3.259 \(\int x^2 \sqrt{c+d x^2} \sqrt{a^2+2 a b x^2+b^2 x^4} \, dx\)

Optimal. Leaf size=243 \[ \frac{c^2 \sqrt{a^2+2 a b x^2+b^2 x^4} (b c-2 a d) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{16 d^{5/2} \left (a+b x^2\right )}-\frac{c x \sqrt{a^2+2 a b x^2+b^2 x^4} \sqrt{c+d x^2} (b c-2 a d)}{16 d^2 \left (a+b x^2\right )}+\frac{b x^3 \sqrt{a^2+2 a b x^2+b^2 x^4} \left (c+d x^2\right )^{3/2}}{6 d \left (a+b x^2\right )}-\frac{x^3 \sqrt{a^2+2 a b x^2+b^2 x^4} \sqrt{c+d x^2} (b c-2 a d)}{8 d \left (a+b x^2\right )} \]

[Out]

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

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Rubi [A]  time = 0.398316, antiderivative size = 243, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.162 \[ \frac{c^2 \sqrt{a^2+2 a b x^2+b^2 x^4} (b c-2 a d) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{16 d^{5/2} \left (a+b x^2\right )}-\frac{c x \sqrt{a^2+2 a b x^2+b^2 x^4} \sqrt{c+d x^2} (b c-2 a d)}{16 d^2 \left (a+b x^2\right )}+\frac{b x^3 \sqrt{a^2+2 a b x^2+b^2 x^4} \left (c+d x^2\right )^{3/2}}{6 d \left (a+b x^2\right )}-\frac{x^3 \sqrt{a^2+2 a b x^2+b^2 x^4} \sqrt{c+d x^2} (b c-2 a d)}{8 d \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[x^2*Sqrt[c + d*x^2]*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4],x]

[Out]

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

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Rubi in Sympy [A]  time = 27.6088, size = 182, normalized size = 0.75 \[ \frac{b x^{3} \left (c + d x^{2}\right )^{\frac{3}{2}} \sqrt{\left (a + b x^{2}\right )^{2}}}{6 d \left (a + b x^{2}\right )} - \frac{c^{2} \left (2 a d - b c\right ) \sqrt{\left (a + b x^{2}\right )^{2}} \operatorname{atanh}{\left (\frac{\sqrt{d} x}{\sqrt{c + d x^{2}}} \right )}}{16 d^{\frac{5}{2}} \left (a + b x^{2}\right )} + \frac{c x \sqrt{c + d x^{2}} \left (2 a d - b c\right ) \sqrt{\left (a + b x^{2}\right )^{2}}}{16 d^{2} \left (a + b x^{2}\right )} + \frac{x^{3} \sqrt{c + d x^{2}} \left (2 a d - b c\right ) \sqrt{\left (a + b x^{2}\right )^{2}}}{8 d \left (a + b x^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.138333, size = 121, normalized size = 0.5 \[ \frac{\sqrt{\left (a+b x^2\right )^2} \left (\sqrt{d} x \sqrt{c+d x^2} \left (6 a d \left (c+2 d x^2\right )+b \left (-3 c^2+2 c d x^2+8 d^2 x^4\right )\right )+3 c^2 (b c-2 a d) \log \left (\sqrt{d} \sqrt{c+d x^2}+d x\right )\right )}{48 d^{5/2} \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2*Sqrt[c + d*x^2]*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4],x]

[Out]

(Sqrt[(a + b*x^2)^2]*(Sqrt[d]*x*Sqrt[c + d*x^2]*(6*a*d*(c + 2*d*x^2) + b*(-3*c^2
+ 2*c*d*x^2 + 8*d^2*x^4)) + 3*c^2*(b*c - 2*a*d)*Log[d*x + Sqrt[d]*Sqrt[c + d*x^
2]]))/(48*d^(5/2)*(a + b*x^2))

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Maple [A]  time = 0.016, size = 164, normalized size = 0.7 \[{\frac{1}{48\,b{x}^{2}+48\,a}\sqrt{ \left ( b{x}^{2}+a \right ) ^{2}} \left ( 8\,b{x}^{3} \left ( d{x}^{2}+c \right ) ^{3/2}{d}^{7/2}+12\,ax \left ( d{x}^{2}+c \right ) ^{3/2}{d}^{7/2}-6\,bcx \left ( d{x}^{2}+c \right ) ^{3/2}{d}^{5/2}-6\,acx\sqrt{d{x}^{2}+c}{d}^{7/2}+3\,b{c}^{2}x\sqrt{d{x}^{2}+c}{d}^{5/2}-6\,a{c}^{2}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ){d}^{3}+3\,b{c}^{3}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ){d}^{2} \right ){d}^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/48*((b*x^2+a)^2)^(1/2)*(8*b*x^3*(d*x^2+c)^(3/2)*d^(7/2)+12*a*x*(d*x^2+c)^(3/2)
*d^(7/2)-6*b*c*x*(d*x^2+c)^(3/2)*d^(5/2)-6*a*c*x*(d*x^2+c)^(1/2)*d^(7/2)+3*b*c^2
*x*(d*x^2+c)^(1/2)*d^(5/2)-6*a*c^2*ln(x*d^(1/2)+(d*x^2+c)^(1/2))*d^3+3*b*c^3*ln(
x*d^(1/2)+(d*x^2+c)^(1/2))*d^2)/(b*x^2+a)/d^(9/2)

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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(sqrt(d*x^2 + c)*sqrt((b*x^2 + a)^2)*x^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.295258, size = 1, normalized size = 0. \[ \left [\frac{2 \,{\left (8 \, b d^{2} x^{5} + 2 \,{\left (b c d + 6 \, a d^{2}\right )} x^{3} - 3 \,{\left (b c^{2} - 2 \, a c d\right )} x\right )} \sqrt{d x^{2} + c} \sqrt{d} - 3 \,{\left (b c^{3} - 2 \, a c^{2} d\right )} \log \left (2 \, \sqrt{d x^{2} + c} d x -{\left (2 \, d x^{2} + c\right )} \sqrt{d}\right )}{96 \, d^{\frac{5}{2}}}, \frac{{\left (8 \, b d^{2} x^{5} + 2 \,{\left (b c d + 6 \, a d^{2}\right )} x^{3} - 3 \,{\left (b c^{2} - 2 \, a c d\right )} x\right )} \sqrt{d x^{2} + c} \sqrt{-d} + 3 \,{\left (b c^{3} - 2 \, a c^{2} d\right )} \arctan \left (\frac{\sqrt{-d} x}{\sqrt{d x^{2} + c}}\right )}{48 \, \sqrt{-d} d^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.277472, size = 211, normalized size = 0.87 \[ \frac{1}{48} \,{\left (2 \,{\left (4 \, b x^{2}{\rm sign}\left (b x^{2} + a\right ) + \frac{b c d^{3}{\rm sign}\left (b x^{2} + a\right ) + 6 \, a d^{4}{\rm sign}\left (b x^{2} + a\right )}{d^{4}}\right )} x^{2} - \frac{3 \,{\left (b c^{2} d^{2}{\rm sign}\left (b x^{2} + a\right ) - 2 \, a c d^{3}{\rm sign}\left (b x^{2} + a\right )\right )}}{d^{4}}\right )} \sqrt{d x^{2} + c} x - \frac{{\left (b c^{3}{\rm sign}\left (b x^{2} + a\right ) - 2 \, a c^{2} d{\rm sign}\left (b x^{2} + a\right )\right )}{\rm ln}\left ({\left | -\sqrt{d} x + \sqrt{d x^{2} + c} \right |}\right )}{16 \, d^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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