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

Optimal. Leaf size=152 \[ \frac{b \sqrt{a^2+2 a b x^2+b^2 x^4} \left (c+d x^2\right )^{3/2}}{3 d \left (a+b x^2\right )}+\frac{a \sqrt{a^2+2 a b x^2+b^2 x^4} \sqrt{c+d x^2}}{a+b x^2}-\frac{a \sqrt{c} \sqrt{a^2+2 a b x^2+b^2 x^4} \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{a+b x^2} \]

[Out]

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

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Rubi [A]  time = 0.305133, antiderivative size = 152, 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 \[ \frac{b \sqrt{a^2+2 a b x^2+b^2 x^4} \left (c+d x^2\right )^{3/2}}{3 d \left (a+b x^2\right )}+\frac{a \sqrt{a^2+2 a b x^2+b^2 x^4} \sqrt{c+d x^2}}{a+b x^2}-\frac{a \sqrt{c} \sqrt{a^2+2 a b x^2+b^2 x^4} \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{a+b x^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 20.0972, size = 105, normalized size = 0.69 \[ - \frac{a \sqrt{c} \sqrt{\left (a + b x^{2}\right )^{2}} \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{2}}}{\sqrt{c}} \right )}}{a + b x^{2}} + \frac{a \sqrt{c + d x^{2}} \sqrt{\left (a + b x^{2}\right )^{2}}}{a + b x^{2}} + \frac{b \left (c + d x^{2}\right )^{\frac{3}{2}} \sqrt{\left (a + b x^{2}\right )^{2}}}{3 d \left (a + b x^{2}\right )} \]

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,x)

[Out]

-a*sqrt(c)*sqrt((a + b*x**2)**2)*atanh(sqrt(c + d*x**2)/sqrt(c))/(a + b*x**2) +
a*sqrt(c + d*x**2)*sqrt((a + b*x**2)**2)/(a + b*x**2) + b*(c + d*x**2)**(3/2)*sq
rt((a + b*x**2)**2)/(3*d*(a + b*x**2))

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Mathematica [A]  time = 0.113774, size = 96, normalized size = 0.63 \[ \frac{\sqrt{\left (a+b x^2\right )^2} \left (\sqrt{c+d x^2} \left (3 a d+b c+b d x^2\right )-3 a \sqrt{c} d \log \left (\sqrt{c} \sqrt{c+d x^2}+c\right )+3 a \sqrt{c} d \log (x)\right )}{3 d \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[(a + b*x^2)^2]*(Sqrt[c + d*x^2]*(b*c + 3*a*d + b*d*x^2) + 3*a*Sqrt[c]*d*Lo
g[x] - 3*a*Sqrt[c]*d*Log[c + Sqrt[c]*Sqrt[c + d*x^2]]))/(3*d*(a + b*x^2))

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Maple [A]  time = 0.01, size = 79, normalized size = 0.5 \[{\frac{1}{ \left ( 3\,b{x}^{2}+3\,a \right ) d}\sqrt{ \left ( b{x}^{2}+a \right ) ^{2}} \left ( b \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}-3\,\sqrt{c}\ln \left ( 2\,{\frac{\sqrt{c}\sqrt{d{x}^{2}+c}+c}{x}} \right ) ad+3\,\sqrt{d{x}^{2}+c}ad \right ) } \]

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,x)

[Out]

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

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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,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.279204, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, a \sqrt{c} d \log \left (-\frac{d x^{2} - 2 \, \sqrt{d x^{2} + c} \sqrt{c} + 2 \, c}{x^{2}}\right ) + 2 \,{\left (b d x^{2} + b c + 3 \, a d\right )} \sqrt{d x^{2} + c}}{6 \, d}, -\frac{3 \, a \sqrt{-c} d \arctan \left (\frac{c}{\sqrt{d x^{2} + c} \sqrt{-c}}\right ) -{\left (b d x^{2} + b c + 3 \, a d\right )} \sqrt{d x^{2} + c}}{3 \, d}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/6*(3*a*sqrt(c)*d*log(-(d*x^2 - 2*sqrt(d*x^2 + c)*sqrt(c) + 2*c)/x^2) + 2*(b*d
*x^2 + b*c + 3*a*d)*sqrt(d*x^2 + c))/d, -1/3*(3*a*sqrt(-c)*d*arctan(c/(sqrt(d*x^
2 + c)*sqrt(-c))) - (b*d*x^2 + b*c + 3*a*d)*sqrt(d*x^2 + c))/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,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.275687, size = 113, normalized size = 0.74 \[ \frac{a c \arctan \left (\frac{\sqrt{d x^{2} + c}}{\sqrt{-c}}\right ){\rm sign}\left (b x^{2} + a\right )}{\sqrt{-c}} + \frac{{\left (d x^{2} + c\right )}^{\frac{3}{2}} b d^{2}{\rm sign}\left (b x^{2} + a\right ) + 3 \, \sqrt{d x^{2} + c} a d^{3}{\rm sign}\left (b x^{2} + a\right )}{3 \, d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*c*arctan(sqrt(d*x^2 + c)/sqrt(-c))*sign(b*x^2 + a)/sqrt(-c) + 1/3*((d*x^2 + c)
^(3/2)*b*d^2*sign(b*x^2 + a) + 3*sqrt(d*x^2 + c)*a*d^3*sign(b*x^2 + a))/d^3