3.1016 \(\int \frac{(a+b x)^2 (A+B x)}{(d+e x)^7} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.234852, 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 \[ \frac{b (-2 a B e-A b e+3 b B d)}{4 e^4 (d+e x)^4}-\frac{(b d-a e) (-a B e-2 A b e+3 b B d)}{5 e^4 (d+e x)^5}+\frac{(b d-a e)^2 (B d-A e)}{6 e^4 (d+e x)^6}-\frac{b^2 B}{3 e^4 (d+e x)^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 36.8376, size = 114, normalized size = 0.95 \[ - \frac{B b^{2}}{3 e^{4} \left (d + e x\right )^{3}} - \frac{b \left (A b e + 2 B a e - 3 B b d\right )}{4 e^{4} \left (d + e x\right )^{4}} - \frac{\left (a e - b d\right ) \left (2 A b e + B a e - 3 B b d\right )}{5 e^{4} \left (d + e x\right )^{5}} - \frac{\left (A e - B d\right ) \left (a e - b d\right )^{2}}{6 e^{4} \left (d + e x\right )^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.107553, size = 126, normalized size = 1.05 \[ -\frac{2 a^2 e^2 (5 A e+B (d+6 e x))+2 a b e \left (2 A e (d+6 e x)+B \left (d^2+6 d e x+15 e^2 x^2\right )\right )+b^2 \left (A e \left (d^2+6 d e x+15 e^2 x^2\right )+B \left (d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right )\right )}{60 e^4 (d+e x)^6} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.36184, size = 281, normalized size = 2.34 \[ -\frac{20 \, B b^{2} e^{3} x^{3} + B b^{2} d^{3} + 10 \, A a^{2} e^{3} +{\left (2 \, B a b + A b^{2}\right )} d^{2} e + 2 \,{\left (B a^{2} + 2 \, A a b\right )} d e^{2} + 15 \,{\left (B b^{2} d e^{2} +{\left (2 \, B a b + A b^{2}\right )} e^{3}\right )} x^{2} + 6 \,{\left (B b^{2} d^{2} e +{\left (2 \, B a b + A b^{2}\right )} d e^{2} + 2 \,{\left (B a^{2} + 2 \, A a b\right )} e^{3}\right )} x}{60 \,{\left (e^{10} x^{6} + 6 \, d e^{9} x^{5} + 15 \, d^{2} e^{8} x^{4} + 20 \, d^{3} e^{7} x^{3} + 15 \, d^{4} e^{6} x^{2} + 6 \, d^{5} e^{5} x + d^{6} e^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.208603, size = 281, normalized size = 2.34 \[ -\frac{20 \, B b^{2} e^{3} x^{3} + B b^{2} d^{3} + 10 \, A a^{2} e^{3} +{\left (2 \, B a b + A b^{2}\right )} d^{2} e + 2 \,{\left (B a^{2} + 2 \, A a b\right )} d e^{2} + 15 \,{\left (B b^{2} d e^{2} +{\left (2 \, B a b + A b^{2}\right )} e^{3}\right )} x^{2} + 6 \,{\left (B b^{2} d^{2} e +{\left (2 \, B a b + A b^{2}\right )} d e^{2} + 2 \,{\left (B a^{2} + 2 \, A a b\right )} e^{3}\right )} x}{60 \,{\left (e^{10} x^{6} + 6 \, d e^{9} x^{5} + 15 \, d^{2} e^{8} x^{4} + 20 \, d^{3} e^{7} x^{3} + 15 \, d^{4} e^{6} x^{2} + 6 \, d^{5} e^{5} x + d^{6} e^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.211438, size = 213, normalized size = 1.78 \[ -\frac{{\left (20 \, B b^{2} x^{3} e^{3} + 15 \, B b^{2} d x^{2} e^{2} + 6 \, B b^{2} d^{2} x e + B b^{2} d^{3} + 30 \, B a b x^{2} e^{3} + 15 \, A b^{2} x^{2} e^{3} + 12 \, B a b d x e^{2} + 6 \, A b^{2} d x e^{2} + 2 \, B a b d^{2} e + A b^{2} d^{2} e + 12 \, B a^{2} x e^{3} + 24 \, A a b x e^{3} + 2 \, B a^{2} d e^{2} + 4 \, A a b d e^{2} + 10 \, A a^{2} e^{3}\right )} e^{\left (-4\right )}}{60 \,{\left (x e + d\right )}^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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