3.484 \(\int \frac{\left (a+c x^2\right )^3}{(d+e x)^9} \, dx\)

Optimal. Leaf size=188 \[ -\frac{3 c^2 \left (a e^2+5 c d^2\right )}{4 e^7 (d+e x)^4}+\frac{4 c^2 d \left (3 a e^2+5 c d^2\right )}{5 e^7 (d+e x)^5}-\frac{c \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{2 e^7 (d+e x)^6}+\frac{6 c d \left (a e^2+c d^2\right )^2}{7 e^7 (d+e x)^7}-\frac{\left (a e^2+c d^2\right )^3}{8 e^7 (d+e x)^8}-\frac{c^3}{2 e^7 (d+e x)^2}+\frac{2 c^3 d}{e^7 (d+e x)^3} \]

[Out]

-(c*d^2 + a*e^2)^3/(8*e^7*(d + e*x)^8) + (6*c*d*(c*d^2 + a*e^2)^2)/(7*e^7*(d + e
*x)^7) - (c*(c*d^2 + a*e^2)*(5*c*d^2 + a*e^2))/(2*e^7*(d + e*x)^6) + (4*c^2*d*(5
*c*d^2 + 3*a*e^2))/(5*e^7*(d + e*x)^5) - (3*c^2*(5*c*d^2 + a*e^2))/(4*e^7*(d + e
*x)^4) + (2*c^3*d)/(e^7*(d + e*x)^3) - c^3/(2*e^7*(d + e*x)^2)

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Rubi [A]  time = 0.383158, antiderivative size = 188, 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 \[ -\frac{3 c^2 \left (a e^2+5 c d^2\right )}{4 e^7 (d+e x)^4}+\frac{4 c^2 d \left (3 a e^2+5 c d^2\right )}{5 e^7 (d+e x)^5}-\frac{c \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{2 e^7 (d+e x)^6}+\frac{6 c d \left (a e^2+c d^2\right )^2}{7 e^7 (d+e x)^7}-\frac{\left (a e^2+c d^2\right )^3}{8 e^7 (d+e x)^8}-\frac{c^3}{2 e^7 (d+e x)^2}+\frac{2 c^3 d}{e^7 (d+e x)^3} \]

Antiderivative was successfully verified.

[In]  Int[(a + c*x^2)^3/(d + e*x)^9,x]

[Out]

-(c*d^2 + a*e^2)^3/(8*e^7*(d + e*x)^8) + (6*c*d*(c*d^2 + a*e^2)^2)/(7*e^7*(d + e
*x)^7) - (c*(c*d^2 + a*e^2)*(5*c*d^2 + a*e^2))/(2*e^7*(d + e*x)^6) + (4*c^2*d*(5
*c*d^2 + 3*a*e^2))/(5*e^7*(d + e*x)^5) - (3*c^2*(5*c*d^2 + a*e^2))/(4*e^7*(d + e
*x)^4) + (2*c^3*d)/(e^7*(d + e*x)^3) - c^3/(2*e^7*(d + e*x)^2)

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Rubi in Sympy [A]  time = 55.3192, size = 180, normalized size = 0.96 \[ \frac{2 c^{3} d}{e^{7} \left (d + e x\right )^{3}} - \frac{c^{3}}{2 e^{7} \left (d + e x\right )^{2}} + \frac{4 c^{2} d \left (3 a e^{2} + 5 c d^{2}\right )}{5 e^{7} \left (d + e x\right )^{5}} - \frac{3 c^{2} \left (a e^{2} + 5 c d^{2}\right )}{4 e^{7} \left (d + e x\right )^{4}} + \frac{6 c d \left (a e^{2} + c d^{2}\right )^{2}}{7 e^{7} \left (d + e x\right )^{7}} - \frac{c \left (a e^{2} + c d^{2}\right ) \left (a e^{2} + 5 c d^{2}\right )}{2 e^{7} \left (d + e x\right )^{6}} - \frac{\left (a e^{2} + c d^{2}\right )^{3}}{8 e^{7} \left (d + e x\right )^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+a)**3/(e*x+d)**9,x)

[Out]

2*c**3*d/(e**7*(d + e*x)**3) - c**3/(2*e**7*(d + e*x)**2) + 4*c**2*d*(3*a*e**2 +
5*c*d**2)/(5*e**7*(d + e*x)**5) - 3*c**2*(a*e**2 + 5*c*d**2)/(4*e**7*(d + e*x)*
*4) + 6*c*d*(a*e**2 + c*d**2)**2/(7*e**7*(d + e*x)**7) - c*(a*e**2 + c*d**2)*(a*
e**2 + 5*c*d**2)/(2*e**7*(d + e*x)**6) - (a*e**2 + c*d**2)**3/(8*e**7*(d + e*x)*
*8)

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Mathematica [A]  time = 0.112582, size = 163, normalized size = 0.87 \[ -\frac{35 a^3 e^6+5 a^2 c e^4 \left (d^2+8 d e x+28 e^2 x^2\right )+3 a c^2 e^2 \left (d^4+8 d^3 e x+28 d^2 e^2 x^2+56 d e^3 x^3+70 e^4 x^4\right )+5 c^3 \left (d^6+8 d^5 e x+28 d^4 e^2 x^2+56 d^3 e^3 x^3+70 d^2 e^4 x^4+56 d e^5 x^5+28 e^6 x^6\right )}{280 e^7 (d+e x)^8} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + c*x^2)^3/(d + e*x)^9,x]

[Out]

-(35*a^3*e^6 + 5*a^2*c*e^4*(d^2 + 8*d*e*x + 28*e^2*x^2) + 3*a*c^2*e^2*(d^4 + 8*d
^3*e*x + 28*d^2*e^2*x^2 + 56*d*e^3*x^3 + 70*e^4*x^4) + 5*c^3*(d^6 + 8*d^5*e*x +
28*d^4*e^2*x^2 + 56*d^3*e^3*x^3 + 70*d^2*e^4*x^4 + 56*d*e^5*x^5 + 28*e^6*x^6))/(
280*e^7*(d + e*x)^8)

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Maple [A]  time = 0.01, size = 218, normalized size = 1.2 \[{\frac{6\,cd \left ({a}^{2}{e}^{4}+2\,ac{d}^{2}{e}^{2}+{c}^{2}{d}^{4} \right ) }{7\,{e}^{7} \left ( ex+d \right ) ^{7}}}+{\frac{4\,{c}^{2}d \left ( 3\,a{e}^{2}+5\,c{d}^{2} \right ) }{5\,{e}^{7} \left ( ex+d \right ) ^{5}}}-{\frac{c \left ({a}^{2}{e}^{4}+6\,ac{d}^{2}{e}^{2}+5\,{c}^{2}{d}^{4} \right ) }{2\,{e}^{7} \left ( ex+d \right ) ^{6}}}-{\frac{{c}^{3}}{2\,{e}^{7} \left ( ex+d \right ) ^{2}}}+2\,{\frac{{c}^{3}d}{{e}^{7} \left ( ex+d \right ) ^{3}}}-{\frac{3\,{c}^{2} \left ( a{e}^{2}+5\,c{d}^{2} \right ) }{4\,{e}^{7} \left ( ex+d \right ) ^{4}}}-{\frac{{a}^{3}{e}^{6}+3\,{a}^{2}c{d}^{2}{e}^{4}+3\,{d}^{4}a{c}^{2}{e}^{2}+{c}^{3}{d}^{6}}{8\,{e}^{7} \left ( ex+d \right ) ^{8}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+a)^3/(e*x+d)^9,x)

[Out]

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

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Maxima [A]  time = 0.726612, size = 381, normalized size = 2.03 \[ -\frac{140 \, c^{3} e^{6} x^{6} + 280 \, c^{3} d e^{5} x^{5} + 5 \, c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 5 \, a^{2} c d^{2} e^{4} + 35 \, a^{3} e^{6} + 70 \,{\left (5 \, c^{3} d^{2} e^{4} + 3 \, a c^{2} e^{6}\right )} x^{4} + 56 \,{\left (5 \, c^{3} d^{3} e^{3} + 3 \, a c^{2} d e^{5}\right )} x^{3} + 28 \,{\left (5 \, c^{3} d^{4} e^{2} + 3 \, a c^{2} d^{2} e^{4} + 5 \, a^{2} c e^{6}\right )} x^{2} + 8 \,{\left (5 \, c^{3} d^{5} e + 3 \, a c^{2} d^{3} e^{3} + 5 \, a^{2} c d e^{5}\right )} x}{280 \,{\left (e^{15} x^{8} + 8 \, d e^{14} x^{7} + 28 \, d^{2} e^{13} x^{6} + 56 \, d^{3} e^{12} x^{5} + 70 \, d^{4} e^{11} x^{4} + 56 \, d^{5} e^{10} x^{3} + 28 \, d^{6} e^{9} x^{2} + 8 \, d^{7} e^{8} x + d^{8} e^{7}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^3/(e*x + d)^9,x, algorithm="maxima")

[Out]

-1/280*(140*c^3*e^6*x^6 + 280*c^3*d*e^5*x^5 + 5*c^3*d^6 + 3*a*c^2*d^4*e^2 + 5*a^
2*c*d^2*e^4 + 35*a^3*e^6 + 70*(5*c^3*d^2*e^4 + 3*a*c^2*e^6)*x^4 + 56*(5*c^3*d^3*
e^3 + 3*a*c^2*d*e^5)*x^3 + 28*(5*c^3*d^4*e^2 + 3*a*c^2*d^2*e^4 + 5*a^2*c*e^6)*x^
2 + 8*(5*c^3*d^5*e + 3*a*c^2*d^3*e^3 + 5*a^2*c*d*e^5)*x)/(e^15*x^8 + 8*d*e^14*x^
7 + 28*d^2*e^13*x^6 + 56*d^3*e^12*x^5 + 70*d^4*e^11*x^4 + 56*d^5*e^10*x^3 + 28*d
^6*e^9*x^2 + 8*d^7*e^8*x + d^8*e^7)

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Fricas [A]  time = 0.221211, size = 381, normalized size = 2.03 \[ -\frac{140 \, c^{3} e^{6} x^{6} + 280 \, c^{3} d e^{5} x^{5} + 5 \, c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 5 \, a^{2} c d^{2} e^{4} + 35 \, a^{3} e^{6} + 70 \,{\left (5 \, c^{3} d^{2} e^{4} + 3 \, a c^{2} e^{6}\right )} x^{4} + 56 \,{\left (5 \, c^{3} d^{3} e^{3} + 3 \, a c^{2} d e^{5}\right )} x^{3} + 28 \,{\left (5 \, c^{3} d^{4} e^{2} + 3 \, a c^{2} d^{2} e^{4} + 5 \, a^{2} c e^{6}\right )} x^{2} + 8 \,{\left (5 \, c^{3} d^{5} e + 3 \, a c^{2} d^{3} e^{3} + 5 \, a^{2} c d e^{5}\right )} x}{280 \,{\left (e^{15} x^{8} + 8 \, d e^{14} x^{7} + 28 \, d^{2} e^{13} x^{6} + 56 \, d^{3} e^{12} x^{5} + 70 \, d^{4} e^{11} x^{4} + 56 \, d^{5} e^{10} x^{3} + 28 \, d^{6} e^{9} x^{2} + 8 \, d^{7} e^{8} x + d^{8} e^{7}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^3/(e*x + d)^9,x, algorithm="fricas")

[Out]

-1/280*(140*c^3*e^6*x^6 + 280*c^3*d*e^5*x^5 + 5*c^3*d^6 + 3*a*c^2*d^4*e^2 + 5*a^
2*c*d^2*e^4 + 35*a^3*e^6 + 70*(5*c^3*d^2*e^4 + 3*a*c^2*e^6)*x^4 + 56*(5*c^3*d^3*
e^3 + 3*a*c^2*d*e^5)*x^3 + 28*(5*c^3*d^4*e^2 + 3*a*c^2*d^2*e^4 + 5*a^2*c*e^6)*x^
2 + 8*(5*c^3*d^5*e + 3*a*c^2*d^3*e^3 + 5*a^2*c*d*e^5)*x)/(e^15*x^8 + 8*d*e^14*x^
7 + 28*d^2*e^13*x^6 + 56*d^3*e^12*x^5 + 70*d^4*e^11*x^4 + 56*d^5*e^10*x^3 + 28*d
^6*e^9*x^2 + 8*d^7*e^8*x + d^8*e^7)

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Sympy [A]  time = 69.3874, size = 296, normalized size = 1.57 \[ - \frac{35 a^{3} e^{6} + 5 a^{2} c d^{2} e^{4} + 3 a c^{2} d^{4} e^{2} + 5 c^{3} d^{6} + 280 c^{3} d e^{5} x^{5} + 140 c^{3} e^{6} x^{6} + x^{4} \left (210 a c^{2} e^{6} + 350 c^{3} d^{2} e^{4}\right ) + x^{3} \left (168 a c^{2} d e^{5} + 280 c^{3} d^{3} e^{3}\right ) + x^{2} \left (140 a^{2} c e^{6} + 84 a c^{2} d^{2} e^{4} + 140 c^{3} d^{4} e^{2}\right ) + x \left (40 a^{2} c d e^{5} + 24 a c^{2} d^{3} e^{3} + 40 c^{3} d^{5} e\right )}{280 d^{8} e^{7} + 2240 d^{7} e^{8} x + 7840 d^{6} e^{9} x^{2} + 15680 d^{5} e^{10} x^{3} + 19600 d^{4} e^{11} x^{4} + 15680 d^{3} e^{12} x^{5} + 7840 d^{2} e^{13} x^{6} + 2240 d e^{14} x^{7} + 280 e^{15} x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+a)**3/(e*x+d)**9,x)

[Out]

-(35*a**3*e**6 + 5*a**2*c*d**2*e**4 + 3*a*c**2*d**4*e**2 + 5*c**3*d**6 + 280*c**
3*d*e**5*x**5 + 140*c**3*e**6*x**6 + x**4*(210*a*c**2*e**6 + 350*c**3*d**2*e**4)
+ x**3*(168*a*c**2*d*e**5 + 280*c**3*d**3*e**3) + x**2*(140*a**2*c*e**6 + 84*a*
c**2*d**2*e**4 + 140*c**3*d**4*e**2) + x*(40*a**2*c*d*e**5 + 24*a*c**2*d**3*e**3
+ 40*c**3*d**5*e))/(280*d**8*e**7 + 2240*d**7*e**8*x + 7840*d**6*e**9*x**2 + 15
680*d**5*e**10*x**3 + 19600*d**4*e**11*x**4 + 15680*d**3*e**12*x**5 + 7840*d**2*
e**13*x**6 + 2240*d*e**14*x**7 + 280*e**15*x**8)

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GIAC/XCAS [A]  time = 0.209958, size = 258, normalized size = 1.37 \[ -\frac{{\left (140 \, c^{3} x^{6} e^{6} + 280 \, c^{3} d x^{5} e^{5} + 350 \, c^{3} d^{2} x^{4} e^{4} + 280 \, c^{3} d^{3} x^{3} e^{3} + 140 \, c^{3} d^{4} x^{2} e^{2} + 40 \, c^{3} d^{5} x e + 5 \, c^{3} d^{6} + 210 \, a c^{2} x^{4} e^{6} + 168 \, a c^{2} d x^{3} e^{5} + 84 \, a c^{2} d^{2} x^{2} e^{4} + 24 \, a c^{2} d^{3} x e^{3} + 3 \, a c^{2} d^{4} e^{2} + 140 \, a^{2} c x^{2} e^{6} + 40 \, a^{2} c d x e^{5} + 5 \, a^{2} c d^{2} e^{4} + 35 \, a^{3} e^{6}\right )} e^{\left (-7\right )}}{280 \,{\left (x e + d\right )}^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^3/(e*x + d)^9,x, algorithm="giac")

[Out]

-1/280*(140*c^3*x^6*e^6 + 280*c^3*d*x^5*e^5 + 350*c^3*d^2*x^4*e^4 + 280*c^3*d^3*
x^3*e^3 + 140*c^3*d^4*x^2*e^2 + 40*c^3*d^5*x*e + 5*c^3*d^6 + 210*a*c^2*x^4*e^6 +
168*a*c^2*d*x^3*e^5 + 84*a*c^2*d^2*x^2*e^4 + 24*a*c^2*d^3*x*e^3 + 3*a*c^2*d^4*e
^2 + 140*a^2*c*x^2*e^6 + 40*a^2*c*d*x*e^5 + 5*a^2*c*d^2*e^4 + 35*a^3*e^6)*e^(-7)
/(x*e + d)^8