3.13.59 \(\int \frac {(a+b x)^6}{(c+d x)^8} \, dx\)

Optimal. Leaf size=28 \[ \frac {(a+b x)^7}{7 (c+d x)^7 (b c-a d)} \]

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Rubi [A]  time = 0.00, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {37} \begin {gather*} \frac {(a+b x)^7}{7 (c+d x)^7 (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^6/(c + d*x)^8,x]

[Out]

(a + b*x)^7/(7*(b*c - a*d)*(c + d*x)^7)

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]

Rubi steps

\begin {align*} \int \frac {(a+b x)^6}{(c+d x)^8} \, dx &=\frac {(a+b x)^7}{7 (b c-a d) (c+d x)^7}\\ \end {align*}

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Mathematica [B]  time = 0.09, size = 271, normalized size = 9.68 \begin {gather*} -\frac {a^6 d^6+a^5 b d^5 (c+7 d x)+a^4 b^2 d^4 \left (c^2+7 c d x+21 d^2 x^2\right )+a^3 b^3 d^3 \left (c^3+7 c^2 d x+21 c d^2 x^2+35 d^3 x^3\right )+a^2 b^4 d^2 \left (c^4+7 c^3 d x+21 c^2 d^2 x^2+35 c d^3 x^3+35 d^4 x^4\right )+a b^5 d \left (c^5+7 c^4 d x+21 c^3 d^2 x^2+35 c^2 d^3 x^3+35 c d^4 x^4+21 d^5 x^5\right )+b^6 \left (c^6+7 c^5 d x+21 c^4 d^2 x^2+35 c^3 d^3 x^3+35 c^2 d^4 x^4+21 c d^5 x^5+7 d^6 x^6\right )}{7 d^7 (c+d x)^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^6/(c + d*x)^8,x]

[Out]

-1/7*(a^6*d^6 + a^5*b*d^5*(c + 7*d*x) + a^4*b^2*d^4*(c^2 + 7*c*d*x + 21*d^2*x^2) + a^3*b^3*d^3*(c^3 + 7*c^2*d*
x + 21*c*d^2*x^2 + 35*d^3*x^3) + a^2*b^4*d^2*(c^4 + 7*c^3*d*x + 21*c^2*d^2*x^2 + 35*c*d^3*x^3 + 35*d^4*x^4) +
a*b^5*d*(c^5 + 7*c^4*d*x + 21*c^3*d^2*x^2 + 35*c^2*d^3*x^3 + 35*c*d^4*x^4 + 21*d^5*x^5) + b^6*(c^6 + 7*c^5*d*x
+ 21*c^4*d^2*x^2 + 35*c^3*d^3*x^3 + 35*c^2*d^4*x^4 + 21*c*d^5*x^5 + 7*d^6*x^6))/(d^7*(c + d*x)^7)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+b x)^6}{(c+d x)^8} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + b*x)^6/(c + d*x)^8,x]

[Out]

IntegrateAlgebraic[(a + b*x)^6/(c + d*x)^8, x]

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fricas [B]  time = 1.38, size = 398, normalized size = 14.21 \begin {gather*} -\frac {7 \, b^{6} d^{6} x^{6} + b^{6} c^{6} + a b^{5} c^{5} d + a^{2} b^{4} c^{4} d^{2} + a^{3} b^{3} c^{3} d^{3} + a^{4} b^{2} c^{2} d^{4} + a^{5} b c d^{5} + a^{6} d^{6} + 21 \, {\left (b^{6} c d^{5} + a b^{5} d^{6}\right )} x^{5} + 35 \, {\left (b^{6} c^{2} d^{4} + a b^{5} c d^{5} + a^{2} b^{4} d^{6}\right )} x^{4} + 35 \, {\left (b^{6} c^{3} d^{3} + a b^{5} c^{2} d^{4} + a^{2} b^{4} c d^{5} + a^{3} b^{3} d^{6}\right )} x^{3} + 21 \, {\left (b^{6} c^{4} d^{2} + a b^{5} c^{3} d^{3} + a^{2} b^{4} c^{2} d^{4} + a^{3} b^{3} c d^{5} + a^{4} b^{2} d^{6}\right )} x^{2} + 7 \, {\left (b^{6} c^{5} d + a b^{5} c^{4} d^{2} + a^{2} b^{4} c^{3} d^{3} + a^{3} b^{3} c^{2} d^{4} + a^{4} b^{2} c d^{5} + a^{5} b d^{6}\right )} x}{7 \, {\left (d^{14} x^{7} + 7 \, c d^{13} x^{6} + 21 \, c^{2} d^{12} x^{5} + 35 \, c^{3} d^{11} x^{4} + 35 \, c^{4} d^{10} x^{3} + 21 \, c^{5} d^{9} x^{2} + 7 \, c^{6} d^{8} x + c^{7} d^{7}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^6/(d*x+c)^8,x, algorithm="fricas")

[Out]

-1/7*(7*b^6*d^6*x^6 + b^6*c^6 + a*b^5*c^5*d + a^2*b^4*c^4*d^2 + a^3*b^3*c^3*d^3 + a^4*b^2*c^2*d^4 + a^5*b*c*d^
5 + a^6*d^6 + 21*(b^6*c*d^5 + a*b^5*d^6)*x^5 + 35*(b^6*c^2*d^4 + a*b^5*c*d^5 + a^2*b^4*d^6)*x^4 + 35*(b^6*c^3*
d^3 + a*b^5*c^2*d^4 + a^2*b^4*c*d^5 + a^3*b^3*d^6)*x^3 + 21*(b^6*c^4*d^2 + a*b^5*c^3*d^3 + a^2*b^4*c^2*d^4 + a
^3*b^3*c*d^5 + a^4*b^2*d^6)*x^2 + 7*(b^6*c^5*d + a*b^5*c^4*d^2 + a^2*b^4*c^3*d^3 + a^3*b^3*c^2*d^4 + a^4*b^2*c
*d^5 + a^5*b*d^6)*x)/(d^14*x^7 + 7*c*d^13*x^6 + 21*c^2*d^12*x^5 + 35*c^3*d^11*x^4 + 35*c^4*d^10*x^3 + 21*c^5*d
^9*x^2 + 7*c^6*d^8*x + c^7*d^7)

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giac [B]  time = 1.28, size = 369, normalized size = 13.18 \begin {gather*} -\frac {7 \, b^{6} d^{6} x^{6} + 21 \, b^{6} c d^{5} x^{5} + 21 \, a b^{5} d^{6} x^{5} + 35 \, b^{6} c^{2} d^{4} x^{4} + 35 \, a b^{5} c d^{5} x^{4} + 35 \, a^{2} b^{4} d^{6} x^{4} + 35 \, b^{6} c^{3} d^{3} x^{3} + 35 \, a b^{5} c^{2} d^{4} x^{3} + 35 \, a^{2} b^{4} c d^{5} x^{3} + 35 \, a^{3} b^{3} d^{6} x^{3} + 21 \, b^{6} c^{4} d^{2} x^{2} + 21 \, a b^{5} c^{3} d^{3} x^{2} + 21 \, a^{2} b^{4} c^{2} d^{4} x^{2} + 21 \, a^{3} b^{3} c d^{5} x^{2} + 21 \, a^{4} b^{2} d^{6} x^{2} + 7 \, b^{6} c^{5} d x + 7 \, a b^{5} c^{4} d^{2} x + 7 \, a^{2} b^{4} c^{3} d^{3} x + 7 \, a^{3} b^{3} c^{2} d^{4} x + 7 \, a^{4} b^{2} c d^{5} x + 7 \, a^{5} b d^{6} x + b^{6} c^{6} + a b^{5} c^{5} d + a^{2} b^{4} c^{4} d^{2} + a^{3} b^{3} c^{3} d^{3} + a^{4} b^{2} c^{2} d^{4} + a^{5} b c d^{5} + a^{6} d^{6}}{7 \, {\left (d x + c\right )}^{7} d^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^6/(d*x+c)^8,x, algorithm="giac")

[Out]

-1/7*(7*b^6*d^6*x^6 + 21*b^6*c*d^5*x^5 + 21*a*b^5*d^6*x^5 + 35*b^6*c^2*d^4*x^4 + 35*a*b^5*c*d^5*x^4 + 35*a^2*b
^4*d^6*x^4 + 35*b^6*c^3*d^3*x^3 + 35*a*b^5*c^2*d^4*x^3 + 35*a^2*b^4*c*d^5*x^3 + 35*a^3*b^3*d^6*x^3 + 21*b^6*c^
4*d^2*x^2 + 21*a*b^5*c^3*d^3*x^2 + 21*a^2*b^4*c^2*d^4*x^2 + 21*a^3*b^3*c*d^5*x^2 + 21*a^4*b^2*d^6*x^2 + 7*b^6*
c^5*d*x + 7*a*b^5*c^4*d^2*x + 7*a^2*b^4*c^3*d^3*x + 7*a^3*b^3*c^2*d^4*x + 7*a^4*b^2*c*d^5*x + 7*a^5*b*d^6*x +
b^6*c^6 + a*b^5*c^5*d + a^2*b^4*c^4*d^2 + a^3*b^3*c^3*d^3 + a^4*b^2*c^2*d^4 + a^5*b*c*d^5 + a^6*d^6)/((d*x + c
)^7*d^7)

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maple [B]  time = 0.01, size = 357, normalized size = 12.75 \begin {gather*} -\frac {b^{6}}{\left (d x +c \right ) d^{7}}-\frac {3 \left (a d -b c \right ) b^{5}}{\left (d x +c \right )^{2} d^{7}}-\frac {5 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) b^{4}}{\left (d x +c \right )^{3} d^{7}}-\frac {5 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) b^{3}}{\left (d x +c \right )^{4} d^{7}}-\frac {3 \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) b^{2}}{\left (d x +c \right )^{5} d^{7}}-\frac {\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 ) b}{\left (d x +c \right )^{6} d^{7}}-\frac {a^{6} d^{6}-6 a^{5} b c \,d^{5}+15 a^{4} b^{2} c^{2} d^{4}-20 a^{3} b^{3} c^{3} d^{3}+15 a^{2} b^{4} c^{4} d^{2}-6 a \,b^{5} c^{5} d +b^{6} c^{6}}{7 \left (d x +c \right )^{7} d^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^6/(d*x+c)^8,x)

[Out]

-1/7*(a^6*d^6-6*a^5*b*c*d^5+15*a^4*b^2*c^2*d^4-20*a^3*b^3*c^3*d^3+15*a^2*b^4*c^4*d^2-6*a*b^5*c^5*d+b^6*c^6)/d^
7/(d*x+c)^7-5*b^4*(a^2*d^2-2*a*b*c*d+b^2*c^2)/d^7/(d*x+c)^3-3*b^2*(a^4*d^4-4*a^3*b*c*d^3+6*a^2*b^2*c^2*d^2-4*a
*b^3*c^3*d+b^4*c^4)/d^7/(d*x+c)^5-b^6/d^7/(d*x+c)-b*(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)/d^7/(d*x+c)^6-5*b^3*(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)/d^7/(d*x+c)^4-3*b^
5*(a*d-b*c)/d^7/(d*x+c)^2

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maxima [B]  time = 1.61, size = 398, normalized size = 14.21 \begin {gather*} -\frac {7 \, b^{6} d^{6} x^{6} + b^{6} c^{6} + a b^{5} c^{5} d + a^{2} b^{4} c^{4} d^{2} + a^{3} b^{3} c^{3} d^{3} + a^{4} b^{2} c^{2} d^{4} + a^{5} b c d^{5} + a^{6} d^{6} + 21 \, {\left (b^{6} c d^{5} + a b^{5} d^{6}\right )} x^{5} + 35 \, {\left (b^{6} c^{2} d^{4} + a b^{5} c d^{5} + a^{2} b^{4} d^{6}\right )} x^{4} + 35 \, {\left (b^{6} c^{3} d^{3} + a b^{5} c^{2} d^{4} + a^{2} b^{4} c d^{5} + a^{3} b^{3} d^{6}\right )} x^{3} + 21 \, {\left (b^{6} c^{4} d^{2} + a b^{5} c^{3} d^{3} + a^{2} b^{4} c^{2} d^{4} + a^{3} b^{3} c d^{5} + a^{4} b^{2} d^{6}\right )} x^{2} + 7 \, {\left (b^{6} c^{5} d + a b^{5} c^{4} d^{2} + a^{2} b^{4} c^{3} d^{3} + a^{3} b^{3} c^{2} d^{4} + a^{4} b^{2} c d^{5} + a^{5} b d^{6}\right )} x}{7 \, {\left (d^{14} x^{7} + 7 \, c d^{13} x^{6} + 21 \, c^{2} d^{12} x^{5} + 35 \, c^{3} d^{11} x^{4} + 35 \, c^{4} d^{10} x^{3} + 21 \, c^{5} d^{9} x^{2} + 7 \, c^{6} d^{8} x + c^{7} d^{7}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^6/(d*x+c)^8,x, algorithm="maxima")

[Out]

-1/7*(7*b^6*d^6*x^6 + b^6*c^6 + a*b^5*c^5*d + a^2*b^4*c^4*d^2 + a^3*b^3*c^3*d^3 + a^4*b^2*c^2*d^4 + a^5*b*c*d^
5 + a^6*d^6 + 21*(b^6*c*d^5 + a*b^5*d^6)*x^5 + 35*(b^6*c^2*d^4 + a*b^5*c*d^5 + a^2*b^4*d^6)*x^4 + 35*(b^6*c^3*
d^3 + a*b^5*c^2*d^4 + a^2*b^4*c*d^5 + a^3*b^3*d^6)*x^3 + 21*(b^6*c^4*d^2 + a*b^5*c^3*d^3 + a^2*b^4*c^2*d^4 + a
^3*b^3*c*d^5 + a^4*b^2*d^6)*x^2 + 7*(b^6*c^5*d + a*b^5*c^4*d^2 + a^2*b^4*c^3*d^3 + a^3*b^3*c^2*d^4 + a^4*b^2*c
*d^5 + a^5*b*d^6)*x)/(d^14*x^7 + 7*c*d^13*x^6 + 21*c^2*d^12*x^5 + 35*c^3*d^11*x^4 + 35*c^4*d^10*x^3 + 21*c^5*d
^9*x^2 + 7*c^6*d^8*x + c^7*d^7)

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mupad [B]  time = 0.15, size = 378, normalized size = 13.50 \begin {gather*} -\frac {\frac {a^6\,d^6+a^5\,b\,c\,d^5+a^4\,b^2\,c^2\,d^4+a^3\,b^3\,c^3\,d^3+a^2\,b^4\,c^4\,d^2+a\,b^5\,c^5\,d+b^6\,c^6}{7\,d^7}+\frac {b^6\,x^6}{d}+\frac {5\,b^3\,x^3\,\left (a^3\,d^3+a^2\,b\,c\,d^2+a\,b^2\,c^2\,d+b^3\,c^3\right )}{d^4}+\frac {b\,x\,\left (a^5\,d^5+a^4\,b\,c\,d^4+a^3\,b^2\,c^2\,d^3+a^2\,b^3\,c^3\,d^2+a\,b^4\,c^4\,d+b^5\,c^5\right )}{d^6}+\frac {3\,b^5\,x^5\,\left (a\,d+b\,c\right )}{d^2}+\frac {3\,b^2\,x^2\,\left (a^4\,d^4+a^3\,b\,c\,d^3+a^2\,b^2\,c^2\,d^2+a\,b^3\,c^3\,d+b^4\,c^4\right )}{d^5}+\frac {5\,b^4\,x^4\,\left (a^2\,d^2+a\,b\,c\,d+b^2\,c^2\right )}{d^3}}{c^7+7\,c^6\,d\,x+21\,c^5\,d^2\,x^2+35\,c^4\,d^3\,x^3+35\,c^3\,d^4\,x^4+21\,c^2\,d^5\,x^5+7\,c\,d^6\,x^6+d^7\,x^7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^6/(c + d*x)^8,x)

[Out]

-((a^6*d^6 + b^6*c^6 + a^2*b^4*c^4*d^2 + a^3*b^3*c^3*d^3 + a^4*b^2*c^2*d^4 + a*b^5*c^5*d + a^5*b*c*d^5)/(7*d^7
) + (b^6*x^6)/d + (5*b^3*x^3*(a^3*d^3 + b^3*c^3 + a*b^2*c^2*d + a^2*b*c*d^2))/d^4 + (b*x*(a^5*d^5 + b^5*c^5 +
a^2*b^3*c^3*d^2 + a^3*b^2*c^2*d^3 + a*b^4*c^4*d + a^4*b*c*d^4))/d^6 + (3*b^5*x^5*(a*d + b*c))/d^2 + (3*b^2*x^2
*(a^4*d^4 + b^4*c^4 + a^2*b^2*c^2*d^2 + a*b^3*c^3*d + a^3*b*c*d^3))/d^5 + (5*b^4*x^4*(a^2*d^2 + b^2*c^2 + a*b*
c*d))/d^3)/(c^7 + d^7*x^7 + 7*c*d^6*x^6 + 21*c^5*d^2*x^2 + 35*c^4*d^3*x^3 + 35*c^3*d^4*x^4 + 21*c^2*d^5*x^5 +
7*c^6*d*x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**6/(d*x+c)**8,x)

[Out]

Timed out

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