3.16.46 \(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^3}{(d+e x)^7} \, dx\)

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

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Rubi [A]  time = 0.08, 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 {3 c^2 d^2 \left (c d^2-a e^2\right )}{e^4 (d+e x)}-\frac {3 c d \left (c d^2-a e^2\right )^2}{2 e^4 (d+e x)^2}+\frac {\left (c d^2-a e^2\right )^3}{3 e^4 (d+e x)^3}+\frac {c^3 d^3 \log (d+e x)}{e^4} \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)^7,x]

[Out]

(c*d^2 - a*e^2)^3/(3*e^4*(d + e*x)^3) - (3*c*d*(c*d^2 - a*e^2)^2)/(2*e^4*(d + e*x)^2) + (3*c^2*d^2*(c*d^2 - a*
e^2))/(e^4*(d + e*x)) + (c^3*d^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)^7} \, dx &=\int \frac {(a e+c d x)^3}{(d+e x)^4} \, dx\\ &=\int \left (\frac {\left (-c d^2+a e^2\right )^3}{e^3 (d+e x)^4}+\frac {3 c d \left (c d^2-a e^2\right )^2}{e^3 (d+e x)^3}-\frac {3 c^2 d^2 \left (c d^2-a e^2\right )}{e^3 (d+e x)^2}+\frac {c^3 d^3}{e^3 (d+e x)}\right ) \, dx\\ &=\frac {\left (c d^2-a e^2\right )^3}{3 e^4 (d+e x)^3}-\frac {3 c d \left (c d^2-a e^2\right )^2}{2 e^4 (d+e x)^2}+\frac {3 c^2 d^2 \left (c d^2-a e^2\right )}{e^4 (d+e x)}+\frac {c^3 d^3 \log (d+e x)}{e^4}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 92, normalized size = 0.88 \begin {gather*} \frac {\frac {\left (c d^2-a e^2\right ) \left (2 a^2 e^4+a c d e^2 (5 d+9 e x)+c^2 d^2 \left (11 d^2+27 d e x+18 e^2 x^2\right )\right )}{(d+e x)^3}+6 c^3 d^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)^7,x]

[Out]

(((c*d^2 - a*e^2)*(2*a^2*e^4 + a*c*d*e^2*(5*d + 9*e*x) + c^2*d^2*(11*d^2 + 27*d*e*x + 18*e^2*x^2)))/(d + e*x)^
3 + 6*c^3*d^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)^7} \, 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)^7,x]

[Out]

IntegrateAlgebraic[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3/(d + e*x)^7, x]

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fricas [A]  time = 0.40, size = 194, normalized size = 1.85 \begin {gather*} \frac {11 \, c^{3} d^{6} - 6 \, a c^{2} d^{4} e^{2} - 3 \, a^{2} c d^{2} e^{4} - 2 \, a^{3} e^{6} + 18 \, {\left (c^{3} d^{4} e^{2} - a c^{2} d^{2} e^{4}\right )} x^{2} + 9 \, {\left (3 \, c^{3} d^{5} e - 2 \, a c^{2} d^{3} e^{3} - a^{2} c d e^{5}\right )} x + 6 \, {\left (c^{3} d^{3} e^{3} x^{3} + 3 \, c^{3} d^{4} e^{2} x^{2} + 3 \, c^{3} d^{5} e x + c^{3} d^{6}\right )} \log \left (e x + d\right )}{6 \, {\left (e^{7} x^{3} + 3 \, d e^{6} x^{2} + 3 \, d^{2} e^{5} x + d^{3} 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)^7,x, algorithm="fricas")

[Out]

1/6*(11*c^3*d^6 - 6*a*c^2*d^4*e^2 - 3*a^2*c*d^2*e^4 - 2*a^3*e^6 + 18*(c^3*d^4*e^2 - a*c^2*d^2*e^4)*x^2 + 9*(3*
c^3*d^5*e - 2*a*c^2*d^3*e^3 - a^2*c*d*e^5)*x + 6*(c^3*d^3*e^3*x^3 + 3*c^3*d^4*e^2*x^2 + 3*c^3*d^5*e*x + c^3*d^
6)*log(e*x + d))/(e^7*x^3 + 3*d*e^6*x^2 + 3*d^2*e^5*x + d^3*e^4)

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giac [B]  time = 0.17, size = 270, normalized size = 2.57 \begin {gather*} c^{3} d^{3} e^{\left (-4\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {{\left (11 \, c^{3} d^{9} - 6 \, a c^{2} d^{7} e^{2} - 3 \, a^{2} c d^{5} e^{4} - 2 \, a^{3} d^{3} e^{6} + 18 \, {\left (c^{3} d^{4} e^{5} - a c^{2} d^{2} e^{7}\right )} x^{5} + 9 \, {\left (9 \, c^{3} d^{5} e^{4} - 8 \, a c^{2} d^{3} e^{6} - a^{2} c d e^{8}\right )} x^{4} + 2 \, {\left (73 \, c^{3} d^{6} e^{3} - 57 \, a c^{2} d^{4} e^{5} - 15 \, a^{2} c d^{2} e^{7} - a^{3} e^{9}\right )} x^{3} + 6 \, {\left (22 \, c^{3} d^{7} e^{2} - 15 \, a c^{2} d^{5} e^{4} - 6 \, a^{2} c d^{3} e^{6} - a^{3} d e^{8}\right )} x^{2} + 6 \, {\left (10 \, c^{3} d^{8} e - 6 \, a c^{2} d^{6} e^{3} - 3 \, a^{2} c d^{4} e^{5} - a^{3} d^{2} e^{7}\right )} x\right )} e^{\left (-4\right )}}{6 \, {\left (x e + d\right )}^{6}} \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)^7,x, algorithm="giac")

[Out]

c^3*d^3*e^(-4)*log(abs(x*e + d)) + 1/6*(11*c^3*d^9 - 6*a*c^2*d^7*e^2 - 3*a^2*c*d^5*e^4 - 2*a^3*d^3*e^6 + 18*(c
^3*d^4*e^5 - a*c^2*d^2*e^7)*x^5 + 9*(9*c^3*d^5*e^4 - 8*a*c^2*d^3*e^6 - a^2*c*d*e^8)*x^4 + 2*(73*c^3*d^6*e^3 -
57*a*c^2*d^4*e^5 - 15*a^2*c*d^2*e^7 - a^3*e^9)*x^3 + 6*(22*c^3*d^7*e^2 - 15*a*c^2*d^5*e^4 - 6*a^2*c*d^3*e^6 -
a^3*d*e^8)*x^2 + 6*(10*c^3*d^8*e - 6*a*c^2*d^6*e^3 - 3*a^2*c*d^4*e^5 - a^3*d^2*e^7)*x)*e^(-4)/(x*e + d)^6

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maple [A]  time = 0.05, size = 173, normalized size = 1.65 \begin {gather*} -\frac {a^{3} e^{2}}{3 \left (e x +d \right )^{3}}+\frac {a^{2} c \,d^{2}}{\left (e x +d \right )^{3}}-\frac {a \,c^{2} d^{4}}{\left (e x +d \right )^{3} e^{2}}+\frac {c^{3} d^{6}}{3 \left (e x +d \right )^{3} e^{4}}-\frac {3 a^{2} c d}{2 \left (e x +d \right )^{2}}+\frac {3 a \,c^{2} d^{3}}{\left (e x +d \right )^{2} e^{2}}-\frac {3 c^{3} d^{5}}{2 \left (e x +d \right )^{2} e^{4}}-\frac {3 a \,c^{2} d^{2}}{\left (e x +d \right ) e^{2}}+\frac {3 c^{3} d^{4}}{\left (e x +d \right ) e^{4}}+\frac {c^{3} d^{3} \ln \left (e x +d \right )}{e^{4}} \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)^7,x)

[Out]

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

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maxima [A]  time = 1.11, size = 158, normalized size = 1.50 \begin {gather*} \frac {c^{3} d^{3} \log \left (e x + d\right )}{e^{4}} + \frac {11 \, c^{3} d^{6} - 6 \, a c^{2} d^{4} e^{2} - 3 \, a^{2} c d^{2} e^{4} - 2 \, a^{3} e^{6} + 18 \, {\left (c^{3} d^{4} e^{2} - a c^{2} d^{2} e^{4}\right )} x^{2} + 9 \, {\left (3 \, c^{3} d^{5} e - 2 \, a c^{2} d^{3} e^{3} - a^{2} c d e^{5}\right )} x}{6 \, {\left (e^{7} x^{3} + 3 \, d e^{6} x^{2} + 3 \, d^{2} e^{5} x + d^{3} 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)^7,x, algorithm="maxima")

[Out]

c^3*d^3*log(e*x + d)/e^4 + 1/6*(11*c^3*d^6 - 6*a*c^2*d^4*e^2 - 3*a^2*c*d^2*e^4 - 2*a^3*e^6 + 18*(c^3*d^4*e^2 -
a*c^2*d^2*e^4)*x^2 + 9*(3*c^3*d^5*e - 2*a*c^2*d^3*e^3 - a^2*c*d*e^5)*x)/(e^7*x^3 + 3*d*e^6*x^2 + 3*d^2*e^5*x
+ d^3*e^4)

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mupad [B]  time = 0.63, size = 157, normalized size = 1.50 \begin {gather*} \frac {c^3\,d^3\,\ln \left (d+e\,x\right )}{e^4}-\frac {\frac {2\,a^3\,e^6+3\,a^2\,c\,d^2\,e^4+6\,a\,c^2\,d^4\,e^2-11\,c^3\,d^6}{6\,e^4}+\frac {3\,x\,\left (a^2\,c\,d\,e^4+2\,a\,c^2\,d^3\,e^2-3\,c^3\,d^5\right )}{2\,e^3}+\frac {3\,c^2\,d^2\,x^2\,\left (a\,e^2-c\,d^2\right )}{e^2}}{d^3+3\,d^2\,e\,x+3\,d\,e^2\,x^2+e^3\,x^3} \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)^7,x)

[Out]

(c^3*d^3*log(d + e*x))/e^4 - ((2*a^3*e^6 - 11*c^3*d^6 + 6*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4)/(6*e^4) + (3*x*(2*a
*c^2*d^3*e^2 - 3*c^3*d^5 + a^2*c*d*e^4))/(2*e^3) + (3*c^2*d^2*x^2*(a*e^2 - c*d^2))/e^2)/(d^3 + e^3*x^3 + 3*d*e
^2*x^2 + 3*d^2*e*x)

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sympy [A]  time = 2.16, size = 163, normalized size = 1.55 \begin {gather*} \frac {c^{3} d^{3} \log {\left (d + e x \right )}}{e^{4}} + \frac {- 2 a^{3} e^{6} - 3 a^{2} c d^{2} e^{4} - 6 a c^{2} d^{4} e^{2} + 11 c^{3} d^{6} + x^{2} \left (- 18 a c^{2} d^{2} e^{4} + 18 c^{3} d^{4} e^{2}\right ) + x \left (- 9 a^{2} c d e^{5} - 18 a c^{2} d^{3} e^{3} + 27 c^{3} d^{5} e\right )}{6 d^{3} e^{4} + 18 d^{2} e^{5} x + 18 d e^{6} x^{2} + 6 e^{7} x^{3}} \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)**7,x)

[Out]

c**3*d**3*log(d + e*x)/e**4 + (-2*a**3*e**6 - 3*a**2*c*d**2*e**4 - 6*a*c**2*d**4*e**2 + 11*c**3*d**6 + x**2*(-
18*a*c**2*d**2*e**4 + 18*c**3*d**4*e**2) + x*(-9*a**2*c*d*e**5 - 18*a*c**2*d**3*e**3 + 27*c**3*d**5*e))/(6*d**
3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3)

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