3.394 \(\int \frac{(d+e x)^3}{a+c x^4} \, dx\)

Optimal. Leaf size=320 \[ -\frac{d \left (\sqrt{c} d^2-3 \sqrt{a} e^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{2} a^{3/4} c^{3/4}}+\frac{d \left (\sqrt{c} d^2-3 \sqrt{a} e^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{2} a^{3/4} c^{3/4}}-\frac{d \left (3 \sqrt{a} e^2+\sqrt{c} d^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{3/4} c^{3/4}}+\frac{d \left (3 \sqrt{a} e^2+\sqrt{c} d^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} a^{3/4} c^{3/4}}+\frac{3 d^2 e \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{2 \sqrt{a} \sqrt{c}}+\frac{e^3 \log \left (a+c x^4\right )}{4 c} \]

[Out]

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

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Rubi [A]  time = 0.583225, antiderivative size = 320, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 10, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.588 \[ -\frac{d \left (\sqrt{c} d^2-3 \sqrt{a} e^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{2} a^{3/4} c^{3/4}}+\frac{d \left (\sqrt{c} d^2-3 \sqrt{a} e^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{2} a^{3/4} c^{3/4}}-\frac{d \left (3 \sqrt{a} e^2+\sqrt{c} d^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{3/4} c^{3/4}}+\frac{d \left (3 \sqrt{a} e^2+\sqrt{c} d^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} a^{3/4} c^{3/4}}+\frac{3 d^2 e \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{2 \sqrt{a} \sqrt{c}}+\frac{e^3 \log \left (a+c x^4\right )}{4 c} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^3/(a + c*x^4),x]

[Out]

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

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Rubi in Sympy [A]  time = 85.5033, size = 306, normalized size = 0.96 \[ \frac{e^{3} \log{\left (a + c x^{4} \right )}}{4 c} + \frac{3 d^{2} e \operatorname{atan}{\left (\frac{\sqrt{c} x^{2}}{\sqrt{a}} \right )}}{2 \sqrt{a} \sqrt{c}} + \frac{\sqrt{2} d \left (3 \sqrt{a} e^{2} - \sqrt{c} d^{2}\right ) \log{\left (- \sqrt{2} \sqrt [4]{a} c^{\frac{3}{4}} x + \sqrt{a} \sqrt{c} + c x^{2} \right )}}{8 a^{\frac{3}{4}} c^{\frac{3}{4}}} - \frac{\sqrt{2} d \left (3 \sqrt{a} e^{2} - \sqrt{c} d^{2}\right ) \log{\left (\sqrt{2} \sqrt [4]{a} c^{\frac{3}{4}} x + \sqrt{a} \sqrt{c} + c x^{2} \right )}}{8 a^{\frac{3}{4}} c^{\frac{3}{4}}} - \frac{\sqrt{2} d \left (3 \sqrt{a} e^{2} + \sqrt{c} d^{2}\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}} \right )}}{4 a^{\frac{3}{4}} c^{\frac{3}{4}}} + \frac{\sqrt{2} d \left (3 \sqrt{a} e^{2} + \sqrt{c} d^{2}\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}} \right )}}{4 a^{\frac{3}{4}} c^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

e**3*log(a + c*x**4)/(4*c) + 3*d**2*e*atan(sqrt(c)*x**2/sqrt(a))/(2*sqrt(a)*sqrt
(c)) + sqrt(2)*d*(3*sqrt(a)*e**2 - sqrt(c)*d**2)*log(-sqrt(2)*a**(1/4)*c**(3/4)*
x + sqrt(a)*sqrt(c) + c*x**2)/(8*a**(3/4)*c**(3/4)) - sqrt(2)*d*(3*sqrt(a)*e**2
- sqrt(c)*d**2)*log(sqrt(2)*a**(1/4)*c**(3/4)*x + sqrt(a)*sqrt(c) + c*x**2)/(8*a
**(3/4)*c**(3/4)) - sqrt(2)*d*(3*sqrt(a)*e**2 + sqrt(c)*d**2)*atan(1 - sqrt(2)*c
**(1/4)*x/a**(1/4))/(4*a**(3/4)*c**(3/4)) + sqrt(2)*d*(3*sqrt(a)*e**2 + sqrt(c)*
d**2)*atan(1 + sqrt(2)*c**(1/4)*x/a**(1/4))/(4*a**(3/4)*c**(3/4))

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Mathematica [A]  time = 0.541466, size = 322, normalized size = 1.01 \[ \frac{-\sqrt{2} \sqrt [4]{c} \left (\sqrt [4]{a} \sqrt{c} d^3-3 a^{3/4} d e^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )+\sqrt{2} \sqrt [4]{c} \left (\sqrt [4]{a} \sqrt{c} d^3-3 a^{3/4} d e^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )-2 \sqrt [4]{a} \sqrt [4]{c} d \left (6 \sqrt [4]{a} \sqrt [4]{c} d e+3 \sqrt{2} \sqrt{a} e^2+\sqrt{2} \sqrt{c} d^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )+2 \sqrt [4]{a} \sqrt [4]{c} d \left (-6 \sqrt [4]{a} \sqrt [4]{c} d e+3 \sqrt{2} \sqrt{a} e^2+\sqrt{2} \sqrt{c} d^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )+2 a e^3 \log \left (a+c x^4\right )}{8 a c} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.011, size = 314, normalized size = 1. \[{\frac{{d}^{3}\sqrt{2}}{8\,a}\sqrt [4]{{\frac{a}{c}}}\ln \left ({1 \left ({x}^{2}+\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) \left ({x}^{2}-\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) ^{-1}} \right ) }+{\frac{{d}^{3}\sqrt{2}}{4\,a}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({\sqrt{2}x{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ) }+{\frac{{d}^{3}\sqrt{2}}{4\,a}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({\sqrt{2}x{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ) }+{\frac{3\,{d}^{2}e}{2}\arctan \left ({x}^{2}\sqrt{{\frac{c}{a}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{3\,{e}^{2}d\sqrt{2}}{8\,c}\ln \left ({1 \left ({x}^{2}-\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) \left ({x}^{2}+\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+{\frac{3\,{e}^{2}d\sqrt{2}}{4\,c}\arctan \left ({\sqrt{2}x{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+{\frac{3\,{e}^{2}d\sqrt{2}}{4\,c}\arctan \left ({\sqrt{2}x{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+{\frac{{e}^{3}\ln \left ( c{x}^{4}+a \right ) }{4\,c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^3/(c*x^4+a),x)

[Out]

1/8*d^3*(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)))+1/4*d^3*(1/c*a)^(1/4)/a*2^(1/2)*arct
an(2^(1/2)/(1/c*a)^(1/4)*x+1)+1/4*d^3*(1/c*a)^(1/4)/a*2^(1/2)*arctan(2^(1/2)/(1/
c*a)^(1/4)*x-1)+3/2*d^2*e/(a*c)^(1/2)*arctan(x^2*(c/a)^(1/2))+3/8*e^2*d/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)))+3/4*e^2*d/c/(1/c*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/
c*a)^(1/4)*x+1)+3/4*e^2*d/c/(1/c*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/c*a)^(1/4)*x
-1)+1/4*e^3*ln(c*x^4+a)/c

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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((e*x + d)^3/(c*x^4 + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: NotImplementedError

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Sympy [A]  time = 13.5301, size = 384, normalized size = 1.2 \[ \operatorname{RootSum}{\left (256 t^{4} a^{3} c^{4} - 256 t^{3} a^{3} c^{3} e^{3} + t^{2} \left (96 a^{3} c^{2} e^{6} + 480 a^{2} c^{3} d^{4} e^{2}\right ) + t \left (- 16 a^{3} c e^{9} + 192 a^{2} c^{2} d^{4} e^{5} - 48 a c^{3} d^{8} e\right ) + a^{3} e^{12} + 3 a^{2} c d^{4} e^{8} + 3 a c^{2} d^{8} e^{4} + c^{3} d^{12}, \left ( t \mapsto t \log{\left (x + \frac{1728 t^{3} a^{4} c^{3} e^{6} + 960 t^{3} a^{3} c^{4} d^{4} e^{2} - 1296 t^{2} a^{4} c^{2} e^{9} - 2016 t^{2} a^{3} c^{3} d^{4} e^{5} + 48 t^{2} a^{2} c^{4} d^{8} e + 324 t a^{4} c e^{12} + 4716 t a^{3} c^{2} d^{4} e^{8} + 1452 t a^{2} c^{3} d^{8} e^{4} + 4 t a c^{4} d^{12} - 27 a^{4} e^{15} + 1119 a^{3} c d^{4} e^{11} - 609 a^{2} c^{2} d^{8} e^{7} - 91 a c^{3} d^{12} e^{3}}{729 a^{3} c d^{3} e^{12} - 1053 a^{2} c^{2} d^{7} e^{8} - 117 a c^{3} d^{11} e^{4} + c^{4} d^{15}} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

RootSum(256*_t**4*a**3*c**4 - 256*_t**3*a**3*c**3*e**3 + _t**2*(96*a**3*c**2*e**
6 + 480*a**2*c**3*d**4*e**2) + _t*(-16*a**3*c*e**9 + 192*a**2*c**2*d**4*e**5 - 4
8*a*c**3*d**8*e) + a**3*e**12 + 3*a**2*c*d**4*e**8 + 3*a*c**2*d**8*e**4 + c**3*d
**12, Lambda(_t, _t*log(x + (1728*_t**3*a**4*c**3*e**6 + 960*_t**3*a**3*c**4*d**
4*e**2 - 1296*_t**2*a**4*c**2*e**9 - 2016*_t**2*a**3*c**3*d**4*e**5 + 48*_t**2*a
**2*c**4*d**8*e + 324*_t*a**4*c*e**12 + 4716*_t*a**3*c**2*d**4*e**8 + 1452*_t*a*
*2*c**3*d**8*e**4 + 4*_t*a*c**4*d**12 - 27*a**4*e**15 + 1119*a**3*c*d**4*e**11 -
609*a**2*c**2*d**8*e**7 - 91*a*c**3*d**12*e**3)/(729*a**3*c*d**3*e**12 - 1053*a
**2*c**2*d**7*e**8 - 117*a*c**3*d**11*e**4 + c**4*d**15))))

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GIAC/XCAS [A]  time = 0.273609, size = 420, normalized size = 1.31 \[ \frac{e^{3}{\rm ln}\left ({\left | c x^{4} + a \right |}\right )}{4 \, c} + \frac{\sqrt{2}{\left (3 \, \sqrt{2} \sqrt{a c} c^{2} d^{2} e + \left (a c^{3}\right )^{\frac{1}{4}} c^{2} d^{3} + 3 \, \left (a c^{3}\right )^{\frac{3}{4}} d e^{2}\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{4 \, a c^{3}} + \frac{\sqrt{2}{\left (3 \, \sqrt{2} \sqrt{a c} c^{2} d^{2} e + \left (a c^{3}\right )^{\frac{1}{4}} c^{2} d^{3} + 3 \, \left (a c^{3}\right )^{\frac{3}{4}} d e^{2}\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{4 \, a c^{3}} + \frac{\sqrt{2}{\left (\left (a c^{3}\right )^{\frac{1}{4}} c^{2} d^{3} - 3 \, \left (a c^{3}\right )^{\frac{3}{4}} d e^{2}\right )}{\rm ln}\left (x^{2} + \sqrt{2} x \left (\frac{a}{c}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{c}}\right )}{8 \, a c^{3}} - \frac{\sqrt{2}{\left (\left (a c^{3}\right )^{\frac{1}{4}} c^{2} d^{3} - 3 \, \left (a c^{3}\right )^{\frac{3}{4}} d e^{2}\right )}{\rm ln}\left (x^{2} - \sqrt{2} x \left (\frac{a}{c}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{c}}\right )}{8 \, a c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*e^3*ln(abs(c*x^4 + a))/c + 1/4*sqrt(2)*(3*sqrt(2)*sqrt(a*c)*c^2*d^2*e + (a*c
^3)^(1/4)*c^2*d^3 + 3*(a*c^3)^(3/4)*d*e^2)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*(a/
c)^(1/4))/(a/c)^(1/4))/(a*c^3) + 1/4*sqrt(2)*(3*sqrt(2)*sqrt(a*c)*c^2*d^2*e + (a
*c^3)^(1/4)*c^2*d^3 + 3*(a*c^3)^(3/4)*d*e^2)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(
a/c)^(1/4))/(a/c)^(1/4))/(a*c^3) + 1/8*sqrt(2)*((a*c^3)^(1/4)*c^2*d^3 - 3*(a*c^3
)^(3/4)*d*e^2)*ln(x^2 + sqrt(2)*x*(a/c)^(1/4) + sqrt(a/c))/(a*c^3) - 1/8*sqrt(2)
*((a*c^3)^(1/4)*c^2*d^3 - 3*(a*c^3)^(3/4)*d*e^2)*ln(x^2 - sqrt(2)*x*(a/c)^(1/4)
+ sqrt(a/c))/(a*c^3)