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

Optimal. Leaf size=107 \[ \frac{\left (8 a^2 d^2-8 a b c d+3 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{8 d^{5/2}}-\frac{3 b x \sqrt{c+d x^2} (b c-2 a d)}{8 d^2}+\frac{b x \left (a+b x^2\right ) \sqrt{c+d x^2}}{4 d} \]

[Out]

(-3*b*(b*c - 2*a*d)*x*Sqrt[c + d*x^2])/(8*d^2) + (b*x*(a + b*x^2)*Sqrt[c + d*x^2
])/(4*d) + ((3*b^2*c^2 - 8*a*b*c*d + 8*a^2*d^2)*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x
^2]])/(8*d^(5/2))

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Rubi [A]  time = 0.150103, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19 \[ \frac{\left (8 a^2 d^2-8 a b c d+3 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{8 d^{5/2}}-\frac{3 b x \sqrt{c+d x^2} (b c-2 a d)}{8 d^2}+\frac{b x \left (a+b x^2\right ) \sqrt{c+d x^2}}{4 d} \]

Antiderivative was successfully verified.

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

[Out]

(-3*b*(b*c - 2*a*d)*x*Sqrt[c + d*x^2])/(8*d^2) + (b*x*(a + b*x^2)*Sqrt[c + d*x^2
])/(4*d) + ((3*b^2*c^2 - 8*a*b*c*d + 8*a^2*d^2)*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x
^2]])/(8*d^(5/2))

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Rubi in Sympy [A]  time = 20.3841, size = 102, normalized size = 0.95 \[ \frac{b x \left (a + b x^{2}\right ) \sqrt{c + d x^{2}}}{4 d} + \frac{3 b x \sqrt{c + d x^{2}} \left (2 a d - b c\right )}{8 d^{2}} + \frac{\left (8 a^{2} d^{2} - 8 a b c d + 3 b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} x}{\sqrt{c + d x^{2}}} \right )}}{8 d^{\frac{5}{2}}} \]

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)

[Out]

b*x*(a + b*x**2)*sqrt(c + d*x**2)/(4*d) + 3*b*x*sqrt(c + d*x**2)*(2*a*d - b*c)/(
8*d**2) + (8*a**2*d**2 - 8*a*b*c*d + 3*b**2*c**2)*atanh(sqrt(d)*x/sqrt(c + d*x**
2))/(8*d**(5/2))

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Mathematica [A]  time = 0.088147, size = 91, normalized size = 0.85 \[ \frac{\left (8 a^2 d^2-8 a b c d+3 b^2 c^2\right ) \log \left (\sqrt{d} \sqrt{c+d x^2}+d x\right )+b \sqrt{d} x \sqrt{c+d x^2} \left (8 a d-3 b c+2 b d x^2\right )}{8 d^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.011, size = 131, normalized size = 1.2 \[{{a}^{2}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ){\frac{1}{\sqrt{d}}}}+{\frac{{b}^{2}{x}^{3}}{4\,d}\sqrt{d{x}^{2}+c}}-{\frac{3\,{b}^{2}cx}{8\,{d}^{2}}\sqrt{d{x}^{2}+c}}+{\frac{3\,{b}^{2}{c}^{2}}{8}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ){d}^{-{\frac{5}{2}}}}+{\frac{abx}{d}\sqrt{d{x}^{2}+c}}-{abc\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ){d}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a^2*ln(x*d^(1/2)+(d*x^2+c)^(1/2))/d^(1/2)+1/4*b^2*x^3/d*(d*x^2+c)^(1/2)-3/8*b^2*
c/d^2*x*(d*x^2+c)^(1/2)+3/8*b^2*c^2/d^(5/2)*ln(x*d^(1/2)+(d*x^2+c)^(1/2))+a*b*x/
d*(d*x^2+c)^(1/2)-a*b*c/d^(3/2)*ln(x*d^(1/2)+(d*x^2+c)^(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((b*x^2 + a)^2/sqrt(d*x^2 + c),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.264206, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (2 \, b^{2} d x^{3} -{\left (3 \, b^{2} c - 8 \, a b d\right )} x\right )} \sqrt{d x^{2} + c} \sqrt{d} +{\left (3 \, b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} \log \left (-2 \, \sqrt{d x^{2} + c} d x -{\left (2 \, d x^{2} + c\right )} \sqrt{d}\right )}{16 \, d^{\frac{5}{2}}}, \frac{{\left (2 \, b^{2} d x^{3} -{\left (3 \, b^{2} c - 8 \, a b d\right )} x\right )} \sqrt{d x^{2} + c} \sqrt{-d} +{\left (3 \, b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} \arctan \left (\frac{\sqrt{-d} x}{\sqrt{d x^{2} + c}}\right )}{8 \, \sqrt{-d} d^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/16*(2*(2*b^2*d*x^3 - (3*b^2*c - 8*a*b*d)*x)*sqrt(d*x^2 + c)*sqrt(d) + (3*b^2*
c^2 - 8*a*b*c*d + 8*a^2*d^2)*log(-2*sqrt(d*x^2 + c)*d*x - (2*d*x^2 + c)*sqrt(d))
)/d^(5/2), 1/8*((2*b^2*d*x^3 - (3*b^2*c - 8*a*b*d)*x)*sqrt(d*x^2 + c)*sqrt(-d) +
(3*b^2*c^2 - 8*a*b*c*d + 8*a^2*d^2)*arctan(sqrt(-d)*x/sqrt(d*x^2 + c)))/(sqrt(-
d)*d^2)]

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Sympy [A]  time = 20.3893, size = 238, normalized size = 2.22 \[ a^{2} \left (\begin{cases} \frac{\sqrt{- \frac{c}{d}} \operatorname{asin}{\left (x \sqrt{- \frac{d}{c}} \right )}}{\sqrt{c}} & \text{for}\: c > 0 \wedge d < 0 \\\frac{\sqrt{\frac{c}{d}} \operatorname{asinh}{\left (x \sqrt{\frac{d}{c}} \right )}}{\sqrt{c}} & \text{for}\: c > 0 \wedge d > 0 \\\frac{\sqrt{- \frac{c}{d}} \operatorname{acosh}{\left (x \sqrt{- \frac{d}{c}} \right )}}{\sqrt{- c}} & \text{for}\: d > 0 \wedge c < 0 \end{cases}\right ) + \frac{a b \sqrt{c} x \sqrt{1 + \frac{d x^{2}}{c}}}{d} - \frac{a b c \operatorname{asinh}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{d^{\frac{3}{2}}} - \frac{3 b^{2} c^{\frac{3}{2}} x}{8 d^{2} \sqrt{1 + \frac{d x^{2}}{c}}} - \frac{b^{2} \sqrt{c} x^{3}}{8 d \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{3 b^{2} c^{2} \operatorname{asinh}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{8 d^{\frac{5}{2}}} + \frac{b^{2} x^{5}}{4 \sqrt{c} \sqrt{1 + \frac{d x^{2}}{c}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**2*Piecewise((sqrt(-c/d)*asin(x*sqrt(-d/c))/sqrt(c), (c > 0) & (d < 0)), (sqrt
(c/d)*asinh(x*sqrt(d/c))/sqrt(c), (c > 0) & (d > 0)), (sqrt(-c/d)*acosh(x*sqrt(-
d/c))/sqrt(-c), (d > 0) & (c < 0))) + a*b*sqrt(c)*x*sqrt(1 + d*x**2/c)/d - a*b*c
*asinh(sqrt(d)*x/sqrt(c))/d**(3/2) - 3*b**2*c**(3/2)*x/(8*d**2*sqrt(1 + d*x**2/c
)) - b**2*sqrt(c)*x**3/(8*d*sqrt(1 + d*x**2/c)) + 3*b**2*c**2*asinh(sqrt(d)*x/sq
rt(c))/(8*d**(5/2)) + b**2*x**5/(4*sqrt(c)*sqrt(1 + d*x**2/c))

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GIAC/XCAS [A]  time = 0.242061, size = 123, normalized size = 1.15 \[ \frac{1}{8} \,{\left (\frac{2 \, b^{2} x^{2}}{d} - \frac{3 \, b^{2} c d - 8 \, a b d^{2}}{d^{3}}\right )} \sqrt{d x^{2} + c} x - \frac{{\left (3 \, b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )}{\rm ln}\left ({\left | -\sqrt{d} x + \sqrt{d x^{2} + c} \right |}\right )}{8 \, d^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*(2*b^2*x^2/d - (3*b^2*c*d - 8*a*b*d^2)/d^3)*sqrt(d*x^2 + c)*x - 1/8*(3*b^2*c
^2 - 8*a*b*c*d + 8*a^2*d^2)*ln(abs(-sqrt(d)*x + sqrt(d*x^2 + c)))/d^(5/2)