3.1778 \(\int \frac {(a c+(b c+a d) x+b d x^2)^2}{(a+b x)^6} \, dx\)

Optimal. Leaf size=28 \[ -\frac {(c+d x)^3}{3 (a+b x)^3 (b c-a d)} \]

[Out]

-1/3*(d*x+c)^3/(-a*d+b*c)/(b*x+a)^3

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Rubi [A]  time = 0.01, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {626, 37} \[ -\frac {(c+d x)^3}{3 (a+b x)^3 (b c-a d)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(c + d*x)^3/(3*(b*c - a*d)*(a + b*x)^3)

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

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

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Mathematica [A]  time = 0.02, size = 53, normalized size = 1.89 \[ -\frac {a^2 d^2+a b d (c+3 d x)+b^2 \left (c^2+3 c d x+3 d^2 x^2\right )}{3 b^3 (a+b x)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 1.01, size = 84, normalized size = 3.00 \[ -\frac {3 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + a b c d + a^{2} d^{2} + 3 \, {\left (b^{2} c d + a b d^{2}\right )} x}{3 \, {\left (b^{6} x^{3} + 3 \, a b^{5} x^{2} + 3 \, a^{2} b^{4} x + a^{3} b^{3}\right )}} \]

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)^6,x, algorithm="fricas")

[Out]

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

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giac [B]  time = 0.16, size = 59, normalized size = 2.11 \[ -\frac {3 \, b^{2} d^{2} x^{2} + 3 \, b^{2} c d x + 3 \, a b d^{2} x + b^{2} c^{2} + a b c d + a^{2} d^{2}}{3 \, {\left (b x + a\right )}^{3} b^{3}} \]

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)^6,x, algorithm="giac")

[Out]

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

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maple [B]  time = 0.06, size = 70, normalized size = 2.50 \[ -\frac {d^{2}}{\left (b x +a \right ) b^{3}}+\frac {\left (a d -b c \right ) d}{\left (b x +a \right )^{2} b^{3}}-\frac {a^{2} d^{2}-2 a b c d +b^{2} c^{2}}{3 \left (b x +a \right )^{3} b^{3}} \]

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)^6,x)

[Out]

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

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maxima [B]  time = 1.06, size = 84, normalized size = 3.00 \[ -\frac {3 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + a b c d + a^{2} d^{2} + 3 \, {\left (b^{2} c d + a b d^{2}\right )} x}{3 \, {\left (b^{6} x^{3} + 3 \, a b^{5} x^{2} + 3 \, a^{2} b^{4} x + a^{3} b^{3}\right )}} \]

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)^6,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.56, size = 80, normalized size = 2.86 \[ -\frac {\frac {a^2\,d^2+a\,b\,c\,d+b^2\,c^2}{3\,b^3}+\frac {d^2\,x^2}{b}+\frac {d\,x\,\left (a\,d+b\,c\right )}{b^2}}{a^3+3\,a^2\,b\,x+3\,a\,b^2\,x^2+b^3\,x^3} \]

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)^6,x)

[Out]

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

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sympy [B]  time = 0.70, size = 88, normalized size = 3.14 \[ \frac {- a^{2} d^{2} - a b c d - b^{2} c^{2} - 3 b^{2} d^{2} x^{2} + x \left (- 3 a b d^{2} - 3 b^{2} c d\right )}{3 a^{3} b^{3} + 9 a^{2} b^{4} x + 9 a b^{5} x^{2} + 3 b^{6} x^{3}} \]

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)**6,x)

[Out]

(-a**2*d**2 - a*b*c*d - b**2*c**2 - 3*b**2*d**2*x**2 + x*(-3*a*b*d**2 - 3*b**2*c*d))/(3*a**3*b**3 + 9*a**2*b**
4*x + 9*a*b**5*x**2 + 3*b**6*x**3)

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