3.468 \(\int \frac{\left (a+c x^2\right )^2}{(d+e x)^5} \, dx\)

Optimal. Leaf size=109 \[ -\frac{c \left (a e^2+3 c d^2\right )}{e^5 (d+e x)^2}+\frac{4 c d \left (a e^2+c d^2\right )}{3 e^5 (d+e x)^3}-\frac{\left (a e^2+c d^2\right )^2}{4 e^5 (d+e x)^4}+\frac{4 c^2 d}{e^5 (d+e x)}+\frac{c^2 \log (d+e x)}{e^5} \]

[Out]

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

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Rubi [A]  time = 0.190368, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059 \[ -\frac{c \left (a e^2+3 c d^2\right )}{e^5 (d+e x)^2}+\frac{4 c d \left (a e^2+c d^2\right )}{3 e^5 (d+e x)^3}-\frac{\left (a e^2+c d^2\right )^2}{4 e^5 (d+e x)^4}+\frac{4 c^2 d}{e^5 (d+e x)}+\frac{c^2 \log (d+e x)}{e^5} \]

Antiderivative was successfully verified.

[In]  Int[(a + c*x^2)^2/(d + e*x)^5,x]

[Out]

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

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Rubi in Sympy [A]  time = 30.7833, size = 102, normalized size = 0.94 \[ \frac{4 c^{2} d}{e^{5} \left (d + e x\right )} + \frac{c^{2} \log{\left (d + e x \right )}}{e^{5}} + \frac{4 c d \left (a e^{2} + c d^{2}\right )}{3 e^{5} \left (d + e x\right )^{3}} - \frac{c \left (a e^{2} + 3 c d^{2}\right )}{e^{5} \left (d + e x\right )^{2}} - \frac{\left (a e^{2} + c d^{2}\right )^{2}}{4 e^{5} \left (d + e x\right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+a)**2/(e*x+d)**5,x)

[Out]

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

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Mathematica [A]  time = 0.0677074, size = 100, normalized size = 0.92 \[ \frac{-3 a^2 e^4-2 a c e^2 \left (d^2+4 d e x+6 e^2 x^2\right )+c^2 d \left (25 d^3+88 d^2 e x+108 d e^2 x^2+48 e^3 x^3\right )+12 c^2 (d+e x)^4 \log (d+e x)}{12 e^5 (d+e x)^4} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + c*x^2)^2/(d + e*x)^5,x]

[Out]

(-3*a^2*e^4 - 2*a*c*e^2*(d^2 + 4*d*e*x + 6*e^2*x^2) + c^2*d*(25*d^3 + 88*d^2*e*x
+ 108*d*e^2*x^2 + 48*e^3*x^3) + 12*c^2*(d + e*x)^4*Log[d + e*x])/(12*e^5*(d + e
*x)^4)

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Maple [A]  time = 0.011, size = 146, normalized size = 1.3 \[ -{\frac{ac}{{e}^{3} \left ( ex+d \right ) ^{2}}}-3\,{\frac{{c}^{2}{d}^{2}}{{e}^{5} \left ( ex+d \right ) ^{2}}}+{\frac{{c}^{2}\ln \left ( ex+d \right ) }{{e}^{5}}}+{\frac{4\,acd}{3\,{e}^{3} \left ( ex+d \right ) ^{3}}}+{\frac{4\,{c}^{2}{d}^{3}}{3\,{e}^{5} \left ( ex+d \right ) ^{3}}}+4\,{\frac{{c}^{2}d}{{e}^{5} \left ( ex+d \right ) }}-{\frac{{a}^{2}}{4\,e \left ( ex+d \right ) ^{4}}}-{\frac{{d}^{2}ac}{2\,{e}^{3} \left ( ex+d \right ) ^{4}}}-{\frac{{c}^{2}{d}^{4}}{4\,{e}^{5} \left ( ex+d \right ) ^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+a)^2/(e*x+d)^5,x)

[Out]

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

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Maxima [A]  time = 0.710795, size = 197, normalized size = 1.81 \[ \frac{48 \, c^{2} d e^{3} x^{3} + 25 \, c^{2} d^{4} - 2 \, a c d^{2} e^{2} - 3 \, a^{2} e^{4} + 12 \,{\left (9 \, c^{2} d^{2} e^{2} - a c e^{4}\right )} x^{2} + 8 \,{\left (11 \, c^{2} d^{3} e - a c d e^{3}\right )} x}{12 \,{\left (e^{9} x^{4} + 4 \, d e^{8} x^{3} + 6 \, d^{2} e^{7} x^{2} + 4 \, d^{3} e^{6} x + d^{4} e^{5}\right )}} + \frac{c^{2} \log \left (e x + d\right )}{e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^2/(e*x + d)^5,x, algorithm="maxima")

[Out]

1/12*(48*c^2*d*e^3*x^3 + 25*c^2*d^4 - 2*a*c*d^2*e^2 - 3*a^2*e^4 + 12*(9*c^2*d^2*
e^2 - a*c*e^4)*x^2 + 8*(11*c^2*d^3*e - a*c*d*e^3)*x)/(e^9*x^4 + 4*d*e^8*x^3 + 6*
d^2*e^7*x^2 + 4*d^3*e^6*x + d^4*e^5) + c^2*log(e*x + d)/e^5

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Fricas [A]  time = 0.20927, size = 262, normalized size = 2.4 \[ \frac{48 \, c^{2} d e^{3} x^{3} + 25 \, c^{2} d^{4} - 2 \, a c d^{2} e^{2} - 3 \, a^{2} e^{4} + 12 \,{\left (9 \, c^{2} d^{2} e^{2} - a c e^{4}\right )} x^{2} + 8 \,{\left (11 \, c^{2} d^{3} e - a c d e^{3}\right )} x + 12 \,{\left (c^{2} e^{4} x^{4} + 4 \, c^{2} d e^{3} x^{3} + 6 \, c^{2} d^{2} e^{2} x^{2} + 4 \, c^{2} d^{3} e x + c^{2} d^{4}\right )} \log \left (e x + d\right )}{12 \,{\left (e^{9} x^{4} + 4 \, d e^{8} x^{3} + 6 \, d^{2} e^{7} x^{2} + 4 \, d^{3} e^{6} x + d^{4} e^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^2/(e*x + d)^5,x, algorithm="fricas")

[Out]

1/12*(48*c^2*d*e^3*x^3 + 25*c^2*d^4 - 2*a*c*d^2*e^2 - 3*a^2*e^4 + 12*(9*c^2*d^2*
e^2 - a*c*e^4)*x^2 + 8*(11*c^2*d^3*e - a*c*d*e^3)*x + 12*(c^2*e^4*x^4 + 4*c^2*d*
e^3*x^3 + 6*c^2*d^2*e^2*x^2 + 4*c^2*d^3*e*x + c^2*d^4)*log(e*x + d))/(e^9*x^4 +
4*d*e^8*x^3 + 6*d^2*e^7*x^2 + 4*d^3*e^6*x + d^4*e^5)

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Sympy [A]  time = 6.27568, size = 150, normalized size = 1.38 \[ \frac{c^{2} \log{\left (d + e x \right )}}{e^{5}} + \frac{- 3 a^{2} e^{4} - 2 a c d^{2} e^{2} + 25 c^{2} d^{4} + 48 c^{2} d e^{3} x^{3} + x^{2} \left (- 12 a c e^{4} + 108 c^{2} d^{2} e^{2}\right ) + x \left (- 8 a c d e^{3} + 88 c^{2} d^{3} e\right )}{12 d^{4} e^{5} + 48 d^{3} e^{6} x + 72 d^{2} e^{7} x^{2} + 48 d e^{8} x^{3} + 12 e^{9} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+a)**2/(e*x+d)**5,x)

[Out]

c**2*log(d + e*x)/e**5 + (-3*a**2*e**4 - 2*a*c*d**2*e**2 + 25*c**2*d**4 + 48*c**
2*d*e**3*x**3 + x**2*(-12*a*c*e**4 + 108*c**2*d**2*e**2) + x*(-8*a*c*d*e**3 + 88
*c**2*d**3*e))/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x*
*3 + 12*e**9*x**4)

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GIAC/XCAS [A]  time = 0.21474, size = 220, normalized size = 2.02 \[ -c^{2} e^{\left (-5\right )}{\rm ln}\left (\frac{{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) + \frac{1}{12} \,{\left (\frac{48 \, c^{2} d e^{15}}{x e + d} - \frac{36 \, c^{2} d^{2} e^{15}}{{\left (x e + d\right )}^{2}} + \frac{16 \, c^{2} d^{3} e^{15}}{{\left (x e + d\right )}^{3}} - \frac{3 \, c^{2} d^{4} e^{15}}{{\left (x e + d\right )}^{4}} - \frac{12 \, a c e^{17}}{{\left (x e + d\right )}^{2}} + \frac{16 \, a c d e^{17}}{{\left (x e + d\right )}^{3}} - \frac{6 \, a c d^{2} e^{17}}{{\left (x e + d\right )}^{4}} - \frac{3 \, a^{2} e^{19}}{{\left (x e + d\right )}^{4}}\right )} e^{\left (-20\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^2/(e*x + d)^5,x, algorithm="giac")

[Out]

-c^2*e^(-5)*ln(abs(x*e + d)*e^(-1)/(x*e + d)^2) + 1/12*(48*c^2*d*e^15/(x*e + d)
- 36*c^2*d^2*e^15/(x*e + d)^2 + 16*c^2*d^3*e^15/(x*e + d)^3 - 3*c^2*d^4*e^15/(x*
e + d)^4 - 12*a*c*e^17/(x*e + d)^2 + 16*a*c*d*e^17/(x*e + d)^3 - 6*a*c*d^2*e^17/
(x*e + d)^4 - 3*a^2*e^19/(x*e + d)^4)*e^(-20)