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

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

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Rubi [A]  time = 0.10, antiderivative size = 110, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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} \end {gather*}

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*}

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Mathematica [A]  time = 0.23, size = 107, normalized size = 0.97 \begin {gather*} \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} \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^4}{(a+b x) (c+d x)^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 1.37, size = 285, normalized size = 2.59 \begin {gather*} \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 {gather*}

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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giac [A]  time = 1.07, size = 185, normalized size = 1.68 \begin {gather*} \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 {gather*}

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)

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maple [A]  time = 0.01, size = 131, normalized size = 1.19 \begin {gather*} \frac {a^{4} \ln \left (b x +a \right )}{\left (a d -b c \right )^{2} b^{3}}-\frac {4 a \,c^{3} \ln \left (d x +c \right )}{\left (a d -b c \right )^{2} d^{3}}+\frac {3 b \,c^{4} \ln \left (d x +c \right )}{\left (a d -b c \right )^{2} d^{4}}-\frac {c^{4}}{\left (a d -b c \right ) \left (d x +c \right ) d^{4}}+\frac {x^{2}}{2 b \,d^{2}}-\frac {a x}{b^{2} d^{2}}-\frac {2 c x}{b \,d^{3}} \end {gather*}

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*x*a-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 = 1.00, size = 151, normalized size = 1.37 \begin {gather*} \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 {gather*}

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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mupad [B]  time = 0.55, size = 144, normalized size = 1.31 \begin {gather*} \frac {\ln \left (c+d\,x\right )\,\left (3\,b\,c^4-4\,a\,c^3\,d\right )}{a^2\,d^6-2\,a\,b\,c\,d^5+b^2\,c^2\,d^4}+\frac {x^2}{2\,b\,d^2}+\frac {a^4\,\ln \left (a+b\,x\right )}{b^3\,{\left (a\,d-b\,c\right )}^2}-\frac {x\,\left (a\,d^2+2\,b\,c\,d\right )}{b^2\,d^4}-\frac {b^2\,c^4}{d\,\left (x\,b^2\,d^4+c\,b^2\,d^3\right )\,\left (a\,d-b\,c\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/((a + b*x)*(c + d*x)^2),x)

[Out]

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

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sympy [B]  time = 3.80, size = 428, normalized size = 3.89 \begin {gather*} \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}} + x \left (- \frac {a}{b^{2} d^{2}} - \frac {2 c}{b d^{3}}\right ) + \frac {x^{2}}{2 b d^{2}} \end {gather*}

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

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