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

Optimal. Leaf size=178 \[ -\frac{c \sqrt{a^2+2 a b x^2+b^2 x^4} (b c-4 a d) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{8 d^{3/2} \left (a+b x^2\right )}+\frac{b x \sqrt{a^2+2 a b x^2+b^2 x^4} \left (c+d x^2\right )^{3/2}}{4 d \left (a+b x^2\right )}-\frac{x \sqrt{a^2+2 a b x^2+b^2 x^4} \sqrt{c+d x^2} (b c-4 a d)}{8 d \left (a+b x^2\right )} \]

[Out]

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

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Rubi [A]  time = 0.190048, antiderivative size = 178, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.147 \[ -\frac{c \sqrt{a^2+2 a b x^2+b^2 x^4} (b c-4 a d) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{8 d^{3/2} \left (a+b x^2\right )}+\frac{b x \sqrt{a^2+2 a b x^2+b^2 x^4} \left (c+d x^2\right )^{3/2}}{4 d \left (a+b x^2\right )}-\frac{x \sqrt{a^2+2 a b x^2+b^2 x^4} \sqrt{c+d x^2} (b c-4 a d)}{8 d \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{c + d x^{2}} \sqrt{\left (a + b x^{2}\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt(c + d*x**2)*sqrt((a + b*x**2)**2), x)

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Mathematica [A]  time = 0.084479, size = 98, normalized size = 0.55 \[ \frac{\sqrt{\left (a+b x^2\right )^2} \left (\sqrt{d} x \sqrt{c+d x^2} \left (4 a d+b \left (c+2 d x^2\right )\right )+c (4 a d-b c) \log \left (\sqrt{d} \sqrt{c+d x^2}+d x\right )\right )}{8 d^{3/2} \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

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

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Maple [A]  time = 0.01, size = 122, normalized size = 0.7 \[{\frac{1}{8\,b{x}^{2}+8\,a}\sqrt{ \left ( b{x}^{2}+a \right ) ^{2}} \left ( 2\,bx \left ( d{x}^{2}+c \right ) ^{3/2}{d}^{3/2}+4\,ax\sqrt{d{x}^{2}+c}{d}^{5/2}-bcx\sqrt{d{x}^{2}+c}{d}^{{\frac{3}{2}}}+4\,ac\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ){d}^{2}-b{c}^{2}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ) d \right ){d}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*((b*x^2+a)^2)^(1/2)*(2*b*x*(d*x^2+c)^(3/2)*d^(3/2)+4*a*x*(d*x^2+c)^(1/2)*d^(
5/2)-b*c*x*(d*x^2+c)^(1/2)*d^(3/2)+4*a*c*ln(x*d^(1/2)+(d*x^2+c)^(1/2))*d^2-b*c^2
*ln(x*d^(1/2)+(d*x^2+c)^(1/2))*d)/(b*x^2+a)/d^(5/2)

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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, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.2857, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (2 \, b d x^{3} +{\left (b c + 4 \, a d\right )} x\right )} \sqrt{d x^{2} + c} \sqrt{d} -{\left (b c^{2} - 4 \, a c d\right )} \log \left (-2 \, \sqrt{d x^{2} + c} d x -{\left (2 \, d x^{2} + c\right )} \sqrt{d}\right )}{16 \, d^{\frac{3}{2}}}, \frac{{\left (2 \, b d x^{3} +{\left (b c + 4 \, a d\right )} x\right )} \sqrt{d x^{2} + c} \sqrt{-d} -{\left (b c^{2} - 4 \, a c d\right )} \arctan \left (\frac{\sqrt{-d} x}{\sqrt{d x^{2} + c}}\right )}{8 \, \sqrt{-d} d}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/16*(2*(2*b*d*x^3 + (b*c + 4*a*d)*x)*sqrt(d*x^2 + c)*sqrt(d) - (b*c^2 - 4*a*c*
d)*log(-2*sqrt(d*x^2 + c)*d*x - (2*d*x^2 + c)*sqrt(d)))/d^(3/2), 1/8*((2*b*d*x^3
+ (b*c + 4*a*d)*x)*sqrt(d*x^2 + c)*sqrt(-d) - (b*c^2 - 4*a*c*d)*arctan(sqrt(-d)
*x/sqrt(d*x^2 + c)))/(sqrt(-d)*d)]

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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((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 = 0.276751, size = 147, normalized size = 0.83 \[ \frac{1}{8} \,{\left (2 \, b x^{2}{\rm sign}\left (b x^{2} + a\right ) + \frac{b c d{\rm sign}\left (b x^{2} + a\right ) + 4 \, a d^{2}{\rm sign}\left (b x^{2} + a\right )}{d^{2}}\right )} \sqrt{d x^{2} + c} x + \frac{{\left (b c^{2}{\rm sign}\left (b x^{2} + a\right ) - 4 \, a c d{\rm sign}\left (b x^{2} + a\right )\right )}{\rm ln}\left ({\left | -\sqrt{d} x + \sqrt{d x^{2} + c} \right |}\right )}{8 \, d^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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