3.138 \(\int \frac{a+b x^2+c x^4}{x^7 \sqrt{d-e x} \sqrt{d+e x}} \, dx\)

Optimal. Leaf size=212 \[ -\frac{e^2 \sqrt{d^2-e^2 x^2} \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right ) \left (5 a e^4+6 b d^2 e^2+8 c d^4\right )}{16 d^7 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\sqrt{d-e x} \sqrt{d+e x} \left (5 a e^4+6 b d^2 e^2+8 c d^4\right )}{16 d^6 x^2}-\frac{\sqrt{d-e x} \sqrt{d+e x} \left (5 a e^2+6 b d^2\right )}{24 d^4 x^4}-\frac{a \sqrt{d-e x} \sqrt{d+e x}}{6 d^2 x^6} \]

[Out]

-(a*Sqrt[d - e*x]*Sqrt[d + e*x])/(6*d^2*x^6) - ((6*b*d^2 + 5*a*e^2)*Sqrt[d - e*x
]*Sqrt[d + e*x])/(24*d^4*x^4) - ((8*c*d^4 + 6*b*d^2*e^2 + 5*a*e^4)*Sqrt[d - e*x]
*Sqrt[d + e*x])/(16*d^6*x^2) - (e^2*(8*c*d^4 + 6*b*d^2*e^2 + 5*a*e^4)*Sqrt[d^2 -
e^2*x^2]*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/(16*d^7*Sqrt[d - e*x]*Sqrt[d + e*x])

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Rubi [A]  time = 0.758547, antiderivative size = 248, normalized size of antiderivative = 1.17, number of steps used = 7, number of rules used = 7, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{e^2 \sqrt{d^2-e^2 x^2} \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right ) \left (5 a e^4+6 b d^2 e^2+8 c d^4\right )}{16 d^7 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left (d^2-e^2 x^2\right ) \left (5 a e^4+6 b d^2 e^2+8 c d^4\right )}{16 d^6 x^2 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left (d^2-e^2 x^2\right ) \left (5 a e^2+6 b d^2\right )}{24 d^4 x^4 \sqrt{d-e x} \sqrt{d+e x}}-\frac{a \left (d^2-e^2 x^2\right )}{6 d^2 x^6 \sqrt{d-e x} \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2 + c*x^4)/(x^7*Sqrt[d - e*x]*Sqrt[d + e*x]),x]

[Out]

-(a*(d^2 - e^2*x^2))/(6*d^2*x^6*Sqrt[d - e*x]*Sqrt[d + e*x]) - ((6*b*d^2 + 5*a*e
^2)*(d^2 - e^2*x^2))/(24*d^4*x^4*Sqrt[d - e*x]*Sqrt[d + e*x]) - ((8*c*d^4 + 6*b*
d^2*e^2 + 5*a*e^4)*(d^2 - e^2*x^2))/(16*d^6*x^2*Sqrt[d - e*x]*Sqrt[d + e*x]) - (
e^2*(8*c*d^4 + 6*b*d^2*e^2 + 5*a*e^4)*Sqrt[d^2 - e^2*x^2]*ArcTanh[Sqrt[d^2 - e^2
*x^2]/d])/(16*d^7*Sqrt[d - e*x]*Sqrt[d + e*x])

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Rubi in Sympy [A]  time = 42.9818, size = 196, normalized size = 0.92 \[ - \frac{a \sqrt{d - e x} \sqrt{d + e x}}{6 d^{2} x^{6}} - \frac{\sqrt{d - e x} \sqrt{d + e x} \left (5 a e^{2} + 6 b d^{2}\right )}{24 d^{4} x^{4}} - \frac{\sqrt{d - e x} \sqrt{d + e x} \left (5 a e^{4} + 6 b d^{2} e^{2} + 8 c d^{4}\right )}{16 d^{6} x^{2}} - \frac{e^{2} \sqrt{d - e x} \sqrt{d + e x} \left (5 a e^{4} + 6 b d^{2} e^{2} + 8 c d^{4}\right ) \operatorname{atanh}{\left (\frac{\sqrt{d^{2} - e^{2} x^{2}}}{d} \right )}}{16 d^{7} \sqrt{d^{2} - e^{2} x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**4+b*x**2+a)/x**7/(-e*x+d)**(1/2)/(e*x+d)**(1/2),x)

[Out]

-a*sqrt(d - e*x)*sqrt(d + e*x)/(6*d**2*x**6) - sqrt(d - e*x)*sqrt(d + e*x)*(5*a*
e**2 + 6*b*d**2)/(24*d**4*x**4) - sqrt(d - e*x)*sqrt(d + e*x)*(5*a*e**4 + 6*b*d*
*2*e**2 + 8*c*d**4)/(16*d**6*x**2) - e**2*sqrt(d - e*x)*sqrt(d + e*x)*(5*a*e**4
+ 6*b*d**2*e**2 + 8*c*d**4)*atanh(sqrt(d**2 - e**2*x**2)/d)/(16*d**7*sqrt(d**2 -
e**2*x**2))

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Mathematica [A]  time = 0.409857, size = 177, normalized size = 0.83 \[ \frac{e^2 \log (x) \left (5 a e^4+6 b d^2 e^2+8 c d^4\right )}{16 d^7}-\frac{e^2 \log \left (\sqrt{d-e x} \sqrt{d+e x}+d\right ) \left (5 a e^4+6 b d^2 e^2+8 c d^4\right )}{16 d^7}+\sqrt{d-e x} \sqrt{d+e x} \left (\frac{-5 a e^4-6 b d^2 e^2-8 c d^4}{16 d^6 x^2}+\frac{-5 a e^2-6 b d^2}{24 d^4 x^4}-\frac{a}{6 d^2 x^6}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2 + c*x^4)/(x^7*Sqrt[d - e*x]*Sqrt[d + e*x]),x]

[Out]

(-a/(6*d^2*x^6) + (-6*b*d^2 - 5*a*e^2)/(24*d^4*x^4) + (-8*c*d^4 - 6*b*d^2*e^2 -
5*a*e^4)/(16*d^6*x^2))*Sqrt[d - e*x]*Sqrt[d + e*x] + (e^2*(8*c*d^4 + 6*b*d^2*e^2
+ 5*a*e^4)*Log[x])/(16*d^7) - (e^2*(8*c*d^4 + 6*b*d^2*e^2 + 5*a*e^4)*Log[d + Sq
rt[d - e*x]*Sqrt[d + e*x]])/(16*d^7)

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Maple [C]  time = 0.052, size = 306, normalized size = 1.4 \[ -{\frac{{\it csgn} \left ( d \right ) }{48\,{d}^{7}{x}^{6}}\sqrt{-ex+d}\sqrt{ex+d} \left ( 15\,\ln \left ( 2\,{\frac{d \left ( \sqrt{-{e}^{2}{x}^{2}+{d}^{2}}{\it csgn} \left ( d \right ) +d \right ) }{x}} \right ){x}^{6}a{e}^{6}+18\,\ln \left ( 2\,{\frac{d \left ( \sqrt{-{e}^{2}{x}^{2}+{d}^{2}}{\it csgn} \left ( d \right ) +d \right ) }{x}} \right ){x}^{6}b{d}^{2}{e}^{4}+24\,\ln \left ( 2\,{\frac{d \left ( \sqrt{-{e}^{2}{x}^{2}+{d}^{2}}{\it csgn} \left ( d \right ) +d \right ) }{x}} \right ){x}^{6}c{d}^{4}{e}^{2}+15\,{\it csgn} \left ( d \right ){x}^{4}ad{e}^{4}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}+18\,{\it csgn} \left ( d \right ){x}^{4}b{d}^{3}{e}^{2}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}+24\,{\it csgn} \left ( d \right ){x}^{4}c{d}^{5}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}+10\,{\it csgn} \left ( d \right ){x}^{2}a{d}^{3}{e}^{2}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}+12\,{\it csgn} \left ( d \right ){x}^{2}b{d}^{5}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}+8\,{\it csgn} \left ( d \right ) a{d}^{5}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}} \right ){\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^4+b*x^2+a)/x^7/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x)

[Out]

-1/48*(-e*x+d)^(1/2)*(e*x+d)^(1/2)/d^7*(15*ln(2*d*((-e^2*x^2+d^2)^(1/2)*csgn(d)+
d)/x)*x^6*a*e^6+18*ln(2*d*((-e^2*x^2+d^2)^(1/2)*csgn(d)+d)/x)*x^6*b*d^2*e^4+24*l
n(2*d*((-e^2*x^2+d^2)^(1/2)*csgn(d)+d)/x)*x^6*c*d^4*e^2+15*csgn(d)*x^4*a*d*e^4*(
-e^2*x^2+d^2)^(1/2)+18*csgn(d)*x^4*b*d^3*e^2*(-e^2*x^2+d^2)^(1/2)+24*csgn(d)*x^4
*c*d^5*(-e^2*x^2+d^2)^(1/2)+10*csgn(d)*x^2*a*d^3*e^2*(-e^2*x^2+d^2)^(1/2)+12*csg
n(d)*x^2*b*d^5*(-e^2*x^2+d^2)^(1/2)+8*csgn(d)*a*d^5*(-e^2*x^2+d^2)^(1/2))*csgn(d
)/(-e^2*x^2+d^2)^(1/2)/x^6

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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((c*x^4 + b*x^2 + a)/(sqrt(e*x + d)*sqrt(-e*x + d)*x^7),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.483568, size = 923, normalized size = 4.35 \[ -\frac{256 \, a d^{12} - 18 \,{\left (8 \, c d^{6} e^{6} + 6 \, b d^{4} e^{8} + 5 \, a d^{2} e^{10}\right )} x^{10} + 6 \,{\left (152 \, c d^{8} e^{4} + 102 \, b d^{6} e^{6} + 85 \, a d^{4} e^{8}\right )} x^{8} - 4 \,{\left (384 \, c d^{10} e^{2} + 174 \, b d^{8} e^{4} + 157 \, a d^{6} e^{6}\right )} x^{6} + 48 \,{\left (16 \, c d^{12} - 4 \, b d^{10} e^{2} + 3 \, a d^{8} e^{4}\right )} x^{4} + 192 \,{\left (2 \, b d^{12} - a d^{10} e^{2}\right )} x^{2} -{\left (256 \, a d^{11} - 3 \,{\left (8 \, c d^{5} e^{6} + 6 \, b d^{3} e^{8} + 5 \, a d e^{10}\right )} x^{10} + 4 \,{\left (108 \, c d^{7} e^{4} + 78 \, b d^{5} e^{6} + 65 \, a d^{3} e^{8}\right )} x^{8} - 4 \,{\left (288 \, c d^{9} e^{2} + 162 \, b d^{7} e^{4} + 137 \, a d^{5} e^{6}\right )} x^{6} + 48 \,{\left (16 \, c d^{11} + 3 \, a d^{7} e^{4}\right )} x^{4} + 64 \,{\left (6 \, b d^{11} - a d^{9} e^{2}\right )} x^{2}\right )} \sqrt{e x + d} \sqrt{-e x + d} - 3 \,{\left ({\left (8 \, c d^{4} e^{8} + 6 \, b d^{2} e^{10} + 5 \, a e^{12}\right )} x^{12} - 18 \,{\left (8 \, c d^{6} e^{6} + 6 \, b d^{4} e^{8} + 5 \, a d^{2} e^{10}\right )} x^{10} + 48 \,{\left (8 \, c d^{8} e^{4} + 6 \, b d^{6} e^{6} + 5 \, a d^{4} e^{8}\right )} x^{8} - 32 \,{\left (8 \, c d^{10} e^{2} + 6 \, b d^{8} e^{4} + 5 \, a d^{6} e^{6}\right )} x^{6} + 2 \,{\left (3 \,{\left (8 \, c d^{5} e^{6} + 6 \, b d^{3} e^{8} + 5 \, a d e^{10}\right )} x^{10} - 16 \,{\left (8 \, c d^{7} e^{4} + 6 \, b d^{5} e^{6} + 5 \, a d^{3} e^{8}\right )} x^{8} + 16 \,{\left (8 \, c d^{9} e^{2} + 6 \, b d^{7} e^{4} + 5 \, a d^{5} e^{6}\right )} x^{6}\right )} \sqrt{e x + d} \sqrt{-e x + d}\right )} \log \left (\frac{\sqrt{e x + d} \sqrt{-e x + d} - d}{x}\right )}{48 \,{\left (d^{7} e^{6} x^{12} - 18 \, d^{9} e^{4} x^{10} + 48 \, d^{11} e^{2} x^{8} - 32 \, d^{13} x^{6} + 2 \,{\left (3 \, d^{8} e^{4} x^{10} - 16 \, d^{10} e^{2} x^{8} + 16 \, d^{12} x^{6}\right )} \sqrt{e x + d} \sqrt{-e x + d}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)/(sqrt(e*x + d)*sqrt(-e*x + d)*x^7),x, algorithm="fricas")

[Out]

-1/48*(256*a*d^12 - 18*(8*c*d^6*e^6 + 6*b*d^4*e^8 + 5*a*d^2*e^10)*x^10 + 6*(152*
c*d^8*e^4 + 102*b*d^6*e^6 + 85*a*d^4*e^8)*x^8 - 4*(384*c*d^10*e^2 + 174*b*d^8*e^
4 + 157*a*d^6*e^6)*x^6 + 48*(16*c*d^12 - 4*b*d^10*e^2 + 3*a*d^8*e^4)*x^4 + 192*(
2*b*d^12 - a*d^10*e^2)*x^2 - (256*a*d^11 - 3*(8*c*d^5*e^6 + 6*b*d^3*e^8 + 5*a*d*
e^10)*x^10 + 4*(108*c*d^7*e^4 + 78*b*d^5*e^6 + 65*a*d^3*e^8)*x^8 - 4*(288*c*d^9*
e^2 + 162*b*d^7*e^4 + 137*a*d^5*e^6)*x^6 + 48*(16*c*d^11 + 3*a*d^7*e^4)*x^4 + 64
*(6*b*d^11 - a*d^9*e^2)*x^2)*sqrt(e*x + d)*sqrt(-e*x + d) - 3*((8*c*d^4*e^8 + 6*
b*d^2*e^10 + 5*a*e^12)*x^12 - 18*(8*c*d^6*e^6 + 6*b*d^4*e^8 + 5*a*d^2*e^10)*x^10
+ 48*(8*c*d^8*e^4 + 6*b*d^6*e^6 + 5*a*d^4*e^8)*x^8 - 32*(8*c*d^10*e^2 + 6*b*d^8
*e^4 + 5*a*d^6*e^6)*x^6 + 2*(3*(8*c*d^5*e^6 + 6*b*d^3*e^8 + 5*a*d*e^10)*x^10 - 1
6*(8*c*d^7*e^4 + 6*b*d^5*e^6 + 5*a*d^3*e^8)*x^8 + 16*(8*c*d^9*e^2 + 6*b*d^7*e^4
+ 5*a*d^5*e^6)*x^6)*sqrt(e*x + d)*sqrt(-e*x + d))*log((sqrt(e*x + d)*sqrt(-e*x +
d) - d)/x))/(d^7*e^6*x^12 - 18*d^9*e^4*x^10 + 48*d^11*e^2*x^8 - 32*d^13*x^6 + 2
*(3*d^8*e^4*x^10 - 16*d^10*e^2*x^8 + 16*d^12*x^6)*sqrt(e*x + d)*sqrt(-e*x + d))

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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((c*x**4+b*x**2+a)/x**7/(-e*x+d)**(1/2)/(e*x+d)**(1/2),x)

[Out]

Exception raised: MellinTransformStripError

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GIAC/XCAS [A]  time = 0.743374, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)/(sqrt(e*x + d)*sqrt(-e*x + d)*x^7),x, algorithm="giac")

[Out]

sage0*x