3.1124 \(\int \frac{(A+B x) (b x+c x^2)^2}{(d+e x)^6} \, dx\)

Optimal. Leaf size=248 \[ \frac{2 A c e (2 c d-b e)-B \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )}{2 e^6 (d+e x)^2}-\frac{A e \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )-B d \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )}{3 e^6 (d+e x)^3}+\frac{d^2 (B d-A e) (c d-b e)^2}{5 e^6 (d+e x)^5}+\frac{c (-A c e-2 b B e+5 B c d)}{e^6 (d+e x)}-\frac{d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{4 e^6 (d+e x)^4}+\frac{B c^2 \log (d+e x)}{e^6} \]

[Out]

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

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Rubi [A]  time = 0.239904, antiderivative size = 248, normalized size of antiderivative = 1., 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} \[ \frac{2 A c e (2 c d-b e)-B \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )}{2 e^6 (d+e x)^2}-\frac{A e \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )-B d \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )}{3 e^6 (d+e x)^3}+\frac{d^2 (B d-A e) (c d-b e)^2}{5 e^6 (d+e x)^5}+\frac{c (-A c e-2 b B e+5 B c d)}{e^6 (d+e x)}-\frac{d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{4 e^6 (d+e x)^4}+\frac{B c^2 \log (d+e x)}{e^6} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(b*x + c*x^2)^2)/(d + e*x)^6,x]

[Out]

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

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

Mathematica [A]  time = 0.137004, size = 269, normalized size = 1.08 \[ \frac{-2 A e \left (b^2 e^2 \left (d^2+5 d e x+10 e^2 x^2\right )+3 b c e \left (5 d^2 e x+d^3+10 d e^2 x^2+10 e^3 x^3\right )+6 c^2 \left (10 d^2 e^2 x^2+5 d^3 e x+d^4+10 d e^3 x^3+5 e^4 x^4\right )\right )+B \left (-3 b^2 e^2 \left (5 d^2 e x+d^3+10 d e^2 x^2+10 e^3 x^3\right )-24 b c e \left (10 d^2 e^2 x^2+5 d^3 e x+d^4+10 d e^3 x^3+5 e^4 x^4\right )+c^2 d \left (1100 d^2 e^2 x^2+625 d^3 e x+137 d^4+900 d e^3 x^3+300 e^4 x^4\right )\right )+60 B c^2 (d+e x)^5 \log (d+e x)}{60 e^6 (d+e x)^5} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(b*x + c*x^2)^2)/(d + e*x)^6,x]

[Out]

(-2*A*e*(b^2*e^2*(d^2 + 5*d*e*x + 10*e^2*x^2) + 3*b*c*e*(d^3 + 5*d^2*e*x + 10*d*e^2*x^2 + 10*e^3*x^3) + 6*c^2*
(d^4 + 5*d^3*e*x + 10*d^2*e^2*x^2 + 10*d*e^3*x^3 + 5*e^4*x^4)) + B*(-3*b^2*e^2*(d^3 + 5*d^2*e*x + 10*d*e^2*x^2
+ 10*e^3*x^3) - 24*b*c*e*(d^4 + 5*d^3*e*x + 10*d^2*e^2*x^2 + 10*d*e^3*x^3 + 5*e^4*x^4) + c^2*d*(137*d^4 + 625
*d^3*e*x + 1100*d^2*e^2*x^2 + 900*d*e^3*x^3 + 300*e^4*x^4)) + 60*B*c^2*(d + e*x)^5*Log[d + e*x])/(60*e^6*(d +
e*x)^5)

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Maple [A]  time = 0.012, size = 472, normalized size = 1.9 \begin{align*} -{\frac{A{d}^{2}{b}^{2}}{5\,{e}^{3} \left ( ex+d \right ) ^{5}}}-{\frac{A{d}^{4}{c}^{2}}{5\,{e}^{5} \left ( ex+d \right ) ^{5}}}-{\frac{{b}^{2}B}{2\,{e}^{4} \left ( ex+d \right ) ^{2}}}-{\frac{2\,B{d}^{4}bc}{5\,{e}^{5} \left ( ex+d \right ) ^{5}}}+{\frac{2\,A{d}^{3}bc}{5\,{e}^{4} \left ( ex+d \right ) ^{5}}}-4\,{\frac{B{d}^{2}bc}{{e}^{5} \left ( ex+d \right ) ^{3}}}+4\,{\frac{Bdbc}{{e}^{5} \left ( ex+d \right ) ^{2}}}+2\,{\frac{Abcd}{{e}^{4} \left ( ex+d \right ) ^{3}}}-{\frac{3\,A{d}^{2}bc}{2\,{e}^{4} \left ( ex+d \right ) ^{4}}}+2\,{\frac{B{d}^{3}bc}{{e}^{5} \left ( ex+d \right ) ^{4}}}-{\frac{A{c}^{2}}{{e}^{5} \left ( ex+d \right ) }}-{\frac{A{b}^{2}}{3\,{e}^{3} \left ( ex+d \right ) ^{3}}}+{\frac{B{c}^{2}\ln \left ( ex+d \right ) }{{e}^{6}}}+{\frac{{b}^{2}Bd}{{e}^{4} \left ( ex+d \right ) ^{3}}}+{\frac{10\,B{c}^{2}{d}^{3}}{3\,{e}^{6} \left ( ex+d \right ) ^{3}}}-2\,{\frac{Bcb}{{e}^{5} \left ( ex+d \right ) }}+5\,{\frac{B{c}^{2}d}{{e}^{6} \left ( ex+d \right ) }}+{\frac{B{c}^{2}{d}^{5}}{5\,{e}^{6} \left ( ex+d \right ) ^{5}}}-{\frac{3\,{b}^{2}B{d}^{2}}{4\,{e}^{4} \left ( ex+d \right ) ^{4}}}-{\frac{5\,B{c}^{2}{d}^{4}}{4\,{e}^{6} \left ( ex+d \right ) ^{4}}}-2\,{\frac{A{c}^{2}{d}^{2}}{{e}^{5} \left ( ex+d \right ) ^{3}}}+{\frac{A{d}^{3}{c}^{2}}{{e}^{5} \left ( ex+d \right ) ^{4}}}-5\,{\frac{B{c}^{2}{d}^{2}}{{e}^{6} \left ( ex+d \right ) ^{2}}}+{\frac{Ad{b}^{2}}{2\,{e}^{3} \left ( ex+d \right ) ^{4}}}+{\frac{{b}^{2}B{d}^{3}}{5\,{e}^{4} \left ( ex+d \right ) ^{5}}}-{\frac{Abc}{{e}^{4} \left ( ex+d \right ) ^{2}}}+2\,{\frac{A{c}^{2}d}{{e}^{5} \left ( ex+d \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+b*x)^2/(e*x+d)^6,x)

[Out]

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

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Maxima [A]  time = 1.19591, size = 459, normalized size = 1.85 \begin{align*} \frac{137 \, B c^{2} d^{5} - 2 \, A b^{2} d^{2} e^{3} - 12 \,{\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} + 60 \,{\left (5 \, B c^{2} d e^{4} -{\left (2 \, B b c + A c^{2}\right )} e^{5}\right )} x^{4} + 30 \,{\left (30 \, B c^{2} d^{2} e^{3} - 4 \,{\left (2 \, B b c + A c^{2}\right )} d e^{4} -{\left (B b^{2} + 2 \, A b c\right )} e^{5}\right )} x^{3} + 10 \,{\left (110 \, B c^{2} d^{3} e^{2} - 2 \, A b^{2} e^{5} - 12 \,{\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} + 5 \,{\left (125 \, B c^{2} d^{4} e - 2 \, A b^{2} d e^{4} - 12 \,{\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}{60 \,{\left (e^{11} x^{5} + 5 \, d e^{10} x^{4} + 10 \, d^{2} e^{9} x^{3} + 10 \, d^{3} e^{8} x^{2} + 5 \, d^{4} e^{7} x + d^{5} e^{6}\right )}} + \frac{B c^{2} \log \left (e x + d\right )}{e^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^2/(e*x+d)^6,x, algorithm="maxima")

[Out]

1/60*(137*B*c^2*d^5 - 2*A*b^2*d^2*e^3 - 12*(2*B*b*c + A*c^2)*d^4*e - 3*(B*b^2 + 2*A*b*c)*d^3*e^2 + 60*(5*B*c^2
*d*e^4 - (2*B*b*c + A*c^2)*e^5)*x^4 + 30*(30*B*c^2*d^2*e^3 - 4*(2*B*b*c + A*c^2)*d*e^4 - (B*b^2 + 2*A*b*c)*e^5
)*x^3 + 10*(110*B*c^2*d^3*e^2 - 2*A*b^2*e^5 - 12*(2*B*b*c + A*c^2)*d^2*e^3 - 3*(B*b^2 + 2*A*b*c)*d*e^4)*x^2 +
5*(125*B*c^2*d^4*e - 2*A*b^2*d*e^4 - 12*(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^11*x^5
+ 5*d*e^10*x^4 + 10*d^2*e^9*x^3 + 10*d^3*e^8*x^2 + 5*d^4*e^7*x + d^5*e^6) + B*c^2*log(e*x + d)/e^6

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Fricas [A]  time = 1.50852, size = 871, normalized size = 3.51 \begin{align*} \frac{137 \, B c^{2} d^{5} - 2 \, A b^{2} d^{2} e^{3} - 12 \,{\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} + 60 \,{\left (5 \, B c^{2} d e^{4} -{\left (2 \, B b c + A c^{2}\right )} e^{5}\right )} x^{4} + 30 \,{\left (30 \, B c^{2} d^{2} e^{3} - 4 \,{\left (2 \, B b c + A c^{2}\right )} d e^{4} -{\left (B b^{2} + 2 \, A b c\right )} e^{5}\right )} x^{3} + 10 \,{\left (110 \, B c^{2} d^{3} e^{2} - 2 \, A b^{2} e^{5} - 12 \,{\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} + 5 \,{\left (125 \, B c^{2} d^{4} e - 2 \, A b^{2} d e^{4} - 12 \,{\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 + 60 \,{\left (B c^{2} e^{5} x^{5} + 5 \, B c^{2} d e^{4} x^{4} + 10 \, B c^{2} d^{2} e^{3} x^{3} + 10 \, B c^{2} d^{3} e^{2} x^{2} + 5 \, B c^{2} d^{4} e x + B c^{2} d^{5}\right )} \log \left (e x + d\right )}{60 \,{\left (e^{11} x^{5} + 5 \, d e^{10} x^{4} + 10 \, d^{2} e^{9} x^{3} + 10 \, d^{3} e^{8} x^{2} + 5 \, d^{4} e^{7} x + d^{5} e^{6}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^2/(e*x+d)^6,x, algorithm="fricas")

[Out]

1/60*(137*B*c^2*d^5 - 2*A*b^2*d^2*e^3 - 12*(2*B*b*c + A*c^2)*d^4*e - 3*(B*b^2 + 2*A*b*c)*d^3*e^2 + 60*(5*B*c^2
*d*e^4 - (2*B*b*c + A*c^2)*e^5)*x^4 + 30*(30*B*c^2*d^2*e^3 - 4*(2*B*b*c + A*c^2)*d*e^4 - (B*b^2 + 2*A*b*c)*e^5
)*x^3 + 10*(110*B*c^2*d^3*e^2 - 2*A*b^2*e^5 - 12*(2*B*b*c + A*c^2)*d^2*e^3 - 3*(B*b^2 + 2*A*b*c)*d*e^4)*x^2 +
5*(125*B*c^2*d^4*e - 2*A*b^2*d*e^4 - 12*(2*B*b*c + A*c^2)*d^3*e^2 - 3*(B*b^2 + 2*A*b*c)*d^2*e^3)*x + 60*(B*c^2
*e^5*x^5 + 5*B*c^2*d*e^4*x^4 + 10*B*c^2*d^2*e^3*x^3 + 10*B*c^2*d^3*e^2*x^2 + 5*B*c^2*d^4*e*x + B*c^2*d^5)*log(
e*x + d))/(e^11*x^5 + 5*d*e^10*x^4 + 10*d^2*e^9*x^3 + 10*d^3*e^8*x^2 + 5*d^4*e^7*x + d^5*e^6)

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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)*(c*x**2+b*x)**2/(e*x+d)**6,x)

[Out]

Timed out

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Giac [A]  time = 1.33111, size = 406, normalized size = 1.64 \begin{align*} B c^{2} e^{\left (-6\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{{\left (60 \,{\left (5 \, B c^{2} d e^{3} - 2 \, B b c e^{4} - A c^{2} e^{4}\right )} x^{4} + 30 \,{\left (30 \, B c^{2} d^{2} e^{2} - 8 \, B b c d e^{3} - 4 \, A c^{2} d e^{3} - B b^{2} e^{4} - 2 \, A b c e^{4}\right )} x^{3} + 10 \,{\left (110 \, B c^{2} d^{3} e - 24 \, B b c d^{2} e^{2} - 12 \, A c^{2} d^{2} e^{2} - 3 \, B b^{2} d e^{3} - 6 \, A b c d e^{3} - 2 \, A b^{2} e^{4}\right )} x^{2} + 5 \,{\left (125 \, B c^{2} d^{4} - 24 \, B b c d^{3} e - 12 \, A c^{2} d^{3} e - 3 \, B b^{2} d^{2} e^{2} - 6 \, A b c d^{2} e^{2} - 2 \, A b^{2} d e^{3}\right )} x +{\left (137 \, B c^{2} d^{5} - 24 \, B b c d^{4} e - 12 \, A c^{2} d^{4} e - 3 \, B b^{2} d^{3} e^{2} - 6 \, A b c d^{3} e^{2} - 2 \, A b^{2} d^{2} e^{3}\right )} e^{\left (-1\right )}\right )} e^{\left (-5\right )}}{60 \,{\left (x e + d\right )}^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^2/(e*x+d)^6,x, algorithm="giac")

[Out]

B*c^2*e^(-6)*log(abs(x*e + d)) + 1/60*(60*(5*B*c^2*d*e^3 - 2*B*b*c*e^4 - A*c^2*e^4)*x^4 + 30*(30*B*c^2*d^2*e^2
- 8*B*b*c*d*e^3 - 4*A*c^2*d*e^3 - B*b^2*e^4 - 2*A*b*c*e^4)*x^3 + 10*(110*B*c^2*d^3*e - 24*B*b*c*d^2*e^2 - 12*
A*c^2*d^2*e^2 - 3*B*b^2*d*e^3 - 6*A*b*c*d*e^3 - 2*A*b^2*e^4)*x^2 + 5*(125*B*c^2*d^4 - 24*B*b*c*d^3*e - 12*A*c^
2*d^3*e - 3*B*b^2*d^2*e^2 - 6*A*b*c*d^2*e^2 - 2*A*b^2*d*e^3)*x + (137*B*c^2*d^5 - 24*B*b*c*d^4*e - 12*A*c^2*d^
4*e - 3*B*b^2*d^3*e^2 - 6*A*b*c*d^3*e^2 - 2*A*b^2*d^2*e^3)*e^(-1))*e^(-5)/(x*e + d)^5