3.1125 \(\int \frac{(A+B x) (b x+c x^2)^2}{(d+e x)^7} \, dx\)

Optimal. Leaf size=253 \[ \frac{2 A c e (2 c d-b e)-B \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )}{3 e^6 (d+e x)^3}-\frac{A e \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )-B d \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )}{4 e^6 (d+e x)^4}+\frac{d^2 (B d-A e) (c d-b e)^2}{6 e^6 (d+e x)^6}+\frac{c (-A c e-2 b B e+5 B c d)}{2 e^6 (d+e x)^2}-\frac{d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{5 e^6 (d+e x)^5}-\frac{B c^2}{e^6 (d+e x)} \]

[Out]

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

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Rubi [A]  time = 0.226653, antiderivative size = 253, 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} \[ \frac{2 A c e (2 c d-b e)-B \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )}{3 e^6 (d+e x)^3}-\frac{A e \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )-B d \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )}{4 e^6 (d+e x)^4}+\frac{d^2 (B d-A e) (c d-b e)^2}{6 e^6 (d+e x)^6}+\frac{c (-A c e-2 b B e+5 B c d)}{2 e^6 (d+e x)^2}-\frac{d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{5 e^6 (d+e x)^5}-\frac{B c^2}{e^6 (d+e x)} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(b*x + c*x^2)^2)/(d + e*x)^7,x]

[Out]

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

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

Mathematica [A]  time = 0.120679, size = 257, normalized size = 1.02 \[ -\frac{A e \left (b^2 e^2 \left (d^2+6 d e x+15 e^2 x^2\right )+2 b c e \left (6 d^2 e x+d^3+15 d e^2 x^2+20 e^3 x^3\right )+2 c^2 \left (15 d^2 e^2 x^2+6 d^3 e x+d^4+20 d e^3 x^3+15 e^4 x^4\right )\right )+B \left (b^2 e^2 \left (6 d^2 e x+d^3+15 d e^2 x^2+20 e^3 x^3\right )+4 b c e \left (15 d^2 e^2 x^2+6 d^3 e x+d^4+20 d e^3 x^3+15 e^4 x^4\right )+10 c^2 \left (15 d^3 e^2 x^2+20 d^2 e^3 x^3+6 d^4 e x+d^5+15 d e^4 x^4+6 e^5 x^5\right )\right )}{60 e^6 (d+e x)^6} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(b*x + c*x^2)^2)/(d + e*x)^7,x]

[Out]

-(A*e*(b^2*e^2*(d^2 + 6*d*e*x + 15*e^2*x^2) + 2*b*c*e*(d^3 + 6*d^2*e*x + 15*d*e^2*x^2 + 20*e^3*x^3) + 2*c^2*(d
^4 + 6*d^3*e*x + 15*d^2*e^2*x^2 + 20*d*e^3*x^3 + 15*e^4*x^4)) + B*(b^2*e^2*(d^3 + 6*d^2*e*x + 15*d*e^2*x^2 + 2
0*e^3*x^3) + 4*b*c*e*(d^4 + 6*d^3*e*x + 15*d^2*e^2*x^2 + 20*d*e^3*x^3 + 15*e^4*x^4) + 10*c^2*(d^5 + 6*d^4*e*x
+ 15*d^3*e^2*x^2 + 20*d^2*e^3*x^3 + 15*d*e^4*x^4 + 6*e^5*x^5)))/(60*e^6*(d + e*x)^6)

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Maple [A]  time = 0.008, size = 307, normalized size = 1.2 \begin{align*} -{\frac{{d}^{2} \left ( A{b}^{2}{e}^{3}-2\,Abcd{e}^{2}+A{c}^{2}{d}^{2}e-{b}^{2}Bd{e}^{2}+2\,B{d}^{2}bce-B{c}^{2}{d}^{3} \right ) }{6\,{e}^{6} \left ( ex+d \right ) ^{6}}}-{\frac{A{b}^{2}{e}^{3}-6\,Abcd{e}^{2}+6\,A{c}^{2}{d}^{2}e-3\,{b}^{2}Bd{e}^{2}+12\,B{d}^{2}bce-10\,B{c}^{2}{d}^{3}}{4\,{e}^{6} \left ( ex+d \right ) ^{4}}}-{\frac{2\,Abc{e}^{2}-4\,A{c}^{2}de+B{e}^{2}{b}^{2}-8\,Bdbce+10\,B{c}^{2}{d}^{2}}{3\,{e}^{6} \left ( ex+d \right ) ^{3}}}-{\frac{c \left ( Ace+2\,bBe-5\,Bcd \right ) }{2\,{e}^{6} \left ( ex+d \right ) ^{2}}}+{\frac{d \left ( 2\,A{b}^{2}{e}^{3}-6\,Abcd{e}^{2}+4\,A{c}^{2}{d}^{2}e-3\,{b}^{2}Bd{e}^{2}+8\,B{d}^{2}bce-5\,B{c}^{2}{d}^{3} \right ) }{5\,{e}^{6} \left ( ex+d \right ) ^{5}}}-{\frac{B{c}^{2}}{{e}^{6} \left ( ex+d \right ) }} \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)^7,x)

[Out]

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

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Maxima [A]  time = 1.14904, size = 459, normalized size = 1.81 \begin{align*} -\frac{60 \, B c^{2} e^{5} x^{5} + 10 \, B c^{2} d^{5} + A b^{2} d^{2} e^{3} + 2 \,{\left (2 \, B b c + A c^{2}\right )} d^{4} e +{\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{2} + 30 \,{\left (5 \, B c^{2} d e^{4} +{\left (2 \, B b c + A c^{2}\right )} e^{5}\right )} x^{4} + 20 \,{\left (10 \, B c^{2} d^{2} e^{3} + 2 \,{\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} + 15 \,{\left (10 \, B c^{2} d^{3} e^{2} + A b^{2} e^{5} + 2 \,{\left (2 \, B b c + A c^{2}\right )} d^{2} e^{3} +{\left (B b^{2} + 2 \, A b c\right )} d e^{4}\right )} x^{2} + 6 \,{\left (10 \, B c^{2} d^{4} e + A b^{2} d e^{4} + 2 \,{\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} +{\left (B b^{2} + 2 \, A b c\right )} d^{2} e^{3}\right )} x}{60 \,{\left (e^{12} x^{6} + 6 \, d e^{11} x^{5} + 15 \, d^{2} e^{10} x^{4} + 20 \, d^{3} e^{9} x^{3} + 15 \, d^{4} e^{8} x^{2} + 6 \, d^{5} e^{7} x + d^{6} 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)^7,x, algorithm="maxima")

[Out]

-1/60*(60*B*c^2*e^5*x^5 + 10*B*c^2*d^5 + A*b^2*d^2*e^3 + 2*(2*B*b*c + A*c^2)*d^4*e + (B*b^2 + 2*A*b*c)*d^3*e^2
+ 30*(5*B*c^2*d*e^4 + (2*B*b*c + A*c^2)*e^5)*x^4 + 20*(10*B*c^2*d^2*e^3 + 2*(2*B*b*c + A*c^2)*d*e^4 + (B*b^2
+ 2*A*b*c)*e^5)*x^3 + 15*(10*B*c^2*d^3*e^2 + A*b^2*e^5 + 2*(2*B*b*c + A*c^2)*d^2*e^3 + (B*b^2 + 2*A*b*c)*d*e^4
)*x^2 + 6*(10*B*c^2*d^4*e + A*b^2*d*e^4 + 2*(2*B*b*c + A*c^2)*d^3*e^2 + (B*b^2 + 2*A*b*c)*d^2*e^3)*x)/(e^12*x^
6 + 6*d*e^11*x^5 + 15*d^2*e^10*x^4 + 20*d^3*e^9*x^3 + 15*d^4*e^8*x^2 + 6*d^5*e^7*x + d^6*e^6)

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Fricas [A]  time = 1.81625, size = 722, normalized size = 2.85 \begin{align*} -\frac{60 \, B c^{2} e^{5} x^{5} + 10 \, B c^{2} d^{5} + A b^{2} d^{2} e^{3} + 2 \,{\left (2 \, B b c + A c^{2}\right )} d^{4} e +{\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{2} + 30 \,{\left (5 \, B c^{2} d e^{4} +{\left (2 \, B b c + A c^{2}\right )} e^{5}\right )} x^{4} + 20 \,{\left (10 \, B c^{2} d^{2} e^{3} + 2 \,{\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} + 15 \,{\left (10 \, B c^{2} d^{3} e^{2} + A b^{2} e^{5} + 2 \,{\left (2 \, B b c + A c^{2}\right )} d^{2} e^{3} +{\left (B b^{2} + 2 \, A b c\right )} d e^{4}\right )} x^{2} + 6 \,{\left (10 \, B c^{2} d^{4} e + A b^{2} d e^{4} + 2 \,{\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} +{\left (B b^{2} + 2 \, A b c\right )} d^{2} e^{3}\right )} x}{60 \,{\left (e^{12} x^{6} + 6 \, d e^{11} x^{5} + 15 \, d^{2} e^{10} x^{4} + 20 \, d^{3} e^{9} x^{3} + 15 \, d^{4} e^{8} x^{2} + 6 \, d^{5} e^{7} x + d^{6} 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)^7,x, algorithm="fricas")

[Out]

-1/60*(60*B*c^2*e^5*x^5 + 10*B*c^2*d^5 + A*b^2*d^2*e^3 + 2*(2*B*b*c + A*c^2)*d^4*e + (B*b^2 + 2*A*b*c)*d^3*e^2
+ 30*(5*B*c^2*d*e^4 + (2*B*b*c + A*c^2)*e^5)*x^4 + 20*(10*B*c^2*d^2*e^3 + 2*(2*B*b*c + A*c^2)*d*e^4 + (B*b^2
+ 2*A*b*c)*e^5)*x^3 + 15*(10*B*c^2*d^3*e^2 + A*b^2*e^5 + 2*(2*B*b*c + A*c^2)*d^2*e^3 + (B*b^2 + 2*A*b*c)*d*e^4
)*x^2 + 6*(10*B*c^2*d^4*e + A*b^2*d*e^4 + 2*(2*B*b*c + A*c^2)*d^3*e^2 + (B*b^2 + 2*A*b*c)*d^2*e^3)*x)/(e^12*x^
6 + 6*d*e^11*x^5 + 15*d^2*e^10*x^4 + 20*d^3*e^9*x^3 + 15*d^4*e^8*x^2 + 6*d^5*e^7*x + d^6*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)**7,x)

[Out]

Timed out

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Giac [A]  time = 1.33705, size = 429, normalized size = 1.7 \begin{align*} -\frac{{\left (60 \, B c^{2} x^{5} e^{5} + 150 \, B c^{2} d x^{4} e^{4} + 200 \, B c^{2} d^{2} x^{3} e^{3} + 150 \, B c^{2} d^{3} x^{2} e^{2} + 60 \, B c^{2} d^{4} x e + 10 \, B c^{2} d^{5} + 60 \, B b c x^{4} e^{5} + 30 \, A c^{2} x^{4} e^{5} + 80 \, B b c d x^{3} e^{4} + 40 \, A c^{2} d x^{3} e^{4} + 60 \, B b c d^{2} x^{2} e^{3} + 30 \, A c^{2} d^{2} x^{2} e^{3} + 24 \, B b c d^{3} x e^{2} + 12 \, A c^{2} d^{3} x e^{2} + 4 \, B b c d^{4} e + 2 \, A c^{2} d^{4} e + 20 \, B b^{2} x^{3} e^{5} + 40 \, A b c x^{3} e^{5} + 15 \, B b^{2} d x^{2} e^{4} + 30 \, A b c d x^{2} e^{4} + 6 \, B b^{2} d^{2} x e^{3} + 12 \, A b c d^{2} x e^{3} + B b^{2} d^{3} e^{2} + 2 \, A b c d^{3} e^{2} + 15 \, A b^{2} x^{2} e^{5} + 6 \, A b^{2} d x e^{4} + A b^{2} d^{2} e^{3}\right )} e^{\left (-6\right )}}{60 \,{\left (x e + d\right )}^{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)^7,x, algorithm="giac")

[Out]

-1/60*(60*B*c^2*x^5*e^5 + 150*B*c^2*d*x^4*e^4 + 200*B*c^2*d^2*x^3*e^3 + 150*B*c^2*d^3*x^2*e^2 + 60*B*c^2*d^4*x
*e + 10*B*c^2*d^5 + 60*B*b*c*x^4*e^5 + 30*A*c^2*x^4*e^5 + 80*B*b*c*d*x^3*e^4 + 40*A*c^2*d*x^3*e^4 + 60*B*b*c*d
^2*x^2*e^3 + 30*A*c^2*d^2*x^2*e^3 + 24*B*b*c*d^3*x*e^2 + 12*A*c^2*d^3*x*e^2 + 4*B*b*c*d^4*e + 2*A*c^2*d^4*e +
20*B*b^2*x^3*e^5 + 40*A*b*c*x^3*e^5 + 15*B*b^2*d*x^2*e^4 + 30*A*b*c*d*x^2*e^4 + 6*B*b^2*d^2*x*e^3 + 12*A*b*c*d
^2*x*e^3 + B*b^2*d^3*e^2 + 2*A*b*c*d^3*e^2 + 15*A*b^2*x^2*e^5 + 6*A*b^2*d*x*e^4 + A*b^2*d^2*e^3)*e^(-6)/(x*e +
d)^6