3.1794 \(\int \frac {(a c+(b c+a d) x+b d x^2)^3}{(a+b x)^8} \, dx\)

Optimal. Leaf size=28 \[ -\frac {(c+d x)^4}{4 (a+b x)^4 (b c-a d)} \]

[Out]

-1/4*(d*x+c)^4/(-a*d+b*c)/(b*x+a)^4

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Rubi [A]  time = 0.01, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {626, 37} \[ -\frac {(c+d x)^4}{4 (a+b x)^4 (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Int[(a*c + (b*c + a*d)*x + b*d*x^2)^3/(a + b*x)^8,x]

[Out]

-(c + d*x)^4/(4*(b*c - a*d)*(a + b*x)^4)

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]

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 {\left (a c+(b c+a d) x+b d x^2\right )^3}{(a+b x)^8} \, dx &=\int \frac {(c+d x)^3}{(a+b x)^5} \, dx\\ &=-\frac {(c+d x)^4}{4 (b c-a d) (a+b x)^4}\\ \end {align*}

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Mathematica [B]  time = 0.03, size = 91, normalized size = 3.25 \[ -\frac {a^3 d^3+a^2 b d^2 (c+4 d x)+a b^2 d \left (c^2+4 c d x+6 d^2 x^2\right )+b^3 \left (c^3+4 c^2 d x+6 c d^2 x^2+4 d^3 x^3\right )}{4 b^4 (a+b x)^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*c + (b*c + a*d)*x + b*d*x^2)^3/(a + b*x)^8,x]

[Out]

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

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fricas [B]  time = 0.95, size = 143, normalized size = 5.11 \[ -\frac {4 \, b^{3} d^{3} x^{3} + b^{3} c^{3} + a b^{2} c^{2} d + a^{2} b c d^{2} + a^{3} d^{3} + 6 \, {\left (b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{2} + 4 \, {\left (b^{3} c^{2} d + a b^{2} c d^{2} + a^{2} b d^{3}\right )} x}{4 \, {\left (b^{8} x^{4} + 4 \, a b^{7} x^{3} + 6 \, a^{2} b^{6} x^{2} + 4 \, a^{3} b^{5} x + a^{4} b^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*c+(a*d+b*c)*x+b*d*x^2)^3/(b*x+a)^8,x, algorithm="fricas")

[Out]

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

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giac [B]  time = 0.16, size = 111, normalized size = 3.96 \[ -\frac {4 \, b^{3} d^{3} x^{3} + 6 \, b^{3} c d^{2} x^{2} + 6 \, a b^{2} d^{3} x^{2} + 4 \, b^{3} c^{2} d x + 4 \, a b^{2} c d^{2} x + 4 \, a^{2} b d^{3} x + b^{3} c^{3} + a b^{2} c^{2} d + a^{2} b c d^{2} + a^{3} d^{3}}{4 \, {\left (b x + a\right )}^{4} b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*c+(a*d+b*c)*x+b*d*x^2)^3/(b*x+a)^8,x, algorithm="giac")

[Out]

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

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maple [B]  time = 0.05, size = 122, normalized size = 4.36 \[ -\frac {d^{3}}{\left (b x +a \right ) b^{4}}+\frac {3 \left (a d -b c \right ) d^{2}}{2 \left (b x +a \right )^{2} b^{4}}-\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) d}{\left (b x +a \right )^{3} b^{4}}-\frac {-a^{3} d^{3}+3 a^{2} b c \,d^{2}-3 a \,c^{2} d \,b^{2}+c^{3} b^{3}}{4 \left (b x +a \right )^{4} b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*c+(a*d+b*c)*x+b*d*x^2)^3/(b*x+a)^8,x)

[Out]

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

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maxima [B]  time = 1.17, size = 143, normalized size = 5.11 \[ -\frac {4 \, b^{3} d^{3} x^{3} + b^{3} c^{3} + a b^{2} c^{2} d + a^{2} b c d^{2} + a^{3} d^{3} + 6 \, {\left (b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{2} + 4 \, {\left (b^{3} c^{2} d + a b^{2} c d^{2} + a^{2} b d^{3}\right )} x}{4 \, {\left (b^{8} x^{4} + 4 \, a b^{7} x^{3} + 6 \, a^{2} b^{6} x^{2} + 4 \, a^{3} b^{5} x + a^{4} b^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*c+(a*d+b*c)*x+b*d*x^2)^3/(b*x+a)^8,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.60, size = 135, normalized size = 4.82 \[ -\frac {\frac {a^3\,d^3+a^2\,b\,c\,d^2+a\,b^2\,c^2\,d+b^3\,c^3}{4\,b^4}+\frac {d^3\,x^3}{b}+\frac {d\,x\,\left (a^2\,d^2+a\,b\,c\,d+b^2\,c^2\right )}{b^3}+\frac {3\,d^2\,x^2\,\left (a\,d+b\,c\right )}{2\,b^2}}{a^4+4\,a^3\,b\,x+6\,a^2\,b^2\,x^2+4\,a\,b^3\,x^3+b^4\,x^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*c + x*(a*d + b*c) + b*d*x^2)^3/(a + b*x)^8,x)

[Out]

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

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sympy [B]  time = 1.97, size = 155, normalized size = 5.54 \[ \frac {- a^{3} d^{3} - a^{2} b c d^{2} - a b^{2} c^{2} d - b^{3} c^{3} - 4 b^{3} d^{3} x^{3} + x^{2} \left (- 6 a b^{2} d^{3} - 6 b^{3} c d^{2}\right ) + x \left (- 4 a^{2} b d^{3} - 4 a b^{2} c d^{2} - 4 b^{3} c^{2} d\right )}{4 a^{4} b^{4} + 16 a^{3} b^{5} x + 24 a^{2} b^{6} x^{2} + 16 a b^{7} x^{3} + 4 b^{8} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*c+(a*d+b*c)*x+b*d*x**2)**3/(b*x+a)**8,x)

[Out]

(-a**3*d**3 - a**2*b*c*d**2 - a*b**2*c**2*d - b**3*c**3 - 4*b**3*d**3*x**3 + x**2*(-6*a*b**2*d**3 - 6*b**3*c*d
**2) + x*(-4*a**2*b*d**3 - 4*a*b**2*c*d**2 - 4*b**3*c**2*d))/(4*a**4*b**4 + 16*a**3*b**5*x + 24*a**2*b**6*x**2
+ 16*a*b**7*x**3 + 4*b**8*x**4)

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