3.470 \(\int \frac{\left (a+c x^2\right )^2}{(d+e x)^7} \, dx\)

Optimal. Leaf size=117 \[ -\frac{c \left (a e^2+3 c d^2\right )}{2 e^5 (d+e x)^4}+\frac{4 c d \left (a e^2+c d^2\right )}{5 e^5 (d+e x)^5}-\frac{\left (a e^2+c d^2\right )^2}{6 e^5 (d+e x)^6}-\frac{c^2}{2 e^5 (d+e x)^2}+\frac{4 c^2 d}{3 e^5 (d+e x)^3} \]

[Out]

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

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Rubi [A]  time = 0.190076, antiderivative size = 117, 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 \[ -\frac{c \left (a e^2+3 c d^2\right )}{2 e^5 (d+e x)^4}+\frac{4 c d \left (a e^2+c d^2\right )}{5 e^5 (d+e x)^5}-\frac{\left (a e^2+c d^2\right )^2}{6 e^5 (d+e x)^6}-\frac{c^2}{2 e^5 (d+e x)^2}+\frac{4 c^2 d}{3 e^5 (d+e x)^3} \]

Antiderivative was successfully verified.

[In]  Int[(a + c*x^2)^2/(d + e*x)^7,x]

[Out]

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

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Rubi in Sympy [A]  time = 30.8884, size = 109, normalized size = 0.93 \[ \frac{4 c^{2} d}{3 e^{5} \left (d + e x\right )^{3}} - \frac{c^{2}}{2 e^{5} \left (d + e x\right )^{2}} + \frac{4 c d \left (a e^{2} + c d^{2}\right )}{5 e^{5} \left (d + e x\right )^{5}} - \frac{c \left (a e^{2} + 3 c d^{2}\right )}{2 e^{5} \left (d + e x\right )^{4}} - \frac{\left (a e^{2} + c d^{2}\right )^{2}}{6 e^{5} \left (d + e x\right )^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+a)**2/(e*x+d)**7,x)

[Out]

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

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Mathematica [A]  time = 0.0601878, size = 89, normalized size = 0.76 \[ -\frac{5 a^2 e^4+a c e^2 \left (d^2+6 d e x+15 e^2 x^2\right )+c^2 \left (d^4+6 d^3 e x+15 d^2 e^2 x^2+20 d e^3 x^3+15 e^4 x^4\right )}{30 e^5 (d+e x)^6} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + c*x^2)^2/(d + e*x)^7,x]

[Out]

-(5*a^2*e^4 + a*c*e^2*(d^2 + 6*d*e*x + 15*e^2*x^2) + c^2*(d^4 + 6*d^3*e*x + 15*d
^2*e^2*x^2 + 20*d*e^3*x^3 + 15*e^4*x^4))/(30*e^5*(d + e*x)^6)

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Maple [A]  time = 0.008, size = 120, normalized size = 1. \[ -{\frac{{a}^{2}{e}^{4}+2\,ac{d}^{2}{e}^{2}+{c}^{2}{d}^{4}}{6\,{e}^{5} \left ( ex+d \right ) ^{6}}}+{\frac{4\,cd \left ( a{e}^{2}+c{d}^{2} \right ) }{5\,{e}^{5} \left ( ex+d \right ) ^{5}}}-{\frac{{c}^{2}}{2\,{e}^{5} \left ( ex+d \right ) ^{2}}}+{\frac{4\,{c}^{2}d}{3\,{e}^{5} \left ( ex+d \right ) ^{3}}}-{\frac{c \left ( a{e}^{2}+3\,c{d}^{2} \right ) }{2\,{e}^{5} \left ( ex+d \right ) ^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+a)^2/(e*x+d)^7,x)

[Out]

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

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Maxima [A]  time = 0.710664, size = 215, normalized size = 1.84 \[ -\frac{15 \, c^{2} e^{4} x^{4} + 20 \, c^{2} d e^{3} x^{3} + c^{2} d^{4} + a c d^{2} e^{2} + 5 \, a^{2} e^{4} + 15 \,{\left (c^{2} d^{2} e^{2} + a c e^{4}\right )} x^{2} + 6 \,{\left (c^{2} d^{3} e + a c d e^{3}\right )} x}{30 \,{\left (e^{11} x^{6} + 6 \, d e^{10} x^{5} + 15 \, d^{2} e^{9} x^{4} + 20 \, d^{3} e^{8} x^{3} + 15 \, d^{4} e^{7} x^{2} + 6 \, d^{5} e^{6} x + d^{6} e^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^2/(e*x + d)^7,x, algorithm="maxima")

[Out]

-1/30*(15*c^2*e^4*x^4 + 20*c^2*d*e^3*x^3 + c^2*d^4 + a*c*d^2*e^2 + 5*a^2*e^4 + 1
5*(c^2*d^2*e^2 + a*c*e^4)*x^2 + 6*(c^2*d^3*e + a*c*d*e^3)*x)/(e^11*x^6 + 6*d*e^1
0*x^5 + 15*d^2*e^9*x^4 + 20*d^3*e^8*x^3 + 15*d^4*e^7*x^2 + 6*d^5*e^6*x + d^6*e^5
)

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Fricas [A]  time = 0.198346, size = 215, normalized size = 1.84 \[ -\frac{15 \, c^{2} e^{4} x^{4} + 20 \, c^{2} d e^{3} x^{3} + c^{2} d^{4} + a c d^{2} e^{2} + 5 \, a^{2} e^{4} + 15 \,{\left (c^{2} d^{2} e^{2} + a c e^{4}\right )} x^{2} + 6 \,{\left (c^{2} d^{3} e + a c d e^{3}\right )} x}{30 \,{\left (e^{11} x^{6} + 6 \, d e^{10} x^{5} + 15 \, d^{2} e^{9} x^{4} + 20 \, d^{3} e^{8} x^{3} + 15 \, d^{4} e^{7} x^{2} + 6 \, d^{5} e^{6} x + d^{6} e^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^2/(e*x + d)^7,x, algorithm="fricas")

[Out]

-1/30*(15*c^2*e^4*x^4 + 20*c^2*d*e^3*x^3 + c^2*d^4 + a*c*d^2*e^2 + 5*a^2*e^4 + 1
5*(c^2*d^2*e^2 + a*c*e^4)*x^2 + 6*(c^2*d^3*e + a*c*d*e^3)*x)/(e^11*x^6 + 6*d*e^1
0*x^5 + 15*d^2*e^9*x^4 + 20*d^3*e^8*x^3 + 15*d^4*e^7*x^2 + 6*d^5*e^6*x + d^6*e^5
)

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Sympy [A]  time = 10.0697, size = 170, normalized size = 1.45 \[ - \frac{5 a^{2} e^{4} + a c d^{2} e^{2} + c^{2} d^{4} + 20 c^{2} d e^{3} x^{3} + 15 c^{2} e^{4} x^{4} + x^{2} \left (15 a c e^{4} + 15 c^{2} d^{2} e^{2}\right ) + x \left (6 a c d e^{3} + 6 c^{2} d^{3} e\right )}{30 d^{6} e^{5} + 180 d^{5} e^{6} x + 450 d^{4} e^{7} x^{2} + 600 d^{3} e^{8} x^{3} + 450 d^{2} e^{9} x^{4} + 180 d e^{10} x^{5} + 30 e^{11} x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+a)**2/(e*x+d)**7,x)

[Out]

-(5*a**2*e**4 + a*c*d**2*e**2 + c**2*d**4 + 20*c**2*d*e**3*x**3 + 15*c**2*e**4*x
**4 + x**2*(15*a*c*e**4 + 15*c**2*d**2*e**2) + x*(6*a*c*d*e**3 + 6*c**2*d**3*e))
/(30*d**6*e**5 + 180*d**5*e**6*x + 450*d**4*e**7*x**2 + 600*d**3*e**8*x**3 + 450
*d**2*e**9*x**4 + 180*d*e**10*x**5 + 30*e**11*x**6)

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GIAC/XCAS [A]  time = 0.209588, size = 131, normalized size = 1.12 \[ -\frac{{\left (15 \, c^{2} x^{4} e^{4} + 20 \, c^{2} d x^{3} e^{3} + 15 \, c^{2} d^{2} x^{2} e^{2} + 6 \, c^{2} d^{3} x e + c^{2} d^{4} + 15 \, a c x^{2} e^{4} + 6 \, a c d x e^{3} + a c d^{2} e^{2} + 5 \, a^{2} e^{4}\right )} e^{\left (-5\right )}}{30 \,{\left (x e + d\right )}^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^2/(e*x + d)^7,x, algorithm="giac")

[Out]

-1/30*(15*c^2*x^4*e^4 + 20*c^2*d*x^3*e^3 + 15*c^2*d^2*x^2*e^2 + 6*c^2*d^3*x*e +
c^2*d^4 + 15*a*c*x^2*e^4 + 6*a*c*d*x*e^3 + a*c*d^2*e^2 + 5*a^2*e^4)*e^(-5)/(x*e
+ d)^6