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

Optimal. Leaf size=121 \[ -\frac{\sqrt{d x-c} \sqrt{c+d x} \left (a d^2+4 b c^2\right )}{8 c^2 x^2}+\frac{d^2 \left (a d^2+4 b c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d x-c} \sqrt{c+d x}}{c}\right )}{8 c^3}+\frac{a (d x-c)^{3/2} (c+d x)^{3/2}}{4 c^2 x^4} \]

[Out]

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

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Rubi [A]  time = 0.398807, antiderivative size = 164, normalized size of antiderivative = 1.36, number of steps used = 5, number of rules used = 4, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.129 \[ -\frac{\sqrt{d x-c} (c+d x)^{3/2} \left (a d^2+4 b c^2\right )}{8 c^3 x^2}+\frac{d \sqrt{d x-c} \sqrt{c+d x} \left (a d^2+4 b c^2\right )}{8 c^3 x}+\frac{d^2 \left (a d^2+4 b c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d x-c} \sqrt{c+d x}}{c}\right )}{8 c^3}+\frac{a (d x-c)^{3/2} (c+d x)^{3/2}}{4 c^2 x^4} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 22.6634, size = 141, normalized size = 1.17 \[ \frac{a \left (- c + d x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{3}{2}}}{4 c^{2} x^{4}} + \frac{d^{2} \left (a d^{2} + 4 b c^{2}\right ) \operatorname{atan}{\left (\frac{\sqrt{- c + d x} \sqrt{c + d x}}{c} \right )}}{8 c^{3}} + \frac{d \sqrt{- c + d x} \sqrt{c + d x} \left (a d^{2} + 4 b c^{2}\right )}{8 c^{3} x} - \frac{\sqrt{- c + d x} \left (c + d x\right )^{\frac{3}{2}} \left (a d^{2} + 4 b c^{2}\right )}{8 c^{3} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.179346, size = 132, normalized size = 1.09 \[ \frac{c \sqrt{d x-c} \sqrt{c+d x} \left (-2 a c^2+a d^2 x^2-4 b c^2 x^2\right )-i d^2 x^4 \left (a d^2+4 b c^2\right ) \log \left (\frac{16 c^2 \left (\sqrt{d x-c} \sqrt{c+d x}-i c\right )}{d^2 x \left (a d^2+4 b c^2\right )}\right )}{8 c^3 x^4} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.023, size = 226, normalized size = 1.9 \[ -{\frac{1}{8\,{c}^{2}{x}^{4}}\sqrt{dx-c}\sqrt{dx+c} \left ( \ln \left ( -2\,{\frac{{c}^{2}-\sqrt{-{c}^{2}}\sqrt{{d}^{2}{x}^{2}-{c}^{2}}}{x}} \right ){x}^{4}a{d}^{4}+4\,\ln \left ( -2\,{\frac{{c}^{2}-\sqrt{-{c}^{2}}\sqrt{{d}^{2}{x}^{2}-{c}^{2}}}{x}} \right ){x}^{4}b{c}^{2}{d}^{2}-a{d}^{2}\sqrt{{d}^{2}{x}^{2}-{c}^{2}}{x}^{2}\sqrt{-{c}^{2}}+4\,b\sqrt{{d}^{2}{x}^{2}-{c}^{2}}{c}^{2}{x}^{2}\sqrt{-{c}^{2}}+2\,a\sqrt{{d}^{2}{x}^{2}-{c}^{2}}{c}^{2}\sqrt{-{c}^{2}} \right ){\frac{1}{\sqrt{-{c}^{2}}}}{\frac{1}{\sqrt{{d}^{2}{x}^{2}-{c}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8*(d*x-c)^(1/2)*(d*x+c)^(1/2)/c^2*(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-
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 = 0.263067, size = 539, normalized size = 4.45 \[ \frac{8 \, a c^{7} d x + 8 \,{\left (4 \, b c^{3} d^{5} - a c d^{7}\right )} x^{7} - 4 \,{\left (12 \, b c^{5} d^{3} - 7 \, a c^{3} d^{5}\right )} x^{5} + 4 \,{\left (4 \, b c^{7} d - 7 \, a c^{5} d^{3}\right )} x^{3} -{\left (2 \, a c^{7} + 8 \,{\left (4 \, b c^{3} d^{4} - a c d^{6}\right )} x^{6} - 8 \,{\left (4 \, b c^{5} d^{2} - 3 \, a c^{3} d^{4}\right )} x^{4} +{\left (4 \, b c^{7} - 17 \, a c^{5} d^{2}\right )} x^{2}\right )} \sqrt{d x + c} \sqrt{d x - c} + 2 \,{\left (8 \,{\left (4 \, b c^{2} d^{6} + a d^{8}\right )} x^{8} - 8 \,{\left (4 \, b c^{4} d^{4} + a c^{2} d^{6}\right )} x^{6} +{\left (4 \, b c^{6} d^{2} + a c^{4} d^{4}\right )} x^{4} - 4 \,{\left (2 \,{\left (4 \, b c^{2} d^{5} + a d^{7}\right )} x^{7} -{\left (4 \, b c^{4} d^{3} + a c^{2} d^{5}\right )} x^{5}\right )} \sqrt{d x + c} \sqrt{d x - c}\right )} \arctan \left (-\frac{d x - \sqrt{d x + c} \sqrt{d x - c}}{c}\right )}{8 \,{\left (8 \, c^{3} d^{4} x^{8} - 8 \, c^{5} d^{2} x^{6} + c^{7} x^{4} - 4 \,{\left (2 \, c^{3} d^{3} x^{7} - c^{5} d x^{5}\right )} \sqrt{d x + c} \sqrt{d x - c}\right )}} \]

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 - a*c*d^7)*x^7 - 4*(12*b*c^5*d^3 - 7*a*c^3*d^5
)*x^5 + 4*(4*b*c^7*d - 7*a*c^5*d^3)*x^3 - (2*a*c^7 + 8*(4*b*c^3*d^4 - a*c*d^6)*x
^6 - 8*(4*b*c^5*d^2 - 3*a*c^3*d^4)*x^4 + (4*b*c^7 - 17*a*c^5*d^2)*x^2)*sqrt(d*x
+ c)*sqrt(d*x - c) + 2*(8*(4*b*c^2*d^6 + a*d^8)*x^8 - 8*(4*b*c^4*d^4 + a*c^2*d^6
)*x^6 + (4*b*c^6*d^2 + a*c^4*d^4)*x^4 - 4*(2*(4*b*c^2*d^5 + a*d^7)*x^7 - (4*b*c^
4*d^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^3*d^4*x^8 - 8*c^5*d^2*x^6 + c^7*x^4 - 4*(2*c^3*d^3*x^7
- c^5*d*x^5)*sqrt(d*x + c)*sqrt(d*x - c))

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: MellinTransformStripError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: MellinTransformStripError

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GIAC/XCAS [A]  time = 0.244135, size = 437, normalized size = 3.61 \[ -\frac{\frac{{\left (4 \, b c^{2} d^{3} + a d^{5}\right )} \arctan \left (\frac{{\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{2}}{2 \, c}\right )}{c^{3}} - \frac{2 \,{\left (4 \, b c^{2} d^{3}{\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{14} - a d^{5}{\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{14} + 16 \, b c^{4} d^{3}{\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{10} + 28 \, a c^{2} d^{5}{\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{10} - 64 \, b c^{6} d^{3}{\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{6} - 112 \, a c^{4} d^{5}{\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{6} - 256 \, b c^{8} d^{3}{\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{2} + 64 \, a c^{6} d^{5}{\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{2}\right )}}{{\left ({\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{4} + 4 \, c^{2}\right )}^{4} c^{2}}}{4 \, d} \]

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 + a*d^5)*arctan(1/2*(sqrt(d*x + c) - sqrt(d*x - c))^2/c)/c^3
- 2*(4*b*c^2*d^3*(sqrt(d*x + c) - sqrt(d*x - c))^14 - a*d^5*(sqrt(d*x + c) - sqr
t(d*x - c))^14 + 16*b*c^4*d^3*(sqrt(d*x + c) - sqrt(d*x - c))^10 + 28*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 - 112*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 + 64*a*c^6*d^5*(sqrt(d*x + c) - sqrt(d*x - c))^2)/(((sqr
t(d*x + c) - sqrt(d*x - c))^4 + 4*c^2)^4*c^2))/d