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

Optimal. Leaf size=166 \[ -\frac{3 d^2 \left (5 a d^2+4 b c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d x-c} \sqrt{c+d x}}{c}\right )}{8 c^7}-\frac{3 d^2 \left (5 a d^2+4 b c^2\right )}{8 c^6 \sqrt{d x-c} \sqrt{c+d x}}+\frac{5 a d^2+4 b c^2}{8 c^4 x^2 \sqrt{d x-c} \sqrt{c+d x}}+\frac{a}{4 c^2 x^4 \sqrt{d x-c} \sqrt{c+d x}} \]

[Out]

(-3*d^2*(4*b*c^2 + 5*a*d^2))/(8*c^6*Sqrt[-c + d*x]*Sqrt[c + d*x]) + a/(4*c^2*x^4
*Sqrt[-c + d*x]*Sqrt[c + d*x]) + (4*b*c^2 + 5*a*d^2)/(8*c^4*x^2*Sqrt[-c + d*x]*S
qrt[c + d*x]) - (3*d^2*(4*b*c^2 + 5*a*d^2)*ArcTan[(Sqrt[-c + d*x]*Sqrt[c + d*x])
/c])/(8*c^7)

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Rubi [A]  time = 0.459333, antiderivative size = 166, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.226 \[ -\frac{3 d^2 \left (5 a d^2+4 b c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d x-c} \sqrt{c+d x}}{c}\right )}{8 c^7}-\frac{3 d^2 \left (5 a d^2+4 b c^2\right )}{8 c^6 \sqrt{d x-c} \sqrt{c+d x}}+\frac{5 a d^2+4 b c^2}{8 c^4 x^2 \sqrt{d x-c} \sqrt{c+d x}}+\frac{a}{4 c^2 x^4 \sqrt{d x-c} \sqrt{c+d x}} \]

Antiderivative was successfully verified.

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

[Out]

(-3*d^2*(4*b*c^2 + 5*a*d^2))/(8*c^6*Sqrt[-c + d*x]*Sqrt[c + d*x]) + a/(4*c^2*x^4
*Sqrt[-c + d*x]*Sqrt[c + d*x]) + (4*b*c^2 + 5*a*d^2)/(8*c^4*x^2*Sqrt[-c + d*x]*S
qrt[c + d*x]) - (3*d^2*(4*b*c^2 + 5*a*d^2)*ArcTan[(Sqrt[-c + d*x]*Sqrt[c + d*x])
/c])/(8*c^7)

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Rubi in Sympy [A]  time = 30.9584, size = 150, normalized size = 0.9 \[ \frac{a}{4 c^{2} x^{4} \sqrt{- c + d x} \sqrt{c + d x}} + \frac{5 a d^{2} + 4 b c^{2}}{8 c^{4} x^{2} \sqrt{- c + d x} \sqrt{c + d x}} - \frac{3 d^{2} \left (5 a d^{2} + 4 b c^{2}\right )}{8 c^{6} \sqrt{- c + d x} \sqrt{c + d x}} - \frac{3 d^{2} \left (5 a d^{2} + 4 b c^{2}\right ) \operatorname{atan}{\left (\frac{\sqrt{- c + d x} \sqrt{c + d x}}{c} \right )}}{8 c^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a/(4*c**2*x**4*sqrt(-c + d*x)*sqrt(c + d*x)) + (5*a*d**2 + 4*b*c**2)/(8*c**4*x**
2*sqrt(-c + d*x)*sqrt(c + d*x)) - 3*d**2*(5*a*d**2 + 4*b*c**2)/(8*c**6*sqrt(-c +
d*x)*sqrt(c + d*x)) - 3*d**2*(5*a*d**2 + 4*b*c**2)*atan(sqrt(-c + d*x)*sqrt(c +
d*x)/c)/(8*c**7)

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Mathematica [C]  time = 0.38189, size = 157, normalized size = 0.95 \[ \frac{\frac{a \left (2 c^5+5 c^3 d^2 x^2-15 c d^4 x^4\right )+4 b c^3 x^2 \left (c^2-3 d^2 x^2\right )}{x^4 \sqrt{d x-c} \sqrt{c+d x}}+3 i \left (5 a d^4+4 b c^2 d^2\right ) \log \left (\frac{-16 c^6 \sqrt{d x-c} \sqrt{c+d x}+16 i c^7}{15 a d^4 x+12 b c^2 d^2 x}\right )}{8 c^7} \]

Antiderivative was successfully verified.

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

[Out]

((4*b*c^3*x^2*(c^2 - 3*d^2*x^2) + a*(2*c^5 + 5*c^3*d^2*x^2 - 15*c*d^4*x^4))/(x^4
*Sqrt[-c + d*x]*Sqrt[c + d*x]) + (3*I)*(4*b*c^2*d^2 + 5*a*d^4)*Log[((16*I)*c^7 -
16*c^6*Sqrt[-c + d*x]*Sqrt[c + d*x])/(12*b*c^2*d^2*x + 15*a*d^4*x)])/(8*c^7)

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Maple [B]  time = 0.038, size = 387, normalized size = 2.3 \[{\frac{1}{8\,{c}^{6}{x}^{4}} \left ( 15\,\ln \left ( -2\,{\frac{{c}^{2}-\sqrt{-{c}^{2}}\sqrt{{d}^{2}{x}^{2}-{c}^{2}}}{x}} \right ){x}^{6}a{d}^{6}+12\,\ln \left ( -2\,{\frac{{c}^{2}-\sqrt{-{c}^{2}}\sqrt{{d}^{2}{x}^{2}-{c}^{2}}}{x}} \right ){x}^{6}b{c}^{2}{d}^{4}-15\,\ln \left ( -2\,{\frac{{c}^{2}-\sqrt{-{c}^{2}}\sqrt{{d}^{2}{x}^{2}-{c}^{2}}}{x}} \right ){x}^{4}a{c}^{2}{d}^{4}-12\,\ln \left ( -2\,{\frac{{c}^{2}-\sqrt{-{c}^{2}}\sqrt{{d}^{2}{x}^{2}-{c}^{2}}}{x}} \right ){x}^{4}b{c}^{4}{d}^{2}-15\,{x}^{4}a{d}^{4}\sqrt{-{c}^{2}}\sqrt{{d}^{2}{x}^{2}-{c}^{2}}-12\,{x}^{4}b{c}^{2}{d}^{2}\sqrt{-{c}^{2}}\sqrt{{d}^{2}{x}^{2}-{c}^{2}}+5\,{x}^{2}a{c}^{2}{d}^{2}\sqrt{-{c}^{2}}\sqrt{{d}^{2}{x}^{2}-{c}^{2}}+4\,{x}^{2}b{c}^{4}\sqrt{-{c}^{2}}\sqrt{{d}^{2}{x}^{2}-{c}^{2}}+2\,a{c}^{4}\sqrt{-{c}^{2}}\sqrt{{d}^{2}{x}^{2}-{c}^{2}} \right ){\frac{1}{\sqrt{-{c}^{2}}}}{\frac{1}{\sqrt{{d}^{2}{x}^{2}-{c}^{2}}}}{\frac{1}{\sqrt{dx+c}}}{\frac{1}{\sqrt{dx-c}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8/c^6*(15*ln(-2*(c^2-(-c^2)^(1/2)*(d^2*x^2-c^2)^(1/2))/x)*x^6*a*d^6+12*ln(-2*(
c^2-(-c^2)^(1/2)*(d^2*x^2-c^2)^(1/2))/x)*x^6*b*c^2*d^4-15*ln(-2*(c^2-(-c^2)^(1/2
)*(d^2*x^2-c^2)^(1/2))/x)*x^4*a*c^2*d^4-12*ln(-2*(c^2-(-c^2)^(1/2)*(d^2*x^2-c^2)
^(1/2))/x)*x^4*b*c^4*d^2-15*x^4*a*d^4*(-c^2)^(1/2)*(d^2*x^2-c^2)^(1/2)-12*x^4*b*
c^2*d^2*(-c^2)^(1/2)*(d^2*x^2-c^2)^(1/2)+5*x^2*a*c^2*d^2*(-c^2)^(1/2)*(d^2*x^2-c
^2)^(1/2)+4*x^2*b*c^4*(-c^2)^(1/2)*(d^2*x^2-c^2)^(1/2)+2*a*c^4*(-c^2)^(1/2)*(d^2
*x^2-c^2)^(1/2))/(-c^2)^(1/2)/x^4/(d^2*x^2-c^2)^(1/2)/(d*x+c)^(1/2)/(d*x-c)^(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)/((d*x + c)^(3/2)*(d*x - c)^(3/2)*x^5),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.304714, size = 707, normalized size = 4.26 \[ -\frac{10 \, a c^{9} d x - 48 \,{\left (4 \, b c^{3} d^{7} + 5 \, a c d^{9}\right )} x^{9} + 76 \,{\left (4 \, b c^{5} d^{5} + 5 \, a c^{3} d^{7}\right )} x^{7} -{\left (140 \, b c^{7} d^{3} + 143 \, a c^{5} d^{5}\right )} x^{5} + 5 \,{\left (4 \, b c^{9} d - 3 \, a c^{7} d^{3}\right )} x^{3} -{\left (2 \, a c^{9} - 48 \,{\left (4 \, b c^{3} d^{6} + 5 \, a c d^{8}\right )} x^{8} + 52 \,{\left (4 \, b c^{5} d^{4} + 5 \, a c^{3} d^{6}\right )} x^{6} -{\left (60 \, b c^{7} d^{2} + 43 \, a c^{5} d^{4}\right )} x^{4} +{\left (4 \, b c^{9} - 19 \, a c^{7} d^{2}\right )} x^{2}\right )} \sqrt{d x + c} \sqrt{d x - c} + 6 \,{\left (16 \,{\left (4 \, b c^{2} d^{8} + 5 \, a d^{10}\right )} x^{10} - 28 \,{\left (4 \, b c^{4} d^{6} + 5 \, a c^{2} d^{8}\right )} x^{8} + 13 \,{\left (4 \, b c^{6} d^{4} + 5 \, a c^{4} d^{6}\right )} x^{6} -{\left (4 \, b c^{8} d^{2} + 5 \, a c^{6} d^{4}\right )} x^{4} -{\left (16 \,{\left (4 \, b c^{2} d^{7} + 5 \, a d^{9}\right )} x^{9} - 20 \,{\left (4 \, b c^{4} d^{5} + 5 \, a c^{2} d^{7}\right )} x^{7} + 5 \,{\left (4 \, b c^{6} d^{3} + 5 \, a c^{4} d^{5}\right )} x^{5}\right )} \sqrt{d x + c} \sqrt{d x - c}\right )} \arctan \left (-\frac{d x - \sqrt{d x + c} \sqrt{d x - c}}{c}\right )}{8 \,{\left (16 \, c^{7} d^{6} x^{10} - 28 \, c^{9} d^{4} x^{8} + 13 \, c^{11} d^{2} x^{6} - c^{13} x^{4} -{\left (16 \, c^{7} d^{5} x^{9} - 20 \, c^{9} d^{3} x^{7} + 5 \, c^{11} d x^{5}\right )} \sqrt{d x + c} \sqrt{d x - c}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8*(10*a*c^9*d*x - 48*(4*b*c^3*d^7 + 5*a*c*d^9)*x^9 + 76*(4*b*c^5*d^5 + 5*a*c^
3*d^7)*x^7 - (140*b*c^7*d^3 + 143*a*c^5*d^5)*x^5 + 5*(4*b*c^9*d - 3*a*c^7*d^3)*x
^3 - (2*a*c^9 - 48*(4*b*c^3*d^6 + 5*a*c*d^8)*x^8 + 52*(4*b*c^5*d^4 + 5*a*c^3*d^6
)*x^6 - (60*b*c^7*d^2 + 43*a*c^5*d^4)*x^4 + (4*b*c^9 - 19*a*c^7*d^2)*x^2)*sqrt(d
*x + c)*sqrt(d*x - c) + 6*(16*(4*b*c^2*d^8 + 5*a*d^10)*x^10 - 28*(4*b*c^4*d^6 +
5*a*c^2*d^8)*x^8 + 13*(4*b*c^6*d^4 + 5*a*c^4*d^6)*x^6 - (4*b*c^8*d^2 + 5*a*c^6*d
^4)*x^4 - (16*(4*b*c^2*d^7 + 5*a*d^9)*x^9 - 20*(4*b*c^4*d^5 + 5*a*c^2*d^7)*x^7 +
5*(4*b*c^6*d^3 + 5*a*c^4*d^5)*x^5)*sqrt(d*x + c)*sqrt(d*x - c))*arctan(-(d*x -
sqrt(d*x + c)*sqrt(d*x - c))/c))/(16*c^7*d^6*x^10 - 28*c^9*d^4*x^8 + 13*c^11*d^2
*x^6 - c^13*x^4 - (16*c^7*d^5*x^9 - 20*c^9*d^3*x^7 + 5*c^11*d*x^5)*sqrt(d*x + c)
*sqrt(d*x - 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)/x**5/(d*x-c)**(3/2)/(d*x+c)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.345015, size = 543, normalized size = 3.27 \[ \frac{3 \,{\left (4 \, b c^{2} d^{2} + 5 \, a d^{4}\right )} \arctan \left (\frac{{\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{2}}{2 \, c}\right )}{4 \, c^{7}} - \frac{{\left (b c^{2} d^{2} + a d^{4}\right )} \sqrt{d x + c}}{2 \, \sqrt{d x - c} c^{7}} + \frac{2 \,{\left (b c^{2} d^{2} + a d^{4}\right )}}{{\left ({\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{2} + 2 \, c\right )} c^{6}} + \frac{4 \, b c^{2} d^{2}{\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{14} + 7 \, a d^{4}{\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{14} + 16 \, b c^{4} d^{2}{\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{10} + 60 \, a c^{2} d^{4}{\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{10} - 64 \, b c^{6} d^{2}{\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{6} - 240 \, a c^{4} d^{4}{\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{6} - 256 \, b c^{8} d^{2}{\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{2} - 448 \, a c^{6} d^{4}{\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{2}}{2 \,{\left ({\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{4} + 4 \, c^{2}\right )}^{4} c^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/4*(4*b*c^2*d^2 + 5*a*d^4)*arctan(1/2*(sqrt(d*x + c) - sqrt(d*x - c))^2/c)/c^7
- 1/2*(b*c^2*d^2 + a*d^4)*sqrt(d*x + c)/(sqrt(d*x - c)*c^7) + 2*(b*c^2*d^2 + a*d
^4)/(((sqrt(d*x + c) - sqrt(d*x - c))^2 + 2*c)*c^6) + 1/2*(4*b*c^2*d^2*(sqrt(d*x
+ c) - sqrt(d*x - c))^14 + 7*a*d^4*(sqrt(d*x + c) - sqrt(d*x - c))^14 + 16*b*c^
4*d^2*(sqrt(d*x + c) - sqrt(d*x - c))^10 + 60*a*c^2*d^4*(sqrt(d*x + c) - sqrt(d*
x - c))^10 - 64*b*c^6*d^2*(sqrt(d*x + c) - sqrt(d*x - c))^6 - 240*a*c^4*d^4*(sqr
t(d*x + c) - sqrt(d*x - c))^6 - 256*b*c^8*d^2*(sqrt(d*x + c) - sqrt(d*x - c))^2
- 448*a*c^6*d^4*(sqrt(d*x + c) - sqrt(d*x - c))^2)/(((sqrt(d*x + c) - sqrt(d*x -
c))^4 + 4*c^2)^4*c^6)