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

Optimal. Leaf size=212 \[ -\frac{a c^2 \sqrt{a^2+2 a b x+b^2 x^2} \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{8 d^{3/2} (a+b x)}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} \left (c+d x^2\right )^{3/2} (8 b c-15 a d x)}{60 d^2 (a+b x)}-\frac{a c x \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{c+d x^2}}{8 d (a+b x)}+\frac{b x^2 \sqrt{a^2+2 a b x+b^2 x^2} \left (c+d x^2\right )^{3/2}}{5 d (a+b x)} \]

[Out]

-(a*c*x*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*Sqrt[c + d*x^2])/(8*d*(a + b*x)) + (b*x^2*
Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(c + d*x^2)^(3/2))/(5*d*(a + b*x)) - ((8*b*c - 15*
a*d*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(c + d*x^2)^(3/2))/(60*d^2*(a + b*x)) - (a*
c^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/(8*d^(3/
2)*(a + b*x))

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Rubi [A]  time = 0.393442, antiderivative size = 212, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.171 \[ -\frac{a c^2 \sqrt{a^2+2 a b x+b^2 x^2} \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{8 d^{3/2} (a+b x)}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} \left (c+d x^2\right )^{3/2} (8 b c-15 a d x)}{60 d^2 (a+b x)}-\frac{a c x \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{c+d x^2}}{8 d (a+b x)}+\frac{b x^2 \sqrt{a^2+2 a b x+b^2 x^2} \left (c+d x^2\right )^{3/2}}{5 d (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

-(a*c*x*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*Sqrt[c + d*x^2])/(8*d*(a + b*x)) + (b*x^2*
Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(c + d*x^2)^(3/2))/(5*d*(a + b*x)) - ((8*b*c - 15*
a*d*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(c + d*x^2)^(3/2))/(60*d^2*(a + b*x)) - (a*
c^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/(8*d^(3/
2)*(a + b*x))

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Rubi in Sympy [A]  time = 40.4385, size = 196, normalized size = 0.92 \[ - \frac{a c^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \operatorname{atanh}{\left (\frac{\sqrt{d} x}{\sqrt{c + d x^{2}}} \right )}}{8 d^{\frac{3}{2}} \left (a + b x\right )} - \frac{a c x \sqrt{c + d x^{2}} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{8 d \left (a + b x\right )} + \frac{b x^{2} \left (c + d x^{2}\right )^{\frac{3}{2}} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{5 d \left (a + b x\right )} - \frac{\left (c + d x^{2}\right )^{\frac{3}{2}} \left (- 30 a d x + 16 b c\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{120 d^{2} \left (a + b x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a*c**2*sqrt(a**2 + 2*a*b*x + b**2*x**2)*atanh(sqrt(d)*x/sqrt(c + d*x**2))/(8*d*
*(3/2)*(a + b*x)) - a*c*x*sqrt(c + d*x**2)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(8*d
*(a + b*x)) + b*x**2*(c + d*x**2)**(3/2)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(5*d*(
a + b*x)) - (c + d*x**2)**(3/2)*(-30*a*d*x + 16*b*c)*sqrt(a**2 + 2*a*b*x + b**2*
x**2)/(120*d**2*(a + b*x))

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Mathematica [A]  time = 0.128794, size = 108, normalized size = 0.51 \[ \frac{\sqrt{(a+b x)^2} \left (\sqrt{c+d x^2} \left (15 a d x \left (c+2 d x^2\right )+8 b \left (-2 c^2+c d x^2+3 d^2 x^4\right )\right )-15 a c^2 \sqrt{d} \log \left (\sqrt{d} \sqrt{c+d x^2}+d x\right )\right )}{120 d^2 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2*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]*(15*a*d*x*(c + 2*d*x^2) + 8*b*(-2*c^2 + c*d*
x^2 + 3*d^2*x^4)) - 15*a*c^2*Sqrt[d]*Log[d*x + Sqrt[d]*Sqrt[c + d*x^2]]))/(120*d
^2*(a + b*x))

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Maple [C]  time = 0.043, size = 105, normalized size = 0.5 \[{\frac{{\it csgn} \left ( bx+a \right ) }{120} \left ( 24\,{d}^{5/2} \left ( d{x}^{2}+c \right ) ^{3/2}{x}^{2}b+30\,ax \left ( d{x}^{2}+c \right ) ^{3/2}{d}^{5/2}-16\,{d}^{3/2} \left ( d{x}^{2}+c \right ) ^{3/2}bc-15\,acx\sqrt{d{x}^{2}+c}{d}^{5/2}-15\,a{c}^{2}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ){d}^{2} \right ){d}^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/120*csgn(b*x+a)*(24*d^(5/2)*(d*x^2+c)^(3/2)*x^2*b+30*a*x*(d*x^2+c)^(3/2)*d^(5/
2)-16*d^(3/2)*(d*x^2+c)^(3/2)*b*c-15*a*c*x*(d*x^2+c)^(1/2)*d^(5/2)-15*a*c^2*ln(x
*d^(1/2)+(d*x^2+c)^(1/2))*d^2)/d^(7/2)

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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^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.323893, size = 1, normalized size = 0. \[ \left [\frac{15 \, a c^{2} d \log \left (2 \, \sqrt{d x^{2} + c} d x -{\left (2 \, d x^{2} + c\right )} \sqrt{d}\right ) + 2 \,{\left (24 \, b d^{2} x^{4} + 30 \, a d^{2} x^{3} + 8 \, b c d x^{2} + 15 \, a c d x - 16 \, b c^{2}\right )} \sqrt{d x^{2} + c} \sqrt{d}}{240 \, d^{\frac{5}{2}}}, -\frac{15 \, a c^{2} d \arctan \left (\frac{\sqrt{-d} x}{\sqrt{d x^{2} + c}}\right ) -{\left (24 \, b d^{2} x^{4} + 30 \, a d^{2} x^{3} + 8 \, b c d x^{2} + 15 \, a c d x - 16 \, b c^{2}\right )} \sqrt{d x^{2} + c} \sqrt{-d}}{120 \, \sqrt{-d} d^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/240*(15*a*c^2*d*log(2*sqrt(d*x^2 + c)*d*x - (2*d*x^2 + c)*sqrt(d)) + 2*(24*b*
d^2*x^4 + 30*a*d^2*x^3 + 8*b*c*d*x^2 + 15*a*c*d*x - 16*b*c^2)*sqrt(d*x^2 + c)*sq
rt(d))/d^(5/2), -1/120*(15*a*c^2*d*arctan(sqrt(-d)*x/sqrt(d*x^2 + c)) - (24*b*d^
2*x^4 + 30*a*d^2*x^3 + 8*b*c*d*x^2 + 15*a*c*d*x - 16*b*c^2)*sqrt(d*x^2 + c)*sqrt
(-d))/(sqrt(-d)*d^2)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.269735, size = 158, normalized size = 0.75 \[ \frac{a c^{2}{\rm ln}\left ({\left | -\sqrt{d} x + \sqrt{d x^{2} + c} \right |}\right ){\rm sign}\left (b x + a\right )}{8 \, d^{\frac{3}{2}}} + \frac{1}{120} \, \sqrt{d x^{2} + c}{\left ({\left (2 \,{\left (3 \,{\left (4 \, b x{\rm sign}\left (b x + a\right ) + 5 \, a{\rm sign}\left (b x + a\right )\right )} x + \frac{4 \, b c{\rm sign}\left (b x + a\right )}{d}\right )} x + \frac{15 \, a c{\rm sign}\left (b x + a\right )}{d}\right )} x - \frac{16 \, b c^{2}{\rm sign}\left (b x + a\right )}{d^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*a*c^2*ln(abs(-sqrt(d)*x + sqrt(d*x^2 + c)))*sign(b*x + a)/d^(3/2) + 1/120*sq
rt(d*x^2 + c)*((2*(3*(4*b*x*sign(b*x + a) + 5*a*sign(b*x + a))*x + 4*b*c*sign(b*
x + a)/d)*x + 15*a*c*sign(b*x + a)/d)*x - 16*b*c^2*sign(b*x + a)/d^2)