3.1017 \(\int \frac{(a+b x)^2 (A+B x)}{(d+e x)^8} \, dx\)

Optimal. Leaf size=120 \[ \frac{b (-2 a B e-A b e+3 b B d)}{5 e^4 (d+e x)^5}-\frac{(b d-a e) (-a B e-2 A b e+3 b B d)}{6 e^4 (d+e x)^6}+\frac{(b d-a e)^2 (B d-A e)}{7 e^4 (d+e x)^7}-\frac{b^2 B}{4 e^4 (d+e x)^4} \]

[Out]

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

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Rubi [A]  time = 0.215853, antiderivative size = 120, 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 \[ \frac{b (-2 a B e-A b e+3 b B d)}{5 e^4 (d+e x)^5}-\frac{(b d-a e) (-a B e-2 A b e+3 b B d)}{6 e^4 (d+e x)^6}+\frac{(b d-a e)^2 (B d-A e)}{7 e^4 (d+e x)^7}-\frac{b^2 B}{4 e^4 (d+e x)^4} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^2*(A + B*x))/(d + e*x)^8,x]

[Out]

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

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Rubi in Sympy [A]  time = 37.637, size = 114, normalized size = 0.95 \[ - \frac{B b^{2}}{4 e^{4} \left (d + e x\right )^{4}} - \frac{b \left (A b e + 2 B a e - 3 B b d\right )}{5 e^{4} \left (d + e x\right )^{5}} - \frac{\left (a e - b d\right ) \left (2 A b e + B a e - 3 B b d\right )}{6 e^{4} \left (d + e x\right )^{6}} - \frac{\left (A e - B d\right ) \left (a e - b d\right )^{2}}{7 e^{4} \left (d + e x\right )^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**2*(B*x+A)/(e*x+d)**8,x)

[Out]

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

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Mathematica [A]  time = 0.103556, size = 129, normalized size = 1.08 \[ -\frac{10 a^2 e^2 (6 A e+B (d+7 e x))+4 a b e \left (5 A e (d+7 e x)+2 B \left (d^2+7 d e x+21 e^2 x^2\right )\right )+b^2 \left (4 A e \left (d^2+7 d e x+21 e^2 x^2\right )+3 B \left (d^3+7 d^2 e x+21 d e^2 x^2+35 e^3 x^3\right )\right )}{420 e^4 (d+e x)^7} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^2*(A + B*x))/(d + e*x)^8,x]

[Out]

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

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Maple [A]  time = 0.009, size = 166, normalized size = 1.4 \[ -{\frac{2\,Aab{e}^{2}-2\,Ad{b}^{2}e+B{a}^{2}{e}^{2}-4\,Bdabe+3\,{b}^{2}B{d}^{2}}{6\,{e}^{4} \left ( ex+d \right ) ^{6}}}-{\frac{{a}^{2}A{e}^{3}-2\,Aabd{e}^{2}+A{d}^{2}{b}^{2}e-B{a}^{2}d{e}^{2}+2\,B{d}^{2}abe-{b}^{2}B{d}^{3}}{7\,{e}^{4} \left ( ex+d \right ) ^{7}}}-{\frac{B{b}^{2}}{4\,{e}^{4} \left ( ex+d \right ) ^{4}}}-{\frac{b \left ( Abe+2\,Bae-3\,Bbd \right ) }{5\,{e}^{4} \left ( ex+d \right ) ^{5}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^2*(B*x+A)/(e*x+d)^8,x)

[Out]

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

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Maxima [A]  time = 1.3652, size = 304, normalized size = 2.53 \[ -\frac{105 \, B b^{2} e^{3} x^{3} + 3 \, B b^{2} d^{3} + 60 \, A a^{2} e^{3} + 4 \,{\left (2 \, B a b + A b^{2}\right )} d^{2} e + 10 \,{\left (B a^{2} + 2 \, A a b\right )} d e^{2} + 21 \,{\left (3 \, B b^{2} d e^{2} + 4 \,{\left (2 \, B a b + A b^{2}\right )} e^{3}\right )} x^{2} + 7 \,{\left (3 \, B b^{2} d^{2} e + 4 \,{\left (2 \, B a b + A b^{2}\right )} d e^{2} + 10 \,{\left (B a^{2} + 2 \, A a b\right )} e^{3}\right )} x}{420 \,{\left (e^{11} x^{7} + 7 \, d e^{10} x^{6} + 21 \, d^{2} e^{9} x^{5} + 35 \, d^{3} e^{8} x^{4} + 35 \, d^{4} e^{7} x^{3} + 21 \, d^{5} e^{6} x^{2} + 7 \, d^{6} e^{5} x + d^{7} e^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^2/(e*x + d)^8,x, algorithm="maxima")

[Out]

-1/420*(105*B*b^2*e^3*x^3 + 3*B*b^2*d^3 + 60*A*a^2*e^3 + 4*(2*B*a*b + A*b^2)*d^2
*e + 10*(B*a^2 + 2*A*a*b)*d*e^2 + 21*(3*B*b^2*d*e^2 + 4*(2*B*a*b + A*b^2)*e^3)*x
^2 + 7*(3*B*b^2*d^2*e + 4*(2*B*a*b + A*b^2)*d*e^2 + 10*(B*a^2 + 2*A*a*b)*e^3)*x)
/(e^11*x^7 + 7*d*e^10*x^6 + 21*d^2*e^9*x^5 + 35*d^3*e^8*x^4 + 35*d^4*e^7*x^3 + 2
1*d^5*e^6*x^2 + 7*d^6*e^5*x + d^7*e^4)

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Fricas [A]  time = 0.219693, size = 304, normalized size = 2.53 \[ -\frac{105 \, B b^{2} e^{3} x^{3} + 3 \, B b^{2} d^{3} + 60 \, A a^{2} e^{3} + 4 \,{\left (2 \, B a b + A b^{2}\right )} d^{2} e + 10 \,{\left (B a^{2} + 2 \, A a b\right )} d e^{2} + 21 \,{\left (3 \, B b^{2} d e^{2} + 4 \,{\left (2 \, B a b + A b^{2}\right )} e^{3}\right )} x^{2} + 7 \,{\left (3 \, B b^{2} d^{2} e + 4 \,{\left (2 \, B a b + A b^{2}\right )} d e^{2} + 10 \,{\left (B a^{2} + 2 \, A a b\right )} e^{3}\right )} x}{420 \,{\left (e^{11} x^{7} + 7 \, d e^{10} x^{6} + 21 \, d^{2} e^{9} x^{5} + 35 \, d^{3} e^{8} x^{4} + 35 \, d^{4} e^{7} x^{3} + 21 \, d^{5} e^{6} x^{2} + 7 \, d^{6} e^{5} x + d^{7} e^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^2/(e*x + d)^8,x, algorithm="fricas")

[Out]

-1/420*(105*B*b^2*e^3*x^3 + 3*B*b^2*d^3 + 60*A*a^2*e^3 + 4*(2*B*a*b + A*b^2)*d^2
*e + 10*(B*a^2 + 2*A*a*b)*d*e^2 + 21*(3*B*b^2*d*e^2 + 4*(2*B*a*b + A*b^2)*e^3)*x
^2 + 7*(3*B*b^2*d^2*e + 4*(2*B*a*b + A*b^2)*d*e^2 + 10*(B*a^2 + 2*A*a*b)*e^3)*x)
/(e^11*x^7 + 7*d*e^10*x^6 + 21*d^2*e^9*x^5 + 35*d^3*e^8*x^4 + 35*d^4*e^7*x^3 + 2
1*d^5*e^6*x^2 + 7*d^6*e^5*x + d^7*e^4)

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.218609, size = 216, normalized size = 1.8 \[ -\frac{{\left (105 \, B b^{2} x^{3} e^{3} + 63 \, B b^{2} d x^{2} e^{2} + 21 \, B b^{2} d^{2} x e + 3 \, B b^{2} d^{3} + 168 \, B a b x^{2} e^{3} + 84 \, A b^{2} x^{2} e^{3} + 56 \, B a b d x e^{2} + 28 \, A b^{2} d x e^{2} + 8 \, B a b d^{2} e + 4 \, A b^{2} d^{2} e + 70 \, B a^{2} x e^{3} + 140 \, A a b x e^{3} + 10 \, B a^{2} d e^{2} + 20 \, A a b d e^{2} + 60 \, A a^{2} e^{3}\right )} e^{\left (-4\right )}}{420 \,{\left (x e + d\right )}^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^2/(e*x + d)^8,x, algorithm="giac")

[Out]

-1/420*(105*B*b^2*x^3*e^3 + 63*B*b^2*d*x^2*e^2 + 21*B*b^2*d^2*x*e + 3*B*b^2*d^3
+ 168*B*a*b*x^2*e^3 + 84*A*b^2*x^2*e^3 + 56*B*a*b*d*x*e^2 + 28*A*b^2*d*x*e^2 + 8
*B*a*b*d^2*e + 4*A*b^2*d^2*e + 70*B*a^2*x*e^3 + 140*A*a*b*x*e^3 + 10*B*a^2*d*e^2
+ 20*A*a*b*d*e^2 + 60*A*a^2*e^3)*e^(-4)/(x*e + d)^7