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

Optimal. Leaf size=122 \[ -\frac {(b c-a d)^5 \log (c+d x)}{d^6}+\frac {b x (b c-a d)^4}{d^5}-\frac {(a+b x)^2 (b c-a d)^3}{2 d^4}+\frac {(a+b x)^3 (b c-a d)^2}{3 d^3}-\frac {(a+b x)^4 (b c-a d)}{4 d^2}+\frac {(a+b x)^5}{5 d} \]

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Rubi [A]  time = 0.06, antiderivative size = 122, 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 {b x (b c-a d)^4}{d^5}-\frac {(a+b x)^2 (b c-a d)^3}{2 d^4}+\frac {(a+b x)^3 (b c-a d)^2}{3 d^3}-\frac {(a+b x)^4 (b c-a d)}{4 d^2}-\frac {(b c-a d)^5 \log (c+d x)}{d^6}+\frac {(a+b x)^5}{5 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]  time = 0.07, size = 167, normalized size = 1.37 \begin {gather*} \frac {b d x \left (300 a^4 d^4+300 a^3 b d^3 (d x-2 c)+100 a^2 b^2 d^2 \left (6 c^2-3 c d x+2 d^2 x^2\right )+25 a b^3 d \left (-12 c^3+6 c^2 d x-4 c d^2 x^2+3 d^3 x^3\right )+b^4 \left (60 c^4-30 c^3 d x+20 c^2 d^2 x^2-15 c d^3 x^3+12 d^4 x^4\right )\right )-60 (b c-a d)^5 \log (c+d x)}{60 d^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(b*d*x*(300*a^4*d^4 + 300*a^3*b*d^3*(-2*c + d*x) + 100*a^2*b^2*d^2*(6*c^2 - 3*c*d*x + 2*d^2*x^2) + 25*a*b^3*d*
(-12*c^3 + 6*c^2*d*x - 4*c*d^2*x^2 + 3*d^3*x^3) + b^4*(60*c^4 - 30*c^3*d*x + 20*c^2*d^2*x^2 - 15*c*d^3*x^3 + 1
2*d^4*x^4)) - 60*(b*c - a*d)^5*Log[c + d*x])/(60*d^6)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+b x)^6}{a c+(b c+a d) x+b d x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 0.40, size = 259, normalized size = 2.12 \begin {gather*} \frac {12 \, b^{5} d^{5} x^{5} - 15 \, {\left (b^{5} c d^{4} - 5 \, a b^{4} d^{5}\right )} x^{4} + 20 \, {\left (b^{5} c^{2} d^{3} - 5 \, a b^{4} c d^{4} + 10 \, a^{2} b^{3} d^{5}\right )} x^{3} - 30 \, {\left (b^{5} c^{3} d^{2} - 5 \, a b^{4} c^{2} d^{3} + 10 \, a^{2} b^{3} c d^{4} - 10 \, a^{3} b^{2} d^{5}\right )} x^{2} + 60 \, {\left (b^{5} c^{4} d - 5 \, a b^{4} c^{3} d^{2} + 10 \, a^{2} b^{3} c^{2} d^{3} - 10 \, a^{3} b^{2} c d^{4} + 5 \, a^{4} b d^{5}\right )} x - 60 \, {\left (b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5}\right )} \log \left (d x + c\right )}{60 \, d^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^6/(a*c+(a*d+b*c)*x+b*d*x^2),x, algorithm="fricas")

[Out]

1/60*(12*b^5*d^5*x^5 - 15*(b^5*c*d^4 - 5*a*b^4*d^5)*x^4 + 20*(b^5*c^2*d^3 - 5*a*b^4*c*d^4 + 10*a^2*b^3*d^5)*x^
3 - 30*(b^5*c^3*d^2 - 5*a*b^4*c^2*d^3 + 10*a^2*b^3*c*d^4 - 10*a^3*b^2*d^5)*x^2 + 60*(b^5*c^4*d - 5*a*b^4*c^3*d
^2 + 10*a^2*b^3*c^2*d^3 - 10*a^3*b^2*c*d^4 + 5*a^4*b*d^5)*x - 60*(b^5*c^5 - 5*a*b^4*c^4*d + 10*a^2*b^3*c^3*d^2
- 10*a^3*b^2*c^2*d^3 + 5*a^4*b*c*d^4 - a^5*d^5)*log(d*x + c))/d^6

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giac [B]  time = 0.16, size = 273, normalized size = 2.24 \begin {gather*} \frac {12 \, b^{5} d^{4} x^{5} - 15 \, b^{5} c d^{3} x^{4} + 75 \, a b^{4} d^{4} x^{4} + 20 \, b^{5} c^{2} d^{2} x^{3} - 100 \, a b^{4} c d^{3} x^{3} + 200 \, a^{2} b^{3} d^{4} x^{3} - 30 \, b^{5} c^{3} d x^{2} + 150 \, a b^{4} c^{2} d^{2} x^{2} - 300 \, a^{2} b^{3} c d^{3} x^{2} + 300 \, a^{3} b^{2} d^{4} x^{2} + 60 \, b^{5} c^{4} x - 300 \, a b^{4} c^{3} d x + 600 \, a^{2} b^{3} c^{2} d^{2} x - 600 \, a^{3} b^{2} c d^{3} x + 300 \, a^{4} b d^{4} x}{60 \, d^{5}} - \frac {{\left (b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5}\right )} \log \left ({\left | d x + c \right |}\right )}{d^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^6/(a*c+(a*d+b*c)*x+b*d*x^2),x, algorithm="giac")

[Out]

1/60*(12*b^5*d^4*x^5 - 15*b^5*c*d^3*x^4 + 75*a*b^4*d^4*x^4 + 20*b^5*c^2*d^2*x^3 - 100*a*b^4*c*d^3*x^3 + 200*a^
2*b^3*d^4*x^3 - 30*b^5*c^3*d*x^2 + 150*a*b^4*c^2*d^2*x^2 - 300*a^2*b^3*c*d^3*x^2 + 300*a^3*b^2*d^4*x^2 + 60*b^
5*c^4*x - 300*a*b^4*c^3*d*x + 600*a^2*b^3*c^2*d^2*x - 600*a^3*b^2*c*d^3*x + 300*a^4*b*d^4*x)/d^5 - (b^5*c^5 -
5*a*b^4*c^4*d + 10*a^2*b^3*c^3*d^2 - 10*a^3*b^2*c^2*d^3 + 5*a^4*b*c*d^4 - a^5*d^5)*log(abs(d*x + c))/d^6

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maple [B]  time = 0.05, size = 302, normalized size = 2.48 \begin {gather*} \frac {b^{5} x^{5}}{5 d}+\frac {5 a \,b^{4} x^{4}}{4 d}-\frac {b^{5} c \,x^{4}}{4 d^{2}}+\frac {10 a^{2} b^{3} x^{3}}{3 d}-\frac {5 a \,b^{4} c \,x^{3}}{3 d^{2}}+\frac {b^{5} c^{2} x^{3}}{3 d^{3}}+\frac {5 a^{3} b^{2} x^{2}}{d}-\frac {5 a^{2} b^{3} c \,x^{2}}{d^{2}}+\frac {5 a \,b^{4} c^{2} x^{2}}{2 d^{3}}-\frac {b^{5} c^{3} x^{2}}{2 d^{4}}+\frac {a^{5} \ln \left (d x +c \right )}{d}-\frac {5 a^{4} b c \ln \left (d x +c \right )}{d^{2}}+\frac {5 a^{4} b x}{d}+\frac {10 a^{3} b^{2} c^{2} \ln \left (d x +c \right )}{d^{3}}-\frac {10 a^{3} b^{2} c x}{d^{2}}-\frac {10 a^{2} b^{3} c^{3} \ln \left (d x +c \right )}{d^{4}}+\frac {10 a^{2} b^{3} c^{2} x}{d^{3}}+\frac {5 a \,b^{4} c^{4} \ln \left (d x +c \right )}{d^{5}}-\frac {5 a \,b^{4} c^{3} x}{d^{4}}-\frac {b^{5} c^{5} \ln \left (d x +c \right )}{d^{6}}+\frac {b^{5} c^{4} x}{d^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^6/(a*c+(a*d+b*c)*x+b*d*x^2),x)

[Out]

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

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maxima [B]  time = 1.01, size = 258, normalized size = 2.11 \begin {gather*} \frac {12 \, b^{5} d^{4} x^{5} - 15 \, {\left (b^{5} c d^{3} - 5 \, a b^{4} d^{4}\right )} x^{4} + 20 \, {\left (b^{5} c^{2} d^{2} - 5 \, a b^{4} c d^{3} + 10 \, a^{2} b^{3} d^{4}\right )} x^{3} - 30 \, {\left (b^{5} c^{3} d - 5 \, a b^{4} c^{2} d^{2} + 10 \, a^{2} b^{3} c d^{3} - 10 \, a^{3} b^{2} d^{4}\right )} x^{2} + 60 \, {\left (b^{5} c^{4} - 5 \, a b^{4} c^{3} d + 10 \, a^{2} b^{3} c^{2} d^{2} - 10 \, a^{3} b^{2} c d^{3} + 5 \, a^{4} b d^{4}\right )} x}{60 \, d^{5}} - \frac {{\left (b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5}\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)^6/(a*c+(a*d+b*c)*x+b*d*x^2),x, algorithm="maxima")

[Out]

1/60*(12*b^5*d^4*x^5 - 15*(b^5*c*d^3 - 5*a*b^4*d^4)*x^4 + 20*(b^5*c^2*d^2 - 5*a*b^4*c*d^3 + 10*a^2*b^3*d^4)*x^
3 - 30*(b^5*c^3*d - 5*a*b^4*c^2*d^2 + 10*a^2*b^3*c*d^3 - 10*a^3*b^2*d^4)*x^2 + 60*(b^5*c^4 - 5*a*b^4*c^3*d + 1
0*a^2*b^3*c^2*d^2 - 10*a^3*b^2*c*d^3 + 5*a^4*b*d^4)*x)/d^5 - (b^5*c^5 - 5*a*b^4*c^4*d + 10*a^2*b^3*c^3*d^2 - 1
0*a^3*b^2*c^2*d^3 + 5*a^4*b*c*d^4 - a^5*d^5)*log(d*x + c)/d^6

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mupad [B]  time = 0.60, size = 280, normalized size = 2.30 \begin {gather*} x\,\left (\frac {5\,a^4\,b}{d}-\frac {c\,\left (\frac {10\,a^3\,b^2}{d}+\frac {c\,\left (\frac {c\,\left (\frac {5\,a\,b^4}{d}-\frac {b^5\,c}{d^2}\right )}{d}-\frac {10\,a^2\,b^3}{d}\right )}{d}\right )}{d}\right )+x^4\,\left (\frac {5\,a\,b^4}{4\,d}-\frac {b^5\,c}{4\,d^2}\right )+x^2\,\left (\frac {5\,a^3\,b^2}{d}+\frac {c\,\left (\frac {c\,\left (\frac {5\,a\,b^4}{d}-\frac {b^5\,c}{d^2}\right )}{d}-\frac {10\,a^2\,b^3}{d}\right )}{2\,d}\right )-x^3\,\left (\frac {c\,\left (\frac {5\,a\,b^4}{d}-\frac {b^5\,c}{d^2}\right )}{3\,d}-\frac {10\,a^2\,b^3}{3\,d}\right )+\frac {b^5\,x^5}{5\,d}+\frac {\ln \left (c+d\,x\right )\,\left (a^5\,d^5-5\,a^4\,b\,c\,d^4+10\,a^3\,b^2\,c^2\,d^3-10\,a^2\,b^3\,c^3\,d^2+5\,a\,b^4\,c^4\,d-b^5\,c^5\right )}{d^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^6/(a*c + x*(a*d + b*c) + b*d*x^2),x)

[Out]

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

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sympy [B]  time = 0.58, size = 209, normalized size = 1.71 \begin {gather*} \frac {b^{5} x^{5}}{5 d} + x^{4} \left (\frac {5 a b^{4}}{4 d} - \frac {b^{5} c}{4 d^{2}}\right ) + x^{3} \left (\frac {10 a^{2} b^{3}}{3 d} - \frac {5 a b^{4} c}{3 d^{2}} + \frac {b^{5} c^{2}}{3 d^{3}}\right ) + x^{2} \left (\frac {5 a^{3} b^{2}}{d} - \frac {5 a^{2} b^{3} c}{d^{2}} + \frac {5 a b^{4} c^{2}}{2 d^{3}} - \frac {b^{5} c^{3}}{2 d^{4}}\right ) + x \left (\frac {5 a^{4} b}{d} - \frac {10 a^{3} b^{2} c}{d^{2}} + \frac {10 a^{2} b^{3} c^{2}}{d^{3}} - \frac {5 a b^{4} c^{3}}{d^{4}} + \frac {b^{5} c^{4}}{d^{5}}\right ) + \frac {\left (a d - b c\right )^{5} \log {\left (c + d x \right )}}{d^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**6/(a*c+(a*d+b*c)*x+b*d*x**2),x)

[Out]

b**5*x**5/(5*d) + x**4*(5*a*b**4/(4*d) - b**5*c/(4*d**2)) + x**3*(10*a**2*b**3/(3*d) - 5*a*b**4*c/(3*d**2) + b
**5*c**2/(3*d**3)) + x**2*(5*a**3*b**2/d - 5*a**2*b**3*c/d**2 + 5*a*b**4*c**2/(2*d**3) - b**5*c**3/(2*d**4)) +
x*(5*a**4*b/d - 10*a**3*b**2*c/d**2 + 10*a**2*b**3*c**2/d**3 - 5*a*b**4*c**3/d**4 + b**5*c**4/d**5) + (a*d -
b*c)**5*log(c + d*x)/d**6

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