3.243 \(\int \frac{x^4}{(a+b x) (c+d x)^2} \, dx\)

Optimal. Leaf size=110 \[ \frac{a^4 \log (a+b x)}{b^3 (b c-a d)^2}-\frac{x (a d+2 b c)}{b^2 d^3}+\frac{c^4}{d^4 (c+d x) (b c-a d)}+\frac{c^3 (3 b c-4 a d) \log (c+d x)}{d^4 (b c-a d)^2}+\frac{x^2}{2 b d^2} \]

[Out]

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

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Rubi [A]  time = 0.103468, antiderivative size = 110, 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} \[ \frac{a^4 \log (a+b x)}{b^3 (b c-a d)^2}-\frac{x (a d+2 b c)}{b^2 d^3}+\frac{c^4}{d^4 (c+d x) (b c-a d)}+\frac{c^3 (3 b c-4 a d) \log (c+d x)}{d^4 (b c-a d)^2}+\frac{x^2}{2 b d^2} \]

Antiderivative was successfully verified.

[In]

Int[x^4/((a + b*x)*(c + d*x)^2),x]

[Out]

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

Mathematica [A]  time = 0.187839, size = 107, normalized size = 0.97 \[ \frac{a^4 \log (a+b x)}{b^3 (b c-a d)^2}+\frac{-\frac{2 a d^2 x}{b^2}+\frac{2 c^4}{(c+d x) (b c-a d)}+\frac{2 c^3 (3 b c-4 a d) \log (c+d x)}{(b c-a d)^2}+\frac{d x (d x-4 c)}{b}}{2 d^4} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4/((a + b*x)*(c + d*x)^2),x]

[Out]

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

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Maple [A]  time = 0.01, size = 131, normalized size = 1.2 \begin{align*}{\frac{{x}^{2}}{2\,b{d}^{2}}}-{\frac{ax}{{d}^{2}{b}^{2}}}-2\,{\frac{cx}{b{d}^{3}}}-{\frac{{c}^{4}}{{d}^{4} \left ( ad-bc \right ) \left ( dx+c \right ) }}-4\,{\frac{{c}^{3}\ln \left ( dx+c \right ) a}{{d}^{3} \left ( ad-bc \right ) ^{2}}}+3\,{\frac{{c}^{4}\ln \left ( dx+c \right ) b}{{d}^{4} \left ( ad-bc \right ) ^{2}}}+{\frac{{a}^{4}\ln \left ( bx+a \right ) }{{b}^{3} \left ( ad-bc \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/(b*x+a)/(d*x+c)^2,x)

[Out]

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

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Maxima [A]  time = 1.2406, size = 204, normalized size = 1.85 \begin{align*} \frac{a^{4} \log \left (b x + a\right )}{b^{5} c^{2} - 2 \, a b^{4} c d + a^{2} b^{3} d^{2}} + \frac{c^{4}}{b c^{2} d^{4} - a c d^{5} +{\left (b c d^{5} - a d^{6}\right )} x} + \frac{{\left (3 \, b c^{4} - 4 \, a c^{3} d\right )} \log \left (d x + c\right )}{b^{2} c^{2} d^{4} - 2 \, a b c d^{5} + a^{2} d^{6}} + \frac{b d x^{2} - 2 \,{\left (2 \, b c + a d\right )} x}{2 \, b^{2} d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(b*x+a)/(d*x+c)^2,x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 2.46288, size = 571, normalized size = 5.19 \begin{align*} \frac{2 \, b^{4} c^{5} - 2 \, a b^{3} c^{4} d +{\left (b^{4} c^{2} d^{3} - 2 \, a b^{3} c d^{4} + a^{2} b^{2} d^{5}\right )} x^{3} -{\left (3 \, b^{4} c^{3} d^{2} - 4 \, a b^{3} c^{2} d^{3} - a^{2} b^{2} c d^{4} + 2 \, a^{3} b d^{5}\right )} x^{2} - 2 \,{\left (2 \, b^{4} c^{4} d - 3 \, a b^{3} c^{3} d^{2} + a^{3} b c d^{4}\right )} x + 2 \,{\left (a^{4} d^{5} x + a^{4} c d^{4}\right )} \log \left (b x + a\right ) + 2 \,{\left (3 \, b^{4} c^{5} - 4 \, a b^{3} c^{4} d +{\left (3 \, b^{4} c^{4} d - 4 \, a b^{3} c^{3} d^{2}\right )} x\right )} \log \left (d x + c\right )}{2 \,{\left (b^{5} c^{3} d^{4} - 2 \, a b^{4} c^{2} d^{5} + a^{2} b^{3} c d^{6} +{\left (b^{5} c^{2} d^{5} - 2 \, a b^{4} c d^{6} + a^{2} b^{3} d^{7}\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)^2,x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 3.25104, size = 425, normalized size = 3.86 \begin{align*} \frac{a^{4} \log{\left (x + \frac{\frac{a^{7} d^{6}}{b \left (a d - b c\right )^{2}} - \frac{3 a^{6} c d^{5}}{\left (a d - b c\right )^{2}} + \frac{3 a^{5} b c^{2} d^{4}}{\left (a d - b c\right )^{2}} - \frac{a^{4} b^{2} c^{3} d^{3}}{\left (a d - b c\right )^{2}} + a^{4} c d^{3} + 4 a^{2} b^{2} c^{3} d - 3 a b^{3} c^{4}}{a^{4} d^{4} + 4 a b^{3} c^{3} d - 3 b^{4} c^{4}} \right )}}{b^{3} \left (a d - b c\right )^{2}} - \frac{c^{4}}{a c d^{5} - b c^{2} d^{4} + x \left (a d^{6} - b c d^{5}\right )} - \frac{c^{3} \left (4 a d - 3 b c\right ) \log{\left (x + \frac{a^{4} c d^{3} - \frac{a^{3} b^{2} c^{3} d^{2} \left (4 a d - 3 b c\right )}{\left (a d - b c\right )^{2}} + \frac{3 a^{2} b^{3} c^{4} d \left (4 a d - 3 b c\right )}{\left (a d - b c\right )^{2}} + 4 a^{2} b^{2} c^{3} d - \frac{3 a b^{4} c^{5} \left (4 a d - 3 b c\right )}{\left (a d - b c\right )^{2}} - 3 a b^{3} c^{4} + \frac{b^{5} c^{6} \left (4 a d - 3 b c\right )}{d \left (a d - b c\right )^{2}}}{a^{4} d^{4} + 4 a b^{3} c^{3} d - 3 b^{4} c^{4}} \right )}}{d^{4} \left (a d - b c\right )^{2}} + \frac{x^{2}}{2 b d^{2}} - \frac{x \left (a d + 2 b c\right )}{b^{2} d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4/(b*x+a)/(d*x+c)**2,x)

[Out]

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

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Giac [A]  time = 1.21139, size = 250, normalized size = 2.27 \begin{align*} \frac{c^{4} d^{3}}{{\left (b c d^{7} - a d^{8}\right )}{\left (d x + c\right )}} + \frac{a^{4} d \log \left ({\left | b - \frac{b c}{d x + c} + \frac{a d}{d x + c} \right |}\right )}{b^{5} c^{2} d - 2 \, a b^{4} c d^{2} + a^{2} b^{3} d^{3}} + \frac{{\left (b^{2} - \frac{2 \,{\left (3 \, b^{2} c d + a b d^{2}\right )}}{{\left (d x + c\right )} d}\right )}{\left (d x + c\right )}^{2}}{2 \, b^{3} d^{4}} - \frac{{\left (3 \, b^{2} c^{2} + 2 \, a b c d + a^{2} d^{2}\right )} \log \left (\frac{{\left | d x + c \right |}}{{\left (d x + c\right )}^{2}{\left | d \right |}}\right )}{b^{3} d^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(b*x+a)/(d*x+c)^2,x, algorithm="giac")

[Out]

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