3.1781 \(\int \frac {(a c+(b c+a d) x+b d x^2)^2}{(a+b x)^9} \, dx\)

Optimal. Leaf size=65 \[ -\frac {2 d (b c-a d)}{5 b^3 (a+b x)^5}-\frac {(b c-a d)^2}{6 b^3 (a+b x)^6}-\frac {d^2}{4 b^3 (a+b x)^4} \]

[Out]

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

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Rubi [A]  time = 0.04, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {626, 43} \[ -\frac {2 d (b c-a d)}{5 b^3 (a+b x)^5}-\frac {(b c-a d)^2}{6 b^3 (a+b x)^6}-\frac {d^2}{4 b^3 (a+b x)^4} \]

Antiderivative was successfully verified.

[In]

Int[(a*c + (b*c + a*d)*x + b*d*x^2)^2/(a + b*x)^9,x]

[Out]

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 626

Int[((d_) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a
/d + (c*x)/e)^p, x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] &&
IntegerQ[p]

Rubi steps

\begin {align*} \int \frac {\left (a c+(b c+a d) x+b d x^2\right )^2}{(a+b x)^9} \, dx &=\int \frac {(c+d x)^2}{(a+b x)^7} \, dx\\ &=\int \left (\frac {(b c-a d)^2}{b^2 (a+b x)^7}+\frac {2 d (b c-a d)}{b^2 (a+b x)^6}+\frac {d^2}{b^2 (a+b x)^5}\right ) \, dx\\ &=-\frac {(b c-a d)^2}{6 b^3 (a+b x)^6}-\frac {2 d (b c-a d)}{5 b^3 (a+b x)^5}-\frac {d^2}{4 b^3 (a+b x)^4}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 58, normalized size = 0.89 \[ -\frac {a^2 d^2+2 a b d (2 c+3 d x)+b^2 \left (10 c^2+24 c d x+15 d^2 x^2\right )}{60 b^3 (a+b x)^6} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*c + (b*c + a*d)*x + b*d*x^2)^2/(a + b*x)^9,x]

[Out]

-1/60*(a^2*d^2 + 2*a*b*d*(2*c + 3*d*x) + b^2*(10*c^2 + 24*c*d*x + 15*d^2*x^2))/(b^3*(a + b*x)^6)

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fricas [B]  time = 0.89, size = 120, normalized size = 1.85 \[ -\frac {15 \, b^{2} d^{2} x^{2} + 10 \, b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2} + 6 \, {\left (4 \, b^{2} c d + a b d^{2}\right )} x}{60 \, {\left (b^{9} x^{6} + 6 \, a b^{8} x^{5} + 15 \, a^{2} b^{7} x^{4} + 20 \, a^{3} b^{6} x^{3} + 15 \, a^{4} b^{5} x^{2} + 6 \, a^{5} b^{4} x + a^{6} b^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*c+(a*d+b*c)*x+b*d*x^2)^2/(b*x+a)^9,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.16, size = 61, normalized size = 0.94 \[ -\frac {15 \, b^{2} d^{2} x^{2} + 24 \, b^{2} c d x + 6 \, a b d^{2} x + 10 \, b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}}{60 \, {\left (b x + a\right )}^{6} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*c+(a*d+b*c)*x+b*d*x^2)^2/(b*x+a)^9,x, algorithm="giac")

[Out]

-1/60*(15*b^2*d^2*x^2 + 24*b^2*c*d*x + 6*a*b*d^2*x + 10*b^2*c^2 + 4*a*b*c*d + a^2*d^2)/((b*x + a)^6*b^3)

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maple [A]  time = 0.06, size = 71, normalized size = 1.09 \[ -\frac {d^{2}}{4 \left (b x +a \right )^{4} b^{3}}+\frac {2 \left (a d -b c \right ) d}{5 \left (b x +a \right )^{5} b^{3}}-\frac {a^{2} d^{2}-2 a b c d +b^{2} c^{2}}{6 \left (b x +a \right )^{6} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*c+(a*d+b*c)*x+b*d*x^2)^2/(b*x+a)^9,x)

[Out]

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

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maxima [B]  time = 1.12, size = 120, normalized size = 1.85 \[ -\frac {15 \, b^{2} d^{2} x^{2} + 10 \, b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2} + 6 \, {\left (4 \, b^{2} c d + a b d^{2}\right )} x}{60 \, {\left (b^{9} x^{6} + 6 \, a b^{8} x^{5} + 15 \, a^{2} b^{7} x^{4} + 20 \, a^{3} b^{6} x^{3} + 15 \, a^{4} b^{5} x^{2} + 6 \, a^{5} b^{4} x + a^{6} b^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*c+(a*d+b*c)*x+b*d*x^2)^2/(b*x+a)^9,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.63, size = 118, normalized size = 1.82 \[ -\frac {\frac {a^2\,d^2+4\,a\,b\,c\,d+10\,b^2\,c^2}{60\,b^3}+\frac {d^2\,x^2}{4\,b}+\frac {d\,x\,\left (a\,d+4\,b\,c\right )}{10\,b^2}}{a^6+6\,a^5\,b\,x+15\,a^4\,b^2\,x^2+20\,a^3\,b^3\,x^3+15\,a^2\,b^4\,x^4+6\,a\,b^5\,x^5+b^6\,x^6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*c + x*(a*d + b*c) + b*d*x^2)^2/(a + b*x)^9,x)

[Out]

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

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sympy [B]  time = 1.36, size = 128, normalized size = 1.97 \[ \frac {- a^{2} d^{2} - 4 a b c d - 10 b^{2} c^{2} - 15 b^{2} d^{2} x^{2} + x \left (- 6 a b d^{2} - 24 b^{2} c d\right )}{60 a^{6} b^{3} + 360 a^{5} b^{4} x + 900 a^{4} b^{5} x^{2} + 1200 a^{3} b^{6} x^{3} + 900 a^{2} b^{7} x^{4} + 360 a b^{8} x^{5} + 60 b^{9} x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*c+(a*d+b*c)*x+b*d*x**2)**2/(b*x+a)**9,x)

[Out]

(-a**2*d**2 - 4*a*b*c*d - 10*b**2*c**2 - 15*b**2*d**2*x**2 + x*(-6*a*b*d**2 - 24*b**2*c*d))/(60*a**6*b**3 + 36
0*a**5*b**4*x + 900*a**4*b**5*x**2 + 1200*a**3*b**6*x**3 + 900*a**2*b**7*x**4 + 360*a*b**8*x**5 + 60*b**9*x**6
)

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