3.1365 \(\int \frac{(a+b x)^6}{(c+d x)^8} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0177008, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \frac{(a+b x)^7}{7 (c+d x)^7 (b c-a d)} \]

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)

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Rubi in Sympy [A]  time = 4.15254, size = 22, normalized size = 0.79 \[ - \frac{\left (a + b x\right )^{7}}{7 \left (c + d x\right )^{7} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**6/(d*x+c)**8,x)

[Out]

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

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Mathematica [B]  time = 0.172697, size = 271, normalized size = 9.68 \[ -\frac{a^6 d^6+a^5 b d^5 (c+7 d x)+a^4 b^2 d^4 \left (c^2+7 c d x+21 d^2 x^2\right )+a^3 b^3 d^3 \left (c^3+7 c^2 d x+21 c d^2 x^2+35 d^3 x^3\right )+a^2 b^4 d^2 \left (c^4+7 c^3 d x+21 c^2 d^2 x^2+35 c d^3 x^3+35 d^4 x^4\right )+a b^5 d \left (c^5+7 c^4 d x+21 c^3 d^2 x^2+35 c^2 d^3 x^3+35 c d^4 x^4+21 d^5 x^5\right )+b^6 \left (c^6+7 c^5 d x+21 c^4 d^2 x^2+35 c^3 d^3 x^3+35 c^2 d^4 x^4+21 c d^5 x^5+7 d^6 x^6\right )}{7 d^7 (c+d x)^7} \]

Antiderivative was successfully verified.

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

[Out]

-(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))/(7*d^7*(c + d*x)^7)

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Maple [B]  time = 0.011, size = 357, normalized size = 12.8 \[ -{\frac{b \left ({a}^{5}{d}^{5}-5\,{a}^{4}bc{d}^{4}+10\,{a}^{3}{b}^{2}{c}^{2}{d}^{3}-10\,{a}^{2}{b}^{3}{c}^{3}{d}^{2}+5\,a{b}^{4}{c}^{4}d-{b}^{5}{c}^{5} \right ) }{{d}^{7} \left ( dx+c \right ) ^{6}}}-{\frac{{a}^{6}{d}^{6}-6\,{a}^{5}bc{d}^{5}+15\,{a}^{4}{b}^{2}{c}^{2}{d}^{4}-20\,{a}^{3}{b}^{3}{c}^{3}{d}^{3}+15\,{a}^{2}{b}^{4}{c}^{4}{d}^{2}-6\,a{b}^{5}{c}^{5}d+{b}^{6}{c}^{6}}{7\,{d}^{7} \left ( dx+c \right ) ^{7}}}-3\,{\frac{{b}^{5} \left ( ad-bc \right ) }{{d}^{7} \left ( dx+c \right ) ^{2}}}-5\,{\frac{{b}^{3} \left ({a}^{3}{d}^{3}-3\,{a}^{2}bc{d}^{2}+3\,a{b}^{2}{c}^{2}d-{b}^{3}{c}^{3} \right ) }{{d}^{7} \left ( dx+c \right ) ^{4}}}-3\,{\frac{{b}^{2} \left ({a}^{4}{d}^{4}-4\,{a}^{3}bc{d}^{3}+6\,{a}^{2}{b}^{2}{c}^{2}{d}^{2}-4\,a{b}^{3}{c}^{3}d+{b}^{4}{c}^{4} \right ) }{{d}^{7} \left ( dx+c \right ) ^{5}}}-{\frac{{b}^{6}}{{d}^{7} \left ( dx+c \right ) }}-5\,{\frac{{b}^{4} \left ({a}^{2}{d}^{2}-2\,abcd+{b}^{2}{c}^{2} \right ) }{{d}^{7} \left ( dx+c \right ) ^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^6/(d*x+c)^8,x)

[Out]

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

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Maxima [A]  time = 1.36492, size = 537, normalized size = 19.18 \[ -\frac{7 \, b^{6} d^{6} x^{6} + b^{6} c^{6} + a b^{5} c^{5} d + a^{2} b^{4} c^{4} d^{2} + a^{3} b^{3} c^{3} d^{3} + a^{4} b^{2} c^{2} d^{4} + a^{5} b c d^{5} + a^{6} d^{6} + 21 \,{\left (b^{6} c d^{5} + a b^{5} d^{6}\right )} x^{5} + 35 \,{\left (b^{6} c^{2} d^{4} + a b^{5} c d^{5} + a^{2} b^{4} d^{6}\right )} x^{4} + 35 \,{\left (b^{6} c^{3} d^{3} + a b^{5} c^{2} d^{4} + a^{2} b^{4} c d^{5} + a^{3} b^{3} d^{6}\right )} x^{3} + 21 \,{\left (b^{6} c^{4} d^{2} + a b^{5} c^{3} d^{3} + a^{2} b^{4} c^{2} d^{4} + a^{3} b^{3} c d^{5} + a^{4} b^{2} d^{6}\right )} x^{2} + 7 \,{\left (b^{6} c^{5} d + a b^{5} c^{4} d^{2} + a^{2} b^{4} c^{3} d^{3} + a^{3} b^{3} c^{2} d^{4} + a^{4} b^{2} c d^{5} + a^{5} b d^{6}\right )} x}{7 \,{\left (d^{14} x^{7} + 7 \, c d^{13} x^{6} + 21 \, c^{2} d^{12} x^{5} + 35 \, c^{3} d^{11} x^{4} + 35 \, c^{4} d^{10} x^{3} + 21 \, c^{5} d^{9} x^{2} + 7 \, c^{6} d^{8} x + c^{7} d^{7}\right )}} \]

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^1
3*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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Fricas [A]  time = 0.203525, size = 537, normalized size = 19.18 \[ -\frac{7 \, b^{6} d^{6} x^{6} + b^{6} c^{6} + a b^{5} c^{5} d + a^{2} b^{4} c^{4} d^{2} + a^{3} b^{3} c^{3} d^{3} + a^{4} b^{2} c^{2} d^{4} + a^{5} b c d^{5} + a^{6} d^{6} + 21 \,{\left (b^{6} c d^{5} + a b^{5} d^{6}\right )} x^{5} + 35 \,{\left (b^{6} c^{2} d^{4} + a b^{5} c d^{5} + a^{2} b^{4} d^{6}\right )} x^{4} + 35 \,{\left (b^{6} c^{3} d^{3} + a b^{5} c^{2} d^{4} + a^{2} b^{4} c d^{5} + a^{3} b^{3} d^{6}\right )} x^{3} + 21 \,{\left (b^{6} c^{4} d^{2} + a b^{5} c^{3} d^{3} + a^{2} b^{4} c^{2} d^{4} + a^{3} b^{3} c d^{5} + a^{4} b^{2} d^{6}\right )} x^{2} + 7 \,{\left (b^{6} c^{5} d + a b^{5} c^{4} d^{2} + a^{2} b^{4} c^{3} d^{3} + a^{3} b^{3} c^{2} d^{4} + a^{4} b^{2} c d^{5} + a^{5} b d^{6}\right )} x}{7 \,{\left (d^{14} x^{7} + 7 \, c d^{13} x^{6} + 21 \, c^{2} d^{12} x^{5} + 35 \, c^{3} d^{11} x^{4} + 35 \, c^{4} d^{10} x^{3} + 21 \, c^{5} d^{9} x^{2} + 7 \, c^{6} d^{8} x + c^{7} d^{7}\right )}} \]

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^1
3*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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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

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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GIAC/XCAS [A]  time = 0.218323, size = 498, normalized size = 17.79 \[ -\frac{7 \, b^{6} d^{6} x^{6} + 21 \, b^{6} c d^{5} x^{5} + 21 \, a b^{5} d^{6} x^{5} + 35 \, b^{6} c^{2} d^{4} x^{4} + 35 \, a b^{5} c d^{5} x^{4} + 35 \, a^{2} b^{4} d^{6} x^{4} + 35 \, b^{6} c^{3} d^{3} x^{3} + 35 \, a b^{5} c^{2} d^{4} x^{3} + 35 \, a^{2} b^{4} c d^{5} x^{3} + 35 \, a^{3} b^{3} d^{6} x^{3} + 21 \, b^{6} c^{4} d^{2} x^{2} + 21 \, a b^{5} c^{3} d^{3} x^{2} + 21 \, a^{2} b^{4} c^{2} d^{4} x^{2} + 21 \, a^{3} b^{3} c d^{5} x^{2} + 21 \, a^{4} b^{2} d^{6} x^{2} + 7 \, b^{6} c^{5} d x + 7 \, a b^{5} c^{4} d^{2} x + 7 \, a^{2} b^{4} c^{3} d^{3} x + 7 \, a^{3} b^{3} c^{2} d^{4} x + 7 \, a^{4} b^{2} c d^{5} x + 7 \, a^{5} b d^{6} x + b^{6} c^{6} + a b^{5} c^{5} d + a^{2} b^{4} c^{4} d^{2} + a^{3} b^{3} c^{3} d^{3} + a^{4} b^{2} c^{2} d^{4} + a^{5} b c d^{5} + a^{6} d^{6}}{7 \,{\left (d x + c\right )}^{7} d^{7}} \]

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)