3.1369 \(\int \frac{(a+b x)^2}{(c+d x)^8} \, dx\)

Optimal. Leaf size=65 \[ \frac{b (b c-a d)}{3 d^3 (c+d x)^6}-\frac{(b c-a d)^2}{7 d^3 (c+d x)^7}-\frac{b^2}{5 d^3 (c+d x)^5} \]

[Out]

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

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Rubi [A]  time = 0.0951281, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \frac{b (b c-a d)}{3 d^3 (c+d x)^6}-\frac{(b c-a d)^2}{7 d^3 (c+d x)^7}-\frac{b^2}{5 d^3 (c+d x)^5} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^2/(c + d*x)^8,x]

[Out]

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

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Rubi in Sympy [A]  time = 18.787, size = 56, normalized size = 0.86 \[ - \frac{b^{2}}{5 d^{3} \left (c + d x\right )^{5}} - \frac{b \left (a d - b c\right )}{3 d^{3} \left (c + d x\right )^{6}} - \frac{\left (a d - b c\right )^{2}}{7 d^{3} \left (c + d x\right )^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**2/(d*x+c)**8,x)

[Out]

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

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Mathematica [A]  time = 0.0399563, size = 55, normalized size = 0.85 \[ -\frac{15 a^2 d^2+5 a b d (c+7 d x)+b^2 \left (c^2+7 c d x+21 d^2 x^2\right )}{105 d^3 (c+d x)^7} \]

Antiderivative was successfully verified.

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

[Out]

-(15*a^2*d^2 + 5*a*b*d*(c + 7*d*x) + b^2*(c^2 + 7*c*d*x + 21*d^2*x^2))/(105*d^3*
(c + d*x)^7)

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Maple [A]  time = 0.007, size = 71, normalized size = 1.1 \[ -{\frac{{b}^{2}}{5\,{d}^{3} \left ( dx+c \right ) ^{5}}}-{\frac{{a}^{2}{d}^{2}-2\,abcd+{b}^{2}{c}^{2}}{7\,{d}^{3} \left ( dx+c \right ) ^{7}}}-{\frac{b \left ( ad-bc \right ) }{3\,{d}^{3} \left ( dx+c \right ) ^{6}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^2/(d*x+c)^8,x)

[Out]

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

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Maxima [A]  time = 1.35257, size = 177, normalized size = 2.72 \[ -\frac{21 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 5 \, a b c d + 15 \, a^{2} d^{2} + 7 \,{\left (b^{2} c d + 5 \, a b d^{2}\right )} x}{105 \,{\left (d^{10} x^{7} + 7 \, c d^{9} x^{6} + 21 \, c^{2} d^{8} x^{5} + 35 \, c^{3} d^{7} x^{4} + 35 \, c^{4} d^{6} x^{3} + 21 \, c^{5} d^{5} x^{2} + 7 \, c^{6} d^{4} x + c^{7} d^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/105*(21*b^2*d^2*x^2 + b^2*c^2 + 5*a*b*c*d + 15*a^2*d^2 + 7*(b^2*c*d + 5*a*b*d
^2)*x)/(d^10*x^7 + 7*c*d^9*x^6 + 21*c^2*d^8*x^5 + 35*c^3*d^7*x^4 + 35*c^4*d^6*x^
3 + 21*c^5*d^5*x^2 + 7*c^6*d^4*x + c^7*d^3)

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Fricas [A]  time = 0.205564, size = 177, normalized size = 2.72 \[ -\frac{21 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 5 \, a b c d + 15 \, a^{2} d^{2} + 7 \,{\left (b^{2} c d + 5 \, a b d^{2}\right )} x}{105 \,{\left (d^{10} x^{7} + 7 \, c d^{9} x^{6} + 21 \, c^{2} d^{8} x^{5} + 35 \, c^{3} d^{7} x^{4} + 35 \, c^{4} d^{6} x^{3} + 21 \, c^{5} d^{5} x^{2} + 7 \, c^{6} d^{4} x + c^{7} d^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/105*(21*b^2*d^2*x^2 + b^2*c^2 + 5*a*b*c*d + 15*a^2*d^2 + 7*(b^2*c*d + 5*a*b*d
^2)*x)/(d^10*x^7 + 7*c*d^9*x^6 + 21*c^2*d^8*x^5 + 35*c^3*d^7*x^4 + 35*c^4*d^6*x^
3 + 21*c^5*d^5*x^2 + 7*c^6*d^4*x + c^7*d^3)

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Sympy [A]  time = 2.84875, size = 139, normalized size = 2.14 \[ - \frac{15 a^{2} d^{2} + 5 a b c d + b^{2} c^{2} + 21 b^{2} d^{2} x^{2} + x \left (35 a b d^{2} + 7 b^{2} c d\right )}{105 c^{7} d^{3} + 735 c^{6} d^{4} x + 2205 c^{5} d^{5} x^{2} + 3675 c^{4} d^{6} x^{3} + 3675 c^{3} d^{7} x^{4} + 2205 c^{2} d^{8} x^{5} + 735 c d^{9} x^{6} + 105 d^{10} x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**2/(d*x+c)**8,x)

[Out]

-(15*a**2*d**2 + 5*a*b*c*d + b**2*c**2 + 21*b**2*d**2*x**2 + x*(35*a*b*d**2 + 7*
b**2*c*d))/(105*c**7*d**3 + 735*c**6*d**4*x + 2205*c**5*d**5*x**2 + 3675*c**4*d*
*6*x**3 + 3675*c**3*d**7*x**4 + 2205*c**2*d**8*x**5 + 735*c*d**9*x**6 + 105*d**1
0*x**7)

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GIAC/XCAS [A]  time = 0.22137, size = 82, normalized size = 1.26 \[ -\frac{21 \, b^{2} d^{2} x^{2} + 7 \, b^{2} c d x + 35 \, a b d^{2} x + b^{2} c^{2} + 5 \, a b c d + 15 \, a^{2} d^{2}}{105 \,{\left (d x + c\right )}^{7} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/105*(21*b^2*d^2*x^2 + 7*b^2*c*d*x + 35*a*b*d^2*x + b^2*c^2 + 5*a*b*c*d + 15*a
^2*d^2)/((d*x + c)^7*d^3)