3.1308 \(\int \frac{(A+B x) \left (a+c x^2\right )^2}{(d+e x)^6} \, dx\)

Optimal. Leaf size=197 \[ -\frac{c \left (a B e^2-2 A c d e+5 B c d^2\right )}{e^6 (d+e x)^2}-\frac{\left (a e^2+c d^2\right ) \left (a B e^2-4 A c d e+5 B c d^2\right )}{4 e^6 (d+e x)^4}+\frac{\left (a e^2+c d^2\right )^2 (B d-A e)}{5 e^6 (d+e x)^5}+\frac{2 c \left (-a A e^3+3 a B d e^2-3 A c d^2 e+5 B c d^3\right )}{3 e^6 (d+e x)^3}+\frac{c^2 (5 B d-A e)}{e^6 (d+e x)}+\frac{B c^2 \log (d+e x)}{e^6} \]

[Out]

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

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Rubi [A]  time = 0.405927, antiderivative size = 197, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045 \[ -\frac{c \left (a B e^2-2 A c d e+5 B c d^2\right )}{e^6 (d+e x)^2}-\frac{\left (a e^2+c d^2\right ) \left (a B e^2-4 A c d e+5 B c d^2\right )}{4 e^6 (d+e x)^4}+\frac{\left (a e^2+c d^2\right )^2 (B d-A e)}{5 e^6 (d+e x)^5}+\frac{2 c \left (-a A e^3+3 a B d e^2-3 A c d^2 e+5 B c d^3\right )}{3 e^6 (d+e x)^3}+\frac{c^2 (5 B d-A e)}{e^6 (d+e x)}+\frac{B c^2 \log (d+e x)}{e^6} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(a + c*x^2)^2)/(d + e*x)^6,x]

[Out]

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

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Rubi in Sympy [A]  time = 66.8843, size = 197, normalized size = 1. \[ \frac{B c^{2} \log{\left (d + e x \right )}}{e^{6}} - \frac{c^{2} \left (A e - 5 B d\right )}{e^{6} \left (d + e x\right )} - \frac{c \left (- 2 A c d e + B a e^{2} + 5 B c d^{2}\right )}{e^{6} \left (d + e x\right )^{2}} - \frac{2 c \left (A a e^{3} + 3 A c d^{2} e - 3 B a d e^{2} - 5 B c d^{3}\right )}{3 e^{6} \left (d + e x\right )^{3}} - \frac{\left (a e^{2} + c d^{2}\right ) \left (- 4 A c d e + B a e^{2} + 5 B c d^{2}\right )}{4 e^{6} \left (d + e x\right )^{4}} - \frac{\left (A e - B d\right ) \left (a e^{2} + c d^{2}\right )^{2}}{5 e^{6} \left (d + e x\right )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(c*x**2+a)**2/(e*x+d)**6,x)

[Out]

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

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Mathematica [A]  time = 0.196909, size = 212, normalized size = 1.08 \[ \frac{-4 A e \left (3 a^2 e^4+a c e^2 \left (d^2+5 d e x+10 e^2 x^2\right )+3 c^2 \left (d^4+5 d^3 e x+10 d^2 e^2 x^2+10 d e^3 x^3+5 e^4 x^4\right )\right )+B \left (-3 a^2 e^4 (d+5 e x)-6 a c e^2 \left (d^3+5 d^2 e x+10 d e^2 x^2+10 e^3 x^3\right )+c^2 d \left (137 d^4+625 d^3 e x+1100 d^2 e^2 x^2+900 d e^3 x^3+300 e^4 x^4\right )\right )+60 B c^2 (d+e x)^5 \log (d+e x)}{60 e^6 (d+e x)^5} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(a + c*x^2)^2)/(d + e*x)^6,x]

[Out]

(-4*A*e*(3*a^2*e^4 + a*c*e^2*(d^2 + 5*d*e*x + 10*e^2*x^2) + 3*c^2*(d^4 + 5*d^3*e
*x + 10*d^2*e^2*x^2 + 10*d*e^3*x^3 + 5*e^4*x^4)) + B*(-3*a^2*e^4*(d + 5*e*x) - 6
*a*c*e^2*(d^3 + 5*d^2*e*x + 10*d*e^2*x^2 + 10*e^3*x^3) + c^2*d*(137*d^4 + 625*d^
3*e*x + 1100*d^2*e^2*x^2 + 900*d*e^3*x^3 + 300*e^4*x^4)) + 60*B*c^2*(d + e*x)^5*
Log[d + e*x])/(60*e^6*(d + e*x)^5)

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Maple [A]  time = 0.012, size = 362, normalized size = 1.8 \[{\frac{Adac}{{e}^{3} \left ( ex+d \right ) ^{4}}}+{\frac{A{d}^{3}{c}^{2}}{{e}^{5} \left ( ex+d \right ) ^{4}}}-{\frac{{a}^{2}B}{4\,{e}^{2} \left ( ex+d \right ) ^{4}}}-{\frac{3\,aBc{d}^{2}}{2\,{e}^{4} \left ( ex+d \right ) ^{4}}}-{\frac{5\,B{c}^{2}{d}^{4}}{4\,{e}^{6} \left ( ex+d \right ) ^{4}}}-{\frac{2\,aAc}{3\,{e}^{3} \left ( ex+d \right ) ^{3}}}-2\,{\frac{A{c}^{2}{d}^{2}}{{e}^{5} \left ( ex+d \right ) ^{3}}}+2\,{\frac{aBcd}{{e}^{4} \left ( ex+d \right ) ^{3}}}+{\frac{10\,B{c}^{2}{d}^{3}}{3\,{e}^{6} \left ( ex+d \right ) ^{3}}}+2\,{\frac{A{c}^{2}d}{{e}^{5} \left ( ex+d \right ) ^{2}}}-{\frac{aBc}{{e}^{4} \left ( ex+d \right ) ^{2}}}-5\,{\frac{B{c}^{2}{d}^{2}}{{e}^{6} \left ( ex+d \right ) ^{2}}}+{\frac{B{c}^{2}\ln \left ( ex+d \right ) }{{e}^{6}}}-{\frac{A{a}^{2}}{5\,e \left ( ex+d \right ) ^{5}}}-{\frac{2\,A{d}^{2}ac}{5\,{e}^{3} \left ( ex+d \right ) ^{5}}}-{\frac{A{d}^{4}{c}^{2}}{5\,{e}^{5} \left ( ex+d \right ) ^{5}}}+{\frac{Bd{a}^{2}}{5\,{e}^{2} \left ( ex+d \right ) ^{5}}}+{\frac{2\,aBc{d}^{3}}{5\,{e}^{4} \left ( ex+d \right ) ^{5}}}+{\frac{B{c}^{2}{d}^{5}}{5\,{e}^{6} \left ( ex+d \right ) ^{5}}}-{\frac{A{c}^{2}}{{e}^{5} \left ( ex+d \right ) }}+5\,{\frac{B{c}^{2}d}{{e}^{6} \left ( ex+d \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+a)^2/(e*x+d)^6,x)

[Out]

1/e^3/(e*x+d)^4*A*d*a*c+d^3/e^5/(e*x+d)^4*A*c^2-1/4/e^2/(e*x+d)^4*B*a^2-3/2/e^4/
(e*x+d)^4*a*B*c*d^2-5/4*d^4/e^6/(e*x+d)^4*B*c^2-2/3*c/e^3/(e*x+d)^3*a*A-2/e^5/(e
*x+d)^3*A*c^2*d^2+2*c/e^4/(e*x+d)^3*a*B*d+10/3/e^6/(e*x+d)^3*B*c^2*d^3+2/e^5/(e*
x+d)^2*A*c^2*d-c/e^4/(e*x+d)^2*a*B-5/e^6/(e*x+d)^2*B*c^2*d^2+B*c^2*ln(e*x+d)/e^6
-1/5/e/(e*x+d)^5*A*a^2-2/5/e^3/(e*x+d)^5*A*d^2*a*c-1/5*d^4/e^5/(e*x+d)^5*A*c^2+1
/5/e^2/(e*x+d)^5*B*d*a^2+2/5/e^4/(e*x+d)^5*a*B*c*d^3+1/5*d^5/e^6/(e*x+d)^5*B*c^2
-c^2/e^5/(e*x+d)*A+5*c^2/e^6/(e*x+d)*B*d

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Maxima [A]  time = 0.708965, size = 402, normalized size = 2.04 \[ \frac{137 \, B c^{2} d^{5} - 12 \, A c^{2} d^{4} e - 6 \, B a c d^{3} e^{2} - 4 \, A a c d^{2} e^{3} - 3 \, B a^{2} d e^{4} - 12 \, A a^{2} e^{5} + 60 \,{\left (5 \, B c^{2} d e^{4} - A c^{2} e^{5}\right )} x^{4} + 60 \,{\left (15 \, B c^{2} d^{2} e^{3} - 2 \, A c^{2} d e^{4} - B a c e^{5}\right )} x^{3} + 20 \,{\left (55 \, B c^{2} d^{3} e^{2} - 6 \, A c^{2} d^{2} e^{3} - 3 \, B a c d e^{4} - 2 \, A a c e^{5}\right )} x^{2} + 5 \,{\left (125 \, B c^{2} d^{4} e - 12 \, A c^{2} d^{3} e^{2} - 6 \, B a c d^{2} e^{3} - 4 \, A a c d e^{4} - 3 \, B a^{2} e^{5}\right )} x}{60 \,{\left (e^{11} x^{5} + 5 \, d e^{10} x^{4} + 10 \, d^{2} e^{9} x^{3} + 10 \, d^{3} e^{8} x^{2} + 5 \, d^{4} e^{7} x + d^{5} e^{6}\right )}} + \frac{B c^{2} \log \left (e x + d\right )}{e^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^2*(B*x + A)/(e*x + d)^6,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.270291, size = 493, normalized size = 2.5 \[ \frac{137 \, B c^{2} d^{5} - 12 \, A c^{2} d^{4} e - 6 \, B a c d^{3} e^{2} - 4 \, A a c d^{2} e^{3} - 3 \, B a^{2} d e^{4} - 12 \, A a^{2} e^{5} + 60 \,{\left (5 \, B c^{2} d e^{4} - A c^{2} e^{5}\right )} x^{4} + 60 \,{\left (15 \, B c^{2} d^{2} e^{3} - 2 \, A c^{2} d e^{4} - B a c e^{5}\right )} x^{3} + 20 \,{\left (55 \, B c^{2} d^{3} e^{2} - 6 \, A c^{2} d^{2} e^{3} - 3 \, B a c d e^{4} - 2 \, A a c e^{5}\right )} x^{2} + 5 \,{\left (125 \, B c^{2} d^{4} e - 12 \, A c^{2} d^{3} e^{2} - 6 \, B a c d^{2} e^{3} - 4 \, A a c d e^{4} - 3 \, B a^{2} e^{5}\right )} x + 60 \,{\left (B c^{2} e^{5} x^{5} + 5 \, B c^{2} d e^{4} x^{4} + 10 \, B c^{2} d^{2} e^{3} x^{3} + 10 \, B c^{2} d^{3} e^{2} x^{2} + 5 \, B c^{2} d^{4} e x + B c^{2} d^{5}\right )} \log \left (e x + d\right )}{60 \,{\left (e^{11} x^{5} + 5 \, d e^{10} x^{4} + 10 \, d^{2} e^{9} x^{3} + 10 \, d^{3} e^{8} x^{2} + 5 \, d^{4} e^{7} x + d^{5} e^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^2*(B*x + A)/(e*x + d)^6,x, algorithm="fricas")

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(c*x**2+a)**2/(e*x+d)**6,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.280244, size = 323, normalized size = 1.64 \[ B c^{2} e^{\left (-6\right )}{\rm ln}\left ({\left | x e + d \right |}\right ) + \frac{{\left (60 \,{\left (5 \, B c^{2} d e^{3} - A c^{2} e^{4}\right )} x^{4} + 60 \,{\left (15 \, B c^{2} d^{2} e^{2} - 2 \, A c^{2} d e^{3} - B a c e^{4}\right )} x^{3} + 20 \,{\left (55 \, B c^{2} d^{3} e - 6 \, A c^{2} d^{2} e^{2} - 3 \, B a c d e^{3} - 2 \, A a c e^{4}\right )} x^{2} + 5 \,{\left (125 \, B c^{2} d^{4} - 12 \, A c^{2} d^{3} e - 6 \, B a c d^{2} e^{2} - 4 \, A a c d e^{3} - 3 \, B a^{2} e^{4}\right )} x +{\left (137 \, B c^{2} d^{5} - 12 \, A c^{2} d^{4} e - 6 \, B a c d^{3} e^{2} - 4 \, A a c d^{2} e^{3} - 3 \, B a^{2} d e^{4} - 12 \, A a^{2} e^{5}\right )} e^{\left (-1\right )}\right )} e^{\left (-5\right )}}{60 \,{\left (x e + d\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^2*(B*x + A)/(e*x + d)^6,x, algorithm="giac")

[Out]

B*c^2*e^(-6)*ln(abs(x*e + d)) + 1/60*(60*(5*B*c^2*d*e^3 - A*c^2*e^4)*x^4 + 60*(1
5*B*c^2*d^2*e^2 - 2*A*c^2*d*e^3 - B*a*c*e^4)*x^3 + 20*(55*B*c^2*d^3*e - 6*A*c^2*
d^2*e^2 - 3*B*a*c*d*e^3 - 2*A*a*c*e^4)*x^2 + 5*(125*B*c^2*d^4 - 12*A*c^2*d^3*e -
6*B*a*c*d^2*e^2 - 4*A*a*c*d*e^3 - 3*B*a^2*e^4)*x + (137*B*c^2*d^5 - 12*A*c^2*d^
4*e - 6*B*a*c*d^3*e^2 - 4*A*a*c*d^2*e^3 - 3*B*a^2*d*e^4 - 12*A*a^2*e^5)*e^(-1))*
e^(-5)/(x*e + d)^5