3.757 \(\int \frac{\left (c+d x^2\right )^{5/2}}{x^4 \left (a+b x^2\right )^2} \, dx\)

Optimal. Leaf size=176 \[ \frac{5 c (b c-a d)^{3/2} \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{2 a^{7/2}}-\frac{c \sqrt{c+d x^2} (5 b c-3 a d)}{6 a^2 b x^3}+\frac{\sqrt{c+d x^2} \left (3 a^2 d^2-20 a b c d+15 b^2 c^2\right )}{6 a^3 b x}+\frac{\left (c+d x^2\right )^{3/2} (b c-a d)}{2 a b x^3 \left (a+b x^2\right )} \]

[Out]

-(c*(5*b*c - 3*a*d)*Sqrt[c + d*x^2])/(6*a^2*b*x^3) + ((15*b^2*c^2 - 20*a*b*c*d +
3*a^2*d^2)*Sqrt[c + d*x^2])/(6*a^3*b*x) + ((b*c - a*d)*(c + d*x^2)^(3/2))/(2*a*
b*x^3*(a + b*x^2)) + (5*c*(b*c - a*d)^(3/2)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*
Sqrt[c + d*x^2])])/(2*a^(7/2))

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Rubi [A]  time = 0.701997, antiderivative size = 176, 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 \[ \frac{5 c (b c-a d)^{3/2} \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{2 a^{7/2}}-\frac{c \sqrt{c+d x^2} (5 b c-3 a d)}{6 a^2 b x^3}+\frac{\sqrt{c+d x^2} \left (3 a^2 d^2-20 a b c d+15 b^2 c^2\right )}{6 a^3 b x}+\frac{\left (c+d x^2\right )^{3/2} (b c-a d)}{2 a b x^3 \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x^2)^(5/2)/(x^4*(a + b*x^2)^2),x]

[Out]

-(c*(5*b*c - 3*a*d)*Sqrt[c + d*x^2])/(6*a^2*b*x^3) + ((15*b^2*c^2 - 20*a*b*c*d +
3*a^2*d^2)*Sqrt[c + d*x^2])/(6*a^3*b*x) + ((b*c - a*d)*(c + d*x^2)^(3/2))/(2*a*
b*x^3*(a + b*x^2)) + (5*c*(b*c - a*d)^(3/2)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*
Sqrt[c + d*x^2])])/(2*a^(7/2))

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Rubi in Sympy [A]  time = 106.82, size = 156, normalized size = 0.89 \[ - \frac{\left (c + d x^{2}\right )^{\frac{3}{2}} \left (a d - b c\right )}{2 a b x^{3} \left (a + b x^{2}\right )} + \frac{c \sqrt{c + d x^{2}} \left (3 a d - 5 b c\right )}{6 a^{2} b x^{3}} + \frac{\sqrt{c + d x^{2}} \left (3 a^{2} d^{2} - 20 a b c d + 15 b^{2} c^{2}\right )}{6 a^{3} b x} + \frac{5 c \left (a d - b c\right )^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{x \sqrt{a d - b c}}{\sqrt{a} \sqrt{c + d x^{2}}} \right )}}{2 a^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x**2+c)**(5/2)/x**4/(b*x**2+a)**2,x)

[Out]

-(c + d*x**2)**(3/2)*(a*d - b*c)/(2*a*b*x**3*(a + b*x**2)) + c*sqrt(c + d*x**2)*
(3*a*d - 5*b*c)/(6*a**2*b*x**3) + sqrt(c + d*x**2)*(3*a**2*d**2 - 20*a*b*c*d + 1
5*b**2*c**2)/(6*a**3*b*x) + 5*c*(a*d - b*c)**(3/2)*atanh(x*sqrt(a*d - b*c)/(sqrt
(a)*sqrt(c + d*x**2)))/(2*a**(7/2))

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Mathematica [A]  time = 0.276535, size = 121, normalized size = 0.69 \[ \frac{5 c (b c-a d)^{3/2} \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{2 a^{7/2}}+\frac{\sqrt{c+d x^2} \left (2 c x^2 (6 b c-7 a d)+\frac{3 x^4 (b c-a d)^2}{a+b x^2}-2 a c^2\right )}{6 a^3 x^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x^2)^(5/2)/(x^4*(a + b*x^2)^2),x]

[Out]

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

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Maple [B]  time = 0.032, size = 7705, normalized size = 43.8 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x^2+c)^(5/2)/x^4/(b*x^2+a)^2,x)

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x^{2} + c\right )}^{\frac{5}{2}}}{{\left (b x^{2} + a\right )}^{2} x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)^(5/2)/((b*x^2 + a)^2*x^4),x, algorithm="maxima")

[Out]

integrate((d*x^2 + c)^(5/2)/((b*x^2 + a)^2*x^4), x)

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Fricas [A]  time = 0.355838, size = 1, normalized size = 0.01 \[ \left [-\frac{15 \,{\left ({\left (b^{2} c^{2} - a b c d\right )} x^{5} +{\left (a b c^{2} - a^{2} c d\right )} x^{3}\right )} \sqrt{-\frac{b c - a d}{a}} \log \left (\frac{{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} - 2 \,{\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{2} + 4 \,{\left (a^{2} c x -{\left (a b c - 2 \, a^{2} d\right )} x^{3}\right )} \sqrt{d x^{2} + c} \sqrt{-\frac{b c - a d}{a}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) - 4 \,{\left ({\left (15 \, b^{2} c^{2} - 20 \, a b c d + 3 \, a^{2} d^{2}\right )} x^{4} - 2 \, a^{2} c^{2} + 2 \,{\left (5 \, a b c^{2} - 7 \, a^{2} c d\right )} x^{2}\right )} \sqrt{d x^{2} + c}}{24 \,{\left (a^{3} b x^{5} + a^{4} x^{3}\right )}}, -\frac{15 \,{\left ({\left (b^{2} c^{2} - a b c d\right )} x^{5} +{\left (a b c^{2} - a^{2} c d\right )} x^{3}\right )} \sqrt{\frac{b c - a d}{a}} \arctan \left (-\frac{{\left (b c - 2 \, a d\right )} x^{2} - a c}{2 \, \sqrt{d x^{2} + c} a x \sqrt{\frac{b c - a d}{a}}}\right ) - 2 \,{\left ({\left (15 \, b^{2} c^{2} - 20 \, a b c d + 3 \, a^{2} d^{2}\right )} x^{4} - 2 \, a^{2} c^{2} + 2 \,{\left (5 \, a b c^{2} - 7 \, a^{2} c d\right )} x^{2}\right )} \sqrt{d x^{2} + c}}{12 \,{\left (a^{3} b x^{5} + a^{4} x^{3}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)^(5/2)/((b*x^2 + a)^2*x^4),x, algorithm="fricas")

[Out]

[-1/24*(15*((b^2*c^2 - a*b*c*d)*x^5 + (a*b*c^2 - a^2*c*d)*x^3)*sqrt(-(b*c - a*d)
/a)*log(((b^2*c^2 - 8*a*b*c*d + 8*a^2*d^2)*x^4 + a^2*c^2 - 2*(3*a*b*c^2 - 4*a^2*
c*d)*x^2 + 4*(a^2*c*x - (a*b*c - 2*a^2*d)*x^3)*sqrt(d*x^2 + c)*sqrt(-(b*c - a*d)
/a))/(b^2*x^4 + 2*a*b*x^2 + a^2)) - 4*((15*b^2*c^2 - 20*a*b*c*d + 3*a^2*d^2)*x^4
- 2*a^2*c^2 + 2*(5*a*b*c^2 - 7*a^2*c*d)*x^2)*sqrt(d*x^2 + c))/(a^3*b*x^5 + a^4*
x^3), -1/12*(15*((b^2*c^2 - a*b*c*d)*x^5 + (a*b*c^2 - a^2*c*d)*x^3)*sqrt((b*c -
a*d)/a)*arctan(-1/2*((b*c - 2*a*d)*x^2 - a*c)/(sqrt(d*x^2 + c)*a*x*sqrt((b*c - a
*d)/a))) - 2*((15*b^2*c^2 - 20*a*b*c*d + 3*a^2*d^2)*x^4 - 2*a^2*c^2 + 2*(5*a*b*c
^2 - 7*a^2*c*d)*x^2)*sqrt(d*x^2 + c))/(a^3*b*x^5 + a^4*x^3)]

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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((d*x**2+c)**(5/2)/x**4/(b*x**2+a)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 4.99204, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)^(5/2)/((b*x^2 + a)^2*x^4),x, algorithm="giac")

[Out]

sage0*x