3.248 \(\int \frac{\sqrt{-c+d x} \sqrt{c+d x} \left (a+b x^2\right )}{x^4} \, dx\)

Optimal. Leaf size=84 \[ \frac{a (d x-c)^{3/2} (c+d x)^{3/2}}{3 c^2 x^3}-\frac{b \sqrt{d x-c} \sqrt{c+d x}}{x}+2 b d \tanh ^{-1}\left (\frac{\sqrt{d x-c}}{\sqrt{c+d x}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.259594, antiderivative size = 84, 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 \[ \frac{a (d x-c)^{3/2} (c+d x)^{3/2}}{3 c^2 x^3}-\frac{b \sqrt{d x-c} \sqrt{c+d x}}{x}+2 b d \tanh ^{-1}\left (\frac{\sqrt{d x-c}}{\sqrt{c+d x}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 15.5248, size = 70, normalized size = 0.83 \[ \frac{a \left (- c + d x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{3}{2}}}{3 c^{2} x^{3}} + 2 b d \operatorname{atanh}{\left (\frac{\sqrt{- c + d x}}{\sqrt{c + d x}} \right )} - \frac{b \sqrt{- c + d x} \sqrt{c + d x}}{x} \]

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**4,x)

[Out]

a*(-c + d*x)**(3/2)*(c + d*x)**(3/2)/(3*c**2*x**3) + 2*b*d*atanh(sqrt(-c + d*x)/
sqrt(c + d*x)) - b*sqrt(-c + d*x)*sqrt(c + d*x)/x

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Mathematica [A]  time = 0.162109, size = 78, normalized size = 0.93 \[ \frac{\sqrt{d x-c} \sqrt{c+d x} \left (a \left (\frac{d^2 x^2}{c^2}-1\right )-3 b x^2\right )}{3 x^3}+b d \log \left (\sqrt{d x-c} \sqrt{c+d x}+d x\right ) \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[-c + d*x]*Sqrt[c + d*x]*(-3*b*x^2 + a*(-1 + (d^2*x^2)/c^2)))/(3*x^3) + b*d
*Log[d*x + Sqrt[-c + d*x]*Sqrt[c + d*x]]

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Maple [C]  time = 0.022, size = 153, normalized size = 1.8 \[{\frac{{\it csgn} \left ( d \right ) }{3\,{c}^{2}{x}^{3}}\sqrt{dx-c}\sqrt{dx+c} \left ( 3\,\ln \left ( \left ({\it csgn} \left ( d \right ) \sqrt{{d}^{2}{x}^{2}-{c}^{2}}+dx \right ){\it csgn} \left ( d \right ) \right ){x}^{3}b{c}^{2}d+{\it csgn} \left ( d \right ){x}^{2}a{d}^{2}\sqrt{{d}^{2}{x}^{2}-{c}^{2}}-3\,{\it csgn} \left ( d \right ){x}^{2}b{c}^{2}\sqrt{{d}^{2}{x}^{2}-{c}^{2}}-{\it csgn} \left ( d \right ) a{c}^{2}\sqrt{{d}^{2}{x}^{2}-{c}^{2}} \right ){\frac{1}{\sqrt{{d}^{2}{x}^{2}-{c}^{2}}}}} \]

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^4,x)

[Out]

1/3*(d*x-c)^(1/2)*(d*x+c)^(1/2)*(3*ln((csgn(d)*(d^2*x^2-c^2)^(1/2)+d*x)*csgn(d))
*x^3*b*c^2*d+csgn(d)*x^2*a*d^2*(d^2*x^2-c^2)^(1/2)-3*csgn(d)*x^2*b*c^2*(d^2*x^2-
c^2)^(1/2)-csgn(d)*a*c^2*(d^2*x^2-c^2)^(1/2))*csgn(d)/(d^2*x^2-c^2)^(1/2)/c^2/x^
3

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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^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.237663, size = 313, normalized size = 3.73 \[ \frac{a c^{4} - 6 \,{\left (b c^{2} d^{2} - a d^{4}\right )} x^{4} + 3 \,{\left (b c^{4} - 2 \, a c^{2} d^{2}\right )} x^{2} + 3 \,{\left (a c^{2} d x + 2 \,{\left (b c^{2} d - a d^{3}\right )} x^{3}\right )} \sqrt{d x + c} \sqrt{d x - c} - 3 \,{\left (4 \, b d^{4} x^{6} - 3 \, b c^{2} d^{2} x^{4} -{\left (4 \, b d^{3} x^{5} - b c^{2} d x^{3}\right )} \sqrt{d x + c} \sqrt{d x - c}\right )} \log \left (-d x + \sqrt{d x + c} \sqrt{d x - c}\right )}{3 \,{\left (4 \, d^{3} x^{6} - 3 \, c^{2} d x^{4} -{\left (4 \, d^{2} x^{5} - c^{2} x^{3}\right )} \sqrt{d x + c} \sqrt{d x - c}\right )}} \]

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^4,x, algorithm="fricas")

[Out]

1/3*(a*c^4 - 6*(b*c^2*d^2 - a*d^4)*x^4 + 3*(b*c^4 - 2*a*c^2*d^2)*x^2 + 3*(a*c^2*
d*x + 2*(b*c^2*d - a*d^3)*x^3)*sqrt(d*x + c)*sqrt(d*x - c) - 3*(4*b*d^4*x^6 - 3*
b*c^2*d^2*x^4 - (4*b*d^3*x^5 - b*c^2*d*x^3)*sqrt(d*x + c)*sqrt(d*x - c))*log(-d*
x + sqrt(d*x + c)*sqrt(d*x - c)))/(4*d^3*x^6 - 3*c^2*d*x^4 - (4*d^2*x^5 - c^2*x^
3)*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**4,x)

[Out]

Exception raised: MellinTransformStripError

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GIAC/XCAS [A]  time = 0.23387, size = 231, normalized size = 2.75 \[ -\frac{3 \, b d^{2}{\rm ln}\left ({\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{4}\right ) + \frac{16 \,{\left (3 \, b c^{2} d^{2}{\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{8} - 3 \, a d^{4}{\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{8} + 24 \, b c^{4} d^{2}{\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{4} + 48 \, b c^{6} d^{2} - 16 \, a c^{4} d^{4}\right )}}{{\left ({\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{4} + 4 \, c^{2}\right )}^{3}}}{6 \, d} \]

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^4,x, algorithm="giac")

[Out]

-1/6*(3*b*d^2*ln((sqrt(d*x + c) - sqrt(d*x - c))^4) + 16*(3*b*c^2*d^2*(sqrt(d*x
+ c) - sqrt(d*x - c))^8 - 3*a*d^4*(sqrt(d*x + c) - sqrt(d*x - c))^8 + 24*b*c^4*d
^2*(sqrt(d*x + c) - sqrt(d*x - c))^4 + 48*b*c^6*d^2 - 16*a*c^4*d^4)/((sqrt(d*x +
c) - sqrt(d*x - c))^4 + 4*c^2)^3)/d