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

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

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Rubi [A]  time = 0.13, 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)}{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} \end {gather*}

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)

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

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Mathematica [A]  time = 0.10, size = 215, normalized size = 1.32 \begin {gather*} -\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} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/420*(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 + 21*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 + 3
5*e^4*x^4)))/(e^5*(d + e*x)^7)

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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)^8} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 1.14, size = 332, normalized size = 2.04 \begin {gather*} -\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 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(B*x+A)/(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^1
0*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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giac [A]  time = 1.36, size = 283, normalized size = 1.74 \begin {gather*} -\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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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

[Out]

-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)^5-1/6*(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

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maxima [B]  time = 0.60, size = 332, normalized size = 2.04 \begin {gather*} -\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 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(B*x+A)/(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^1
0*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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mupad [B]  time = 1.13, size = 336, normalized size = 2.06 \begin {gather*} -\frac {\frac {10\,B\,a^3\,d\,e^3+60\,A\,a^3\,e^4+12\,B\,a^2\,b\,d^2\,e^2+30\,A\,a^2\,b\,d\,e^3+9\,B\,a\,b^2\,d^3\,e+12\,A\,a\,b^2\,d^2\,e^2+4\,B\,b^3\,d^4+3\,A\,b^3\,d^3\,e}{420\,e^5}+\frac {x\,\left (10\,B\,a^3\,e^3+12\,B\,a^2\,b\,d\,e^2+30\,A\,a^2\,b\,e^3+9\,B\,a\,b^2\,d^2\,e+12\,A\,a\,b^2\,d\,e^2+4\,B\,b^3\,d^3+3\,A\,b^3\,d^2\,e\right )}{60\,e^4}+\frac {b^2\,x^3\,\left (3\,A\,b\,e+9\,B\,a\,e+4\,B\,b\,d\right )}{12\,e^2}+\frac {b\,x^2\,\left (12\,B\,a^2\,e^2+9\,B\,a\,b\,d\,e+12\,A\,a\,b\,e^2+4\,B\,b^2\,d^2+3\,A\,b^2\,d\,e\right )}{20\,e^3}+\frac {B\,b^3\,x^4}{3\,e}}{d^7+7\,d^6\,e\,x+21\,d^5\,e^2\,x^2+35\,d^4\,e^3\,x^3+35\,d^3\,e^4\,x^4+21\,d^2\,e^5\,x^5+7\,d\,e^6\,x^6+e^7\,x^7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((60*A*a^3*e^4 + 4*B*b^3*d^4 + 3*A*b^3*d^3*e + 10*B*a^3*d*e^3 + 12*A*a*b^2*d^2*e^2 + 12*B*a^2*b*d^2*e^2 + 30*
A*a^2*b*d*e^3 + 9*B*a*b^2*d^3*e)/(420*e^5) + (x*(10*B*a^3*e^3 + 4*B*b^3*d^3 + 30*A*a^2*b*e^3 + 3*A*b^3*d^2*e +
12*A*a*b^2*d*e^2 + 9*B*a*b^2*d^2*e + 12*B*a^2*b*d*e^2))/(60*e^4) + (b^2*x^3*(3*A*b*e + 9*B*a*e + 4*B*b*d))/(1
2*e^2) + (b*x^2*(12*B*a^2*e^2 + 4*B*b^2*d^2 + 12*A*a*b*e^2 + 3*A*b^2*d*e + 9*B*a*b*d*e))/(20*e^3) + (B*b^3*x^4
)/(3*e))/(d^7 + e^7*x^7 + 7*d*e^6*x^6 + 21*d^5*e^2*x^2 + 35*d^4*e^3*x^3 + 35*d^3*e^4*x^4 + 21*d^2*e^5*x^5 + 7*
d^6*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)**8,x)

[Out]

Timed out

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