3.2329 \(\int \frac{(A+B x) \left (a+b x+c x^2\right )^2}{(d+e x)^7} \, dx\)

Optimal. Leaf size=302 \[ \frac{2 A c e (2 c d-b e)-B \left (-2 c e (4 b d-a e)+b^2 e^2+10 c^2 d^2\right )}{3 e^6 (d+e x)^3}+\frac{B \left (-6 c d e (2 b d-a e)+b e^2 (3 b d-2 a e)+10 c^2 d^3\right )-A e \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{4 e^6 (d+e x)^4}+\frac{\left (a e^2-b d e+c d^2\right ) \left (2 A e (2 c d-b e)-B \left (5 c d^2-e (3 b d-a e)\right )\right )}{5 e^6 (d+e x)^5}+\frac{(B d-A e) \left (a e^2-b d e+c d^2\right )^2}{6 e^6 (d+e x)^6}+\frac{c (-A c e-2 b B e+5 B c d)}{2 e^6 (d+e x)^2}-\frac{B c^2}{e^6 (d+e x)} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 1.05292, antiderivative size = 300, normalized size of antiderivative = 0.99, number of steps used = 2, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04 \[ \frac{2 A c e (2 c d-b e)-B \left (-2 c e (4 b d-a e)+b^2 e^2+10 c^2 d^2\right )}{3 e^6 (d+e x)^3}+\frac{B \left (-6 c d e (2 b d-a e)+b e^2 (3 b d-2 a e)+10 c^2 d^3\right )-A e \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{4 e^6 (d+e x)^4}-\frac{\left (a e^2-b d e+c d^2\right ) \left (-B e (3 b d-a e)-2 A e (2 c d-b e)+5 B c d^2\right )}{5 e^6 (d+e x)^5}+\frac{(B d-A e) \left (a e^2-b d e+c d^2\right )^2}{6 e^6 (d+e x)^6}+\frac{c (-A c e-2 b B e+5 B c d)}{2 e^6 (d+e x)^2}-\frac{B c^2}{e^6 (d+e x)} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

_______________________________________________________________________________________

Mathematica [A]  time = 0.596757, size = 372, normalized size = 1.23 \[ -\frac{A e \left (e^2 \left (10 a^2 e^2+4 a b e (d+6 e x)+b^2 \left (d^2+6 d e x+15 e^2 x^2\right )\right )+2 c e \left (a e \left (d^2+6 d e x+15 e^2 x^2\right )+b \left (d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right )\right )+2 c^2 \left (d^4+6 d^3 e x+15 d^2 e^2 x^2+20 d e^3 x^3+15 e^4 x^4\right )\right )+B \left (e^2 \left (2 a^2 e^2 (d+6 e x)+2 a b e \left (d^2+6 d e x+15 e^2 x^2\right )+b^2 \left (d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right )\right )+2 c e \left (a e \left (d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right )+2 b \left (d^4+6 d^3 e x+15 d^2 e^2 x^2+20 d e^3 x^3+15 e^4 x^4\right )\right )+10 c^2 \left (d^5+6 d^4 e x+15 d^3 e^2 x^2+20 d^2 e^3 x^3+15 d e^4 x^4+6 e^5 x^5\right )\right )}{60 e^6 (d+e x)^6} \]

Antiderivative was successfully verified.

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

[Out]

-(A*e*(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) + e^
2*(10*a^2*e^2 + 4*a*b*e*(d + 6*e*x) + b^2*(d^2 + 6*d*e*x + 15*e^2*x^2)) + 2*c*e*
(a*e*(d^2 + 6*d*e*x + 15*e^2*x^2) + b*(d^3 + 6*d^2*e*x + 15*d*e^2*x^2 + 20*e^3*x
^3))) + B*(10*c^2*(d^5 + 6*d^4*e*x + 15*d^3*e^2*x^2 + 20*d^2*e^3*x^3 + 15*d*e^4*
x^4 + 6*e^5*x^5) + e^2*(2*a^2*e^2*(d + 6*e*x) + 2*a*b*e*(d^2 + 6*d*e*x + 15*e^2*
x^2) + b^2*(d^3 + 6*d^2*e*x + 15*d*e^2*x^2 + 20*e^3*x^3)) + 2*c*e*(a*e*(d^3 + 6*
d^2*e*x + 15*d*e^2*x^2 + 20*e^3*x^3) + 2*b*(d^4 + 6*d^3*e*x + 15*d^2*e^2*x^2 + 2
0*d*e^3*x^3 + 15*e^4*x^4))))/(60*e^6*(d + e*x)^6)

_______________________________________________________________________________________

Maple [A]  time = 0.01, size = 453, normalized size = 1.5 \[ -{\frac{2\,Abc{e}^{2}-4\,A{c}^{2}de+2\,aBc{e}^{2}+{b}^{2}B{e}^{2}-8\,Bdbce+10\,B{c}^{2}{d}^{2}}{3\,{e}^{6} \left ( ex+d \right ) ^{3}}}-{\frac{B{c}^{2}}{{e}^{6} \left ( ex+d \right ) }}-{\frac{2\,aAc{e}^{3}+A{b}^{2}{e}^{3}-6\,Abcd{e}^{2}+6\,A{c}^{2}{d}^{2}e+2\,B{e}^{3}ab-6\,Bdac{e}^{2}-3\,Bd{b}^{2}{e}^{2}+12\,B{d}^{2}bce-10\,B{c}^{2}{d}^{3}}{4\,{e}^{6} \left ( ex+d \right ) ^{4}}}-{\frac{c \left ( Ace+2\,bBe-5\,Bcd \right ) }{2\,{e}^{6} \left ( ex+d \right ) ^{2}}}-{\frac{2\,Aab{e}^{4}-4\,Adac{e}^{3}-2\,Ad{b}^{2}{e}^{3}+6\,A{d}^{2}bc{e}^{2}-4\,A{c}^{2}{d}^{3}e+B{e}^{4}{a}^{2}-4\,Bdab{e}^{3}+6\,B{d}^{2}ac{e}^{2}+3\,B{d}^{2}{b}^{2}{e}^{2}-8\,B{d}^{3}bce+5\,B{c}^{2}{d}^{4}}{5\,{e}^{6} \left ( ex+d \right ) ^{5}}}-{\frac{A{a}^{2}{e}^{5}-2\,Adab{e}^{4}+2\,A{d}^{2}ac{e}^{3}+A{d}^{2}{b}^{2}{e}^{3}-2\,A{d}^{3}bc{e}^{2}+A{d}^{4}{c}^{2}e-Bd{a}^{2}{e}^{4}+2\,B{d}^{2}ab{e}^{3}-2\,B{d}^{3}ac{e}^{2}-B{d}^{3}{b}^{2}{e}^{2}+2\,B{d}^{4}bce-B{c}^{2}{d}^{5}}{6\,{e}^{6} \left ( ex+d \right ) ^{6}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+b*x+a)^2/(e*x+d)^7,x)

[Out]

-1/3*(2*A*b*c*e^2-4*A*c^2*d*e+2*B*a*c*e^2+B*b^2*e^2-8*B*b*c*d*e+10*B*c^2*d^2)/e^
6/(e*x+d)^3-B*c^2/e^6/(e*x+d)-1/4*(2*A*a*c*e^3+A*b^2*e^3-6*A*b*c*d*e^2+6*A*c^2*d
^2*e+2*B*a*b*e^3-6*B*a*c*d*e^2-3*B*b^2*d*e^2+12*B*b*c*d^2*e-10*B*c^2*d^3)/e^6/(e
*x+d)^4-1/2*c*(A*c*e+2*B*b*e-5*B*c*d)/e^6/(e*x+d)^2-1/5*(2*A*a*b*e^4-4*A*a*c*d*e
^3-2*A*b^2*d*e^3+6*A*b*c*d^2*e^2-4*A*c^2*d^3*e+B*a^2*e^4-4*B*a*b*d*e^3+6*B*a*c*d
^2*e^2+3*B*b^2*d^2*e^2-8*B*b*c*d^3*e+5*B*c^2*d^4)/e^6/(e*x+d)^5-1/6*(A*a^2*e^5-2
*A*a*b*d*e^4+2*A*a*c*d^2*e^3+A*b^2*d^2*e^3-2*A*b*c*d^3*e^2+A*c^2*d^4*e-B*a^2*d*e
^4+2*B*a*b*d^2*e^3-2*B*a*c*d^3*e^2-B*b^2*d^3*e^2+2*B*b*c*d^4*e-B*c^2*d^5)/e^6/(e
*x+d)^6

_______________________________________________________________________________________

Maxima [A]  time = 0.712573, size = 591, normalized size = 1.96 \[ -\frac{60 \, B c^{2} e^{5} x^{5} + 10 \, B c^{2} d^{5} + 10 \, A a^{2} e^{5} + 2 \,{\left (2 \, B b c + A c^{2}\right )} d^{4} e +{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} d^{3} e^{2} +{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d^{2} e^{3} + 2 \,{\left (B a^{2} + 2 \, A a b\right )} d e^{4} + 30 \,{\left (5 \, B c^{2} d e^{4} +{\left (2 \, B b c + A c^{2}\right )} e^{5}\right )} x^{4} + 20 \,{\left (10 \, B c^{2} d^{2} e^{3} + 2 \,{\left (2 \, B b c + A c^{2}\right )} d e^{4} +{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} e^{5}\right )} x^{3} + 15 \,{\left (10 \, B c^{2} d^{3} e^{2} + 2 \,{\left (2 \, B b c + A c^{2}\right )} d^{2} e^{3} +{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} d e^{4} +{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} e^{5}\right )} x^{2} + 6 \,{\left (10 \, B c^{2} d^{4} e + 2 \,{\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} +{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} d^{2} e^{3} +{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d e^{4} + 2 \,{\left (B a^{2} + 2 \, A a b\right )} e^{5}\right )} x}{60 \,{\left (e^{12} x^{6} + 6 \, d e^{11} x^{5} + 15 \, d^{2} e^{10} x^{4} + 20 \, d^{3} e^{9} x^{3} + 15 \, d^{4} e^{8} x^{2} + 6 \, d^{5} e^{7} x + d^{6} e^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/60*(60*B*c^2*e^5*x^5 + 10*B*c^2*d^5 + 10*A*a^2*e^5 + 2*(2*B*b*c + A*c^2)*d^4*
e + (B*b^2 + 2*(B*a + A*b)*c)*d^3*e^2 + (2*B*a*b + A*b^2 + 2*A*a*c)*d^2*e^3 + 2*
(B*a^2 + 2*A*a*b)*d*e^4 + 30*(5*B*c^2*d*e^4 + (2*B*b*c + A*c^2)*e^5)*x^4 + 20*(1
0*B*c^2*d^2*e^3 + 2*(2*B*b*c + A*c^2)*d*e^4 + (B*b^2 + 2*(B*a + A*b)*c)*e^5)*x^3
+ 15*(10*B*c^2*d^3*e^2 + 2*(2*B*b*c + A*c^2)*d^2*e^3 + (B*b^2 + 2*(B*a + A*b)*c
)*d*e^4 + (2*B*a*b + A*b^2 + 2*A*a*c)*e^5)*x^2 + 6*(10*B*c^2*d^4*e + 2*(2*B*b*c
+ A*c^2)*d^3*e^2 + (B*b^2 + 2*(B*a + A*b)*c)*d^2*e^3 + (2*B*a*b + A*b^2 + 2*A*a*
c)*d*e^4 + 2*(B*a^2 + 2*A*a*b)*e^5)*x)/(e^12*x^6 + 6*d*e^11*x^5 + 15*d^2*e^10*x^
4 + 20*d^3*e^9*x^3 + 15*d^4*e^8*x^2 + 6*d^5*e^7*x + d^6*e^6)

_______________________________________________________________________________________

Fricas [A]  time = 0.257843, size = 591, normalized size = 1.96 \[ -\frac{60 \, B c^{2} e^{5} x^{5} + 10 \, B c^{2} d^{5} + 10 \, A a^{2} e^{5} + 2 \,{\left (2 \, B b c + A c^{2}\right )} d^{4} e +{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} d^{3} e^{2} +{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d^{2} e^{3} + 2 \,{\left (B a^{2} + 2 \, A a b\right )} d e^{4} + 30 \,{\left (5 \, B c^{2} d e^{4} +{\left (2 \, B b c + A c^{2}\right )} e^{5}\right )} x^{4} + 20 \,{\left (10 \, B c^{2} d^{2} e^{3} + 2 \,{\left (2 \, B b c + A c^{2}\right )} d e^{4} +{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} e^{5}\right )} x^{3} + 15 \,{\left (10 \, B c^{2} d^{3} e^{2} + 2 \,{\left (2 \, B b c + A c^{2}\right )} d^{2} e^{3} +{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} d e^{4} +{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} e^{5}\right )} x^{2} + 6 \,{\left (10 \, B c^{2} d^{4} e + 2 \,{\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} +{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} d^{2} e^{3} +{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d e^{4} + 2 \,{\left (B a^{2} + 2 \, A a b\right )} e^{5}\right )} x}{60 \,{\left (e^{12} x^{6} + 6 \, d e^{11} x^{5} + 15 \, d^{2} e^{10} x^{4} + 20 \, d^{3} e^{9} x^{3} + 15 \, d^{4} e^{8} x^{2} + 6 \, d^{5} e^{7} x + d^{6} e^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/60*(60*B*c^2*e^5*x^5 + 10*B*c^2*d^5 + 10*A*a^2*e^5 + 2*(2*B*b*c + A*c^2)*d^4*
e + (B*b^2 + 2*(B*a + A*b)*c)*d^3*e^2 + (2*B*a*b + A*b^2 + 2*A*a*c)*d^2*e^3 + 2*
(B*a^2 + 2*A*a*b)*d*e^4 + 30*(5*B*c^2*d*e^4 + (2*B*b*c + A*c^2)*e^5)*x^4 + 20*(1
0*B*c^2*d^2*e^3 + 2*(2*B*b*c + A*c^2)*d*e^4 + (B*b^2 + 2*(B*a + A*b)*c)*e^5)*x^3
+ 15*(10*B*c^2*d^3*e^2 + 2*(2*B*b*c + A*c^2)*d^2*e^3 + (B*b^2 + 2*(B*a + A*b)*c
)*d*e^4 + (2*B*a*b + A*b^2 + 2*A*a*c)*e^5)*x^2 + 6*(10*B*c^2*d^4*e + 2*(2*B*b*c
+ A*c^2)*d^3*e^2 + (B*b^2 + 2*(B*a + A*b)*c)*d^2*e^3 + (2*B*a*b + A*b^2 + 2*A*a*
c)*d*e^4 + 2*(B*a^2 + 2*A*a*b)*e^5)*x)/(e^12*x^6 + 6*d*e^11*x^5 + 15*d^2*e^10*x^
4 + 20*d^3*e^9*x^3 + 15*d^4*e^8*x^2 + 6*d^5*e^7*x + d^6*e^6)

_______________________________________________________________________________________

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+b*x+a)**2/(e*x+d)**7,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.258165, size = 621, normalized size = 2.06 \[ -\frac{{\left (60 \, B c^{2} x^{5} e^{5} + 150 \, B c^{2} d x^{4} e^{4} + 200 \, B c^{2} d^{2} x^{3} e^{3} + 150 \, B c^{2} d^{3} x^{2} e^{2} + 60 \, B c^{2} d^{4} x e + 10 \, B c^{2} d^{5} + 60 \, B b c x^{4} e^{5} + 30 \, A c^{2} x^{4} e^{5} + 80 \, B b c d x^{3} e^{4} + 40 \, A c^{2} d x^{3} e^{4} + 60 \, B b c d^{2} x^{2} e^{3} + 30 \, A c^{2} d^{2} x^{2} e^{3} + 24 \, B b c d^{3} x e^{2} + 12 \, A c^{2} d^{3} x e^{2} + 4 \, B b c d^{4} e + 2 \, A c^{2} d^{4} e + 20 \, B b^{2} x^{3} e^{5} + 40 \, B a c x^{3} e^{5} + 40 \, A b c x^{3} e^{5} + 15 \, B b^{2} d x^{2} e^{4} + 30 \, B a c d x^{2} e^{4} + 30 \, A b c d x^{2} e^{4} + 6 \, B b^{2} d^{2} x e^{3} + 12 \, B a c d^{2} x e^{3} + 12 \, A b c d^{2} x e^{3} + B b^{2} d^{3} e^{2} + 2 \, B a c d^{3} e^{2} + 2 \, A b c d^{3} e^{2} + 30 \, B a b x^{2} e^{5} + 15 \, A b^{2} x^{2} e^{5} + 30 \, A a c x^{2} e^{5} + 12 \, B a b d x e^{4} + 6 \, A b^{2} d x e^{4} + 12 \, A a c d x e^{4} + 2 \, B a b d^{2} e^{3} + A b^{2} d^{2} e^{3} + 2 \, A a c d^{2} e^{3} + 12 \, B a^{2} x e^{5} + 24 \, A a b x e^{5} + 2 \, B a^{2} d e^{4} + 4 \, A a b d e^{4} + 10 \, A a^{2} e^{5}\right )} e^{\left (-6\right )}}{60 \,{\left (x e + d\right )}^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/60*(60*B*c^2*x^5*e^5 + 150*B*c^2*d*x^4*e^4 + 200*B*c^2*d^2*x^3*e^3 + 150*B*c^
2*d^3*x^2*e^2 + 60*B*c^2*d^4*x*e + 10*B*c^2*d^5 + 60*B*b*c*x^4*e^5 + 30*A*c^2*x^
4*e^5 + 80*B*b*c*d*x^3*e^4 + 40*A*c^2*d*x^3*e^4 + 60*B*b*c*d^2*x^2*e^3 + 30*A*c^
2*d^2*x^2*e^3 + 24*B*b*c*d^3*x*e^2 + 12*A*c^2*d^3*x*e^2 + 4*B*b*c*d^4*e + 2*A*c^
2*d^4*e + 20*B*b^2*x^3*e^5 + 40*B*a*c*x^3*e^5 + 40*A*b*c*x^3*e^5 + 15*B*b^2*d*x^
2*e^4 + 30*B*a*c*d*x^2*e^4 + 30*A*b*c*d*x^2*e^4 + 6*B*b^2*d^2*x*e^3 + 12*B*a*c*d
^2*x*e^3 + 12*A*b*c*d^2*x*e^3 + B*b^2*d^3*e^2 + 2*B*a*c*d^3*e^2 + 2*A*b*c*d^3*e^
2 + 30*B*a*b*x^2*e^5 + 15*A*b^2*x^2*e^5 + 30*A*a*c*x^2*e^5 + 12*B*a*b*d*x*e^4 +
6*A*b^2*d*x*e^4 + 12*A*a*c*d*x*e^4 + 2*B*a*b*d^2*e^3 + A*b^2*d^2*e^3 + 2*A*a*c*d
^2*e^3 + 12*B*a^2*x*e^5 + 24*A*a*b*x*e^5 + 2*B*a^2*d*e^4 + 4*A*a*b*d*e^4 + 10*A*
a^2*e^5)*e^(-6)/(x*e + d)^6