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

Optimal. Leaf size=91 \[ -\frac{a^3 \log (a+b x)}{b^2 (b c-a d)^2}-\frac{c^3}{d^3 (c+d x) (b c-a d)}-\frac{c^2 (2 b c-3 a d) \log (c+d x)}{d^3 (b c-a d)^2}+\frac{x}{b d^2} \]

[Out]

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

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Rubi [A]  time = 0.0762005, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {88} \[ -\frac{a^3 \log (a+b x)}{b^2 (b c-a d)^2}-\frac{c^3}{d^3 (c+d x) (b c-a d)}-\frac{c^2 (2 b c-3 a d) \log (c+d x)}{d^3 (b c-a d)^2}+\frac{x}{b d^2} \]

Antiderivative was successfully verified.

[In]

Int[x^3/((a + b*x)*(c + d*x)^2),x]

[Out]

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

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps

\begin{align*} \int \frac{x^3}{(a+b x) (c+d x)^2} \, dx &=\int \left (\frac{1}{b d^2}-\frac{a^3}{b (b c-a d)^2 (a+b x)}-\frac{c^3}{d^2 (-b c+a d) (c+d x)^2}-\frac{c^2 (2 b c-3 a d)}{d^2 (-b c+a d)^2 (c+d x)}\right ) \, dx\\ &=\frac{x}{b d^2}-\frac{c^3}{d^3 (b c-a d) (c+d x)}-\frac{a^3 \log (a+b x)}{b^2 (b c-a d)^2}-\frac{c^2 (2 b c-3 a d) \log (c+d x)}{d^3 (b c-a d)^2}\\ \end{align*}

Mathematica [A]  time = 0.111856, size = 87, normalized size = 0.96 \[ \frac{\frac{c^3}{(c+d x) (a d-b c)}-\frac{c^2 (2 b c-3 a d) \log (c+d x)}{(b c-a d)^2}+\frac{d x}{b}}{d^3}-\frac{a^3 \log (a+b x)}{b^2 (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.009, size = 108, normalized size = 1.2 \begin{align*}{\frac{x}{b{d}^{2}}}+3\,{\frac{{c}^{2}\ln \left ( dx+c \right ) a}{{d}^{2} \left ( ad-bc \right ) ^{2}}}-2\,{\frac{{c}^{3}\ln \left ( dx+c \right ) b}{{d}^{3} \left ( ad-bc \right ) ^{2}}}+{\frac{{c}^{3}}{{d}^{3} \left ( ad-bc \right ) \left ( dx+c \right ) }}-{\frac{{a}^{3}\ln \left ( bx+a \right ) }{{b}^{2} \left ( ad-bc \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.17937, size = 184, normalized size = 2.02 \begin{align*} -\frac{a^{3} \log \left (b x + a\right )}{b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}} - \frac{c^{3}}{b c^{2} d^{3} - a c d^{4} +{\left (b c d^{4} - a d^{5}\right )} x} - \frac{{\left (2 \, b c^{3} - 3 \, a c^{2} d\right )} \log \left (d x + c\right )}{b^{2} c^{2} d^{3} - 2 \, a b c d^{4} + a^{2} d^{5}} + \frac{x}{b d^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 2.3801, size = 454, normalized size = 4.99 \begin{align*} -\frac{b^{3} c^{4} - a b^{2} c^{3} d -{\left (b^{3} c^{2} d^{2} - 2 \, a b^{2} c d^{3} + a^{2} b d^{4}\right )} x^{2} -{\left (b^{3} c^{3} d - 2 \, a b^{2} c^{2} d^{2} + a^{2} b c d^{3}\right )} x +{\left (a^{3} d^{4} x + a^{3} c d^{3}\right )} \log \left (b x + a\right ) +{\left (2 \, b^{3} c^{4} - 3 \, a b^{2} c^{3} d +{\left (2 \, b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2}\right )} x\right )} \log \left (d x + c\right )}{b^{4} c^{3} d^{3} - 2 \, a b^{3} c^{2} d^{4} + a^{2} b^{2} c d^{5} +{\left (b^{4} c^{2} d^{4} - 2 \, a b^{3} c d^{5} + a^{2} b^{2} d^{6}\right )} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 2.59482, size = 400, normalized size = 4.4 \begin{align*} - \frac{a^{3} \log{\left (x + \frac{\frac{a^{6} d^{5}}{b \left (a d - b c\right )^{2}} - \frac{3 a^{5} c d^{4}}{\left (a d - b c\right )^{2}} + \frac{3 a^{4} b c^{2} d^{3}}{\left (a d - b c\right )^{2}} - \frac{a^{3} b^{2} c^{3} d^{2}}{\left (a d - b c\right )^{2}} + a^{3} c d^{2} + 3 a^{2} b c^{2} d - 2 a b^{2} c^{3}}{a^{3} d^{3} + 3 a b^{2} c^{2} d - 2 b^{3} c^{3}} \right )}}{b^{2} \left (a d - b c\right )^{2}} + \frac{c^{3}}{a c d^{4} - b c^{2} d^{3} + x \left (a d^{5} - b c d^{4}\right )} + \frac{c^{2} \left (3 a d - 2 b c\right ) \log{\left (x + \frac{- \frac{a^{3} b c^{2} d^{2} \left (3 a d - 2 b c\right )}{\left (a d - b c\right )^{2}} + a^{3} c d^{2} + \frac{3 a^{2} b^{2} c^{3} d \left (3 a d - 2 b c\right )}{\left (a d - b c\right )^{2}} + 3 a^{2} b c^{2} d - \frac{3 a b^{3} c^{4} \left (3 a d - 2 b c\right )}{\left (a d - b c\right )^{2}} - 2 a b^{2} c^{3} + \frac{b^{4} c^{5} \left (3 a d - 2 b c\right )}{d \left (a d - b c\right )^{2}}}{a^{3} d^{3} + 3 a b^{2} c^{2} d - 2 b^{3} c^{3}} \right )}}{d^{3} \left (a d - b c\right )^{2}} + \frac{x}{b d^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.18299, size = 188, normalized size = 2.07 \begin{align*} -\frac{a^{3} d \log \left ({\left | b - \frac{b c}{d x + c} + \frac{a d}{d x + c} \right |}\right )}{b^{4} c^{2} d - 2 \, a b^{3} c d^{2} + a^{2} b^{2} d^{3}} - \frac{c^{3} d^{2}}{{\left (b c d^{5} - a d^{6}\right )}{\left (d x + c\right )}} + \frac{d x + c}{b d^{3}} + \frac{{\left (2 \, b c + a d\right )} \log \left (\frac{{\left | d x + c \right |}}{{\left (d x + c\right )}^{2}{\left | d \right |}}\right )}{b^{2} d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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