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

Optimal. Leaf size=89 \[ \frac {1}{2} \left (a-\frac {c d^2}{e^2}\right ) (a e+c d x)^2-\frac {\left (c d^2-a e^2\right )^3 \log (d+e x)}{e^4}+\frac {c d x \left (c d^2-a e^2\right )^2}{e^3}+\frac {(a e+c d x)^3}{3 e} \]

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Rubi [A]  time = 0.05, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {626, 43} \begin {gather*} \frac {c d x \left (c d^2-a e^2\right )^2}{e^3}+\frac {1}{2} \left (a-\frac {c d^2}{e^2}\right ) (a e+c d x)^2-\frac {\left (c d^2-a e^2\right )^3 \log (d+e x)}{e^4}+\frac {(a e+c d x)^3}{3 e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 626

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

Rubi steps

\begin {align*} \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^4} \, dx &=\int \frac {(a e+c d x)^3}{d+e x} \, dx\\ &=\int \left (\frac {c d \left (c d^2-a e^2\right )^2}{e^3}-\frac {c d \left (c d^2-a e^2\right ) (a e+c d x)}{e^2}+\frac {c d (a e+c d x)^2}{e}+\frac {\left (-c d^2+a e^2\right )^3}{e^3 (d+e x)}\right ) \, dx\\ &=\frac {c d \left (c d^2-a e^2\right )^2 x}{e^3}+\frac {1}{2} \left (a-\frac {c d^2}{e^2}\right ) (a e+c d x)^2+\frac {(a e+c d x)^3}{3 e}-\frac {\left (c d^2-a e^2\right )^3 \log (d+e x)}{e^4}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 85, normalized size = 0.96 \begin {gather*} \frac {c d e x \left (18 a^2 e^4+9 a c d e^2 (e x-2 d)+c^2 d^2 \left (6 d^2-3 d e x+2 e^2 x^2\right )\right )-6 \left (c d^2-a e^2\right )^3 \log (d+e x)}{6 e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(c*d*e*x*(18*a^2*e^4 + 9*a*c*d*e^2*(-2*d + e*x) + c^2*d^2*(6*d^2 - 3*d*e*x + 2*e^2*x^2)) - 6*(c*d^2 - a*e^2)^3
*Log[d + e*x])/(6*e^4)

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.39, size = 130, normalized size = 1.46 \begin {gather*} \frac {2 \, c^{3} d^{3} e^{3} x^{3} - 3 \, {\left (c^{3} d^{4} e^{2} - 3 \, a c^{2} d^{2} e^{4}\right )} x^{2} + 6 \, {\left (c^{3} d^{5} e - 3 \, a c^{2} d^{3} e^{3} + 3 \, a^{2} c d e^{5}\right )} x - 6 \, {\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} \log \left (e x + d\right )}{6 \, e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.17, size = 128, normalized size = 1.44 \begin {gather*} -{\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} e^{\left (-4\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {1}{6} \, {\left (2 \, c^{3} d^{3} x^{3} e^{11} - 3 \, c^{3} d^{4} x^{2} e^{10} + 6 \, c^{3} d^{5} x e^{9} + 9 \, a c^{2} d^{2} x^{2} e^{12} - 18 \, a c^{2} d^{3} x e^{11} + 18 \, a^{2} c d x e^{13}\right )} e^{\left (-12\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(c^3*d^6 - 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 - a^3*e^6)*e^(-4)*log(abs(x*e + d)) + 1/6*(2*c^3*d^3*x^3*e^11 -
3*c^3*d^4*x^2*e^10 + 6*c^3*d^5*x*e^9 + 9*a*c^2*d^2*x^2*e^12 - 18*a*c^2*d^3*x*e^11 + 18*a^2*c*d*x*e^13)*e^(-12)

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maple [A]  time = 0.04, size = 138, normalized size = 1.55 \begin {gather*} \frac {c^{3} d^{3} x^{3}}{3 e}+\frac {3 a \,c^{2} d^{2} x^{2}}{2}-\frac {c^{3} d^{4} x^{2}}{2 e^{2}}+a^{3} e^{2} \ln \left (e x +d \right )-3 a^{2} c \,d^{2} \ln \left (e x +d \right )+3 a^{2} c d e x +\frac {3 a \,c^{2} d^{4} \ln \left (e x +d \right )}{e^{2}}-\frac {3 a \,c^{2} d^{3} x}{e}-\frac {c^{3} d^{6} \ln \left (e x +d \right )}{e^{4}}+\frac {c^{3} d^{5} x}{e^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.03, size = 131, normalized size = 1.47 \begin {gather*} \frac {2 \, c^{3} d^{3} e^{2} x^{3} - 3 \, {\left (c^{3} d^{4} e - 3 \, a c^{2} d^{2} e^{3}\right )} x^{2} + 6 \, {\left (c^{3} d^{5} - 3 \, a c^{2} d^{3} e^{2} + 3 \, a^{2} c d e^{4}\right )} x}{6 \, e^{3}} - \frac {{\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} \log \left (e x + d\right )}{e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.06, size = 128, normalized size = 1.44 \begin {gather*} x^2\,\left (\frac {3\,a\,c^2\,d^2}{2}-\frac {c^3\,d^4}{2\,e^2}\right )-x\,\left (\frac {d\,\left (3\,a\,c^2\,d^2-\frac {c^3\,d^4}{e^2}\right )}{e}-3\,a^2\,c\,d\,e\right )+\frac {\ln \left (d+e\,x\right )\,\left (a^3\,e^6-3\,a^2\,c\,d^2\,e^4+3\,a\,c^2\,d^4\,e^2-c^3\,d^6\right )}{e^4}+\frac {c^3\,d^3\,x^3}{3\,e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.41, size = 95, normalized size = 1.07 \begin {gather*} \frac {c^{3} d^{3} x^{3}}{3 e} + x^{2} \left (\frac {3 a c^{2} d^{2}}{2} - \frac {c^{3} d^{4}}{2 e^{2}}\right ) + x \left (3 a^{2} c d e - \frac {3 a c^{2} d^{3}}{e} + \frac {c^{3} d^{5}}{e^{3}}\right ) + \frac {\left (a e^{2} - c d^{2}\right )^{3} \log {\left (d + e x \right )}}{e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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