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

Optimal. Leaf size=118 \[ \frac{a^2 (6 A b-a B) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{16 b^{3/2}}+\frac{x \left (a+b x^2\right )^{3/2} (6 A b-a B)}{24 b}+\frac{a x \sqrt{a+b x^2} (6 A b-a B)}{16 b}+\frac{B x \left (a+b x^2\right )^{5/2}}{6 b} \]

[Out]

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

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Rubi [A]  time = 0.110344, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ \frac{a^2 (6 A b-a B) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{16 b^{3/2}}+\frac{x \left (a+b x^2\right )^{3/2} (6 A b-a B)}{24 b}+\frac{a x \sqrt{a+b x^2} (6 A b-a B)}{16 b}+\frac{B x \left (a+b x^2\right )^{5/2}}{6 b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 12.0059, size = 102, normalized size = 0.86 \[ \frac{B x \left (a + b x^{2}\right )^{\frac{5}{2}}}{6 b} + \frac{a^{2} \left (6 A b - B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )}}{16 b^{\frac{3}{2}}} + \frac{a x \sqrt{a + b x^{2}} \left (6 A b - B a\right )}{16 b} + \frac{x \left (a + b x^{2}\right )^{\frac{3}{2}} \left (6 A b - B a\right )}{24 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.124259, size = 98, normalized size = 0.83 \[ \sqrt{a+b x^2} \left (\frac{1}{24} x^3 (7 a B+6 A b)+\frac{a x (a B+10 A b)}{16 b}+\frac{1}{6} b B x^5\right )-\frac{a^2 (a B-6 A b) \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )}{16 b^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[a + b*x^2]*((a*(10*A*b + a*B)*x)/(16*b) + ((6*A*b + 7*a*B)*x^3)/24 + (b*B*x
^5)/6) - (a^2*(-6*A*b + a*B)*Log[b*x + Sqrt[b]*Sqrt[a + b*x^2]])/(16*b^(3/2))

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Maple [A]  time = 0.008, size = 131, normalized size = 1.1 \[{\frac{Ax}{4} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{3\,aAx}{8}\sqrt{b{x}^{2}+a}}+{\frac{3\,A{a}^{2}}{8}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){\frac{1}{\sqrt{b}}}}+{\frac{Bx}{6\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{Bxa}{24\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{Bx{a}^{2}}{16\,b}\sqrt{b{x}^{2}+a}}-{\frac{B{a}^{3}}{16}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.238467, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (8 \, B b^{2} x^{5} + 2 \,{\left (7 \, B a b + 6 \, A b^{2}\right )} x^{3} + 3 \,{\left (B a^{2} + 10 \, A a b\right )} x\right )} \sqrt{b x^{2} + a} \sqrt{b} - 3 \,{\left (B a^{3} - 6 \, A a^{2} b\right )} \log \left (-2 \, \sqrt{b x^{2} + a} b x -{\left (2 \, b x^{2} + a\right )} \sqrt{b}\right )}{96 \, b^{\frac{3}{2}}}, \frac{{\left (8 \, B b^{2} x^{5} + 2 \,{\left (7 \, B a b + 6 \, A b^{2}\right )} x^{3} + 3 \,{\left (B a^{2} + 10 \, A a b\right )} x\right )} \sqrt{b x^{2} + a} \sqrt{-b} - 3 \,{\left (B a^{3} - 6 \, A a^{2} b\right )} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right )}{48 \, \sqrt{-b} b}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/96*(2*(8*B*b^2*x^5 + 2*(7*B*a*b + 6*A*b^2)*x^3 + 3*(B*a^2 + 10*A*a*b)*x)*sqrt
(b*x^2 + a)*sqrt(b) - 3*(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^(3/2), 1/48*((8*B*b^2*x^5 + 2*(7*B*a*b + 6*A*b^2)*x^3 + 3*(B*
a^2 + 10*A*a*b)*x)*sqrt(b*x^2 + a)*sqrt(-b) - 3*(B*a^3 - 6*A*a^2*b)*arctan(sqrt(
-b)*x/sqrt(b*x^2 + a)))/(sqrt(-b)*b)]

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Sympy [A]  time = 43.7418, size = 253, normalized size = 2.14 \[ \frac{A a^{\frac{3}{2}} x \sqrt{1 + \frac{b x^{2}}{a}}}{2} + \frac{A a^{\frac{3}{2}} x}{8 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{3 A \sqrt{a} b x^{3}}{8 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{3 A a^{2} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{8 \sqrt{b}} + \frac{A b^{2} x^{5}}{4 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{B a^{\frac{5}{2}} x}{16 b \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{17 B a^{\frac{3}{2}} x^{3}}{48 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{11 B \sqrt{a} b x^{5}}{24 \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{B a^{3} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{16 b^{\frac{3}{2}}} + \frac{B b^{2} x^{7}}{6 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.248828, size = 138, normalized size = 1.17 \[ \frac{1}{48} \,{\left (2 \,{\left (4 \, B b x^{2} + \frac{7 \, B a b^{4} + 6 \, A b^{5}}{b^{4}}\right )} x^{2} + \frac{3 \,{\left (B a^{2} b^{3} + 10 \, A a b^{4}\right )}}{b^{4}}\right )} \sqrt{b x^{2} + a} x + \frac{{\left (B a^{3} - 6 \, A a^{2} b\right )}{\rm ln}\left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{16 \, b^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/48*(2*(4*B*b*x^2 + (7*B*a*b^4 + 6*A*b^5)/b^4)*x^2 + 3*(B*a^2*b^3 + 10*A*a*b^4)
/b^4)*sqrt(b*x^2 + a)*x + 1/16*(B*a^3 - 6*A*a^2*b)*ln(abs(-sqrt(b)*x + sqrt(b*x^
2 + a)))/b^(3/2)