3.19.60 \(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^3}{(d+e x)^7} \, dx\) [1860]

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

[Out]

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

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Rubi [A]
time = 0.06, 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 = {640, 45} \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 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)^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 = 0.03, 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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Maple [A]
time = 0.79, size = 139, normalized size = 1.32

method	result	size
risch	\(\frac {-\frac {3 c^{2} d^{2} \left (e^{2} a -c \,d^{2}\right ) x^{2}}{e^{2}}-\frac {3 c d \left (a^{2} e^{4}+2 a c \,d^{2} e^{2}-3 c^{2} d^{4}\right ) x}{2 e^{3}}-\frac {2 e^{6} a^{3}+3 e^{4} d^{2} a^{2} c +6 d^{4} e^{2} c^{2} a -11 d^{6} c^{3}}{6 e^{4}}}{\left (e x +d \right )^{3}}+\frac {c^{3} d^{3} \ln \left (e x +d \right )}{e^{4}}\)	\(133\)
default	\(-\frac {e^{6} a^{3}-3 e^{4} d^{2} a^{2} c +3 d^{4} e^{2} c^{2} a -d^{6} c^{3}}{3 e^{4} \left (e x +d \right )^{3}}-\frac {3 c^{2} d^{2} \left (e^{2} a -c \,d^{2}\right )}{e^{4} \left (e x +d \right )}+\frac {c^{3} d^{3} \ln \left (e x +d \right )}{e^{4}}-\frac {3 c d \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right )}{2 e^{4} \left (e x +d \right )^{2}}\)	\(139\)
norman	\(\frac {-\frac {d^{3} \left (2 e^{8} a^{3}+3 d^{2} e^{6} a^{2} c +6 c^{2} d^{4} a \,e^{4}-11 c^{3} d^{6} e^{2}\right )}{6 e^{6}}-\frac {\left (e^{8} a^{3}+15 d^{2} e^{6} a^{2} c +57 c^{2} d^{4} a \,e^{4}-73 c^{3} d^{6} e^{2}\right ) x^{3}}{3 e^{3}}-\frac {3 d \left (a \,c^{2} d \,e^{4}-e^{2} c^{3} d^{3}\right ) x^{5}}{e}-\frac {3 d \left (e^{6} a^{2} c +8 e^{4} d^{2} c^{2} a -9 d^{4} e^{2} c^{3}\right ) x^{4}}{2 e^{2}}-\frac {d \left (e^{8} a^{3}+6 d^{2} e^{6} a^{2} c +15 c^{2} d^{4} a \,e^{4}-22 c^{3} d^{6} e^{2}\right ) x^{2}}{e^{4}}-\frac {d^{2} \left (e^{8} a^{3}+3 d^{2} e^{6} a^{2} c +6 c^{2} d^{4} a \,e^{4}-10 c^{3} d^{6} e^{2}\right ) x}{e^{5}}}{\left (e x +d \right )^{6}}+\frac {c^{3} d^{3} \ln \left (e x +d \right )}{e^{4}}\)	\(305\)

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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.31, size = 148, normalized size = 1.41 \begin {gather*} c^{3} d^{3} e^{\left (-4\right )} \log \left (x e + d\right ) + \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 (x^{3} e^{7} + 3 \, d x^{2} e^{6} + 3 \, d^{2} x e^{5} + 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*e^(-4)*log(x*e + d) + 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)/(x^3*e^7 + 3*d*x^2*e^6 + 3*d^2*x*e
^5 + d^3*e^4)

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Fricas [A]
time = 2.56, size = 185, normalized size = 1.76 \begin {gather*} \frac {27 \, c^{3} d^{5} x e + 11 \, c^{3} d^{6} - 18 \, a c^{2} d^{3} x e^{3} - 9 \, a^{2} c d x e^{5} - 2 \, a^{3} e^{6} - 3 \, {\left (6 \, a c^{2} d^{2} x^{2} + a^{2} c d^{2}\right )} e^{4} + 6 \, {\left (3 \, c^{3} d^{4} x^{2} - a c^{2} d^{4}\right )} e^{2} + 6 \, {\left (c^{3} d^{3} x^{3} e^{3} + 3 \, c^{3} d^{4} x^{2} e^{2} + 3 \, c^{3} d^{5} x e + c^{3} d^{6}\right )} \log \left (x e + d\right )}{6 \, {\left (x^{3} e^{7} + 3 \, d x^{2} e^{6} + 3 \, d^{2} x e^{5} + 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*(27*c^3*d^5*x*e + 11*c^3*d^6 - 18*a*c^2*d^3*x*e^3 - 9*a^2*c*d*x*e^5 - 2*a^3*e^6 - 3*(6*a*c^2*d^2*x^2 + a^2
*c*d^2)*e^4 + 6*(3*c^3*d^4*x^2 - a*c^2*d^4)*e^2 + 6*(c^3*d^3*x^3*e^3 + 3*c^3*d^4*x^2*e^2 + 3*c^3*d^5*x*e + c^3
*d^6)*log(x*e + d))/(x^3*e^7 + 3*d*x^2*e^6 + 3*d^2*x*e^5 + d^3*e^4)

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Sympy [A]
time = 7.06, 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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Giac [A]
time = 1.79, size = 130, normalized size = 1.24 \begin {gather*} c^{3} d^{3} e^{\left (-4\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {{\left (18 \, {\left (c^{3} d^{4} e - a c^{2} d^{2} e^{3}\right )} x^{2} + 9 \, {\left (3 \, c^{3} d^{5} - 2 \, a c^{2} d^{3} e^{2} - a^{2} c d e^{4}\right )} x + {\left (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}\right )} e^{\left (-1\right )}\right )} e^{\left (-3\right )}}{6 \, {\left (x e + d\right )}^{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, algorithm="giac")

[Out]

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

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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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