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

Optimal. Leaf size=155 \[ \frac{a^3 (8 A b-5 a B) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{128 b^{7/2}}-\frac{a^2 x \sqrt{a+b x^2} (8 A b-5 a B)}{128 b^3}+\frac{a x^3 \sqrt{a+b x^2} (8 A b-5 a B)}{192 b^2}+\frac{x^5 \sqrt{a+b x^2} (8 A b-5 a B)}{48 b}+\frac{B x^5 \left (a+b x^2\right )^{3/2}}{8 b} \]

[Out]

-(a^2*(8*A*b - 5*a*B)*x*Sqrt[a + b*x^2])/(128*b^3) + (a*(8*A*b - 5*a*B)*x^3*Sqrt
[a + b*x^2])/(192*b^2) + ((8*A*b - 5*a*B)*x^5*Sqrt[a + b*x^2])/(48*b) + (B*x^5*(
a + b*x^2)^(3/2))/(8*b) + (a^3*(8*A*b - 5*a*B)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^
2]])/(128*b^(7/2))

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Rubi [A]  time = 0.223511, antiderivative size = 155, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ \frac{a^3 (8 A b-5 a B) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{128 b^{7/2}}-\frac{a^2 x \sqrt{a+b x^2} (8 A b-5 a B)}{128 b^3}+\frac{a x^3 \sqrt{a+b x^2} (8 A b-5 a B)}{192 b^2}+\frac{x^5 \sqrt{a+b x^2} (8 A b-5 a B)}{48 b}+\frac{B x^5 \left (a+b x^2\right )^{3/2}}{8 b} \]

Antiderivative was successfully verified.

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

[Out]

-(a^2*(8*A*b - 5*a*B)*x*Sqrt[a + b*x^2])/(128*b^3) + (a*(8*A*b - 5*a*B)*x^3*Sqrt
[a + b*x^2])/(192*b^2) + ((8*A*b - 5*a*B)*x^5*Sqrt[a + b*x^2])/(48*b) + (B*x^5*(
a + b*x^2)^(3/2))/(8*b) + (a^3*(8*A*b - 5*a*B)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^
2]])/(128*b^(7/2))

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Rubi in Sympy [A]  time = 22.8884, size = 144, normalized size = 0.93 \[ \frac{B x^{5} \left (a + b x^{2}\right )^{\frac{3}{2}}}{8 b} + \frac{a^{3} \left (8 A b - 5 B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )}}{128 b^{\frac{7}{2}}} - \frac{a^{2} x \sqrt{a + b x^{2}} \left (8 A b - 5 B a\right )}{128 b^{3}} + \frac{a x^{3} \sqrt{a + b x^{2}} \left (8 A b - 5 B a\right )}{192 b^{2}} + \frac{x^{5} \sqrt{a + b x^{2}} \left (8 A b - 5 B a\right )}{48 b} \]

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*(a + b*x**2)**(3/2)/(8*b) + a**3*(8*A*b - 5*B*a)*atanh(sqrt(b)*x/sqrt(a +
b*x**2))/(128*b**(7/2)) - a**2*x*sqrt(a + b*x**2)*(8*A*b - 5*B*a)/(128*b**3) +
a*x**3*sqrt(a + b*x**2)*(8*A*b - 5*B*a)/(192*b**2) + x**5*sqrt(a + b*x**2)*(8*A*
b - 5*B*a)/(48*b)

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Mathematica [A]  time = 0.131558, size = 123, normalized size = 0.79 \[ \sqrt{a+b x^2} \left (\frac{a^2 x (5 a B-8 A b)}{128 b^3}-\frac{a x^3 (5 a B-8 A b)}{192 b^2}+\frac{x^5 (a B+8 A b)}{48 b}+\frac{B x^7}{8}\right )-\frac{a^3 (5 a B-8 A b) \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )}{128 b^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[a + b*x^2]*((a^2*(-8*A*b + 5*a*B)*x)/(128*b^3) - (a*(-8*A*b + 5*a*B)*x^3)/(
192*b^2) + ((8*A*b + a*B)*x^5)/(48*b) + (B*x^7)/8) - (a^3*(-8*A*b + 5*a*B)*Log[b
*x + Sqrt[b]*Sqrt[a + b*x^2]])/(128*b^(7/2))

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Maple [A]  time = 0.011, size = 181, normalized size = 1.2 \[{\frac{A{x}^{3}}{6\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{aAx}{8\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{{a}^{2}Ax}{16\,{b}^{2}}\sqrt{b{x}^{2}+a}}+{\frac{A{a}^{3}}{16}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{5}{2}}}}+{\frac{B{x}^{5}}{8\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{5\,Ba{x}^{3}}{48\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{5\,Bx{a}^{2}}{64\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{5\,B{a}^{3}x}{128\,{b}^{3}}\sqrt{b{x}^{2}+a}}-{\frac{5\,B{a}^{4}}{128}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{7}{2}}}} \]

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]

1/6*A*x^3*(b*x^2+a)^(3/2)/b-1/8*A*a/b^2*x*(b*x^2+a)^(3/2)+1/16*A*a^2/b^2*x*(b*x^
2+a)^(1/2)+1/16*A*a^3/b^(5/2)*ln(x*b^(1/2)+(b*x^2+a)^(1/2))+1/8*B*x^5*(b*x^2+a)^
(3/2)/b-5/48*B*a/b^2*x^3*(b*x^2+a)^(3/2)+5/64*B*a^2/b^3*x*(b*x^2+a)^(3/2)-5/128*
B*a^3/b^3*x*(b*x^2+a)^(1/2)-5/128*B*a^4/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)*sqrt(b*x^2 + a)*x^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.305985, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (48 \, B b^{3} x^{7} + 8 \,{\left (B a b^{2} + 8 \, A b^{3}\right )} x^{5} - 2 \,{\left (5 \, B a^{2} b - 8 \, A a b^{2}\right )} x^{3} + 3 \,{\left (5 \, B a^{3} - 8 \, A a^{2} b\right )} x\right )} \sqrt{b x^{2} + a} \sqrt{b} - 3 \,{\left (5 \, B a^{4} - 8 \, A a^{3} b\right )} \log \left (-2 \, \sqrt{b x^{2} + a} b x -{\left (2 \, b x^{2} + a\right )} \sqrt{b}\right )}{768 \, b^{\frac{7}{2}}}, \frac{{\left (48 \, B b^{3} x^{7} + 8 \,{\left (B a b^{2} + 8 \, A b^{3}\right )} x^{5} - 2 \,{\left (5 \, B a^{2} b - 8 \, A a b^{2}\right )} x^{3} + 3 \,{\left (5 \, B a^{3} - 8 \, A a^{2} b\right )} x\right )} \sqrt{b x^{2} + a} \sqrt{-b} - 3 \,{\left (5 \, B a^{4} - 8 \, A a^{3} b\right )} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right )}{384 \, \sqrt{-b} b^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/768*(2*(48*B*b^3*x^7 + 8*(B*a*b^2 + 8*A*b^3)*x^5 - 2*(5*B*a^2*b - 8*A*a*b^2)*
x^3 + 3*(5*B*a^3 - 8*A*a^2*b)*x)*sqrt(b*x^2 + a)*sqrt(b) - 3*(5*B*a^4 - 8*A*a^3*
b)*log(-2*sqrt(b*x^2 + a)*b*x - (2*b*x^2 + a)*sqrt(b)))/b^(7/2), 1/384*((48*B*b^
3*x^7 + 8*(B*a*b^2 + 8*A*b^3)*x^5 - 2*(5*B*a^2*b - 8*A*a*b^2)*x^3 + 3*(5*B*a^3 -
8*A*a^2*b)*x)*sqrt(b*x^2 + a)*sqrt(-b) - 3*(5*B*a^4 - 8*A*a^3*b)*arctan(sqrt(-b
)*x/sqrt(b*x^2 + a)))/(sqrt(-b)*b^3)]

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Sympy [A]  time = 46.6973, size = 286, normalized size = 1.85 \[ - \frac{A a^{\frac{5}{2}} x}{16 b^{2} \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{A a^{\frac{3}{2}} x^{3}}{48 b \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{5 A \sqrt{a} x^{5}}{24 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{A a^{3} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{16 b^{\frac{5}{2}}} + \frac{A b x^{7}}{6 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{5 B a^{\frac{7}{2}} x}{128 b^{3} \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{5 B a^{\frac{5}{2}} x^{3}}{384 b^{2} \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{B a^{\frac{3}{2}} x^{5}}{192 b \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{7 B \sqrt{a} x^{7}}{48 \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{5 B a^{4} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{128 b^{\frac{7}{2}}} + \frac{B b x^{9}}{8 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} \]

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]

-A*a**(5/2)*x/(16*b**2*sqrt(1 + b*x**2/a)) - A*a**(3/2)*x**3/(48*b*sqrt(1 + b*x*
*2/a)) + 5*A*sqrt(a)*x**5/(24*sqrt(1 + b*x**2/a)) + A*a**3*asinh(sqrt(b)*x/sqrt(
a))/(16*b**(5/2)) + A*b*x**7/(6*sqrt(a)*sqrt(1 + b*x**2/a)) + 5*B*a**(7/2)*x/(12
8*b**3*sqrt(1 + b*x**2/a)) + 5*B*a**(5/2)*x**3/(384*b**2*sqrt(1 + b*x**2/a)) - B
*a**(3/2)*x**5/(192*b*sqrt(1 + b*x**2/a)) + 7*B*sqrt(a)*x**7/(48*sqrt(1 + b*x**2
/a)) - 5*B*a**4*asinh(sqrt(b)*x/sqrt(a))/(128*b**(7/2)) + B*b*x**9/(8*sqrt(a)*sq
rt(1 + b*x**2/a))

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GIAC/XCAS [A]  time = 0.236525, size = 178, normalized size = 1.15 \[ \frac{1}{384} \,{\left (2 \,{\left (4 \,{\left (6 \, B x^{2} + \frac{B a b^{5} + 8 \, A b^{6}}{b^{6}}\right )} x^{2} - \frac{5 \, B a^{2} b^{4} - 8 \, A a b^{5}}{b^{6}}\right )} x^{2} + \frac{3 \,{\left (5 \, B a^{3} b^{3} - 8 \, A a^{2} b^{4}\right )}}{b^{6}}\right )} \sqrt{b x^{2} + a} x + \frac{{\left (5 \, B a^{4} - 8 \, A a^{3} b\right )}{\rm ln}\left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{128 \, b^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/384*(2*(4*(6*B*x^2 + (B*a*b^5 + 8*A*b^6)/b^6)*x^2 - (5*B*a^2*b^4 - 8*A*a*b^5)/
b^6)*x^2 + 3*(5*B*a^3*b^3 - 8*A*a^2*b^4)/b^6)*sqrt(b*x^2 + a)*x + 1/128*(5*B*a^4
- 8*A*a^3*b)*ln(abs(-sqrt(b)*x + sqrt(b*x^2 + a)))/b^(7/2)