3.91 \(\int \frac{\left (c+d x^2\right )^2}{\left (a+b x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=105 \[ \frac{x (b c-a d) (3 a d+2 b c)}{3 a^2 b^2 \sqrt{a+b x^2}}+\frac{d^2 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{b^{5/2}}+\frac{x \left (c+d x^2\right ) (b c-a d)}{3 a b \left (a+b x^2\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.133348, antiderivative size = 105, 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 \[ \frac{x (b c-a d) (3 a d+2 b c)}{3 a^2 b^2 \sqrt{a+b x^2}}+\frac{d^2 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{b^{5/2}}+\frac{x \left (c+d x^2\right ) (b c-a d)}{3 a b \left (a+b x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x^2)^2/(a + b*x^2)^(5/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 23.8377, size = 94, normalized size = 0.9 \[ \frac{d^{2} \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )}}{b^{\frac{5}{2}}} - \frac{x \left (c + d x^{2}\right ) \left (a d - b c\right )}{3 a b \left (a + b x^{2}\right )^{\frac{3}{2}}} - \frac{x \left (a d - b c\right ) \left (3 a d + 2 b c\right )}{3 a^{2} b^{2} \sqrt{a + b x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x**2+c)**2/(b*x**2+a)**(5/2),x)

[Out]

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

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Mathematica [A]  time = 0.219513, size = 97, normalized size = 0.92 \[ \frac{x \left (2 \left (a+b x^2\right ) \left (-2 a^2 d^2+a b c d+b^2 c^2\right )+a (b c-a d)^2\right )}{3 a^2 b^2 \left (a+b x^2\right )^{3/2}}+\frac{d^2 \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )}{b^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x^2)^2/(a + b*x^2)^(5/2),x]

[Out]

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

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Maple [A]  time = 0.01, size = 136, normalized size = 1.3 \[{\frac{{c}^{2}x}{3\,a} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}+{\frac{2\,{c}^{2}x}{3\,{a}^{2}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}-{\frac{{d}^{2}{x}^{3}}{3\,b} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}-{\frac{{d}^{2}x}{{b}^{2}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}+{{d}^{2}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{5}{2}}}}-{\frac{2\,cdx}{3\,b} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}+{\frac{2\,cdx}{3\,ab}{\frac{1}{\sqrt{b{x}^{2}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x^2+c)^2/(b*x^2+a)^(5/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.236081, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (2 \,{\left (b^{3} c^{2} + a b^{2} c d - 2 \, a^{2} b d^{2}\right )} x^{3} + 3 \,{\left (a b^{2} c^{2} - a^{3} d^{2}\right )} x\right )} \sqrt{b x^{2} + a} \sqrt{b} + 3 \,{\left (a^{2} b^{2} d^{2} x^{4} + 2 \, a^{3} b d^{2} x^{2} + a^{4} d^{2}\right )} \log \left (-2 \, \sqrt{b x^{2} + a} b x -{\left (2 \, b x^{2} + a\right )} \sqrt{b}\right )}{6 \,{\left (a^{2} b^{4} x^{4} + 2 \, a^{3} b^{3} x^{2} + a^{4} b^{2}\right )} \sqrt{b}}, \frac{{\left (2 \,{\left (b^{3} c^{2} + a b^{2} c d - 2 \, a^{2} b d^{2}\right )} x^{3} + 3 \,{\left (a b^{2} c^{2} - a^{3} d^{2}\right )} x\right )} \sqrt{b x^{2} + a} \sqrt{-b} + 3 \,{\left (a^{2} b^{2} d^{2} x^{4} + 2 \, a^{3} b d^{2} x^{2} + a^{4} d^{2}\right )} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right )}{3 \,{\left (a^{2} b^{4} x^{4} + 2 \, a^{3} b^{3} x^{2} + a^{4} b^{2}\right )} \sqrt{-b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)^2/(b*x^2 + a)^(5/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (c + d x^{2}\right )^{2}}{\left (a + b x^{2}\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x**2+c)**2/(b*x**2+a)**(5/2),x)

[Out]

Integral((c + d*x**2)**2/(a + b*x**2)**(5/2), x)

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GIAC/XCAS [A]  time = 0.232037, size = 139, normalized size = 1.32 \[ \frac{x{\left (\frac{2 \,{\left (b^{4} c^{2} + a b^{3} c d - 2 \, a^{2} b^{2} d^{2}\right )} x^{2}}{a^{2} b^{3}} + \frac{3 \,{\left (a b^{3} c^{2} - a^{3} b d^{2}\right )}}{a^{2} b^{3}}\right )}}{3 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}}} - \frac{d^{2}{\rm ln}\left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{b^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)^2/(b*x^2 + a)^(5/2),x, algorithm="giac")

[Out]

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