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

Optimal. Leaf size=160 \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2} (2 a+b x) \sqrt{c+d x^2}}{2 (a+b x)}+\frac{b c \sqrt{a^2+2 a b x+b^2 x^2} \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{2 \sqrt{d} (a+b x)}-\frac{a \sqrt{c} \sqrt{a^2+2 a b x+b^2 x^2} \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{a+b x} \]

[Out]

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

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Rubi [A]  time = 0.377451, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.229 \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2} (2 a+b x) \sqrt{c+d x^2}}{2 (a+b x)}+\frac{b c \sqrt{a^2+2 a b x+b^2 x^2} \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{2 \sqrt{d} (a+b x)}-\frac{a \sqrt{c} \sqrt{a^2+2 a b x+b^2 x^2} \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{a+b x} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 42.1211, size = 150, normalized size = 0.94 \[ - \frac{a \sqrt{c} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{2}}}{\sqrt{c}} \right )}}{a + b x} + \frac{b c \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \operatorname{atanh}{\left (\frac{\sqrt{d} x}{\sqrt{c + d x^{2}}} \right )}}{2 \sqrt{d} \left (a + b x\right )} + \frac{\left (4 a + 2 b x\right ) \sqrt{c + d x^{2}} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{4 \left (a + b x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a*sqrt(c)*sqrt(a**2 + 2*a*b*x + b**2*x**2)*atanh(sqrt(c + d*x**2)/sqrt(c))/(a +
b*x) + b*c*sqrt(a**2 + 2*a*b*x + b**2*x**2)*atanh(sqrt(d)*x/sqrt(c + d*x**2))/(
2*sqrt(d)*(a + b*x)) + (4*a + 2*b*x)*sqrt(c + d*x**2)*sqrt(a**2 + 2*a*b*x + b**2
*x**2)/(4*(a + b*x))

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Mathematica [A]  time = 0.106102, size = 139, normalized size = 0.87 \[ \frac{\sqrt{(a+b x)^2} \left (2 a \sqrt{d} \sqrt{c+d x^2}-2 a \sqrt{c} \sqrt{d} \log \left (\sqrt{c} \sqrt{c+d x^2}+c\right )+2 a \sqrt{c} \sqrt{d} \log (x)+b \sqrt{d} x \sqrt{c+d x^2}+b c \log \left (\sqrt{d} \sqrt{c+d x^2}+d x\right )\right )}{2 \sqrt{d} (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [C]  time = 0.014, size = 94, normalized size = 0.6 \[ -{\frac{{\it csgn} \left ( bx+a \right ) }{2} \left ( 2\,\sqrt{c}\ln \left ( 2\,{\frac{\sqrt{c}\sqrt{d{x}^{2}+c}+c}{x}} \right ) a\sqrt{d}-bx\sqrt{d{x}^{2}+c}\sqrt{d}-2\,\sqrt{d{x}^{2}+c}a\sqrt{d}-bc\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ) \right ){\frac{1}{\sqrt{d}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*csgn(b*x+a)*(2*c^(1/2)*ln(2*(c^(1/2)*(d*x^2+c)^(1/2)+c)/x)*a*d^(1/2)-b*x*(d
*x^2+c)^(1/2)*d^(1/2)-2*(d*x^2+c)^(1/2)*a*d^(1/2)-b*c*ln(x*d^(1/2)+(d*x^2+c)^(1/
2)))/d^(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(sqrt(d*x^2 + c)*sqrt((b*x + a)^2)/x,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.341941, size = 1, normalized size = 0.01 \[ \left [\frac{b c \log \left (-2 \, \sqrt{d x^{2} + c} d x -{\left (2 \, d x^{2} + c\right )} \sqrt{d}\right ) + 2 \, a \sqrt{c} \sqrt{d} \log \left (-\frac{d x^{2} - 2 \, \sqrt{d x^{2} + c} \sqrt{c} + 2 \, c}{x^{2}}\right ) + 2 \, \sqrt{d x^{2} + c}{\left (b x + 2 \, a\right )} \sqrt{d}}{4 \, \sqrt{d}}, \frac{b c \arctan \left (\frac{\sqrt{-d} x}{\sqrt{d x^{2} + c}}\right ) + a \sqrt{c} \sqrt{-d} \log \left (-\frac{d x^{2} - 2 \, \sqrt{d x^{2} + c} \sqrt{c} + 2 \, c}{x^{2}}\right ) + \sqrt{d x^{2} + c}{\left (b x + 2 \, a\right )} \sqrt{-d}}{2 \, \sqrt{-d}}, -\frac{4 \, a \sqrt{-c} \sqrt{d} \arctan \left (\frac{c}{\sqrt{d x^{2} + c} \sqrt{-c}}\right ) - b c \log \left (-2 \, \sqrt{d x^{2} + c} d x -{\left (2 \, d x^{2} + c\right )} \sqrt{d}\right ) - 2 \, \sqrt{d x^{2} + c}{\left (b x + 2 \, a\right )} \sqrt{d}}{4 \, \sqrt{d}}, \frac{b c \arctan \left (\frac{\sqrt{-d} x}{\sqrt{d x^{2} + c}}\right ) - 2 \, a \sqrt{-c} \sqrt{-d} \arctan \left (\frac{c}{\sqrt{d x^{2} + c} \sqrt{-c}}\right ) + \sqrt{d x^{2} + c}{\left (b x + 2 \, a\right )} \sqrt{-d}}{2 \, \sqrt{-d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.274964, size = 138, normalized size = 0.86 \[ \frac{2 \, a c \arctan \left (-\frac{\sqrt{d} x - \sqrt{d x^{2} + c}}{\sqrt{-c}}\right ){\rm sign}\left (b x + a\right )}{\sqrt{-c}} - \frac{b c{\rm ln}\left ({\left | -\sqrt{d} x + \sqrt{d x^{2} + c} \right |}\right ){\rm sign}\left (b x + a\right )}{2 \, \sqrt{d}} + \frac{1}{2} \, \sqrt{d x^{2} + c}{\left (b x{\rm sign}\left (b x + a\right ) + 2 \, a{\rm sign}\left (b x + a\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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