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

Optimal. Leaf size=109 \[ \frac {a^4 \log (a+b x)}{b^4 (b c-a d)}+\frac {x \left (a^2 d^2+a b c d+b^2 c^2\right )}{b^3 d^3}-\frac {x^2 (a d+b c)}{2 b^2 d^2}-\frac {c^4 \log (c+d x)}{d^4 (b c-a d)}+\frac {x^3}{3 b d} \]

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Rubi [A]  time = 0.08, antiderivative size = 109, 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 = {72} \begin {gather*} \frac {x \left (a^2 d^2+a b c d+b^2 c^2\right )}{b^3 d^3}+\frac {a^4 \log (a+b x)}{b^4 (b c-a d)}-\frac {x^2 (a d+b c)}{2 b^2 d^2}-\frac {c^4 \log (c+d x)}{d^4 (b c-a d)}+\frac {x^3}{3 b d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^4/((a + b*x)*(c + d*x)),x]

[Out]

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

Rule 72

Int[((e_.) + (f_.)*(x_))^(p_.)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Int[ExpandIntegrand[(
e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[p]

Rubi steps

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

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Mathematica [A]  time = 0.07, size = 105, normalized size = 0.96 \begin {gather*} \frac {6 a^4 d^4 \log (a+b x)+b d x (b c-a d) \left (6 a^2 d^2-3 a b d (d x-2 c)+b^2 \left (6 c^2-3 c d x+2 d^2 x^2\right )\right )-6 b^4 c^4 \log (c+d x)}{6 b^4 d^4 (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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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)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.34, size = 122, normalized size = 1.12 \begin {gather*} \frac {6 \, a^{4} d^{4} \log \left (b x + a\right ) - 6 \, b^{4} c^{4} \log \left (d x + c\right ) + 2 \, {\left (b^{4} c d^{3} - a b^{3} d^{4}\right )} x^{3} - 3 \, {\left (b^{4} c^{2} d^{2} - a^{2} b^{2} d^{4}\right )} x^{2} + 6 \, {\left (b^{4} c^{3} d - a^{3} b d^{4}\right )} x}{6 \, {\left (b^{5} c d^{4} - a b^{4} d^{5}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 1.00, size = 119, normalized size = 1.09 \begin {gather*} \frac {a^{4} \log \left ({\left | b x + a \right |}\right )}{b^{5} c - a b^{4} d} - \frac {c^{4} \log \left ({\left | d x + c \right |}\right )}{b c d^{4} - a d^{5}} + \frac {2 \, b^{2} d^{2} x^{3} - 3 \, b^{2} c d x^{2} - 3 \, a b d^{2} x^{2} + 6 \, b^{2} c^{2} x + 6 \, a b c d x + 6 \, a^{2} d^{2} x}{6 \, b^{3} d^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.01, size = 116, normalized size = 1.06 \begin {gather*} -\frac {a^{4} \ln \left (b x +a \right )}{\left (a d -b c \right ) b^{4}}+\frac {x^{3}}{3 b d}+\frac {c^{4} \ln \left (d x +c \right )}{\left (a d -b c \right ) d^{4}}-\frac {a \,x^{2}}{2 b^{2} d}-\frac {c \,x^{2}}{2 b \,d^{2}}+\frac {a^{2} x}{b^{3} d}+\frac {a c x}{b^{2} d^{2}}+\frac {c^{2} 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),x)

[Out]

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

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maxima [A]  time = 1.19, size = 113, normalized size = 1.04 \begin {gather*} \frac {a^{4} \log \left (b x + a\right )}{b^{5} c - a b^{4} d} - \frac {c^{4} \log \left (d x + c\right )}{b c d^{4} - a d^{5}} + \frac {2 \, b^{2} d^{2} x^{3} - 3 \, {\left (b^{2} c d + a b d^{2}\right )} x^{2} + 6 \, {\left (b^{2} c^{2} + a b c d + a^{2} d^{2}\right )} x}{6 \, b^{3} d^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.47, size = 106, normalized size = 0.97 \begin {gather*} x\,\left (\frac {{\left (a\,d+b\,c\right )}^2}{b^3\,d^3}-\frac {a\,c}{b^2\,d^2}\right )+\frac {x^3}{3\,b\,d}-\frac {a^4\,\ln \left (a+b\,x\right )}{b^4\,\left (a\,d-b\,c\right )}+\frac {c^4\,\ln \left (c+d\,x\right )}{d^4\,\left (a\,d-b\,c\right )}-\frac {x^2\,\left (a\,d+b\,c\right )}{2\,b^2\,d^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/((a + b*x)*(c + d*x)),x)

[Out]

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

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sympy [B]  time = 2.10, size = 255, normalized size = 2.34 \begin {gather*} - \frac {a^{4} \log {\left (x + \frac {\frac {a^{6} d^{5}}{b \left (a d - b c\right )} - \frac {2 a^{5} c d^{4}}{a d - b c} + \frac {a^{4} b c^{2} d^{3}}{a d - b c} + a^{4} c d^{3} + a b^{3} c^{4}}{a^{4} d^{4} + b^{4} c^{4}} \right )}}{b^{4} \left (a d - b c\right )} + \frac {c^{4} \log {\left (x + \frac {a^{4} c d^{3} - \frac {a^{2} b^{3} c^{4} d}{a d - b c} + \frac {2 a b^{4} c^{5}}{a d - b c} + a b^{3} c^{4} - \frac {b^{5} c^{6}}{d \left (a d - b c\right )}}{a^{4} d^{4} + b^{4} c^{4}} \right )}}{d^{4} \left (a d - b c\right )} + x^{2} \left (- \frac {a}{2 b^{2} d} - \frac {c}{2 b d^{2}}\right ) + x \left (\frac {a^{2}}{b^{3} d} + \frac {a c}{b^{2} d^{2}} + \frac {c^{2}}{b d^{3}}\right ) + \frac {x^{3}}{3 b d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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