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

Optimal. Leaf size=134 \[ \frac {A e (2 c d-b e)-B \left (3 c d^2-e (2 b d-a e)\right )}{5 e^4 (d+e x)^5}+\frac {(B d-A e) \left (a e^2-b d e+c d^2\right )}{6 e^4 (d+e x)^6}+\frac {-A c e-b B e+3 B c d}{4 e^4 (d+e x)^4}-\frac {B c}{3 e^4 (d+e x)^3} \]

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Rubi [A]  time = 0.10, antiderivative size = 133, 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} \begin {gather*} -\frac {-B e (2 b d-a e)-A e (2 c d-b e)+3 B c d^2}{5 e^4 (d+e x)^5}+\frac {(B d-A e) \left (a e^2-b d e+c d^2\right )}{6 e^4 (d+e x)^6}+\frac {-A c e-b B e+3 B c d}{4 e^4 (d+e x)^4}-\frac {B c}{3 e^4 (d+e x)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [A]  time = 0.06, size = 119, normalized size = 0.89 \begin {gather*} -\frac {A e \left (2 e (5 a e+b d+6 b e x)+c \left (d^2+6 d e x+15 e^2 x^2\right )\right )+B \left (e \left (2 a e (d+6 e x)+b \left (d^2+6 d e x+15 e^2 x^2\right )\right )+c \left (d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right )\right )}{60 e^4 (d+e x)^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/60*(A*e*(2*e*(b*d + 5*a*e + 6*b*e*x) + c*(d^2 + 6*d*e*x + 15*e^2*x^2)) + B*(c*(d^3 + 6*d^2*e*x + 15*d*e^2*x
^2 + 20*e^3*x^3) + e*(2*a*e*(d + 6*e*x) + b*(d^2 + 6*d*e*x + 15*e^2*x^2))))/(e^4*(d + e*x)^6)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(A+B x) \left (a+b x+c x^2\right )}{(d+e x)^7} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.36, size = 178, normalized size = 1.33 \begin {gather*} -\frac {20 \, B c e^{3} x^{3} + B c d^{3} + 10 \, A a e^{3} + {\left (B b + A c\right )} d^{2} e + 2 \, {\left (B a + A b\right )} d e^{2} + 15 \, {\left (B c d e^{2} + {\left (B b + A c\right )} e^{3}\right )} x^{2} + 6 \, {\left (B c d^{2} e + {\left (B b + A c\right )} d e^{2} + 2 \, {\left (B a + A b\right )} e^{3}\right )} x}{60 \, {\left (e^{10} x^{6} + 6 \, d e^{9} x^{5} + 15 \, d^{2} e^{8} x^{4} + 20 \, d^{3} e^{7} x^{3} + 15 \, d^{4} e^{6} x^{2} + 6 \, d^{5} e^{5} x + d^{6} e^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x+a)/(e*x+d)^7,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.16, size = 132, normalized size = 0.99 \begin {gather*} -\frac {{\left (20 \, B c x^{3} e^{3} + 15 \, B c d x^{2} e^{2} + 6 \, B c d^{2} x e + B c d^{3} + 15 \, B b x^{2} e^{3} + 15 \, A c x^{2} e^{3} + 6 \, B b d x e^{2} + 6 \, A c d x e^{2} + B b d^{2} e + A c d^{2} e + 12 \, B a x e^{3} + 12 \, A b x e^{3} + 2 \, B a d e^{2} + 2 \, A b d e^{2} + 10 \, A a e^{3}\right )} e^{\left (-4\right )}}{60 \, {\left (x e + d\right )}^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x+a)/(e*x+d)^7,x, algorithm="giac")

[Out]

-1/60*(20*B*c*x^3*e^3 + 15*B*c*d*x^2*e^2 + 6*B*c*d^2*x*e + B*c*d^3 + 15*B*b*x^2*e^3 + 15*A*c*x^2*e^3 + 6*B*b*d
*x*e^2 + 6*A*c*d*x*e^2 + B*b*d^2*e + A*c*d^2*e + 12*B*a*x*e^3 + 12*A*b*x*e^3 + 2*B*a*d*e^2 + 2*A*b*d*e^2 + 10*
A*a*e^3)*e^(-4)/(x*e + d)^6

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maple [A]  time = 0.05, size = 142, normalized size = 1.06 \begin {gather*} -\frac {B c}{3 \left (e x +d \right )^{3} 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}}{6 \left (e x +d \right )^{6} e^{4}}-\frac {A c e +B b e -3 B c d}{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}}{5 \left (e x +d \right )^{5} e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+b*x+a)/(e*x+d)^7,x)

[Out]

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

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maxima [A]  time = 0.52, size = 178, normalized size = 1.33 \begin {gather*} -\frac {20 \, B c e^{3} x^{3} + B c d^{3} + 10 \, A a e^{3} + {\left (B b + A c\right )} d^{2} e + 2 \, {\left (B a + A b\right )} d e^{2} + 15 \, {\left (B c d e^{2} + {\left (B b + A c\right )} e^{3}\right )} x^{2} + 6 \, {\left (B c d^{2} e + {\left (B b + A c\right )} d e^{2} + 2 \, {\left (B a + A b\right )} e^{3}\right )} x}{60 \, {\left (e^{10} x^{6} + 6 \, d e^{9} x^{5} + 15 \, d^{2} e^{8} x^{4} + 20 \, d^{3} e^{7} x^{3} + 15 \, d^{4} e^{6} x^{2} + 6 \, d^{5} e^{5} x + d^{6} e^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x+a)/(e*x+d)^7,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 2.36, size = 182, normalized size = 1.36 \begin {gather*} -\frac {\frac {10\,A\,a\,e^3+B\,c\,d^3+2\,A\,b\,d\,e^2+2\,B\,a\,d\,e^2+A\,c\,d^2\,e+B\,b\,d^2\,e}{60\,e^4}+\frac {x^2\,\left (A\,c\,e+B\,b\,e+B\,c\,d\right )}{4\,e^2}+\frac {x\,\left (2\,A\,b\,e^2+2\,B\,a\,e^2+B\,c\,d^2+A\,c\,d\,e+B\,b\,d\,e\right )}{10\,e^3}+\frac {B\,c\,x^3}{3\,e}}{d^6+6\,d^5\,e\,x+15\,d^4\,e^2\,x^2+20\,d^3\,e^3\,x^3+15\,d^2\,e^4\,x^4+6\,d\,e^5\,x^5+e^6\,x^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a + b*x + c*x^2))/(d + e*x)^7,x)

[Out]

-((10*A*a*e^3 + B*c*d^3 + 2*A*b*d*e^2 + 2*B*a*d*e^2 + A*c*d^2*e + B*b*d^2*e)/(60*e^4) + (x^2*(A*c*e + B*b*e +
B*c*d))/(4*e^2) + (x*(2*A*b*e^2 + 2*B*a*e^2 + B*c*d^2 + A*c*d*e + B*b*d*e))/(10*e^3) + (B*c*x^3)/(3*e))/(d^6 +
e^6*x^6 + 6*d*e^5*x^5 + 15*d^4*e^2*x^2 + 20*d^3*e^3*x^3 + 15*d^2*e^4*x^4 + 6*d^5*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+a)/(e*x+d)**7,x)

[Out]

Timed out

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