3.16.4 \(\int \frac {(a+b x)^8}{(a c+(b c+a d) x+b d x^2)^3} \, dx\)

Optimal. Leaf size=133 \[ -\frac {5 b^4 (c+d x)^2 (b c-a d)}{2 d^6}+\frac {10 b^3 x (b c-a d)^2}{d^5}-\frac {10 b^2 (b c-a d)^3 \log (c+d x)}{d^6}-\frac {5 b (b c-a d)^4}{d^6 (c+d x)}+\frac {(b c-a d)^5}{2 d^6 (c+d x)^2}+\frac {b^5 (c+d x)^3}{3 d^6} \]

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Rubi [A]  time = 0.14, antiderivative size = 133, 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} \begin {gather*} -\frac {5 b^4 (c+d x)^2 (b c-a d)}{2 d^6}+\frac {10 b^3 x (b c-a d)^2}{d^5}-\frac {10 b^2 (b c-a d)^3 \log (c+d x)}{d^6}-\frac {5 b (b c-a d)^4}{d^6 (c+d x)}+\frac {(b c-a d)^5}{2 d^6 (c+d x)^2}+\frac {b^5 (c+d x)^3}{3 d^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^8/(a*c + (b*c + a*d)*x + b*d*x^2)^3,x]

[Out]

(10*b^3*(b*c - a*d)^2*x)/d^5 + (b*c - a*d)^5/(2*d^6*(c + d*x)^2) - (5*b*(b*c - a*d)^4)/(d^6*(c + d*x)) - (5*b^
4*(b*c - a*d)*(c + d*x)^2)/(2*d^6) + (b^5*(c + d*x)^3)/(3*d^6) - (10*b^2*(b*c - a*d)^3*Log[c + d*x])/d^6

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 {(a+b x)^8}{\left (a c+(b c+a d) x+b d x^2\right )^3} \, dx &=\int \frac {(a+b x)^5}{(c+d x)^3} \, dx\\ &=\int \left (\frac {10 b^3 (b c-a d)^2}{d^5}+\frac {(-b c+a d)^5}{d^5 (c+d x)^3}+\frac {5 b (b c-a d)^4}{d^5 (c+d x)^2}-\frac {10 b^2 (b c-a d)^3}{d^5 (c+d x)}-\frac {5 b^4 (b c-a d) (c+d x)}{d^5}+\frac {b^5 (c+d x)^2}{d^5}\right ) \, dx\\ &=\frac {10 b^3 (b c-a d)^2 x}{d^5}+\frac {(b c-a d)^5}{2 d^6 (c+d x)^2}-\frac {5 b (b c-a d)^4}{d^6 (c+d x)}-\frac {5 b^4 (b c-a d) (c+d x)^2}{2 d^6}+\frac {b^5 (c+d x)^3}{3 d^6}-\frac {10 b^2 (b c-a d)^3 \log (c+d x)}{d^6}\\ \end {align*}

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Mathematica [A]  time = 0.08, size = 230, normalized size = 1.73 \begin {gather*} \frac {-3 a^5 d^5-15 a^4 b d^4 (c+2 d x)+30 a^3 b^2 c d^3 (3 c+4 d x)+30 a^2 b^3 d^2 \left (-5 c^3-4 c^2 d x+4 c d^2 x^2+2 d^3 x^3\right )+15 a b^4 d \left (7 c^4+2 c^3 d x-11 c^2 d^2 x^2-4 c d^3 x^3+d^4 x^4\right )-60 b^2 (c+d x)^2 (b c-a d)^3 \log (c+d x)+b^5 \left (-27 c^5+6 c^4 d x+63 c^3 d^2 x^2+20 c^2 d^3 x^3-5 c d^4 x^4+2 d^5 x^5\right )}{6 d^6 (c+d x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^8/(a*c + (b*c + a*d)*x + b*d*x^2)^3,x]

[Out]

(-3*a^5*d^5 - 15*a^4*b*d^4*(c + 2*d*x) + 30*a^3*b^2*c*d^3*(3*c + 4*d*x) + 30*a^2*b^3*d^2*(-5*c^3 - 4*c^2*d*x +
4*c*d^2*x^2 + 2*d^3*x^3) + 15*a*b^4*d*(7*c^4 + 2*c^3*d*x - 11*c^2*d^2*x^2 - 4*c*d^3*x^3 + d^4*x^4) + b^5*(-27
*c^5 + 6*c^4*d*x + 63*c^3*d^2*x^2 + 20*c^2*d^3*x^3 - 5*c*d^4*x^4 + 2*d^5*x^5) - 60*b^2*(b*c - a*d)^3*(c + d*x)
^2*Log[c + d*x])/(6*d^6*(c + d*x)^2)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+b x)^8}{\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)^8/(a*c + (b*c + a*d)*x + b*d*x^2)^3,x]

[Out]

IntegrateAlgebraic[(a + b*x)^8/(a*c + (b*c + a*d)*x + b*d*x^2)^3, x]

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fricas [B]  time = 0.41, size = 416, normalized size = 3.13 \begin {gather*} \frac {2 \, b^{5} d^{5} x^{5} - 27 \, b^{5} c^{5} + 105 \, a b^{4} c^{4} d - 150 \, a^{2} b^{3} c^{3} d^{2} + 90 \, a^{3} b^{2} c^{2} d^{3} - 15 \, a^{4} b c d^{4} - 3 \, a^{5} d^{5} - 5 \, {\left (b^{5} c d^{4} - 3 \, a b^{4} d^{5}\right )} x^{4} + 20 \, {\left (b^{5} c^{2} d^{3} - 3 \, a b^{4} c d^{4} + 3 \, a^{2} b^{3} d^{5}\right )} x^{3} + 3 \, {\left (21 \, b^{5} c^{3} d^{2} - 55 \, a b^{4} c^{2} d^{3} + 40 \, a^{2} b^{3} c d^{4}\right )} x^{2} + 6 \, {\left (b^{5} c^{4} d + 5 \, a b^{4} c^{3} d^{2} - 20 \, a^{2} b^{3} c^{2} d^{3} + 20 \, a^{3} b^{2} c d^{4} - 5 \, a^{4} b d^{5}\right )} x - 60 \, {\left (b^{5} c^{5} - 3 \, a b^{4} c^{4} d + 3 \, a^{2} b^{3} c^{3} d^{2} - a^{3} b^{2} c^{2} d^{3} + {\left (b^{5} c^{3} d^{2} - 3 \, a b^{4} c^{2} d^{3} + 3 \, a^{2} b^{3} c d^{4} - a^{3} b^{2} d^{5}\right )} x^{2} + 2 \, {\left (b^{5} c^{4} d - 3 \, a b^{4} c^{3} d^{2} + 3 \, a^{2} b^{3} c^{2} d^{3} - a^{3} b^{2} c d^{4}\right )} x\right )} \log \left (d x + c\right )}{6 \, {\left (d^{8} x^{2} + 2 \, c d^{7} x + c^{2} d^{6}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^8/(a*c+(a*d+b*c)*x+b*d*x^2)^3,x, algorithm="fricas")

[Out]

1/6*(2*b^5*d^5*x^5 - 27*b^5*c^5 + 105*a*b^4*c^4*d - 150*a^2*b^3*c^3*d^2 + 90*a^3*b^2*c^2*d^3 - 15*a^4*b*c*d^4
- 3*a^5*d^5 - 5*(b^5*c*d^4 - 3*a*b^4*d^5)*x^4 + 20*(b^5*c^2*d^3 - 3*a*b^4*c*d^4 + 3*a^2*b^3*d^5)*x^3 + 3*(21*b
^5*c^3*d^2 - 55*a*b^4*c^2*d^3 + 40*a^2*b^3*c*d^4)*x^2 + 6*(b^5*c^4*d + 5*a*b^4*c^3*d^2 - 20*a^2*b^3*c^2*d^3 +
20*a^3*b^2*c*d^4 - 5*a^4*b*d^5)*x - 60*(b^5*c^5 - 3*a*b^4*c^4*d + 3*a^2*b^3*c^3*d^2 - a^3*b^2*c^2*d^3 + (b^5*c
^3*d^2 - 3*a*b^4*c^2*d^3 + 3*a^2*b^3*c*d^4 - a^3*b^2*d^5)*x^2 + 2*(b^5*c^4*d - 3*a*b^4*c^3*d^2 + 3*a^2*b^3*c^2
*d^3 - a^3*b^2*c*d^4)*x)*log(d*x + c))/(d^8*x^2 + 2*c*d^7*x + c^2*d^6)

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giac [B]  time = 0.18, size = 264, normalized size = 1.98 \begin {gather*} -\frac {10 \, {\left (b^{5} c^{3} - 3 \, a b^{4} c^{2} d + 3 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} \log \left ({\left | d x + c \right |}\right )}{d^{6}} - \frac {9 \, b^{5} c^{5} - 35 \, a b^{4} c^{4} d + 50 \, a^{2} b^{3} c^{3} d^{2} - 30 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} + a^{5} d^{5} + 10 \, {\left (b^{5} c^{4} d - 4 \, a b^{4} c^{3} d^{2} + 6 \, a^{2} b^{3} c^{2} d^{3} - 4 \, a^{3} b^{2} c d^{4} + a^{4} b d^{5}\right )} x}{2 \, {\left (d x + c\right )}^{2} d^{6}} + \frac {2 \, b^{5} d^{6} x^{3} - 9 \, b^{5} c d^{5} x^{2} + 15 \, a b^{4} d^{6} x^{2} + 36 \, b^{5} c^{2} d^{4} x - 90 \, a b^{4} c d^{5} x + 60 \, a^{2} b^{3} d^{6} x}{6 \, d^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^8/(a*c+(a*d+b*c)*x+b*d*x^2)^3,x, algorithm="giac")

[Out]

-10*(b^5*c^3 - 3*a*b^4*c^2*d + 3*a^2*b^3*c*d^2 - a^3*b^2*d^3)*log(abs(d*x + c))/d^6 - 1/2*(9*b^5*c^5 - 35*a*b^
4*c^4*d + 50*a^2*b^3*c^3*d^2 - 30*a^3*b^2*c^2*d^3 + 5*a^4*b*c*d^4 + a^5*d^5 + 10*(b^5*c^4*d - 4*a*b^4*c^3*d^2
+ 6*a^2*b^3*c^2*d^3 - 4*a^3*b^2*c*d^4 + a^4*b*d^5)*x)/((d*x + c)^2*d^6) + 1/6*(2*b^5*d^6*x^3 - 9*b^5*c*d^5*x^2
+ 15*a*b^4*d^6*x^2 + 36*b^5*c^2*d^4*x - 90*a*b^4*c*d^5*x + 60*a^2*b^3*d^6*x)/d^9

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maple [B]  time = 0.06, size = 346, normalized size = 2.60 \begin {gather*} \frac {b^{5} x^{3}}{3 d^{3}}-\frac {a^{5}}{2 \left (d x +c \right )^{2} d}+\frac {5 a^{4} b c}{2 \left (d x +c \right )^{2} d^{2}}-\frac {5 a^{3} b^{2} c^{2}}{\left (d x +c \right )^{2} d^{3}}+\frac {5 a^{2} b^{3} c^{3}}{\left (d x +c \right )^{2} d^{4}}-\frac {5 a \,b^{4} c^{4}}{2 \left (d x +c \right )^{2} d^{5}}+\frac {5 a \,b^{4} x^{2}}{2 d^{3}}+\frac {b^{5} c^{5}}{2 \left (d x +c \right )^{2} d^{6}}-\frac {3 b^{5} c \,x^{2}}{2 d^{4}}-\frac {5 a^{4} b}{\left (d x +c \right ) d^{2}}+\frac {20 a^{3} b^{2} c}{\left (d x +c \right ) d^{3}}+\frac {10 a^{3} b^{2} \ln \left (d x +c \right )}{d^{3}}-\frac {30 a^{2} b^{3} c^{2}}{\left (d x +c \right ) d^{4}}-\frac {30 a^{2} b^{3} c \ln \left (d x +c \right )}{d^{4}}+\frac {10 a^{2} b^{3} x}{d^{3}}+\frac {20 a \,b^{4} c^{3}}{\left (d x +c \right ) d^{5}}+\frac {30 a \,b^{4} c^{2} \ln \left (d x +c \right )}{d^{5}}-\frac {15 a \,b^{4} c x}{d^{4}}-\frac {5 b^{5} c^{4}}{\left (d x +c \right ) d^{6}}-\frac {10 b^{5} c^{3} \ln \left (d x +c \right )}{d^{6}}+\frac {6 b^{5} c^{2} x}{d^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^8/(a*c+(a*d+b*c)*x+b*d*x^2)^3,x)

[Out]

1/3*b^5/d^3*x^3+5/2*b^4/d^3*x^2*a-3/2*b^5/d^4*x^2*c+10*b^3/d^3*a^2*x-15*b^4/d^4*a*c*x+6*b^5/d^5*c^2*x-5*b/d^2/
(d*x+c)*a^4+20*b^2/d^3/(d*x+c)*a^3*c-30*b^3/d^4/(d*x+c)*a^2*c^2+20*b^4/d^5/(d*x+c)*a*c^3-5*b^5/d^6/(d*x+c)*c^4
+10*b^2/d^3*ln(d*x+c)*a^3-30*b^3/d^4*ln(d*x+c)*a^2*c+30*b^4/d^5*ln(d*x+c)*a*c^2-10*b^5/d^6*ln(d*x+c)*c^3-1/2/d
/(d*x+c)^2*a^5+5/2/d^2/(d*x+c)^2*c*a^4*b-5/d^3/(d*x+c)^2*c^2*a^3*b^2+5/d^4/(d*x+c)^2*a^2*b^3*c^3-5/2/d^5/(d*x+
c)^2*a*b^4*c^4+1/2/d^6/(d*x+c)^2*b^5*c^5

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maxima [B]  time = 1.28, size = 271, normalized size = 2.04 \begin {gather*} -\frac {9 \, b^{5} c^{5} - 35 \, a b^{4} c^{4} d + 50 \, a^{2} b^{3} c^{3} d^{2} - 30 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} + a^{5} d^{5} + 10 \, {\left (b^{5} c^{4} d - 4 \, a b^{4} c^{3} d^{2} + 6 \, a^{2} b^{3} c^{2} d^{3} - 4 \, a^{3} b^{2} c d^{4} + a^{4} b d^{5}\right )} x}{2 \, {\left (d^{8} x^{2} + 2 \, c d^{7} x + c^{2} d^{6}\right )}} + \frac {2 \, b^{5} d^{2} x^{3} - 3 \, {\left (3 \, b^{5} c d - 5 \, a b^{4} d^{2}\right )} x^{2} + 6 \, {\left (6 \, b^{5} c^{2} - 15 \, a b^{4} c d + 10 \, a^{2} b^{3} d^{2}\right )} x}{6 \, d^{5}} - \frac {10 \, {\left (b^{5} c^{3} - 3 \, a b^{4} c^{2} d + 3 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} \log \left (d x + c\right )}{d^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^8/(a*c+(a*d+b*c)*x+b*d*x^2)^3,x, algorithm="maxima")

[Out]

-1/2*(9*b^5*c^5 - 35*a*b^4*c^4*d + 50*a^2*b^3*c^3*d^2 - 30*a^3*b^2*c^2*d^3 + 5*a^4*b*c*d^4 + a^5*d^5 + 10*(b^5
*c^4*d - 4*a*b^4*c^3*d^2 + 6*a^2*b^3*c^2*d^3 - 4*a^3*b^2*c*d^4 + a^4*b*d^5)*x)/(d^8*x^2 + 2*c*d^7*x + c^2*d^6)
+ 1/6*(2*b^5*d^2*x^3 - 3*(3*b^5*c*d - 5*a*b^4*d^2)*x^2 + 6*(6*b^5*c^2 - 15*a*b^4*c*d + 10*a^2*b^3*d^2)*x)/d^5
- 10*(b^5*c^3 - 3*a*b^4*c^2*d + 3*a^2*b^3*c*d^2 - a^3*b^2*d^3)*log(d*x + c)/d^6

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mupad [B]  time = 0.62, size = 291, normalized size = 2.19 \begin {gather*} x^2\,\left (\frac {5\,a\,b^4}{2\,d^3}-\frac {3\,b^5\,c}{2\,d^4}\right )-\frac {\frac {a^5\,d^5+5\,a^4\,b\,c\,d^4-30\,a^3\,b^2\,c^2\,d^3+50\,a^2\,b^3\,c^3\,d^2-35\,a\,b^4\,c^4\,d+9\,b^5\,c^5}{2\,d}+x\,\left (5\,a^4\,b\,d^4-20\,a^3\,b^2\,c\,d^3+30\,a^2\,b^3\,c^2\,d^2-20\,a\,b^4\,c^3\,d+5\,b^5\,c^4\right )}{c^2\,d^5+2\,c\,d^6\,x+d^7\,x^2}-x\,\left (\frac {3\,c\,\left (\frac {5\,a\,b^4}{d^3}-\frac {3\,b^5\,c}{d^4}\right )}{d}-\frac {10\,a^2\,b^3}{d^3}+\frac {3\,b^5\,c^2}{d^5}\right )-\frac {\ln \left (c+d\,x\right )\,\left (-10\,a^3\,b^2\,d^3+30\,a^2\,b^3\,c\,d^2-30\,a\,b^4\,c^2\,d+10\,b^5\,c^3\right )}{d^6}+\frac {b^5\,x^3}{3\,d^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^8/(a*c + x*(a*d + b*c) + b*d*x^2)^3,x)

[Out]

x^2*((5*a*b^4)/(2*d^3) - (3*b^5*c)/(2*d^4)) - ((a^5*d^5 + 9*b^5*c^5 + 50*a^2*b^3*c^3*d^2 - 30*a^3*b^2*c^2*d^3
- 35*a*b^4*c^4*d + 5*a^4*b*c*d^4)/(2*d) + x*(5*b^5*c^4 + 5*a^4*b*d^4 - 20*a^3*b^2*c*d^3 + 30*a^2*b^3*c^2*d^2 -
20*a*b^4*c^3*d))/(c^2*d^5 + d^7*x^2 + 2*c*d^6*x) - x*((3*c*((5*a*b^4)/d^3 - (3*b^5*c)/d^4))/d - (10*a^2*b^3)/
d^3 + (3*b^5*c^2)/d^5) - (log(c + d*x)*(10*b^5*c^3 - 10*a^3*b^2*d^3 + 30*a^2*b^3*c*d^2 - 30*a*b^4*c^2*d))/d^6
+ (b^5*x^3)/(3*d^3)

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sympy [B]  time = 2.04, size = 258, normalized size = 1.94 \begin {gather*} \frac {b^{5} x^{3}}{3 d^{3}} + \frac {10 b^{2} \left (a d - b c\right )^{3} \log {\left (c + d x \right )}}{d^{6}} + x^{2} \left (\frac {5 a b^{4}}{2 d^{3}} - \frac {3 b^{5} c}{2 d^{4}}\right ) + x \left (\frac {10 a^{2} b^{3}}{d^{3}} - \frac {15 a b^{4} c}{d^{4}} + \frac {6 b^{5} c^{2}}{d^{5}}\right ) + \frac {- a^{5} d^{5} - 5 a^{4} b c d^{4} + 30 a^{3} b^{2} c^{2} d^{3} - 50 a^{2} b^{3} c^{3} d^{2} + 35 a b^{4} c^{4} d - 9 b^{5} c^{5} + x \left (- 10 a^{4} b d^{5} + 40 a^{3} b^{2} c d^{4} - 60 a^{2} b^{3} c^{2} d^{3} + 40 a b^{4} c^{3} d^{2} - 10 b^{5} c^{4} d\right )}{2 c^{2} d^{6} + 4 c d^{7} x + 2 d^{8} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**8/(a*c+(a*d+b*c)*x+b*d*x**2)**3,x)

[Out]

b**5*x**3/(3*d**3) + 10*b**2*(a*d - b*c)**3*log(c + d*x)/d**6 + x**2*(5*a*b**4/(2*d**3) - 3*b**5*c/(2*d**4)) +
x*(10*a**2*b**3/d**3 - 15*a*b**4*c/d**4 + 6*b**5*c**2/d**5) + (-a**5*d**5 - 5*a**4*b*c*d**4 + 30*a**3*b**2*c*
*2*d**3 - 50*a**2*b**3*c**3*d**2 + 35*a*b**4*c**4*d - 9*b**5*c**5 + x*(-10*a**4*b*d**5 + 40*a**3*b**2*c*d**4 -
60*a**2*b**3*c**2*d**3 + 40*a*b**4*c**3*d**2 - 10*b**5*c**4*d))/(2*c**2*d**6 + 4*c*d**7*x + 2*d**8*x**2)

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