3.19.59 \(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^3}{(d+e x)^6} \, dx\) [1859]

Optimal. Leaf size=97 \[ \frac {c^3 d^3 x}{e^3}+\frac {\left (c d^2-a e^2\right )^3}{2 e^4 (d+e x)^2}-\frac {3 c d \left (c d^2-a e^2\right )^2}{e^4 (d+e x)}-\frac {3 c^2 d^2 \left (c d^2-a e^2\right ) \log (d+e x)}{e^4} \]

[Out]

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

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Rubi [A]
time = 0.06, antiderivative size = 97, 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 = {640, 45} \begin {gather*} -\frac {3 c^2 d^2 \left (c d^2-a e^2\right ) \log (d+e x)}{e^4}-\frac {3 c d \left (c d^2-a e^2\right )^2}{e^4 (d+e x)}+\frac {\left (c d^2-a e^2\right )^3}{2 e^4 (d+e x)^2}+\frac {c^3 d^3 x}{e^3} \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)^6,x]

[Out]

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

Rule 45

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 640

Int[((d_) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a
/d + (c/e)*x)^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)^6} \, dx &=\int \frac {(a e+c d x)^3}{(d+e x)^3} \, dx\\ &=\int \left (\frac {c^3 d^3}{e^3}+\frac {\left (-c d^2+a e^2\right )^3}{e^3 (d+e x)^3}+\frac {3 c d \left (c d^2-a e^2\right )^2}{e^3 (d+e x)^2}-\frac {3 c^2 d^2 \left (c d^2-a e^2\right )}{e^3 (d+e x)}\right ) \, dx\\ &=\frac {c^3 d^3 x}{e^3}+\frac {\left (c d^2-a e^2\right )^3}{2 e^4 (d+e x)^2}-\frac {3 c d \left (c d^2-a e^2\right )^2}{e^4 (d+e x)}-\frac {3 c^2 d^2 \left (c d^2-a e^2\right ) \log (d+e x)}{e^4}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 129, normalized size = 1.33 \begin {gather*} \frac {-a^3 e^6-3 a^2 c d e^4 (d+2 e x)+3 a c^2 d^3 e^2 (3 d+4 e x)+c^3 d^3 \left (-5 d^3-4 d^2 e x+4 d e^2 x^2+2 e^3 x^3\right )-6 c^2 d^2 \left (c d^2-a e^2\right ) (d+e x)^2 \log (d+e x)}{2 e^4 (d+e x)^2} \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)^6,x]

[Out]

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

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Maple [A]
time = 0.68, size = 133, normalized size = 1.37

method	result	size
default	\(\frac {c^{3} d^{3} x}{e^{3}}-\frac {3 c d \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right )}{e^{4} \left (e x +d \right )}+\frac {3 c^{2} d^{2} \left (e^{2} a -c \,d^{2}\right ) \ln \left (e x +d \right )}{e^{4}}-\frac {e^{6} a^{3}-3 e^{4} d^{2} a^{2} c +3 d^{4} e^{2} c^{2} a -d^{6} c^{3}}{2 e^{4} \left (e x +d \right )^{2}}\)	\(133\)
risch	\(\frac {c^{3} d^{3} x}{e^{3}}+\frac {\left (-3 d \,e^{4} a^{2} c +6 d^{3} e^{2} c^{2} a -3 d^{5} c^{3}\right ) x -\frac {e^{6} a^{3}+3 e^{4} d^{2} a^{2} c -9 d^{4} e^{2} c^{2} a +5 d^{6} c^{3}}{2 e}}{e^{3} \left (e x +d \right )^{2}}+\frac {3 c^{2} d^{2} \ln \left (e x +d \right ) a}{e^{2}}-\frac {3 c^{3} d^{4} \ln \left (e x +d \right )}{e^{4}}\)	\(138\)
norman	\(\frac {e^{2} c^{3} d^{3} x^{6}-\frac {d^{3} \left (e^{7} a^{3}+3 d^{2} e^{5} a^{2} c -9 c^{2} d^{4} a \,e^{3}+15 c^{3} d^{6} e \right )}{2 e^{5}}-\frac {\left (e^{7} a^{3}+21 d^{2} e^{5} a^{2} c -45 c^{2} d^{4} a \,e^{3}+103 c^{3} d^{6} e \right ) x^{3}}{2 e^{2}}-\frac {d \left (3 a^{2} c \,e^{5}-6 e^{3} c^{2} d^{2} a +18 d^{4} e \,c^{3}\right ) x^{4}}{e}-\frac {d \left (3 e^{7} a^{3}+27 d^{2} e^{5} a^{2} c -63 c^{2} d^{4} a \,e^{3}+123 c^{3} d^{6} e \right ) x^{2}}{2 e^{3}}-\frac {d^{2} \left (3 e^{7} a^{3}+15 d^{2} e^{5} a^{2} c -39 c^{2} d^{4} a \,e^{3}+69 c^{3} d^{6} e \right ) x}{2 e^{4}}}{\left (e x +d \right )^{5}}+\frac {3 c^{2} d^{2} \left (e^{2} a -c \,d^{2}\right ) \ln \left (e x +d \right )}{e^{4}}\)	\(293\)

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)^6,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.32, size = 133, normalized size = 1.37 \begin {gather*} c^{3} d^{3} x e^{\left (-3\right )} - 3 \, {\left (c^{3} d^{4} - a c^{2} d^{2} e^{2}\right )} e^{\left (-4\right )} \log \left (x e + d\right ) - \frac {5 \, c^{3} d^{6} - 9 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6} + 6 \, {\left (c^{3} d^{5} e - 2 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x}{2 \, {\left (x^{2} e^{6} + 2 \, d x e^{5} + d^{2} e^{4}\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)^6,x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 197 vs. \(2 (91) = 182\).
time = 3.21, size = 197, normalized size = 2.03 \begin {gather*} -\frac {4 \, c^{3} d^{5} x e + 5 \, c^{3} d^{6} + 6 \, a^{2} c d x e^{5} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6} - 2 \, {\left (c^{3} d^{3} x^{3} + 6 \, a c^{2} d^{3} x\right )} e^{3} - {\left (4 \, c^{3} d^{4} x^{2} + 9 \, a c^{2} d^{4}\right )} e^{2} + 6 \, {\left (2 \, c^{3} d^{5} x e + c^{3} d^{6} - a c^{2} d^{2} x^{2} e^{4} - 2 \, a c^{2} d^{3} x e^{3} + {\left (c^{3} d^{4} x^{2} - a c^{2} d^{4}\right )} e^{2}\right )} \log \left (x e + d\right )}{2 \, {\left (x^{2} e^{6} + 2 \, d x e^{5} + d^{2} e^{4}\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)^6,x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 0.75, size = 144, normalized size = 1.48 \begin {gather*} \frac {c^{3} d^{3} x}{e^{3}} + \frac {3 c^{2} d^{2} \left (a e^{2} - c d^{2}\right ) \log {\left (d + e x \right )}}{e^{4}} + \frac {- a^{3} e^{6} - 3 a^{2} c d^{2} e^{4} + 9 a c^{2} d^{4} e^{2} - 5 c^{3} d^{6} + x \left (- 6 a^{2} c d e^{5} + 12 a c^{2} d^{3} e^{3} - 6 c^{3} d^{5} e\right )}{2 d^{2} e^{4} + 4 d e^{5} x + 2 e^{6} x^{2}} \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)**6,x)

[Out]

c**3*d**3*x/e**3 + 3*c**2*d**2*(a*e**2 - c*d**2)*log(d + e*x)/e**4 + (-a**3*e**6 - 3*a**2*c*d**2*e**4 + 9*a*c*
*2*d**4*e**2 - 5*c**3*d**6 + x*(-6*a**2*c*d*e**5 + 12*a*c**2*d**3*e**3 - 6*c**3*d**5*e))/(2*d**2*e**4 + 4*d*e*
*5*x + 2*e**6*x**2)

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Giac [A]
time = 1.45, size = 123, normalized size = 1.27 \begin {gather*} c^{3} d^{3} x e^{\left (-3\right )} - 3 \, {\left (c^{3} d^{4} - a c^{2} d^{2} e^{2}\right )} e^{\left (-4\right )} \log \left ({\left | x e + d \right |}\right ) - \frac {{\left (5 \, c^{3} d^{6} - 9 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6} + 6 \, {\left (c^{3} d^{5} e - 2 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x\right )} e^{\left (-4\right )}}{2 \, {\left (x e + d\right )}^{2}} \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)^6,x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 0.12, size = 149, normalized size = 1.54 \begin {gather*} \frac {c^3\,d^3\,x}{e^3}-\frac {\ln \left (d+e\,x\right )\,\left (3\,c^3\,d^4-3\,a\,c^2\,d^2\,e^2\right )}{e^4}-\frac {\frac {a^3\,e^6+3\,a^2\,c\,d^2\,e^4-9\,a\,c^2\,d^4\,e^2+5\,c^3\,d^6}{2\,e}+x\,\left (3\,a^2\,c\,d\,e^4-6\,a\,c^2\,d^3\,e^2+3\,c^3\,d^5\right )}{d^2\,e^3+2\,d\,e^4\,x+e^5\,x^2} \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)^6,x)

[Out]

(c^3*d^3*x)/e^3 - (log(d + e*x)*(3*c^3*d^4 - 3*a*c^2*d^2*e^2))/e^4 - ((a^3*e^6 + 5*c^3*d^6 - 9*a*c^2*d^4*e^2 +
3*a^2*c*d^2*e^4)/(2*e) + x*(3*c^3*d^5 - 6*a*c^2*d^3*e^2 + 3*a^2*c*d*e^4))/(d^2*e^3 + e^5*x^2 + 2*d*e^4*x)

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