3.606 \(\int \left (a+b x^2\right )^2 \sqrt{c+d x^2} \, dx\)

Optimal. Leaf size=149 \[ \frac{x \sqrt{c+d x^2} \left (8 a^2 d^2-4 a b c d+b^2 c^2\right )}{16 d^2}+\frac{c \left (8 a^2 d^2-4 a b c d+b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{16 d^{5/2}}-\frac{b x \left (c+d x^2\right )^{3/2} (3 b c-8 a d)}{24 d^2}+\frac{b x \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}{6 d} \]

[Out]

((b^2*c^2 - 4*a*b*c*d + 8*a^2*d^2)*x*Sqrt[c + d*x^2])/(16*d^2) - (b*(3*b*c - 8*a
*d)*x*(c + d*x^2)^(3/2))/(24*d^2) + (b*x*(a + b*x^2)*(c + d*x^2)^(3/2))/(6*d) +
(c*(b^2*c^2 - 4*a*b*c*d + 8*a^2*d^2)*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/(16*d
^(5/2))

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Rubi [A]  time = 0.210795, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238 \[ \frac{x \sqrt{c+d x^2} \left (8 a^2 d^2-4 a b c d+b^2 c^2\right )}{16 d^2}+\frac{c \left (8 a^2 d^2-4 a b c d+b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{16 d^{5/2}}-\frac{b x \left (c+d x^2\right )^{3/2} (3 b c-8 a d)}{24 d^2}+\frac{b x \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}{6 d} \]

Antiderivative was successfully verified.

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

[Out]

((b^2*c^2 - 4*a*b*c*d + 8*a^2*d^2)*x*Sqrt[c + d*x^2])/(16*d^2) - (b*(3*b*c - 8*a
*d)*x*(c + d*x^2)^(3/2))/(24*d^2) + (b*x*(a + b*x^2)*(c + d*x^2)^(3/2))/(6*d) +
(c*(b^2*c^2 - 4*a*b*c*d + 8*a^2*d^2)*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/(16*d
^(5/2))

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Rubi in Sympy [A]  time = 23.3195, size = 143, normalized size = 0.96 \[ \frac{b x \left (a + b x^{2}\right ) \left (c + d x^{2}\right )^{\frac{3}{2}}}{6 d} + \frac{b x \left (c + d x^{2}\right )^{\frac{3}{2}} \left (8 a d - 3 b c\right )}{24 d^{2}} + \frac{c \left (8 a^{2} d^{2} - 4 a b c d + b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} x}{\sqrt{c + d x^{2}}} \right )}}{16 d^{\frac{5}{2}}} + \frac{x \sqrt{c + d x^{2}} \left (8 a^{2} d^{2} - 4 a b c d + b^{2} c^{2}\right )}{16 d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**2*(d*x**2+c)**(1/2),x)

[Out]

b*x*(a + b*x**2)*(c + d*x**2)**(3/2)/(6*d) + b*x*(c + d*x**2)**(3/2)*(8*a*d - 3*
b*c)/(24*d**2) + c*(8*a**2*d**2 - 4*a*b*c*d + b**2*c**2)*atanh(sqrt(d)*x/sqrt(c
+ d*x**2))/(16*d**(5/2)) + x*sqrt(c + d*x**2)*(8*a**2*d**2 - 4*a*b*c*d + b**2*c*
*2)/(16*d**2)

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Mathematica [A]  time = 0.111786, size = 122, normalized size = 0.82 \[ \frac{3 c \left (8 a^2 d^2-4 a b c d+b^2 c^2\right ) \log \left (\sqrt{d} \sqrt{c+d x^2}+d x\right )+\sqrt{d} x \sqrt{c+d x^2} \left (24 a^2 d^2+12 a b d \left (c+2 d x^2\right )+b^2 \left (-3 c^2+2 c d x^2+8 d^2 x^4\right )\right )}{48 d^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[d]*x*Sqrt[c + d*x^2]*(24*a^2*d^2 + 12*a*b*d*(c + 2*d*x^2) + b^2*(-3*c^2 +
2*c*d*x^2 + 8*d^2*x^4)) + 3*c*(b^2*c^2 - 4*a*b*c*d + 8*a^2*d^2)*Log[d*x + Sqrt[d
]*Sqrt[c + d*x^2]])/(48*d^(5/2))

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Maple [A]  time = 0.012, size = 190, normalized size = 1.3 \[{\frac{{a}^{2}x}{2}\sqrt{d{x}^{2}+c}}+{\frac{{a}^{2}c}{2}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ){\frac{1}{\sqrt{d}}}}+{\frac{{b}^{2}{x}^{3}}{6\,d} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}-{\frac{{b}^{2}cx}{8\,{d}^{2}} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}+{\frac{{b}^{2}{c}^{2}x}{16\,{d}^{2}}\sqrt{d{x}^{2}+c}}+{\frac{{b}^{2}{c}^{3}}{16}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ){d}^{-{\frac{5}{2}}}}+{\frac{abx}{2\,d} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}-{\frac{abcx}{4\,d}\sqrt{d{x}^{2}+c}}-{\frac{ab{c}^{2}}{4}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ){d}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^2*(d*x^2+c)^(1/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.257659, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (8 \, b^{2} d^{2} x^{5} + 2 \,{\left (b^{2} c d + 12 \, a b d^{2}\right )} x^{3} - 3 \,{\left (b^{2} c^{2} - 4 \, a b c d - 8 \, a^{2} d^{2}\right )} x\right )} \sqrt{d x^{2} + c} \sqrt{d} + 3 \,{\left (b^{2} c^{3} - 4 \, a b c^{2} d + 8 \, a^{2} c d^{2}\right )} \log \left (-2 \, \sqrt{d x^{2} + c} d x -{\left (2 \, d x^{2} + c\right )} \sqrt{d}\right )}{96 \, d^{\frac{5}{2}}}, \frac{{\left (8 \, b^{2} d^{2} x^{5} + 2 \,{\left (b^{2} c d + 12 \, a b d^{2}\right )} x^{3} - 3 \,{\left (b^{2} c^{2} - 4 \, a b c d - 8 \, a^{2} d^{2}\right )} x\right )} \sqrt{d x^{2} + c} \sqrt{-d} + 3 \,{\left (b^{2} c^{3} - 4 \, a b c^{2} d + 8 \, a^{2} c d^{2}\right )} \arctan \left (\frac{\sqrt{-d} x}{\sqrt{d x^{2} + c}}\right )}{48 \, \sqrt{-d} d^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/96*(2*(8*b^2*d^2*x^5 + 2*(b^2*c*d + 12*a*b*d^2)*x^3 - 3*(b^2*c^2 - 4*a*b*c*d
- 8*a^2*d^2)*x)*sqrt(d*x^2 + c)*sqrt(d) + 3*(b^2*c^3 - 4*a*b*c^2*d + 8*a^2*c*d^2
)*log(-2*sqrt(d*x^2 + c)*d*x - (2*d*x^2 + c)*sqrt(d)))/d^(5/2), 1/48*((8*b^2*d^2
*x^5 + 2*(b^2*c*d + 12*a*b*d^2)*x^3 - 3*(b^2*c^2 - 4*a*b*c*d - 8*a^2*d^2)*x)*sqr
t(d*x^2 + c)*sqrt(-d) + 3*(b^2*c^3 - 4*a*b*c^2*d + 8*a^2*c*d^2)*arctan(sqrt(-d)*
x/sqrt(d*x^2 + c)))/(sqrt(-d)*d^2)]

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Sympy [A]  time = 34.6228, size = 291, normalized size = 1.95 \[ \frac{a^{2} \sqrt{c} x \sqrt{1 + \frac{d x^{2}}{c}}}{2} + \frac{a^{2} c \operatorname{asinh}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{2 \sqrt{d}} + \frac{a b c^{\frac{3}{2}} x}{4 d \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{3 a b \sqrt{c} x^{3}}{4 \sqrt{1 + \frac{d x^{2}}{c}}} - \frac{a b c^{2} \operatorname{asinh}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{4 d^{\frac{3}{2}}} + \frac{a b d x^{5}}{2 \sqrt{c} \sqrt{1 + \frac{d x^{2}}{c}}} - \frac{b^{2} c^{\frac{5}{2}} x}{16 d^{2} \sqrt{1 + \frac{d x^{2}}{c}}} - \frac{b^{2} c^{\frac{3}{2}} x^{3}}{48 d \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{5 b^{2} \sqrt{c} x^{5}}{24 \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{b^{2} c^{3} \operatorname{asinh}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{16 d^{\frac{5}{2}}} + \frac{b^{2} d x^{7}}{6 \sqrt{c} \sqrt{1 + \frac{d x^{2}}{c}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**2*(d*x**2+c)**(1/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.246155, size = 173, normalized size = 1.16 \[ \frac{1}{48} \,{\left (2 \,{\left (4 \, b^{2} x^{2} + \frac{b^{2} c d^{3} + 12 \, a b d^{4}}{d^{4}}\right )} x^{2} - \frac{3 \,{\left (b^{2} c^{2} d^{2} - 4 \, a b c d^{3} - 8 \, a^{2} d^{4}\right )}}{d^{4}}\right )} \sqrt{d x^{2} + c} x - \frac{{\left (b^{2} c^{3} - 4 \, a b c^{2} d + 8 \, a^{2} c d^{2}\right )}{\rm ln}\left ({\left | -\sqrt{d} x + \sqrt{d x^{2} + c} \right |}\right )}{16 \, d^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/48*(2*(4*b^2*x^2 + (b^2*c*d^3 + 12*a*b*d^4)/d^4)*x^2 - 3*(b^2*c^2*d^2 - 4*a*b*
c*d^3 - 8*a^2*d^4)/d^4)*sqrt(d*x^2 + c)*x - 1/16*(b^2*c^3 - 4*a*b*c^2*d + 8*a^2*
c*d^2)*ln(abs(-sqrt(d)*x + sqrt(d*x^2 + c)))/d^(5/2)