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

Optimal. Leaf size=165 \[ -\frac{4 c^2 d \left (3 a e^2+5 c d^2\right ) \log (d+e x)}{e^7}+\frac{c^2 x \left (3 a e^2+10 c d^2\right )}{e^6}-\frac{3 c \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{e^7 (d+e x)}+\frac{3 c d \left (a e^2+c d^2\right )^2}{e^7 (d+e x)^2}-\frac{\left (a e^2+c d^2\right )^3}{3 e^7 (d+e x)^3}-\frac{2 c^3 d x^2}{e^5}+\frac{c^3 x^3}{3 e^4} \]

[Out]

(c^2*(10*c*d^2 + 3*a*e^2)*x)/e^6 - (2*c^3*d*x^2)/e^5 + (c^3*x^3)/(3*e^4) - (c*d^
2 + a*e^2)^3/(3*e^7*(d + e*x)^3) + (3*c*d*(c*d^2 + a*e^2)^2)/(e^7*(d + e*x)^2) -
(3*c*(c*d^2 + a*e^2)*(5*c*d^2 + a*e^2))/(e^7*(d + e*x)) - (4*c^2*d*(5*c*d^2 + 3
*a*e^2)*Log[d + e*x])/e^7

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Rubi [A]  time = 0.365263, antiderivative size = 165, 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 \[ -\frac{4 c^2 d \left (3 a e^2+5 c d^2\right ) \log (d+e x)}{e^7}+\frac{c^2 x \left (3 a e^2+10 c d^2\right )}{e^6}-\frac{3 c \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{e^7 (d+e x)}+\frac{3 c d \left (a e^2+c d^2\right )^2}{e^7 (d+e x)^2}-\frac{\left (a e^2+c d^2\right )^3}{3 e^7 (d+e x)^3}-\frac{2 c^3 d x^2}{e^5}+\frac{c^3 x^3}{3 e^4} \]

Antiderivative was successfully verified.

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

[Out]

(c^2*(10*c*d^2 + 3*a*e^2)*x)/e^6 - (2*c^3*d*x^2)/e^5 + (c^3*x^3)/(3*e^4) - (c*d^
2 + a*e^2)^3/(3*e^7*(d + e*x)^3) + (3*c*d*(c*d^2 + a*e^2)^2)/(e^7*(d + e*x)^2) -
(3*c*(c*d^2 + a*e^2)*(5*c*d^2 + a*e^2))/(e^7*(d + e*x)) - (4*c^2*d*(5*c*d^2 + 3
*a*e^2)*Log[d + e*x])/e^7

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{4 c^{3} d \int x\, dx}{e^{5}} + \frac{c^{3} x^{3}}{3 e^{4}} - \frac{4 c^{2} d \left (3 a e^{2} + 5 c d^{2}\right ) \log{\left (d + e x \right )}}{e^{7}} + \frac{3 c d \left (a e^{2} + c d^{2}\right )^{2}}{e^{7} \left (d + e x\right )^{2}} - \frac{3 c \left (a e^{2} + c d^{2}\right ) \left (a e^{2} + 5 c d^{2}\right )}{e^{7} \left (d + e x\right )} + \frac{\left (3 a e^{2} + 10 c d^{2}\right ) \int c^{2}\, dx}{e^{6}} - \frac{\left (a e^{2} + c d^{2}\right )^{3}}{3 e^{7} \left (d + e x\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4*c**3*d*Integral(x, x)/e**5 + c**3*x**3/(3*e**4) - 4*c**2*d*(3*a*e**2 + 5*c*d*
*2)*log(d + e*x)/e**7 + 3*c*d*(a*e**2 + c*d**2)**2/(e**7*(d + e*x)**2) - 3*c*(a*
e**2 + c*d**2)*(a*e**2 + 5*c*d**2)/(e**7*(d + e*x)) + (3*a*e**2 + 10*c*d**2)*Int
egral(c**2, x)/e**6 - (a*e**2 + c*d**2)**3/(3*e**7*(d + e*x)**3)

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Mathematica [A]  time = 0.139308, size = 197, normalized size = 1.19 \[ \frac{-a^3 e^6-3 a^2 c e^4 \left (d^2+3 d e x+3 e^2 x^2\right )-12 c^2 d (d+e x)^3 \left (3 a e^2+5 c d^2\right ) \log (d+e x)+3 a c^2 e^2 \left (-13 d^4-27 d^3 e x-9 d^2 e^2 x^2+9 d e^3 x^3+3 e^4 x^4\right )+c^3 \left (-37 d^6-51 d^5 e x+39 d^4 e^2 x^2+73 d^3 e^3 x^3+15 d^2 e^4 x^4-3 d e^5 x^5+e^6 x^6\right )}{3 e^7 (d+e x)^3} \]

Antiderivative was successfully verified.

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

[Out]

(-(a^3*e^6) - 3*a^2*c*e^4*(d^2 + 3*d*e*x + 3*e^2*x^2) + 3*a*c^2*e^2*(-13*d^4 - 2
7*d^3*e*x - 9*d^2*e^2*x^2 + 9*d*e^3*x^3 + 3*e^4*x^4) + c^3*(-37*d^6 - 51*d^5*e*x
+ 39*d^4*e^2*x^2 + 73*d^3*e^3*x^3 + 15*d^2*e^4*x^4 - 3*d*e^5*x^5 + e^6*x^6) - 1
2*c^2*d*(5*c*d^2 + 3*a*e^2)*(d + e*x)^3*Log[d + e*x])/(3*e^7*(d + e*x)^3)

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Maple [A]  time = 0.014, size = 258, normalized size = 1.6 \[{\frac{{x}^{3}{c}^{3}}{3\,{e}^{4}}}-2\,{\frac{{c}^{3}d{x}^{2}}{{e}^{5}}}+3\,{\frac{a{c}^{2}x}{{e}^{4}}}+10\,{\frac{x{c}^{3}{d}^{2}}{{e}^{6}}}+3\,{\frac{{a}^{2}cd}{{e}^{3} \left ( ex+d \right ) ^{2}}}+6\,{\frac{{d}^{3}a{c}^{2}}{{e}^{5} \left ( ex+d \right ) ^{2}}}+3\,{\frac{{c}^{3}{d}^{5}}{{e}^{7} \left ( ex+d \right ) ^{2}}}-12\,{\frac{{c}^{2}d\ln \left ( ex+d \right ) a}{{e}^{5}}}-20\,{\frac{\ln \left ( ex+d \right ){c}^{3}{d}^{3}}{{e}^{7}}}-{\frac{{a}^{3}}{3\,e \left ( ex+d \right ) ^{3}}}-{\frac{{a}^{2}c{d}^{2}}{{e}^{3} \left ( ex+d \right ) ^{3}}}-{\frac{{d}^{4}a{c}^{2}}{{e}^{5} \left ( ex+d \right ) ^{3}}}-{\frac{{c}^{3}{d}^{6}}{3\,{e}^{7} \left ( ex+d \right ) ^{3}}}-3\,{\frac{{a}^{2}c}{{e}^{3} \left ( ex+d \right ) }}-18\,{\frac{{d}^{2}a{c}^{2}}{{e}^{5} \left ( ex+d \right ) }}-15\,{\frac{{d}^{4}{c}^{3}}{{e}^{7} \left ( ex+d \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+a)^3/(e*x+d)^4,x)

[Out]

1/3*c^3*x^3/e^4-2*c^3*d*x^2/e^5+3*c^2/e^4*a*x+10*c^3/e^6*d^2*x+3*c*d/e^3/(e*x+d)
^2*a^2+6*c^2*d^3/e^5/(e*x+d)^2*a+3*d^5/e^7/(e*x+d)^2*c^3-12*c^2*d/e^5*ln(e*x+d)*
a-20/e^7*ln(e*x+d)*c^3*d^3-1/3/e/(e*x+d)^3*a^3-1/e^3/(e*x+d)^3*d^2*a^2*c-1/e^5/(
e*x+d)^3*a*c^2*d^4-1/3*d^6/e^7/(e*x+d)^3*c^3-3*c/e^3/(e*x+d)*a^2-18*c^2/e^5/(e*x
+d)*a*d^2-15*d^4/e^7/(e*x+d)*c^3

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Maxima [A]  time = 0.717417, size = 305, normalized size = 1.85 \[ -\frac{37 \, c^{3} d^{6} + 39 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6} + 9 \,{\left (5 \, c^{3} d^{4} e^{2} + 6 \, a c^{2} d^{2} e^{4} + a^{2} c e^{6}\right )} x^{2} + 9 \,{\left (9 \, c^{3} d^{5} e + 10 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x}{3 \,{\left (e^{10} x^{3} + 3 \, d e^{9} x^{2} + 3 \, d^{2} e^{8} x + d^{3} e^{7}\right )}} + \frac{c^{3} e^{2} x^{3} - 6 \, c^{3} d e x^{2} + 3 \,{\left (10 \, c^{3} d^{2} + 3 \, a c^{2} e^{2}\right )} x}{3 \, e^{6}} - \frac{4 \,{\left (5 \, c^{3} d^{3} + 3 \, a c^{2} d e^{2}\right )} \log \left (e x + d\right )}{e^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(37*c^3*d^6 + 39*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + a^3*e^6 + 9*(5*c^3*d^4*e
^2 + 6*a*c^2*d^2*e^4 + a^2*c*e^6)*x^2 + 9*(9*c^3*d^5*e + 10*a*c^2*d^3*e^3 + a^2*
c*d*e^5)*x)/(e^10*x^3 + 3*d*e^9*x^2 + 3*d^2*e^8*x + d^3*e^7) + 1/3*(c^3*e^2*x^3
- 6*c^3*d*e*x^2 + 3*(10*c^3*d^2 + 3*a*c^2*e^2)*x)/e^6 - 4*(5*c^3*d^3 + 3*a*c^2*d
*e^2)*log(e*x + d)/e^7

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Fricas [A]  time = 0.211379, size = 451, normalized size = 2.73 \[ \frac{c^{3} e^{6} x^{6} - 3 \, c^{3} d e^{5} x^{5} - 37 \, c^{3} d^{6} - 39 \, a c^{2} d^{4} e^{2} - 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6} + 3 \,{\left (5 \, c^{3} d^{2} e^{4} + 3 \, a c^{2} e^{6}\right )} x^{4} +{\left (73 \, c^{3} d^{3} e^{3} + 27 \, a c^{2} d e^{5}\right )} x^{3} + 3 \,{\left (13 \, c^{3} d^{4} e^{2} - 9 \, a c^{2} d^{2} e^{4} - 3 \, a^{2} c e^{6}\right )} x^{2} - 3 \,{\left (17 \, c^{3} d^{5} e + 27 \, a c^{2} d^{3} e^{3} + 3 \, a^{2} c d e^{5}\right )} x - 12 \,{\left (5 \, c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} +{\left (5 \, c^{3} d^{3} e^{3} + 3 \, a c^{2} d e^{5}\right )} x^{3} + 3 \,{\left (5 \, c^{3} d^{4} e^{2} + 3 \, a c^{2} d^{2} e^{4}\right )} x^{2} + 3 \,{\left (5 \, c^{3} d^{5} e + 3 \, a c^{2} d^{3} e^{3}\right )} x\right )} \log \left (e x + d\right )}{3 \,{\left (e^{10} x^{3} + 3 \, d e^{9} x^{2} + 3 \, d^{2} e^{8} x + d^{3} e^{7}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(c^3*e^6*x^6 - 3*c^3*d*e^5*x^5 - 37*c^3*d^6 - 39*a*c^2*d^4*e^2 - 3*a^2*c*d^2
*e^4 - a^3*e^6 + 3*(5*c^3*d^2*e^4 + 3*a*c^2*e^6)*x^4 + (73*c^3*d^3*e^3 + 27*a*c^
2*d*e^5)*x^3 + 3*(13*c^3*d^4*e^2 - 9*a*c^2*d^2*e^4 - 3*a^2*c*e^6)*x^2 - 3*(17*c^
3*d^5*e + 27*a*c^2*d^3*e^3 + 3*a^2*c*d*e^5)*x - 12*(5*c^3*d^6 + 3*a*c^2*d^4*e^2
+ (5*c^3*d^3*e^3 + 3*a*c^2*d*e^5)*x^3 + 3*(5*c^3*d^4*e^2 + 3*a*c^2*d^2*e^4)*x^2
+ 3*(5*c^3*d^5*e + 3*a*c^2*d^3*e^3)*x)*log(e*x + d))/(e^10*x^3 + 3*d*e^9*x^2 + 3
*d^2*e^8*x + d^3*e^7)

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Sympy [A]  time = 8.84097, size = 235, normalized size = 1.42 \[ - \frac{2 c^{3} d x^{2}}{e^{5}} + \frac{c^{3} x^{3}}{3 e^{4}} - \frac{4 c^{2} d \left (3 a e^{2} + 5 c d^{2}\right ) \log{\left (d + e x \right )}}{e^{7}} - \frac{a^{3} e^{6} + 3 a^{2} c d^{2} e^{4} + 39 a c^{2} d^{4} e^{2} + 37 c^{3} d^{6} + x^{2} \left (9 a^{2} c e^{6} + 54 a c^{2} d^{2} e^{4} + 45 c^{3} d^{4} e^{2}\right ) + x \left (9 a^{2} c d e^{5} + 90 a c^{2} d^{3} e^{3} + 81 c^{3} d^{5} e\right )}{3 d^{3} e^{7} + 9 d^{2} e^{8} x + 9 d e^{9} x^{2} + 3 e^{10} x^{3}} + \frac{x \left (3 a c^{2} e^{2} + 10 c^{3} d^{2}\right )}{e^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*c**3*d*x**2/e**5 + c**3*x**3/(3*e**4) - 4*c**2*d*(3*a*e**2 + 5*c*d**2)*log(d
+ e*x)/e**7 - (a**3*e**6 + 3*a**2*c*d**2*e**4 + 39*a*c**2*d**4*e**2 + 37*c**3*d*
*6 + x**2*(9*a**2*c*e**6 + 54*a*c**2*d**2*e**4 + 45*c**3*d**4*e**2) + x*(9*a**2*
c*d*e**5 + 90*a*c**2*d**3*e**3 + 81*c**3*d**5*e))/(3*d**3*e**7 + 9*d**2*e**8*x +
9*d*e**9*x**2 + 3*e**10*x**3) + x*(3*a*c**2*e**2 + 10*c**3*d**2)/e**6

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GIAC/XCAS [A]  time = 0.212201, size = 259, normalized size = 1.57 \[ -4 \,{\left (5 \, c^{3} d^{3} + 3 \, a c^{2} d e^{2}\right )} e^{\left (-7\right )}{\rm ln}\left ({\left | x e + d \right |}\right ) + \frac{1}{3} \,{\left (c^{3} x^{3} e^{8} - 6 \, c^{3} d x^{2} e^{7} + 30 \, c^{3} d^{2} x e^{6} + 9 \, a c^{2} x e^{8}\right )} e^{\left (-12\right )} - \frac{{\left (37 \, c^{3} d^{6} + 39 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6} + 9 \,{\left (5 \, c^{3} d^{4} e^{2} + 6 \, a c^{2} d^{2} e^{4} + a^{2} c e^{6}\right )} x^{2} + 9 \,{\left (9 \, c^{3} d^{5} e + 10 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x\right )} e^{\left (-7\right )}}{3 \,{\left (x e + d\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4*(5*c^3*d^3 + 3*a*c^2*d*e^2)*e^(-7)*ln(abs(x*e + d)) + 1/3*(c^3*x^3*e^8 - 6*c^
3*d*x^2*e^7 + 30*c^3*d^2*x*e^6 + 9*a*c^2*x*e^8)*e^(-12) - 1/3*(37*c^3*d^6 + 39*a
*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + a^3*e^6 + 9*(5*c^3*d^4*e^2 + 6*a*c^2*d^2*e^4 +
a^2*c*e^6)*x^2 + 9*(9*c^3*d^5*e + 10*a*c^2*d^3*e^3 + a^2*c*d*e^5)*x)*e^(-7)/(x*e
+ d)^3