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

Optimal. Leaf size=100 \[ \frac {\left (c d^2-a e^2\right )^3 \log (a e+c d x)}{c^4 d^4}+\frac {e x \left (c d^2-a e^2\right )^2}{c^3 d^3}+\frac {(d+e x)^2 \left (c d^2-a e^2\right )}{2 c^2 d^2}+\frac {(d+e x)^3}{3 c d} \]

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Rubi [A]  time = 0.05, antiderivative size = 100, 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 {e x \left (c d^2-a e^2\right )^2}{c^3 d^3}+\frac {(d+e x)^2 \left (c d^2-a e^2\right )}{2 c^2 d^2}+\frac {\left (c d^2-a e^2\right )^3 \log (a e+c d x)}{c^4 d^4}+\frac {(d+e x)^3}{3 c d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]  time = 0.03, size = 91, normalized size = 0.91 \begin {gather*} \frac {c d e x \left (6 a^2 e^4-3 a c d e^2 (6 d+e x)+c^2 d^2 \left (18 d^2+9 d e x+2 e^2 x^2\right )\right )+6 \left (c d^2-a e^2\right )^3 \log (a e+c d x)}{6 c^4 d^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.39, size = 137, normalized size = 1.37 \begin {gather*} \frac {2 \, c^{3} d^{3} e^{3} x^{3} + 3 \, {\left (3 \, c^{3} d^{4} e^{2} - a c^{2} d^{2} e^{4}\right )} x^{2} + 6 \, {\left (3 \, c^{3} d^{5} e - 3 \, a c^{2} d^{3} e^{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 (c d x + a e\right )}{6 \, c^{4} d^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 0.19, size = 277, normalized size = 2.77 \begin {gather*} \frac {{\left (2 \, c^{2} d^{2} x^{3} e^{6} + 9 \, c^{2} d^{3} x^{2} e^{5} + 18 \, c^{2} d^{4} x e^{4} - 3 \, a c d x^{2} e^{7} - 18 \, a c d^{2} x e^{6} + 6 \, a^{2} x e^{8}\right )} e^{\left (-3\right )}}{6 \, c^{3} d^{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 (c d x^{2} e + c d^{2} x + a x e^{2} + a d e\right )}{2 \, c^{4} d^{4}} + \frac {{\left (c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}\right )} \arctan \left (\frac {2 \, c d x e + c d^{2} + a e^{2}}{\sqrt {-c^{2} d^{4} + 2 \, a c d^{2} e^{2} - a^{2} e^{4}}}\right )}{\sqrt {-c^{2} d^{4} + 2 \, a c d^{2} e^{2} - a^{2} e^{4}} c^{4} d^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(2*c^2*d^2*x^3*e^6 + 9*c^2*d^3*x^2*e^5 + 18*c^2*d^4*x*e^4 - 3*a*c*d*x^2*e^7 - 18*a*c*d^2*x*e^6 + 6*a^2*x*e
^8)*e^(-3)/(c^3*d^3) + 1/2*(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(c*d*x^2*e + c*d^2*x + a
*x*e^2 + a*d*e)/(c^4*d^4) + (c^4*d^8 - 4*a*c^3*d^6*e^2 + 6*a^2*c^2*d^4*e^4 - 4*a^3*c*d^2*e^6 + a^4*e^8)*arctan
((2*c*d*x*e + c*d^2 + a*e^2)/sqrt(-c^2*d^4 + 2*a*c*d^2*e^2 - a^2*e^4))/(sqrt(-c^2*d^4 + 2*a*c*d^2*e^2 - a^2*e^
4)*c^4*d^4)

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maple [A]  time = 0.04, size = 157, normalized size = 1.57 \begin {gather*} \frac {e^{3} x^{3}}{3 c d}-\frac {a \,e^{4} x^{2}}{2 c^{2} d^{2}}+\frac {3 e^{2} x^{2}}{2 c}-\frac {a^{3} e^{6} \ln \left (c d x +a e \right )}{c^{4} d^{4}}+\frac {3 a^{2} e^{4} \ln \left (c d x +a e \right )}{c^{3} d^{2}}+\frac {a^{2} e^{5} x}{c^{3} d^{3}}-\frac {3 a \,e^{3} x}{c^{2} d}-\frac {3 a \,e^{2} \ln \left (c d x +a e \right )}{c^{2}}+\frac {d^{2} \ln \left (c d x +a e \right )}{c}+\frac {3 d e x}{c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.07, size = 135, normalized size = 1.35 \begin {gather*} \frac {2 \, c^{2} d^{2} e^{3} x^{3} + 3 \, {\left (3 \, c^{2} d^{3} e^{2} - a c d e^{4}\right )} x^{2} + 6 \, {\left (3 \, c^{2} d^{4} e - 3 \, a c d^{2} e^{3} + a^{2} e^{5}\right )} x}{6 \, c^{3} d^{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 (c d x + a e\right )}{c^{4} d^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.59, size = 138, normalized size = 1.38 \begin {gather*} x\,\left (\frac {3\,d\,e}{c}-\frac {a\,e\,\left (\frac {3\,e^2}{c}-\frac {a\,e^4}{c^2\,d^2}\right )}{c\,d}\right )+x^2\,\left (\frac {3\,e^2}{2\,c}-\frac {a\,e^4}{2\,c^2\,d^2}\right )+\frac {e^3\,x^3}{3\,c\,d}-\frac {\ln \left (a\,e+c\,d\,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 )}{c^4\,d^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.38, size = 99, normalized size = 0.99 \begin {gather*} x^{2} \left (- \frac {a e^{4}}{2 c^{2} d^{2}} + \frac {3 e^{2}}{2 c}\right ) + x \left (\frac {a^{2} e^{5}}{c^{3} d^{3}} - \frac {3 a e^{3}}{c^{2} d} + \frac {3 d e}{c}\right ) + \frac {e^{3} x^{3}}{3 c d} - \frac {\left (a e^{2} - c d^{2}\right )^{3} \log {\left (a e + c d x \right )}}{c^{4} d^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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