3.1860 \(\int \frac{(a d e+(c d^2+a e^2) x+c d e x^2)^3}{(d+e x)^7} \, dx\)

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

[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

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Rubi [A]  time = 0.0750259, antiderivative size = 105, normalized size of antiderivative = 1., 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} \[ \frac{3 c^2 d^2 \left (c d^2-a e^2\right )}{e^4 (d+e x)}-\frac{3 c d \left (c d^2-a e^2\right )^2}{2 e^4 (d+e x)^2}+\frac{\left (c d^2-a e^2\right )^3}{3 e^4 (d+e x)^3}+\frac{c^3 d^3 \log (d+e x)}{e^4} \]

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 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]

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])

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*}

Mathematica [A]  time = 0.0474225, size = 92, normalized size = 0.88 \[ \frac{\frac{\left (c d^2-a e^2\right ) \left (2 a^2 e^4+a c d e^2 (5 d+9 e x)+c^2 d^2 \left (11 d^2+27 d e x+18 e^2 x^2\right )\right )}{(d+e x)^3}+6 c^3 d^3 \log (d+e x)}{6 e^4} \]

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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Maple [A]  time = 0.044, size = 173, normalized size = 1.7 \begin{align*} -{\frac{3\,cd{a}^{2}}{2\, \left ( ex+d \right ) ^{2}}}+3\,{\frac{a{c}^{2}{d}^{3}}{{e}^{2} \left ( ex+d \right ) ^{2}}}-{\frac{3\,{c}^{3}{d}^{5}}{2\,{e}^{4} \left ( ex+d \right ) ^{2}}}+{\frac{{c}^{3}{d}^{3}\ln \left ( ex+d \right ) }{{e}^{4}}}-3\,{\frac{a{c}^{2}{d}^{2}}{{e}^{2} \left ( ex+d \right ) }}+3\,{\frac{{c}^{3}{d}^{4}}{{e}^{4} \left ( ex+d \right ) }}-{\frac{{e}^{2}{a}^{3}}{3\, \left ( ex+d \right ) ^{3}}}+{\frac{{a}^{2}c{d}^{2}}{ \left ( ex+d \right ) ^{3}}}-{\frac{a{c}^{2}{d}^{4}}{{e}^{2} \left ( ex+d \right ) ^{3}}}+{\frac{{c}^{3}{d}^{6}}{3\,{e}^{4} \left ( ex+d \right ) ^{3}}} \end{align*}

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]

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

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Maxima [A]  time = 1.07755, size = 213, normalized size = 2.03 \begin{align*} \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{align*}

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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Fricas [A]  time = 1.60749, size = 386, normalized size = 3.68 \begin{align*} \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{align*}

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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Sympy [A]  time = 3.2611, size = 163, normalized size = 1.55 \begin{align*} \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{align*}

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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Giac [B]  time = 1.22466, size = 365, normalized size = 3.48 \begin{align*} 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{align*}

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