3 + 3*
(5*B*a^2 - 6*A*a*b)*x)*sqrt(b*x^2 + a)*sqrt(-b) - 3*(5*B*a^3 - 6*A*a^2*b)*arctan
(sqrt(-b)*x/sqrt(b*x^2 + a)))/(sqrt(-b)*b^3)]

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Sympy [A]  time = 33.1091, size = 235, normalized size = 1.93 \[ - \frac{3 A a^{\frac{3}{2}} x}{8 b^{2} \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{A \sqrt{a} x^{3}}{8 b \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{3 A a^{2} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{8 b^{\frac{5}{2}}} + \frac{A x^{5}}{4 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{5 B a^{\frac{5}{2}} x}{16 b^{3} \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{5 B a^{\frac{3}{2}} x^{3}}{48 b^{2} \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{B \sqrt{a} x^{5}}{24 b \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{5 B a^{3} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{16 b^{\frac{7}{2}}} + \frac{B x^{7}}{6 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4*(B*x**2+A)/(b*x**2+a)**(1/2),x)

[Out]

-3*A*a**(3/2)*x/(8*b**2*sqrt(1 + b*x**2/a)) - A*sqrt(a)*x**3/(8*b*sqrt(1 + b*x**
2/a)) + 3*A*a**2*asinh(sqrt(b)*x/sqrt(a))/(8*b**(5/2)) + A*x**5/(4*sqrt(a)*sqrt(
1 + b*x**2/a)) + 5*B*a**(5/2)*x/(16*b**3*sqrt(1 + b*x**2/a)) + 5*B*a**(3/2)*x**3
/(48*b**2*sqrt(1 + b*x**2/a)) - B*sqrt(a)*x**5/(24*b*sqrt(1 + b*x**2/a)) - 5*B*a
**3*asinh(sqrt(b)*x/sqrt(a))/(16*b**(7/2)) + B*x**7/(6*sqrt(a)*sqrt(1 + b*x**2/a
))

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GIAC/XCAS [A]  time = 0.234347, size = 144, normalized size = 1.18 \[ \frac{1}{48} \,{\left (2 \,{\left (\frac{4 \, B x^{2}}{b} - \frac{5 \, B a b^{3} - 6 \, A b^{4}}{b^{5}}\right )} x^{2} + \frac{3 \,{\left (5 \, B a^{2} b^{2} - 6 \, A a b^{3}\right )}}{b^{5}}\right )} \sqrt{b x^{2} + a} x + \frac{{\left (5 \, B a^{3} - 6 \, A a^{2} b\right )}{\rm ln}\left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{16 \, b^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/48*(2*(4*B*x^2/b - (5*B*a*b^3 - 6*A*b^4)/b^5)*x^2 + 3*(5*B*a^2*b^2 - 6*A*a*b^3
)/b^5)*sqrt(b*x^2 + a)*x + 1/16*(5*B*a^3 - 6*A*a^2*b)*ln(abs(-sqrt(b)*x + sqrt(b
*x^2 + a)))/b^(7/2)