3.16.8 \(\int \frac {(a+b x)^4}{(a c+(b c+a d) x+b d x^2)^3} \, dx\)

Optimal. Leaf size=28 \[ \frac {(a+b x)^2}{2 (c+d x)^2 (b c-a d)} \]

________________________________________________________________________________________

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} \begin {gather*} \frac {(a+b x)^2}{2 (c+d x)^2 (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^4/(a*c + (b*c + a*d)*x + b*d*x^2)^3,x]

[Out]

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

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

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 26, normalized size = 0.93 \begin {gather*} -\frac {a d+b (c+2 d x)}{2 d^2 (c+d x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^4/(a*c + (b*c + a*d)*x + b*d*x^2)^3,x]

[Out]

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

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+b x)^4}{\left (a c+(b c+a d) x+b d x^2\right )^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + b*x)^4/(a*c + (b*c + a*d)*x + b*d*x^2)^3,x]

[Out]

IntegrateAlgebraic[(a + b*x)^4/(a*c + (b*c + a*d)*x + b*d*x^2)^3, x]

________________________________________________________________________________________

fricas [A]  time = 0.40, size = 38, normalized size = 1.36 \begin {gather*} -\frac {2 \, b d x + b c + a d}{2 \, {\left (d^{4} x^{2} + 2 \, c d^{3} x + c^{2} d^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^4/(a*c+(a*d+b*c)*x+b*d*x^2)^3,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

giac [A]  time = 0.21, size = 24, normalized size = 0.86 \begin {gather*} -\frac {2 \, b d x + b c + a d}{2 \, {\left (d x + c\right )}^{2} d^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^4/(a*c+(a*d+b*c)*x+b*d*x^2)^3,x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.06, size = 35, normalized size = 1.25 \begin {gather*} -\frac {b}{\left (d x +c \right ) d^{2}}-\frac {a d -b c}{2 \left (d x +c \right )^{2} d^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^4/(a*c+(a*d+b*c)*x+b*d*x^2)^3,x)

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 1.03, size = 38, normalized size = 1.36 \begin {gather*} -\frac {2 \, b d x + b c + a d}{2 \, {\left (d^{4} x^{2} + 2 \, c d^{3} x + c^{2} d^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^4/(a*c+(a*d+b*c)*x+b*d*x^2)^3,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 0.03, size = 39, normalized size = 1.39 \begin {gather*} -\frac {\frac {a\,d+b\,c}{2\,d^2}+\frac {b\,x}{d}}{c^2+2\,c\,d\,x+d^2\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^4/(a*c + x*(a*d + b*c) + b*d*x^2)^3,x)

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 0.34, size = 39, normalized size = 1.39 \begin {gather*} \frac {- a d - b c - 2 b d x}{2 c^{2} d^{2} + 4 c d^{3} x + 2 d^{4} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**4/(a*c+(a*d+b*c)*x+b*d*x**2)**3,x)

[Out]

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

________________________________________________________________________________________