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

Optimal. Leaf size=112 \[ \frac{a^3}{b^2 (a+b x) (b c-a d)^2}+\frac{a^2 (3 b c-a d) \log (a+b x)}{b^2 (b c-a d)^3}+\frac{c^3}{d^2 (c+d x) (b c-a d)^2}+\frac{c^2 (b c-3 a d) \log (c+d x)}{d^2 (b c-a d)^3} \]

[Out]

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

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Rubi [A]  time = 0.112276, antiderivative size = 112, 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} \[ \frac{a^3}{b^2 (a+b x) (b c-a d)^2}+\frac{a^2 (3 b c-a d) \log (a+b x)}{b^2 (b c-a d)^3}+\frac{c^3}{d^2 (c+d x) (b c-a d)^2}+\frac{c^2 (b c-3 a d) \log (c+d x)}{d^2 (b c-a d)^3} \]

Antiderivative was successfully verified.

[In]

Int[x^3/((a + b*x)^2*(c + d*x)^2),x]

[Out]

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

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

Mathematica [A]  time = 0.15532, size = 105, normalized size = 0.94 \[ \frac{\frac{a^3}{b^2 (a+b x)}+\frac{c^3}{d^2 (c+d x)}}{(b c-a d)^2}+\frac{a^2 (3 b c-a d) \log (a+b x)}{b^2 (b c-a d)^3}+\frac{c^2 (3 a d-b c) \log (c+d x)}{d^2 (a d-b c)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.01, size = 149, normalized size = 1.3 \begin{align*} 3\,{\frac{{c}^{2}\ln \left ( dx+c \right ) a}{ \left ( ad-bc \right ) ^{3}d}}-{\frac{{c}^{3}\ln \left ( dx+c \right ) b}{ \left ( ad-bc \right ) ^{3}{d}^{2}}}+{\frac{{c}^{3}}{{d}^{2} \left ( ad-bc \right ) ^{2} \left ( dx+c \right ) }}+{\frac{{a}^{3}\ln \left ( bx+a \right ) d}{ \left ( ad-bc \right ) ^{3}{b}^{2}}}-3\,{\frac{{a}^{2}\ln \left ( bx+a \right ) c}{ \left ( ad-bc \right ) ^{3}b}}+{\frac{{a}^{3}}{{b}^{2} \left ( ad-bc \right ) ^{2} \left ( bx+a \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.20512, size = 383, normalized size = 3.42 \begin{align*} \frac{{\left (3 \, a^{2} b c - a^{3} d\right )} \log \left (b x + a\right )}{b^{5} c^{3} - 3 \, a b^{4} c^{2} d + 3 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}} + \frac{{\left (b c^{3} - 3 \, a c^{2} d\right )} \log \left (d x + c\right )}{b^{3} c^{3} d^{2} - 3 \, a b^{2} c^{2} d^{3} + 3 \, a^{2} b c d^{4} - a^{3} d^{5}} + \frac{a b^{2} c^{3} + a^{3} c d^{2} +{\left (b^{3} c^{3} + a^{3} d^{3}\right )} x}{a b^{4} c^{3} d^{2} - 2 \, a^{2} b^{3} c^{2} d^{3} + a^{3} b^{2} c d^{4} +{\left (b^{5} c^{2} d^{3} - 2 \, a b^{4} c d^{4} + a^{2} b^{3} d^{5}\right )} x^{2} +{\left (b^{5} c^{3} d^{2} - a b^{4} c^{2} d^{3} - a^{2} b^{3} c d^{4} + a^{3} b^{2} d^{5}\right )} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 2.32947, size = 784, normalized size = 7. \begin{align*} \frac{a b^{3} c^{4} - a^{2} b^{2} c^{3} d + a^{3} b c^{2} d^{2} - a^{4} c d^{3} +{\left (b^{4} c^{4} - a b^{3} c^{3} d + a^{3} b c d^{3} - a^{4} d^{4}\right )} x +{\left (3 \, a^{3} b c^{2} d^{2} - a^{4} c d^{3} +{\left (3 \, a^{2} b^{2} c d^{3} - a^{3} b d^{4}\right )} x^{2} +{\left (3 \, a^{2} b^{2} c^{2} d^{2} + 2 \, a^{3} b c d^{3} - a^{4} d^{4}\right )} x\right )} \log \left (b x + a\right ) +{\left (a b^{3} c^{4} - 3 \, a^{2} b^{2} c^{3} d +{\left (b^{4} c^{3} d - 3 \, a b^{3} c^{2} d^{2}\right )} x^{2} +{\left (b^{4} c^{4} - 2 \, a b^{3} c^{3} d - 3 \, a^{2} b^{2} c^{2} d^{2}\right )} x\right )} \log \left (d x + c\right )}{a b^{5} c^{4} d^{2} - 3 \, a^{2} b^{4} c^{3} d^{3} + 3 \, a^{3} b^{3} c^{2} d^{4} - a^{4} b^{2} c d^{5} +{\left (b^{6} c^{3} d^{3} - 3 \, a b^{5} c^{2} d^{4} + 3 \, a^{2} b^{4} c d^{5} - a^{3} b^{3} d^{6}\right )} x^{2} +{\left (b^{6} c^{4} d^{2} - 2 \, a b^{5} c^{3} d^{3} + 2 \, a^{3} b^{3} c d^{5} - a^{4} 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)^2/(d*x+c)^2,x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 3.37856, size = 627, normalized size = 5.6 \begin{align*} \frac{a^{2} \left (a d - 3 b c\right ) \log{\left (x + \frac{\frac{a^{6} d^{5} \left (a d - 3 b c\right )}{b \left (a d - b c\right )^{3}} - \frac{4 a^{5} c d^{4} \left (a d - 3 b c\right )}{\left (a d - b c\right )^{3}} + \frac{6 a^{4} b c^{2} d^{3} \left (a d - 3 b c\right )}{\left (a d - b c\right )^{3}} - \frac{4 a^{3} b^{2} c^{3} d^{2} \left (a d - 3 b c\right )}{\left (a d - b c\right )^{3}} + a^{3} c d^{2} + \frac{a^{2} b^{3} c^{4} d \left (a d - 3 b c\right )}{\left (a d - b c\right )^{3}} - 6 a^{2} b c^{2} d + a b^{2} c^{3}}{a^{3} d^{3} - 3 a^{2} b c d^{2} - 3 a b^{2} c^{2} d + b^{3} c^{3}} \right )}}{b^{2} \left (a d - b c\right )^{3}} + \frac{c^{2} \left (3 a d - b c\right ) \log{\left (x + \frac{\frac{a^{4} b c^{2} d^{3} \left (3 a d - b c\right )}{\left (a d - b c\right )^{3}} - \frac{4 a^{3} b^{2} c^{3} d^{2} \left (3 a d - b c\right )}{\left (a d - b c\right )^{3}} + a^{3} c d^{2} + \frac{6 a^{2} b^{3} c^{4} d \left (3 a d - b c\right )}{\left (a d - b c\right )^{3}} - 6 a^{2} b c^{2} d - \frac{4 a b^{4} c^{5} \left (3 a d - b c\right )}{\left (a d - b c\right )^{3}} + a b^{2} c^{3} + \frac{b^{5} c^{6} \left (3 a d - b c\right )}{d \left (a d - b c\right )^{3}}}{a^{3} d^{3} - 3 a^{2} b c d^{2} - 3 a b^{2} c^{2} d + b^{3} c^{3}} \right )}}{d^{2} \left (a d - b c\right )^{3}} + \frac{a^{3} c d^{2} + a b^{2} c^{3} + x \left (a^{3} d^{3} + b^{3} c^{3}\right )}{a^{3} b^{2} c d^{4} - 2 a^{2} b^{3} c^{2} d^{3} + a b^{4} c^{3} d^{2} + x^{2} \left (a^{2} b^{3} d^{5} - 2 a b^{4} c d^{4} + b^{5} c^{2} d^{3}\right ) + x \left (a^{3} b^{2} d^{5} - a^{2} b^{3} c d^{4} - a b^{4} c^{2} d^{3} + b^{5} c^{3} d^{2}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.21585, size = 273, normalized size = 2.44 \begin{align*} \frac{a^{3} b^{2}}{{\left (b^{6} c^{2} - 2 \, a b^{5} c d + a^{2} b^{4} d^{2}\right )}{\left (b x + a\right )}} + \frac{{\left (b^{2} c^{3} - 3 \, a b c^{2} d\right )} \log \left ({\left | \frac{b c}{b x + a} - \frac{a d}{b x + a} + d \right |}\right )}{b^{4} c^{3} d^{2} - 3 \, a b^{3} c^{2} d^{3} + 3 \, a^{2} b^{2} c d^{4} - a^{3} b d^{5}} - \frac{b c^{3}}{{\left (b c - a d\right )}^{3}{\left (\frac{b c}{b x + a} - \frac{a d}{b x + a} + d\right )} d} - \frac{\log \left (\frac{{\left | b x + a \right |}}{{\left (b x + a\right )}^{2}{\left | b \right |}}\right )}{b^{2} d^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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