3.16.63 \(\int \frac {(d+e x)^5}{(a d e+(c d^2+a e^2) x+c d e x^2)^2} \, dx\)

Optimal. Leaf size=105 \[ -\frac {\left (c d^2-a e^2\right )^3}{c^4 d^4 (a e+c d x)}+\frac {3 e \left (c d^2-a e^2\right )^2 \log (a e+c d x)}{c^4 d^4}+\frac {e^2 x \left (3 c d^2-2 a e^2\right )}{c^3 d^3}+\frac {e^3 x^2}{2 c^2 d^2} \]

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Rubi [A]  time = 0.09, antiderivative size = 105, 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^2 x \left (3 c d^2-2 a e^2\right )}{c^3 d^3}-\frac {\left (c d^2-a e^2\right )^3}{c^4 d^4 (a e+c d x)}+\frac {3 e \left (c d^2-a e^2\right )^2 \log (a e+c d x)}{c^4 d^4}+\frac {e^3 x^2}{2 c^2 d^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^5/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2,x]

[Out]

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

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Mathematica [A]  time = 0.05, size = 142, normalized size = 1.35 \begin {gather*} \frac {3 \left (a^2 e^5-2 a c d^2 e^3+c^2 d^4 e\right ) \log (a e+c d x)}{c^4 d^4}+\frac {a^3 e^6-3 a^2 c d^2 e^4+3 a c^2 d^4 e^2-c^3 d^6}{c^4 d^4 (a e+c d x)}-\frac {e^2 x \left (2 a e^2-3 c d^2\right )}{c^3 d^3}+\frac {e^3 x^2}{2 c^2 d^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^5/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2,x]

[Out]

-((e^2*(-3*c*d^2 + 2*a*e^2)*x)/(c^3*d^3)) + (e^3*x^2)/(2*c^2*d^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)/(c^4*d^4*(a*e + c*d*x)) + (3*(c^2*d^4*e - 2*a*c*d^2*e^3 + a^2*e^5)*Log[a*e + c*d*x])/(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)^5}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(d + e*x)^5/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2,x]

[Out]

IntegrateAlgebraic[(d + e*x)^5/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2, x]

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fricas [A]  time = 0.39, size = 205, normalized size = 1.95 \begin {gather*} \frac {c^{3} d^{3} e^{3} x^{3} - 2 \, c^{3} d^{6} + 6 \, a c^{2} d^{4} e^{2} - 6 \, a^{2} c d^{2} e^{4} + 2 \, a^{3} e^{6} + 3 \, {\left (2 \, c^{3} d^{4} e^{2} - a c^{2} d^{2} e^{4}\right )} x^{2} + 2 \, {\left (3 \, a c^{2} d^{3} e^{3} - 2 \, a^{2} c d e^{5}\right )} x + 6 \, {\left (a c^{2} d^{4} e^{2} - 2 \, a^{2} c d^{2} e^{4} + a^{3} e^{6} + {\left (c^{3} d^{5} e - 2 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x\right )} \log \left (c d x + a e\right )}{2 \, {\left (c^{5} d^{5} x + a c^{4} d^{4} e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^5/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2,x, algorithm="fricas")

[Out]

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

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giac [B]  time = 0.34, size = 466, normalized size = 4.44 \begin {gather*} \frac {3 \, {\left (c^{5} d^{10} e - 5 \, a c^{4} d^{8} e^{3} + 10 \, a^{2} c^{3} d^{6} e^{5} - 10 \, a^{3} c^{2} d^{4} e^{7} + 5 \, a^{4} c d^{2} e^{9} - a^{5} e^{11}\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 )}{{\left (c^{6} d^{8} - 2 \, a c^{5} d^{6} e^{2} + a^{2} c^{4} d^{4} e^{4}\right )} \sqrt {-c^{2} d^{4} + 2 \, a c d^{2} e^{2} - a^{2} e^{4}}} + \frac {{\left (c^{2} d^{2} x^{2} e^{7} + 6 \, c^{2} d^{3} x e^{6} - 4 \, a c d x e^{8}\right )} e^{\left (-4\right )}}{2 \, c^{4} d^{4}} + \frac {3 \, {\left (c^{2} d^{4} e - 2 \, a c d^{2} e^{3} + a^{2} e^{5}\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 {c^{5} d^{11} - 5 \, a c^{4} d^{9} e^{2} + 10 \, a^{2} c^{3} d^{7} e^{4} - 10 \, a^{3} c^{2} d^{5} e^{6} + 5 \, a^{4} c d^{3} e^{8} - a^{5} d e^{10} + {\left (c^{5} d^{10} e - 5 \, a c^{4} d^{8} e^{3} + 10 \, a^{2} c^{3} d^{6} e^{5} - 10 \, a^{3} c^{2} d^{4} e^{7} + 5 \, a^{4} c d^{2} e^{9} - a^{5} e^{11}\right )} x}{{\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} {\left (c d x^{2} e + c d^{2} x + a x e^{2} + a d e\right )} c^{4} d^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^5/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2,x, algorithm="giac")

[Out]

3*(c^5*d^10*e - 5*a*c^4*d^8*e^3 + 10*a^2*c^3*d^6*e^5 - 10*a^3*c^2*d^4*e^7 + 5*a^4*c*d^2*e^9 - a^5*e^11)*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))/((c^6*d^8 - 2*a*c^5*d^6*e^2 + a^2*c^4*d
^4*e^4)*sqrt(-c^2*d^4 + 2*a*c*d^2*e^2 - a^2*e^4)) + 1/2*(c^2*d^2*x^2*e^7 + 6*c^2*d^3*x*e^6 - 4*a*c*d*x*e^8)*e^
(-4)/(c^4*d^4) + 3/2*(c^2*d^4*e - 2*a*c*d^2*e^3 + a^2*e^5)*log(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e)/(c^4*d^4
) - (c^5*d^11 - 5*a*c^4*d^9*e^2 + 10*a^2*c^3*d^7*e^4 - 10*a^3*c^2*d^5*e^6 + 5*a^4*c*d^3*e^8 - a^5*d*e^10 + (c^
5*d^10*e - 5*a*c^4*d^8*e^3 + 10*a^2*c^3*d^6*e^5 - 10*a^3*c^2*d^4*e^7 + 5*a^4*c*d^2*e^9 - a^5*e^11)*x)/((c^2*d^
4 - 2*a*c*d^2*e^2 + a^2*e^4)*(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e)*c^4*d^4)

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maple [A]  time = 0.05, size = 184, normalized size = 1.75 \begin {gather*} \frac {a^{3} e^{6}}{\left (c d x +a e \right ) c^{4} d^{4}}-\frac {3 a^{2} e^{4}}{\left (c d x +a e \right ) c^{3} d^{2}}+\frac {3 a \,e^{2}}{\left (c d x +a e \right ) c^{2}}-\frac {d^{2}}{\left (c d x +a e \right ) c}+\frac {e^{3} x^{2}}{2 c^{2} d^{2}}+\frac {3 a^{2} e^{5} \ln \left (c d x +a e \right )}{c^{4} d^{4}}-\frac {6 a \,e^{3} \ln \left (c d x +a e \right )}{c^{3} d^{2}}-\frac {2 a \,e^{4} x}{c^{3} d^{3}}+\frac {3 e^{2} x}{c^{2} d}+\frac {3 e \ln \left (c d x +a e \right )}{c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^5/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2,x)

[Out]

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

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maxima [A]  time = 1.10, size = 143, normalized size = 1.36 \begin {gather*} -\frac {c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}}{c^{5} d^{5} x + a c^{4} d^{4} e} + \frac {c d e^{3} x^{2} + 2 \, {\left (3 \, c d^{2} e^{2} - 2 \, a e^{4}\right )} x}{2 \, c^{3} d^{3}} + \frac {3 \, {\left (c^{2} d^{4} e - 2 \, a c d^{2} e^{3} + a^{2} e^{5}\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)^5/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2,x, algorithm="maxima")

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

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mupad [B]  time = 0.08, size = 152, normalized size = 1.45 \begin {gather*} x\,\left (\frac {3\,e^2}{c^2\,d}-\frac {2\,a\,e^4}{c^3\,d^3}\right )+\frac {e^3\,x^2}{2\,c^2\,d^2}+\frac {\ln \left (a\,e+c\,d\,x\right )\,\left (3\,a^2\,e^5-6\,a\,c\,d^2\,e^3+3\,c^2\,d^4\,e\right )}{c^4\,d^4}+\frac {a^3\,e^6-3\,a^2\,c\,d^2\,e^4+3\,a\,c^2\,d^4\,e^2-c^3\,d^6}{c\,d\,\left (x\,c^4\,d^4+a\,e\,c^3\,d^3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^5/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^2,x)

[Out]

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

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sympy [A]  time = 0.64, size = 131, normalized size = 1.25 \begin {gather*} x \left (- \frac {2 a e^{4}}{c^{3} d^{3}} + \frac {3 e^{2}}{c^{2} d}\right ) + \frac {a^{3} e^{6} - 3 a^{2} c d^{2} e^{4} + 3 a c^{2} d^{4} e^{2} - c^{3} d^{6}}{a c^{4} d^{4} e + c^{5} d^{5} x} + \frac {e^{3} x^{2}}{2 c^{2} d^{2}} + \frac {3 e \left (a e^{2} - c d^{2}\right )^{2} \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)**5/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**2,x)

[Out]

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

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