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

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

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Rubi [A]  time = 0.11, antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {77} \begin {gather*} \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} \end {gather*}

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*}

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Mathematica [A]  time = 0.11, size = 214, normalized size = 1.31 \begin {gather*} -\frac {35 a^3 e^3 (8 A e+B (d+9 e x))+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 )+15 a b^2 e \left (2 A e \left (d^2+9 d e x+36 e^2 x^2\right )+B \left (d^3+9 d^2 e x+36 d e^2 x^2+84 e^3 x^3\right )\right )+b^3 \left (5 A e \left (d^3+9 d^2 e x+36 d e^2 x^2+84 e^3 x^3\right )+4 B \left (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\right )\right )}{2520 e^5 (d+e x)^9} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2520*(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)) + 15*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)))/(e^5*(d + e*x)^9)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^{10}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 0.59, size = 354, normalized size = 2.17 \begin {gather*} -\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 {gather*}

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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giac [A]  time = 1.21, size = 283, normalized size = 1.74 \begin {gather*} -\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 {gather*}

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

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maple [A]  time = 0.01, size = 281, normalized size = 1.72 \begin {gather*} -\frac {B \,b^{3}}{5 \left (e x +d \right )^{5} e^{5}}-\frac {\left (A b e +3 B a e -4 B b d \right ) b^{2}}{6 \left (e x +d \right )^{6} e^{5}}-\frac {3 \left (A a b \,e^{2}-A d \,b^{2} e +B \,a^{2} e^{2}-3 B d a b e +2 B \,b^{2} d^{2}\right ) b}{7 \left (e x +d \right )^{7} e^{5}}-\frac {3 A \,a^{2} b \,e^{3}-6 A d a \,b^{2} e^{2}+3 A \,d^{2} b^{3} e +B \,a^{3} e^{3}-6 B d \,a^{2} b \,e^{2}+9 B \,d^{2} a \,b^{2} e -4 B \,b^{3} d^{3}}{8 \left (e x +d \right )^{8} e^{5}}-\frac {A \,a^{3} e^{4}-3 A d \,a^{2} b \,e^{3}+3 A \,d^{2} a \,b^{2} e^{2}-A \,d^{3} b^{3} e -B d \,a^{3} e^{3}+3 B \,d^{2} a^{2} b \,e^{2}-3 B \,d^{3} a \,b^{2} e +B \,b^{3} d^{4}}{9 \left (e x +d \right )^{9} e^{5}} \end {gather*}

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]

-1/5*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)^8-1/9*(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)^9-1/6*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

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maxima [B]  time = 0.75, size = 354, normalized size = 2.17 \begin {gather*} -\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 {gather*}

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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mupad [B]  time = 1.17, size = 358, normalized size = 2.20 \begin {gather*} -\frac {\frac {35\,B\,a^3\,d\,e^3+280\,A\,a^3\,e^4+30\,B\,a^2\,b\,d^2\,e^2+105\,A\,a^2\,b\,d\,e^3+15\,B\,a\,b^2\,d^3\,e+30\,A\,a\,b^2\,d^2\,e^2+4\,B\,b^3\,d^4+5\,A\,b^3\,d^3\,e}{2520\,e^5}+\frac {x\,\left (35\,B\,a^3\,e^3+30\,B\,a^2\,b\,d\,e^2+105\,A\,a^2\,b\,e^3+15\,B\,a\,b^2\,d^2\,e+30\,A\,a\,b^2\,d\,e^2+4\,B\,b^3\,d^3+5\,A\,b^3\,d^2\,e\right )}{280\,e^4}+\frac {b^2\,x^3\,\left (5\,A\,b\,e+15\,B\,a\,e+4\,B\,b\,d\right )}{30\,e^2}+\frac {b\,x^2\,\left (30\,B\,a^2\,e^2+15\,B\,a\,b\,d\,e+30\,A\,a\,b\,e^2+4\,B\,b^2\,d^2+5\,A\,b^2\,d\,e\right )}{70\,e^3}+\frac {B\,b^3\,x^4}{5\,e}}{d^9+9\,d^8\,e\,x+36\,d^7\,e^2\,x^2+84\,d^6\,e^3\,x^3+126\,d^5\,e^4\,x^4+126\,d^4\,e^5\,x^5+84\,d^3\,e^6\,x^6+36\,d^2\,e^7\,x^7+9\,d\,e^8\,x^8+e^9\,x^9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((280*A*a^3*e^4 + 4*B*b^3*d^4 + 5*A*b^3*d^3*e + 35*B*a^3*d*e^3 + 30*A*a*b^2*d^2*e^2 + 30*B*a^2*b*d^2*e^2 + 10
5*A*a^2*b*d*e^3 + 15*B*a*b^2*d^3*e)/(2520*e^5) + (x*(35*B*a^3*e^3 + 4*B*b^3*d^3 + 105*A*a^2*b*e^3 + 5*A*b^3*d^
2*e + 30*A*a*b^2*d*e^2 + 15*B*a*b^2*d^2*e + 30*B*a^2*b*d*e^2))/(280*e^4) + (b^2*x^3*(5*A*b*e + 15*B*a*e + 4*B*
b*d))/(30*e^2) + (b*x^2*(30*B*a^2*e^2 + 4*B*b^2*d^2 + 30*A*a*b*e^2 + 5*A*b^2*d*e + 15*B*a*b*d*e))/(70*e^3) + (
B*b^3*x^4)/(5*e))/(d^9 + e^9*x^9 + 9*d*e^8*x^8 + 36*d^7*e^2*x^2 + 84*d^6*e^3*x^3 + 126*d^5*e^4*x^4 + 126*d^4*e
^5*x^5 + 84*d^3*e^6*x^6 + 36*d^2*e^7*x^7 + 9*d^8*e*x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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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