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

Optimal. Leaf size=88 \[ \frac{\frac{a^2}{c^2}-\frac{b^2}{d^2}}{\sqrt{c+d x^2}}-\frac{a^2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{c^{5/2}}+\frac{(b c-a d)^2}{3 c d^2 \left (c+d x^2\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.239524, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{\frac{a^2}{c^2}-\frac{b^2}{d^2}}{\sqrt{c+d x^2}}-\frac{a^2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{c^{5/2}}+\frac{(b c-a d)^2}{3 c d^2 \left (c+d x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 40.5884, size = 73, normalized size = 0.83 \[ - \frac{a^{2} \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{2}}}{\sqrt{c}} \right )}}{c^{\frac{5}{2}}} + \frac{\frac{a^{2}}{c^{2}} - \frac{b^{2}}{d^{2}}}{\sqrt{c + d x^{2}}} + \frac{\left (a d - b c\right )^{2}}{3 c d^{2} \left (c + d x^{2}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.444759, size = 106, normalized size = 1.2 \[ \frac{\frac{\sqrt{c} \left (a^2 d^2 \left (4 c+3 d x^2\right )-2 a b c^2 d-b^2 c^2 \left (2 c+3 d x^2\right )\right )}{d^2 \left (c+d x^2\right )^{3/2}}-3 a^2 \log \left (\sqrt{c} \sqrt{c+d x^2}+c\right )+3 a^2 \log (x)}{3 c^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.014, size = 120, normalized size = 1.4 \[{\frac{{a}^{2}}{3\,c} \left ( d{x}^{2}+c \right ) ^{-{\frac{3}{2}}}}+{\frac{{a}^{2}}{{c}^{2}}{\frac{1}{\sqrt{d{x}^{2}+c}}}}-{{a}^{2}\ln \left ({\frac{1}{x} \left ( 2\,c+2\,\sqrt{c}\sqrt{d{x}^{2}+c} \right ) } \right ){c}^{-{\frac{5}{2}}}}-{\frac{{b}^{2}{x}^{2}}{d} \left ( d{x}^{2}+c \right ) ^{-{\frac{3}{2}}}}-{\frac{2\,{b}^{2}c}{3\,{d}^{2}} \left ( d{x}^{2}+c \right ) ^{-{\frac{3}{2}}}}-{\frac{2\,ab}{3\,d} \left ( d{x}^{2}+c \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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/((d*x^2 + c)^(5/2)*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.231119, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \,{\left (2 \, b^{2} c^{3} + 2 \, a b c^{2} d - 4 \, a^{2} c d^{2} + 3 \,{\left (b^{2} c^{2} d - a^{2} d^{3}\right )} x^{2}\right )} \sqrt{d x^{2} + c} \sqrt{c} - 3 \,{\left (a^{2} d^{4} x^{4} + 2 \, a^{2} c d^{3} x^{2} + a^{2} c^{2} d^{2}\right )} \log \left (-\frac{{\left (d x^{2} + 2 \, c\right )} \sqrt{c} - 2 \, \sqrt{d x^{2} + c} c}{x^{2}}\right )}{6 \,{\left (c^{2} d^{4} x^{4} + 2 \, c^{3} d^{3} x^{2} + c^{4} d^{2}\right )} \sqrt{c}}, -\frac{{\left (2 \, b^{2} c^{3} + 2 \, a b c^{2} d - 4 \, a^{2} c d^{2} + 3 \,{\left (b^{2} c^{2} d - a^{2} d^{3}\right )} x^{2}\right )} \sqrt{d x^{2} + c} \sqrt{-c} + 3 \,{\left (a^{2} d^{4} x^{4} + 2 \, a^{2} c d^{3} x^{2} + a^{2} c^{2} d^{2}\right )} \arctan \left (\frac{\sqrt{-c}}{\sqrt{d x^{2} + c}}\right )}{3 \,{\left (c^{2} d^{4} x^{4} + 2 \, c^{3} d^{3} x^{2} + c^{4} d^{2}\right )} \sqrt{-c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.238783, size = 138, normalized size = 1.57 \[ \frac{a^{2} \arctan \left (\frac{\sqrt{d x^{2} + c}}{\sqrt{-c}}\right )}{\sqrt{-c} c^{2}} - \frac{3 \,{\left (d x^{2} + c\right )} b^{2} c^{2} - b^{2} c^{3} + 2 \, a b c^{2} d - 3 \,{\left (d x^{2} + c\right )} a^{2} d^{2} - a^{2} c d^{2}}{3 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} c^{2} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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