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

Optimal. Leaf size=108 \[ \frac{\left (3 a^2 d^2-8 a b c d+8 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{8 b^{5/2}}+\frac{3 d x \sqrt{a+b x^2} (2 b c-a d)}{8 b^2}+\frac{d x \sqrt{a+b x^2} \left (c+d x^2\right )}{4 b} \]

[Out]

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

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Rubi [A]  time = 0.144331, antiderivative size = 108, 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 \[ \frac{\left (3 a^2 d^2-8 a b c d+8 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{8 b^{5/2}}+\frac{3 d x \sqrt{a+b x^2} (2 b c-a d)}{8 b^2}+\frac{d x \sqrt{a+b x^2} \left (c+d x^2\right )}{4 b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 20.4416, size = 102, normalized size = 0.94 \[ \frac{d x \sqrt{a + b x^{2}} \left (c + d x^{2}\right )}{4 b} - \frac{3 d x \sqrt{a + b x^{2}} \left (a d - 2 b c\right )}{8 b^{2}} + \frac{\left (3 a^{2} d^{2} - 8 a b c d + 8 b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )}}{8 b^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0895376, size = 91, normalized size = 0.84 \[ \frac{\left (3 a^2 d^2-8 a b c d+8 b^2 c^2\right ) \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )+\sqrt{b} d x \sqrt{a+b x^2} \left (-3 a d+8 b c+2 b d x^2\right )}{8 b^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.01, size = 131, normalized size = 1.2 \[{{c}^{2}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){\frac{1}{\sqrt{b}}}}+{\frac{{d}^{2}{x}^{3}}{4\,b}\sqrt{b{x}^{2}+a}}-{\frac{3\,a{d}^{2}x}{8\,{b}^{2}}\sqrt{b{x}^{2}+a}}+{\frac{3\,{a}^{2}{d}^{2}}{8}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{5}{2}}}}+{\frac{cdx}{b}\sqrt{b{x}^{2}+a}}-{cda\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.228464, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (2 \, b d^{2} x^{3} +{\left (8 \, b c d - 3 \, a d^{2}\right )} x\right )} \sqrt{b x^{2} + a} \sqrt{b} +{\left (8 \, b^{2} c^{2} - 8 \, a b c d + 3 \, a^{2} d^{2}\right )} \log \left (-2 \, \sqrt{b x^{2} + a} b x -{\left (2 \, b x^{2} + a\right )} \sqrt{b}\right )}{16 \, b^{\frac{5}{2}}}, \frac{{\left (2 \, b d^{2} x^{3} +{\left (8 \, b c d - 3 \, a d^{2}\right )} x\right )} \sqrt{b x^{2} + a} \sqrt{-b} +{\left (8 \, b^{2} c^{2} - 8 \, a b c d + 3 \, a^{2} d^{2}\right )} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right )}{8 \, \sqrt{-b} b^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 20.5179, size = 238, normalized size = 2.2 \[ - \frac{3 a^{\frac{3}{2}} d^{2} x}{8 b^{2} \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{\sqrt{a} c d x \sqrt{1 + \frac{b x^{2}}{a}}}{b} - \frac{\sqrt{a} d^{2} x^{3}}{8 b \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{3 a^{2} d^{2} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{8 b^{\frac{5}{2}}} - \frac{a c d \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{b^{\frac{3}{2}}} + c^{2} \left (\begin{cases} \frac{\sqrt{- \frac{a}{b}} \operatorname{asin}{\left (x \sqrt{- \frac{b}{a}} \right )}}{\sqrt{a}} & \text{for}\: a > 0 \wedge b < 0 \\\frac{\sqrt{\frac{a}{b}} \operatorname{asinh}{\left (x \sqrt{\frac{b}{a}} \right )}}{\sqrt{a}} & \text{for}\: a > 0 \wedge b > 0 \\\frac{\sqrt{- \frac{a}{b}} \operatorname{acosh}{\left (x \sqrt{- \frac{b}{a}} \right )}}{\sqrt{- a}} & \text{for}\: b > 0 \wedge a < 0 \end{cases}\right ) + \frac{d^{2} x^{5}}{4 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.239524, size = 122, normalized size = 1.13 \[ \frac{1}{8} \, \sqrt{b x^{2} + a}{\left (\frac{2 \, d^{2} x^{2}}{b} + \frac{8 \, b^{2} c d - 3 \, a b d^{2}}{b^{3}}\right )} x - \frac{{\left (8 \, b^{2} c^{2} - 8 \, a b c d + 3 \, a^{2} d^{2}\right )}{\rm ln}\left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{8 \, b^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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