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

Optimal. Leaf size=77 \[ -\frac {a^3 \log (a+b x)}{b^3 (b c-a d)}-\frac {x (a d+b c)}{b^2 d^2}+\frac {c^3 \log (c+d x)}{d^3 (b c-a d)}+\frac {x^2}{2 b d} \]

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Rubi [A]  time = 0.06, antiderivative size = 77, 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 {a^3 \log (a+b x)}{b^3 (b c-a d)}-\frac {x (a d+b c)}{b^2 d^2}+\frac {c^3 \log (c+d x)}{d^3 (b c-a d)}+\frac {x^2}{2 b d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3/((a + b*x)*(c + d*x)),x]

[Out]

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

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Mathematica [A]  time = 0.05, size = 74, normalized size = 0.96 \begin {gather*} \frac {-2 a^3 d^3 \log (a+b x)+b d x (b c-a d) (-2 a d-2 b c+b d x)+2 b^3 c^3 \log (c+d x)}{2 b^3 d^3 (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^3}{(a+b x) (c+d x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.35, size = 95, normalized size = 1.23 \begin {gather*} -\frac {2 \, a^{3} d^{3} \log \left (b x + a\right ) - 2 \, b^{3} c^{3} \log \left (d x + c\right ) - {\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{2} + 2 \, {\left (b^{3} c^{2} d - a^{2} b d^{3}\right )} x}{2 \, {\left (b^{4} c d^{3} - a b^{3} d^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.82, size = 79, normalized size = 1.03 \begin {gather*} -\frac {a^{3} \log \left ({\left | b x + a \right |}\right )}{b^{4} c - a b^{3} d} + \frac {c^{3} \log \left ({\left | d x + c \right |}\right )}{b c d^{3} - a d^{4}} + \frac {b d x^{2} - 2 \, b c x - 2 \, a d x}{2 \, b^{2} d^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.01, size = 80, normalized size = 1.04 \begin {gather*} \frac {a^{3} \ln \left (b x +a \right )}{\left (a d -b c \right ) b^{3}}-\frac {c^{3} \ln \left (d x +c \right )}{\left (a d -b c \right ) d^{3}}+\frac {x^{2}}{2 b d}-\frac {a x}{b^{2} d}-\frac {c x}{b \,d^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.07, size = 77, normalized size = 1.00 \begin {gather*} -\frac {a^{3} \log \left (b x + a\right )}{b^{4} c - a b^{3} d} + \frac {c^{3} \log \left (d x + c\right )}{b c d^{3} - a d^{4}} + \frac {b d x^{2} - 2 \, {\left (b c + a d\right )} x}{2 \, b^{2} d^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.19, size = 78, normalized size = 1.01 \begin {gather*} \frac {x^2}{2\,b\,d}-\frac {a^3\,\ln \left (a+b\,x\right )}{b^4\,c-a\,b^3\,d}-\frac {x\,\left (a\,d+b\,c\right )}{b^2\,d^2}-\frac {c^3\,\ln \left (c+d\,x\right )}{d^3\,\left (a\,d-b\,c\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/((a + b*x)*(c + d*x)),x)

[Out]

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

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sympy [B]  time = 1.63, size = 221, normalized size = 2.87 \begin {gather*} \frac {a^{3} \log {\left (x + \frac {\frac {a^{5} d^{4}}{b \left (a d - b c\right )} - \frac {2 a^{4} c d^{3}}{a d - b c} + \frac {a^{3} b c^{2} d^{2}}{a d - b c} + a^{3} c d^{2} + a b^{2} c^{3}}{a^{3} d^{3} + b^{3} c^{3}} \right )}}{b^{3} \left (a d - b c\right )} - \frac {c^{3} \log {\left (x + \frac {a^{3} c d^{2} - \frac {a^{2} b^{2} c^{3} d}{a d - b c} + \frac {2 a b^{3} c^{4}}{a d - b c} + a b^{2} c^{3} - \frac {b^{4} c^{5}}{d \left (a d - b c\right )}}{a^{3} d^{3} + b^{3} c^{3}} \right )}}{d^{3} \left (a d - b c\right )} + x \left (- \frac {a}{b^{2} d} - \frac {c}{b d^{2}}\right ) + \frac {x^{2}}{2 b d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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