3.4.3 \(\int \frac {x (c+d x)^3}{(a+b x)^3} \, dx\) [303]

Optimal. Leaf size=114 \[ \frac {3 d^2 (b c-a d) x}{b^4}+\frac {d^3 x^2}{2 b^3}+\frac {a (b c-a d)^3}{2 b^5 (a+b x)^2}-\frac {(b c-4 a d) (b c-a d)^2}{b^5 (a+b x)}+\frac {3 d (b c-2 a d) (b c-a d) \log (a+b x)}{b^5} \]

[Out]

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

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Rubi [A]
time = 0.07, antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {78} \begin {gather*} -\frac {(b c-4 a d) (b c-a d)^2}{b^5 (a+b x)}+\frac {a (b c-a d)^3}{2 b^5 (a+b x)^2}+\frac {3 d (b c-2 a d) (b c-a d) \log (a+b x)}{b^5}+\frac {3 d^2 x (b c-a d)}{b^4}+\frac {d^3 x^2}{2 b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x*(c + d*x)^3)/(a + b*x)^3,x]

[Out]

(3*d^2*(b*c - a*d)*x)/b^4 + (d^3*x^2)/(2*b^3) + (a*(b*c - a*d)^3)/(2*b^5*(a + b*x)^2) - ((b*c - 4*a*d)*(b*c -
a*d)^2)/(b^5*(a + b*x)) + (3*d*(b*c - 2*a*d)*(b*c - a*d)*Log[a + b*x])/b^5

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

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

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Mathematica [A]
time = 0.04, size = 165, normalized size = 1.45 \begin {gather*} \frac {7 a^4 d^3+a^3 b d^2 (-15 c+2 d x)+a^2 b^2 d \left (9 c^2-12 c d x-11 d^2 x^2\right )+b^4 x \left (-2 c^3+6 c d^2 x^2+d^3 x^3\right )-a b^3 \left (c^3-12 c^2 d x-12 c d^2 x^2+4 d^3 x^3\right )+6 d \left (b^2 c^2-3 a b c d+2 a^2 d^2\right ) (a+b x)^2 \log (a+b x)}{2 b^5 (a+b x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(7*a^4*d^3 + a^3*b*d^2*(-15*c + 2*d*x) + a^2*b^2*d*(9*c^2 - 12*c*d*x - 11*d^2*x^2) + b^4*x*(-2*c^3 + 6*c*d^2*x
^2 + d^3*x^3) - a*b^3*(c^3 - 12*c^2*d*x - 12*c*d^2*x^2 + 4*d^3*x^3) + 6*d*(b^2*c^2 - 3*a*b*c*d + 2*a^2*d^2)*(a
+ b*x)^2*Log[a + b*x])/(2*b^5*(a + b*x)^2)

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Maple [A]
time = 0.08, size = 159, normalized size = 1.39

method	result	size
default	\(-\frac {d^{2} \left (-\frac {1}{2} b d \,x^{2}+3 a d x -3 b c x \right )}{b^{4}}+\frac {3 d \left (2 a^{2} d^{2}-3 a b c d +b^{2} c^{2}\right ) \ln \left (b x +a \right )}{b^{5}}-\frac {-4 a^{3} d^{3}+9 a^{2} b c \,d^{2}-6 a \,b^{2} c^{2} d +b^{3} c^{3}}{b^{5} \left (b x +a \right )}-\frac {a \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}{2 b^{5} \left (b x +a \right )^{2}}\)	\(159\)
norman	\(\frac {\frac {\left (12 a^{3} d^{3}-18 a^{2} b c \,d^{2}+6 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) x}{b^{4}}+\frac {a \left (18 a^{3} d^{3}-27 a^{2} b c \,d^{2}+9 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}{2 b^{5}}+\frac {d^{3} x^{4}}{2 b}-\frac {d^{2} \left (2 a d -3 b c \right ) x^{3}}{b^{2}}}{\left (b x +a \right )^{2}}+\frac {3 d \left (2 a^{2} d^{2}-3 a b c d +b^{2} c^{2}\right ) \ln \left (b x +a \right )}{b^{5}}\)	\(161\)
risch	\(\frac {d^{3} x^{2}}{2 b^{3}}-\frac {3 d^{3} a x}{b^{4}}+\frac {3 d^{2} c x}{b^{3}}+\frac {\left (4 a^{3} d^{3}-9 a^{2} b c \,d^{2}+6 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) x +\frac {a \left (7 a^{3} d^{3}-15 a^{2} b c \,d^{2}+9 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}{2 b}}{b^{4} \left (b x +a \right )^{2}}+\frac {6 d^{3} \ln \left (b x +a \right ) a^{2}}{b^{5}}-\frac {9 d^{2} \ln \left (b x +a \right ) a c}{b^{4}}+\frac {3 d \ln \left (b x +a \right ) c^{2}}{b^{3}}\)	\(175\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(d*x+c)^3/(b*x+a)^3,x,method=_RETURNVERBOSE)

[Out]

-d^2/b^4*(-1/2*b*d*x^2+3*a*d*x-3*b*c*x)+3/b^5*d*(2*a^2*d^2-3*a*b*c*d+b^2*c^2)*ln(b*x+a)-(-4*a^3*d^3+9*a^2*b*c*
d^2-6*a*b^2*c^2*d+b^3*c^3)/b^5/(b*x+a)-1/2*a/b^5*(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)/(b*x+a)^2

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Maxima [A]
time = 0.27, size = 174, normalized size = 1.53 \begin {gather*} -\frac {a b^{3} c^{3} - 9 \, a^{2} b^{2} c^{2} d + 15 \, a^{3} b c d^{2} - 7 \, a^{4} d^{3} + 2 \, {\left (b^{4} c^{3} - 6 \, a b^{3} c^{2} d + 9 \, a^{2} b^{2} c d^{2} - 4 \, a^{3} b d^{3}\right )} x}{2 \, {\left (b^{7} x^{2} + 2 \, a b^{6} x + a^{2} b^{5}\right )}} + \frac {b d^{3} x^{2} + 6 \, {\left (b c d^{2} - a d^{3}\right )} x}{2 \, b^{4}} + \frac {3 \, {\left (b^{2} c^{2} d - 3 \, a b c d^{2} + 2 \, a^{2} d^{3}\right )} \log \left (b x + a\right )}{b^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 274 vs. \(2 (110) = 220\).
time = 0.93, size = 274, normalized size = 2.40 \begin {gather*} \frac {b^{4} d^{3} x^{4} - a b^{3} c^{3} + 9 \, a^{2} b^{2} c^{2} d - 15 \, a^{3} b c d^{2} + 7 \, a^{4} d^{3} + 2 \, {\left (3 \, b^{4} c d^{2} - 2 \, a b^{3} d^{3}\right )} x^{3} + {\left (12 \, a b^{3} c d^{2} - 11 \, a^{2} b^{2} d^{3}\right )} x^{2} - 2 \, {\left (b^{4} c^{3} - 6 \, a b^{3} c^{2} d + 6 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} x + 6 \, {\left (a^{2} b^{2} c^{2} d - 3 \, a^{3} b c d^{2} + 2 \, a^{4} d^{3} + {\left (b^{4} c^{2} d - 3 \, a b^{3} c d^{2} + 2 \, a^{2} b^{2} d^{3}\right )} x^{2} + 2 \, {\left (a b^{3} c^{2} d - 3 \, a^{2} b^{2} c d^{2} + 2 \, a^{3} b d^{3}\right )} x\right )} \log \left (b x + a\right )}{2 \, {\left (b^{7} x^{2} + 2 \, a b^{6} x + a^{2} b^{5}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(b^4*d^3*x^4 - a*b^3*c^3 + 9*a^2*b^2*c^2*d - 15*a^3*b*c*d^2 + 7*a^4*d^3 + 2*(3*b^4*c*d^2 - 2*a*b^3*d^3)*x^
3 + (12*a*b^3*c*d^2 - 11*a^2*b^2*d^3)*x^2 - 2*(b^4*c^3 - 6*a*b^3*c^2*d + 6*a^2*b^2*c*d^2 - a^3*b*d^3)*x + 6*(a
^2*b^2*c^2*d - 3*a^3*b*c*d^2 + 2*a^4*d^3 + (b^4*c^2*d - 3*a*b^3*c*d^2 + 2*a^2*b^2*d^3)*x^2 + 2*(a*b^3*c^2*d -
3*a^2*b^2*c*d^2 + 2*a^3*b*d^3)*x)*log(b*x + a))/(b^7*x^2 + 2*a*b^6*x + a^2*b^5)

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Sympy [A]
time = 0.69, size = 175, normalized size = 1.54 \begin {gather*} x \left (- \frac {3 a d^{3}}{b^{4}} + \frac {3 c d^{2}}{b^{3}}\right ) + \frac {7 a^{4} d^{3} - 15 a^{3} b c d^{2} + 9 a^{2} b^{2} c^{2} d - a b^{3} c^{3} + x \left (8 a^{3} b d^{3} - 18 a^{2} b^{2} c d^{2} + 12 a b^{3} c^{2} d - 2 b^{4} c^{3}\right )}{2 a^{2} b^{5} + 4 a b^{6} x + 2 b^{7} x^{2}} + \frac {d^{3} x^{2}}{2 b^{3}} + \frac {3 d \left (a d - b c\right ) \left (2 a d - b c\right ) \log {\left (a + b x \right )}}{b^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*(-3*a*d**3/b**4 + 3*c*d**2/b**3) + (7*a**4*d**3 - 15*a**3*b*c*d**2 + 9*a**2*b**2*c**2*d - a*b**3*c**3 + x*(8
*a**3*b*d**3 - 18*a**2*b**2*c*d**2 + 12*a*b**3*c**2*d - 2*b**4*c**3))/(2*a**2*b**5 + 4*a*b**6*x + 2*b**7*x**2)
+ d**3*x**2/(2*b**3) + 3*d*(a*d - b*c)*(2*a*d - b*c)*log(a + b*x)/b**5

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Giac [A]
time = 0.76, size = 167, normalized size = 1.46 \begin {gather*} \frac {3 \, {\left (b^{2} c^{2} d - 3 \, a b c d^{2} + 2 \, a^{2} d^{3}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{5}} + \frac {b^{3} d^{3} x^{2} + 6 \, b^{3} c d^{2} x - 6 \, a b^{2} d^{3} x}{2 \, b^{6}} - \frac {a b^{3} c^{3} - 9 \, a^{2} b^{2} c^{2} d + 15 \, a^{3} b c d^{2} - 7 \, a^{4} d^{3} + 2 \, {\left (b^{4} c^{3} - 6 \, a b^{3} c^{2} d + 9 \, a^{2} b^{2} c d^{2} - 4 \, a^{3} b d^{3}\right )} x}{2 \, {\left (b x + a\right )}^{2} b^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*(b^2*c^2*d - 3*a*b*c*d^2 + 2*a^2*d^3)*log(abs(b*x + a))/b^5 + 1/2*(b^3*d^3*x^2 + 6*b^3*c*d^2*x - 6*a*b^2*d^3
*x)/b^6 - 1/2*(a*b^3*c^3 - 9*a^2*b^2*c^2*d + 15*a^3*b*c*d^2 - 7*a^4*d^3 + 2*(b^4*c^3 - 6*a*b^3*c^2*d + 9*a^2*b
^2*c*d^2 - 4*a^3*b*d^3)*x)/((b*x + a)^2*b^5)

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Mupad [B]
time = 0.39, size = 180, normalized size = 1.58 \begin {gather*} \frac {x\,\left (4\,a^3\,d^3-9\,a^2\,b\,c\,d^2+6\,a\,b^2\,c^2\,d-b^3\,c^3\right )+\frac {7\,a^4\,d^3-15\,a^3\,b\,c\,d^2+9\,a^2\,b^2\,c^2\,d-a\,b^3\,c^3}{2\,b}}{a^2\,b^4+2\,a\,b^5\,x+b^6\,x^2}-x\,\left (\frac {3\,a\,d^3}{b^4}-\frac {3\,c\,d^2}{b^3}\right )+\frac {\ln \left (a+b\,x\right )\,\left (6\,a^2\,d^3-9\,a\,b\,c\,d^2+3\,b^2\,c^2\,d\right )}{b^5}+\frac {d^3\,x^2}{2\,b^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x*(c + d*x)^3)/(a + b*x)^3,x)

[Out]

(x*(4*a^3*d^3 - b^3*c^3 + 6*a*b^2*c^2*d - 9*a^2*b*c*d^2) + (7*a^4*d^3 - a*b^3*c^3 + 9*a^2*b^2*c^2*d - 15*a^3*b
*c*d^2)/(2*b))/(a^2*b^4 + b^6*x^2 + 2*a*b^5*x) - x*((3*a*d^3)/b^4 - (3*c*d^2)/b^3) + (log(a + b*x)*(6*a^2*d^3
+ 3*b^2*c^2*d - 9*a*b*c*d^2))/b^5 + (d^3*x^2)/(2*b^3)

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