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

Optimal. Leaf size=349 \[ -\frac{3 d \left (\sqrt{c} d^2-\sqrt{a} e^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{16 \sqrt{2} a^{7/4} c^{3/4}}+\frac{3 d \left (\sqrt{c} d^2-\sqrt{a} e^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{16 \sqrt{2} a^{7/4} c^{3/4}}-\frac{3 d \left (\sqrt{a} e^2+\sqrt{c} d^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{7/4} c^{3/4}}+\frac{3 d \left (\sqrt{a} e^2+\sqrt{c} d^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{8 \sqrt{2} a^{7/4} c^{3/4}}+\frac{3 d^2 e \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{4 a^{3/2} \sqrt{c}}-\frac{a e^3-c x \left (d^3+3 d^2 e x+3 d e^2 x^2\right )}{4 a c \left (a+c x^4\right )} \]

[Out]

-(a*e^3 - c*x*(d^3 + 3*d^2*e*x + 3*d*e^2*x^2))/(4*a*c*(a + c*x^4)) + (3*d^2*e*Ar
cTan[(Sqrt[c]*x^2)/Sqrt[a]])/(4*a^(3/2)*Sqrt[c]) - (3*d*(Sqrt[c]*d^2 + Sqrt[a]*e
^2)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(8*Sqrt[2]*a^(7/4)*c^(3/4)) + (3*d*
(Sqrt[c]*d^2 + Sqrt[a]*e^2)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(8*Sqrt[2]*
a^(7/4)*c^(3/4)) - (3*d*(Sqrt[c]*d^2 - Sqrt[a]*e^2)*Log[Sqrt[a] - Sqrt[2]*a^(1/4
)*c^(1/4)*x + Sqrt[c]*x^2])/(16*Sqrt[2]*a^(7/4)*c^(3/4)) + (3*d*(Sqrt[c]*d^2 - S
qrt[a]*e^2)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(16*Sqrt[2]*
a^(7/4)*c^(3/4))

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Rubi [A]  time = 0.712593, antiderivative size = 349, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 12, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.706 \[ -\frac{3 d \left (\sqrt{c} d^2-\sqrt{a} e^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{16 \sqrt{2} a^{7/4} c^{3/4}}+\frac{3 d \left (\sqrt{c} d^2-\sqrt{a} e^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{16 \sqrt{2} a^{7/4} c^{3/4}}-\frac{3 d \left (\sqrt{a} e^2+\sqrt{c} d^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{7/4} c^{3/4}}+\frac{3 d \left (\sqrt{a} e^2+\sqrt{c} d^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{8 \sqrt{2} a^{7/4} c^{3/4}}+\frac{3 d^2 e \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{4 a^{3/2} \sqrt{c}}-\frac{a e^3-c x \left (d^3+3 d^2 e x+3 d e^2 x^2\right )}{4 a c \left (a+c x^4\right )} \]

Antiderivative was successfully verified.

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

[Out]

-(a*e^3 - c*x*(d^3 + 3*d^2*e*x + 3*d*e^2*x^2))/(4*a*c*(a + c*x^4)) + (3*d^2*e*Ar
cTan[(Sqrt[c]*x^2)/Sqrt[a]])/(4*a^(3/2)*Sqrt[c]) - (3*d*(Sqrt[c]*d^2 + Sqrt[a]*e
^2)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(8*Sqrt[2]*a^(7/4)*c^(3/4)) + (3*d*
(Sqrt[c]*d^2 + Sqrt[a]*e^2)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(8*Sqrt[2]*
a^(7/4)*c^(3/4)) - (3*d*(Sqrt[c]*d^2 - Sqrt[a]*e^2)*Log[Sqrt[a] - Sqrt[2]*a^(1/4
)*c^(1/4)*x + Sqrt[c]*x^2])/(16*Sqrt[2]*a^(7/4)*c^(3/4)) + (3*d*(Sqrt[c]*d^2 - S
qrt[a]*e^2)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(16*Sqrt[2]*
a^(7/4)*c^(3/4))

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Rubi in Sympy [A]  time = 145.379, size = 333, normalized size = 0.95 \[ - \frac{a e^{3} - c x \left (d^{3} + 3 d^{2} e x + 3 d e^{2} x^{2}\right )}{4 a c \left (a + c x^{4}\right )} + \frac{3 d^{2} e \operatorname{atan}{\left (\frac{\sqrt{c} x^{2}}{\sqrt{a}} \right )}}{4 a^{\frac{3}{2}} \sqrt{c}} + \frac{3 \sqrt{2} d \left (\sqrt{a} e^{2} - \sqrt{c} d^{2}\right ) \log{\left (- \sqrt{2} \sqrt [4]{a} c^{\frac{3}{4}} x + \sqrt{a} \sqrt{c} + c x^{2} \right )}}{32 a^{\frac{7}{4}} c^{\frac{3}{4}}} - \frac{3 \sqrt{2} d \left (\sqrt{a} e^{2} - \sqrt{c} d^{2}\right ) \log{\left (\sqrt{2} \sqrt [4]{a} c^{\frac{3}{4}} x + \sqrt{a} \sqrt{c} + c x^{2} \right )}}{32 a^{\frac{7}{4}} c^{\frac{3}{4}}} - \frac{3 \sqrt{2} d \left (\sqrt{a} e^{2} + \sqrt{c} d^{2}\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}} \right )}}{16 a^{\frac{7}{4}} c^{\frac{3}{4}}} + \frac{3 \sqrt{2} d \left (\sqrt{a} e^{2} + \sqrt{c} d^{2}\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}} \right )}}{16 a^{\frac{7}{4}} c^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(a*e**3 - c*x*(d**3 + 3*d**2*e*x + 3*d*e**2*x**2))/(4*a*c*(a + c*x**4)) + 3*d**
2*e*atan(sqrt(c)*x**2/sqrt(a))/(4*a**(3/2)*sqrt(c)) + 3*sqrt(2)*d*(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)/(32
*a**(7/4)*c**(3/4)) - 3*sqrt(2)*d*(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)/(32*a**(7/4)*c**(3/4)) - 3*sqrt(2)*d
*(sqrt(a)*e**2 + sqrt(c)*d**2)*atan(1 - sqrt(2)*c**(1/4)*x/a**(1/4))/(16*a**(7/4
)*c**(3/4)) + 3*sqrt(2)*d*(sqrt(a)*e**2 + sqrt(c)*d**2)*atan(1 + sqrt(2)*c**(1/4
)*x/a**(1/4))/(16*a**(7/4)*c**(3/4))

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Mathematica [A]  time = 0.726295, size = 347, normalized size = 0.99 \[ \frac{3 \sqrt{2} \sqrt [4]{c} \left (a^{3/4} d e^2-\sqrt [4]{a} \sqrt{c} d^3\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )+3 \sqrt{2} \sqrt [4]{c} \left (\sqrt [4]{a} \sqrt{c} d^3-a^{3/4} d e^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )-6 \sqrt [4]{a} \sqrt [4]{c} d \left (4 \sqrt [4]{a} \sqrt [4]{c} d e+\sqrt{2} \sqrt{a} e^2+\sqrt{2} \sqrt{c} d^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )+6 \sqrt [4]{a} \sqrt [4]{c} d \left (-4 \sqrt [4]{a} \sqrt [4]{c} d e+\sqrt{2} \sqrt{a} e^2+\sqrt{2} \sqrt{c} d^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )-\frac{8 a \left (a e^3-c d x \left (d^2+3 d e x+3 e^2 x^2\right )\right )}{a+c x^4}}{32 a^2 c} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.008, size = 390, normalized size = 1.1 \[{\frac{{d}^{3}x}{4\,a \left ( c{x}^{4}+a \right ) }}+{\frac{3\,{d}^{3}\sqrt{2}}{32\,{a}^{2}}\sqrt [4]{{\frac{a}{c}}}\ln \left ({1 \left ({x}^{2}+\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) \left ({x}^{2}-\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) ^{-1}} \right ) }+{\frac{3\,{d}^{3}\sqrt{2}}{16\,{a}^{2}}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({\sqrt{2}x{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ) }+{\frac{3\,{d}^{3}\sqrt{2}}{16\,{a}^{2}}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({\sqrt{2}x{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ) }+{\frac{3\,{d}^{2}e{x}^{2}}{4\,a \left ( c{x}^{4}+a \right ) }}+{\frac{3\,{d}^{2}e}{4\,a}\arctan \left ({x}^{2}\sqrt{{\frac{c}{a}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{3\,d{e}^{2}{x}^{3}}{4\,a \left ( c{x}^{4}+a \right ) }}+{\frac{3\,{e}^{2}d\sqrt{2}}{32\,ac}\ln \left ({1 \left ({x}^{2}-\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) \left ({x}^{2}+\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+{\frac{3\,{e}^{2}d\sqrt{2}}{16\,ac}\arctan \left ({\sqrt{2}x{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+{\frac{3\,{e}^{2}d\sqrt{2}}{16\,ac}\arctan \left ({\sqrt{2}x{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+{\frac{{e}^{3}{x}^{4}}{4\,a \left ( c{x}^{4}+a \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^3/(c*x^4+a)^2,x)

[Out]

1/4*d^3*x/a/(c*x^4+a)+3/32*d^3/a^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)))+3/16*d^3/a^
2*(1/c*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/c*a)^(1/4)*x+1)+3/16*d^3/a^2*(1/c*a)^(
1/4)*2^(1/2)*arctan(2^(1/2)/(1/c*a)^(1/4)*x-1)+3/4*d^2*e*x^2/a/(c*x^4+a)+3/4*d^2
*e/a/(a*c)^(1/2)*arctan(x^2*(c/a)^(1/2))+3/4*e^2*d*x^3/a/(c*x^4+a)+3/32*e^2*d/a/
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)))+3/16*e^2*d/a/c/(1/c*a)^(1/4)*2^(1/2)*arctan(
2^(1/2)/(1/c*a)^(1/4)*x+1)+3/16*e^2*d/a/c/(1/c*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(
1/c*a)^(1/4)*x-1)+1/4*e^3*x^4/a/(c*x^4+a)

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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)^2,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)^2,x, algorithm="fricas")

[Out]

Exception raised: NotImplementedError

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Sympy [A]  time = 22.8319, size = 350, normalized size = 1. \[ \operatorname{RootSum}{\left (65536 t^{4} a^{7} c^{3} + 27648 t^{2} a^{4} c^{2} d^{4} e^{2} + t \left (3456 a^{3} c d^{4} e^{5} - 3456 a^{2} c^{2} d^{8} e\right ) + 81 a^{2} d^{4} e^{8} + 162 a c d^{8} e^{4} + 81 c^{2} d^{12}, \left ( t \mapsto t \log{\left (x + \frac{4096 t^{3} a^{7} c^{2} e^{6} + 28672 t^{3} a^{6} c^{3} d^{4} e^{2} - 7680 t^{2} a^{5} c^{2} d^{4} e^{5} + 1536 t^{2} a^{4} c^{3} d^{8} e + 2160 t a^{4} c d^{4} e^{8} + 9216 t a^{3} c^{2} d^{8} e^{4} + 144 t a^{2} c^{3} d^{12} + 162 a^{3} d^{4} e^{11} - 648 a^{2} c d^{8} e^{7} - 810 a c^{2} d^{12} e^{3}}{27 a^{3} d^{3} e^{12} - 891 a^{2} c d^{7} e^{8} - 891 a c^{2} d^{11} e^{4} + 27 c^{3} d^{15}} \right )} \right )\right )} + \frac{- a e^{3} + c d^{3} x + 3 c d^{2} e x^{2} + 3 c d e^{2} x^{3}}{4 a^{2} c + 4 a c^{2} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

RootSum(65536*_t**4*a**7*c**3 + 27648*_t**2*a**4*c**2*d**4*e**2 + _t*(3456*a**3*
c*d**4*e**5 - 3456*a**2*c**2*d**8*e) + 81*a**2*d**4*e**8 + 162*a*c*d**8*e**4 + 8
1*c**2*d**12, Lambda(_t, _t*log(x + (4096*_t**3*a**7*c**2*e**6 + 28672*_t**3*a**
6*c**3*d**4*e**2 - 7680*_t**2*a**5*c**2*d**4*e**5 + 1536*_t**2*a**4*c**3*d**8*e
+ 2160*_t*a**4*c*d**4*e**8 + 9216*_t*a**3*c**2*d**8*e**4 + 144*_t*a**2*c**3*d**1
2 + 162*a**3*d**4*e**11 - 648*a**2*c*d**8*e**7 - 810*a*c**2*d**12*e**3)/(27*a**3
*d**3*e**12 - 891*a**2*c*d**7*e**8 - 891*a*c**2*d**11*e**4 + 27*c**3*d**15)))) +
(-a*e**3 + c*d**3*x + 3*c*d**2*e*x**2 + 3*c*d*e**2*x**3)/(4*a**2*c + 4*a*c**2*x
**4)

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GIAC/XCAS [A]  time = 0.270418, size = 462, normalized size = 1.32 \[ \frac{3 \, c d x^{3} e^{2} + 3 \, c d^{2} x^{2} e + c d^{3} x - a e^{3}}{4 \,{\left (c x^{4} + a\right )} a c} + \frac{3 \, \sqrt{2}{\left (2 \, \sqrt{2} \sqrt{a c} c^{2} d^{2} e + \left (a c^{3}\right )^{\frac{1}{4}} c^{2} d^{3} + \left (a c^{3}\right )^{\frac{3}{4}} d e^{2}\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{16 \, a^{2} c^{3}} + \frac{3 \, \sqrt{2}{\left (2 \, \sqrt{2} \sqrt{a c} c^{2} d^{2} e + \left (a c^{3}\right )^{\frac{1}{4}} c^{2} d^{3} + \left (a c^{3}\right )^{\frac{3}{4}} d e^{2}\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{16 \, a^{2} c^{3}} + \frac{3 \, \sqrt{2}{\left (\left (a c^{3}\right )^{\frac{1}{4}} c^{2} d^{3} - \left (a c^{3}\right )^{\frac{3}{4}} d e^{2}\right )}{\rm ln}\left (x^{2} + \sqrt{2} x \left (\frac{a}{c}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{c}}\right )}{32 \, a^{2} c^{3}} - \frac{3 \, \sqrt{2}{\left (\left (a c^{3}\right )^{\frac{1}{4}} c^{2} d^{3} - \left (a c^{3}\right )^{\frac{3}{4}} d e^{2}\right )}{\rm ln}\left (x^{2} - \sqrt{2} x \left (\frac{a}{c}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{c}}\right )}{32 \, a^{2} c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*(3*c*d*x^3*e^2 + 3*c*d^2*x^2*e + c*d^3*x - a*e^3)/((c*x^4 + a)*a*c) + 3/16*s
qrt(2)*(2*sqrt(2)*sqrt(a*c)*c^2*d^2*e + (a*c^3)^(1/4)*c^2*d^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^2*c^3) + 3/1
6*sqrt(2)*(2*sqrt(2)*sqrt(a*c)*c^2*d^2*e + (a*c^3)^(1/4)*c^2*d^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^2*c^3) +
3/32*sqrt(2)*((a*c^3)^(1/4)*c^2*d^3 - (a*c^3)^(3/4)*d*e^2)*ln(x^2 + sqrt(2)*x*(a
/c)^(1/4) + sqrt(a/c))/(a^2*c^3) - 3/32*sqrt(2)*((a*c^3)^(1/4)*c^2*d^3 - (a*c^3)
^(3/4)*d*e^2)*ln(x^2 - sqrt(2)*x*(a/c)^(1/4) + sqrt(a/c))/(a^2*c^3)