3.2315 \(\int \frac {(A+B x) (a+b x+c x^2)}{(d+e x)^5} \, dx\)

Optimal. Leaf size=132 \[ \frac {A e (2 c d-b e)-B \left (3 c d^2-e (2 b d-a e)\right )}{3 e^4 (d+e x)^3}+\frac {(B d-A e) \left (a e^2-b d e+c d^2\right )}{4 e^4 (d+e x)^4}+\frac {-A c e-b B e+3 B c d}{2 e^4 (d+e x)^2}-\frac {B c}{e^4 (d+e x)} \]

[Out]

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

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Rubi [A]  time = 0.12, antiderivative size = 131, normalized size of antiderivative = 0.99, number of steps used = 2, number of rules used = 1, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {771} \[ -\frac {-B e (2 b d-a e)-A e (2 c d-b e)+3 B c d^2}{3 e^4 (d+e x)^3}+\frac {(B d-A e) \left (a e^2-b d e+c d^2\right )}{4 e^4 (d+e x)^4}+\frac {-A c e-b B e+3 B c d}{2 e^4 (d+e x)^2}-\frac {B c}{e^4 (d+e x)} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a + b*x + c*x^2))/(d + e*x)^5,x]

[Out]

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

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Mathematica [A]  time = 0.06, size = 118, normalized size = 0.89 \[ -\frac {A e \left (e (3 a e+b d+4 b e x)+c \left (d^2+4 d e x+6 e^2 x^2\right )\right )+B \left (e \left (a e (d+4 e x)+b \left (d^2+4 d e x+6 e^2 x^2\right )\right )+3 c \left (d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right )\right )}{12 e^4 (d+e x)^4} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a + b*x + c*x^2))/(d + e*x)^5,x]

[Out]

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

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fricas [A]  time = 1.10, size = 157, normalized size = 1.19 \[ -\frac {12 \, B c e^{3} x^{3} + 3 \, B c d^{3} + 3 \, A a e^{3} + {\left (B b + A c\right )} d^{2} e + {\left (B a + A b\right )} d e^{2} + 6 \, {\left (3 \, B c d e^{2} + {\left (B b + A c\right )} e^{3}\right )} x^{2} + 4 \, {\left (3 \, B c d^{2} e + {\left (B b + A c\right )} d e^{2} + {\left (B a + A b\right )} e^{3}\right )} x}{12 \, {\left (e^{8} x^{4} + 4 \, d e^{7} x^{3} + 6 \, d^{2} e^{6} x^{2} + 4 \, d^{3} e^{5} x + d^{4} e^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x+a)/(e*x+d)^5,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.18, size = 220, normalized size = 1.67 \[ -\frac {1}{12} \, {\left (\frac {12 \, B c e^{\left (-1\right )}}{x e + d} - \frac {18 \, B c d e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}} + \frac {12 \, B c d^{2} e^{\left (-1\right )}}{{\left (x e + d\right )}^{3}} - \frac {3 \, B c d^{3} e^{\left (-1\right )}}{{\left (x e + d\right )}^{4}} + \frac {6 \, B b}{{\left (x e + d\right )}^{2}} + \frac {6 \, A c}{{\left (x e + d\right )}^{2}} - \frac {8 \, B b d}{{\left (x e + d\right )}^{3}} - \frac {8 \, A c d}{{\left (x e + d\right )}^{3}} + \frac {3 \, B b d^{2}}{{\left (x e + d\right )}^{4}} + \frac {3 \, A c d^{2}}{{\left (x e + d\right )}^{4}} + \frac {4 \, B a e}{{\left (x e + d\right )}^{3}} + \frac {4 \, A b e}{{\left (x e + d\right )}^{3}} - \frac {3 \, B a d e}{{\left (x e + d\right )}^{4}} - \frac {3 \, A b d e}{{\left (x e + d\right )}^{4}} + \frac {3 \, A a e^{2}}{{\left (x e + d\right )}^{4}}\right )} e^{\left (-3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x+a)/(e*x+d)^5,x, algorithm="giac")

[Out]

-1/12*(12*B*c*e^(-1)/(x*e + d) - 18*B*c*d*e^(-1)/(x*e + d)^2 + 12*B*c*d^2*e^(-1)/(x*e + d)^3 - 3*B*c*d^3*e^(-1
)/(x*e + d)^4 + 6*B*b/(x*e + d)^2 + 6*A*c/(x*e + d)^2 - 8*B*b*d/(x*e + d)^3 - 8*A*c*d/(x*e + d)^3 + 3*B*b*d^2/
(x*e + d)^4 + 3*A*c*d^2/(x*e + d)^4 + 4*B*a*e/(x*e + d)^3 + 4*A*b*e/(x*e + d)^3 - 3*B*a*d*e/(x*e + d)^4 - 3*A*
b*d*e/(x*e + d)^4 + 3*A*a*e^2/(x*e + d)^4)*e^(-3)

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maple [A]  time = 0.05, size = 142, normalized size = 1.08 \[ -\frac {B c}{\left (e x +d \right ) e^{4}}-\frac {A c e +B b e -3 B c d}{2 \left (e x +d \right )^{2} e^{4}}-\frac {a A \,e^{3}-A b d \,e^{2}+A c \,d^{2} e -a B d \,e^{2}+B \,d^{2} b e -B c \,d^{3}}{4 \left (e x +d \right )^{4} e^{4}}-\frac {A b \,e^{2}-2 A c d e +B a \,e^{2}-2 B b d e +3 B c \,d^{2}}{3 \left (e x +d \right )^{3} e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+b*x+a)/(e*x+d)^5,x)

[Out]

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

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maxima [A]  time = 0.65, size = 157, normalized size = 1.19 \[ -\frac {12 \, B c e^{3} x^{3} + 3 \, B c d^{3} + 3 \, A a e^{3} + {\left (B b + A c\right )} d^{2} e + {\left (B a + A b\right )} d e^{2} + 6 \, {\left (3 \, B c d e^{2} + {\left (B b + A c\right )} e^{3}\right )} x^{2} + 4 \, {\left (3 \, B c d^{2} e + {\left (B b + A c\right )} d e^{2} + {\left (B a + A b\right )} e^{3}\right )} x}{12 \, {\left (e^{8} x^{4} + 4 \, d e^{7} x^{3} + 6 \, d^{2} e^{6} x^{2} + 4 \, d^{3} e^{5} x + d^{4} e^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x+a)/(e*x+d)^5,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.08, size = 158, normalized size = 1.20 \[ -\frac {\frac {3\,A\,a\,e^3+3\,B\,c\,d^3+A\,b\,d\,e^2+B\,a\,d\,e^2+A\,c\,d^2\,e+B\,b\,d^2\,e}{12\,e^4}+\frac {x^2\,\left (A\,c\,e+B\,b\,e+3\,B\,c\,d\right )}{2\,e^2}+\frac {x\,\left (A\,b\,e^2+B\,a\,e^2+3\,B\,c\,d^2+A\,c\,d\,e+B\,b\,d\,e\right )}{3\,e^3}+\frac {B\,c\,x^3}{e}}{d^4+4\,d^3\,e\,x+6\,d^2\,e^2\,x^2+4\,d\,e^3\,x^3+e^4\,x^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a + b*x + c*x^2))/(d + e*x)^5,x)

[Out]

-((3*A*a*e^3 + 3*B*c*d^3 + A*b*d*e^2 + B*a*d*e^2 + A*c*d^2*e + B*b*d^2*e)/(12*e^4) + (x^2*(A*c*e + B*b*e + 3*B
*c*d))/(2*e^2) + (x*(A*b*e^2 + B*a*e^2 + 3*B*c*d^2 + A*c*d*e + B*b*d*e))/(3*e^3) + (B*c*x^3)/e)/(d^4 + e^4*x^4
+ 4*d*e^3*x^3 + 6*d^2*e^2*x^2 + 4*d^3*e*x)

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sympy [A]  time = 45.73, size = 194, normalized size = 1.47 \[ \frac {- 3 A a e^{3} - A b d e^{2} - A c d^{2} e - B a d e^{2} - B b d^{2} e - 3 B c d^{3} - 12 B c e^{3} x^{3} + x^{2} \left (- 6 A c e^{3} - 6 B b e^{3} - 18 B c d e^{2}\right ) + x \left (- 4 A b e^{3} - 4 A c d e^{2} - 4 B a e^{3} - 4 B b d e^{2} - 12 B c d^{2} e\right )}{12 d^{4} e^{4} + 48 d^{3} e^{5} x + 72 d^{2} e^{6} x^{2} + 48 d e^{7} x^{3} + 12 e^{8} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+b*x+a)/(e*x+d)**5,x)

[Out]

(-3*A*a*e**3 - A*b*d*e**2 - A*c*d**2*e - B*a*d*e**2 - B*b*d**2*e - 3*B*c*d**3 - 12*B*c*e**3*x**3 + x**2*(-6*A*
c*e**3 - 6*B*b*e**3 - 18*B*c*d*e**2) + x*(-4*A*b*e**3 - 4*A*c*d*e**2 - 4*B*a*e**3 - 4*B*b*d*e**2 - 12*B*c*d**2
*e))/(12*d**4*e**4 + 48*d**3*e**5*x + 72*d**2*e**6*x**2 + 48*d*e**7*x**3 + 12*e**8*x**4)

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