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

Optimal. Leaf size=185 \[ -\frac{a^2}{4 c x^4 \left (c+d x^2\right )^{3/2}}-\frac{\left (8 b^2 c^2-5 a d (8 b c-7 a d)\right ) \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{8 c^{9/2}}+\frac{8 b^2 c^2-5 a d (8 b c-7 a d)}{8 c^4 \sqrt{c+d x^2}}+\frac{8 b^2 c^2-5 a d (8 b c-7 a d)}{24 c^3 \left (c+d x^2\right )^{3/2}}-\frac{a (8 b c-7 a d)}{8 c^2 x^2 \left (c+d x^2\right )^{3/2}} \]

[Out]

(8*b^2*c^2 - 5*a*d*(8*b*c - 7*a*d))/(24*c^3*(c + d*x^2)^(3/2)) - a^2/(4*c*x^4*(c
+ d*x^2)^(3/2)) - (a*(8*b*c - 7*a*d))/(8*c^2*x^2*(c + d*x^2)^(3/2)) + (8*b^2*c^
2 - 5*a*d*(8*b*c - 7*a*d))/(8*c^4*Sqrt[c + d*x^2]) - ((8*b^2*c^2 - 5*a*d*(8*b*c
- 7*a*d))*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/(8*c^(9/2))

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Rubi [A]  time = 0.533453, antiderivative size = 185, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{a^2}{4 c x^4 \left (c+d x^2\right )^{3/2}}+\frac{8 b^2-\frac{5 a d (8 b c-7 a d)}{c^2}}{24 c \left (c+d x^2\right )^{3/2}}-\frac{\left (8 b^2 c^2-5 a d (8 b c-7 a d)\right ) \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{8 c^{9/2}}+\frac{8 b^2 c^2-5 a d (8 b c-7 a d)}{8 c^4 \sqrt{c+d x^2}}-\frac{a (8 b c-7 a d)}{8 c^2 x^2 \left (c+d x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(8*b^2 - (5*a*d*(8*b*c - 7*a*d))/c^2)/(24*c*(c + d*x^2)^(3/2)) - a^2/(4*c*x^4*(c
+ d*x^2)^(3/2)) - (a*(8*b*c - 7*a*d))/(8*c^2*x^2*(c + d*x^2)^(3/2)) + (8*b^2*c^
2 - 5*a*d*(8*b*c - 7*a*d))/(8*c^4*Sqrt[c + d*x^2]) - ((8*b^2*c^2 - 5*a*d*(8*b*c
- 7*a*d))*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/(8*c^(9/2))

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Rubi in Sympy [A]  time = 35.6733, size = 175, normalized size = 0.95 \[ - \frac{a^{2}}{4 c x^{4} \left (c + d x^{2}\right )^{\frac{3}{2}}} + \frac{a \left (7 a d - 8 b c\right )}{8 c^{2} x^{2} \left (c + d x^{2}\right )^{\frac{3}{2}}} + \frac{5 a d \left (7 a d - 8 b c\right ) + 8 b^{2} c^{2}}{24 c^{3} \left (c + d x^{2}\right )^{\frac{3}{2}}} + \frac{5 a d \left (7 a d - 8 b c\right ) + 8 b^{2} c^{2}}{8 c^{4} \sqrt{c + d x^{2}}} - \frac{\left (5 a d \left (7 a d - 8 b c\right ) + 8 b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{2}}}{\sqrt{c}} \right )}}{8 c^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**2/(4*c*x**4*(c + d*x**2)**(3/2)) + a*(7*a*d - 8*b*c)/(8*c**2*x**2*(c + d*x**
2)**(3/2)) + (5*a*d*(7*a*d - 8*b*c) + 8*b**2*c**2)/(24*c**3*(c + d*x**2)**(3/2))
+ (5*a*d*(7*a*d - 8*b*c) + 8*b**2*c**2)/(8*c**4*sqrt(c + d*x**2)) - (5*a*d*(7*a
*d - 8*b*c) + 8*b**2*c**2)*atanh(sqrt(c + d*x**2)/sqrt(c))/(8*c**(9/2))

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Mathematica [A]  time = 0.479322, size = 179, normalized size = 0.97 \[ \frac{-3 \left (35 a^2 d^2-40 a b c d+8 b^2 c^2\right ) \log \left (\sqrt{c} \sqrt{c+d x^2}+c\right )+\sqrt{c} \sqrt{c+d x^2} \left (\frac{24 \left (3 a^2 d^2-4 a b c d+b^2 c^2\right )}{c+d x^2}-\frac{6 a^2 c}{x^4}+\frac{3 a (11 a d-8 b c)}{x^2}+\frac{8 c (b c-a d)^2}{\left (c+d x^2\right )^2}\right )+3 \log (x) \left (35 a^2 d^2-40 a b c d+8 b^2 c^2\right )}{24 c^{9/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[c]*Sqrt[c + d*x^2]*((-6*a^2*c)/x^4 + (3*a*(-8*b*c + 11*a*d))/x^2 + (8*c*(b
*c - a*d)^2)/(c + d*x^2)^2 + (24*(b^2*c^2 - 4*a*b*c*d + 3*a^2*d^2))/(c + d*x^2))
+ 3*(8*b^2*c^2 - 40*a*b*c*d + 35*a^2*d^2)*Log[x] - 3*(8*b^2*c^2 - 40*a*b*c*d +
35*a^2*d^2)*Log[c + Sqrt[c]*Sqrt[c + d*x^2]])/(24*c^(9/2))

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Maple [A]  time = 0.02, size = 265, normalized size = 1.4 \[ -{\frac{{a}^{2}}{4\,c{x}^{4}} \left ( d{x}^{2}+c \right ) ^{-{\frac{3}{2}}}}+{\frac{7\,{a}^{2}d}{8\,{c}^{2}{x}^{2}} \left ( d{x}^{2}+c \right ) ^{-{\frac{3}{2}}}}+{\frac{35\,{a}^{2}{d}^{2}}{24\,{c}^{3}} \left ( d{x}^{2}+c \right ) ^{-{\frac{3}{2}}}}+{\frac{35\,{a}^{2}{d}^{2}}{8\,{c}^{4}}{\frac{1}{\sqrt{d{x}^{2}+c}}}}-{\frac{35\,{a}^{2}{d}^{2}}{8}\ln \left ({\frac{1}{x} \left ( 2\,c+2\,\sqrt{c}\sqrt{d{x}^{2}+c} \right ) } \right ){c}^{-{\frac{9}{2}}}}+{\frac{{b}^{2}}{3\,c} \left ( d{x}^{2}+c \right ) ^{-{\frac{3}{2}}}}+{\frac{{b}^{2}}{{c}^{2}}{\frac{1}{\sqrt{d{x}^{2}+c}}}}-{{b}^{2}\ln \left ({\frac{1}{x} \left ( 2\,c+2\,\sqrt{c}\sqrt{d{x}^{2}+c} \right ) } \right ){c}^{-{\frac{5}{2}}}}-{\frac{ab}{c{x}^{2}} \left ( d{x}^{2}+c \right ) ^{-{\frac{3}{2}}}}-{\frac{5\,abd}{3\,{c}^{2}} \left ( d{x}^{2}+c \right ) ^{-{\frac{3}{2}}}}-5\,{\frac{abd}{{c}^{3}\sqrt{d{x}^{2}+c}}}+5\,{\frac{abd}{{c}^{7/2}}\ln \left ({\frac{2\,c+2\,\sqrt{c}\sqrt{d{x}^{2}+c}}{x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*a^2/c/x^4/(d*x^2+c)^(3/2)+7/8*a^2*d/c^2/x^2/(d*x^2+c)^(3/2)+35/24*a^2*d^2/c
^3/(d*x^2+c)^(3/2)+35/8*a^2*d^2/c^4/(d*x^2+c)^(1/2)-35/8*a^2*d^2/c^(9/2)*ln((2*c
+2*c^(1/2)*(d*x^2+c)^(1/2))/x)+1/3*b^2/c/(d*x^2+c)^(3/2)+b^2/c^2/(d*x^2+c)^(1/2)
-b^2/c^(5/2)*ln((2*c+2*c^(1/2)*(d*x^2+c)^(1/2))/x)-a*b/c/x^2/(d*x^2+c)^(3/2)-5/3
*a*b*d/c^2/(d*x^2+c)^(3/2)-5*a*b*d/c^3/(d*x^2+c)^(1/2)+5*a*b*d/c^(7/2)*ln((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^5),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.255123, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (3 \,{\left (8 \, b^{2} c^{2} d - 40 \, a b c d^{2} + 35 \, a^{2} d^{3}\right )} x^{6} - 6 \, a^{2} c^{3} + 4 \,{\left (8 \, b^{2} c^{3} - 40 \, a b c^{2} d + 35 \, a^{2} c d^{2}\right )} x^{4} - 3 \,{\left (8 \, a b c^{3} - 7 \, a^{2} c^{2} d\right )} x^{2}\right )} \sqrt{d x^{2} + c} \sqrt{c} + 3 \,{\left ({\left (8 \, b^{2} c^{2} d^{2} - 40 \, a b c d^{3} + 35 \, a^{2} d^{4}\right )} x^{8} + 2 \,{\left (8 \, b^{2} c^{3} d - 40 \, a b c^{2} d^{2} + 35 \, a^{2} c d^{3}\right )} x^{6} +{\left (8 \, b^{2} c^{4} - 40 \, a b c^{3} d + 35 \, a^{2} c^{2} d^{2}\right )} x^{4}\right )} \log \left (-\frac{{\left (d x^{2} + 2 \, c\right )} \sqrt{c} - 2 \, \sqrt{d x^{2} + c} c}{x^{2}}\right )}{48 \,{\left (c^{4} d^{2} x^{8} + 2 \, c^{5} d x^{6} + c^{6} x^{4}\right )} \sqrt{c}}, \frac{{\left (3 \,{\left (8 \, b^{2} c^{2} d - 40 \, a b c d^{2} + 35 \, a^{2} d^{3}\right )} x^{6} - 6 \, a^{2} c^{3} + 4 \,{\left (8 \, b^{2} c^{3} - 40 \, a b c^{2} d + 35 \, a^{2} c d^{2}\right )} x^{4} - 3 \,{\left (8 \, a b c^{3} - 7 \, a^{2} c^{2} d\right )} x^{2}\right )} \sqrt{d x^{2} + c} \sqrt{-c} - 3 \,{\left ({\left (8 \, b^{2} c^{2} d^{2} - 40 \, a b c d^{3} + 35 \, a^{2} d^{4}\right )} x^{8} + 2 \,{\left (8 \, b^{2} c^{3} d - 40 \, a b c^{2} d^{2} + 35 \, a^{2} c d^{3}\right )} x^{6} +{\left (8 \, b^{2} c^{4} - 40 \, a b c^{3} d + 35 \, a^{2} c^{2} d^{2}\right )} x^{4}\right )} \arctan \left (\frac{\sqrt{-c}}{\sqrt{d x^{2} + c}}\right )}{24 \,{\left (c^{4} d^{2} x^{8} + 2 \, c^{5} d x^{6} + c^{6} x^{4}\right )} \sqrt{-c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/48*(2*(3*(8*b^2*c^2*d - 40*a*b*c*d^2 + 35*a^2*d^3)*x^6 - 6*a^2*c^3 + 4*(8*b^2
*c^3 - 40*a*b*c^2*d + 35*a^2*c*d^2)*x^4 - 3*(8*a*b*c^3 - 7*a^2*c^2*d)*x^2)*sqrt(
d*x^2 + c)*sqrt(c) + 3*((8*b^2*c^2*d^2 - 40*a*b*c*d^3 + 35*a^2*d^4)*x^8 + 2*(8*b
^2*c^3*d - 40*a*b*c^2*d^2 + 35*a^2*c*d^3)*x^6 + (8*b^2*c^4 - 40*a*b*c^3*d + 35*a
^2*c^2*d^2)*x^4)*log(-((d*x^2 + 2*c)*sqrt(c) - 2*sqrt(d*x^2 + c)*c)/x^2))/((c^4*
d^2*x^8 + 2*c^5*d*x^6 + c^6*x^4)*sqrt(c)), 1/24*((3*(8*b^2*c^2*d - 40*a*b*c*d^2
+ 35*a^2*d^3)*x^6 - 6*a^2*c^3 + 4*(8*b^2*c^3 - 40*a*b*c^2*d + 35*a^2*c*d^2)*x^4
- 3*(8*a*b*c^3 - 7*a^2*c^2*d)*x^2)*sqrt(d*x^2 + c)*sqrt(-c) - 3*((8*b^2*c^2*d^2
- 40*a*b*c*d^3 + 35*a^2*d^4)*x^8 + 2*(8*b^2*c^3*d - 40*a*b*c^2*d^2 + 35*a^2*c*d^
3)*x^6 + (8*b^2*c^4 - 40*a*b*c^3*d + 35*a^2*c^2*d^2)*x^4)*arctan(sqrt(-c)/sqrt(d
*x^2 + c)))/((c^4*d^2*x^8 + 2*c^5*d*x^6 + c^6*x^4)*sqrt(-c))]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.252295, size = 284, normalized size = 1.54 \[ \frac{{\left (8 \, b^{2} c^{2} - 40 \, a b c d + 35 \, a^{2} d^{2}\right )} \arctan \left (\frac{\sqrt{d x^{2} + c}}{\sqrt{-c}}\right )}{8 \, \sqrt{-c} c^{4}} + \frac{3 \,{\left (d x^{2} + c\right )} b^{2} c^{2} + b^{2} c^{3} - 12 \,{\left (d x^{2} + c\right )} a b c d - 2 \, a b c^{2} d + 9 \,{\left (d x^{2} + c\right )} a^{2} d^{2} + a^{2} c d^{2}}{3 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} c^{4}} - \frac{8 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} a b c d - 8 \, \sqrt{d x^{2} + c} a b c^{2} d - 11 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} a^{2} d^{2} + 13 \, \sqrt{d x^{2} + c} a^{2} c d^{2}}{8 \, c^{4} d^{2} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*(8*b^2*c^2 - 40*a*b*c*d + 35*a^2*d^2)*arctan(sqrt(d*x^2 + c)/sqrt(-c))/(sqrt
(-c)*c^4) + 1/3*(3*(d*x^2 + c)*b^2*c^2 + b^2*c^3 - 12*(d*x^2 + c)*a*b*c*d - 2*a*
b*c^2*d + 9*(d*x^2 + c)*a^2*d^2 + a^2*c*d^2)/((d*x^2 + c)^(3/2)*c^4) - 1/8*(8*(d
*x^2 + c)^(3/2)*a*b*c*d - 8*sqrt(d*x^2 + c)*a*b*c^2*d - 11*(d*x^2 + c)^(3/2)*a^2
*d^2 + 13*sqrt(d*x^2 + c)*a^2*c*d^2)/(c^4*d^2*x^4)