3.20.41 \(\int \frac {(a+b x) (a c+b c x)^m}{(a^2+2 a b x+b^2 x^2)^3} \, dx\)

Optimal. Leaf size=27 \[ -\frac {c^4 (a c+b c x)^{m-4}}{b (4-m)} \]

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Rubi [A]  time = 0.02, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.088, Rules used = {21, 27, 32} \begin {gather*} -\frac {c^4 (a c+b c x)^{m-4}}{b (4-m)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((c^4*(a*c + b*c*x)^(-4 + m))/(b*(4 - m)))

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

\begin {align*} \int \frac {(a+b x) (a c+b c x)^m}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\frac {\int \frac {(a c+b c x)^{1+m}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx}{c}\\ &=\frac {\int \frac {(a c+b c x)^{1+m}}{(a+b x)^6} \, dx}{c}\\ &=c^5 \int (a c+b c x)^{-5+m} \, dx\\ &=-\frac {c^4 (a c+b c x)^{-4+m}}{b (4-m)}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 25, normalized size = 0.93 \begin {gather*} \frac {(c (a+b x))^m}{b (m-4) (a+b x)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(c*(a + b*x))^m/(b*(-4 + m)*(a + b*x)^4)

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IntegrateAlgebraic [F]  time = 0.15, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+b x) (a c+b c x)^m}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 0.43, size = 101, normalized size = 3.74 \begin {gather*} \frac {{\left (b c x + a c\right )}^{m}}{a^{4} b m - 4 \, a^{4} b + {\left (b^{5} m - 4 \, b^{5}\right )} x^{4} + 4 \, {\left (a b^{4} m - 4 \, a b^{4}\right )} x^{3} + 6 \, {\left (a^{2} b^{3} m - 4 \, a^{2} b^{3}\right )} x^{2} + 4 \, {\left (a^{3} b^{2} m - 4 \, a^{3} b^{2}\right )} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*x + a)*(b*c*x + a*c)^m/(b^2*x^2 + 2*a*b*x + a^2)^3, x)

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maple [A]  time = 0.05, size = 45, normalized size = 1.67 \begin {gather*} \frac {\left (b c x +a c \right )^{m}}{\left (b x +a \right )^{2} \left (b^{2} x^{2}+2 a b x +a^{2}\right ) \left (m -4\right ) b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*c*x+a*c)^m/(b*x+a)^2/(b^2*x^2+2*a*b*x+a^2)/b/(-4+m)

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maxima [B]  time = 0.85, size = 216, normalized size = 8.00 \begin {gather*} \frac {{\left (b c^{m} {\left (m - 5\right )} x - a c^{m}\right )} {\left (b x + a\right )}^{m} b}{{\left (m^{2} - 9 \, m + 20\right )} b^{7} x^{5} + 5 \, {\left (m^{2} - 9 \, m + 20\right )} a b^{6} x^{4} + 10 \, {\left (m^{2} - 9 \, m + 20\right )} a^{2} b^{5} x^{3} + 10 \, {\left (m^{2} - 9 \, m + 20\right )} a^{3} b^{4} x^{2} + 5 \, {\left (m^{2} - 9 \, m + 20\right )} a^{4} b^{3} x + {\left (m^{2} - 9 \, m + 20\right )} a^{5} b^{2}} + \frac {{\left (b x + a\right )}^{m} a c^{m}}{b^{6} {\left (m - 5\right )} x^{5} + 5 \, a b^{5} {\left (m - 5\right )} x^{4} + 10 \, a^{2} b^{4} {\left (m - 5\right )} x^{3} + 10 \, a^{3} b^{3} {\left (m - 5\right )} x^{2} + 5 \, a^{4} b^{2} {\left (m - 5\right )} x + a^{5} b {\left (m - 5\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*c^m*(m - 5)*x - a*c^m)*(b*x + a)^m*b/((m^2 - 9*m + 20)*b^7*x^5 + 5*(m^2 - 9*m + 20)*a*b^6*x^4 + 10*(m^2 - 9
*m + 20)*a^2*b^5*x^3 + 10*(m^2 - 9*m + 20)*a^3*b^4*x^2 + 5*(m^2 - 9*m + 20)*a^4*b^3*x + (m^2 - 9*m + 20)*a^5*b
^2) + (b*x + a)^m*a*c^m/(b^6*(m - 5)*x^5 + 5*a*b^5*(m - 5)*x^4 + 10*a^2*b^4*(m - 5)*x^3 + 10*a^3*b^3*(m - 5)*x
^2 + 5*a^4*b^2*(m - 5)*x + a^5*b*(m - 5))

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mupad [B]  time = 2.18, size = 61, normalized size = 2.26 \begin {gather*} \frac {{\left (a\,c+b\,c\,x\right )}^m}{b^5\,\left (m-4\right )\,\left (x^4+\frac {a^4}{b^4}+\frac {4\,a\,x^3}{b}+\frac {4\,a^3\,x}{b^3}+\frac {6\,a^2\,x^2}{b^2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 4.18, size = 136, normalized size = 5.04 \begin {gather*} \begin {cases} \frac {c^{4} x}{a} & \text {for}\: b = 0 \wedge m = 4 \\\frac {x \left (a c\right )^{m}}{a^{5}} & \text {for}\: b = 0 \\\frac {c^{4} \log {\left (\frac {a}{b} + x \right )}}{b} & \text {for}\: m = 4 \\\frac {\left (a c + b c x\right )^{m}}{a^{4} b m - 4 a^{4} b + 4 a^{3} b^{2} m x - 16 a^{3} b^{2} x + 6 a^{2} b^{3} m x^{2} - 24 a^{2} b^{3} x^{2} + 4 a b^{4} m x^{3} - 16 a b^{4} x^{3} + b^{5} m x^{4} - 4 b^{5} x^{4}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((c**4*x/a, Eq(b, 0) & Eq(m, 4)), (x*(a*c)**m/a**5, Eq(b, 0)), (c**4*log(a/b + x)/b, Eq(m, 4)), ((a*c
+ b*c*x)**m/(a**4*b*m - 4*a**4*b + 4*a**3*b**2*m*x - 16*a**3*b**2*x + 6*a**2*b**3*m*x**2 - 24*a**2*b**3*x**2
+ 4*a*b**4*m*x**3 - 16*a*b**4*x**3 + b**5*m*x**4 - 4*b**5*x**4), True))

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