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

Optimal. Leaf size=155 \[ -\frac{a^3 (8 A b-3 a B) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{128 b^{5/2}}+\frac{a^2 x \sqrt{a+b x^2} (8 A b-3 a B)}{128 b^2}+\frac{a x^3 \sqrt{a+b x^2} (8 A b-3 a B)}{64 b}+\frac{x^3 \left (a+b x^2\right )^{3/2} (8 A b-3 a B)}{48 b}+\frac{B x^3 \left (a+b x^2\right )^{5/2}}{8 b} \]

[Out]

(a^2*(8*A*b - 3*a*B)*x*Sqrt[a + b*x^2])/(128*b^2) + (a*(8*A*b - 3*a*B)*x^3*Sqrt[
a + b*x^2])/(64*b) + ((8*A*b - 3*a*B)*x^3*(a + b*x^2)^(3/2))/(48*b) + (B*x^3*(a
+ b*x^2)^(5/2))/(8*b) - (a^3*(8*A*b - 3*a*B)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]
])/(128*b^(5/2))

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Rubi [A]  time = 0.210596, 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 \[ -\frac{a^3 (8 A b-3 a B) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{128 b^{5/2}}+\frac{a^2 x \sqrt{a+b x^2} (8 A b-3 a B)}{128 b^2}+\frac{a x^3 \sqrt{a+b x^2} (8 A b-3 a B)}{64 b}+\frac{x^3 \left (a+b x^2\right )^{3/2} (8 A b-3 a B)}{48 b}+\frac{B x^3 \left (a+b x^2\right )^{5/2}}{8 b} \]

Antiderivative was successfully verified.

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

[Out]

(a^2*(8*A*b - 3*a*B)*x*Sqrt[a + b*x^2])/(128*b^2) + (a*(8*A*b - 3*a*B)*x^3*Sqrt[
a + b*x^2])/(64*b) + ((8*A*b - 3*a*B)*x^3*(a + b*x^2)^(3/2))/(48*b) + (B*x^3*(a
+ b*x^2)^(5/2))/(8*b) - (a^3*(8*A*b - 3*a*B)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]
])/(128*b^(5/2))

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Rubi in Sympy [A]  time = 24.4759, size = 143, normalized size = 0.92 \[ \frac{B x^{3} \left (a + b x^{2}\right )^{\frac{5}{2}}}{8 b} - \frac{a^{3} \left (8 A b - 3 B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )}}{128 b^{\frac{5}{2}}} + \frac{a^{2} x \sqrt{a + b x^{2}} \left (8 A b - 3 B a\right )}{128 b^{2}} + \frac{a x^{3} \sqrt{a + b x^{2}} \left (8 A b - 3 B a\right )}{64 b} + \frac{x^{3} \left (a + b x^{2}\right )^{\frac{3}{2}} \left (8 A b - 3 B a\right )}{48 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*x**3*(a + b*x**2)**(5/2)/(8*b) - a**3*(8*A*b - 3*B*a)*atanh(sqrt(b)*x/sqrt(a +
b*x**2))/(128*b**(5/2)) + a**2*x*sqrt(a + b*x**2)*(8*A*b - 3*B*a)/(128*b**2) +
a*x**3*sqrt(a + b*x**2)*(8*A*b - 3*B*a)/(64*b) + x**3*(a + b*x**2)**(3/2)*(8*A*b
- 3*B*a)/(48*b)

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Mathematica [A]  time = 0.125664, size = 122, normalized size = 0.79 \[ \frac{a^3 (3 a B-8 A b) \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )}{128 b^{5/2}}+\sqrt{a+b x^2} \left (-\frac{a^2 x (3 a B-8 A b)}{128 b^2}+\frac{1}{48} x^5 (9 a B+8 A b)+\frac{a x^3 (3 a B+56 A b)}{192 b}+\frac{1}{8} b B x^7\right ) \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[a + b*x^2]*(-(a^2*(-8*A*b + 3*a*B)*x)/(128*b^2) + (a*(56*A*b + 3*a*B)*x^3)/
(192*b) + ((8*A*b + 9*a*B)*x^5)/48 + (b*B*x^7)/8) + (a^3*(-8*A*b + 3*a*B)*Log[b*
x + Sqrt[b]*Sqrt[a + b*x^2]])/(128*b^(5/2))

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Maple [A]  time = 0.01, size = 177, normalized size = 1.1 \[{\frac{Ax}{6\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{aAx}{24\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{{a}^{2}Ax}{16\,b}\sqrt{b{x}^{2}+a}}-{\frac{A{a}^{3}}{16}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{3}{2}}}}+{\frac{B{x}^{3}}{8\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{Bxa}{16\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{Bx{a}^{2}}{64\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{3\,B{a}^{3}x}{128\,{b}^{2}}\sqrt{b{x}^{2}+a}}+{\frac{3\,B{a}^{4}}{128}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*A*x*(b*x^2+a)^(5/2)/b-1/24*A*a/b*x*(b*x^2+a)^(3/2)-1/16*A*a^2/b*x*(b*x^2+a)^
(1/2)-1/16*A*a^3/b^(3/2)*ln(x*b^(1/2)+(b*x^2+a)^(1/2))+1/8*B*x^3*(b*x^2+a)^(5/2)
/b-1/16*B*a/b^2*x*(b*x^2+a)^(5/2)+1/64*B*a^2/b^2*x*(b*x^2+a)^(3/2)+3/128*B*a^3/b
^2*x*(b*x^2+a)^(1/2)+3/128*B*a^4/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^2 + A)*(b*x^2 + a)^(3/2)*x^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.279023, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (48 \, B b^{3} x^{7} + 8 \,{\left (9 \, B a b^{2} + 8 \, A b^{3}\right )} x^{5} + 2 \,{\left (3 \, B a^{2} b + 56 \, A a b^{2}\right )} x^{3} - 3 \,{\left (3 \, B a^{3} - 8 \, A a^{2} b\right )} x\right )} \sqrt{b x^{2} + a} \sqrt{b} - 3 \,{\left (3 \, B a^{4} - 8 \, A a^{3} b\right )} \log \left (2 \, \sqrt{b x^{2} + a} b x -{\left (2 \, b x^{2} + a\right )} \sqrt{b}\right )}{768 \, b^{\frac{5}{2}}}, \frac{{\left (48 \, B b^{3} x^{7} + 8 \,{\left (9 \, B a b^{2} + 8 \, A b^{3}\right )} x^{5} + 2 \,{\left (3 \, B a^{2} b + 56 \, A a b^{2}\right )} x^{3} - 3 \,{\left (3 \, B a^{3} - 8 \, A a^{2} b\right )} x\right )} \sqrt{b x^{2} + a} \sqrt{-b} + 3 \,{\left (3 \, B a^{4} - 8 \, A a^{3} b\right )} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right )}{384 \, \sqrt{-b} b^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/768*(2*(48*B*b^3*x^7 + 8*(9*B*a*b^2 + 8*A*b^3)*x^5 + 2*(3*B*a^2*b + 56*A*a*b^
2)*x^3 - 3*(3*B*a^3 - 8*A*a^2*b)*x)*sqrt(b*x^2 + a)*sqrt(b) - 3*(3*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^(5/2), 1/384*((48*B*
b^3*x^7 + 8*(9*B*a*b^2 + 8*A*b^3)*x^5 + 2*(3*B*a^2*b + 56*A*a*b^2)*x^3 - 3*(3*B*
a^3 - 8*A*a^2*b)*x)*sqrt(b*x^2 + a)*sqrt(-b) + 3*(3*B*a^4 - 8*A*a^3*b)*arctan(sq
rt(-b)*x/sqrt(b*x^2 + a)))/(sqrt(-b)*b^2)]

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Sympy [A]  time = 73.7097, size = 287, normalized size = 1.85 \[ \frac{A a^{\frac{5}{2}} x}{16 b \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{17 A a^{\frac{3}{2}} x^{3}}{48 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{11 A \sqrt{a} b x^{5}}{24 \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{A a^{3} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{16 b^{\frac{3}{2}}} + \frac{A b^{2} x^{7}}{6 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{3 B a^{\frac{7}{2}} x}{128 b^{2} \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{B a^{\frac{5}{2}} x^{3}}{128 b \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{13 B a^{\frac{3}{2}} x^{5}}{64 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{5 B \sqrt{a} b x^{7}}{16 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{3 B a^{4} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{128 b^{\frac{5}{2}}} + \frac{B b^{2} x^{9}}{8 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*a**(5/2)*x/(16*b*sqrt(1 + b*x**2/a)) + 17*A*a**(3/2)*x**3/(48*sqrt(1 + b*x**2/
a)) + 11*A*sqrt(a)*b*x**5/(24*sqrt(1 + b*x**2/a)) - A*a**3*asinh(sqrt(b)*x/sqrt(
a))/(16*b**(3/2)) + A*b**2*x**7/(6*sqrt(a)*sqrt(1 + b*x**2/a)) - 3*B*a**(7/2)*x/
(128*b**2*sqrt(1 + b*x**2/a)) - B*a**(5/2)*x**3/(128*b*sqrt(1 + b*x**2/a)) + 13*
B*a**(3/2)*x**5/(64*sqrt(1 + b*x**2/a)) + 5*B*sqrt(a)*b*x**7/(16*sqrt(1 + b*x**2
/a)) + 3*B*a**4*asinh(sqrt(b)*x/sqrt(a))/(128*b**(5/2)) + B*b**2*x**9/(8*sqrt(a)
*sqrt(1 + b*x**2/a))

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GIAC/XCAS [A]  time = 0.23891, size = 180, normalized size = 1.16 \[ \frac{1}{384} \,{\left (2 \,{\left (4 \,{\left (6 \, B b x^{2} + \frac{9 \, B a b^{6} + 8 \, A b^{7}}{b^{6}}\right )} x^{2} + \frac{3 \, B a^{2} b^{5} + 56 \, A a b^{6}}{b^{6}}\right )} x^{2} - \frac{3 \,{\left (3 \, B a^{3} b^{4} - 8 \, A a^{2} b^{5}\right )}}{b^{6}}\right )} \sqrt{b x^{2} + a} x - \frac{{\left (3 \, B a^{4} - 8 \, A a^{3} b\right )}{\rm ln}\left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{128 \, b^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/384*(2*(4*(6*B*b*x^2 + (9*B*a*b^6 + 8*A*b^7)/b^6)*x^2 + (3*B*a^2*b^5 + 56*A*a*
b^6)/b^6)*x^2 - 3*(3*B*a^3*b^4 - 8*A*a^2*b^5)/b^6)*sqrt(b*x^2 + a)*x - 1/128*(3*
B*a^4 - 8*A*a^3*b)*ln(abs(-sqrt(b)*x + sqrt(b*x^2 + a)))/b^(5/2)