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

Optimal. Leaf size=148 \[ \frac{b \sqrt{a^2+2 a b x+b^2 x^2} \left (c+d x^2\right )^{3/2}}{3 d (a+b x)}+\frac{a x \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{c+d x^2}}{2 (a+b x)}+\frac{a c \sqrt{a^2+2 a b x+b^2 x^2} \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{2 \sqrt{d} (a+b x)} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.149308, antiderivative size = 148, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.156 \[ \frac{b \sqrt{a^2+2 a b x+b^2 x^2} \left (c+d x^2\right )^{3/2}}{3 d (a+b x)}+\frac{a x \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{c+d x^2}}{2 (a+b x)}+\frac{a c \sqrt{a^2+2 a b x+b^2 x^2} \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{2 \sqrt{d} (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 26.1897, size = 136, normalized size = 0.92 \[ \frac{a c \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \operatorname{atanh}{\left (\frac{\sqrt{d} x}{\sqrt{c + d x^{2}}} \right )}}{2 \sqrt{d} \left (a + b x\right )} + \frac{a x \sqrt{c + d x^{2}} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{2 \left (a + b x\right )} + \frac{b \left (c + d x^{2}\right )^{\frac{3}{2}} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{3 d \left (a + b x\right )} \]

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)

[Out]

a*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)) + a*x*sqrt(c + d*x**2)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(2*(a + b*x
)) + b*(c + d*x**2)**(3/2)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(3*d*(a + b*x))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0746258, size = 85, normalized size = 0.57 \[ \frac{\sqrt{(a+b x)^2} \left (\sqrt{c+d x^2} \left (3 a d x+2 b \left (c+d x^2\right )\right )+3 a c \sqrt{d} \log \left (\sqrt{d} \sqrt{c+d x^2}+d x\right )\right )}{6 d (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[(a + b*x)^2]*(Sqrt[c + d*x^2]*(3*a*d*x + 2*b*(c + d*x^2)) + 3*a*c*Sqrt[d]*
Log[d*x + Sqrt[d]*Sqrt[c + d*x^2]]))/(6*d*(a + b*x))

_______________________________________________________________________________________

Maple [C]  time = 0.013, size = 65, normalized size = 0.4 \[{\frac{{\it csgn} \left ( bx+a \right ) }{6} \left ( 2\,b \left ( d{x}^{2}+c \right ) ^{3/2}\sqrt{d}+3\,ax\sqrt{d{x}^{2}+c}{d}^{3/2}+3\,ac\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ) d \right ){d}^{-{\frac{3}{2}}}} \]

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)

[Out]

1/6*csgn(b*x+a)*(2*b*(d*x^2+c)^(3/2)*d^(1/2)+3*a*x*(d*x^2+c)^(1/2)*d^(3/2)+3*a*c
*ln(x*d^(1/2)+(d*x^2+c)^(1/2))*d)/d^(3/2)

_______________________________________________________________________________________

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, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.339508, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, a c d \log \left (-2 \, \sqrt{d x^{2} + c} d x -{\left (2 \, d x^{2} + c\right )} \sqrt{d}\right ) + 2 \,{\left (2 \, b d x^{2} + 3 \, a d x + 2 \, b c\right )} \sqrt{d x^{2} + c} \sqrt{d}}{12 \, d^{\frac{3}{2}}}, \frac{3 \, a c d \arctan \left (\frac{\sqrt{-d} x}{\sqrt{d x^{2} + c}}\right ) +{\left (2 \, b d x^{2} + 3 \, a d x + 2 \, b c\right )} \sqrt{d x^{2} + c} \sqrt{-d}}{6 \, \sqrt{-d} d}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/12*(3*a*c*d*log(-2*sqrt(d*x^2 + c)*d*x - (2*d*x^2 + c)*sqrt(d)) + 2*(2*b*d*x^
2 + 3*a*d*x + 2*b*c)*sqrt(d*x^2 + c)*sqrt(d))/d^(3/2), 1/6*(3*a*c*d*arctan(sqrt(
-d)*x/sqrt(d*x^2 + c)) + (2*b*d*x^2 + 3*a*d*x + 2*b*c)*sqrt(d*x^2 + c)*sqrt(-d))
/(sqrt(-d)*d)]

_______________________________________________________________________________________

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

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)

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.271617, size = 107, normalized size = 0.72 \[ -\frac{a c{\rm ln}\left ({\left | -\sqrt{d} x + \sqrt{d x^{2} + c} \right |}\right ){\rm sign}\left (b x + a\right )}{2 \, \sqrt{d}} + \frac{1}{6} \, \sqrt{d x^{2} + c}{\left ({\left (2 \, b x{\rm sign}\left (b x + a\right ) + 3 \, a{\rm sign}\left (b x + a\right )\right )} x + \frac{2 \, b c{\rm sign}\left (b x + a\right )}{d}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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