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

Optimal. Leaf size=552 \[ -\frac{\sqrt [4]{c} \left (\sqrt{c} d^2 \left (c d^4-3 a e^4\right )-\sqrt{a} e^2 \left (3 c d^4-a e^4\right )\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{2} a^{3/4} \left (a e^4+c d^4\right )^2}+\frac{\sqrt [4]{c} \left (\sqrt{c} d^2 \left (c d^4-3 a e^4\right )-\sqrt{a} e^2 \left (3 c d^4-a e^4\right )\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{2} a^{3/4} \left (a e^4+c d^4\right )^2}-\frac{\sqrt [4]{c} \left (\sqrt{a} e^2 \left (3 c d^4-a e^4\right )+\sqrt{c} d^2 \left (c d^4-3 a e^4\right )\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{3/4} \left (a e^4+c d^4\right )^2}+\frac{\sqrt [4]{c} \left (\sqrt{a} e^2 \left (3 c d^4-a e^4\right )+\sqrt{c} d^2 \left (c d^4-3 a e^4\right )\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} a^{3/4} \left (a e^4+c d^4\right )^2}-\frac{\sqrt{c} d e \left (c d^4-a e^4\right ) \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{\sqrt{a} \left (a e^4+c d^4\right )^2}-\frac{e^3}{(d+e x) \left (a e^4+c d^4\right )}-\frac{c d^3 e^3 \log \left (a+c x^4\right )}{\left (a e^4+c d^4\right )^2}+\frac{4 c d^3 e^3 \log (d+e x)}{\left (a e^4+c d^4\right )^2} \]

[Out]

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

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Rubi [A]  time = 1.82237, antiderivative size = 552, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 11, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.647 \[ -\frac{\sqrt [4]{c} \left (\sqrt{c} d^2 \left (c d^4-3 a e^4\right )-\sqrt{a} e^2 \left (3 c d^4-a e^4\right )\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{2} a^{3/4} \left (a e^4+c d^4\right )^2}+\frac{\sqrt [4]{c} \left (\sqrt{c} d^2 \left (c d^4-3 a e^4\right )-\sqrt{a} e^2 \left (3 c d^4-a e^4\right )\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{2} a^{3/4} \left (a e^4+c d^4\right )^2}-\frac{\sqrt [4]{c} \left (\sqrt{a} e^2 \left (3 c d^4-a e^4\right )+\sqrt{c} d^2 \left (c d^4-3 a e^4\right )\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{3/4} \left (a e^4+c d^4\right )^2}+\frac{\sqrt [4]{c} \left (\sqrt{a} e^2 \left (3 c d^4-a e^4\right )+\sqrt{c} d^2 \left (c d^4-3 a e^4\right )\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} a^{3/4} \left (a e^4+c d^4\right )^2}-\frac{\sqrt{c} d e \left (c d^4-a e^4\right ) \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{\sqrt{a} \left (a e^4+c d^4\right )^2}-\frac{e^3}{(d+e x) \left (a e^4+c d^4\right )}-\frac{c d^3 e^3 \log \left (a+c x^4\right )}{\left (a e^4+c d^4\right )^2}+\frac{4 c d^3 e^3 \log (d+e x)}{\left (a e^4+c d^4\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [A]  time = 1.37711, size = 524, normalized size = 0.95 \[ \frac{-\frac{\sqrt{2} \sqrt [4]{c} \left (a^{3/2} e^6-3 \sqrt{a} c d^4 e^2-3 a \sqrt{c} d^2 e^4+c^{3/2} d^6\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{a^{3/4}}+\frac{\sqrt{2} \sqrt [4]{c} \left (a^{3/2} e^6-3 \sqrt{a} c d^4 e^2-3 a \sqrt{c} d^2 e^4+c^{3/2} d^6\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{a^{3/4}}+\frac{2 \sqrt [4]{c} \left (\sqrt{a} e^2-\sqrt{c} d^2\right ) \left (-4 a^{3/4} \sqrt [4]{c} d e^3-4 \sqrt [4]{a} c^{3/4} d^3 e+4 \sqrt{2} \sqrt{a} \sqrt{c} d^2 e^2+\sqrt{2} a e^4+\sqrt{2} c d^4\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{a^{3/4}}+\frac{2 \sqrt [4]{c} \left (\sqrt{c} d^2-\sqrt{a} e^2\right ) \left (4 a^{3/4} \sqrt [4]{c} d e^3+4 \sqrt [4]{a} c^{3/4} d^3 e+4 \sqrt{2} \sqrt{a} \sqrt{c} d^2 e^2+\sqrt{2} a e^4+\sqrt{2} c d^4\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{a^{3/4}}-\frac{8 e^3 \left (a e^4+c d^4\right )}{d+e x}-8 c d^3 e^3 \log \left (a+c x^4\right )+32 c d^3 e^3 \log (d+e x)}{8 \left (a e^4+c d^4\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

((-8*e^3*(c*d^4 + a*e^4))/(d + e*x) + (2*c^(1/4)*(-(Sqrt[c]*d^2) + Sqrt[a]*e^2)*
(Sqrt[2]*c*d^4 - 4*a^(1/4)*c^(3/4)*d^3*e + 4*Sqrt[2]*Sqrt[a]*Sqrt[c]*d^2*e^2 - 4
*a^(3/4)*c^(1/4)*d*e^3 + Sqrt[2]*a*e^4)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])
/a^(3/4) + (2*c^(1/4)*(Sqrt[c]*d^2 - Sqrt[a]*e^2)*(Sqrt[2]*c*d^4 + 4*a^(1/4)*c^(
3/4)*d^3*e + 4*Sqrt[2]*Sqrt[a]*Sqrt[c]*d^2*e^2 + 4*a^(3/4)*c^(1/4)*d*e^3 + Sqrt[
2]*a*e^4)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/a^(3/4) + 32*c*d^3*e^3*Log[d
+ e*x] - (Sqrt[2]*c^(1/4)*(c^(3/2)*d^6 - 3*Sqrt[a]*c*d^4*e^2 - 3*a*Sqrt[c]*d^2*e
^4 + a^(3/2)*e^6)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/a^(3/4
) + (Sqrt[2]*c^(1/4)*(c^(3/2)*d^6 - 3*Sqrt[a]*c*d^4*e^2 - 3*a*Sqrt[c]*d^2*e^4 +
a^(3/2)*e^6)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/a^(3/4) - 8
*c*d^3*e^3*Log[a + c*x^4])/(8*(c*d^4 + a*e^4)^2)

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Maple [A]  time = 0.016, size = 866, normalized size = 1.6 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)^2/(c*x^4+a),x)

[Out]

-3/4/(a*e^4+c*d^4)^2*c*(1/c*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/c*a)^(1/4)*x+1)*d
^2*e^4+1/4/(a*e^4+c*d^4)^2*c^2*(1/c*a)^(1/4)/a*2^(1/2)*arctan(2^(1/2)/(1/c*a)^(1
/4)*x+1)*d^6-3/4/(a*e^4+c*d^4)^2*c*(1/c*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/c*a)^
(1/4)*x-1)*d^2*e^4+1/4/(a*e^4+c*d^4)^2*c^2*(1/c*a)^(1/4)/a*2^(1/2)*arctan(2^(1/2
)/(1/c*a)^(1/4)*x-1)*d^6-3/8/(a*e^4+c*d^4)^2*c*(1/c*a)^(1/4)*2^(1/2)*ln((x^2+(1/
c*a)^(1/4)*x*2^(1/2)+(1/c*a)^(1/2))/(x^2-(1/c*a)^(1/4)*x*2^(1/2)+(1/c*a)^(1/2)))
*d^2*e^4+1/8/(a*e^4+c*d^4)^2*c^2*(1/c*a)^(1/4)/a*2^(1/2)*ln((x^2+(1/c*a)^(1/4)*x
*2^(1/2)+(1/c*a)^(1/2))/(x^2-(1/c*a)^(1/4)*x*2^(1/2)+(1/c*a)^(1/2)))*d^6+1/(a*e^
4+c*d^4)^2*c/(a*c)^(1/2)*arctan(x^2*(c/a)^(1/2))*a*d*e^5-1/(a*e^4+c*d^4)^2*c^2/(
a*c)^(1/2)*arctan(x^2*(c/a)^(1/2))*d^5*e-1/8/(a*e^4+c*d^4)^2/(1/c*a)^(1/4)*2^(1/
2)*ln((x^2-(1/c*a)^(1/4)*x*2^(1/2)+(1/c*a)^(1/2))/(x^2+(1/c*a)^(1/4)*x*2^(1/2)+(
1/c*a)^(1/2)))*a*e^6+3/8/(a*e^4+c*d^4)^2*c/(1/c*a)^(1/4)*2^(1/2)*ln((x^2-(1/c*a)
^(1/4)*x*2^(1/2)+(1/c*a)^(1/2))/(x^2+(1/c*a)^(1/4)*x*2^(1/2)+(1/c*a)^(1/2)))*d^4
*e^2-1/4/(a*e^4+c*d^4)^2/(1/c*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/c*a)^(1/4)*x+1)
*a*e^6+3/4/(a*e^4+c*d^4)^2*c/(1/c*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/c*a)^(1/4)*
x+1)*d^4*e^2-1/4/(a*e^4+c*d^4)^2/(1/c*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/c*a)^(1
/4)*x-1)*a*e^6+3/4/(a*e^4+c*d^4)^2*c/(1/c*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/c*a
)^(1/4)*x-1)*d^4*e^2-c*d^3*e^3*ln(c*x^4+a)/(a*e^4+c*d^4)^2-e^3/(a*e^4+c*d^4)/(e*
x+d)+4*c*d^3*e^3*ln(e*x+d)/(a*e^4+c*d^4)^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)*(e*x + d)^2),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)*(e*x + d)^2),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/(e*x+d)**2/(c*x**4+a),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (c x^{4} + a\right )}{\left (e x + d\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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