3.1050 \(\int \frac{(a+b x)^3 (A+B x)}{(d+e x)^{10}} \, dx\)

Optimal. Leaf size=163 \[ \frac{b^2 (-3 a B e-A b e+4 b B d)}{6 e^5 (d+e x)^6}-\frac{3 b (b d-a e) (-a B e-A b e+2 b B d)}{7 e^5 (d+e x)^7}+\frac{(b d-a e)^2 (-a B e-3 A b e+4 b B d)}{8 e^5 (d+e x)^8}-\frac{(b d-a e)^3 (B d-A e)}{9 e^5 (d+e x)^9}-\frac{b^3 B}{5 e^5 (d+e x)^5} \]

[Out]

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

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Rubi [A]  time = 0.107964, 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, Rules used = {77} \[ \frac{b^2 (-3 a B e-A b e+4 b B d)}{6 e^5 (d+e x)^6}-\frac{3 b (b d-a e) (-a B e-A b e+2 b B d)}{7 e^5 (d+e x)^7}+\frac{(b d-a e)^2 (-a B e-3 A b e+4 b B d)}{8 e^5 (d+e x)^8}-\frac{(b d-a e)^3 (B d-A e)}{9 e^5 (d+e x)^9}-\frac{b^3 B}{5 e^5 (d+e x)^5} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^3*(A + B*x))/(d + e*x)^10,x]

[Out]

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

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin{align*} \int \frac{(a+b x)^3 (A+B x)}{(d+e x)^{10}} \, dx &=\int \left (\frac{(-b d+a e)^3 (-B d+A e)}{e^4 (d+e x)^{10}}+\frac{(-b d+a e)^2 (-4 b B d+3 A b e+a B e)}{e^4 (d+e x)^9}-\frac{3 b (b d-a e) (-2 b B d+A b e+a B e)}{e^4 (d+e x)^8}+\frac{b^2 (-4 b B d+A b e+3 a B e)}{e^4 (d+e x)^7}+\frac{b^3 B}{e^4 (d+e x)^6}\right ) \, dx\\ &=-\frac{(b d-a e)^3 (B d-A e)}{9 e^5 (d+e x)^9}+\frac{(b d-a e)^2 (4 b B d-3 A b e-a B e)}{8 e^5 (d+e x)^8}-\frac{3 b (b d-a e) (2 b B d-A b e-a B e)}{7 e^5 (d+e x)^7}+\frac{b^2 (4 b B d-A b e-3 a B e)}{6 e^5 (d+e x)^6}-\frac{b^3 B}{5 e^5 (d+e x)^5}\\ \end{align*}

Mathematica [A]  time = 0.0975933, size = 214, normalized size = 1.31 \[ -\frac{15 a^2 b e^2 \left (7 A e (d+9 e x)+2 B \left (d^2+9 d e x+36 e^2 x^2\right )\right )+35 a^3 e^3 (8 A e+B (d+9 e x))+15 a b^2 e \left (2 A e \left (d^2+9 d e x+36 e^2 x^2\right )+B \left (9 d^2 e x+d^3+36 d e^2 x^2+84 e^3 x^3\right )\right )+b^3 \left (5 A e \left (9 d^2 e x+d^3+36 d e^2 x^2+84 e^3 x^3\right )+4 B \left (36 d^2 e^2 x^2+9 d^3 e x+d^4+84 d e^3 x^3+126 e^4 x^4\right )\right )}{2520 e^5 (d+e x)^9} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^3*(A + B*x))/(d + e*x)^10,x]

[Out]

-(35*a^3*e^3*(8*A*e + B*(d + 9*e*x)) + 15*a^2*b*e^2*(7*A*e*(d + 9*e*x) + 2*B*(d^2 + 9*d*e*x + 36*e^2*x^2)) + 1
5*a*b^2*e*(2*A*e*(d^2 + 9*d*e*x + 36*e^2*x^2) + B*(d^3 + 9*d^2*e*x + 36*d*e^2*x^2 + 84*e^3*x^3)) + b^3*(5*A*e*
(d^3 + 9*d^2*e*x + 36*d*e^2*x^2 + 84*e^3*x^3) + 4*B*(d^4 + 9*d^3*e*x + 36*d^2*e^2*x^2 + 84*d*e^3*x^3 + 126*e^4
*x^4)))/(2520*e^5*(d + e*x)^9)

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Maple [A]  time = 0.007, size = 281, normalized size = 1.7 \begin{align*} -{\frac{{b}^{2} \left ( Abe+3\,Bae-4\,Bbd \right ) }{6\,{e}^{5} \left ( ex+d \right ) ^{6}}}-{\frac{3\,b \left ( Aba{e}^{2}-A{b}^{2}de+B{a}^{2}{e}^{2}-3\,Bdabe+2\,{b}^{2}B{d}^{2} \right ) }{7\,{e}^{5} \left ( ex+d \right ) ^{7}}}-{\frac{{a}^{3}A{e}^{4}-3\,Ad{a}^{2}b{e}^{3}+3\,A{d}^{2}a{b}^{2}{e}^{2}-A{d}^{3}{b}^{3}e-Bd{a}^{3}{e}^{3}+3\,B{d}^{2}{a}^{2}b{e}^{2}-3\,B{d}^{3}a{b}^{2}e+{b}^{3}B{d}^{4}}{9\,{e}^{5} \left ( ex+d \right ) ^{9}}}-{\frac{B{b}^{3}}{5\,{e}^{5} \left ( ex+d \right ) ^{5}}}-{\frac{3\,Ab{a}^{2}{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}}{8\,{e}^{5} \left ( ex+d \right ) ^{8}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^3*(B*x+A)/(e*x+d)^10,x)

[Out]

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

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Maxima [B]  time = 1.19532, size = 478, normalized size = 2.93 \begin{align*} -\frac{504 \, B b^{3} e^{4} x^{4} + 4 \, B b^{3} d^{4} + 280 \, A a^{3} e^{4} + 5 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 30 \,{\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} + 35 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3} + 84 \,{\left (4 \, B b^{3} d e^{3} + 5 \,{\left (3 \, B a b^{2} + A b^{3}\right )} e^{4}\right )} x^{3} + 36 \,{\left (4 \, B b^{3} d^{2} e^{2} + 5 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d e^{3} + 30 \,{\left (B a^{2} b + A a b^{2}\right )} e^{4}\right )} x^{2} + 9 \,{\left (4 \, B b^{3} d^{3} e + 5 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2} + 30 \,{\left (B a^{2} b + A a b^{2}\right )} d e^{3} + 35 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} e^{4}\right )} x}{2520 \,{\left (e^{14} x^{9} + 9 \, d e^{13} x^{8} + 36 \, d^{2} e^{12} x^{7} + 84 \, d^{3} e^{11} x^{6} + 126 \, d^{4} e^{10} x^{5} + 126 \, d^{5} e^{9} x^{4} + 84 \, d^{6} e^{8} x^{3} + 36 \, d^{7} e^{7} x^{2} + 9 \, d^{8} e^{6} x + d^{9} e^{5}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(B*x+A)/(e*x+d)^10,x, algorithm="maxima")

[Out]

-1/2520*(504*B*b^3*e^4*x^4 + 4*B*b^3*d^4 + 280*A*a^3*e^4 + 5*(3*B*a*b^2 + A*b^3)*d^3*e + 30*(B*a^2*b + A*a*b^2
)*d^2*e^2 + 35*(B*a^3 + 3*A*a^2*b)*d*e^3 + 84*(4*B*b^3*d*e^3 + 5*(3*B*a*b^2 + A*b^3)*e^4)*x^3 + 36*(4*B*b^3*d^
2*e^2 + 5*(3*B*a*b^2 + A*b^3)*d*e^3 + 30*(B*a^2*b + A*a*b^2)*e^4)*x^2 + 9*(4*B*b^3*d^3*e + 5*(3*B*a*b^2 + A*b^
3)*d^2*e^2 + 30*(B*a^2*b + A*a*b^2)*d*e^3 + 35*(B*a^3 + 3*A*a^2*b)*e^4)*x)/(e^14*x^9 + 9*d*e^13*x^8 + 36*d^2*e
^12*x^7 + 84*d^3*e^11*x^6 + 126*d^4*e^10*x^5 + 126*d^5*e^9*x^4 + 84*d^6*e^8*x^3 + 36*d^7*e^7*x^2 + 9*d^8*e^6*x
+ d^9*e^5)

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Fricas [B]  time = 1.77593, size = 760, normalized size = 4.66 \begin{align*} -\frac{504 \, B b^{3} e^{4} x^{4} + 4 \, B b^{3} d^{4} + 280 \, A a^{3} e^{4} + 5 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 30 \,{\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} + 35 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3} + 84 \,{\left (4 \, B b^{3} d e^{3} + 5 \,{\left (3 \, B a b^{2} + A b^{3}\right )} e^{4}\right )} x^{3} + 36 \,{\left (4 \, B b^{3} d^{2} e^{2} + 5 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d e^{3} + 30 \,{\left (B a^{2} b + A a b^{2}\right )} e^{4}\right )} x^{2} + 9 \,{\left (4 \, B b^{3} d^{3} e + 5 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2} + 30 \,{\left (B a^{2} b + A a b^{2}\right )} d e^{3} + 35 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} e^{4}\right )} x}{2520 \,{\left (e^{14} x^{9} + 9 \, d e^{13} x^{8} + 36 \, d^{2} e^{12} x^{7} + 84 \, d^{3} e^{11} x^{6} + 126 \, d^{4} e^{10} x^{5} + 126 \, d^{5} e^{9} x^{4} + 84 \, d^{6} e^{8} x^{3} + 36 \, d^{7} e^{7} x^{2} + 9 \, d^{8} e^{6} x + d^{9} e^{5}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(B*x+A)/(e*x+d)^10,x, algorithm="fricas")

[Out]

-1/2520*(504*B*b^3*e^4*x^4 + 4*B*b^3*d^4 + 280*A*a^3*e^4 + 5*(3*B*a*b^2 + A*b^3)*d^3*e + 30*(B*a^2*b + A*a*b^2
)*d^2*e^2 + 35*(B*a^3 + 3*A*a^2*b)*d*e^3 + 84*(4*B*b^3*d*e^3 + 5*(3*B*a*b^2 + A*b^3)*e^4)*x^3 + 36*(4*B*b^3*d^
2*e^2 + 5*(3*B*a*b^2 + A*b^3)*d*e^3 + 30*(B*a^2*b + A*a*b^2)*e^4)*x^2 + 9*(4*B*b^3*d^3*e + 5*(3*B*a*b^2 + A*b^
3)*d^2*e^2 + 30*(B*a^2*b + A*a*b^2)*d*e^3 + 35*(B*a^3 + 3*A*a^2*b)*e^4)*x)/(e^14*x^9 + 9*d*e^13*x^8 + 36*d^2*e
^12*x^7 + 84*d^3*e^11*x^6 + 126*d^4*e^10*x^5 + 126*d^5*e^9*x^4 + 84*d^6*e^8*x^3 + 36*d^7*e^7*x^2 + 9*d^8*e^6*x
+ d^9*e^5)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**3*(B*x+A)/(e*x+d)**10,x)

[Out]

Timed out

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Giac [A]  time = 2.07934, size = 382, normalized size = 2.34 \begin{align*} -\frac{{\left (504 \, B b^{3} x^{4} e^{4} + 336 \, B b^{3} d x^{3} e^{3} + 144 \, B b^{3} d^{2} x^{2} e^{2} + 36 \, B b^{3} d^{3} x e + 4 \, B b^{3} d^{4} + 1260 \, B a b^{2} x^{3} e^{4} + 420 \, A b^{3} x^{3} e^{4} + 540 \, B a b^{2} d x^{2} e^{3} + 180 \, A b^{3} d x^{2} e^{3} + 135 \, B a b^{2} d^{2} x e^{2} + 45 \, A b^{3} d^{2} x e^{2} + 15 \, B a b^{2} d^{3} e + 5 \, A b^{3} d^{3} e + 1080 \, B a^{2} b x^{2} e^{4} + 1080 \, A a b^{2} x^{2} e^{4} + 270 \, B a^{2} b d x e^{3} + 270 \, A a b^{2} d x e^{3} + 30 \, B a^{2} b d^{2} e^{2} + 30 \, A a b^{2} d^{2} e^{2} + 315 \, B a^{3} x e^{4} + 945 \, A a^{2} b x e^{4} + 35 \, B a^{3} d e^{3} + 105 \, A a^{2} b d e^{3} + 280 \, A a^{3} e^{4}\right )} e^{\left (-5\right )}}{2520 \,{\left (x e + d\right )}^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(B*x+A)/(e*x+d)^10,x, algorithm="giac")

[Out]

-1/2520*(504*B*b^3*x^4*e^4 + 336*B*b^3*d*x^3*e^3 + 144*B*b^3*d^2*x^2*e^2 + 36*B*b^3*d^3*x*e + 4*B*b^3*d^4 + 12
60*B*a*b^2*x^3*e^4 + 420*A*b^3*x^3*e^4 + 540*B*a*b^2*d*x^2*e^3 + 180*A*b^3*d*x^2*e^3 + 135*B*a*b^2*d^2*x*e^2 +
45*A*b^3*d^2*x*e^2 + 15*B*a*b^2*d^3*e + 5*A*b^3*d^3*e + 1080*B*a^2*b*x^2*e^4 + 1080*A*a*b^2*x^2*e^4 + 270*B*a
^2*b*d*x*e^3 + 270*A*a*b^2*d*x*e^3 + 30*B*a^2*b*d^2*e^2 + 30*A*a*b^2*d^2*e^2 + 315*B*a^3*x*e^4 + 945*A*a^2*b*x
*e^4 + 35*B*a^3*d*e^3 + 105*A*a^2*b*d*e^3 + 280*A*a^3*e^4)*e^(-5)/(x*e + d)^9