3.485 \(\int \frac{\left (a+c x^2\right )^3}{(d+e x)^{10}} \, dx\)

Optimal. Leaf size=190 \[ -\frac{3 c^2 \left (a e^2+5 c d^2\right )}{5 e^7 (d+e x)^5}+\frac{2 c^2 d \left (3 a e^2+5 c d^2\right )}{3 e^7 (d+e x)^6}-\frac{3 c \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{7 e^7 (d+e x)^7}+\frac{3 c d \left (a e^2+c d^2\right )^2}{4 e^7 (d+e x)^8}-\frac{\left (a e^2+c d^2\right )^3}{9 e^7 (d+e x)^9}-\frac{c^3}{3 e^7 (d+e x)^3}+\frac{3 c^3 d}{2 e^7 (d+e x)^4} \]

[Out]

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

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Rubi [A]  time = 0.381207, antiderivative size = 190, 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 \[ -\frac{3 c^2 \left (a e^2+5 c d^2\right )}{5 e^7 (d+e x)^5}+\frac{2 c^2 d \left (3 a e^2+5 c d^2\right )}{3 e^7 (d+e x)^6}-\frac{3 c \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{7 e^7 (d+e x)^7}+\frac{3 c d \left (a e^2+c d^2\right )^2}{4 e^7 (d+e x)^8}-\frac{\left (a e^2+c d^2\right )^3}{9 e^7 (d+e x)^9}-\frac{c^3}{3 e^7 (d+e x)^3}+\frac{3 c^3 d}{2 e^7 (d+e x)^4} \]

Antiderivative was successfully verified.

[In]  Int[(a + c*x^2)^3/(d + e*x)^10,x]

[Out]

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

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Rubi in Sympy [A]  time = 56.9993, size = 184, normalized size = 0.97 \[ \frac{3 c^{3} d}{2 e^{7} \left (d + e x\right )^{4}} - \frac{c^{3}}{3 e^{7} \left (d + e x\right )^{3}} + \frac{2 c^{2} d \left (3 a e^{2} + 5 c d^{2}\right )}{3 e^{7} \left (d + e x\right )^{6}} - \frac{3 c^{2} \left (a e^{2} + 5 c d^{2}\right )}{5 e^{7} \left (d + e x\right )^{5}} + \frac{3 c d \left (a e^{2} + c d^{2}\right )^{2}}{4 e^{7} \left (d + e x\right )^{8}} - \frac{3 c \left (a e^{2} + c d^{2}\right ) \left (a e^{2} + 5 c d^{2}\right )}{7 e^{7} \left (d + e x\right )^{7}} - \frac{\left (a e^{2} + c d^{2}\right )^{3}}{9 e^{7} \left (d + e x\right )^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+a)**3/(e*x+d)**10,x)

[Out]

3*c**3*d/(2*e**7*(d + e*x)**4) - c**3/(3*e**7*(d + e*x)**3) + 2*c**2*d*(3*a*e**2
+ 5*c*d**2)/(3*e**7*(d + e*x)**6) - 3*c**2*(a*e**2 + 5*c*d**2)/(5*e**7*(d + e*x
)**5) + 3*c*d*(a*e**2 + c*d**2)**2/(4*e**7*(d + e*x)**8) - 3*c*(a*e**2 + c*d**2)
*(a*e**2 + 5*c*d**2)/(7*e**7*(d + e*x)**7) - (a*e**2 + c*d**2)**3/(9*e**7*(d + e
*x)**9)

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Mathematica [A]  time = 0.0994677, size = 163, normalized size = 0.86 \[ -\frac{140 a^3 e^6+15 a^2 c e^4 \left (d^2+9 d e x+36 e^2 x^2\right )+6 a c^2 e^2 \left (d^4+9 d^3 e x+36 d^2 e^2 x^2+84 d e^3 x^3+126 e^4 x^4\right )+5 c^3 \left (d^6+9 d^5 e x+36 d^4 e^2 x^2+84 d^3 e^3 x^3+126 d^2 e^4 x^4+126 d e^5 x^5+84 e^6 x^6\right )}{1260 e^7 (d+e x)^9} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + c*x^2)^3/(d + e*x)^10,x]

[Out]

-(140*a^3*e^6 + 15*a^2*c*e^4*(d^2 + 9*d*e*x + 36*e^2*x^2) + 6*a*c^2*e^2*(d^4 + 9
*d^3*e*x + 36*d^2*e^2*x^2 + 84*d*e^3*x^3 + 126*e^4*x^4) + 5*c^3*(d^6 + 9*d^5*e*x
+ 36*d^4*e^2*x^2 + 84*d^3*e^3*x^3 + 126*d^2*e^4*x^4 + 126*d*e^5*x^5 + 84*e^6*x^
6))/(1260*e^7*(d + e*x)^9)

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Maple [A]  time = 0.01, size = 218, normalized size = 1.2 \[ -{\frac{3\,c \left ({a}^{2}{e}^{4}+6\,ac{d}^{2}{e}^{2}+5\,{c}^{2}{d}^{4} \right ) }{7\,{e}^{7} \left ( ex+d \right ) ^{7}}}-{\frac{{a}^{3}{e}^{6}+3\,{a}^{2}c{d}^{2}{e}^{4}+3\,{d}^{4}a{c}^{2}{e}^{2}+{c}^{3}{d}^{6}}{9\,{e}^{7} \left ( ex+d \right ) ^{9}}}-{\frac{3\,{c}^{2} \left ( a{e}^{2}+5\,c{d}^{2} \right ) }{5\,{e}^{7} \left ( ex+d \right ) ^{5}}}+{\frac{2\,{c}^{2}d \left ( 3\,a{e}^{2}+5\,c{d}^{2} \right ) }{3\,{e}^{7} \left ( ex+d \right ) ^{6}}}-{\frac{{c}^{3}}{3\,{e}^{7} \left ( ex+d \right ) ^{3}}}+{\frac{3\,{c}^{3}d}{2\,{e}^{7} \left ( ex+d \right ) ^{4}}}+{\frac{3\,cd \left ({a}^{2}{e}^{4}+2\,ac{d}^{2}{e}^{2}+{c}^{2}{d}^{4} \right ) }{4\,{e}^{7} \left ( ex+d \right ) ^{8}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+a)^3/(e*x+d)^10,x)

[Out]

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

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Maxima [A]  time = 0.717239, size = 396, normalized size = 2.08 \[ -\frac{420 \, c^{3} e^{6} x^{6} + 630 \, c^{3} d e^{5} x^{5} + 5 \, c^{3} d^{6} + 6 \, a c^{2} d^{4} e^{2} + 15 \, a^{2} c d^{2} e^{4} + 140 \, a^{3} e^{6} + 126 \,{\left (5 \, c^{3} d^{2} e^{4} + 6 \, a c^{2} e^{6}\right )} x^{4} + 84 \,{\left (5 \, c^{3} d^{3} e^{3} + 6 \, a c^{2} d e^{5}\right )} x^{3} + 36 \,{\left (5 \, c^{3} d^{4} e^{2} + 6 \, a c^{2} d^{2} e^{4} + 15 \, a^{2} c e^{6}\right )} x^{2} + 9 \,{\left (5 \, c^{3} d^{5} e + 6 \, a c^{2} d^{3} e^{3} + 15 \, a^{2} c d e^{5}\right )} x}{1260 \,{\left (e^{16} x^{9} + 9 \, d e^{15} x^{8} + 36 \, d^{2} e^{14} x^{7} + 84 \, d^{3} e^{13} x^{6} + 126 \, d^{4} e^{12} x^{5} + 126 \, d^{5} e^{11} x^{4} + 84 \, d^{6} e^{10} x^{3} + 36 \, d^{7} e^{9} x^{2} + 9 \, d^{8} e^{8} x + d^{9} e^{7}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^3/(e*x + d)^10,x, algorithm="maxima")

[Out]

-1/1260*(420*c^3*e^6*x^6 + 630*c^3*d*e^5*x^5 + 5*c^3*d^6 + 6*a*c^2*d^4*e^2 + 15*
a^2*c*d^2*e^4 + 140*a^3*e^6 + 126*(5*c^3*d^2*e^4 + 6*a*c^2*e^6)*x^4 + 84*(5*c^3*
d^3*e^3 + 6*a*c^2*d*e^5)*x^3 + 36*(5*c^3*d^4*e^2 + 6*a*c^2*d^2*e^4 + 15*a^2*c*e^
6)*x^2 + 9*(5*c^3*d^5*e + 6*a*c^2*d^3*e^3 + 15*a^2*c*d*e^5)*x)/(e^16*x^9 + 9*d*e
^15*x^8 + 36*d^2*e^14*x^7 + 84*d^3*e^13*x^6 + 126*d^4*e^12*x^5 + 126*d^5*e^11*x^
4 + 84*d^6*e^10*x^3 + 36*d^7*e^9*x^2 + 9*d^8*e^8*x + d^9*e^7)

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Fricas [A]  time = 0.222485, size = 396, normalized size = 2.08 \[ -\frac{420 \, c^{3} e^{6} x^{6} + 630 \, c^{3} d e^{5} x^{5} + 5 \, c^{3} d^{6} + 6 \, a c^{2} d^{4} e^{2} + 15 \, a^{2} c d^{2} e^{4} + 140 \, a^{3} e^{6} + 126 \,{\left (5 \, c^{3} d^{2} e^{4} + 6 \, a c^{2} e^{6}\right )} x^{4} + 84 \,{\left (5 \, c^{3} d^{3} e^{3} + 6 \, a c^{2} d e^{5}\right )} x^{3} + 36 \,{\left (5 \, c^{3} d^{4} e^{2} + 6 \, a c^{2} d^{2} e^{4} + 15 \, a^{2} c e^{6}\right )} x^{2} + 9 \,{\left (5 \, c^{3} d^{5} e + 6 \, a c^{2} d^{3} e^{3} + 15 \, a^{2} c d e^{5}\right )} x}{1260 \,{\left (e^{16} x^{9} + 9 \, d e^{15} x^{8} + 36 \, d^{2} e^{14} x^{7} + 84 \, d^{3} e^{13} x^{6} + 126 \, d^{4} e^{12} x^{5} + 126 \, d^{5} e^{11} x^{4} + 84 \, d^{6} e^{10} x^{3} + 36 \, d^{7} e^{9} x^{2} + 9 \, d^{8} e^{8} x + d^{9} e^{7}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^3/(e*x + d)^10,x, algorithm="fricas")

[Out]

-1/1260*(420*c^3*e^6*x^6 + 630*c^3*d*e^5*x^5 + 5*c^3*d^6 + 6*a*c^2*d^4*e^2 + 15*
a^2*c*d^2*e^4 + 140*a^3*e^6 + 126*(5*c^3*d^2*e^4 + 6*a*c^2*e^6)*x^4 + 84*(5*c^3*
d^3*e^3 + 6*a*c^2*d*e^5)*x^3 + 36*(5*c^3*d^4*e^2 + 6*a*c^2*d^2*e^4 + 15*a^2*c*e^
6)*x^2 + 9*(5*c^3*d^5*e + 6*a*c^2*d^3*e^3 + 15*a^2*c*d*e^5)*x)/(e^16*x^9 + 9*d*e
^15*x^8 + 36*d^2*e^14*x^7 + 84*d^3*e^13*x^6 + 126*d^4*e^12*x^5 + 126*d^5*e^11*x^
4 + 84*d^6*e^10*x^3 + 36*d^7*e^9*x^2 + 9*d^8*e^8*x + d^9*e^7)

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Sympy [A]  time = 114.337, size = 308, normalized size = 1.62 \[ - \frac{140 a^{3} e^{6} + 15 a^{2} c d^{2} e^{4} + 6 a c^{2} d^{4} e^{2} + 5 c^{3} d^{6} + 630 c^{3} d e^{5} x^{5} + 420 c^{3} e^{6} x^{6} + x^{4} \left (756 a c^{2} e^{6} + 630 c^{3} d^{2} e^{4}\right ) + x^{3} \left (504 a c^{2} d e^{5} + 420 c^{3} d^{3} e^{3}\right ) + x^{2} \left (540 a^{2} c e^{6} + 216 a c^{2} d^{2} e^{4} + 180 c^{3} d^{4} e^{2}\right ) + x \left (135 a^{2} c d e^{5} + 54 a c^{2} d^{3} e^{3} + 45 c^{3} d^{5} e\right )}{1260 d^{9} e^{7} + 11340 d^{8} e^{8} x + 45360 d^{7} e^{9} x^{2} + 105840 d^{6} e^{10} x^{3} + 158760 d^{5} e^{11} x^{4} + 158760 d^{4} e^{12} x^{5} + 105840 d^{3} e^{13} x^{6} + 45360 d^{2} e^{14} x^{7} + 11340 d e^{15} x^{8} + 1260 e^{16} x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+a)**3/(e*x+d)**10,x)

[Out]

-(140*a**3*e**6 + 15*a**2*c*d**2*e**4 + 6*a*c**2*d**4*e**2 + 5*c**3*d**6 + 630*c
**3*d*e**5*x**5 + 420*c**3*e**6*x**6 + x**4*(756*a*c**2*e**6 + 630*c**3*d**2*e**
4) + x**3*(504*a*c**2*d*e**5 + 420*c**3*d**3*e**3) + x**2*(540*a**2*c*e**6 + 216
*a*c**2*d**2*e**4 + 180*c**3*d**4*e**2) + x*(135*a**2*c*d*e**5 + 54*a*c**2*d**3*
e**3 + 45*c**3*d**5*e))/(1260*d**9*e**7 + 11340*d**8*e**8*x + 45360*d**7*e**9*x*
*2 + 105840*d**6*e**10*x**3 + 158760*d**5*e**11*x**4 + 158760*d**4*e**12*x**5 +
105840*d**3*e**13*x**6 + 45360*d**2*e**14*x**7 + 11340*d*e**15*x**8 + 1260*e**16
*x**9)

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GIAC/XCAS [A]  time = 0.21135, size = 258, normalized size = 1.36 \[ -\frac{{\left (420 \, c^{3} x^{6} e^{6} + 630 \, c^{3} d x^{5} e^{5} + 630 \, c^{3} d^{2} x^{4} e^{4} + 420 \, c^{3} d^{3} x^{3} e^{3} + 180 \, c^{3} d^{4} x^{2} e^{2} + 45 \, c^{3} d^{5} x e + 5 \, c^{3} d^{6} + 756 \, a c^{2} x^{4} e^{6} + 504 \, a c^{2} d x^{3} e^{5} + 216 \, a c^{2} d^{2} x^{2} e^{4} + 54 \, a c^{2} d^{3} x e^{3} + 6 \, a c^{2} d^{4} e^{2} + 540 \, a^{2} c x^{2} e^{6} + 135 \, a^{2} c d x e^{5} + 15 \, a^{2} c d^{2} e^{4} + 140 \, a^{3} e^{6}\right )} e^{\left (-7\right )}}{1260 \,{\left (x e + d\right )}^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^3/(e*x + d)^10,x, algorithm="giac")

[Out]

-1/1260*(420*c^3*x^6*e^6 + 630*c^3*d*x^5*e^5 + 630*c^3*d^2*x^4*e^4 + 420*c^3*d^3
*x^3*e^3 + 180*c^3*d^4*x^2*e^2 + 45*c^3*d^5*x*e + 5*c^3*d^6 + 756*a*c^2*x^4*e^6
+ 504*a*c^2*d*x^3*e^5 + 216*a*c^2*d^2*x^2*e^4 + 54*a*c^2*d^3*x*e^3 + 6*a*c^2*d^4
*e^2 + 540*a^2*c*x^2*e^6 + 135*a^2*c*d*x*e^5 + 15*a^2*c*d^2*e^4 + 140*a^3*e^6)*e
^(-7)/(x*e + d)^9