3.602 \(\int \frac{\left (a+b x^2\right )^2 \sqrt{c+d x^2}}{x^3} \, dx\)

Optimal. Leaf size=109 \[ -\frac{a^2 \left (c+d x^2\right )^{3/2}}{2 c x^2}+\frac{a \sqrt{c+d x^2} (a d+4 b c)}{2 c}-\frac{a (a d+4 b c) \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{2 \sqrt{c}}+\frac{b^2 \left (c+d x^2\right )^{3/2}}{3 d} \]

[Out]

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

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Rubi [A]  time = 0.264589, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{a^2 \left (c+d x^2\right )^{3/2}}{2 c x^2}+\frac{a \sqrt{c+d x^2} (a d+4 b c)}{2 c}-\frac{a (a d+4 b c) \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{2 \sqrt{c}}+\frac{b^2 \left (c+d x^2\right )^{3/2}}{3 d} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 24.7308, size = 94, normalized size = 0.86 \[ - \frac{a^{2} \left (c + d x^{2}\right )^{\frac{3}{2}}}{2 c x^{2}} + \frac{a \sqrt{c + d x^{2}} \left (a d + 4 b c\right )}{2 c} - \frac{a \left (a d + 4 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{2}}}{\sqrt{c}} \right )}}{2 \sqrt{c}} + \frac{b^{2} \left (c + d x^{2}\right )^{\frac{3}{2}}}{3 d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**2*(d*x**2+c)**(1/2)/x**3,x)

[Out]

-a**2*(c + d*x**2)**(3/2)/(2*c*x**2) + a*sqrt(c + d*x**2)*(a*d + 4*b*c)/(2*c) -
a*(a*d + 4*b*c)*atanh(sqrt(c + d*x**2)/sqrt(c))/(2*sqrt(c)) + b**2*(c + d*x**2)*
*(3/2)/(3*d)

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Mathematica [A]  time = 0.223724, size = 103, normalized size = 0.94 \[ \sqrt{c+d x^2} \left (-\frac{a^2}{2 x^2}+2 a b+\frac{b^2 \left (c+d x^2\right )}{3 d}\right )-\frac{a (a d+4 b c) \log \left (\sqrt{c} \sqrt{c+d x^2}+c\right )}{2 \sqrt{c}}+\frac{a \log (x) (a d+4 b c)}{2 \sqrt{c}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.016, size = 132, normalized size = 1.2 \[{\frac{{b}^{2}}{3\,d} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}-{\frac{{a}^{2}}{2\,c{x}^{2}} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}-{\frac{{a}^{2}d}{2}\ln \left ({\frac{1}{x} \left ( 2\,c+2\,\sqrt{c}\sqrt{d{x}^{2}+c} \right ) } \right ){\frac{1}{\sqrt{c}}}}+{\frac{{a}^{2}d}{2\,c}\sqrt{d{x}^{2}+c}}-2\,\ln \left ({\frac{2\,c+2\,\sqrt{c}\sqrt{d{x}^{2}+c}}{x}} \right ) \sqrt{c}ab+2\,\sqrt{d{x}^{2}+c}ab \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^2*(d*x^2+c)^(1/2)/x^3,x)

[Out]

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

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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((b*x^2 + a)^2*sqrt(d*x^2 + c)/x^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.240473, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (4 \, a b c d + a^{2} d^{2}\right )} x^{2} \log \left (-\frac{{\left (d x^{2} + 2 \, c\right )} \sqrt{c} - 2 \, \sqrt{d x^{2} + c} c}{x^{2}}\right ) + 2 \,{\left (2 \, b^{2} d x^{4} - 3 \, a^{2} d + 2 \,{\left (b^{2} c + 6 \, a b d\right )} x^{2}\right )} \sqrt{d x^{2} + c} \sqrt{c}}{12 \, \sqrt{c} d x^{2}}, -\frac{3 \,{\left (4 \, a b c d + a^{2} d^{2}\right )} x^{2} \arctan \left (\frac{\sqrt{-c}}{\sqrt{d x^{2} + c}}\right ) -{\left (2 \, b^{2} d x^{4} - 3 \, a^{2} d + 2 \,{\left (b^{2} c + 6 \, a b d\right )} x^{2}\right )} \sqrt{d x^{2} + c} \sqrt{-c}}{6 \, \sqrt{-c} d x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2*sqrt(d*x^2 + c)/x^3,x, algorithm="fricas")

[Out]

[1/12*(3*(4*a*b*c*d + a^2*d^2)*x^2*log(-((d*x^2 + 2*c)*sqrt(c) - 2*sqrt(d*x^2 +
c)*c)/x^2) + 2*(2*b^2*d*x^4 - 3*a^2*d + 2*(b^2*c + 6*a*b*d)*x^2)*sqrt(d*x^2 + c)
*sqrt(c))/(sqrt(c)*d*x^2), -1/6*(3*(4*a*b*c*d + a^2*d^2)*x^2*arctan(sqrt(-c)/sqr
t(d*x^2 + c)) - (2*b^2*d*x^4 - 3*a^2*d + 2*(b^2*c + 6*a*b*d)*x^2)*sqrt(d*x^2 + c
)*sqrt(-c))/(sqrt(-c)*d*x^2)]

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Sympy [A]  time = 31.6659, size = 148, normalized size = 1.36 \[ - \frac{a^{2} \sqrt{d} \sqrt{\frac{c}{d x^{2}} + 1}}{2 x} - \frac{a^{2} d \operatorname{asinh}{\left (\frac{\sqrt{c}}{\sqrt{d} x} \right )}}{2 \sqrt{c}} - 2 a b \sqrt{c} \operatorname{asinh}{\left (\frac{\sqrt{c}}{\sqrt{d} x} \right )} + \frac{2 a b c}{\sqrt{d} x \sqrt{\frac{c}{d x^{2}} + 1}} + \frac{2 a b \sqrt{d} x}{\sqrt{\frac{c}{d x^{2}} + 1}} + b^{2} \left (\begin{cases} \frac{\sqrt{c} x^{2}}{2} & \text{for}\: d = 0 \\\frac{\left (c + d x^{2}\right )^{\frac{3}{2}}}{3 d} & \text{otherwise} \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**2*(d*x**2+c)**(1/2)/x**3,x)

[Out]

-a**2*sqrt(d)*sqrt(c/(d*x**2) + 1)/(2*x) - a**2*d*asinh(sqrt(c)/(sqrt(d)*x))/(2*
sqrt(c)) - 2*a*b*sqrt(c)*asinh(sqrt(c)/(sqrt(d)*x)) + 2*a*b*c/(sqrt(d)*x*sqrt(c/
(d*x**2) + 1)) + 2*a*b*sqrt(d)*x/sqrt(c/(d*x**2) + 1) + b**2*Piecewise((sqrt(c)*
x**2/2, Eq(d, 0)), ((c + d*x**2)**(3/2)/(3*d), True))

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GIAC/XCAS [A]  time = 0.229893, size = 120, normalized size = 1.1 \[ \frac{2 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} b^{2} + 12 \, \sqrt{d x^{2} + c} a b d - \frac{3 \, \sqrt{d x^{2} + c} a^{2} d}{x^{2}} + \frac{3 \,{\left (4 \, a b c d + a^{2} d^{2}\right )} \arctan \left (\frac{\sqrt{d x^{2} + c}}{\sqrt{-c}}\right )}{\sqrt{-c}}}{6 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2*sqrt(d*x^2 + c)/x^3,x, algorithm="giac")

[Out]

1/6*(2*(d*x^2 + c)^(3/2)*b^2 + 12*sqrt(d*x^2 + c)*a*b*d - 3*sqrt(d*x^2 + c)*a^2*
d/x^2 + 3*(4*a*b*c*d + a^2*d^2)*arctan(sqrt(d*x^2 + c)/sqrt(-c))/sqrt(-c))/d