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

Optimal. Leaf size=255 \[ \frac {2 A c e (2 c d-b e)-B \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )}{4 e^6 (d+e x)^4}-\frac {A e \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )-B d \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )}{5 e^6 (d+e x)^5}+\frac {d^2 (B d-A e) (c d-b e)^2}{7 e^6 (d+e x)^7}+\frac {c (-A c e-2 b B e+5 B c d)}{3 e^6 (d+e x)^3}-\frac {d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{6 e^6 (d+e x)^6}-\frac {B c^2}{2 e^6 (d+e x)^2} \]

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Rubi [A]  time = 0.22, antiderivative size = 255, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {771} \begin {gather*} \frac {2 A c e (2 c d-b e)-B \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )}{4 e^6 (d+e x)^4}-\frac {A e \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )-B d \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )}{5 e^6 (d+e x)^5}+\frac {d^2 (B d-A e) (c d-b e)^2}{7 e^6 (d+e x)^7}+\frac {c (-A c e-2 b B e+5 B c d)}{3 e^6 (d+e x)^3}-\frac {d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{6 e^6 (d+e x)^6}-\frac {B c^2}{2 e^6 (d+e x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(d^2*(B*d - A*e)*(c*d - b*e)^2)/(7*e^6*(d + e*x)^7) - (d*(c*d - b*e)*(B*d*(5*c*d - 3*b*e) - 2*A*e*(2*c*d - b*e
)))/(6*e^6*(d + e*x)^6) - (A*e*(6*c^2*d^2 - 6*b*c*d*e + b^2*e^2) - B*d*(10*c^2*d^2 - 12*b*c*d*e + 3*b^2*e^2))/
(5*e^6*(d + e*x)^5) + (2*A*c*e*(2*c*d - b*e) - B*(10*c^2*d^2 - 8*b*c*d*e + b^2*e^2))/(4*e^6*(d + e*x)^4) + (c*
(5*B*c*d - 2*b*B*e - A*c*e))/(3*e^6*(d + e*x)^3) - (B*c^2)/(2*e^6*(d + e*x)^2)

Rule 771

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(d + e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && N
eQ[b^2 - 4*a*c, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin {align*} \int \frac {(A+B x) \left (b x+c x^2\right )^2}{(d+e x)^8} \, dx &=\int \left (-\frac {d^2 (B d-A e) (c d-b e)^2}{e^5 (d+e x)^8}+\frac {d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{e^5 (d+e x)^7}+\frac {A e \left (6 c^2 d^2-6 b c d e+b^2 e^2\right )-B d \left (10 c^2 d^2-12 b c d e+3 b^2 e^2\right )}{e^5 (d+e x)^6}+\frac {-2 A c e (2 c d-b e)+B \left (10 c^2 d^2-8 b c d e+b^2 e^2\right )}{e^5 (d+e x)^5}+\frac {c (-5 B c d+2 b B e+A c e)}{e^5 (d+e x)^4}+\frac {B c^2}{e^5 (d+e x)^3}\right ) \, dx\\ &=\frac {d^2 (B d-A e) (c d-b e)^2}{7 e^6 (d+e x)^7}-\frac {d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{6 e^6 (d+e x)^6}-\frac {A e \left (6 c^2 d^2-6 b c d e+b^2 e^2\right )-B d \left (10 c^2 d^2-12 b c d e+3 b^2 e^2\right )}{5 e^6 (d+e x)^5}+\frac {2 A c e (2 c d-b e)-B \left (10 c^2 d^2-8 b c d e+b^2 e^2\right )}{4 e^6 (d+e x)^4}+\frac {c (5 B c d-2 b B e-A c e)}{3 e^6 (d+e x)^3}-\frac {B c^2}{2 e^6 (d+e x)^2}\\ \end {align*}

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Mathematica [A]  time = 0.11, size = 260, normalized size = 1.02 \begin {gather*} -\frac {2 A e \left (2 b^2 e^2 \left (d^2+7 d e x+21 e^2 x^2\right )+3 b c e \left (d^3+7 d^2 e x+21 d e^2 x^2+35 e^3 x^3\right )+2 c^2 \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 )+B \left (3 b^2 e^2 \left (d^3+7 d^2 e x+21 d e^2 x^2+35 e^3 x^3\right )+8 b c e \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 )+10 c^2 \left (d^5+7 d^4 e x+21 d^3 e^2 x^2+35 d^2 e^3 x^3+35 d e^4 x^4+21 e^5 x^5\right )\right )}{420 e^6 (d+e x)^7} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/420*(2*A*e*(2*b^2*e^2*(d^2 + 7*d*e*x + 21*e^2*x^2) + 3*b*c*e*(d^3 + 7*d^2*e*x + 21*d*e^2*x^2 + 35*e^3*x^3)
+ 2*c^2*(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)) + B*(3*b^2*e^2*(d^3 + 7*d^2*e*x + 21*d
*e^2*x^2 + 35*e^3*x^3) + 8*b*c*e*(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) + 10*c^2*(d^5
+ 7*d^4*e*x + 21*d^3*e^2*x^2 + 35*d^2*e^3*x^3 + 35*d*e^4*x^4 + 21*e^5*x^5)))/(e^6*(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) \left (b x+c x^2\right )^2}{(d+e x)^8} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((A + B*x)*(b*x + c*x^2)^2)/(d + e*x)^8,x]

[Out]

IntegrateAlgebraic[((A + B*x)*(b*x + c*x^2)^2)/(d + e*x)^8, x]

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fricas [A]  time = 0.38, size = 359, normalized size = 1.41 \begin {gather*} -\frac {210 \, B c^{2} e^{5} x^{5} + 10 \, B c^{2} d^{5} + 4 \, A b^{2} d^{2} e^{3} + 4 \, {\left (2 \, B b c + A c^{2}\right )} d^{4} e + 3 \, {\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{2} + 70 \, {\left (5 \, B c^{2} d e^{4} + 2 \, {\left (2 \, B b c + A c^{2}\right )} e^{5}\right )} x^{4} + 35 \, {\left (10 \, B c^{2} d^{2} e^{3} + 4 \, {\left (2 \, B b c + A c^{2}\right )} d e^{4} + 3 \, {\left (B b^{2} + 2 \, A b c\right )} e^{5}\right )} x^{3} + 21 \, {\left (10 \, B c^{2} d^{3} e^{2} + 4 \, A b^{2} e^{5} + 4 \, {\left (2 \, B b c + A c^{2}\right )} d^{2} e^{3} + 3 \, {\left (B b^{2} + 2 \, A b c\right )} d e^{4}\right )} x^{2} + 7 \, {\left (10 \, B c^{2} d^{4} e + 4 \, A b^{2} d e^{4} + 4 \, {\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} + 3 \, {\left (B b^{2} + 2 \, A b c\right )} d^{2} e^{3}\right )} x}{420 \, {\left (e^{13} x^{7} + 7 \, d e^{12} x^{6} + 21 \, d^{2} e^{11} x^{5} + 35 \, d^{3} e^{10} x^{4} + 35 \, d^{4} e^{9} x^{3} + 21 \, d^{5} e^{8} x^{2} + 7 \, d^{6} e^{7} x + d^{7} e^{6}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/420*(210*B*c^2*e^5*x^5 + 10*B*c^2*d^5 + 4*A*b^2*d^2*e^3 + 4*(2*B*b*c + A*c^2)*d^4*e + 3*(B*b^2 + 2*A*b*c)*d
^3*e^2 + 70*(5*B*c^2*d*e^4 + 2*(2*B*b*c + A*c^2)*e^5)*x^4 + 35*(10*B*c^2*d^2*e^3 + 4*(2*B*b*c + A*c^2)*d*e^4 +
3*(B*b^2 + 2*A*b*c)*e^5)*x^3 + 21*(10*B*c^2*d^3*e^2 + 4*A*b^2*e^5 + 4*(2*B*b*c + A*c^2)*d^2*e^3 + 3*(B*b^2 +
2*A*b*c)*d*e^4)*x^2 + 7*(10*B*c^2*d^4*e + 4*A*b^2*d*e^4 + 4*(2*B*b*c + A*c^2)*d^3*e^2 + 3*(B*b^2 + 2*A*b*c)*d^
2*e^3)*x)/(e^13*x^7 + 7*d*e^12*x^6 + 21*d^2*e^11*x^5 + 35*d^3*e^10*x^4 + 35*d^4*e^9*x^3 + 21*d^5*e^8*x^2 + 7*d
^6*e^7*x + d^7*e^6)

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giac [A]  time = 0.16, size = 320, normalized size = 1.25 \begin {gather*} -\frac {{\left (210 \, B c^{2} x^{5} e^{5} + 350 \, B c^{2} d x^{4} e^{4} + 350 \, B c^{2} d^{2} x^{3} e^{3} + 210 \, B c^{2} d^{3} x^{2} e^{2} + 70 \, B c^{2} d^{4} x e + 10 \, B c^{2} d^{5} + 280 \, B b c x^{4} e^{5} + 140 \, A c^{2} x^{4} e^{5} + 280 \, B b c d x^{3} e^{4} + 140 \, A c^{2} d x^{3} e^{4} + 168 \, B b c d^{2} x^{2} e^{3} + 84 \, A c^{2} d^{2} x^{2} e^{3} + 56 \, B b c d^{3} x e^{2} + 28 \, A c^{2} d^{3} x e^{2} + 8 \, B b c d^{4} e + 4 \, A c^{2} d^{4} e + 105 \, B b^{2} x^{3} e^{5} + 210 \, A b c x^{3} e^{5} + 63 \, B b^{2} d x^{2} e^{4} + 126 \, A b c d x^{2} e^{4} + 21 \, B b^{2} d^{2} x e^{3} + 42 \, A b c d^{2} x e^{3} + 3 \, B b^{2} d^{3} e^{2} + 6 \, A b c d^{3} e^{2} + 84 \, A b^{2} x^{2} e^{5} + 28 \, A b^{2} d x e^{4} + 4 \, A b^{2} d^{2} e^{3}\right )} e^{\left (-6\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)*(c*x^2+b*x)^2/(e*x+d)^8,x, algorithm="giac")

[Out]

-1/420*(210*B*c^2*x^5*e^5 + 350*B*c^2*d*x^4*e^4 + 350*B*c^2*d^2*x^3*e^3 + 210*B*c^2*d^3*x^2*e^2 + 70*B*c^2*d^4
*x*e + 10*B*c^2*d^5 + 280*B*b*c*x^4*e^5 + 140*A*c^2*x^4*e^5 + 280*B*b*c*d*x^3*e^4 + 140*A*c^2*d*x^3*e^4 + 168*
B*b*c*d^2*x^2*e^3 + 84*A*c^2*d^2*x^2*e^3 + 56*B*b*c*d^3*x*e^2 + 28*A*c^2*d^3*x*e^2 + 8*B*b*c*d^4*e + 4*A*c^2*d
^4*e + 105*B*b^2*x^3*e^5 + 210*A*b*c*x^3*e^5 + 63*B*b^2*d*x^2*e^4 + 126*A*b*c*d*x^2*e^4 + 21*B*b^2*d^2*x*e^3 +
42*A*b*c*d^2*x*e^3 + 3*B*b^2*d^3*e^2 + 6*A*b*c*d^3*e^2 + 84*A*b^2*x^2*e^5 + 28*A*b^2*d*x*e^4 + 4*A*b^2*d^2*e^
3)*e^(-6)/(x*e + d)^7

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maple [A]  time = 0.05, size = 307, normalized size = 1.20 \begin {gather*} -\frac {B \,c^{2}}{2 \left (e x +d \right )^{2} e^{6}}-\frac {\left (A \,b^{2} e^{3}-2 A b c d \,e^{2}+A \,c^{2} d^{2} e -B d \,b^{2} e^{2}+2 B \,d^{2} b c e -B \,c^{2} d^{3}\right ) d^{2}}{7 \left (e x +d \right )^{7} e^{6}}-\frac {\left (A c e +2 B b e -5 B c d \right ) c}{3 \left (e x +d \right )^{3} e^{6}}+\frac {\left (2 A \,b^{2} e^{3}-6 A b c d \,e^{2}+4 A \,c^{2} d^{2} e -3 B d \,b^{2} e^{2}+8 B \,d^{2} b c e -5 B \,c^{2} d^{3}\right ) d}{6 \left (e x +d \right )^{6} e^{6}}-\frac {2 A b c \,e^{2}-4 A \,c^{2} d e +B \,b^{2} e^{2}-8 B d b c e +10 B \,d^{2} c^{2}}{4 \left (e x +d \right )^{4} e^{6}}-\frac {A \,b^{2} e^{3}-6 A b c d \,e^{2}+6 A \,c^{2} d^{2} e -3 B d \,b^{2} e^{2}+12 B \,d^{2} b c e -10 B \,c^{2} d^{3}}{5 \left (e x +d \right )^{5} e^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+b*x)^2/(e*x+d)^8,x)

[Out]

-1/4*(2*A*b*c*e^2-4*A*c^2*d*e+B*b^2*e^2-8*B*b*c*d*e+10*B*c^2*d^2)/e^6/(e*x+d)^4-1/5*(A*b^2*e^3-6*A*b*c*d*e^2+6
*A*c^2*d^2*e-3*B*b^2*d*e^2+12*B*b*c*d^2*e-10*B*c^2*d^3)/e^6/(e*x+d)^5+1/6*d*(2*A*b^2*e^3-6*A*b*c*d*e^2+4*A*c^2
*d^2*e-3*B*b^2*d*e^2+8*B*b*c*d^2*e-5*B*c^2*d^3)/e^6/(e*x+d)^6-1/2*B*c^2/e^6/(e*x+d)^2-1/7*d^2*(A*b^2*e^3-2*A*b
*c*d*e^2+A*c^2*d^2*e-B*b^2*d*e^2+2*B*b*c*d^2*e-B*c^2*d^3)/e^6/(e*x+d)^7-1/3*c*(A*c*e+2*B*b*e-5*B*c*d)/e^6/(e*x
+d)^3

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maxima [A]  time = 0.70, size = 359, normalized size = 1.41 \begin {gather*} -\frac {210 \, B c^{2} e^{5} x^{5} + 10 \, B c^{2} d^{5} + 4 \, A b^{2} d^{2} e^{3} + 4 \, {\left (2 \, B b c + A c^{2}\right )} d^{4} e + 3 \, {\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{2} + 70 \, {\left (5 \, B c^{2} d e^{4} + 2 \, {\left (2 \, B b c + A c^{2}\right )} e^{5}\right )} x^{4} + 35 \, {\left (10 \, B c^{2} d^{2} e^{3} + 4 \, {\left (2 \, B b c + A c^{2}\right )} d e^{4} + 3 \, {\left (B b^{2} + 2 \, A b c\right )} e^{5}\right )} x^{3} + 21 \, {\left (10 \, B c^{2} d^{3} e^{2} + 4 \, A b^{2} e^{5} + 4 \, {\left (2 \, B b c + A c^{2}\right )} d^{2} e^{3} + 3 \, {\left (B b^{2} + 2 \, A b c\right )} d e^{4}\right )} x^{2} + 7 \, {\left (10 \, B c^{2} d^{4} e + 4 \, A b^{2} d e^{4} + 4 \, {\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} + 3 \, {\left (B b^{2} + 2 \, A b c\right )} d^{2} e^{3}\right )} x}{420 \, {\left (e^{13} x^{7} + 7 \, d e^{12} x^{6} + 21 \, d^{2} e^{11} x^{5} + 35 \, d^{3} e^{10} x^{4} + 35 \, d^{4} e^{9} x^{3} + 21 \, d^{5} e^{8} x^{2} + 7 \, d^{6} e^{7} x + d^{7} e^{6}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/420*(210*B*c^2*e^5*x^5 + 10*B*c^2*d^5 + 4*A*b^2*d^2*e^3 + 4*(2*B*b*c + A*c^2)*d^4*e + 3*(B*b^2 + 2*A*b*c)*d
^3*e^2 + 70*(5*B*c^2*d*e^4 + 2*(2*B*b*c + A*c^2)*e^5)*x^4 + 35*(10*B*c^2*d^2*e^3 + 4*(2*B*b*c + A*c^2)*d*e^4 +
3*(B*b^2 + 2*A*b*c)*e^5)*x^3 + 21*(10*B*c^2*d^3*e^2 + 4*A*b^2*e^5 + 4*(2*B*b*c + A*c^2)*d^2*e^3 + 3*(B*b^2 +
2*A*b*c)*d*e^4)*x^2 + 7*(10*B*c^2*d^4*e + 4*A*b^2*d*e^4 + 4*(2*B*b*c + A*c^2)*d^3*e^2 + 3*(B*b^2 + 2*A*b*c)*d^
2*e^3)*x)/(e^13*x^7 + 7*d*e^12*x^6 + 21*d^2*e^11*x^5 + 35*d^3*e^10*x^4 + 35*d^4*e^9*x^3 + 21*d^5*e^8*x^2 + 7*d
^6*e^7*x + d^7*e^6)

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mupad [B]  time = 0.12, size = 357, normalized size = 1.40 \begin {gather*} -\frac {\frac {x^3\,\left (3\,B\,b^2\,e^2+8\,B\,b\,c\,d\,e+6\,A\,b\,c\,e^2+10\,B\,c^2\,d^2+4\,A\,c^2\,d\,e\right )}{12\,e^3}+\frac {d^2\,\left (3\,B\,b^2\,d\,e^2+4\,A\,b^2\,e^3+8\,B\,b\,c\,d^2\,e+6\,A\,b\,c\,d\,e^2+10\,B\,c^2\,d^3+4\,A\,c^2\,d^2\,e\right )}{420\,e^6}+\frac {x^2\,\left (3\,B\,b^2\,d\,e^2+4\,A\,b^2\,e^3+8\,B\,b\,c\,d^2\,e+6\,A\,b\,c\,d\,e^2+10\,B\,c^2\,d^3+4\,A\,c^2\,d^2\,e\right )}{20\,e^4}+\frac {d\,x\,\left (3\,B\,b^2\,d\,e^2+4\,A\,b^2\,e^3+8\,B\,b\,c\,d^2\,e+6\,A\,b\,c\,d\,e^2+10\,B\,c^2\,d^3+4\,A\,c^2\,d^2\,e\right )}{60\,e^5}+\frac {c\,x^4\,\left (2\,A\,c\,e+4\,B\,b\,e+5\,B\,c\,d\right )}{6\,e^2}+\frac {B\,c^2\,x^5}{2\,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(((b*x + c*x^2)^2*(A + B*x))/(d + e*x)^8,x)

[Out]

-((x^3*(3*B*b^2*e^2 + 10*B*c^2*d^2 + 6*A*b*c*e^2 + 4*A*c^2*d*e + 8*B*b*c*d*e))/(12*e^3) + (d^2*(4*A*b^2*e^3 +
10*B*c^2*d^3 + 4*A*c^2*d^2*e + 3*B*b^2*d*e^2 + 6*A*b*c*d*e^2 + 8*B*b*c*d^2*e))/(420*e^6) + (x^2*(4*A*b^2*e^3 +
10*B*c^2*d^3 + 4*A*c^2*d^2*e + 3*B*b^2*d*e^2 + 6*A*b*c*d*e^2 + 8*B*b*c*d^2*e))/(20*e^4) + (d*x*(4*A*b^2*e^3 +
10*B*c^2*d^3 + 4*A*c^2*d^2*e + 3*B*b^2*d*e^2 + 6*A*b*c*d*e^2 + 8*B*b*c*d^2*e))/(60*e^5) + (c*x^4*(2*A*c*e + 4
*B*b*e + 5*B*c*d))/(6*e^2) + (B*c^2*x^5)/(2*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)*(c*x**2+b*x)**2/(e*x+d)**8,x)

[Out]

Timed out

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