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

Optimal. Leaf size=90 \[ \frac{d (4 b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{2 b^{5/2}}+\frac{x (b c-a d)^2}{a b^2 \sqrt{a+b x^2}}+\frac{d^2 x \sqrt{a+b x^2}}{2 b^2} \]

[Out]

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

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Rubi [A]  time = 0.154494, antiderivative size = 105, normalized size of antiderivative = 1.17, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19 \[ \frac{d (4 b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{2 b^{5/2}}-\frac{d x \sqrt{a+b x^2} (2 b c-3 a d)}{2 a b^2}+\frac{x \left (c+d x^2\right ) (b c-a d)}{a b \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 20.4238, size = 95, normalized size = 1.06 \[ - \frac{d \left (3 a d - 4 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )}}{2 b^{\frac{5}{2}}} - \frac{x \left (c + d x^{2}\right ) \left (a d - b c\right )}{a b \sqrt{a + b x^{2}}} + \frac{d x \sqrt{a + b x^{2}} \left (3 a d - 2 b c\right )}{2 a b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.153965, size = 89, normalized size = 0.99 \[ \frac{\sqrt{b} x \sqrt{a+b x^2} \left (\frac{2 (b c-a d)^2}{a \left (a+b x^2\right )}+d^2\right )+d (4 b c-3 a d) \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )}{2 b^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.01, size = 123, normalized size = 1.4 \[{\frac{{c}^{2}x}{a}{\frac{1}{\sqrt{b{x}^{2}+a}}}}+{\frac{{d}^{2}{x}^{3}}{2\,b}{\frac{1}{\sqrt{b{x}^{2}+a}}}}+{\frac{3\,a{d}^{2}x}{2\,{b}^{2}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}-{\frac{3\,a{d}^{2}}{2}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{5}{2}}}}-2\,{\frac{cdx}{b\sqrt{b{x}^{2}+a}}}+2\,{\frac{cd\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ) }{{b}^{3/2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.229807, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (a b d^{2} x^{3} +{\left (2 \, b^{2} c^{2} - 4 \, a b c d + 3 \, a^{2} d^{2}\right )} x\right )} \sqrt{b x^{2} + a} \sqrt{b} -{\left (4 \, a^{2} b c d - 3 \, a^{3} d^{2} +{\left (4 \, a b^{2} c d - 3 \, a^{2} b d^{2}\right )} x^{2}\right )} \log \left (2 \, \sqrt{b x^{2} + a} b x -{\left (2 \, b x^{2} + a\right )} \sqrt{b}\right )}{4 \,{\left (a b^{3} x^{2} + a^{2} b^{2}\right )} \sqrt{b}}, \frac{{\left (a b d^{2} x^{3} +{\left (2 \, b^{2} c^{2} - 4 \, a b c d + 3 \, a^{2} d^{2}\right )} x\right )} \sqrt{b x^{2} + a} \sqrt{-b} +{\left (4 \, a^{2} b c d - 3 \, a^{3} d^{2} +{\left (4 \, a b^{2} c d - 3 \, a^{2} b d^{2}\right )} x^{2}\right )} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right )}{2 \,{\left (a b^{3} x^{2} + a^{2} b^{2}\right )} \sqrt{-b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (c + d x^{2}\right )^{2}}{\left (a + b x^{2}\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.244329, size = 124, normalized size = 1.38 \[ \frac{{\left (\frac{d^{2} x^{2}}{b} + \frac{2 \, b^{3} c^{2} - 4 \, a b^{2} c d + 3 \, a^{2} b d^{2}}{a b^{3}}\right )} x}{2 \, \sqrt{b x^{2} + a}} - \frac{{\left (4 \, b c d - 3 \, a d^{2}\right )}{\rm ln}\left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{2 \, b^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(d^2*x^2/b + (2*b^3*c^2 - 4*a*b^2*c*d + 3*a^2*b*d^2)/(a*b^3))*x/sqrt(b*x^2 +
a) - 1/2*(4*b*c*d - 3*a*d^2)*ln(abs(-sqrt(b)*x + sqrt(b*x^2 + a)))/b^(5/2)