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

Optimal. Leaf size=173 \[ -\frac{c d x \left (3 a^2 e^4+3 a c d^2 e^2+c^2 d^4\right )}{e^6}+\frac{c x^2 \left (3 a^2 e^4+3 a c d^2 e^2+c^2 d^4\right )}{2 e^5}-\frac{c^2 d x^3 \left (3 a e^2+c d^2\right )}{3 e^4}+\frac{c^2 x^4 \left (3 a e^2+c d^2\right )}{4 e^3}+\frac{\left (a e^2+c d^2\right )^3 \log (d+e x)}{e^7}-\frac{c^3 d x^5}{5 e^2}+\frac{c^3 x^6}{6 e} \]

[Out]

-((c*d*(c^2*d^4 + 3*a*c*d^2*e^2 + 3*a^2*e^4)*x)/e^6) + (c*(c^2*d^4 + 3*a*c*d^2*e
^2 + 3*a^2*e^4)*x^2)/(2*e^5) - (c^2*d*(c*d^2 + 3*a*e^2)*x^3)/(3*e^4) + (c^2*(c*d
^2 + 3*a*e^2)*x^4)/(4*e^3) - (c^3*d*x^5)/(5*e^2) + (c^3*x^6)/(6*e) + ((c*d^2 + a
*e^2)^3*Log[d + e*x])/e^7

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Rubi [A]  time = 0.314589, antiderivative size = 173, 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 \[ -\frac{c d x \left (3 a^2 e^4+3 a c d^2 e^2+c^2 d^4\right )}{e^6}+\frac{c x^2 \left (3 a^2 e^4+3 a c d^2 e^2+c^2 d^4\right )}{2 e^5}-\frac{c^2 d x^3 \left (3 a e^2+c d^2\right )}{3 e^4}+\frac{c^2 x^4 \left (3 a e^2+c d^2\right )}{4 e^3}+\frac{\left (a e^2+c d^2\right )^3 \log (d+e x)}{e^7}-\frac{c^3 d x^5}{5 e^2}+\frac{c^3 x^6}{6 e} \]

Antiderivative was successfully verified.

[In]  Int[(a + c*x^2)^3/(d + e*x),x]

[Out]

-((c*d*(c^2*d^4 + 3*a*c*d^2*e^2 + 3*a^2*e^4)*x)/e^6) + (c*(c^2*d^4 + 3*a*c*d^2*e
^2 + 3*a^2*e^4)*x^2)/(2*e^5) - (c^2*d*(c*d^2 + 3*a*e^2)*x^3)/(3*e^4) + (c^2*(c*d
^2 + 3*a*e^2)*x^4)/(4*e^3) - (c^3*d*x^5)/(5*e^2) + (c^3*x^6)/(6*e) + ((c*d^2 + a
*e^2)^3*Log[d + e*x])/e^7

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{c^{3} d x^{5}}{5 e^{2}} + \frac{c^{3} x^{6}}{6 e} - \frac{c^{2} d x^{3} \left (3 a e^{2} + c d^{2}\right )}{3 e^{4}} + \frac{c^{2} x^{4} \left (3 a e^{2} + c d^{2}\right )}{4 e^{3}} + \frac{c \left (3 a^{2} e^{4} + 3 a c d^{2} e^{2} + c^{2} d^{4}\right ) \int x\, dx}{e^{5}} - \frac{d \left (3 a^{2} e^{4} + 3 a c d^{2} e^{2} + c^{2} d^{4}\right ) \int c\, dx}{e^{6}} + \frac{\left (a e^{2} + c d^{2}\right )^{3} \log{\left (d + e x \right )}}{e^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-c**3*d*x**5/(5*e**2) + c**3*x**6/(6*e) - c**2*d*x**3*(3*a*e**2 + c*d**2)/(3*e**
4) + c**2*x**4*(3*a*e**2 + c*d**2)/(4*e**3) + c*(3*a**2*e**4 + 3*a*c*d**2*e**2 +
c**2*d**4)*Integral(x, x)/e**5 - d*(3*a**2*e**4 + 3*a*c*d**2*e**2 + c**2*d**4)*
Integral(c, x)/e**6 + (a*e**2 + c*d**2)**3*log(d + e*x)/e**7

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Mathematica [A]  time = 0.105839, size = 142, normalized size = 0.82 \[ \frac{c e x \left (90 a^2 e^4 (e x-2 d)+15 a c e^2 \left (-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right )+c^2 \left (-60 d^5+30 d^4 e x-20 d^3 e^2 x^2+15 d^2 e^3 x^3-12 d e^4 x^4+10 e^5 x^5\right )\right )+60 \left (a e^2+c d^2\right )^3 \log (d+e x)}{60 e^7} \]

Antiderivative was successfully verified.

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

[Out]

(c*e*x*(90*a^2*e^4*(-2*d + e*x) + 15*a*c*e^2*(-12*d^3 + 6*d^2*e*x - 4*d*e^2*x^2
+ 3*e^3*x^3) + c^2*(-60*d^5 + 30*d^4*e*x - 20*d^3*e^2*x^2 + 15*d^2*e^3*x^3 - 12*
d*e^4*x^4 + 10*e^5*x^5)) + 60*(c*d^2 + a*e^2)^3*Log[d + e*x])/(60*e^7)

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Maple [A]  time = 0.006, size = 220, normalized size = 1.3 \[{\frac{{c}^{3}{x}^{6}}{6\,e}}-{\frac{{c}^{3}d{x}^{5}}{5\,{e}^{2}}}+{\frac{3\,{c}^{2}{x}^{4}a}{4\,e}}+{\frac{{x}^{4}{c}^{3}{d}^{2}}{4\,{e}^{3}}}-{\frac{{c}^{2}{x}^{3}ad}{{e}^{2}}}-{\frac{{x}^{3}{c}^{3}{d}^{3}}{3\,{e}^{4}}}+{\frac{3\,c{x}^{2}{a}^{2}}{2\,e}}+{\frac{3\,{c}^{2}{x}^{2}a{d}^{2}}{2\,{e}^{3}}}+{\frac{{x}^{2}{c}^{3}{d}^{4}}{2\,{e}^{5}}}-3\,{\frac{{a}^{2}cdx}{{e}^{2}}}-3\,{\frac{{d}^{3}a{c}^{2}x}{{e}^{4}}}-{\frac{{c}^{3}{d}^{5}x}{{e}^{6}}}+{\frac{\ln \left ( ex+d \right ){a}^{3}}{e}}+3\,{\frac{\ln \left ( ex+d \right ){a}^{2}c{d}^{2}}{{e}^{3}}}+3\,{\frac{\ln \left ( ex+d \right ) a{c}^{2}{d}^{4}}{{e}^{5}}}+{\frac{{d}^{6}\ln \left ( ex+d \right ){c}^{3}}{{e}^{7}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+a)^3/(e*x+d),x)

[Out]

1/6*c^3*x^6/e-1/5*c^3*d*x^5/e^2+3/4*c^2/e*x^4*a+1/4/e^3*x^4*c^3*d^2-c^2/e^2*x^3*
a*d-1/3/e^4*x^3*c^3*d^3+3/2*c/e*x^2*a^2+3/2*c^2/e^3*x^2*a*d^2+1/2/e^5*x^2*c^3*d^
4-3*c/e^2*a^2*d*x-3*c^2/e^4*a*d^3*x-1/e^6*c^3*d^5*x+1/e*ln(e*x+d)*a^3+3/e^3*ln(e
*x+d)*a^2*c*d^2+3/e^5*ln(e*x+d)*a*c^2*d^4+d^6/e^7*ln(e*x+d)*c^3

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Maxima [A]  time = 0.708925, size = 267, normalized size = 1.54 \[ \frac{10 \, c^{3} e^{5} x^{6} - 12 \, c^{3} d e^{4} x^{5} + 15 \,{\left (c^{3} d^{2} e^{3} + 3 \, a c^{2} e^{5}\right )} x^{4} - 20 \,{\left (c^{3} d^{3} e^{2} + 3 \, a c^{2} d e^{4}\right )} x^{3} + 30 \,{\left (c^{3} d^{4} e + 3 \, a c^{2} d^{2} e^{3} + 3 \, a^{2} c e^{5}\right )} x^{2} - 60 \,{\left (c^{3} d^{5} + 3 \, a c^{2} d^{3} e^{2} + 3 \, a^{2} c d e^{4}\right )} x}{60 \, e^{6}} + \frac{{\left (c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )} \log \left (e x + d\right )}{e^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/60*(10*c^3*e^5*x^6 - 12*c^3*d*e^4*x^5 + 15*(c^3*d^2*e^3 + 3*a*c^2*e^5)*x^4 - 2
0*(c^3*d^3*e^2 + 3*a*c^2*d*e^4)*x^3 + 30*(c^3*d^4*e + 3*a*c^2*d^2*e^3 + 3*a^2*c*
e^5)*x^2 - 60*(c^3*d^5 + 3*a*c^2*d^3*e^2 + 3*a^2*c*d*e^4)*x)/e^6 + (c^3*d^6 + 3*
a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + a^3*e^6)*log(e*x + d)/e^7

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Fricas [A]  time = 0.205923, size = 267, normalized size = 1.54 \[ \frac{10 \, c^{3} e^{6} x^{6} - 12 \, c^{3} d e^{5} x^{5} + 15 \,{\left (c^{3} d^{2} e^{4} + 3 \, a c^{2} e^{6}\right )} x^{4} - 20 \,{\left (c^{3} d^{3} e^{3} + 3 \, a c^{2} d e^{5}\right )} x^{3} + 30 \,{\left (c^{3} d^{4} e^{2} + 3 \, a c^{2} d^{2} e^{4} + 3 \, a^{2} c e^{6}\right )} x^{2} - 60 \,{\left (c^{3} d^{5} e + 3 \, a c^{2} d^{3} e^{3} + 3 \, a^{2} c d e^{5}\right )} x + 60 \,{\left (c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )} \log \left (e x + d\right )}{60 \, e^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/60*(10*c^3*e^6*x^6 - 12*c^3*d*e^5*x^5 + 15*(c^3*d^2*e^4 + 3*a*c^2*e^6)*x^4 - 2
0*(c^3*d^3*e^3 + 3*a*c^2*d*e^5)*x^3 + 30*(c^3*d^4*e^2 + 3*a*c^2*d^2*e^4 + 3*a^2*
c*e^6)*x^2 - 60*(c^3*d^5*e + 3*a*c^2*d^3*e^3 + 3*a^2*c*d*e^5)*x + 60*(c^3*d^6 +
3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + a^3*e^6)*log(e*x + d))/e^7

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Sympy [A]  time = 2.11584, size = 173, normalized size = 1. \[ - \frac{c^{3} d x^{5}}{5 e^{2}} + \frac{c^{3} x^{6}}{6 e} + \frac{x^{4} \left (3 a c^{2} e^{2} + c^{3} d^{2}\right )}{4 e^{3}} - \frac{x^{3} \left (3 a c^{2} d e^{2} + c^{3} d^{3}\right )}{3 e^{4}} + \frac{x^{2} \left (3 a^{2} c e^{4} + 3 a c^{2} d^{2} e^{2} + c^{3} d^{4}\right )}{2 e^{5}} - \frac{x \left (3 a^{2} c d e^{4} + 3 a c^{2} d^{3} e^{2} + c^{3} d^{5}\right )}{e^{6}} + \frac{\left (a e^{2} + c d^{2}\right )^{3} \log{\left (d + e x \right )}}{e^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-c**3*d*x**5/(5*e**2) + c**3*x**6/(6*e) + x**4*(3*a*c**2*e**2 + c**3*d**2)/(4*e*
*3) - x**3*(3*a*c**2*d*e**2 + c**3*d**3)/(3*e**4) + x**2*(3*a**2*c*e**4 + 3*a*c*
*2*d**2*e**2 + c**3*d**4)/(2*e**5) - x*(3*a**2*c*d*e**4 + 3*a*c**2*d**3*e**2 + c
**3*d**5)/e**6 + (a*e**2 + c*d**2)**3*log(d + e*x)/e**7

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GIAC/XCAS [A]  time = 0.210761, size = 259, normalized size = 1.5 \[{\left (c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )} e^{\left (-7\right )}{\rm ln}\left ({\left | x e + d \right |}\right ) + \frac{1}{60} \,{\left (10 \, c^{3} x^{6} e^{5} - 12 \, c^{3} d x^{5} e^{4} + 15 \, c^{3} d^{2} x^{4} e^{3} - 20 \, c^{3} d^{3} x^{3} e^{2} + 30 \, c^{3} d^{4} x^{2} e - 60 \, c^{3} d^{5} x + 45 \, a c^{2} x^{4} e^{5} - 60 \, a c^{2} d x^{3} e^{4} + 90 \, a c^{2} d^{2} x^{2} e^{3} - 180 \, a c^{2} d^{3} x e^{2} + 90 \, a^{2} c x^{2} e^{5} - 180 \, a^{2} c d x e^{4}\right )} e^{\left (-6\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(c^3*d^6 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + a^3*e^6)*e^(-7)*ln(abs(x*e + d))
+ 1/60*(10*c^3*x^6*e^5 - 12*c^3*d*x^5*e^4 + 15*c^3*d^2*x^4*e^3 - 20*c^3*d^3*x^3*
e^2 + 30*c^3*d^4*x^2*e - 60*c^3*d^5*x + 45*a*c^2*x^4*e^5 - 60*a*c^2*d*x^3*e^4 +
90*a*c^2*d^2*x^2*e^3 - 180*a*c^2*d^3*x*e^2 + 90*a^2*c*x^2*e^5 - 180*a^2*c*d*x*e^
4)*e^(-6)