3.1031 \(\int \frac{(a+b x)^3 (A+B x)}{(d+e x)^8} \, dx\)

Optimal. Leaf size=163 \[ \frac{b^2 (-3 a B e-A b e+4 b B d)}{4 e^5 (d+e x)^4}-\frac{3 b (b d-a e) (-a B e-A b e+2 b B d)}{5 e^5 (d+e x)^5}+\frac{(b d-a e)^2 (-a B e-3 A b e+4 b B d)}{6 e^5 (d+e x)^6}-\frac{(b d-a e)^3 (B d-A e)}{7 e^5 (d+e x)^7}-\frac{b^3 B}{3 e^5 (d+e x)^3} \]

[Out]

-((b*d - a*e)^3*(B*d - A*e))/(7*e^5*(d + e*x)^7) + ((b*d - a*e)^2*(4*b*B*d - 3*A
*b*e - a*B*e))/(6*e^5*(d + e*x)^6) - (3*b*(b*d - a*e)*(2*b*B*d - A*b*e - a*B*e))
/(5*e^5*(d + e*x)^5) + (b^2*(4*b*B*d - A*b*e - 3*a*B*e))/(4*e^5*(d + e*x)^4) - (
b^3*B)/(3*e^5*(d + e*x)^3)

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Rubi [A]  time = 0.393306, antiderivative size = 163, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05 \[ \frac{b^2 (-3 a B e-A b e+4 b B d)}{4 e^5 (d+e x)^4}-\frac{3 b (b d-a e) (-a B e-A b e+2 b B d)}{5 e^5 (d+e x)^5}+\frac{(b d-a e)^2 (-a B e-3 A b e+4 b B d)}{6 e^5 (d+e x)^6}-\frac{(b d-a e)^3 (B d-A e)}{7 e^5 (d+e x)^7}-\frac{b^3 B}{3 e^5 (d+e x)^3} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^3*(A + B*x))/(d + e*x)^8,x]

[Out]

-((b*d - a*e)^3*(B*d - A*e))/(7*e^5*(d + e*x)^7) + ((b*d - a*e)^2*(4*b*B*d - 3*A
*b*e - a*B*e))/(6*e^5*(d + e*x)^6) - (3*b*(b*d - a*e)*(2*b*B*d - A*b*e - a*B*e))
/(5*e^5*(d + e*x)^5) + (b^2*(4*b*B*d - A*b*e - 3*a*B*e))/(4*e^5*(d + e*x)^4) - (
b^3*B)/(3*e^5*(d + e*x)^3)

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Rubi in Sympy [A]  time = 57.3834, size = 156, normalized size = 0.96 \[ - \frac{B b^{3}}{3 e^{5} \left (d + e x\right )^{3}} - \frac{b^{2} \left (A b e + 3 B a e - 4 B b d\right )}{4 e^{5} \left (d + e x\right )^{4}} - \frac{3 b \left (a e - b d\right ) \left (A b e + B a e - 2 B b d\right )}{5 e^{5} \left (d + e x\right )^{5}} - \frac{\left (a e - b d\right )^{2} \left (3 A b e + B a e - 4 B b d\right )}{6 e^{5} \left (d + e x\right )^{6}} - \frac{\left (A e - B d\right ) \left (a e - b d\right )^{3}}{7 e^{5} \left (d + e x\right )^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**3*(B*x+A)/(e*x+d)**8,x)

[Out]

-B*b**3/(3*e**5*(d + e*x)**3) - b**2*(A*b*e + 3*B*a*e - 4*B*b*d)/(4*e**5*(d + e*
x)**4) - 3*b*(a*e - b*d)*(A*b*e + B*a*e - 2*B*b*d)/(5*e**5*(d + e*x)**5) - (a*e
- b*d)**2*(3*A*b*e + B*a*e - 4*B*b*d)/(6*e**5*(d + e*x)**6) - (A*e - B*d)*(a*e -
b*d)**3/(7*e**5*(d + e*x)**7)

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Mathematica [A]  time = 0.160947, size = 215, normalized size = 1.32 \[ -\frac{10 a^3 e^3 (6 A e+B (d+7 e x))+6 a^2 b e^2 \left (5 A e (d+7 e x)+2 B \left (d^2+7 d e x+21 e^2 x^2\right )\right )+3 a b^2 e \left (4 A e \left (d^2+7 d e x+21 e^2 x^2\right )+3 B \left (d^3+7 d^2 e x+21 d e^2 x^2+35 e^3 x^3\right )\right )+b^3 \left (3 A e \left (d^3+7 d^2 e x+21 d e^2 x^2+35 e^3 x^3\right )+4 B \left (d^4+7 d^3 e x+21 d^2 e^2 x^2+35 d e^3 x^3+35 e^4 x^4\right )\right )}{420 e^5 (d+e x)^7} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^3*(A + B*x))/(d + e*x)^8,x]

[Out]

-(10*a^3*e^3*(6*A*e + B*(d + 7*e*x)) + 6*a^2*b*e^2*(5*A*e*(d + 7*e*x) + 2*B*(d^2
+ 7*d*e*x + 21*e^2*x^2)) + 3*a*b^2*e*(4*A*e*(d^2 + 7*d*e*x + 21*e^2*x^2) + 3*B*
(d^3 + 7*d^2*e*x + 21*d*e^2*x^2 + 35*e^3*x^3)) + b^3*(3*A*e*(d^3 + 7*d^2*e*x + 2
1*d*e^2*x^2 + 35*e^3*x^3) + 4*B*(d^4 + 7*d^3*e*x + 21*d^2*e^2*x^2 + 35*d*e^3*x^3
+ 35*e^4*x^4)))/(420*e^5*(d + e*x)^7)

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Maple [A]  time = 0.01, size = 281, normalized size = 1.7 \[ -{\frac{3\,A{a}^{2}b{e}^{3}-6\,Ada{b}^{2}{e}^{2}+3\,A{d}^{2}{b}^{3}e+B{a}^{3}{e}^{3}-6\,Bd{a}^{2}b{e}^{2}+9\,B{d}^{2}a{b}^{2}e-4\,{b}^{3}B{d}^{3}}{6\,{e}^{5} \left ( ex+d \right ) ^{6}}}-{\frac{{a}^{3}A{e}^{4}-3\,A{a}^{2}bd{e}^{3}+3\,A{d}^{2}a{b}^{2}{e}^{2}-A{d}^{3}{b}^{3}e-B{a}^{3}d{e}^{3}+3\,B{d}^{2}{a}^{2}b{e}^{2}-3\,B{d}^{3}a{b}^{2}e+{b}^{3}B{d}^{4}}{7\,{e}^{5} \left ( ex+d \right ) ^{7}}}-{\frac{B{b}^{3}}{3\,{e}^{5} \left ( ex+d \right ) ^{3}}}-{\frac{{b}^{2} \left ( Abe+3\,Bae-4\,Bbd \right ) }{4\,{e}^{5} \left ( ex+d \right ) ^{4}}}-{\frac{3\,b \left ( Aab{e}^{2}-Ad{b}^{2}e+B{a}^{2}{e}^{2}-3\,Bdabe+2\,{b}^{2}B{d}^{2} \right ) }{5\,{e}^{5} \left ( ex+d \right ) ^{5}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^3*(B*x+A)/(e*x+d)^8,x)

[Out]

-1/6*(3*A*a^2*b*e^3-6*A*a*b^2*d*e^2+3*A*b^3*d^2*e+B*a^3*e^3-6*B*a^2*b*d*e^2+9*B*
a*b^2*d^2*e-4*B*b^3*d^3)/e^5/(e*x+d)^6-1/7*(A*a^3*e^4-3*A*a^2*b*d*e^3+3*A*a*b^2*
d^2*e^2-A*b^3*d^3*e-B*a^3*d*e^3+3*B*a^2*b*d^2*e^2-3*B*a*b^2*d^3*e+B*b^3*d^4)/e^5
/(e*x+d)^7-1/3*b^3*B/e^5/(e*x+d)^3-1/4*b^2*(A*b*e+3*B*a*e-4*B*b*d)/e^5/(e*x+d)^4
-3/5*b*(A*a*b*e^2-A*b^2*d*e+B*a^2*e^2-3*B*a*b*d*e+2*B*b^2*d^2)/e^5/(e*x+d)^5

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Maxima [A]  time = 1.3809, size = 448, normalized size = 2.75 \[ -\frac{140 \, B b^{3} e^{4} x^{4} + 4 \, B b^{3} d^{4} + 60 \, A a^{3} e^{4} + 3 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 12 \,{\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} + 10 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3} + 35 \,{\left (4 \, B b^{3} d e^{3} + 3 \,{\left (3 \, B a b^{2} + A b^{3}\right )} e^{4}\right )} x^{3} + 21 \,{\left (4 \, B b^{3} d^{2} e^{2} + 3 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d e^{3} + 12 \,{\left (B a^{2} b + A a b^{2}\right )} e^{4}\right )} x^{2} + 7 \,{\left (4 \, B b^{3} d^{3} e + 3 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2} + 12 \,{\left (B a^{2} b + A a b^{2}\right )} d e^{3} + 10 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} e^{4}\right )} x}{420 \,{\left (e^{12} x^{7} + 7 \, d e^{11} x^{6} + 21 \, d^{2} e^{10} x^{5} + 35 \, d^{3} e^{9} x^{4} + 35 \, d^{4} e^{8} x^{3} + 21 \, d^{5} e^{7} x^{2} + 7 \, d^{6} e^{6} x + d^{7} e^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^3/(e*x + d)^8,x, algorithm="maxima")

[Out]

-1/420*(140*B*b^3*e^4*x^4 + 4*B*b^3*d^4 + 60*A*a^3*e^4 + 3*(3*B*a*b^2 + A*b^3)*d
^3*e + 12*(B*a^2*b + A*a*b^2)*d^2*e^2 + 10*(B*a^3 + 3*A*a^2*b)*d*e^3 + 35*(4*B*b
^3*d*e^3 + 3*(3*B*a*b^2 + A*b^3)*e^4)*x^3 + 21*(4*B*b^3*d^2*e^2 + 3*(3*B*a*b^2 +
A*b^3)*d*e^3 + 12*(B*a^2*b + A*a*b^2)*e^4)*x^2 + 7*(4*B*b^3*d^3*e + 3*(3*B*a*b^
2 + A*b^3)*d^2*e^2 + 12*(B*a^2*b + A*a*b^2)*d*e^3 + 10*(B*a^3 + 3*A*a^2*b)*e^4)*
x)/(e^12*x^7 + 7*d*e^11*x^6 + 21*d^2*e^10*x^5 + 35*d^3*e^9*x^4 + 35*d^4*e^8*x^3
+ 21*d^5*e^7*x^2 + 7*d^6*e^6*x + d^7*e^5)

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Fricas [A]  time = 0.201955, size = 448, normalized size = 2.75 \[ -\frac{140 \, B b^{3} e^{4} x^{4} + 4 \, B b^{3} d^{4} + 60 \, A a^{3} e^{4} + 3 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 12 \,{\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} + 10 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3} + 35 \,{\left (4 \, B b^{3} d e^{3} + 3 \,{\left (3 \, B a b^{2} + A b^{3}\right )} e^{4}\right )} x^{3} + 21 \,{\left (4 \, B b^{3} d^{2} e^{2} + 3 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d e^{3} + 12 \,{\left (B a^{2} b + A a b^{2}\right )} e^{4}\right )} x^{2} + 7 \,{\left (4 \, B b^{3} d^{3} e + 3 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2} + 12 \,{\left (B a^{2} b + A a b^{2}\right )} d e^{3} + 10 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} e^{4}\right )} x}{420 \,{\left (e^{12} x^{7} + 7 \, d e^{11} x^{6} + 21 \, d^{2} e^{10} x^{5} + 35 \, d^{3} e^{9} x^{4} + 35 \, d^{4} e^{8} x^{3} + 21 \, d^{5} e^{7} x^{2} + 7 \, d^{6} e^{6} x + d^{7} e^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^3/(e*x + d)^8,x, algorithm="fricas")

[Out]

-1/420*(140*B*b^3*e^4*x^4 + 4*B*b^3*d^4 + 60*A*a^3*e^4 + 3*(3*B*a*b^2 + A*b^3)*d
^3*e + 12*(B*a^2*b + A*a*b^2)*d^2*e^2 + 10*(B*a^3 + 3*A*a^2*b)*d*e^3 + 35*(4*B*b
^3*d*e^3 + 3*(3*B*a*b^2 + A*b^3)*e^4)*x^3 + 21*(4*B*b^3*d^2*e^2 + 3*(3*B*a*b^2 +
A*b^3)*d*e^3 + 12*(B*a^2*b + A*a*b^2)*e^4)*x^2 + 7*(4*B*b^3*d^3*e + 3*(3*B*a*b^
2 + A*b^3)*d^2*e^2 + 12*(B*a^2*b + A*a*b^2)*d*e^3 + 10*(B*a^3 + 3*A*a^2*b)*e^4)*
x)/(e^12*x^7 + 7*d*e^11*x^6 + 21*d^2*e^10*x^5 + 35*d^3*e^9*x^4 + 35*d^4*e^8*x^3
+ 21*d^5*e^7*x^2 + 7*d^6*e^6*x + d^7*e^5)

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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)**3*(B*x+A)/(e*x+d)**8,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.231896, size = 382, normalized size = 2.34 \[ -\frac{{\left (140 \, B b^{3} x^{4} e^{4} + 140 \, B b^{3} d x^{3} e^{3} + 84 \, B b^{3} d^{2} x^{2} e^{2} + 28 \, B b^{3} d^{3} x e + 4 \, B b^{3} d^{4} + 315 \, B a b^{2} x^{3} e^{4} + 105 \, A b^{3} x^{3} e^{4} + 189 \, B a b^{2} d x^{2} e^{3} + 63 \, A b^{3} d x^{2} e^{3} + 63 \, B a b^{2} d^{2} x e^{2} + 21 \, A b^{3} d^{2} x e^{2} + 9 \, B a b^{2} d^{3} e + 3 \, A b^{3} d^{3} e + 252 \, B a^{2} b x^{2} e^{4} + 252 \, A a b^{2} x^{2} e^{4} + 84 \, B a^{2} b d x e^{3} + 84 \, A a b^{2} d x e^{3} + 12 \, B a^{2} b d^{2} e^{2} + 12 \, A a b^{2} d^{2} e^{2} + 70 \, B a^{3} x e^{4} + 210 \, A a^{2} b x e^{4} + 10 \, B a^{3} d e^{3} + 30 \, A a^{2} b d e^{3} + 60 \, A a^{3} e^{4}\right )} e^{\left (-5\right )}}{420 \,{\left (x e + d\right )}^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^3/(e*x + d)^8,x, algorithm="giac")

[Out]

-1/420*(140*B*b^3*x^4*e^4 + 140*B*b^3*d*x^3*e^3 + 84*B*b^3*d^2*x^2*e^2 + 28*B*b^
3*d^3*x*e + 4*B*b^3*d^4 + 315*B*a*b^2*x^3*e^4 + 105*A*b^3*x^3*e^4 + 189*B*a*b^2*
d*x^2*e^3 + 63*A*b^3*d*x^2*e^3 + 63*B*a*b^2*d^2*x*e^2 + 21*A*b^3*d^2*x*e^2 + 9*B
*a*b^2*d^3*e + 3*A*b^3*d^3*e + 252*B*a^2*b*x^2*e^4 + 252*A*a*b^2*x^2*e^4 + 84*B*
a^2*b*d*x*e^3 + 84*A*a*b^2*d*x*e^3 + 12*B*a^2*b*d^2*e^2 + 12*A*a*b^2*d^2*e^2 + 7
0*B*a^3*x*e^4 + 210*A*a^2*b*x*e^4 + 10*B*a^3*d*e^3 + 30*A*a^2*b*d*e^3 + 60*A*a^3
*e^4)*e^(-5)/(x*e + d)^7