3.3.56 \(\int \frac {(d+e x)^4}{(b x+c x^2)^3} \, dx\)

Optimal. Leaf size=136 \[ \frac {6 d^2 \log (x) (c d-b e)^2}{b^5}-\frac {6 d^2 (c d-b e)^2 \log (b+c x)}{b^5}+\frac {(c d-b e)^3 (b e+3 c d)}{b^4 c^2 (b+c x)}+\frac {d^3 (3 c d-4 b e)}{b^4 x}+\frac {(c d-b e)^4}{2 b^3 c^2 (b+c x)^2}-\frac {d^4}{2 b^3 x^2} \]

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Rubi [A]  time = 0.15, antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {698} \begin {gather*} \frac {(c d-b e)^3 (b e+3 c d)}{b^4 c^2 (b+c x)}+\frac {(c d-b e)^4}{2 b^3 c^2 (b+c x)^2}+\frac {d^3 (3 c d-4 b e)}{b^4 x}+\frac {6 d^2 \log (x) (c d-b e)^2}{b^5}-\frac {6 d^2 (c d-b e)^2 \log (b+c x)}{b^5}-\frac {d^4}{2 b^3 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^4/(b*x + c*x^2)^3,x]

[Out]

-d^4/(2*b^3*x^2) + (d^3*(3*c*d - 4*b*e))/(b^4*x) + (c*d - b*e)^4/(2*b^3*c^2*(b + c*x)^2) + ((c*d - b*e)^3*(3*c
*d + b*e))/(b^4*c^2*(b + c*x)) + (6*d^2*(c*d - b*e)^2*Log[x])/b^5 - (6*d^2*(c*d - b*e)^2*Log[b + c*x])/b^5

Rule 698

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin {align*} \int \frac {(d+e x)^4}{\left (b x+c x^2\right )^3} \, dx &=\int \left (\frac {d^4}{b^3 x^3}+\frac {d^3 (-3 c d+4 b e)}{b^4 x^2}+\frac {6 d^2 (-c d+b e)^2}{b^5 x}-\frac {(-c d+b e)^4}{b^3 c (b+c x)^3}+\frac {(-c d+b e)^3 (3 c d+b e)}{b^4 c (b+c x)^2}-\frac {6 c d^2 (-c d+b e)^2}{b^5 (b+c x)}\right ) \, dx\\ &=-\frac {d^4}{2 b^3 x^2}+\frac {d^3 (3 c d-4 b e)}{b^4 x}+\frac {(c d-b e)^4}{2 b^3 c^2 (b+c x)^2}+\frac {(c d-b e)^3 (3 c d+b e)}{b^4 c^2 (b+c x)}+\frac {6 d^2 (c d-b e)^2 \log (x)}{b^5}-\frac {6 d^2 (c d-b e)^2 \log (b+c x)}{b^5}\\ \end {align*}

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Mathematica [A]  time = 0.08, size = 130, normalized size = 0.96 \begin {gather*} -\frac {-\frac {b^2 (c d-b e)^4}{c^2 (b+c x)^2}+\frac {b^2 d^4}{x^2}+\frac {2 b (b e-c d)^3 (b e+3 c d)}{c^2 (b+c x)}+\frac {2 b d^3 (4 b e-3 c d)}{x}-12 d^2 \log (x) (c d-b e)^2+12 d^2 (c d-b e)^2 \log (b+c x)}{2 b^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^4/(b*x + c*x^2)^3,x]

[Out]

-1/2*((b^2*d^4)/x^2 + (2*b*d^3*(-3*c*d + 4*b*e))/x - (b^2*(c*d - b*e)^4)/(c^2*(b + c*x)^2) + (2*b*(-(c*d) + b*
e)^3*(3*c*d + b*e))/(c^2*(b + c*x)) - 12*d^2*(c*d - b*e)^2*Log[x] + 12*d^2*(c*d - b*e)^2*Log[b + c*x])/b^5

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(d+e x)^4}{\left (b x+c x^2\right )^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(d + e*x)^4/(b*x + c*x^2)^3,x]

[Out]

IntegrateAlgebraic[(d + e*x)^4/(b*x + c*x^2)^3, x]

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fricas [B]  time = 0.43, size = 426, normalized size = 3.13 \begin {gather*} -\frac {b^{4} c^{2} d^{4} - 2 \, {\left (6 \, b c^{5} d^{4} - 12 \, b^{2} c^{4} d^{3} e + 6 \, b^{3} c^{3} d^{2} e^{2} - b^{5} c e^{4}\right )} x^{3} - {\left (18 \, b^{2} c^{4} d^{4} - 36 \, b^{3} c^{3} d^{3} e + 18 \, b^{4} c^{2} d^{2} e^{2} - 4 \, b^{5} c d e^{3} - b^{6} e^{4}\right )} x^{2} - 4 \, {\left (b^{3} c^{3} d^{4} - 2 \, b^{4} c^{2} d^{3} e\right )} x + 12 \, {\left ({\left (c^{6} d^{4} - 2 \, b c^{5} d^{3} e + b^{2} c^{4} d^{2} e^{2}\right )} x^{4} + 2 \, {\left (b c^{5} d^{4} - 2 \, b^{2} c^{4} d^{3} e + b^{3} c^{3} d^{2} e^{2}\right )} x^{3} + {\left (b^{2} c^{4} d^{4} - 2 \, b^{3} c^{3} d^{3} e + b^{4} c^{2} d^{2} e^{2}\right )} x^{2}\right )} \log \left (c x + b\right ) - 12 \, {\left ({\left (c^{6} d^{4} - 2 \, b c^{5} d^{3} e + b^{2} c^{4} d^{2} e^{2}\right )} x^{4} + 2 \, {\left (b c^{5} d^{4} - 2 \, b^{2} c^{4} d^{3} e + b^{3} c^{3} d^{2} e^{2}\right )} x^{3} + {\left (b^{2} c^{4} d^{4} - 2 \, b^{3} c^{3} d^{3} e + b^{4} c^{2} d^{2} e^{2}\right )} x^{2}\right )} \log \relax (x)}{2 \, {\left (b^{5} c^{4} x^{4} + 2 \, b^{6} c^{3} x^{3} + b^{7} c^{2} x^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^4/(c*x^2+b*x)^3,x, algorithm="fricas")

[Out]

-1/2*(b^4*c^2*d^4 - 2*(6*b*c^5*d^4 - 12*b^2*c^4*d^3*e + 6*b^3*c^3*d^2*e^2 - b^5*c*e^4)*x^3 - (18*b^2*c^4*d^4 -
36*b^3*c^3*d^3*e + 18*b^4*c^2*d^2*e^2 - 4*b^5*c*d*e^3 - b^6*e^4)*x^2 - 4*(b^3*c^3*d^4 - 2*b^4*c^2*d^3*e)*x +
12*((c^6*d^4 - 2*b*c^5*d^3*e + b^2*c^4*d^2*e^2)*x^4 + 2*(b*c^5*d^4 - 2*b^2*c^4*d^3*e + b^3*c^3*d^2*e^2)*x^3 +
(b^2*c^4*d^4 - 2*b^3*c^3*d^3*e + b^4*c^2*d^2*e^2)*x^2)*log(c*x + b) - 12*((c^6*d^4 - 2*b*c^5*d^3*e + b^2*c^4*d
^2*e^2)*x^4 + 2*(b*c^5*d^4 - 2*b^2*c^4*d^3*e + b^3*c^3*d^2*e^2)*x^3 + (b^2*c^4*d^4 - 2*b^3*c^3*d^3*e + b^4*c^2
*d^2*e^2)*x^2)*log(x))/(b^5*c^4*x^4 + 2*b^6*c^3*x^3 + b^7*c^2*x^2)

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giac [A]  time = 0.20, size = 254, normalized size = 1.87 \begin {gather*} \frac {6 \, {\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2}\right )} \log \left ({\left | x \right |}\right )}{b^{5}} - \frac {6 \, {\left (c^{3} d^{4} - 2 \, b c^{2} d^{3} e + b^{2} c d^{2} e^{2}\right )} \log \left ({\left | c x + b \right |}\right )}{b^{5} c} + \frac {12 \, c^{5} d^{4} x^{3} - 24 \, b c^{4} d^{3} x^{3} e + 18 \, b c^{4} d^{4} x^{2} + 12 \, b^{2} c^{3} d^{2} x^{3} e^{2} - 36 \, b^{2} c^{3} d^{3} x^{2} e + 4 \, b^{2} c^{3} d^{4} x + 18 \, b^{3} c^{2} d^{2} x^{2} e^{2} - 8 \, b^{3} c^{2} d^{3} x e - b^{3} c^{2} d^{4} - 2 \, b^{4} c x^{3} e^{4} - 4 \, b^{4} c d x^{2} e^{3} - b^{5} x^{2} e^{4}}{2 \, {\left (c x^{2} + b x\right )}^{2} b^{4} c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^4/(c*x^2+b*x)^3,x, algorithm="giac")

[Out]

6*(c^2*d^4 - 2*b*c*d^3*e + b^2*d^2*e^2)*log(abs(x))/b^5 - 6*(c^3*d^4 - 2*b*c^2*d^3*e + b^2*c*d^2*e^2)*log(abs(
c*x + b))/(b^5*c) + 1/2*(12*c^5*d^4*x^3 - 24*b*c^4*d^3*x^3*e + 18*b*c^4*d^4*x^2 + 12*b^2*c^3*d^2*x^3*e^2 - 36*
b^2*c^3*d^3*x^2*e + 4*b^2*c^3*d^4*x + 18*b^3*c^2*d^2*x^2*e^2 - 8*b^3*c^2*d^3*x*e - b^3*c^2*d^4 - 2*b^4*c*x^3*e
^4 - 4*b^4*c*d*x^2*e^3 - b^5*x^2*e^4)/((c*x^2 + b*x)^2*b^4*c^2)

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maple [B]  time = 0.06, size = 278, normalized size = 2.04 \begin {gather*} \frac {b \,e^{4}}{2 \left (c x +b \right )^{2} c^{2}}+\frac {3 d^{2} e^{2}}{\left (c x +b \right )^{2} b}-\frac {2 c \,d^{3} e}{\left (c x +b \right )^{2} b^{2}}+\frac {c^{2} d^{4}}{2 \left (c x +b \right )^{2} b^{3}}-\frac {2 d \,e^{3}}{\left (c x +b \right )^{2} c}+\frac {6 d^{2} e^{2}}{\left (c x +b \right ) b^{2}}-\frac {8 c \,d^{3} e}{\left (c x +b \right ) b^{3}}+\frac {6 d^{2} e^{2} \ln \relax (x )}{b^{3}}-\frac {6 d^{2} e^{2} \ln \left (c x +b \right )}{b^{3}}+\frac {3 c^{2} d^{4}}{\left (c x +b \right ) b^{4}}-\frac {12 c \,d^{3} e \ln \relax (x )}{b^{4}}+\frac {12 c \,d^{3} e \ln \left (c x +b \right )}{b^{4}}+\frac {6 c^{2} d^{4} \ln \relax (x )}{b^{5}}-\frac {6 c^{2} d^{4} \ln \left (c x +b \right )}{b^{5}}-\frac {e^{4}}{\left (c x +b \right ) c^{2}}-\frac {4 d^{3} e}{b^{3} x}+\frac {3 c \,d^{4}}{b^{4} x}-\frac {d^{4}}{2 b^{3} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^4/(c*x^2+b*x)^3,x)

[Out]

-1/c^2/(c*x+b)*e^4+6/b^2/(c*x+b)*d^2*e^2-8/b^3*c/(c*x+b)*d^3*e+3/b^4*c^2/(c*x+b)*d^4+1/2*b/c^2/(c*x+b)^2*e^4-2
/c/(c*x+b)^2*d*e^3+3/b/(c*x+b)^2*d^2*e^2-2/b^2*c/(c*x+b)^2*d^3*e+1/2/b^3*c^2/(c*x+b)^2*d^4-6*d^2/b^3*ln(c*x+b)
*e^2+12*d^3/b^4*ln(c*x+b)*c*e-6*d^4/b^5*ln(c*x+b)*c^2-1/2*d^4/b^3/x^2-4*d^3/b^3/x*e+3*d^4/b^4/x*c+6*d^2/b^3*ln
(x)*e^2-12*d^3/b^4*ln(x)*c*e+6*d^4/b^5*ln(x)*c^2

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maxima [A]  time = 1.39, size = 250, normalized size = 1.84 \begin {gather*} -\frac {b^{3} c^{2} d^{4} - 2 \, {\left (6 \, c^{5} d^{4} - 12 \, b c^{4} d^{3} e + 6 \, b^{2} c^{3} d^{2} e^{2} - b^{4} c e^{4}\right )} x^{3} - {\left (18 \, b c^{4} d^{4} - 36 \, b^{2} c^{3} d^{3} e + 18 \, b^{3} c^{2} d^{2} e^{2} - 4 \, b^{4} c d e^{3} - b^{5} e^{4}\right )} x^{2} - 4 \, {\left (b^{2} c^{3} d^{4} - 2 \, b^{3} c^{2} d^{3} e\right )} x}{2 \, {\left (b^{4} c^{4} x^{4} + 2 \, b^{5} c^{3} x^{3} + b^{6} c^{2} x^{2}\right )}} - \frac {6 \, {\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2}\right )} \log \left (c x + b\right )}{b^{5}} + \frac {6 \, {\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2}\right )} \log \relax (x)}{b^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^4/(c*x^2+b*x)^3,x, algorithm="maxima")

[Out]

-1/2*(b^3*c^2*d^4 - 2*(6*c^5*d^4 - 12*b*c^4*d^3*e + 6*b^2*c^3*d^2*e^2 - b^4*c*e^4)*x^3 - (18*b*c^4*d^4 - 36*b^
2*c^3*d^3*e + 18*b^3*c^2*d^2*e^2 - 4*b^4*c*d*e^3 - b^5*e^4)*x^2 - 4*(b^2*c^3*d^4 - 2*b^3*c^2*d^3*e)*x)/(b^4*c^
4*x^4 + 2*b^5*c^3*x^3 + b^6*c^2*x^2) - 6*(c^2*d^4 - 2*b*c*d^3*e + b^2*d^2*e^2)*log(c*x + b)/b^5 + 6*(c^2*d^4 -
2*b*c*d^3*e + b^2*d^2*e^2)*log(x)/b^5

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mupad [B]  time = 0.15, size = 238, normalized size = 1.75 \begin {gather*} -\frac {\frac {d^4}{2\,b}+\frac {2\,d^3\,x\,\left (2\,b\,e-c\,d\right )}{b^2}+\frac {x^2\,\left (b^4\,e^4+4\,b^3\,c\,d\,e^3-18\,b^2\,c^2\,d^2\,e^2+36\,b\,c^3\,d^3\,e-18\,c^4\,d^4\right )}{2\,b^3\,c^2}+\frac {x^3\,\left (b^4\,e^4-6\,b^2\,c^2\,d^2\,e^2+12\,b\,c^3\,d^3\,e-6\,c^4\,d^4\right )}{b^4\,c}}{b^2\,x^2+2\,b\,c\,x^3+c^2\,x^4}-\frac {12\,d^2\,\mathrm {atanh}\left (\frac {6\,d^2\,{\left (b\,e-c\,d\right )}^2\,\left (b+2\,c\,x\right )}{b\,\left (6\,b^2\,d^2\,e^2-12\,b\,c\,d^3\,e+6\,c^2\,d^4\right )}\right )\,{\left (b\,e-c\,d\right )}^2}{b^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^4/(b*x + c*x^2)^3,x)

[Out]

- (d^4/(2*b) + (2*d^3*x*(2*b*e - c*d))/b^2 + (x^2*(b^4*e^4 - 18*c^4*d^4 - 18*b^2*c^2*d^2*e^2 + 36*b*c^3*d^3*e
+ 4*b^3*c*d*e^3))/(2*b^3*c^2) + (x^3*(b^4*e^4 - 6*c^4*d^4 - 6*b^2*c^2*d^2*e^2 + 12*b*c^3*d^3*e))/(b^4*c))/(b^2
*x^2 + c^2*x^4 + 2*b*c*x^3) - (12*d^2*atanh((6*d^2*(b*e - c*d)^2*(b + 2*c*x))/(b*(6*c^2*d^4 + 6*b^2*d^2*e^2 -
12*b*c*d^3*e)))*(b*e - c*d)^2)/b^5

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sympy [B]  time = 2.85, size = 389, normalized size = 2.86 \begin {gather*} \frac {- b^{3} c^{2} d^{4} + x^{3} \left (- 2 b^{4} c e^{4} + 12 b^{2} c^{3} d^{2} e^{2} - 24 b c^{4} d^{3} e + 12 c^{5} d^{4}\right ) + x^{2} \left (- b^{5} e^{4} - 4 b^{4} c d e^{3} + 18 b^{3} c^{2} d^{2} e^{2} - 36 b^{2} c^{3} d^{3} e + 18 b c^{4} d^{4}\right ) + x \left (- 8 b^{3} c^{2} d^{3} e + 4 b^{2} c^{3} d^{4}\right )}{2 b^{6} c^{2} x^{2} + 4 b^{5} c^{3} x^{3} + 2 b^{4} c^{4} x^{4}} + \frac {6 d^{2} \left (b e - c d\right )^{2} \log {\left (x + \frac {6 b^{3} d^{2} e^{2} - 12 b^{2} c d^{3} e + 6 b c^{2} d^{4} - 6 b d^{2} \left (b e - c d\right )^{2}}{12 b^{2} c d^{2} e^{2} - 24 b c^{2} d^{3} e + 12 c^{3} d^{4}} \right )}}{b^{5}} - \frac {6 d^{2} \left (b e - c d\right )^{2} \log {\left (x + \frac {6 b^{3} d^{2} e^{2} - 12 b^{2} c d^{3} e + 6 b c^{2} d^{4} + 6 b d^{2} \left (b e - c d\right )^{2}}{12 b^{2} c d^{2} e^{2} - 24 b c^{2} d^{3} e + 12 c^{3} d^{4}} \right )}}{b^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**4/(c*x**2+b*x)**3,x)

[Out]

(-b**3*c**2*d**4 + x**3*(-2*b**4*c*e**4 + 12*b**2*c**3*d**2*e**2 - 24*b*c**4*d**3*e + 12*c**5*d**4) + x**2*(-b
**5*e**4 - 4*b**4*c*d*e**3 + 18*b**3*c**2*d**2*e**2 - 36*b**2*c**3*d**3*e + 18*b*c**4*d**4) + x*(-8*b**3*c**2*
d**3*e + 4*b**2*c**3*d**4))/(2*b**6*c**2*x**2 + 4*b**5*c**3*x**3 + 2*b**4*c**4*x**4) + 6*d**2*(b*e - c*d)**2*l
og(x + (6*b**3*d**2*e**2 - 12*b**2*c*d**3*e + 6*b*c**2*d**4 - 6*b*d**2*(b*e - c*d)**2)/(12*b**2*c*d**2*e**2 -
24*b*c**2*d**3*e + 12*c**3*d**4))/b**5 - 6*d**2*(b*e - c*d)**2*log(x + (6*b**3*d**2*e**2 - 12*b**2*c*d**3*e +
6*b*c**2*d**4 + 6*b*d**2*(b*e - c*d)**2)/(12*b**2*c*d**2*e**2 - 24*b*c**2*d**3*e + 12*c**3*d**4))/b**5

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