3.483 \(\int \frac{\left (a+c x^2\right )^3}{(d+e x)^8} \, dx\)

Optimal. Leaf size=178 \[ -\frac{c^2 \left (a e^2+5 c d^2\right )}{e^7 (d+e x)^3}+\frac{c^2 d \left (3 a e^2+5 c d^2\right )}{e^7 (d+e x)^4}-\frac{3 c \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{5 e^7 (d+e x)^5}+\frac{c d \left (a e^2+c d^2\right )^2}{e^7 (d+e x)^6}-\frac{\left (a e^2+c d^2\right )^3}{7 e^7 (d+e x)^7}-\frac{c^3}{e^7 (d+e x)}+\frac{3 c^3 d}{e^7 (d+e x)^2} \]

[Out]

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

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Rubi [A]  time = 0.379484, antiderivative size = 178, 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 \[ -\frac{c^2 \left (a e^2+5 c d^2\right )}{e^7 (d+e x)^3}+\frac{c^2 d \left (3 a e^2+5 c d^2\right )}{e^7 (d+e x)^4}-\frac{3 c \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{5 e^7 (d+e x)^5}+\frac{c d \left (a e^2+c d^2\right )^2}{e^7 (d+e x)^6}-\frac{\left (a e^2+c d^2\right )^3}{7 e^7 (d+e x)^7}-\frac{c^3}{e^7 (d+e x)}+\frac{3 c^3 d}{e^7 (d+e x)^2} \]

Antiderivative was successfully verified.

[In]  Int[(a + c*x^2)^3/(d + e*x)^8,x]

[Out]

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

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Rubi in Sympy [A]  time = 54.8701, size = 168, normalized size = 0.94 \[ \frac{3 c^{3} d}{e^{7} \left (d + e x\right )^{2}} - \frac{c^{3}}{e^{7} \left (d + e x\right )} + \frac{c^{2} d \left (3 a e^{2} + 5 c d^{2}\right )}{e^{7} \left (d + e x\right )^{4}} - \frac{c^{2} \left (a e^{2} + 5 c d^{2}\right )}{e^{7} \left (d + e x\right )^{3}} + \frac{c d \left (a e^{2} + c d^{2}\right )^{2}}{e^{7} \left (d + e x\right )^{6}} - \frac{3 c \left (a e^{2} + c d^{2}\right ) \left (a e^{2} + 5 c d^{2}\right )}{5 e^{7} \left (d + e x\right )^{5}} - \frac{\left (a e^{2} + c d^{2}\right )^{3}}{7 e^{7} \left (d + e x\right )^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+a)**3/(e*x+d)**8,x)

[Out]

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

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Mathematica [A]  time = 0.111335, size = 161, normalized size = 0.9 \[ -\frac{5 a^3 e^6+a^2 c e^4 \left (d^2+7 d e x+21 e^2 x^2\right )+a c^2 e^2 \left (d^4+7 d^3 e x+21 d^2 e^2 x^2+35 d e^3 x^3+35 e^4 x^4\right )+5 c^3 \left (d^6+7 d^5 e x+21 d^4 e^2 x^2+35 d^3 e^3 x^3+35 d^2 e^4 x^4+21 d e^5 x^5+7 e^6 x^6\right )}{35 e^7 (d+e x)^7} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + c*x^2)^3/(d + e*x)^8,x]

[Out]

-(5*a^3*e^6 + a^2*c*e^4*(d^2 + 7*d*e*x + 21*e^2*x^2) + a*c^2*e^2*(d^4 + 7*d^3*e*
x + 21*d^2*e^2*x^2 + 35*d*e^3*x^3 + 35*e^4*x^4) + 5*c^3*(d^6 + 7*d^5*e*x + 21*d^
4*e^2*x^2 + 35*d^3*e^3*x^3 + 35*d^2*e^4*x^4 + 21*d*e^5*x^5 + 7*e^6*x^6))/(35*e^7
*(d + e*x)^7)

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Maple [A]  time = 0.01, size = 216, normalized size = 1.2 \[ -{\frac{{a}^{3}{e}^{6}+3\,{a}^{2}c{d}^{2}{e}^{4}+3\,{d}^{4}a{c}^{2}{e}^{2}+{c}^{3}{d}^{6}}{7\,{e}^{7} \left ( ex+d \right ) ^{7}}}-{\frac{3\,c \left ({a}^{2}{e}^{4}+6\,ac{d}^{2}{e}^{2}+5\,{c}^{2}{d}^{4} \right ) }{5\,{e}^{7} \left ( ex+d \right ) ^{5}}}+{\frac{cd \left ({a}^{2}{e}^{4}+2\,ac{d}^{2}{e}^{2}+{c}^{2}{d}^{4} \right ) }{{e}^{7} \left ( ex+d \right ) ^{6}}}+3\,{\frac{{c}^{3}d}{{e}^{7} \left ( ex+d \right ) ^{2}}}-{\frac{{c}^{2} \left ( a{e}^{2}+5\,c{d}^{2} \right ) }{{e}^{7} \left ( ex+d \right ) ^{3}}}-{\frac{{c}^{3}}{{e}^{7} \left ( ex+d \right ) }}+{\frac{{c}^{2}d \left ( 3\,a{e}^{2}+5\,c{d}^{2} \right ) }{{e}^{7} \left ( ex+d \right ) ^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+a)^3/(e*x+d)^8,x)

[Out]

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

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Maxima [A]  time = 0.721768, size = 355, normalized size = 1.99 \[ -\frac{35 \, c^{3} e^{6} x^{6} + 105 \, c^{3} d e^{5} x^{5} + 5 \, c^{3} d^{6} + a c^{2} d^{4} e^{2} + a^{2} c d^{2} e^{4} + 5 \, a^{3} e^{6} + 35 \,{\left (5 \, c^{3} d^{2} e^{4} + a c^{2} e^{6}\right )} x^{4} + 35 \,{\left (5 \, c^{3} d^{3} e^{3} + a c^{2} d e^{5}\right )} x^{3} + 21 \,{\left (5 \, c^{3} d^{4} e^{2} + a c^{2} d^{2} e^{4} + a^{2} c e^{6}\right )} x^{2} + 7 \,{\left (5 \, c^{3} d^{5} e + a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x}{35 \,{\left (e^{14} x^{7} + 7 \, d e^{13} x^{6} + 21 \, d^{2} e^{12} x^{5} + 35 \, d^{3} e^{11} x^{4} + 35 \, d^{4} e^{10} x^{3} + 21 \, d^{5} e^{9} x^{2} + 7 \, d^{6} e^{8} x + d^{7} e^{7}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^3/(e*x + d)^8,x, algorithm="maxima")

[Out]

-1/35*(35*c^3*e^6*x^6 + 105*c^3*d*e^5*x^5 + 5*c^3*d^6 + a*c^2*d^4*e^2 + a^2*c*d^
2*e^4 + 5*a^3*e^6 + 35*(5*c^3*d^2*e^4 + a*c^2*e^6)*x^4 + 35*(5*c^3*d^3*e^3 + a*c
^2*d*e^5)*x^3 + 21*(5*c^3*d^4*e^2 + a*c^2*d^2*e^4 + a^2*c*e^6)*x^2 + 7*(5*c^3*d^
5*e + a*c^2*d^3*e^3 + a^2*c*d*e^5)*x)/(e^14*x^7 + 7*d*e^13*x^6 + 21*d^2*e^12*x^5
+ 35*d^3*e^11*x^4 + 35*d^4*e^10*x^3 + 21*d^5*e^9*x^2 + 7*d^6*e^8*x + d^7*e^7)

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Fricas [A]  time = 0.202523, size = 355, normalized size = 1.99 \[ -\frac{35 \, c^{3} e^{6} x^{6} + 105 \, c^{3} d e^{5} x^{5} + 5 \, c^{3} d^{6} + a c^{2} d^{4} e^{2} + a^{2} c d^{2} e^{4} + 5 \, a^{3} e^{6} + 35 \,{\left (5 \, c^{3} d^{2} e^{4} + a c^{2} e^{6}\right )} x^{4} + 35 \,{\left (5 \, c^{3} d^{3} e^{3} + a c^{2} d e^{5}\right )} x^{3} + 21 \,{\left (5 \, c^{3} d^{4} e^{2} + a c^{2} d^{2} e^{4} + a^{2} c e^{6}\right )} x^{2} + 7 \,{\left (5 \, c^{3} d^{5} e + a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x}{35 \,{\left (e^{14} x^{7} + 7 \, d e^{13} x^{6} + 21 \, d^{2} e^{12} x^{5} + 35 \, d^{3} e^{11} x^{4} + 35 \, d^{4} e^{10} x^{3} + 21 \, d^{5} e^{9} x^{2} + 7 \, d^{6} e^{8} x + d^{7} e^{7}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^3/(e*x + d)^8,x, algorithm="fricas")

[Out]

-1/35*(35*c^3*e^6*x^6 + 105*c^3*d*e^5*x^5 + 5*c^3*d^6 + a*c^2*d^4*e^2 + a^2*c*d^
2*e^4 + 5*a^3*e^6 + 35*(5*c^3*d^2*e^4 + a*c^2*e^6)*x^4 + 35*(5*c^3*d^3*e^3 + a*c
^2*d*e^5)*x^3 + 21*(5*c^3*d^4*e^2 + a*c^2*d^2*e^4 + a^2*c*e^6)*x^2 + 7*(5*c^3*d^
5*e + a*c^2*d^3*e^3 + a^2*c*d*e^5)*x)/(e^14*x^7 + 7*d*e^13*x^6 + 21*d^2*e^12*x^5
+ 35*d^3*e^11*x^4 + 35*d^4*e^10*x^3 + 21*d^5*e^9*x^2 + 7*d^6*e^8*x + d^7*e^7)

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Sympy [A]  time = 42.8923, size = 280, normalized size = 1.57 \[ - \frac{5 a^{3} e^{6} + a^{2} c d^{2} e^{4} + a c^{2} d^{4} e^{2} + 5 c^{3} d^{6} + 105 c^{3} d e^{5} x^{5} + 35 c^{3} e^{6} x^{6} + x^{4} \left (35 a c^{2} e^{6} + 175 c^{3} d^{2} e^{4}\right ) + x^{3} \left (35 a c^{2} d e^{5} + 175 c^{3} d^{3} e^{3}\right ) + x^{2} \left (21 a^{2} c e^{6} + 21 a c^{2} d^{2} e^{4} + 105 c^{3} d^{4} e^{2}\right ) + x \left (7 a^{2} c d e^{5} + 7 a c^{2} d^{3} e^{3} + 35 c^{3} d^{5} e\right )}{35 d^{7} e^{7} + 245 d^{6} e^{8} x + 735 d^{5} e^{9} x^{2} + 1225 d^{4} e^{10} x^{3} + 1225 d^{3} e^{11} x^{4} + 735 d^{2} e^{12} x^{5} + 245 d e^{13} x^{6} + 35 e^{14} x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+a)**3/(e*x+d)**8,x)

[Out]

-(5*a**3*e**6 + a**2*c*d**2*e**4 + a*c**2*d**4*e**2 + 5*c**3*d**6 + 105*c**3*d*e
**5*x**5 + 35*c**3*e**6*x**6 + x**4*(35*a*c**2*e**6 + 175*c**3*d**2*e**4) + x**3
*(35*a*c**2*d*e**5 + 175*c**3*d**3*e**3) + x**2*(21*a**2*c*e**6 + 21*a*c**2*d**2
*e**4 + 105*c**3*d**4*e**2) + x*(7*a**2*c*d*e**5 + 7*a*c**2*d**3*e**3 + 35*c**3*
d**5*e))/(35*d**7*e**7 + 245*d**6*e**8*x + 735*d**5*e**9*x**2 + 1225*d**4*e**10*
x**3 + 1225*d**3*e**11*x**4 + 735*d**2*e**12*x**5 + 245*d*e**13*x**6 + 35*e**14*
x**7)

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GIAC/XCAS [A]  time = 0.211213, size = 255, normalized size = 1.43 \[ -\frac{{\left (35 \, c^{3} x^{6} e^{6} + 105 \, c^{3} d x^{5} e^{5} + 175 \, c^{3} d^{2} x^{4} e^{4} + 175 \, c^{3} d^{3} x^{3} e^{3} + 105 \, c^{3} d^{4} x^{2} e^{2} + 35 \, c^{3} d^{5} x e + 5 \, c^{3} d^{6} + 35 \, a c^{2} x^{4} e^{6} + 35 \, a c^{2} d x^{3} e^{5} + 21 \, a c^{2} d^{2} x^{2} e^{4} + 7 \, a c^{2} d^{3} x e^{3} + a c^{2} d^{4} e^{2} + 21 \, a^{2} c x^{2} e^{6} + 7 \, a^{2} c d x e^{5} + a^{2} c d^{2} e^{4} + 5 \, a^{3} e^{6}\right )} e^{\left (-7\right )}}{35 \,{\left (x e + d\right )}^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^3/(e*x + d)^8,x, algorithm="giac")

[Out]

-1/35*(35*c^3*x^6*e^6 + 105*c^3*d*x^5*e^5 + 175*c^3*d^2*x^4*e^4 + 175*c^3*d^3*x^
3*e^3 + 105*c^3*d^4*x^2*e^2 + 35*c^3*d^5*x*e + 5*c^3*d^6 + 35*a*c^2*x^4*e^6 + 35
*a*c^2*d*x^3*e^5 + 21*a*c^2*d^2*x^2*e^4 + 7*a*c^2*d^3*x*e^3 + a*c^2*d^4*e^2 + 21
*a^2*c*x^2*e^6 + 7*a^2*c*d*x*e^5 + a^2*c*d^2*e^4 + 5*a^3*e^6)*e^(-7)/(x*e + d)^7