3.667 \(\int \frac{\left (a+b x^2\right )^2}{x^3 \left (c+d x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=131 \[ \frac{-\frac{5 a^2 d}{c}+4 a b-\frac{2 b^2 c}{d}}{6 c \left (c+d x^2\right )^{3/2}}-\frac{a^2}{2 c x^2 \left (c+d x^2\right )^{3/2}}-\frac{a (4 b c-5 a d) \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{2 c^{7/2}}+\frac{a (4 b c-5 a d)}{2 c^3 \sqrt{c+d x^2}} \]

[Out]

(4*a*b - (2*b^2*c)/d - (5*a^2*d)/c)/(6*c*(c + d*x^2)^(3/2)) - a^2/(2*c*x^2*(c +
d*x^2)^(3/2)) + (a*(4*b*c - 5*a*d))/(2*c^3*Sqrt[c + d*x^2]) - (a*(4*b*c - 5*a*d)
*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/(2*c^(7/2))

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Rubi [A]  time = 0.335154, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{-\frac{5 a^2 d}{c}+4 a b-\frac{2 b^2 c}{d}}{6 c \left (c+d x^2\right )^{3/2}}-\frac{a^2}{2 c x^2 \left (c+d x^2\right )^{3/2}}-\frac{a (4 b c-5 a d) \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{2 c^{7/2}}+\frac{a (4 b c-5 a d)}{2 c^3 \sqrt{c+d x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)^2/(x^3*(c + d*x^2)^(5/2)),x]

[Out]

(4*a*b - (2*b^2*c)/d - (5*a^2*d)/c)/(6*c*(c + d*x^2)^(3/2)) - a^2/(2*c*x^2*(c +
d*x^2)^(3/2)) + (a*(4*b*c - 5*a*d))/(2*c^3*Sqrt[c + d*x^2]) - (a*(4*b*c - 5*a*d)
*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/(2*c^(7/2))

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Rubi in Sympy [A]  time = 29.4304, size = 121, normalized size = 0.92 \[ - \frac{a^{2}}{2 c x^{2} \left (c + d x^{2}\right )^{\frac{3}{2}}} - \frac{a \left (5 a d - 4 b c\right )}{2 c^{3} \sqrt{c + d x^{2}}} + \frac{a \left (5 a d - 4 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{2}}}{\sqrt{c}} \right )}}{2 c^{\frac{7}{2}}} - \frac{\frac{a d \left (5 a d - 4 b c\right )}{2} + b^{2} c^{2}}{3 c^{2} d \left (c + d x^{2}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**2/x**3/(d*x**2+c)**(5/2),x)

[Out]

-a**2/(2*c*x**2*(c + d*x**2)**(3/2)) - a*(5*a*d - 4*b*c)/(2*c**3*sqrt(c + d*x**2
)) + a*(5*a*d - 4*b*c)*atanh(sqrt(c + d*x**2)/sqrt(c))/(2*c**(7/2)) - (a*d*(5*a*
d - 4*b*c)/2 + b**2*c**2)/(3*c**2*d*(c + d*x**2)**(3/2))

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Mathematica [A]  time = 0.437749, size = 128, normalized size = 0.98 \[ \frac{-\sqrt{c} \sqrt{c+d x^2} \left (\frac{3 a^2}{x^2}+\frac{12 a (a d-b c)}{c+d x^2}+\frac{2 c (b c-a d)^2}{d \left (c+d x^2\right )^2}\right )+3 a (5 a d-4 b c) \log \left (\sqrt{c} \sqrt{c+d x^2}+c\right )-3 a \log (x) (5 a d-4 b c)}{6 c^{7/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2)^2/(x^3*(c + d*x^2)^(5/2)),x]

[Out]

(-(Sqrt[c]*Sqrt[c + d*x^2]*((3*a^2)/x^2 + (2*c*(b*c - a*d)^2)/(d*(c + d*x^2)^2)
+ (12*a*(-(b*c) + a*d))/(c + d*x^2))) - 3*a*(-4*b*c + 5*a*d)*Log[x] + 3*a*(-4*b*
c + 5*a*d)*Log[c + Sqrt[c]*Sqrt[c + d*x^2]])/(6*c^(7/2))

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Maple [A]  time = 0.018, size = 169, normalized size = 1.3 \[ -{\frac{{b}^{2}}{3\,d} \left ( d{x}^{2}+c \right ) ^{-{\frac{3}{2}}}}-{\frac{{a}^{2}}{2\,c{x}^{2}} \left ( d{x}^{2}+c \right ) ^{-{\frac{3}{2}}}}-{\frac{5\,{a}^{2}d}{6\,{c}^{2}} \left ( d{x}^{2}+c \right ) ^{-{\frac{3}{2}}}}-{\frac{5\,{a}^{2}d}{2\,{c}^{3}}{\frac{1}{\sqrt{d{x}^{2}+c}}}}+{\frac{5\,{a}^{2}d}{2}\ln \left ({\frac{1}{x} \left ( 2\,c+2\,\sqrt{c}\sqrt{d{x}^{2}+c} \right ) } \right ){c}^{-{\frac{7}{2}}}}+{\frac{2\,ab}{3\,c} \left ( d{x}^{2}+c \right ) ^{-{\frac{3}{2}}}}+2\,{\frac{ab}{{c}^{2}\sqrt{d{x}^{2}+c}}}-2\,{\frac{ab}{{c}^{5/2}}\ln \left ({\frac{2\,c+2\,\sqrt{c}\sqrt{d{x}^{2}+c}}{x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^2/x^3/(d*x^2+c)^(5/2),x)

[Out]

-1/3*b^2/d/(d*x^2+c)^(3/2)-1/2*a^2/c/x^2/(d*x^2+c)^(3/2)-5/6*a^2*d/c^2/(d*x^2+c)
^(3/2)-5/2*a^2*d/c^3/(d*x^2+c)^(1/2)+5/2*a^2*d/c^(7/2)*ln((2*c+2*c^(1/2)*(d*x^2+
c)^(1/2))/x)+2/3*a*b/c/(d*x^2+c)^(3/2)+2*a*b/c^2/(d*x^2+c)^(1/2)-2*a*b/c^(5/2)*l
n((2*c+2*c^(1/2)*(d*x^2+c)^(1/2))/x)

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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^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.238608, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \,{\left (3 \, a^{2} c^{2} d - 3 \,{\left (4 \, a b c d^{2} - 5 \, a^{2} d^{3}\right )} x^{4} + 2 \,{\left (b^{2} c^{3} - 8 \, a b c^{2} d + 10 \, a^{2} c d^{2}\right )} x^{2}\right )} \sqrt{d x^{2} + c} \sqrt{c} + 3 \,{\left ({\left (4 \, a b c d^{3} - 5 \, a^{2} d^{4}\right )} x^{6} + 2 \,{\left (4 \, a b c^{2} d^{2} - 5 \, a^{2} c d^{3}\right )} x^{4} +{\left (4 \, a b c^{3} d - 5 \, a^{2} c^{2} d^{2}\right )} x^{2}\right )} \log \left (-\frac{{\left (d x^{2} + 2 \, c\right )} \sqrt{c} + 2 \, \sqrt{d x^{2} + c} c}{x^{2}}\right )}{12 \,{\left (c^{3} d^{3} x^{6} + 2 \, c^{4} d^{2} x^{4} + c^{5} d x^{2}\right )} \sqrt{c}}, -\frac{{\left (3 \, a^{2} c^{2} d - 3 \,{\left (4 \, a b c d^{2} - 5 \, a^{2} d^{3}\right )} x^{4} + 2 \,{\left (b^{2} c^{3} - 8 \, a b c^{2} d + 10 \, a^{2} c d^{2}\right )} x^{2}\right )} \sqrt{d x^{2} + c} \sqrt{-c} + 3 \,{\left ({\left (4 \, a b c d^{3} - 5 \, a^{2} d^{4}\right )} x^{6} + 2 \,{\left (4 \, a b c^{2} d^{2} - 5 \, a^{2} c d^{3}\right )} x^{4} +{\left (4 \, a b c^{3} d - 5 \, a^{2} c^{2} d^{2}\right )} x^{2}\right )} \arctan \left (\frac{\sqrt{-c}}{\sqrt{d x^{2} + c}}\right )}{6 \,{\left (c^{3} d^{3} x^{6} + 2 \, c^{4} d^{2} x^{4} + c^{5} d x^{2}\right )} \sqrt{-c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2/((d*x^2 + c)^(5/2)*x^3),x, algorithm="fricas")

[Out]

[-1/12*(2*(3*a^2*c^2*d - 3*(4*a*b*c*d^2 - 5*a^2*d^3)*x^4 + 2*(b^2*c^3 - 8*a*b*c^
2*d + 10*a^2*c*d^2)*x^2)*sqrt(d*x^2 + c)*sqrt(c) + 3*((4*a*b*c*d^3 - 5*a^2*d^4)*
x^6 + 2*(4*a*b*c^2*d^2 - 5*a^2*c*d^3)*x^4 + (4*a*b*c^3*d - 5*a^2*c^2*d^2)*x^2)*l
og(-((d*x^2 + 2*c)*sqrt(c) + 2*sqrt(d*x^2 + c)*c)/x^2))/((c^3*d^3*x^6 + 2*c^4*d^
2*x^4 + c^5*d*x^2)*sqrt(c)), -1/6*((3*a^2*c^2*d - 3*(4*a*b*c*d^2 - 5*a^2*d^3)*x^
4 + 2*(b^2*c^3 - 8*a*b*c^2*d + 10*a^2*c*d^2)*x^2)*sqrt(d*x^2 + c)*sqrt(-c) + 3*(
(4*a*b*c*d^3 - 5*a^2*d^4)*x^6 + 2*(4*a*b*c^2*d^2 - 5*a^2*c*d^3)*x^4 + (4*a*b*c^3
*d - 5*a^2*c^2*d^2)*x^2)*arctan(sqrt(-c)/sqrt(d*x^2 + c)))/((c^3*d^3*x^6 + 2*c^4
*d^2*x^4 + c^5*d*x^2)*sqrt(-c))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + b x^{2}\right )^{2}}{x^{3} \left (c + d x^{2}\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**2/x**3/(d*x**2+c)**(5/2),x)

[Out]

Integral((a + b*x**2)**2/(x**3*(c + d*x**2)**(5/2)), x)

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GIAC/XCAS [A]  time = 0.245765, size = 173, normalized size = 1.32 \[ \frac{{\left (4 \, a b c - 5 \, a^{2} d\right )} \arctan \left (\frac{\sqrt{d x^{2} + c}}{\sqrt{-c}}\right )}{2 \, \sqrt{-c} c^{3}} - \frac{\sqrt{d x^{2} + c} a^{2}}{2 \, c^{3} x^{2}} - \frac{b^{2} c^{3} - 6 \,{\left (d x^{2} + c\right )} a b c d - 2 \, a b c^{2} d + 6 \,{\left (d x^{2} + c\right )} a^{2} d^{2} + a^{2} c d^{2}}{3 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} c^{3} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2/((d*x^2 + c)^(5/2)*x^3),x, algorithm="giac")

[Out]

1/2*(4*a*b*c - 5*a^2*d)*arctan(sqrt(d*x^2 + c)/sqrt(-c))/(sqrt(-c)*c^3) - 1/2*sq
rt(d*x^2 + c)*a^2/(c^3*x^2) - 1/3*(b^2*c^3 - 6*(d*x^2 + c)*a*b*c*d - 2*a*b*c^2*d
+ 6*(d*x^2 + c)*a^2*d^2 + a^2*c*d^2)/((d*x^2 + c)^(3/2)*c^3*d)