3.55 \(\int \left (a+b x^2\right )^{3/2} \left (c+d x^2\right ) \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.106461, 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 \[ \frac{a^2 (6 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{16 b^{3/2}}+\frac{x \left (a+b x^2\right )^{3/2} (6 b c-a d)}{24 b}+\frac{a x \sqrt{a+b x^2} (6 b c-a d)}{16 b}+\frac{d x \left (a+b x^2\right )^{5/2}}{6 b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 12.2675, size = 102, normalized size = 0.86 \[ - \frac{a^{2} \left (a d - 6 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )}}{16 b^{\frac{3}{2}}} - \frac{a x \sqrt{a + b x^{2}} \left (a d - 6 b c\right )}{16 b} + \frac{d x \left (a + b x^{2}\right )^{\frac{5}{2}}}{6 b} - \frac{x \left (a + b x^{2}\right )^{\frac{3}{2}} \left (a d - 6 b c\right )}{24 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**(3/2)*(d*x**2+c),x)

[Out]

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

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Mathematica [A]  time = 0.124798, size = 98, normalized size = 0.83 \[ \sqrt{a+b x^2} \left (\frac{1}{24} x^3 (7 a d+6 b c)+\frac{a x (a d+10 b c)}{16 b}+\frac{1}{6} b d x^5\right )-\frac{a^2 (a d-6 b c) \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )}{16 b^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.007, size = 131, normalized size = 1.1 \[{\frac{cx}{4} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{3\,acx}{8}\sqrt{b{x}^{2}+a}}+{\frac{3\,{a}^{2}c}{8}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){\frac{1}{\sqrt{b}}}}+{\frac{dx}{6\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{adx}{24\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{d{a}^{2}x}{16\,b}\sqrt{b{x}^{2}+a}}-{\frac{{a}^{3}d}{16}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.247573, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (8 \, b^{2} d x^{5} + 2 \,{\left (6 \, b^{2} c + 7 \, a b d\right )} x^{3} + 3 \,{\left (10 \, a b c + a^{2} d\right )} x\right )} \sqrt{b x^{2} + a} \sqrt{b} - 3 \,{\left (6 \, a^{2} b c - a^{3} d\right )} \log \left (2 \, \sqrt{b x^{2} + a} b x -{\left (2 \, b x^{2} + a\right )} \sqrt{b}\right )}{96 \, b^{\frac{3}{2}}}, \frac{{\left (8 \, b^{2} d x^{5} + 2 \,{\left (6 \, b^{2} c + 7 \, a b d\right )} x^{3} + 3 \,{\left (10 \, a b c + a^{2} d\right )} x\right )} \sqrt{b x^{2} + a} \sqrt{-b} + 3 \,{\left (6 \, a^{2} b c - a^{3} d\right )} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right )}{48 \, \sqrt{-b} b}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 43.6349, size = 253, normalized size = 2.14 \[ \frac{a^{\frac{5}{2}} d x}{16 b \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{a^{\frac{3}{2}} c x \sqrt{1 + \frac{b x^{2}}{a}}}{2} + \frac{a^{\frac{3}{2}} c x}{8 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{17 a^{\frac{3}{2}} d x^{3}}{48 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{3 \sqrt{a} b c x^{3}}{8 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{11 \sqrt{a} b d x^{5}}{24 \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{a^{3} d \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{16 b^{\frac{3}{2}}} + \frac{3 a^{2} c \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{8 \sqrt{b}} + \frac{b^{2} c x^{5}}{4 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{b^{2} d x^{7}}{6 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**(3/2)*(d*x**2+c),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.28558, size = 139, normalized size = 1.18 \[ \frac{1}{48} \,{\left (2 \,{\left (4 \, b d x^{2} + \frac{6 \, b^{5} c + 7 \, a b^{4} d}{b^{4}}\right )} x^{2} + \frac{3 \,{\left (10 \, a b^{4} c + a^{2} b^{3} d\right )}}{b^{4}}\right )} \sqrt{b x^{2} + a} x - \frac{{\left (6 \, a^{2} b c - a^{3} d\right )}{\rm ln}\left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{16 \, b^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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