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

Optimal. Leaf size=64 \[ -\frac {c^5 (A b-a B) (a c+b c x)^{m-5}}{b^2 (5-m)}-\frac {B c^4 (a c+b c x)^{m-4}}{b^2 (4-m)} \]

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Rubi [A]  time = 0.05, antiderivative size = 64, 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 = {27, 21, 43} \begin {gather*} -\frac {c^5 (A b-a B) (a c+b c x)^{m-5}}{b^2 (5-m)}-\frac {B c^4 (a c+b c x)^{m-4}}{b^2 (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]

-(((A*b - a*B)*c^5*(a*c + b*c*x)^(-5 + m))/(b^2*(5 - m))) - (B*c^4*(a*c + b*c*x)^(-4 + m))/(b^2*(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 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])

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

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Mathematica [A]  time = 0.04, size = 48, normalized size = 0.75 \begin {gather*} \frac {(c (a+b x))^m (-a B+A b (m-4)+b B (m-5) x)}{b^2 (m-5) (m-4) (a+b x)^5} \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*(-(a*B) + A*b*(-4 + m) + b*B*(-5 + m)*x))/(b^2*(-5 + m)*(-4 + m)*(a + b*x)^5)

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IntegrateAlgebraic [F]  time = 0.22, 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 = 0.45, size = 212, normalized size = 3.31 \begin {gather*} \frac {{\left (A b m - B a - 4 \, A b + {\left (B b m - 5 \, B b\right )} x\right )} {\left (b c x + a c\right )}^{m}}{a^{5} b^{2} m^{2} - 9 \, a^{5} b^{2} m + 20 \, a^{5} b^{2} + {\left (b^{7} m^{2} - 9 \, b^{7} m + 20 \, b^{7}\right )} x^{5} + 5 \, {\left (a b^{6} m^{2} - 9 \, a b^{6} m + 20 \, a b^{6}\right )} x^{4} + 10 \, {\left (a^{2} b^{5} m^{2} - 9 \, a^{2} b^{5} m + 20 \, a^{2} b^{5}\right )} x^{3} + 10 \, {\left (a^{3} b^{4} m^{2} - 9 \, a^{3} b^{4} m + 20 \, a^{3} b^{4}\right )} x^{2} + 5 \, {\left (a^{4} b^{3} m^{2} - 9 \, a^{4} b^{3} m + 20 \, a^{4} b^{3}\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]

(A*b*m - B*a - 4*A*b + (B*b*m - 5*B*b)*x)*(b*c*x + a*c)^m/(a^5*b^2*m^2 - 9*a^5*b^2*m + 20*a^5*b^2 + (b^7*m^2 -
9*b^7*m + 20*b^7)*x^5 + 5*(a*b^6*m^2 - 9*a*b^6*m + 20*a*b^6)*x^4 + 10*(a^2*b^5*m^2 - 9*a^2*b^5*m + 20*a^2*b^5
)*x^3 + 10*(a^3*b^4*m^2 - 9*a^3*b^4*m + 20*a^3*b^4)*x^2 + 5*(a^4*b^3*m^2 - 9*a^4*b^3*m + 20*a^4*b^3)*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 = 0.04, size = 73, normalized size = 1.14 \begin {gather*} \frac {\left (B b m x +A b m -5 B b x -4 A b -B a \right ) \left (b c x +a c \right )^{m}}{\left (b x +a \right ) \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{2} \left (m^{2}-9 m +20\right ) b^{2}} \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*b*m*x+A*b*m-5*B*b*x-4*A*b-B*a)*(b*c*x+a*c)^m/(b*x+a)/(b^2*x^2+2*a*b*x+a^2)^2/b^2/(m^2-9*m+20)

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maxima [B]  time = 0.74, size = 216, normalized size = 3.38 \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 = 2.20, size = 113, normalized size = 1.77 \begin {gather*} -\frac {{\left (a\,c+b\,c\,x\right )}^m\,\left (\frac {4\,A\,b+B\,a-A\,b\,m}{b^7\,\left (m^2-9\,m+20\right )}-\frac {B\,x\,\left (m-5\right )}{b^6\,\left (m^2-9\,m+20\right )}\right )}{x^5+\frac {a^5}{b^5}+\frac {5\,a\,x^4}{b}+\frac {5\,a^4\,x}{b^4}+\frac {10\,a^2\,x^3}{b^2}+\frac {10\,a^3\,x^2}{b^3}} \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*((4*A*b + B*a - A*b*m)/(b^7*(m^2 - 9*m + 20)) - (B*x*(m - 5))/(b^6*(m^2 - 9*m + 20))))/(x^5
+ a^5/b^5 + (5*a*x^4)/b + (5*a^4*x)/b^4 + (10*a^2*x^3)/b^2 + (10*a^3*x^2)/b^3)

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sympy [A]  time = 5.73, size = 1268, normalized size = 19.81 \begin {gather*} \begin {cases} \frac {\left (a c\right )^{m} \left (A x + \frac {B x^{2}}{2}\right )}{a^{6}} & \text {for}\: b = 0 \\- \frac {A b c^{4}}{a b^{2} + b^{3} x} + \frac {B a c^{4} \log {\left (\frac {a}{b} + x \right )}}{a b^{2} + b^{3} x} + \frac {B a c^{4}}{a b^{2} + b^{3} x} + \frac {B b c^{4} x \log {\left (\frac {a}{b} + x \right )}}{a b^{2} + b^{3} x} & \text {for}\: m = 4 \\\frac {A c^{5} \log {\left (\frac {a}{b} + x \right )}}{b} - \frac {B a c^{5} \log {\left (\frac {a}{b} + x \right )}}{b^{2}} + \frac {B c^{5} x}{b} & \text {for}\: m = 5 \\\frac {A b m \left (a c + b c x\right )^{m}}{a^{5} b^{2} m^{2} - 9 a^{5} b^{2} m + 20 a^{5} b^{2} + 5 a^{4} b^{3} m^{2} x - 45 a^{4} b^{3} m x + 100 a^{4} b^{3} x + 10 a^{3} b^{4} m^{2} x^{2} - 90 a^{3} b^{4} m x^{2} + 200 a^{3} b^{4} x^{2} + 10 a^{2} b^{5} m^{2} x^{3} - 90 a^{2} b^{5} m x^{3} + 200 a^{2} b^{5} x^{3} + 5 a b^{6} m^{2} x^{4} - 45 a b^{6} m x^{4} + 100 a b^{6} x^{4} + b^{7} m^{2} x^{5} - 9 b^{7} m x^{5} + 20 b^{7} x^{5}} - \frac {4 A b \left (a c + b c x\right )^{m}}{a^{5} b^{2} m^{2} - 9 a^{5} b^{2} m + 20 a^{5} b^{2} + 5 a^{4} b^{3} m^{2} x - 45 a^{4} b^{3} m x + 100 a^{4} b^{3} x + 10 a^{3} b^{4} m^{2} x^{2} - 90 a^{3} b^{4} m x^{2} + 200 a^{3} b^{4} x^{2} + 10 a^{2} b^{5} m^{2} x^{3} - 90 a^{2} b^{5} m x^{3} + 200 a^{2} b^{5} x^{3} + 5 a b^{6} m^{2} x^{4} - 45 a b^{6} m x^{4} + 100 a b^{6} x^{4} + b^{7} m^{2} x^{5} - 9 b^{7} m x^{5} + 20 b^{7} x^{5}} - \frac {B a \left (a c + b c x\right )^{m}}{a^{5} b^{2} m^{2} - 9 a^{5} b^{2} m + 20 a^{5} b^{2} + 5 a^{4} b^{3} m^{2} x - 45 a^{4} b^{3} m x + 100 a^{4} b^{3} x + 10 a^{3} b^{4} m^{2} x^{2} - 90 a^{3} b^{4} m x^{2} + 200 a^{3} b^{4} x^{2} + 10 a^{2} b^{5} m^{2} x^{3} - 90 a^{2} b^{5} m x^{3} + 200 a^{2} b^{5} x^{3} + 5 a b^{6} m^{2} x^{4} - 45 a b^{6} m x^{4} + 100 a b^{6} x^{4} + b^{7} m^{2} x^{5} - 9 b^{7} m x^{5} + 20 b^{7} x^{5}} + \frac {B b m x \left (a c + b c x\right )^{m}}{a^{5} b^{2} m^{2} - 9 a^{5} b^{2} m + 20 a^{5} b^{2} + 5 a^{4} b^{3} m^{2} x - 45 a^{4} b^{3} m x + 100 a^{4} b^{3} x + 10 a^{3} b^{4} m^{2} x^{2} - 90 a^{3} b^{4} m x^{2} + 200 a^{3} b^{4} x^{2} + 10 a^{2} b^{5} m^{2} x^{3} - 90 a^{2} b^{5} m x^{3} + 200 a^{2} b^{5} x^{3} + 5 a b^{6} m^{2} x^{4} - 45 a b^{6} m x^{4} + 100 a b^{6} x^{4} + b^{7} m^{2} x^{5} - 9 b^{7} m x^{5} + 20 b^{7} x^{5}} - \frac {5 B b x \left (a c + b c x\right )^{m}}{a^{5} b^{2} m^{2} - 9 a^{5} b^{2} m + 20 a^{5} b^{2} + 5 a^{4} b^{3} m^{2} x - 45 a^{4} b^{3} m x + 100 a^{4} b^{3} x + 10 a^{3} b^{4} m^{2} x^{2} - 90 a^{3} b^{4} m x^{2} + 200 a^{3} b^{4} x^{2} + 10 a^{2} b^{5} m^{2} x^{3} - 90 a^{2} b^{5} m x^{3} + 200 a^{2} b^{5} x^{3} + 5 a b^{6} m^{2} x^{4} - 45 a b^{6} m x^{4} + 100 a b^{6} x^{4} + b^{7} m^{2} x^{5} - 9 b^{7} m x^{5} + 20 b^{7} x^{5}} & \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(((a*c)**m*(A*x + B*x**2/2)/a**6, Eq(b, 0)), (-A*b*c**4/(a*b**2 + b**3*x) + B*a*c**4*log(a/b + x)/(a*
b**2 + b**3*x) + B*a*c**4/(a*b**2 + b**3*x) + B*b*c**4*x*log(a/b + x)/(a*b**2 + b**3*x), Eq(m, 4)), (A*c**5*lo
g(a/b + x)/b - B*a*c**5*log(a/b + x)/b**2 + B*c**5*x/b, Eq(m, 5)), (A*b*m*(a*c + b*c*x)**m/(a**5*b**2*m**2 - 9
*a**5*b**2*m + 20*a**5*b**2 + 5*a**4*b**3*m**2*x - 45*a**4*b**3*m*x + 100*a**4*b**3*x + 10*a**3*b**4*m**2*x**2
- 90*a**3*b**4*m*x**2 + 200*a**3*b**4*x**2 + 10*a**2*b**5*m**2*x**3 - 90*a**2*b**5*m*x**3 + 200*a**2*b**5*x**
3 + 5*a*b**6*m**2*x**4 - 45*a*b**6*m*x**4 + 100*a*b**6*x**4 + b**7*m**2*x**5 - 9*b**7*m*x**5 + 20*b**7*x**5) -
4*A*b*(a*c + b*c*x)**m/(a**5*b**2*m**2 - 9*a**5*b**2*m + 20*a**5*b**2 + 5*a**4*b**3*m**2*x - 45*a**4*b**3*m*x
+ 100*a**4*b**3*x + 10*a**3*b**4*m**2*x**2 - 90*a**3*b**4*m*x**2 + 200*a**3*b**4*x**2 + 10*a**2*b**5*m**2*x**
3 - 90*a**2*b**5*m*x**3 + 200*a**2*b**5*x**3 + 5*a*b**6*m**2*x**4 - 45*a*b**6*m*x**4 + 100*a*b**6*x**4 + b**7*
m**2*x**5 - 9*b**7*m*x**5 + 20*b**7*x**5) - B*a*(a*c + b*c*x)**m/(a**5*b**2*m**2 - 9*a**5*b**2*m + 20*a**5*b**
2 + 5*a**4*b**3*m**2*x - 45*a**4*b**3*m*x + 100*a**4*b**3*x + 10*a**3*b**4*m**2*x**2 - 90*a**3*b**4*m*x**2 + 2
00*a**3*b**4*x**2 + 10*a**2*b**5*m**2*x**3 - 90*a**2*b**5*m*x**3 + 200*a**2*b**5*x**3 + 5*a*b**6*m**2*x**4 - 4
5*a*b**6*m*x**4 + 100*a*b**6*x**4 + b**7*m**2*x**5 - 9*b**7*m*x**5 + 20*b**7*x**5) + B*b*m*x*(a*c + b*c*x)**m/
(a**5*b**2*m**2 - 9*a**5*b**2*m + 20*a**5*b**2 + 5*a**4*b**3*m**2*x - 45*a**4*b**3*m*x + 100*a**4*b**3*x + 10*
a**3*b**4*m**2*x**2 - 90*a**3*b**4*m*x**2 + 200*a**3*b**4*x**2 + 10*a**2*b**5*m**2*x**3 - 90*a**2*b**5*m*x**3
+ 200*a**2*b**5*x**3 + 5*a*b**6*m**2*x**4 - 45*a*b**6*m*x**4 + 100*a*b**6*x**4 + b**7*m**2*x**5 - 9*b**7*m*x**
5 + 20*b**7*x**5) - 5*B*b*x*(a*c + b*c*x)**m/(a**5*b**2*m**2 - 9*a**5*b**2*m + 20*a**5*b**2 + 5*a**4*b**3*m**2
*x - 45*a**4*b**3*m*x + 100*a**4*b**3*x + 10*a**3*b**4*m**2*x**2 - 90*a**3*b**4*m*x**2 + 200*a**3*b**4*x**2 +
10*a**2*b**5*m**2*x**3 - 90*a**2*b**5*m*x**3 + 200*a**2*b**5*x**3 + 5*a*b**6*m**2*x**4 - 45*a*b**6*m*x**4 + 10
0*a*b**6*x**4 + b**7*m**2*x**5 - 9*b**7*m*x**5 + 20*b**7*x**5), True))

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