3.250 \(\int \frac{1}{x^3 \left (d+e x^2\right ) \left (a+c x^4\right )^2} \, dx\)

Optimal. Leaf size=236 \[ -\frac{c^{3/2} d \left (2 a e^2+c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{2 a^{5/2} \left (a e^2+c d^2\right )^2}-\frac{c^{3/2} d \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{4 a^{5/2} \left (a e^2+c d^2\right )}+\frac{c e \left (2 a e^2+c d^2\right ) \log \left (a+c x^4\right )}{4 a^2 \left (a e^2+c d^2\right )^2}-\frac{c \left (a e+c d x^2\right )}{4 a^2 \left (a+c x^4\right ) \left (a e^2+c d^2\right )}-\frac{e \log (x)}{a^2 d^2}-\frac{1}{2 a^2 d x^2}+\frac{e^5 \log \left (d+e x^2\right )}{2 d^2 \left (a e^2+c d^2\right )^2} \]

[Out]

-1/(2*a^2*d*x^2) - (c*(a*e + c*d*x^2))/(4*a^2*(c*d^2 + a*e^2)*(a + c*x^4)) - (c^
(3/2)*d*ArcTan[(Sqrt[c]*x^2)/Sqrt[a]])/(4*a^(5/2)*(c*d^2 + a*e^2)) - (c^(3/2)*d*
(c*d^2 + 2*a*e^2)*ArcTan[(Sqrt[c]*x^2)/Sqrt[a]])/(2*a^(5/2)*(c*d^2 + a*e^2)^2) -
(e*Log[x])/(a^2*d^2) + (e^5*Log[d + e*x^2])/(2*d^2*(c*d^2 + a*e^2)^2) + (c*e*(c
*d^2 + 2*a*e^2)*Log[a + c*x^4])/(4*a^2*(c*d^2 + a*e^2)^2)

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Rubi [A]  time = 0.53423, antiderivative size = 236, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ -\frac{c^{3/2} d \left (2 a e^2+c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{2 a^{5/2} \left (a e^2+c d^2\right )^2}-\frac{c^{3/2} d \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{4 a^{5/2} \left (a e^2+c d^2\right )}+\frac{c e \left (2 a e^2+c d^2\right ) \log \left (a+c x^4\right )}{4 a^2 \left (a e^2+c d^2\right )^2}-\frac{c \left (a e+c d x^2\right )}{4 a^2 \left (a+c x^4\right ) \left (a e^2+c d^2\right )}-\frac{e \log (x)}{a^2 d^2}-\frac{1}{2 a^2 d x^2}+\frac{e^5 \log \left (d+e x^2\right )}{2 d^2 \left (a e^2+c d^2\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^3*(d + e*x^2)*(a + c*x^4)^2),x]

[Out]

-1/(2*a^2*d*x^2) - (c*(a*e + c*d*x^2))/(4*a^2*(c*d^2 + a*e^2)*(a + c*x^4)) - (c^
(3/2)*d*ArcTan[(Sqrt[c]*x^2)/Sqrt[a]])/(4*a^(5/2)*(c*d^2 + a*e^2)) - (c^(3/2)*d*
(c*d^2 + 2*a*e^2)*ArcTan[(Sqrt[c]*x^2)/Sqrt[a]])/(2*a^(5/2)*(c*d^2 + a*e^2)^2) -
(e*Log[x])/(a^2*d^2) + (e^5*Log[d + e*x^2])/(2*d^2*(c*d^2 + a*e^2)^2) + (c*e*(c
*d^2 + 2*a*e^2)*Log[a + c*x^4])/(4*a^2*(c*d^2 + a*e^2)^2)

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Rubi in Sympy [A]  time = 76.2225, size = 221, normalized size = 0.94 \[ \frac{e^{5} \log{\left (d + e x^{2} \right )}}{2 d^{2} \left (a e^{2} + c d^{2}\right )^{2}} + \frac{c e \left (2 a e^{2} + c d^{2}\right ) \log{\left (a + c x^{4} \right )}}{4 a^{2} \left (a e^{2} + c d^{2}\right )^{2}} - \frac{c \left (a e + c d x^{2}\right )}{4 a^{2} \left (a + c x^{4}\right ) \left (a e^{2} + c d^{2}\right )} - \frac{1}{2 a^{2} d x^{2}} - \frac{e \log{\left (x^{2} \right )}}{2 a^{2} d^{2}} - \frac{c^{\frac{3}{2}} d \operatorname{atan}{\left (\frac{\sqrt{c} x^{2}}{\sqrt{a}} \right )}}{4 a^{\frac{5}{2}} \left (a e^{2} + c d^{2}\right )} - \frac{c^{\frac{3}{2}} d \left (2 a e^{2} + c d^{2}\right ) \operatorname{atan}{\left (\frac{\sqrt{c} x^{2}}{\sqrt{a}} \right )}}{2 a^{\frac{5}{2}} \left (a e^{2} + c d^{2}\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**3/(e*x**2+d)/(c*x**4+a)**2,x)

[Out]

e**5*log(d + e*x**2)/(2*d**2*(a*e**2 + c*d**2)**2) + c*e*(2*a*e**2 + c*d**2)*log
(a + c*x**4)/(4*a**2*(a*e**2 + c*d**2)**2) - c*(a*e + c*d*x**2)/(4*a**2*(a + c*x
**4)*(a*e**2 + c*d**2)) - 1/(2*a**2*d*x**2) - e*log(x**2)/(2*a**2*d**2) - c**(3/
2)*d*atan(sqrt(c)*x**2/sqrt(a))/(4*a**(5/2)*(a*e**2 + c*d**2)) - c**(3/2)*d*(2*a
*e**2 + c*d**2)*atan(sqrt(c)*x**2/sqrt(a))/(2*a**(5/2)*(a*e**2 + c*d**2)**2)

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Mathematica [A]  time = 1.30543, size = 248, normalized size = 1.05 \[ \frac{1}{4} \left (\frac{c^{3/2} d \left (5 a e^2+3 c d^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{a^{5/2} \left (a e^2+c d^2\right )^2}+\frac{c^{3/2} d \left (5 a e^2+3 c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{a^{5/2} \left (a e^2+c d^2\right )^2}-\frac{c \left (a e+c d x^2\right )}{a^2 \left (a+c x^4\right ) \left (a e^2+c d^2\right )}+\frac{c \left (2 a e^3+c d^2 e\right ) \log \left (a+c x^4\right )}{a^2 \left (a e^2+c d^2\right )^2}-\frac{4 e \log (x)}{a^2 d^2}-\frac{2}{a^2 d x^2}+\frac{2 e^5 \log \left (d+e x^2\right )}{\left (a d e^2+c d^3\right )^2}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^3*(d + e*x^2)*(a + c*x^4)^2),x]

[Out]

(-2/(a^2*d*x^2) - (c*(a*e + c*d*x^2))/(a^2*(c*d^2 + a*e^2)*(a + c*x^4)) + (c^(3/
2)*d*(3*c*d^2 + 5*a*e^2)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(a^(5/2)*(c*d^
2 + a*e^2)^2) + (c^(3/2)*d*(3*c*d^2 + 5*a*e^2)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^
(1/4)])/(a^(5/2)*(c*d^2 + a*e^2)^2) - (4*e*Log[x])/(a^2*d^2) + (2*e^5*Log[d + e*
x^2])/(c*d^3 + a*d*e^2)^2 + (c*(c*d^2*e + 2*a*e^3)*Log[a + c*x^4])/(a^2*(c*d^2 +
a*e^2)^2))/4

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Maple [A]  time = 0.032, size = 332, normalized size = 1.4 \[ -{\frac{1}{2\,{a}^{2}d{x}^{2}}}-{\frac{\ln \left ( x \right ) e}{{a}^{2}{d}^{2}}}-{\frac{{c}^{2}{x}^{2}{e}^{2}d}{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}a \left ( c{x}^{4}+a \right ) }}-{\frac{{c}^{3}{x}^{2}{d}^{3}}{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}{a}^{2} \left ( c{x}^{4}+a \right ) }}-{\frac{c{e}^{3}}{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \left ( c{x}^{4}+a \right ) }}-{\frac{{d}^{2}e{c}^{2}}{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}a \left ( c{x}^{4}+a \right ) }}+{\frac{c\ln \left ( c{x}^{4}+a \right ){e}^{3}}{2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}a}}+{\frac{{c}^{2}\ln \left ( c{x}^{4}+a \right ){d}^{2}e}{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}{a}^{2}}}-{\frac{5\,{c}^{2}{e}^{2}d}{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}a}\arctan \left ({c{x}^{2}{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}-{\frac{3\,{c}^{3}{d}^{3}}{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}{a}^{2}}\arctan \left ({c{x}^{2}{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{{e}^{5}\ln \left ( e{x}^{2}+d \right ) }{2\,{d}^{2} \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^3/(e*x^2+d)/(c*x^4+a)^2,x)

[Out]

-1/2/a^2/d/x^2-e*ln(x)/a^2/d^2-1/4*c^2/(a*e^2+c*d^2)^2/a/(c*x^4+a)*x^2*e^2*d-1/4
*c^3/(a*e^2+c*d^2)^2/a^2/(c*x^4+a)*x^2*d^3-1/4*c/(a*e^2+c*d^2)^2/(c*x^4+a)*e^3-1
/4*c^2/(a*e^2+c*d^2)^2/a/(c*x^4+a)*e*d^2+1/2*c/(a*e^2+c*d^2)^2/a*ln(c*x^4+a)*e^3
+1/4*c^2/(a*e^2+c*d^2)^2/a^2*ln(c*x^4+a)*d^2*e-5/4*c^2/(a*e^2+c*d^2)^2/a/(a*c)^(
1/2)*arctan(c*x^2/(a*c)^(1/2))*e^2*d-3/4*c^3/(a*e^2+c*d^2)^2/a^2/(a*c)^(1/2)*arc
tan(c*x^2/(a*c)^(1/2))*d^3+1/2*e^5*ln(e*x^2+d)/d^2/(a*e^2+c*d^2)^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(1/((c*x^4 + a)^2*(e*x^2 + d)*x^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^4 + a)^2*(e*x^2 + d)*x^3),x, algorithm="fricas")

[Out]

Timed out

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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(1/x**3/(e*x**2+d)/(c*x**4+a)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.276167, size = 464, normalized size = 1.97 \[ \frac{{\left (c^{2} d^{2} e + 2 \, a c e^{3}\right )}{\rm ln}\left (c x^{4} + a\right )}{4 \,{\left (a^{2} c^{2} d^{4} + 2 \, a^{3} c d^{2} e^{2} + a^{4} e^{4}\right )}} + \frac{e^{6}{\rm ln}\left ({\left | x^{2} e + d \right |}\right )}{2 \,{\left (c^{2} d^{6} e + 2 \, a c d^{4} e^{3} + a^{2} d^{2} e^{5}\right )}} - \frac{{\left (3 \, c^{3} d^{3} + 5 \, a c^{2} d e^{2}\right )} \arctan \left (\frac{c x^{2}}{\sqrt{a c}}\right )}{4 \,{\left (a^{2} c^{2} d^{4} + 2 \, a^{3} c d^{2} e^{2} + a^{4} e^{4}\right )} \sqrt{a c}} - \frac{9 \, c^{3} d^{5} x^{4} + 15 \, a c^{2} d^{3} x^{4} e^{2} - 2 \, a^{2} c x^{6} e^{5} + 3 \, a c^{2} d^{4} x^{2} e + 6 \, a^{2} c d x^{4} e^{4} + 6 \, a c^{2} d^{5} + 3 \, a^{2} c d^{2} x^{2} e^{3} + 12 \, a^{2} c d^{3} e^{2} - 2 \, a^{3} x^{2} e^{5} + 6 \, a^{3} d e^{4}}{12 \,{\left (a^{2} c^{2} d^{6} + 2 \, a^{3} c d^{4} e^{2} + a^{4} d^{2} e^{4}\right )}{\left (c x^{6} + a x^{2}\right )}} - \frac{e{\rm ln}\left (x^{2}\right )}{2 \, a^{2} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^4 + a)^2*(e*x^2 + d)*x^3),x, algorithm="giac")

[Out]

1/4*(c^2*d^2*e + 2*a*c*e^3)*ln(c*x^4 + a)/(a^2*c^2*d^4 + 2*a^3*c*d^2*e^2 + a^4*e
^4) + 1/2*e^6*ln(abs(x^2*e + d))/(c^2*d^6*e + 2*a*c*d^4*e^3 + a^2*d^2*e^5) - 1/4
*(3*c^3*d^3 + 5*a*c^2*d*e^2)*arctan(c*x^2/sqrt(a*c))/((a^2*c^2*d^4 + 2*a^3*c*d^2
*e^2 + a^4*e^4)*sqrt(a*c)) - 1/12*(9*c^3*d^5*x^4 + 15*a*c^2*d^3*x^4*e^2 - 2*a^2*
c*x^6*e^5 + 3*a*c^2*d^4*x^2*e + 6*a^2*c*d*x^4*e^4 + 6*a*c^2*d^5 + 3*a^2*c*d^2*x^
2*e^3 + 12*a^2*c*d^3*e^2 - 2*a^3*x^2*e^5 + 6*a^3*d*e^4)/((a^2*c^2*d^6 + 2*a^3*c*
d^4*e^2 + a^4*d^2*e^4)*(c*x^6 + a*x^2)) - 1/2*e*ln(x^2)/(a^2*d^2)