3.18.96 \(\int \frac {(a c+(b c+a d) x+b d x^2)^3}{(a+b x)^{10}} \, dx\) [1796]

Optimal. Leaf size=92 \[ -\frac {(b c-a d)^3}{6 b^4 (a+b x)^6}-\frac {3 d (b c-a d)^2}{5 b^4 (a+b x)^5}-\frac {3 d^2 (b c-a d)}{4 b^4 (a+b x)^4}-\frac {d^3}{3 b^4 (a+b x)^3} \]

[Out]

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

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Rubi [A]
time = 0.04, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {640, 45} \begin {gather*} -\frac {3 d^2 (b c-a d)}{4 b^4 (a+b x)^4}-\frac {3 d (b c-a d)^2}{5 b^4 (a+b x)^5}-\frac {(b c-a d)^3}{6 b^4 (a+b x)^6}-\frac {d^3}{3 b^4 (a+b x)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a*c + (b*c + a*d)*x + b*d*x^2)^3/(a + b*x)^10,x]

[Out]

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

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 640

Int[((d_) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a
/d + (c/e)*x)^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)^{10}} \, dx &=\int \frac {(c+d x)^3}{(a+b x)^7} \, dx\\ &=\int \left (\frac {(b c-a d)^3}{b^3 (a+b x)^7}+\frac {3 d (b c-a d)^2}{b^3 (a+b x)^6}+\frac {3 d^2 (b c-a d)}{b^3 (a+b x)^5}+\frac {d^3}{b^3 (a+b x)^4}\right ) \, dx\\ &=-\frac {(b c-a d)^3}{6 b^4 (a+b x)^6}-\frac {3 d (b c-a d)^2}{5 b^4 (a+b x)^5}-\frac {3 d^2 (b c-a d)}{4 b^4 (a+b x)^4}-\frac {d^3}{3 b^4 (a+b x)^3}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 97, normalized size = 1.05 \begin {gather*} -\frac {a^3 d^3+3 a^2 b d^2 (c+2 d x)+3 a b^2 d \left (2 c^2+6 c d x+5 d^2 x^2\right )+b^3 \left (10 c^3+36 c^2 d x+45 c d^2 x^2+20 d^3 x^3\right )}{60 b^4 (a+b x)^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a*c + (b*c + a*d)*x + b*d*x^2)^3/(a + b*x)^10,x]

[Out]

-1/60*(a^3*d^3 + 3*a^2*b*d^2*(c + 2*d*x) + 3*a*b^2*d*(2*c^2 + 6*c*d*x + 5*d^2*x^2) + b^3*(10*c^3 + 36*c^2*d*x
+ 45*c*d^2*x^2 + 20*d^3*x^3))/(b^4*(a + b*x)^6)

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Maple [A]
time = 0.78, size = 122, normalized size = 1.33

method	result	size
risch	\(\frac {-\frac {d^{3} x^{3}}{3 b}-\frac {d^{2} \left (a d +3 b c \right ) x^{2}}{4 b^{2}}-\frac {d \left (a^{2} d^{2}+3 a b c d +6 b^{2} c^{2}\right ) x}{10 b^{3}}-\frac {a^{3} d^{3}+3 a^{2} b c \,d^{2}+6 a \,b^{2} c^{2} d +10 b^{3} c^{3}}{60 b^{4}}}{\left (b x +a \right )^{6}}\)	\(110\)
gosper	\(-\frac {20 d^{3} x^{3} b^{3}+15 a \,b^{2} d^{3} x^{2}+45 b^{3} c \,d^{2} x^{2}+6 a^{2} b \,d^{3} x +18 b^{2} d^{2} x a c +36 b^{3} c^{2} d x +a^{3} d^{3}+3 a^{2} b c \,d^{2}+6 a \,b^{2} c^{2} d +10 b^{3} c^{3}}{60 b^{4} \left (b x +a \right )^{6}}\)	\(115\)
default	\(-\frac {d^{3}}{3 b^{4} \left (b x +a \right )^{3}}-\frac {-a^{3} d^{3}+3 a^{2} b c \,d^{2}-3 a \,b^{2} c^{2} d +b^{3} c^{3}}{6 b^{4} \left (b x +a \right )^{6}}-\frac {3 d \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{5 b^{4} \left (b x +a \right )^{5}}+\frac {3 d^{2} \left (a d -b c \right )}{4 b^{4} \left (b x +a \right )^{4}}\)	\(122\)
norman	\(\frac {\frac {a^{3} \left (-a^{3} b^{5} d^{3}-3 a^{2} b^{6} c \,d^{2}-6 a \,b^{7} c^{2} d -10 b^{8} c^{3}\right )}{60 b^{9}}-\frac {b^{2} d^{3} x^{6}}{3}+\frac {\left (-5 d^{3} a \,b^{5}-3 c \,d^{2} b^{6}\right ) x^{5}}{4 b^{4}}+\frac {\left (-37 a^{2} b^{5} d^{3}-51 a c \,d^{2} b^{6}-12 c^{2} d \,b^{7}\right ) x^{4}}{20 b^{5}}+\frac {\left (-42 a^{3} b^{5} d^{3}-96 a^{2} b^{6} c \,d^{2}-57 a \,b^{7} c^{2} d -5 b^{8} c^{3}\right ) x^{3}}{30 b^{6}}+\frac {a \left (-6 a^{3} b^{5} d^{3}-18 a^{2} b^{6} c \,d^{2}-21 a \,b^{7} c^{2} d -5 b^{8} c^{3}\right ) x^{2}}{10 b^{7}}+\frac {a^{2} \left (-3 a^{3} b^{5} d^{3}-9 a^{2} b^{6} c \,d^{2}-18 a \,b^{7} c^{2} d -10 b^{8} c^{3}\right ) x}{20 b^{8}}}{\left (b x +a \right )^{9}}\)	\(289\)

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)^10,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 171 vs. \(2 (84) = 168\).
time = 0.29, size = 171, normalized size = 1.86 \begin {gather*} -\frac {20 \, b^{3} d^{3} x^{3} + 10 \, b^{3} c^{3} + 6 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} + a^{3} d^{3} + 15 \, {\left (3 \, b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{2} + 6 \, {\left (6 \, b^{3} c^{2} d + 3 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x}{60 \, {\left (b^{10} x^{6} + 6 \, a b^{9} x^{5} + 15 \, a^{2} b^{8} x^{4} + 20 \, a^{3} b^{7} x^{3} + 15 \, a^{4} b^{6} x^{2} + 6 \, a^{5} b^{5} x + a^{6} b^{4}\right )}} \end {gather*}

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)^10,x, algorithm="maxima")

[Out]

-1/60*(20*b^3*d^3*x^3 + 10*b^3*c^3 + 6*a*b^2*c^2*d + 3*a^2*b*c*d^2 + a^3*d^3 + 15*(3*b^3*c*d^2 + a*b^2*d^3)*x^
2 + 6*(6*b^3*c^2*d + 3*a*b^2*c*d^2 + a^2*b*d^3)*x)/(b^10*x^6 + 6*a*b^9*x^5 + 15*a^2*b^8*x^4 + 20*a^3*b^7*x^3 +
15*a^4*b^6*x^2 + 6*a^5*b^5*x + a^6*b^4)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 171 vs. \(2 (84) = 168\).
time = 3.40, size = 171, normalized size = 1.86 \begin {gather*} -\frac {20 \, b^{3} d^{3} x^{3} + 10 \, b^{3} c^{3} + 6 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} + a^{3} d^{3} + 15 \, {\left (3 \, b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{2} + 6 \, {\left (6 \, b^{3} c^{2} d + 3 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x}{60 \, {\left (b^{10} x^{6} + 6 \, a b^{9} x^{5} + 15 \, a^{2} b^{8} x^{4} + 20 \, a^{3} b^{7} x^{3} + 15 \, a^{4} b^{6} x^{2} + 6 \, a^{5} b^{5} x + a^{6} b^{4}\right )}} \end {gather*}

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)^10,x, algorithm="fricas")

[Out]

-1/60*(20*b^3*d^3*x^3 + 10*b^3*c^3 + 6*a*b^2*c^2*d + 3*a^2*b*c*d^2 + a^3*d^3 + 15*(3*b^3*c*d^2 + a*b^2*d^3)*x^
2 + 6*(6*b^3*c^2*d + 3*a*b^2*c*d^2 + a^2*b*d^3)*x)/(b^10*x^6 + 6*a*b^9*x^5 + 15*a^2*b^8*x^4 + 20*a^3*b^7*x^3 +
15*a^4*b^6*x^2 + 6*a^5*b^5*x + a^6*b^4)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 184 vs. \(2 (83) = 166\).
time = 43.39, size = 184, normalized size = 2.00 \begin {gather*} \frac {- a^{3} d^{3} - 3 a^{2} b c d^{2} - 6 a b^{2} c^{2} d - 10 b^{3} c^{3} - 20 b^{3} d^{3} x^{3} + x^{2} \left (- 15 a b^{2} d^{3} - 45 b^{3} c d^{2}\right ) + x \left (- 6 a^{2} b d^{3} - 18 a b^{2} c d^{2} - 36 b^{3} c^{2} d\right )}{60 a^{6} b^{4} + 360 a^{5} b^{5} x + 900 a^{4} b^{6} x^{2} + 1200 a^{3} b^{7} x^{3} + 900 a^{2} b^{8} x^{4} + 360 a b^{9} x^{5} + 60 b^{10} x^{6}} \end {gather*}

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)**10,x)

[Out]

(-a**3*d**3 - 3*a**2*b*c*d**2 - 6*a*b**2*c**2*d - 10*b**3*c**3 - 20*b**3*d**3*x**3 + x**2*(-15*a*b**2*d**3 - 4
5*b**3*c*d**2) + x*(-6*a**2*b*d**3 - 18*a*b**2*c*d**2 - 36*b**3*c**2*d))/(60*a**6*b**4 + 360*a**5*b**5*x + 900
*a**4*b**6*x**2 + 1200*a**3*b**7*x**3 + 900*a**2*b**8*x**4 + 360*a*b**9*x**5 + 60*b**10*x**6)

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Giac [A]
time = 0.87, size = 114, normalized size = 1.24 \begin {gather*} -\frac {20 \, b^{3} d^{3} x^{3} + 45 \, b^{3} c d^{2} x^{2} + 15 \, a b^{2} d^{3} x^{2} + 36 \, b^{3} c^{2} d x + 18 \, a b^{2} c d^{2} x + 6 \, a^{2} b d^{3} x + 10 \, b^{3} c^{3} + 6 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} + a^{3} d^{3}}{60 \, {\left (b x + a\right )}^{6} b^{4}} \end {gather*}

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)^10,x, algorithm="giac")

[Out]

-1/60*(20*b^3*d^3*x^3 + 45*b^3*c*d^2*x^2 + 15*a*b^2*d^3*x^2 + 36*b^3*c^2*d*x + 18*a*b^2*c*d^2*x + 6*a^2*b*d^3*
x + 10*b^3*c^3 + 6*a*b^2*c^2*d + 3*a^2*b*c*d^2 + a^3*d^3)/((b*x + a)^6*b^4)

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Mupad [B]
time = 0.63, size = 165, normalized size = 1.79 \begin {gather*} -\frac {\frac {a^3\,d^3+3\,a^2\,b\,c\,d^2+6\,a\,b^2\,c^2\,d+10\,b^3\,c^3}{60\,b^4}+\frac {d^3\,x^3}{3\,b}+\frac {d\,x\,\left (a^2\,d^2+3\,a\,b\,c\,d+6\,b^2\,c^2\right )}{10\,b^3}+\frac {d^2\,x^2\,\left (a\,d+3\,b\,c\right )}{4\,b^2}}{a^6+6\,a^5\,b\,x+15\,a^4\,b^2\,x^2+20\,a^3\,b^3\,x^3+15\,a^2\,b^4\,x^4+6\,a\,b^5\,x^5+b^6\,x^6} \end {gather*}

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)^10,x)

[Out]

-((a^3*d^3 + 10*b^3*c^3 + 6*a*b^2*c^2*d + 3*a^2*b*c*d^2)/(60*b^4) + (d^3*x^3)/(3*b) + (d*x*(a^2*d^2 + 6*b^2*c^
2 + 3*a*b*c*d))/(10*b^3) + (d^2*x^2*(a*d + 3*b*c))/(4*b^2))/(a^6 + b^6*x^6 + 6*a*b^5*x^5 + 15*a^4*b^2*x^2 + 20
*a^3*b^3*x^3 + 15*a^2*b^4*x^4 + 6*a^5*b*x)

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