3.268 \(\int \frac{a+b x^2}{x^5 \sqrt{-c+d x} \sqrt{c+d x}} \, dx\)

Optimal. Leaf size=123 \[ \frac{d^2 \left (3 a d^2+4 b c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d x-c} \sqrt{c+d x}}{c}\right )}{8 c^5}+\frac{\sqrt{d x-c} \sqrt{c+d x} \left (3 a d^2+4 b c^2\right )}{8 c^4 x^2}+\frac{a \sqrt{d x-c} \sqrt{c+d x}}{4 c^2 x^4} \]

[Out]

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

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Rubi [A]  time = 0.342501, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.161 \[ \frac{d^2 \left (3 a d^2+4 b c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d x-c} \sqrt{c+d x}}{c}\right )}{8 c^5}+\frac{\sqrt{d x-c} \sqrt{c+d x} \left (3 a d^2+4 b c^2\right )}{8 c^4 x^2}+\frac{a \sqrt{d x-c} \sqrt{c+d x}}{4 c^2 x^4} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 19.7361, size = 107, normalized size = 0.87 \[ \frac{a \sqrt{- c + d x} \sqrt{c + d x}}{4 c^{2} x^{4}} + \frac{\sqrt{- c + d x} \sqrt{c + d x} \left (3 a d^{2} + 4 b c^{2}\right )}{8 c^{4} x^{2}} + \frac{d^{2} \left (3 a d^{2} + 4 b c^{2}\right ) \operatorname{atan}{\left (\frac{\sqrt{- c + d x} \sqrt{c + d x}}{c} \right )}}{8 c^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)/x**5/(d*x-c)**(1/2)/(d*x+c)**(1/2),x)

[Out]

a*sqrt(-c + d*x)*sqrt(c + d*x)/(4*c**2*x**4) + sqrt(-c + d*x)*sqrt(c + d*x)*(3*a
*d**2 + 4*b*c**2)/(8*c**4*x**2) + d**2*(3*a*d**2 + 4*b*c**2)*atan(sqrt(-c + d*x)
*sqrt(c + d*x)/c)/(8*c**5)

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Mathematica [C]  time = 0.19576, size = 135, normalized size = 1.1 \[ \frac{c \sqrt{d x-c} \sqrt{c+d x} \left (2 a c^2+3 a d^2 x^2+4 b c^2 x^2\right )-i d^2 x^4 \left (3 a d^2+4 b c^2\right ) \log \left (\frac{16 c^4 \left (\sqrt{d x-c} \sqrt{c+d x}-i c\right )}{d^2 x \left (3 a d^2+4 b c^2\right )}\right )}{8 c^5 x^4} \]

Antiderivative was successfully verified.

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

[Out]

(c*Sqrt[-c + d*x]*Sqrt[c + d*x]*(2*a*c^2 + 4*b*c^2*x^2 + 3*a*d^2*x^2) - I*d^2*(4
*b*c^2 + 3*a*d^2)*x^4*Log[(16*c^4*((-I)*c + Sqrt[-c + d*x]*Sqrt[c + d*x]))/(d^2*
(4*b*c^2 + 3*a*d^2)*x)])/(8*c^5*x^4)

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Maple [B]  time = 0.03, size = 227, normalized size = 1.9 \[ -{\frac{1}{8\,{c}^{4}{x}^{4}}\sqrt{dx-c}\sqrt{dx+c} \left ( 3\,\ln \left ( -2\,{\frac{{c}^{2}-\sqrt{-{c}^{2}}\sqrt{{d}^{2}{x}^{2}-{c}^{2}}}{x}} \right ){x}^{4}a{d}^{4}+4\,\ln \left ( -2\,{\frac{{c}^{2}-\sqrt{-{c}^{2}}\sqrt{{d}^{2}{x}^{2}-{c}^{2}}}{x}} \right ){x}^{4}b{c}^{2}{d}^{2}-3\,a{d}^{2}\sqrt{{d}^{2}{x}^{2}-{c}^{2}}{x}^{2}\sqrt{-{c}^{2}}-4\,b\sqrt{{d}^{2}{x}^{2}-{c}^{2}}{c}^{2}{x}^{2}\sqrt{-{c}^{2}}-2\,a\sqrt{{d}^{2}{x}^{2}-{c}^{2}}{c}^{2}\sqrt{-{c}^{2}} \right ){\frac{1}{\sqrt{-{c}^{2}}}}{\frac{1}{\sqrt{{d}^{2}{x}^{2}-{c}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)/x^5/(d*x-c)^(1/2)/(d*x+c)^(1/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.272273, size = 544, normalized size = 4.42 \[ -\frac{8 \, a c^{7} d x + 8 \,{\left (4 \, b c^{3} d^{5} + 3 \, a c d^{7}\right )} x^{7} - 4 \,{\left (12 \, b c^{5} d^{3} + 5 \, a c^{3} d^{5}\right )} x^{5} + 4 \,{\left (4 \, b c^{7} d - 3 \, a c^{5} d^{3}\right )} x^{3} -{\left (2 \, a c^{7} + 8 \,{\left (4 \, b c^{3} d^{4} + 3 \, a c d^{6}\right )} x^{6} - 8 \,{\left (4 \, b c^{5} d^{2} + a c^{3} d^{4}\right )} x^{4} +{\left (4 \, b c^{7} - 13 \, a c^{5} d^{2}\right )} x^{2}\right )} \sqrt{d x + c} \sqrt{d x - c} - 2 \,{\left (8 \,{\left (4 \, b c^{2} d^{6} + 3 \, a d^{8}\right )} x^{8} - 8 \,{\left (4 \, b c^{4} d^{4} + 3 \, a c^{2} d^{6}\right )} x^{6} +{\left (4 \, b c^{6} d^{2} + 3 \, a c^{4} d^{4}\right )} x^{4} - 4 \,{\left (2 \,{\left (4 \, b c^{2} d^{5} + 3 \, a d^{7}\right )} x^{7} -{\left (4 \, b c^{4} d^{3} + 3 \, a c^{2} d^{5}\right )} x^{5}\right )} \sqrt{d x + c} \sqrt{d x - c}\right )} \arctan \left (-\frac{d x - \sqrt{d x + c} \sqrt{d x - c}}{c}\right )}{8 \,{\left (8 \, c^{5} d^{4} x^{8} - 8 \, c^{7} d^{2} x^{6} + c^{9} x^{4} - 4 \,{\left (2 \, c^{5} d^{3} x^{7} - c^{7} d x^{5}\right )} \sqrt{d x + c} \sqrt{d x - c}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8*(8*a*c^7*d*x + 8*(4*b*c^3*d^5 + 3*a*c*d^7)*x^7 - 4*(12*b*c^5*d^3 + 5*a*c^3*
d^5)*x^5 + 4*(4*b*c^7*d - 3*a*c^5*d^3)*x^3 - (2*a*c^7 + 8*(4*b*c^3*d^4 + 3*a*c*d
^6)*x^6 - 8*(4*b*c^5*d^2 + a*c^3*d^4)*x^4 + (4*b*c^7 - 13*a*c^5*d^2)*x^2)*sqrt(d
*x + c)*sqrt(d*x - c) - 2*(8*(4*b*c^2*d^6 + 3*a*d^8)*x^8 - 8*(4*b*c^4*d^4 + 3*a*
c^2*d^6)*x^6 + (4*b*c^6*d^2 + 3*a*c^4*d^4)*x^4 - 4*(2*(4*b*c^2*d^5 + 3*a*d^7)*x^
7 - (4*b*c^4*d^3 + 3*a*c^2*d^5)*x^5)*sqrt(d*x + c)*sqrt(d*x - c))*arctan(-(d*x -
sqrt(d*x + c)*sqrt(d*x - c))/c))/(8*c^5*d^4*x^8 - 8*c^7*d^2*x^6 + c^9*x^4 - 4*(
2*c^5*d^3*x^7 - c^7*d*x^5)*sqrt(d*x + c)*sqrt(d*x - c))

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Sympy [A]  time = 157.614, size = 172, normalized size = 1.4 \[ - \frac{a d^{4}{G_{6, 6}^{5, 3}\left (\begin{matrix} \frac{11}{4}, \frac{13}{4}, 1 & 3, 3, \frac{7}{2} \\\frac{5}{2}, \frac{11}{4}, 3, \frac{13}{4}, \frac{7}{2} & 0 \end{matrix} \middle |{\frac{c^{2}}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} c^{5}} + \frac{i a d^{4}{G_{6, 6}^{2, 6}\left (\begin{matrix} 2, \frac{9}{4}, \frac{5}{2}, \frac{11}{4}, 3, 1 & \\\frac{9}{4}, \frac{11}{4} & 2, \frac{5}{2}, \frac{5}{2}, 0 \end{matrix} \middle |{\frac{c^{2} e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} c^{5}} - \frac{b d^{2}{G_{6, 6}^{5, 3}\left (\begin{matrix} \frac{7}{4}, \frac{9}{4}, 1 & 2, 2, \frac{5}{2} \\\frac{3}{2}, \frac{7}{4}, 2, \frac{9}{4}, \frac{5}{2} & 0 \end{matrix} \middle |{\frac{c^{2}}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} c^{3}} + \frac{i b d^{2}{G_{6, 6}^{2, 6}\left (\begin{matrix} 1, \frac{5}{4}, \frac{3}{2}, \frac{7}{4}, 2, 1 & \\\frac{5}{4}, \frac{7}{4} & 1, \frac{3}{2}, \frac{3}{2}, 0 \end{matrix} \middle |{\frac{c^{2} e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)/x**5/(d*x-c)**(1/2)/(d*x+c)**(1/2),x)

[Out]

-a*d**4*meijerg(((11/4, 13/4, 1), (3, 3, 7/2)), ((5/2, 11/4, 3, 13/4, 7/2), (0,)
), c**2/(d**2*x**2))/(4*pi**(3/2)*c**5) + I*a*d**4*meijerg(((2, 9/4, 5/2, 11/4,
3, 1), ()), ((9/4, 11/4), (2, 5/2, 5/2, 0)), c**2*exp_polar(2*I*pi)/(d**2*x**2))
/(4*pi**(3/2)*c**5) - b*d**2*meijerg(((7/4, 9/4, 1), (2, 2, 5/2)), ((3/2, 7/4, 2
, 9/4, 5/2), (0,)), c**2/(d**2*x**2))/(4*pi**(3/2)*c**3) + I*b*d**2*meijerg(((1,
5/4, 3/2, 7/4, 2, 1), ()), ((5/4, 7/4), (1, 3/2, 3/2, 0)), c**2*exp_polar(2*I*p
i)/(d**2*x**2))/(4*pi**(3/2)*c**3)

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GIAC/XCAS [A]  time = 0.239979, size = 439, normalized size = 3.57 \[ -\frac{\frac{{\left (4 \, b c^{2} d^{3} + 3 \, a d^{5}\right )} \arctan \left (\frac{{\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{2}}{2 \, c}\right )}{c^{5}} + \frac{2 \,{\left (4 \, b c^{2} d^{3}{\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{14} + 3 \, a d^{5}{\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{14} + 16 \, b c^{4} d^{3}{\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{10} + 44 \, a c^{2} d^{5}{\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{10} - 64 \, b c^{6} d^{3}{\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{6} - 176 \, a c^{4} d^{5}{\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{6} - 256 \, b c^{8} d^{3}{\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{2} - 192 \, a c^{6} d^{5}{\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{2}\right )}}{{\left ({\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{4} + 4 \, c^{2}\right )}^{4} c^{4}}}{4 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*((4*b*c^2*d^3 + 3*a*d^5)*arctan(1/2*(sqrt(d*x + c) - sqrt(d*x - c))^2/c)/c^
5 + 2*(4*b*c^2*d^3*(sqrt(d*x + c) - sqrt(d*x - c))^14 + 3*a*d^5*(sqrt(d*x + c) -
sqrt(d*x - c))^14 + 16*b*c^4*d^3*(sqrt(d*x + c) - sqrt(d*x - c))^10 + 44*a*c^2*
d^5*(sqrt(d*x + c) - sqrt(d*x - c))^10 - 64*b*c^6*d^3*(sqrt(d*x + c) - sqrt(d*x
- c))^6 - 176*a*c^4*d^5*(sqrt(d*x + c) - sqrt(d*x - c))^6 - 256*b*c^8*d^3*(sqrt(
d*x + c) - sqrt(d*x - c))^2 - 192*a*c^6*d^5*(sqrt(d*x + c) - sqrt(d*x - c))^2)/(
((sqrt(d*x + c) - sqrt(d*x - c))^4 + 4*c^2)^4*c^4))/d