3.3.72 \(\int \frac {x (c+d x)^3}{(a+b x)^2} \, dx\) [272]

Optimal. Leaf size=103 \[ \frac {3 d (b c-a d)^2 x}{b^4}+\frac {d^2 (3 b c-2 a d) x^2}{2 b^3}+\frac {d^3 x^3}{3 b^2}+\frac {a (b c-a d)^3}{b^5 (a+b x)}+\frac {(b c-4 a d) (b c-a d)^2 \log (a+b x)}{b^5} \]

[Out]

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

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Rubi [A]
time = 0.07, antiderivative size = 103, 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 {a (b c-a d)^3}{b^5 (a+b x)}+\frac {(b c-4 a d) (b c-a d)^2 \log (a+b x)}{b^5}+\frac {3 d x (b c-a d)^2}{b^4}+\frac {d^2 x^2 (3 b c-2 a d)}{2 b^3}+\frac {d^3 x^3}{3 b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x*(c + d*x)^3)/(a + b*x)^2,x]

[Out]

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

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Mathematica [A]
time = 0.06, size = 100, normalized size = 0.97 \begin {gather*} \frac {18 b d (b c-a d)^2 x+3 b^2 d^2 (3 b c-2 a d) x^2+2 b^3 d^3 x^3-\frac {6 a (-b c+a d)^3}{a+b x}+6 (b c-4 a d) (b c-a d)^2 \log (a+b x)}{6 b^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(18*b*d*(b*c - a*d)^2*x + 3*b^2*d^2*(3*b*c - 2*a*d)*x^2 + 2*b^3*d^3*x^3 - (6*a*(-(b*c) + a*d)^3)/(a + b*x) + 6
*(b*c - 4*a*d)*(b*c - a*d)^2*Log[a + b*x])/(6*b^5)

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Maple [A]
time = 0.07, size = 159, normalized size = 1.54

method	result	size
default	\(\frac {d \left (\frac {1}{3} d^{2} x^{3} b^{2}-a b \,d^{2} x^{2}+\frac {3}{2} b^{2} c d \,x^{2}+3 a^{2} d^{2} x -6 a b c d x +3 b^{2} c^{2} x \right )}{b^{4}}+\frac {\left (-4 a^{3} d^{3}+9 a^{2} b c \,d^{2}-6 a \,b^{2} c^{2} d +b^{3} c^{3}\right ) \ln \left (b x +a \right )}{b^{5}}-\frac {a \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}{b^{5} \left (b x +a \right )}\)	\(159\)
norman	\(\frac {\frac {\left (4 a^{4} d^{3}-9 a^{3} b c \,d^{2}+6 a^{2} b^{2} c^{2} d -a \,b^{3} c^{3}\right ) x}{b^{4} a}+\frac {d^{3} x^{4}}{3 b}+\frac {d \left (4 a^{2} d^{2}-9 a b c d +6 b^{2} c^{2}\right ) x^{2}}{2 b^{3}}-\frac {d^{2} \left (4 a d -9 b c \right ) x^{3}}{6 b^{2}}}{b x +a}-\frac {\left (4 a^{3} d^{3}-9 a^{2} b c \,d^{2}+6 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \ln \left (b x +a \right )}{b^{5}}\)	\(170\)
risch	\(\frac {d^{3} x^{3}}{3 b^{2}}-\frac {d^{3} a \,x^{2}}{b^{3}}+\frac {3 d^{2} c \,x^{2}}{2 b^{2}}+\frac {3 d^{3} a^{2} x}{b^{4}}-\frac {6 d^{2} a c x}{b^{3}}+\frac {3 d \,c^{2} x}{b^{2}}-\frac {a^{4} d^{3}}{b^{5} \left (b x +a \right )}+\frac {3 a^{3} c \,d^{2}}{b^{4} \left (b x +a \right )}-\frac {3 a^{2} c^{2} d}{b^{3} \left (b x +a \right )}+\frac {a \,c^{3}}{b^{2} \left (b x +a \right )}-\frac {4 \ln \left (b x +a \right ) a^{3} d^{3}}{b^{5}}+\frac {9 \ln \left (b x +a \right ) a^{2} c \,d^{2}}{b^{4}}-\frac {6 \ln \left (b x +a \right ) a \,c^{2} d}{b^{3}}+\frac {\ln \left (b x +a \right ) c^{3}}{b^{2}}\)	\(205\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(d*x+c)^3/(b*x+a)^2,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.32, size = 166, normalized size = 1.61 \begin {gather*} \frac {a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}}{b^{6} x + a b^{5}} + \frac {2 \, b^{2} d^{3} x^{3} + 3 \, {\left (3 \, b^{2} c d^{2} - 2 \, a b d^{3}\right )} x^{2} + 18 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} x}{6 \, b^{4}} + \frac {{\left (b^{3} c^{3} - 6 \, a b^{2} c^{2} d + 9 \, a^{2} b c d^{2} - 4 \, a^{3} 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)^2,x, algorithm="maxima")

[Out]

(a*b^3*c^3 - 3*a^2*b^2*c^2*d + 3*a^3*b*c*d^2 - a^4*d^3)/(b^6*x + a*b^5) + 1/6*(2*b^2*d^3*x^3 + 3*(3*b^2*c*d^2
- 2*a*b*d^3)*x^2 + 18*(b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*x)/b^4 + (b^3*c^3 - 6*a*b^2*c^2*d + 9*a^2*b*c*d^2 -
4*a^3*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. 246 vs. \(2 (99) = 198\).
time = 0.60, size = 246, normalized size = 2.39 \begin {gather*} \frac {2 \, b^{4} d^{3} x^{4} + 6 \, a b^{3} c^{3} - 18 \, a^{2} b^{2} c^{2} d + 18 \, a^{3} b c d^{2} - 6 \, a^{4} d^{3} + {\left (9 \, b^{4} c d^{2} - 4 \, a b^{3} d^{3}\right )} x^{3} + 3 \, {\left (6 \, b^{4} c^{2} d - 9 \, a b^{3} c d^{2} + 4 \, a^{2} b^{2} d^{3}\right )} x^{2} + 18 \, {\left (a b^{3} c^{2} d - 2 \, a^{2} b^{2} c d^{2} + a^{3} b d^{3}\right )} x + 6 \, {\left (a b^{3} c^{3} - 6 \, a^{2} b^{2} c^{2} d + 9 \, a^{3} b c d^{2} - 4 \, a^{4} d^{3} + {\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\right )} \log \left (b x + a\right )}{6 \, {\left (b^{6} x + a 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)^2,x, algorithm="fricas")

[Out]

1/6*(2*b^4*d^3*x^4 + 6*a*b^3*c^3 - 18*a^2*b^2*c^2*d + 18*a^3*b*c*d^2 - 6*a^4*d^3 + (9*b^4*c*d^2 - 4*a*b^3*d^3)
*x^3 + 3*(6*b^4*c^2*d - 9*a*b^3*c*d^2 + 4*a^2*b^2*d^3)*x^2 + 18*(a*b^3*c^2*d - 2*a^2*b^2*c*d^2 + a^3*b*d^3)*x
+ 6*(a*b^3*c^3 - 6*a^2*b^2*c^2*d + 9*a^3*b*c*d^2 - 4*a^4*d^3 + (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)*log(b*x + a))/(b^6*x + a*b^5)

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Sympy [A]
time = 0.35, size = 148, normalized size = 1.44 \begin {gather*} x^{2} \left (- \frac {a d^{3}}{b^{3}} + \frac {3 c d^{2}}{2 b^{2}}\right ) + x \left (\frac {3 a^{2} d^{3}}{b^{4}} - \frac {6 a c d^{2}}{b^{3}} + \frac {3 c^{2} d}{b^{2}}\right ) + \frac {- a^{4} d^{3} + 3 a^{3} b c d^{2} - 3 a^{2} b^{2} c^{2} d + a b^{3} c^{3}}{a b^{5} + b^{6} x} + \frac {d^{3} x^{3}}{3 b^{2}} - \frac {\left (a d - b c\right )^{2} \cdot \left (4 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)**2,x)

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 231 vs. \(2 (99) = 198\).
time = 1.38, size = 231, normalized size = 2.24 \begin {gather*} \frac {\frac {{\left (2 \, d^{3} + \frac {3 \, {\left (3 \, b^{2} c d^{2} - 4 \, a b d^{3}\right )}}{{\left (b x + a\right )} b} + \frac {18 \, {\left (b^{4} c^{2} d - 3 \, a b^{3} c d^{2} + 2 \, a^{2} b^{2} d^{3}\right )}}{{\left (b x + a\right )}^{2} b^{2}}\right )} {\left (b x + a\right )}^{3}}{b^{4}} - \frac {6 \, {\left (b^{3} c^{3} - 6 \, a b^{2} c^{2} d + 9 \, a^{2} b c d^{2} - 4 \, a^{3} d^{3}\right )} \log \left (\frac {{\left | b x + a \right |}}{{\left (b x + a\right )}^{2} {\left | b \right |}}\right )}{b^{4}} + \frac {6 \, {\left (\frac {a b^{6} c^{3}}{b x + a} - \frac {3 \, a^{2} b^{5} c^{2} d}{b x + a} + \frac {3 \, a^{3} b^{4} c d^{2}}{b x + a} - \frac {a^{4} b^{3} d^{3}}{b x + a}\right )}}{b^{7}}}{6 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*((2*d^3 + 3*(3*b^2*c*d^2 - 4*a*b*d^3)/((b*x + a)*b) + 18*(b^4*c^2*d - 3*a*b^3*c*d^2 + 2*a^2*b^2*d^3)/((b*x
+ a)^2*b^2))*(b*x + a)^3/b^4 - 6*(b^3*c^3 - 6*a*b^2*c^2*d + 9*a^2*b*c*d^2 - 4*a^3*d^3)*log(abs(b*x + a)/((b*x
+ a)^2*abs(b)))/b^4 + 6*(a*b^6*c^3/(b*x + a) - 3*a^2*b^5*c^2*d/(b*x + a) + 3*a^3*b^4*c*d^2/(b*x + a) - a^4*b^
3*d^3/(b*x + a))/b^7)/b

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Mupad [B]
time = 0.07, size = 188, normalized size = 1.83 \begin {gather*} x\,\left (\frac {3\,c^2\,d}{b^2}+\frac {2\,a\,\left (\frac {2\,a\,d^3}{b^3}-\frac {3\,c\,d^2}{b^2}\right )}{b}-\frac {a^2\,d^3}{b^4}\right )-x^2\,\left (\frac {a\,d^3}{b^3}-\frac {3\,c\,d^2}{2\,b^2}\right )-\frac {a^4\,d^3-3\,a^3\,b\,c\,d^2+3\,a^2\,b^2\,c^2\,d-a\,b^3\,c^3}{b\,\left (x\,b^5+a\,b^4\right )}+\frac {d^3\,x^3}{3\,b^2}-\frac {\ln \left (a+b\,x\right )\,\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 )}{b^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x*(c + d*x)^3)/(a + b*x)^2,x)

[Out]

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

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