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

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

[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 = 0.11, 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 = {77} \[ \frac {3 d^2 x (b c-a d)}{b^4}-\frac {(b c-4 a d) (b c-a d)^2}{b^5 (a+b x)}+\frac {a (b c-a d)^3}{2 b^5 (a+b x)^2}+\frac {3 d (b c-2 a d) (b c-a d) \log (a+b x)}{b^5}+\frac {d^3 x^2}{2 b^3} \]

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 77

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 = 0.07, size = 165, normalized size = 1.45 \[ \frac {7 a^4 d^3+a^3 b d^2 (2 d x-15 c)+a^2 b^2 d \left (9 c^2-12 c d x-11 d^2 x^2\right )+6 d (a+b x)^2 \left (2 a^2 d^2-3 a b c d+b^2 c^2\right ) \log (a+b x)-a b^3 \left (c^3-12 c^2 d x-12 c d^2 x^2+4 d^3 x^3\right )+b^4 x \left (-2 c^3+6 c d^2 x^2+d^3 x^3\right )}{2 b^5 (a+b x)^2} \]

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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fricas [B]  time = 0.80, size = 274, normalized size = 2.40 \[ \frac {b^{4} d^{3} x^{4} - a b^{3} c^{3} + 9 \, a^{2} b^{2} c^{2} d - 15 \, a^{3} b c d^{2} + 7 \, a^{4} d^{3} + 2 \, {\left (3 \, b^{4} c d^{2} - 2 \, a b^{3} d^{3}\right )} x^{3} + {\left (12 \, a b^{3} c d^{2} - 11 \, a^{2} b^{2} d^{3}\right )} x^{2} - 2 \, {\left (b^{4} c^{3} - 6 \, a b^{3} c^{2} d + 6 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} x + 6 \, {\left (a^{2} b^{2} c^{2} d - 3 \, a^{3} b c d^{2} + 2 \, a^{4} d^{3} + {\left (b^{4} c^{2} d - 3 \, a b^{3} c d^{2} + 2 \, a^{2} b^{2} d^{3}\right )} x^{2} + 2 \, {\left (a b^{3} c^{2} d - 3 \, a^{2} b^{2} c d^{2} + 2 \, a^{3} b d^{3}\right )} x\right )} \log \left (b x + a\right )}{2 \, {\left (b^{7} x^{2} + 2 \, a b^{6} x + a^{2} b^{5}\right )}} \]

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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giac [A]  time = 0.94, size = 167, normalized size = 1.46 \[ \frac {3 \, {\left (b^{2} c^{2} d - 3 \, a b c d^{2} + 2 \, a^{2} d^{3}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{5}} + \frac {b^{3} d^{3} x^{2} + 6 \, b^{3} c d^{2} x - 6 \, a b^{2} d^{3} x}{2 \, b^{6}} - \frac {a b^{3} c^{3} - 9 \, a^{2} b^{2} c^{2} d + 15 \, a^{3} b c d^{2} - 7 \, a^{4} d^{3} + 2 \, {\left (b^{4} c^{3} - 6 \, a b^{3} c^{2} d + 9 \, a^{2} b^{2} c d^{2} - 4 \, a^{3} b d^{3}\right )} x}{2 \, {\left (b x + a\right )}^{2} b^{5}} \]

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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maple [B]  time = 0.01, size = 222, normalized size = 1.95 \[ -\frac {a^{4} d^{3}}{2 \left (b x +a \right )^{2} b^{5}}+\frac {3 a^{3} c \,d^{2}}{2 \left (b x +a \right )^{2} b^{4}}-\frac {3 a^{2} c^{2} d}{2 \left (b x +a \right )^{2} b^{3}}+\frac {a \,c^{3}}{2 \left (b x +a \right )^{2} b^{2}}+\frac {d^{3} x^{2}}{2 b^{3}}+\frac {4 a^{3} d^{3}}{\left (b x +a \right ) b^{5}}-\frac {9 a^{2} c \,d^{2}}{\left (b x +a \right ) b^{4}}+\frac {6 a^{2} d^{3} \ln \left (b x +a \right )}{b^{5}}+\frac {6 a \,c^{2} d}{\left (b x +a \right ) b^{3}}-\frac {9 a c \,d^{2} \ln \left (b x +a \right )}{b^{4}}-\frac {3 a \,d^{3} x}{b^{4}}-\frac {c^{3}}{\left (b x +a \right ) b^{2}}+\frac {3 c^{2} d \ln \left (b x +a \right )}{b^{3}}+\frac {3 c \,d^{2} x}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.22, size = 174, normalized size = 1.53 \[ -\frac {a b^{3} c^{3} - 9 \, a^{2} b^{2} c^{2} d + 15 \, a^{3} b c d^{2} - 7 \, a^{4} d^{3} + 2 \, {\left (b^{4} c^{3} - 6 \, a b^{3} c^{2} d + 9 \, a^{2} b^{2} c d^{2} - 4 \, a^{3} b d^{3}\right )} x}{2 \, {\left (b^{7} x^{2} + 2 \, a b^{6} x + a^{2} b^{5}\right )}} + \frac {b d^{3} x^{2} + 6 \, {\left (b c d^{2} - a d^{3}\right )} x}{2 \, b^{4}} + \frac {3 \, {\left (b^{2} c^{2} d - 3 \, a b c d^{2} + 2 \, a^{2} d^{3}\right )} \log \left (b x + a\right )}{b^{5}} \]

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

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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sympy [A]  time = 1.20, size = 175, normalized size = 1.54 \[ x \left (- \frac {3 a d^{3}}{b^{4}} + \frac {3 c d^{2}}{b^{3}}\right ) + \frac {7 a^{4} d^{3} - 15 a^{3} b c d^{2} + 9 a^{2} b^{2} c^{2} d - a b^{3} c^{3} + x \left (8 a^{3} b d^{3} - 18 a^{2} b^{2} c d^{2} + 12 a b^{3} c^{2} d - 2 b^{4} c^{3}\right )}{2 a^{2} b^{5} + 4 a b^{6} x + 2 b^{7} x^{2}} + \frac {d^{3} x^{2}}{2 b^{3}} + \frac {3 d \left (a d - b c\right ) \left (2 a d - b c\right ) \log {\left (a + b x \right )}}{b^{5}} \]

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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