3.252 \(\int \frac{x^4}{(a+b x) (c+d x)^3} \, dx\)

Optimal. Leaf size=140 \[ -\frac{c^2 \left (6 a^2 d^2-8 a b c d+3 b^2 c^2\right ) \log (c+d x)}{d^4 (b c-a d)^3}+\frac{a^4 \log (a+b x)}{b^2 (b c-a d)^3}-\frac{c^3 (3 b c-4 a d)}{d^4 (c+d x) (b c-a d)^2}+\frac{c^4}{2 d^4 (c+d x)^2 (b c-a d)}+\frac{x}{b d^3} \]

[Out]

x/(b*d^3) + c^4/(2*d^4*(b*c - a*d)*(c + d*x)^2) - (c^3*(3*b*c - 4*a*d))/(d^4*(b*c - a*d)^2*(c + d*x)) + (a^4*L
og[a + b*x])/(b^2*(b*c - a*d)^3) - (c^2*(3*b^2*c^2 - 8*a*b*c*d + 6*a^2*d^2)*Log[c + d*x])/(d^4*(b*c - a*d)^3)

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Rubi [A]  time = 0.122569, antiderivative size = 140, 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} \[ -\frac{c^2 \left (6 a^2 d^2-8 a b c d+3 b^2 c^2\right ) \log (c+d x)}{d^4 (b c-a d)^3}+\frac{a^4 \log (a+b x)}{b^2 (b c-a d)^3}-\frac{c^3 (3 b c-4 a d)}{d^4 (c+d x) (b c-a d)^2}+\frac{c^4}{2 d^4 (c+d x)^2 (b c-a d)}+\frac{x}{b d^3} \]

Antiderivative was successfully verified.

[In]

Int[x^4/((a + b*x)*(c + d*x)^3),x]

[Out]

x/(b*d^3) + c^4/(2*d^4*(b*c - a*d)*(c + d*x)^2) - (c^3*(3*b*c - 4*a*d))/(d^4*(b*c - a*d)^2*(c + d*x)) + (a^4*L
og[a + b*x])/(b^2*(b*c - a*d)^3) - (c^2*(3*b^2*c^2 - 8*a*b*c*d + 6*a^2*d^2)*Log[c + d*x])/(d^4*(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^4}{(a+b x) (c+d x)^3} \, dx &=\int \left (\frac{1}{b d^3}+\frac{a^4}{b (b c-a d)^3 (a+b x)}+\frac{c^4}{d^3 (-b c+a d) (c+d x)^3}+\frac{c^3 (3 b c-4 a d)}{d^3 (-b c+a d)^2 (c+d x)^2}+\frac{c^2 \left (3 b^2 c^2-8 a b c d+6 a^2 d^2\right )}{d^3 (-b c+a d)^3 (c+d x)}\right ) \, dx\\ &=\frac{x}{b d^3}+\frac{c^4}{2 d^4 (b c-a d) (c+d x)^2}-\frac{c^3 (3 b c-4 a d)}{d^4 (b c-a d)^2 (c+d x)}+\frac{a^4 \log (a+b x)}{b^2 (b c-a d)^3}-\frac{c^2 \left (3 b^2 c^2-8 a b c d+6 a^2 d^2\right ) \log (c+d x)}{d^4 (b c-a d)^3}\\ \end{align*}

Mathematica [A]  time = 0.151967, size = 138, normalized size = 0.99 \[ \frac{c^2 \left (6 a^2 d^2-8 a b c d+3 b^2 c^2\right ) \log (c+d x)}{d^4 (a d-b c)^3}+\frac{a^4 \log (a+b x)}{b^2 (b c-a d)^3}+\frac{c^3 (4 a d-3 b c)}{d^4 (c+d x) (b c-a d)^2}-\frac{c^4}{2 d^4 (c+d x)^2 (a d-b c)}+\frac{x}{b d^3} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4/((a + b*x)*(c + d*x)^3),x]

[Out]

x/(b*d^3) - c^4/(2*d^4*(-(b*c) + a*d)*(c + d*x)^2) + (c^3*(-3*b*c + 4*a*d))/(d^4*(b*c - a*d)^2*(c + d*x)) + (a
^4*Log[a + b*x])/(b^2*(b*c - a*d)^3) + (c^2*(3*b^2*c^2 - 8*a*b*c*d + 6*a^2*d^2)*Log[c + d*x])/(d^4*(-(b*c) + a
*d)^3)

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Maple [A]  time = 0.01, size = 191, normalized size = 1.4 \begin{align*}{\frac{x}{b{d}^{3}}}-{\frac{{c}^{4}}{2\,{d}^{4} \left ( ad-bc \right ) \left ( dx+c \right ) ^{2}}}+6\,{\frac{{c}^{2}\ln \left ( dx+c \right ){a}^{2}}{{d}^{2} \left ( ad-bc \right ) ^{3}}}-8\,{\frac{{c}^{3}\ln \left ( dx+c \right ) ab}{{d}^{3} \left ( ad-bc \right ) ^{3}}}+3\,{\frac{{c}^{4}\ln \left ( dx+c \right ){b}^{2}}{{d}^{4} \left ( ad-bc \right ) ^{3}}}+4\,{\frac{a{c}^{3}}{{d}^{3} \left ( ad-bc \right ) ^{2} \left ( dx+c \right ) }}-3\,{\frac{{c}^{4}b}{ \left ( ad-bc \right ) ^{2}{d}^{4} \left ( dx+c \right ) }}-{\frac{{a}^{4}\ln \left ( bx+a \right ) }{{b}^{2} \left ( ad-bc \right ) ^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/(b*x+a)/(d*x+c)^3,x)

[Out]

x/b/d^3-1/2/d^4*c^4/(a*d-b*c)/(d*x+c)^2+6/d^2*c^2/(a*d-b*c)^3*ln(d*x+c)*a^2-8/d^3*c^3/(a*d-b*c)^3*ln(d*x+c)*a*
b+3/d^4*c^4/(a*d-b*c)^3*ln(d*x+c)*b^2+4/d^3*c^3/(a*d-b*c)^2/(d*x+c)*a-3/d^4*c^4/(a*d-b*c)^2/(d*x+c)*b-1/b^2*a^
4/(a*d-b*c)^3*ln(b*x+a)

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Maxima [A]  time = 2.17317, size = 367, normalized size = 2.62 \begin{align*} \frac{a^{4} \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 (3 \, b^{2} c^{4} - 8 \, a b c^{3} d + 6 \, a^{2} c^{2} d^{2}\right )} \log \left (d x + c\right )}{b^{3} c^{3} d^{4} - 3 \, a b^{2} c^{2} d^{5} + 3 \, a^{2} b c d^{6} - a^{3} d^{7}} - \frac{5 \, b c^{5} - 7 \, a c^{4} d + 2 \,{\left (3 \, b c^{4} d - 4 \, a c^{3} d^{2}\right )} x}{2 \,{\left (b^{2} c^{4} d^{4} - 2 \, a b c^{3} d^{5} + a^{2} c^{2} d^{6} +{\left (b^{2} c^{2} d^{6} - 2 \, a b c d^{7} + a^{2} d^{8}\right )} x^{2} + 2 \,{\left (b^{2} c^{3} d^{5} - 2 \, a b c^{2} d^{6} + a^{2} c d^{7}\right )} x\right )}} + \frac{x}{b d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(b*x+a)/(d*x+c)^3,x, algorithm="maxima")

[Out]

a^4*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) - (3*b^2*c^4 - 8*a*b*c^3*d + 6*a^2*
c^2*d^2)*log(d*x + c)/(b^3*c^3*d^4 - 3*a*b^2*c^2*d^5 + 3*a^2*b*c*d^6 - a^3*d^7) - 1/2*(5*b*c^5 - 7*a*c^4*d + 2
*(3*b*c^4*d - 4*a*c^3*d^2)*x)/(b^2*c^4*d^4 - 2*a*b*c^3*d^5 + a^2*c^2*d^6 + (b^2*c^2*d^6 - 2*a*b*c*d^7 + a^2*d^
8)*x^2 + 2*(b^2*c^3*d^5 - 2*a*b*c^2*d^6 + a^2*c*d^7)*x) + x/(b*d^3)

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Fricas [B]  time = 2.53158, size = 984, normalized size = 7.03 \begin{align*} -\frac{5 \, b^{4} c^{6} - 12 \, a b^{3} c^{5} d + 7 \, a^{2} b^{2} c^{4} d^{2} - 2 \,{\left (b^{4} c^{3} d^{3} - 3 \, a b^{3} c^{2} d^{4} + 3 \, a^{2} b^{2} c d^{5} - a^{3} b d^{6}\right )} x^{3} - 4 \,{\left (b^{4} c^{4} d^{2} - 3 \, a b^{3} c^{3} d^{3} + 3 \, a^{2} b^{2} c^{2} d^{4} - a^{3} b c d^{5}\right )} x^{2} + 2 \,{\left (2 \, b^{4} c^{5} d - 4 \, a b^{3} c^{4} d^{2} + a^{2} b^{2} c^{3} d^{3} + a^{3} b c^{2} d^{4}\right )} x - 2 \,{\left (a^{4} d^{6} x^{2} + 2 \, a^{4} c d^{5} x + a^{4} c^{2} d^{4}\right )} \log \left (b x + a\right ) + 2 \,{\left (3 \, b^{4} c^{6} - 8 \, a b^{3} c^{5} d + 6 \, a^{2} b^{2} c^{4} d^{2} +{\left (3 \, b^{4} c^{4} d^{2} - 8 \, a b^{3} c^{3} d^{3} + 6 \, a^{2} b^{2} c^{2} d^{4}\right )} x^{2} + 2 \,{\left (3 \, b^{4} c^{5} d - 8 \, a b^{3} c^{4} d^{2} + 6 \, a^{2} b^{2} c^{3} d^{3}\right )} x\right )} \log \left (d x + c\right )}{2 \,{\left (b^{5} c^{5} d^{4} - 3 \, a b^{4} c^{4} d^{5} + 3 \, a^{2} b^{3} c^{3} d^{6} - a^{3} b^{2} c^{2} d^{7} +{\left (b^{5} c^{3} d^{6} - 3 \, a b^{4} c^{2} d^{7} + 3 \, a^{2} b^{3} c d^{8} - a^{3} b^{2} d^{9}\right )} x^{2} + 2 \,{\left (b^{5} c^{4} d^{5} - 3 \, a b^{4} c^{3} d^{6} + 3 \, a^{2} b^{3} c^{2} d^{7} - a^{3} b^{2} c d^{8}\right )} x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(b*x+a)/(d*x+c)^3,x, algorithm="fricas")

[Out]

-1/2*(5*b^4*c^6 - 12*a*b^3*c^5*d + 7*a^2*b^2*c^4*d^2 - 2*(b^4*c^3*d^3 - 3*a*b^3*c^2*d^4 + 3*a^2*b^2*c*d^5 - a^
3*b*d^6)*x^3 - 4*(b^4*c^4*d^2 - 3*a*b^3*c^3*d^3 + 3*a^2*b^2*c^2*d^4 - a^3*b*c*d^5)*x^2 + 2*(2*b^4*c^5*d - 4*a*
b^3*c^4*d^2 + a^2*b^2*c^3*d^3 + a^3*b*c^2*d^4)*x - 2*(a^4*d^6*x^2 + 2*a^4*c*d^5*x + a^4*c^2*d^4)*log(b*x + a)
+ 2*(3*b^4*c^6 - 8*a*b^3*c^5*d + 6*a^2*b^2*c^4*d^2 + (3*b^4*c^4*d^2 - 8*a*b^3*c^3*d^3 + 6*a^2*b^2*c^2*d^4)*x^2
+ 2*(3*b^4*c^5*d - 8*a*b^3*c^4*d^2 + 6*a^2*b^2*c^3*d^3)*x)*log(d*x + c))/(b^5*c^5*d^4 - 3*a*b^4*c^4*d^5 + 3*a
^2*b^3*c^3*d^6 - a^3*b^2*c^2*d^7 + (b^5*c^3*d^6 - 3*a*b^4*c^2*d^7 + 3*a^2*b^3*c*d^8 - a^3*b^2*d^9)*x^2 + 2*(b^
5*c^4*d^5 - 3*a*b^4*c^3*d^6 + 3*a^2*b^3*c^2*d^7 - a^3*b^2*c*d^8)*x)

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Sympy [B]  time = 5.35049, size = 719, normalized size = 5.14 \begin{align*} - \frac{a^{4} \log{\left (x + \frac{\frac{a^{8} d^{7}}{b \left (a d - b c\right )^{3}} - \frac{4 a^{7} c d^{6}}{\left (a d - b c\right )^{3}} + \frac{6 a^{6} b c^{2} d^{5}}{\left (a d - b c\right )^{3}} - \frac{4 a^{5} b^{2} c^{3} d^{4}}{\left (a d - b c\right )^{3}} + \frac{a^{4} b^{3} c^{4} d^{3}}{\left (a d - b c\right )^{3}} + a^{4} c d^{3} + 6 a^{3} b c^{2} d^{2} - 8 a^{2} b^{2} c^{3} d + 3 a b^{3} c^{4}}{a^{4} d^{4} + 6 a^{2} b^{2} c^{2} d^{2} - 8 a b^{3} c^{3} d + 3 b^{4} c^{4}} \right )}}{b^{2} \left (a d - b c\right )^{3}} + \frac{c^{2} \left (6 a^{2} d^{2} - 8 a b c d + 3 b^{2} c^{2}\right ) \log{\left (x + \frac{- \frac{a^{4} b c^{2} d^{3} \left (6 a^{2} d^{2} - 8 a b c d + 3 b^{2} c^{2}\right )}{\left (a d - b c\right )^{3}} + a^{4} c d^{3} + \frac{4 a^{3} b^{2} c^{3} d^{2} \left (6 a^{2} d^{2} - 8 a b c d + 3 b^{2} c^{2}\right )}{\left (a d - b c\right )^{3}} + 6 a^{3} b c^{2} d^{2} - \frac{6 a^{2} b^{3} c^{4} d \left (6 a^{2} d^{2} - 8 a b c d + 3 b^{2} c^{2}\right )}{\left (a d - b c\right )^{3}} - 8 a^{2} b^{2} c^{3} d + \frac{4 a b^{4} c^{5} \left (6 a^{2} d^{2} - 8 a b c d + 3 b^{2} c^{2}\right )}{\left (a d - b c\right )^{3}} + 3 a b^{3} c^{4} - \frac{b^{5} c^{6} \left (6 a^{2} d^{2} - 8 a b c d + 3 b^{2} c^{2}\right )}{d \left (a d - b c\right )^{3}}}{a^{4} d^{4} + 6 a^{2} b^{2} c^{2} d^{2} - 8 a b^{3} c^{3} d + 3 b^{4} c^{4}} \right )}}{d^{4} \left (a d - b c\right )^{3}} + \frac{7 a c^{4} d - 5 b c^{5} + x \left (8 a c^{3} d^{2} - 6 b c^{4} d\right )}{2 a^{2} c^{2} d^{6} - 4 a b c^{3} d^{5} + 2 b^{2} c^{4} d^{4} + x^{2} \left (2 a^{2} d^{8} - 4 a b c d^{7} + 2 b^{2} c^{2} d^{6}\right ) + x \left (4 a^{2} c d^{7} - 8 a b c^{2} d^{6} + 4 b^{2} c^{3} d^{5}\right )} + \frac{x}{b d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4/(b*x+a)/(d*x+c)**3,x)

[Out]

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

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Giac [A]  time = 1.64047, size = 306, normalized size = 2.19 \begin{align*} \frac{a^{4} \log \left ({\left | b x + a \right |}\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 (3 \, b^{2} c^{4} - 8 \, a b c^{3} d + 6 \, a^{2} c^{2} d^{2}\right )} \log \left ({\left | d x + c \right |}\right )}{b^{3} c^{3} d^{4} - 3 \, a b^{2} c^{2} d^{5} + 3 \, a^{2} b c d^{6} - a^{3} d^{7}} + \frac{x}{b d^{3}} - \frac{5 \, b^{2} c^{6} - 12 \, a b c^{5} d + 7 \, a^{2} c^{4} d^{2} + 2 \,{\left (3 \, b^{2} c^{5} d - 7 \, a b c^{4} d^{2} + 4 \, a^{2} c^{3} d^{3}\right )} x}{2 \,{\left (b c - a d\right )}^{3}{\left (d x + c\right )}^{2} d^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(b*x+a)/(d*x+c)^3,x, algorithm="giac")

[Out]

a^4*log(abs(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) - (3*b^2*c^4 - 8*a*b*c^3*d + 6
*a^2*c^2*d^2)*log(abs(d*x + c))/(b^3*c^3*d^4 - 3*a*b^2*c^2*d^5 + 3*a^2*b*c*d^6 - a^3*d^7) + x/(b*d^3) - 1/2*(5
*b^2*c^6 - 12*a*b*c^5*d + 7*a^2*c^4*d^2 + 2*(3*b^2*c^5*d - 7*a*b*c^4*d^2 + 4*a^2*c^3*d^3)*x)/((b*c - a*d)^3*(d
*x + c)^2*d^4)