3.482 \(\int \frac{\left (a+c x^2\right )^3}{(d+e x)^7} \, dx\)

Optimal. Leaf size=184 \[ -\frac{3 c^2 \left (a e^2+5 c d^2\right )}{2 e^7 (d+e x)^2}+\frac{4 c^2 d \left (3 a e^2+5 c d^2\right )}{3 e^7 (d+e x)^3}-\frac{3 c \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{4 e^7 (d+e x)^4}+\frac{6 c d \left (a e^2+c d^2\right )^2}{5 e^7 (d+e x)^5}-\frac{\left (a e^2+c d^2\right )^3}{6 e^7 (d+e x)^6}+\frac{6 c^3 d}{e^7 (d+e x)}+\frac{c^3 \log (d+e x)}{e^7} \]

[Out]

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

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Rubi [A]  time = 0.38374, antiderivative size = 184, 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 \[ -\frac{3 c^2 \left (a e^2+5 c d^2\right )}{2 e^7 (d+e x)^2}+\frac{4 c^2 d \left (3 a e^2+5 c d^2\right )}{3 e^7 (d+e x)^3}-\frac{3 c \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{4 e^7 (d+e x)^4}+\frac{6 c d \left (a e^2+c d^2\right )^2}{5 e^7 (d+e x)^5}-\frac{\left (a e^2+c d^2\right )^3}{6 e^7 (d+e x)^6}+\frac{6 c^3 d}{e^7 (d+e x)}+\frac{c^3 \log (d+e x)}{e^7} \]

Antiderivative was successfully verified.

[In]  Int[(a + c*x^2)^3/(d + e*x)^7,x]

[Out]

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

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Rubi in Sympy [A]  time = 53.9677, size = 178, normalized size = 0.97 \[ \frac{6 c^{3} d}{e^{7} \left (d + e x\right )} + \frac{c^{3} \log{\left (d + e x \right )}}{e^{7}} + \frac{4 c^{2} d \left (3 a e^{2} + 5 c d^{2}\right )}{3 e^{7} \left (d + e x\right )^{3}} - \frac{3 c^{2} \left (a e^{2} + 5 c d^{2}\right )}{2 e^{7} \left (d + e x\right )^{2}} + \frac{6 c d \left (a e^{2} + c d^{2}\right )^{2}}{5 e^{7} \left (d + e x\right )^{5}} - \frac{3 c \left (a e^{2} + c d^{2}\right ) \left (a e^{2} + 5 c d^{2}\right )}{4 e^{7} \left (d + e x\right )^{4}} - \frac{\left (a e^{2} + c d^{2}\right )^{3}}{6 e^{7} \left (d + e x\right )^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+a)**3/(e*x+d)**7,x)

[Out]

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

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Mathematica [A]  time = 0.119445, size = 172, normalized size = 0.93 \[ \frac{-10 a^3 e^6-3 a^2 c e^4 \left (d^2+6 d e x+15 e^2 x^2\right )-6 a c^2 e^2 \left (d^4+6 d^3 e x+15 d^2 e^2 x^2+20 d e^3 x^3+15 e^4 x^4\right )+c^3 d \left (147 d^5+822 d^4 e x+1875 d^3 e^2 x^2+2200 d^2 e^3 x^3+1350 d e^4 x^4+360 e^5 x^5\right )+60 c^3 (d+e x)^6 \log (d+e x)}{60 e^7 (d+e x)^6} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + c*x^2)^3/(d + e*x)^7,x]

[Out]

(-10*a^3*e^6 - 3*a^2*c*e^4*(d^2 + 6*d*e*x + 15*e^2*x^2) - 6*a*c^2*e^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) + c^3*d*(147*d^5 + 822*d^4*
e*x + 1875*d^3*e^2*x^2 + 2200*d^2*e^3*x^3 + 1350*d*e^4*x^4 + 360*e^5*x^5) + 60*c
^3*(d + e*x)^6*Log[d + e*x])/(60*e^7*(d + e*x)^6)

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Maple [A]  time = 0.012, size = 278, normalized size = 1.5 \[{\frac{6\,{a}^{2}cd}{5\,{e}^{3} \left ( ex+d \right ) ^{5}}}+{\frac{12\,{d}^{3}a{c}^{2}}{5\,{e}^{5} \left ( ex+d \right ) ^{5}}}+{\frac{6\,{c}^{3}{d}^{5}}{5\,{e}^{7} \left ( ex+d \right ) ^{5}}}-{\frac{{a}^{3}}{6\,e \left ( ex+d \right ) ^{6}}}-{\frac{{a}^{2}c{d}^{2}}{2\,{e}^{3} \left ( ex+d \right ) ^{6}}}-{\frac{{d}^{4}a{c}^{2}}{2\,{e}^{5} \left ( ex+d \right ) ^{6}}}-{\frac{{c}^{3}{d}^{6}}{6\,{e}^{7} \left ( ex+d \right ) ^{6}}}-{\frac{3\,a{c}^{2}}{2\,{e}^{5} \left ( ex+d \right ) ^{2}}}-{\frac{15\,{d}^{2}{c}^{3}}{2\,{e}^{7} \left ( ex+d \right ) ^{2}}}+{\frac{{c}^{3}\ln \left ( ex+d \right ) }{{e}^{7}}}+4\,{\frac{a{c}^{2}d}{{e}^{5} \left ( ex+d \right ) ^{3}}}+{\frac{20\,{c}^{3}{d}^{3}}{3\,{e}^{7} \left ( ex+d \right ) ^{3}}}+6\,{\frac{{c}^{3}d}{{e}^{7} \left ( ex+d \right ) }}-{\frac{3\,{a}^{2}c}{4\,{e}^{3} \left ( ex+d \right ) ^{4}}}-{\frac{9\,{d}^{2}a{c}^{2}}{2\,{e}^{5} \left ( ex+d \right ) ^{4}}}-{\frac{15\,{d}^{4}{c}^{3}}{4\,{e}^{7} \left ( ex+d \right ) ^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+a)^3/(e*x+d)^7,x)

[Out]

6/5*c*d/e^3/(e*x+d)^5*a^2+12/5*c^2*d^3/e^5/(e*x+d)^5*a+6/5*d^5/e^7/(e*x+d)^5*c^3
-1/6/e/(e*x+d)^6*a^3-1/2/e^3/(e*x+d)^6*a^2*c*d^2-1/2/e^5/(e*x+d)^6*d^4*a*c^2-1/6
*d^6/e^7/(e*x+d)^6*c^3-3/2*c^2/e^5/(e*x+d)^2*a-15/2*c^3/e^7/(e*x+d)^2*d^2+c^3*ln
(e*x+d)/e^7+4*c^2*d/e^5/(e*x+d)^3*a+20/3/e^7/(e*x+d)^3*c^3*d^3+6*c^3*d/e^7/(e*x+
d)-3/4*c/e^3/(e*x+d)^4*a^2-9/2*c^2/e^5/(e*x+d)^4*a*d^2-15/4*d^4/e^7/(e*x+d)^4*c^
3

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Maxima [A]  time = 0.708123, size = 355, normalized size = 1.93 \[ \frac{360 \, c^{3} d e^{5} x^{5} + 147 \, c^{3} d^{6} - 6 \, a c^{2} d^{4} e^{2} - 3 \, a^{2} c d^{2} e^{4} - 10 \, a^{3} e^{6} + 90 \,{\left (15 \, c^{3} d^{2} e^{4} - a c^{2} e^{6}\right )} x^{4} + 40 \,{\left (55 \, c^{3} d^{3} e^{3} - 3 \, a c^{2} d e^{5}\right )} x^{3} + 15 \,{\left (125 \, c^{3} d^{4} e^{2} - 6 \, a c^{2} d^{2} e^{4} - 3 \, a^{2} c e^{6}\right )} x^{2} + 6 \,{\left (137 \, c^{3} d^{5} e - 6 \, a c^{2} d^{3} e^{3} - 3 \, a^{2} c d e^{5}\right )} x}{60 \,{\left (e^{13} x^{6} + 6 \, d e^{12} x^{5} + 15 \, d^{2} e^{11} x^{4} + 20 \, d^{3} e^{10} x^{3} + 15 \, d^{4} e^{9} x^{2} + 6 \, d^{5} e^{8} x + d^{6} e^{7}\right )}} + \frac{c^{3} \log \left (e x + d\right )}{e^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^3/(e*x + d)^7,x, algorithm="maxima")

[Out]

1/60*(360*c^3*d*e^5*x^5 + 147*c^3*d^6 - 6*a*c^2*d^4*e^2 - 3*a^2*c*d^2*e^4 - 10*a
^3*e^6 + 90*(15*c^3*d^2*e^4 - a*c^2*e^6)*x^4 + 40*(55*c^3*d^3*e^3 - 3*a*c^2*d*e^
5)*x^3 + 15*(125*c^3*d^4*e^2 - 6*a*c^2*d^2*e^4 - 3*a^2*c*e^6)*x^2 + 6*(137*c^3*d
^5*e - 6*a*c^2*d^3*e^3 - 3*a^2*c*d*e^5)*x)/(e^13*x^6 + 6*d*e^12*x^5 + 15*d^2*e^1
1*x^4 + 20*d^3*e^10*x^3 + 15*d^4*e^9*x^2 + 6*d^5*e^8*x + d^6*e^7) + c^3*log(e*x
+ d)/e^7

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Fricas [A]  time = 0.203591, size = 458, normalized size = 2.49 \[ \frac{360 \, c^{3} d e^{5} x^{5} + 147 \, c^{3} d^{6} - 6 \, a c^{2} d^{4} e^{2} - 3 \, a^{2} c d^{2} e^{4} - 10 \, a^{3} e^{6} + 90 \,{\left (15 \, c^{3} d^{2} e^{4} - a c^{2} e^{6}\right )} x^{4} + 40 \,{\left (55 \, c^{3} d^{3} e^{3} - 3 \, a c^{2} d e^{5}\right )} x^{3} + 15 \,{\left (125 \, c^{3} d^{4} e^{2} - 6 \, a c^{2} d^{2} e^{4} - 3 \, a^{2} c e^{6}\right )} x^{2} + 6 \,{\left (137 \, c^{3} d^{5} e - 6 \, a c^{2} d^{3} e^{3} - 3 \, a^{2} c d e^{5}\right )} x + 60 \,{\left (c^{3} e^{6} x^{6} + 6 \, c^{3} d e^{5} x^{5} + 15 \, c^{3} d^{2} e^{4} x^{4} + 20 \, c^{3} d^{3} e^{3} x^{3} + 15 \, c^{3} d^{4} e^{2} x^{2} + 6 \, c^{3} d^{5} e x + c^{3} d^{6}\right )} \log \left (e x + d\right )}{60 \,{\left (e^{13} x^{6} + 6 \, d e^{12} x^{5} + 15 \, d^{2} e^{11} x^{4} + 20 \, d^{3} e^{10} x^{3} + 15 \, d^{4} e^{9} x^{2} + 6 \, d^{5} e^{8} x + d^{6} e^{7}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^3/(e*x + d)^7,x, algorithm="fricas")

[Out]

1/60*(360*c^3*d*e^5*x^5 + 147*c^3*d^6 - 6*a*c^2*d^4*e^2 - 3*a^2*c*d^2*e^4 - 10*a
^3*e^6 + 90*(15*c^3*d^2*e^4 - a*c^2*e^6)*x^4 + 40*(55*c^3*d^3*e^3 - 3*a*c^2*d*e^
5)*x^3 + 15*(125*c^3*d^4*e^2 - 6*a*c^2*d^2*e^4 - 3*a^2*c*e^6)*x^2 + 6*(137*c^3*d
^5*e - 6*a*c^2*d^3*e^3 - 3*a^2*c*d*e^5)*x + 60*(c^3*e^6*x^6 + 6*c^3*d*e^5*x^5 +
15*c^3*d^2*e^4*x^4 + 20*c^3*d^3*e^3*x^3 + 15*c^3*d^4*e^2*x^2 + 6*c^3*d^5*e*x + c
^3*d^6)*log(e*x + d))/(e^13*x^6 + 6*d*e^12*x^5 + 15*d^2*e^11*x^4 + 20*d^3*e^10*x
^3 + 15*d^4*e^9*x^2 + 6*d^5*e^8*x + d^6*e^7)

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Sympy [A]  time = 29.4917, size = 272, normalized size = 1.48 \[ \frac{c^{3} \log{\left (d + e x \right )}}{e^{7}} + \frac{- 10 a^{3} e^{6} - 3 a^{2} c d^{2} e^{4} - 6 a c^{2} d^{4} e^{2} + 147 c^{3} d^{6} + 360 c^{3} d e^{5} x^{5} + x^{4} \left (- 90 a c^{2} e^{6} + 1350 c^{3} d^{2} e^{4}\right ) + x^{3} \left (- 120 a c^{2} d e^{5} + 2200 c^{3} d^{3} e^{3}\right ) + x^{2} \left (- 45 a^{2} c e^{6} - 90 a c^{2} d^{2} e^{4} + 1875 c^{3} d^{4} e^{2}\right ) + x \left (- 18 a^{2} c d e^{5} - 36 a c^{2} d^{3} e^{3} + 822 c^{3} d^{5} e\right )}{60 d^{6} e^{7} + 360 d^{5} e^{8} x + 900 d^{4} e^{9} x^{2} + 1200 d^{3} e^{10} x^{3} + 900 d^{2} e^{11} x^{4} + 360 d e^{12} x^{5} + 60 e^{13} x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+a)**3/(e*x+d)**7,x)

[Out]

c**3*log(d + e*x)/e**7 + (-10*a**3*e**6 - 3*a**2*c*d**2*e**4 - 6*a*c**2*d**4*e**
2 + 147*c**3*d**6 + 360*c**3*d*e**5*x**5 + x**4*(-90*a*c**2*e**6 + 1350*c**3*d**
2*e**4) + x**3*(-120*a*c**2*d*e**5 + 2200*c**3*d**3*e**3) + x**2*(-45*a**2*c*e**
6 - 90*a*c**2*d**2*e**4 + 1875*c**3*d**4*e**2) + x*(-18*a**2*c*d*e**5 - 36*a*c**
2*d**3*e**3 + 822*c**3*d**5*e))/(60*d**6*e**7 + 360*d**5*e**8*x + 900*d**4*e**9*
x**2 + 1200*d**3*e**10*x**3 + 900*d**2*e**11*x**4 + 360*d*e**12*x**5 + 60*e**13*
x**6)

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GIAC/XCAS [A]  time = 0.211963, size = 265, normalized size = 1.44 \[ c^{3} e^{\left (-7\right )}{\rm ln}\left ({\left | x e + d \right |}\right ) + \frac{{\left (360 \, c^{3} d x^{5} e^{4} + 90 \,{\left (15 \, c^{3} d^{2} e^{3} - a c^{2} e^{5}\right )} x^{4} + 40 \,{\left (55 \, c^{3} d^{3} e^{2} - 3 \, a c^{2} d e^{4}\right )} x^{3} + 15 \,{\left (125 \, c^{3} d^{4} e - 6 \, a c^{2} d^{2} e^{3} - 3 \, a^{2} c e^{5}\right )} x^{2} + 6 \,{\left (137 \, c^{3} d^{5} - 6 \, a c^{2} d^{3} e^{2} - 3 \, a^{2} c d e^{4}\right )} x +{\left (147 \, c^{3} d^{6} - 6 \, a c^{2} d^{4} e^{2} - 3 \, a^{2} c d^{2} e^{4} - 10 \, a^{3} e^{6}\right )} e^{\left (-1\right )}\right )} e^{\left (-6\right )}}{60 \,{\left (x e + d\right )}^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^3/(e*x + d)^7,x, algorithm="giac")

[Out]

c^3*e^(-7)*ln(abs(x*e + d)) + 1/60*(360*c^3*d*x^5*e^4 + 90*(15*c^3*d^2*e^3 - a*c
^2*e^5)*x^4 + 40*(55*c^3*d^3*e^2 - 3*a*c^2*d*e^4)*x^3 + 15*(125*c^3*d^4*e - 6*a*
c^2*d^2*e^3 - 3*a^2*c*e^5)*x^2 + 6*(137*c^3*d^5 - 6*a*c^2*d^3*e^2 - 3*a^2*c*d*e^
4)*x + (147*c^3*d^6 - 6*a*c^2*d^4*e^2 - 3*a^2*c*d^2*e^4 - 10*a^3*e^6)*e^(-1))*e^
(-6)/(x*e + d)^6