3.1294 \(\int \frac{(c+d x)^7}{(a+b x)^{12}} \, dx\)

Optimal. Leaf size=120 \[ \frac{d^3 (c+d x)^8}{1320 (a+b x)^8 (b c-a d)^4}-\frac{d^2 (c+d x)^8}{165 (a+b x)^9 (b c-a d)^3}+\frac{3 d (c+d x)^8}{110 (a+b x)^{10} (b c-a d)^2}-\frac{(c+d x)^8}{11 (a+b x)^{11} (b c-a d)} \]

[Out]

-(c + d*x)^8/(11*(b*c - a*d)*(a + b*x)^11) + (3*d*(c + d*x)^8)/(110*(b*c - a*d)^
2*(a + b*x)^10) - (d^2*(c + d*x)^8)/(165*(b*c - a*d)^3*(a + b*x)^9) + (d^3*(c +
d*x)^8)/(1320*(b*c - a*d)^4*(a + b*x)^8)

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Rubi [A]  time = 0.0935707, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{d^3 (c+d x)^8}{1320 (a+b x)^8 (b c-a d)^4}-\frac{d^2 (c+d x)^8}{165 (a+b x)^9 (b c-a d)^3}+\frac{3 d (c+d x)^8}{110 (a+b x)^{10} (b c-a d)^2}-\frac{(c+d x)^8}{11 (a+b x)^{11} (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x)^7/(a + b*x)^12,x]

[Out]

-(c + d*x)^8/(11*(b*c - a*d)*(a + b*x)^11) + (3*d*(c + d*x)^8)/(110*(b*c - a*d)^
2*(a + b*x)^10) - (d^2*(c + d*x)^8)/(165*(b*c - a*d)^3*(a + b*x)^9) + (d^3*(c +
d*x)^8)/(1320*(b*c - a*d)^4*(a + b*x)^8)

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Rubi in Sympy [A]  time = 21.8695, size = 102, normalized size = 0.85 \[ \frac{d^{3} \left (c + d x\right )^{8}}{1320 \left (a + b x\right )^{8} \left (a d - b c\right )^{4}} + \frac{d^{2} \left (c + d x\right )^{8}}{165 \left (a + b x\right )^{9} \left (a d - b c\right )^{3}} + \frac{3 d \left (c + d x\right )^{8}}{110 \left (a + b x\right )^{10} \left (a d - b c\right )^{2}} + \frac{\left (c + d x\right )^{8}}{11 \left (a + b x\right )^{11} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x+c)**7/(b*x+a)**12,x)

[Out]

d**3*(c + d*x)**8/(1320*(a + b*x)**8*(a*d - b*c)**4) + d**2*(c + d*x)**8/(165*(a
+ b*x)**9*(a*d - b*c)**3) + 3*d*(c + d*x)**8/(110*(a + b*x)**10*(a*d - b*c)**2)
+ (c + d*x)**8/(11*(a + b*x)**11*(a*d - b*c))

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Mathematica [B]  time = 0.283144, size = 369, normalized size = 3.08 \[ -\frac{a^7 d^7+a^6 b d^6 (4 c+11 d x)+a^5 b^2 d^5 \left (10 c^2+44 c d x+55 d^2 x^2\right )+5 a^4 b^3 d^4 \left (4 c^3+22 c^2 d x+44 c d^2 x^2+33 d^3 x^3\right )+5 a^3 b^4 d^3 \left (7 c^4+44 c^3 d x+110 c^2 d^2 x^2+132 c d^3 x^3+66 d^4 x^4\right )+a^2 b^5 d^2 \left (56 c^5+385 c^4 d x+1100 c^3 d^2 x^2+1650 c^2 d^3 x^3+1320 c d^4 x^4+462 d^5 x^5\right )+a b^6 d \left (84 c^6+616 c^5 d x+1925 c^4 d^2 x^2+3300 c^3 d^3 x^3+3300 c^2 d^4 x^4+1848 c d^5 x^5+462 d^6 x^6\right )+b^7 \left (120 c^7+924 c^6 d x+3080 c^5 d^2 x^2+5775 c^4 d^3 x^3+6600 c^3 d^4 x^4+4620 c^2 d^5 x^5+1848 c d^6 x^6+330 d^7 x^7\right )}{1320 b^8 (a+b x)^{11}} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x)^7/(a + b*x)^12,x]

[Out]

-(a^7*d^7 + a^6*b*d^6*(4*c + 11*d*x) + a^5*b^2*d^5*(10*c^2 + 44*c*d*x + 55*d^2*x
^2) + 5*a^4*b^3*d^4*(4*c^3 + 22*c^2*d*x + 44*c*d^2*x^2 + 33*d^3*x^3) + 5*a^3*b^4
*d^3*(7*c^4 + 44*c^3*d*x + 110*c^2*d^2*x^2 + 132*c*d^3*x^3 + 66*d^4*x^4) + a^2*b
^5*d^2*(56*c^5 + 385*c^4*d*x + 1100*c^3*d^2*x^2 + 1650*c^2*d^3*x^3 + 1320*c*d^4*
x^4 + 462*d^5*x^5) + a*b^6*d*(84*c^6 + 616*c^5*d*x + 1925*c^4*d^2*x^2 + 3300*c^3
*d^3*x^3 + 3300*c^2*d^4*x^4 + 1848*c*d^5*x^5 + 462*d^6*x^6) + b^7*(120*c^7 + 924
*c^6*d*x + 3080*c^5*d^2*x^2 + 5775*c^4*d^3*x^3 + 6600*c^3*d^4*x^4 + 4620*c^2*d^5
*x^5 + 1848*c*d^6*x^6 + 330*d^7*x^7))/(1320*b^8*(a + b*x)^11)

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Maple [B]  time = 0.012, size = 464, normalized size = 3.9 \[{\frac{7\,{d}^{6} \left ( ad-bc \right ) }{5\,{b}^{8} \left ( bx+a \right ) ^{5}}}-{\frac{7\,{d}^{5} \left ({a}^{2}{d}^{2}-2\,abcd+{b}^{2}{c}^{2} \right ) }{2\,{b}^{8} \left ( bx+a \right ) ^{6}}}+5\,{\frac{{d}^{4} \left ({a}^{3}{d}^{3}-3\,{a}^{2}bc{d}^{2}+3\,a{b}^{2}{c}^{2}d-{b}^{3}{c}^{3} \right ) }{{b}^{8} \left ( bx+a \right ) ^{7}}}-{\frac{{d}^{7}}{4\,{b}^{8} \left ( bx+a \right ) ^{4}}}-{\frac{35\,{d}^{3} \left ({a}^{4}{d}^{4}-4\,{a}^{3}bc{d}^{3}+6\,{a}^{2}{b}^{2}{c}^{2}{d}^{2}-4\,a{b}^{3}{c}^{3}d+{b}^{4}{c}^{4} \right ) }{8\,{b}^{8} \left ( bx+a \right ) ^{8}}}-{\frac{7\,d \left ({a}^{6}{d}^{6}-6\,{a}^{5}bc{d}^{5}+15\,{a}^{4}{b}^{2}{c}^{2}{d}^{4}-20\,{a}^{3}{b}^{3}{c}^{3}{d}^{3}+15\,{a}^{2}{b}^{4}{c}^{4}{d}^{2}-6\,a{b}^{5}{c}^{5}d+{b}^{6}{c}^{6} \right ) }{10\,{b}^{8} \left ( bx+a \right ) ^{10}}}-{\frac{-{a}^{7}{d}^{7}+7\,c{d}^{6}{a}^{6}b-21\,{a}^{5}{c}^{2}{d}^{5}{b}^{2}+35\,{a}^{4}{b}^{3}{c}^{3}{d}^{4}-35\,{a}^{3}{b}^{4}{c}^{4}{d}^{3}+21\,{a}^{2}{c}^{5}{d}^{2}{b}^{5}-7\,a{b}^{6}{c}^{6}d+{c}^{7}{b}^{7}}{11\,{b}^{8} \left ( bx+a \right ) ^{11}}}+{\frac{7\,{d}^{2} \left ({a}^{5}{d}^{5}-5\,{a}^{4}bc{d}^{4}+10\,{a}^{3}{b}^{2}{c}^{2}{d}^{3}-10\,{a}^{2}{b}^{3}{c}^{3}{d}^{2}+5\,a{b}^{4}{c}^{4}d-{b}^{5}{c}^{5} \right ) }{3\,{b}^{8} \left ( bx+a \right ) ^{9}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^7/(b*x+a)^12,x)

[Out]

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

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Maxima [A]  time = 1.40959, size = 770, normalized size = 6.42 \[ -\frac{330 \, b^{7} d^{7} x^{7} + 120 \, b^{7} c^{7} + 84 \, a b^{6} c^{6} d + 56 \, a^{2} b^{5} c^{5} d^{2} + 35 \, a^{3} b^{4} c^{4} d^{3} + 20 \, a^{4} b^{3} c^{3} d^{4} + 10 \, a^{5} b^{2} c^{2} d^{5} + 4 \, a^{6} b c d^{6} + a^{7} d^{7} + 462 \,{\left (4 \, b^{7} c d^{6} + a b^{6} d^{7}\right )} x^{6} + 462 \,{\left (10 \, b^{7} c^{2} d^{5} + 4 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{5} + 330 \,{\left (20 \, b^{7} c^{3} d^{4} + 10 \, a b^{6} c^{2} d^{5} + 4 \, a^{2} b^{5} c d^{6} + a^{3} b^{4} d^{7}\right )} x^{4} + 165 \,{\left (35 \, b^{7} c^{4} d^{3} + 20 \, a b^{6} c^{3} d^{4} + 10 \, a^{2} b^{5} c^{2} d^{5} + 4 \, a^{3} b^{4} c d^{6} + a^{4} b^{3} d^{7}\right )} x^{3} + 55 \,{\left (56 \, b^{7} c^{5} d^{2} + 35 \, a b^{6} c^{4} d^{3} + 20 \, a^{2} b^{5} c^{3} d^{4} + 10 \, a^{3} b^{4} c^{2} d^{5} + 4 \, a^{4} b^{3} c d^{6} + a^{5} b^{2} d^{7}\right )} x^{2} + 11 \,{\left (84 \, b^{7} c^{6} d + 56 \, a b^{6} c^{5} d^{2} + 35 \, a^{2} b^{5} c^{4} d^{3} + 20 \, a^{3} b^{4} c^{3} d^{4} + 10 \, a^{4} b^{3} c^{2} d^{5} + 4 \, a^{5} b^{2} c d^{6} + a^{6} b d^{7}\right )} x}{1320 \,{\left (b^{19} x^{11} + 11 \, a b^{18} x^{10} + 55 \, a^{2} b^{17} x^{9} + 165 \, a^{3} b^{16} x^{8} + 330 \, a^{4} b^{15} x^{7} + 462 \, a^{5} b^{14} x^{6} + 462 \, a^{6} b^{13} x^{5} + 330 \, a^{7} b^{12} x^{4} + 165 \, a^{8} b^{11} x^{3} + 55 \, a^{9} b^{10} x^{2} + 11 \, a^{10} b^{9} x + a^{11} b^{8}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^7/(b*x + a)^12,x, algorithm="maxima")

[Out]

-1/1320*(330*b^7*d^7*x^7 + 120*b^7*c^7 + 84*a*b^6*c^6*d + 56*a^2*b^5*c^5*d^2 + 3
5*a^3*b^4*c^4*d^3 + 20*a^4*b^3*c^3*d^4 + 10*a^5*b^2*c^2*d^5 + 4*a^6*b*c*d^6 + a^
7*d^7 + 462*(4*b^7*c*d^6 + a*b^6*d^7)*x^6 + 462*(10*b^7*c^2*d^5 + 4*a*b^6*c*d^6
+ a^2*b^5*d^7)*x^5 + 330*(20*b^7*c^3*d^4 + 10*a*b^6*c^2*d^5 + 4*a^2*b^5*c*d^6 +
a^3*b^4*d^7)*x^4 + 165*(35*b^7*c^4*d^3 + 20*a*b^6*c^3*d^4 + 10*a^2*b^5*c^2*d^5 +
4*a^3*b^4*c*d^6 + a^4*b^3*d^7)*x^3 + 55*(56*b^7*c^5*d^2 + 35*a*b^6*c^4*d^3 + 20
*a^2*b^5*c^3*d^4 + 10*a^3*b^4*c^2*d^5 + 4*a^4*b^3*c*d^6 + a^5*b^2*d^7)*x^2 + 11*
(84*b^7*c^6*d + 56*a*b^6*c^5*d^2 + 35*a^2*b^5*c^4*d^3 + 20*a^3*b^4*c^3*d^4 + 10*
a^4*b^3*c^2*d^5 + 4*a^5*b^2*c*d^6 + a^6*b*d^7)*x)/(b^19*x^11 + 11*a*b^18*x^10 +
55*a^2*b^17*x^9 + 165*a^3*b^16*x^8 + 330*a^4*b^15*x^7 + 462*a^5*b^14*x^6 + 462*a
^6*b^13*x^5 + 330*a^7*b^12*x^4 + 165*a^8*b^11*x^3 + 55*a^9*b^10*x^2 + 11*a^10*b^
9*x + a^11*b^8)

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Fricas [A]  time = 0.214616, size = 770, normalized size = 6.42 \[ -\frac{330 \, b^{7} d^{7} x^{7} + 120 \, b^{7} c^{7} + 84 \, a b^{6} c^{6} d + 56 \, a^{2} b^{5} c^{5} d^{2} + 35 \, a^{3} b^{4} c^{4} d^{3} + 20 \, a^{4} b^{3} c^{3} d^{4} + 10 \, a^{5} b^{2} c^{2} d^{5} + 4 \, a^{6} b c d^{6} + a^{7} d^{7} + 462 \,{\left (4 \, b^{7} c d^{6} + a b^{6} d^{7}\right )} x^{6} + 462 \,{\left (10 \, b^{7} c^{2} d^{5} + 4 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{5} + 330 \,{\left (20 \, b^{7} c^{3} d^{4} + 10 \, a b^{6} c^{2} d^{5} + 4 \, a^{2} b^{5} c d^{6} + a^{3} b^{4} d^{7}\right )} x^{4} + 165 \,{\left (35 \, b^{7} c^{4} d^{3} + 20 \, a b^{6} c^{3} d^{4} + 10 \, a^{2} b^{5} c^{2} d^{5} + 4 \, a^{3} b^{4} c d^{6} + a^{4} b^{3} d^{7}\right )} x^{3} + 55 \,{\left (56 \, b^{7} c^{5} d^{2} + 35 \, a b^{6} c^{4} d^{3} + 20 \, a^{2} b^{5} c^{3} d^{4} + 10 \, a^{3} b^{4} c^{2} d^{5} + 4 \, a^{4} b^{3} c d^{6} + a^{5} b^{2} d^{7}\right )} x^{2} + 11 \,{\left (84 \, b^{7} c^{6} d + 56 \, a b^{6} c^{5} d^{2} + 35 \, a^{2} b^{5} c^{4} d^{3} + 20 \, a^{3} b^{4} c^{3} d^{4} + 10 \, a^{4} b^{3} c^{2} d^{5} + 4 \, a^{5} b^{2} c d^{6} + a^{6} b d^{7}\right )} x}{1320 \,{\left (b^{19} x^{11} + 11 \, a b^{18} x^{10} + 55 \, a^{2} b^{17} x^{9} + 165 \, a^{3} b^{16} x^{8} + 330 \, a^{4} b^{15} x^{7} + 462 \, a^{5} b^{14} x^{6} + 462 \, a^{6} b^{13} x^{5} + 330 \, a^{7} b^{12} x^{4} + 165 \, a^{8} b^{11} x^{3} + 55 \, a^{9} b^{10} x^{2} + 11 \, a^{10} b^{9} x + a^{11} b^{8}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^7/(b*x + a)^12,x, algorithm="fricas")

[Out]

-1/1320*(330*b^7*d^7*x^7 + 120*b^7*c^7 + 84*a*b^6*c^6*d + 56*a^2*b^5*c^5*d^2 + 3
5*a^3*b^4*c^4*d^3 + 20*a^4*b^3*c^3*d^4 + 10*a^5*b^2*c^2*d^5 + 4*a^6*b*c*d^6 + a^
7*d^7 + 462*(4*b^7*c*d^6 + a*b^6*d^7)*x^6 + 462*(10*b^7*c^2*d^5 + 4*a*b^6*c*d^6
+ a^2*b^5*d^7)*x^5 + 330*(20*b^7*c^3*d^4 + 10*a*b^6*c^2*d^5 + 4*a^2*b^5*c*d^6 +
a^3*b^4*d^7)*x^4 + 165*(35*b^7*c^4*d^3 + 20*a*b^6*c^3*d^4 + 10*a^2*b^5*c^2*d^5 +
4*a^3*b^4*c*d^6 + a^4*b^3*d^7)*x^3 + 55*(56*b^7*c^5*d^2 + 35*a*b^6*c^4*d^3 + 20
*a^2*b^5*c^3*d^4 + 10*a^3*b^4*c^2*d^5 + 4*a^4*b^3*c*d^6 + a^5*b^2*d^7)*x^2 + 11*
(84*b^7*c^6*d + 56*a*b^6*c^5*d^2 + 35*a^2*b^5*c^4*d^3 + 20*a^3*b^4*c^3*d^4 + 10*
a^4*b^3*c^2*d^5 + 4*a^5*b^2*c*d^6 + a^6*b*d^7)*x)/(b^19*x^11 + 11*a*b^18*x^10 +
55*a^2*b^17*x^9 + 165*a^3*b^16*x^8 + 330*a^4*b^15*x^7 + 462*a^5*b^14*x^6 + 462*a
^6*b^13*x^5 + 330*a^7*b^12*x^4 + 165*a^8*b^11*x^3 + 55*a^9*b^10*x^2 + 11*a^10*b^
9*x + a^11*b^8)

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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((d*x+c)**7/(b*x+a)**12,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.219888, size = 670, normalized size = 5.58 \[ -\frac{330 \, b^{7} d^{7} x^{7} + 1848 \, b^{7} c d^{6} x^{6} + 462 \, a b^{6} d^{7} x^{6} + 4620 \, b^{7} c^{2} d^{5} x^{5} + 1848 \, a b^{6} c d^{6} x^{5} + 462 \, a^{2} b^{5} d^{7} x^{5} + 6600 \, b^{7} c^{3} d^{4} x^{4} + 3300 \, a b^{6} c^{2} d^{5} x^{4} + 1320 \, a^{2} b^{5} c d^{6} x^{4} + 330 \, a^{3} b^{4} d^{7} x^{4} + 5775 \, b^{7} c^{4} d^{3} x^{3} + 3300 \, a b^{6} c^{3} d^{4} x^{3} + 1650 \, a^{2} b^{5} c^{2} d^{5} x^{3} + 660 \, a^{3} b^{4} c d^{6} x^{3} + 165 \, a^{4} b^{3} d^{7} x^{3} + 3080 \, b^{7} c^{5} d^{2} x^{2} + 1925 \, a b^{6} c^{4} d^{3} x^{2} + 1100 \, a^{2} b^{5} c^{3} d^{4} x^{2} + 550 \, a^{3} b^{4} c^{2} d^{5} x^{2} + 220 \, a^{4} b^{3} c d^{6} x^{2} + 55 \, a^{5} b^{2} d^{7} x^{2} + 924 \, b^{7} c^{6} d x + 616 \, a b^{6} c^{5} d^{2} x + 385 \, a^{2} b^{5} c^{4} d^{3} x + 220 \, a^{3} b^{4} c^{3} d^{4} x + 110 \, a^{4} b^{3} c^{2} d^{5} x + 44 \, a^{5} b^{2} c d^{6} x + 11 \, a^{6} b d^{7} x + 120 \, b^{7} c^{7} + 84 \, a b^{6} c^{6} d + 56 \, a^{2} b^{5} c^{5} d^{2} + 35 \, a^{3} b^{4} c^{4} d^{3} + 20 \, a^{4} b^{3} c^{3} d^{4} + 10 \, a^{5} b^{2} c^{2} d^{5} + 4 \, a^{6} b c d^{6} + a^{7} d^{7}}{1320 \,{\left (b x + a\right )}^{11} b^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^7/(b*x + a)^12,x, algorithm="giac")

[Out]

-1/1320*(330*b^7*d^7*x^7 + 1848*b^7*c*d^6*x^6 + 462*a*b^6*d^7*x^6 + 4620*b^7*c^2
*d^5*x^5 + 1848*a*b^6*c*d^6*x^5 + 462*a^2*b^5*d^7*x^5 + 6600*b^7*c^3*d^4*x^4 + 3
300*a*b^6*c^2*d^5*x^4 + 1320*a^2*b^5*c*d^6*x^4 + 330*a^3*b^4*d^7*x^4 + 5775*b^7*
c^4*d^3*x^3 + 3300*a*b^6*c^3*d^4*x^3 + 1650*a^2*b^5*c^2*d^5*x^3 + 660*a^3*b^4*c*
d^6*x^3 + 165*a^4*b^3*d^7*x^3 + 3080*b^7*c^5*d^2*x^2 + 1925*a*b^6*c^4*d^3*x^2 +
1100*a^2*b^5*c^3*d^4*x^2 + 550*a^3*b^4*c^2*d^5*x^2 + 220*a^4*b^3*c*d^6*x^2 + 55*
a^5*b^2*d^7*x^2 + 924*b^7*c^6*d*x + 616*a*b^6*c^5*d^2*x + 385*a^2*b^5*c^4*d^3*x
+ 220*a^3*b^4*c^3*d^4*x + 110*a^4*b^3*c^2*d^5*x + 44*a^5*b^2*c*d^6*x + 11*a^6*b*
d^7*x + 120*b^7*c^7 + 84*a*b^6*c^6*d + 56*a^2*b^5*c^5*d^2 + 35*a^3*b^4*c^4*d^3 +
20*a^4*b^3*c^3*d^4 + 10*a^5*b^2*c^2*d^5 + 4*a^6*b*c*d^6 + a^7*d^7)/((b*x + a)^1
1*b^8)