3.19.64 \(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^3}{(d+e x)^{11}} \, dx\) [1864]

Optimal. Leaf size=111 \[ \frac {\left (c d^2-a e^2\right )^3}{7 e^4 (d+e x)^7}-\frac {c d \left (c d^2-a e^2\right )^2}{2 e^4 (d+e x)^6}+\frac {3 c^2 d^2 \left (c d^2-a e^2\right )}{5 e^4 (d+e x)^5}-\frac {c^3 d^3}{4 e^4 (d+e x)^4} \]

[Out]

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

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Rubi [A]
time = 0.06, antiderivative size = 111, 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 )}{5 e^4 (d+e x)^5}-\frac {c d \left (c d^2-a e^2\right )^2}{2 e^4 (d+e x)^6}+\frac {\left (c d^2-a e^2\right )^3}{7 e^4 (d+e x)^7}-\frac {c^3 d^3}{4 e^4 (d+e x)^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)^11,x]

[Out]

(c*d^2 - a*e^2)^3/(7*e^4*(d + e*x)^7) - (c*d*(c*d^2 - a*e^2)^2)/(2*e^4*(d + e*x)^6) + (3*c^2*d^2*(c*d^2 - a*e^
2))/(5*e^4*(d + e*x)^5) - (c^3*d^3)/(4*e^4*(d + e*x)^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)^{11}} \, dx &=\int \frac {(a e+c d x)^3}{(d+e x)^8} \, dx\\ &=\int \left (\frac {\left (-c d^2+a e^2\right )^3}{e^3 (d+e x)^8}+\frac {3 c d \left (c d^2-a e^2\right )^2}{e^3 (d+e x)^7}-\frac {3 c^2 d^2 \left (c d^2-a e^2\right )}{e^3 (d+e x)^6}+\frac {c^3 d^3}{e^3 (d+e x)^5}\right ) \, dx\\ &=\frac {\left (c d^2-a e^2\right )^3}{7 e^4 (d+e x)^7}-\frac {c d \left (c d^2-a e^2\right )^2}{2 e^4 (d+e x)^6}+\frac {3 c^2 d^2 \left (c d^2-a e^2\right )}{5 e^4 (d+e x)^5}-\frac {c^3 d^3}{4 e^4 (d+e x)^4}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 103, normalized size = 0.93 \begin {gather*} -\frac {20 a^3 e^6+10 a^2 c d e^4 (d+7 e x)+4 a c^2 d^2 e^2 \left (d^2+7 d e x+21 e^2 x^2\right )+c^3 d^3 \left (d^3+7 d^2 e x+21 d e^2 x^2+35 e^3 x^3\right )}{140 e^4 (d+e x)^7} \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)^11,x]

[Out]

-1/140*(20*a^3*e^6 + 10*a^2*c*d*e^4*(d + 7*e*x) + 4*a*c^2*d^2*e^2*(d^2 + 7*d*e*x + 21*e^2*x^2) + c^3*d^3*(d^3
+ 7*d^2*e*x + 21*d*e^2*x^2 + 35*e^3*x^3))/(e^4*(d + e*x)^7)

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Maple [A]
time = 0.68, size = 141, normalized size = 1.27

method	result	size
risch	\(\frac {-\frac {c^{3} d^{3} x^{3}}{4 e}-\frac {3 d^{2} c^{2} \left (4 e^{2} a +c \,d^{2}\right ) x^{2}}{20 e^{2}}-\frac {d c \left (10 a^{2} e^{4}+4 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) x}{20 e^{3}}-\frac {20 e^{6} a^{3}+10 e^{4} d^{2} a^{2} c +4 d^{4} e^{2} c^{2} a +d^{6} c^{3}}{140 e^{4}}}{\left (e x +d \right )^{7}}\)	\(129\)
gosper	\(-\frac {35 c^{3} d^{3} e^{3} x^{3}+84 a \,c^{2} d^{2} e^{4} x^{2}+21 c^{3} d^{4} e^{2} x^{2}+70 a^{2} c d \,e^{5} x +28 a \,c^{2} d^{3} e^{3} x +7 c^{3} d^{5} e x +20 e^{6} a^{3}+10 e^{4} d^{2} a^{2} c +4 d^{4} e^{2} c^{2} a +d^{6} c^{3}}{140 e^{4} \left (e x +d \right )^{7}}\)	\(130\)
default	\(-\frac {e^{6} a^{3}-3 e^{4} d^{2} a^{2} c +3 d^{4} e^{2} c^{2} a -d^{6} c^{3}}{7 e^{4} \left (e x +d \right )^{7}}-\frac {c d \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right )}{2 e^{4} \left (e x +d \right )^{6}}-\frac {c^{3} d^{3}}{4 e^{4} \left (e x +d \right )^{4}}-\frac {3 c^{2} d^{2} \left (e^{2} a -c \,d^{2}\right )}{5 e^{4} \left (e x +d \right )^{5}}\)	\(141\)
norman	\(\frac {-\frac {d^{3} \left (20 a^{3} e^{12}+10 a^{2} c \,d^{2} e^{10}+4 d^{4} c^{2} a \,e^{8}+c^{3} d^{6} e^{6}\right )}{140 e^{10}}-\frac {\left (a^{3} e^{12}+11 a^{2} c \,d^{2} e^{10}+17 d^{4} c^{2} a \,e^{8}+6 c^{3} d^{6} e^{6}\right ) x^{3}}{7 e^{7}}-\frac {d \left (2 a^{2} c \,e^{10}+8 d^{2} c^{2} a \,e^{8}+5 d^{4} c^{3} e^{6}\right ) x^{4}}{4 e^{6}}-\frac {3 d \left (4 a^{3} e^{12}+16 a^{2} c \,d^{2} e^{10}+12 d^{4} c^{2} a \,e^{8}+3 c^{3} d^{6} e^{6}\right ) x^{2}}{28 e^{8}}-\frac {3 d^{2} \left (2 a \,c^{2} e^{8}+3 c^{3} d^{2} e^{6}\right ) x^{5}}{10 e^{5}}-\frac {d^{2} \left (6 a^{3} e^{12}+10 a^{2} c \,d^{2} e^{10}+4 d^{4} c^{2} a \,e^{8}+c^{3} d^{6} e^{6}\right ) x}{14 e^{9}}-\frac {e^{2} c^{3} d^{3} x^{6}}{4}}{\left (e x +d \right )^{10}}\)	\(305\)

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)^11,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.31, size = 182, normalized size = 1.64 \begin {gather*} -\frac {35 \, c^{3} d^{3} x^{3} e^{3} + c^{3} d^{6} + 4 \, a c^{2} d^{4} e^{2} + 10 \, a^{2} c d^{2} e^{4} + 20 \, a^{3} e^{6} + 21 \, {\left (c^{3} d^{4} e^{2} + 4 \, a c^{2} d^{2} e^{4}\right )} x^{2} + 7 \, {\left (c^{3} d^{5} e + 4 \, a c^{2} d^{3} e^{3} + 10 \, a^{2} c d e^{5}\right )} x}{140 \, {\left (x^{7} e^{11} + 7 \, d x^{6} e^{10} + 21 \, d^{2} x^{5} e^{9} + 35 \, d^{3} x^{4} e^{8} + 35 \, d^{4} x^{3} e^{7} + 21 \, d^{5} x^{2} e^{6} + 7 \, d^{6} x e^{5} + d^{7} 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)^11,x, algorithm="maxima")

[Out]

-1/140*(35*c^3*d^3*x^3*e^3 + c^3*d^6 + 4*a*c^2*d^4*e^2 + 10*a^2*c*d^2*e^4 + 20*a^3*e^6 + 21*(c^3*d^4*e^2 + 4*a
*c^2*d^2*e^4)*x^2 + 7*(c^3*d^5*e + 4*a*c^2*d^3*e^3 + 10*a^2*c*d*e^5)*x)/(x^7*e^11 + 7*d*x^6*e^10 + 21*d^2*x^5*
e^9 + 35*d^3*x^4*e^8 + 35*d^4*x^3*e^7 + 21*d^5*x^2*e^6 + 7*d^6*x*e^5 + d^7*e^4)

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Fricas [A]
time = 2.41, size = 185, normalized size = 1.67 \begin {gather*} -\frac {7 \, c^{3} d^{5} x e + c^{3} d^{6} + 70 \, a^{2} c d x e^{5} + 20 \, a^{3} e^{6} + 2 \, {\left (42 \, a c^{2} d^{2} x^{2} + 5 \, a^{2} c d^{2}\right )} e^{4} + 7 \, {\left (5 \, c^{3} d^{3} x^{3} + 4 \, a c^{2} d^{3} x\right )} e^{3} + {\left (21 \, c^{3} d^{4} x^{2} + 4 \, a c^{2} d^{4}\right )} e^{2}}{140 \, {\left (x^{7} e^{11} + 7 \, d x^{6} e^{10} + 21 \, d^{2} x^{5} e^{9} + 35 \, d^{3} x^{4} e^{8} + 35 \, d^{4} x^{3} e^{7} + 21 \, d^{5} x^{2} e^{6} + 7 \, d^{6} x e^{5} + d^{7} 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)^11,x, algorithm="fricas")

[Out]

-1/140*(7*c^3*d^5*x*e + c^3*d^6 + 70*a^2*c*d*x*e^5 + 20*a^3*e^6 + 2*(42*a*c^2*d^2*x^2 + 5*a^2*c*d^2)*e^4 + 7*(
5*c^3*d^3*x^3 + 4*a*c^2*d^3*x)*e^3 + (21*c^3*d^4*x^2 + 4*a*c^2*d^4)*e^2)/(x^7*e^11 + 7*d*x^6*e^10 + 21*d^2*x^5
*e^9 + 35*d^3*x^4*e^8 + 35*d^4*x^3*e^7 + 21*d^5*x^2*e^6 + 7*d^6*x*e^5 + d^7*e^4)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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)**11,x)

[Out]

Timed out

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Giac [A]
time = 2.47, size = 122, normalized size = 1.10 \begin {gather*} -\frac {{\left (35 \, c^{3} d^{3} x^{3} e^{3} + 21 \, c^{3} d^{4} x^{2} e^{2} + 7 \, c^{3} d^{5} x e + c^{3} d^{6} + 84 \, a c^{2} d^{2} x^{2} e^{4} + 28 \, a c^{2} d^{3} x e^{3} + 4 \, a c^{2} d^{4} e^{2} + 70 \, a^{2} c d x e^{5} + 10 \, a^{2} c d^{2} e^{4} + 20 \, a^{3} e^{6}\right )} e^{\left (-4\right )}}{140 \, {\left (x e + d\right )}^{7}} \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)^11,x, algorithm="giac")

[Out]

-1/140*(35*c^3*d^3*x^3*e^3 + 21*c^3*d^4*x^2*e^2 + 7*c^3*d^5*x*e + c^3*d^6 + 84*a*c^2*d^2*x^2*e^4 + 28*a*c^2*d^
3*x*e^3 + 4*a*c^2*d^4*e^2 + 70*a^2*c*d*x*e^5 + 10*a^2*c*d^2*e^4 + 20*a^3*e^6)*e^(-4)/(x*e + d)^7

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Mupad [B]
time = 0.08, size = 195, normalized size = 1.76 \begin {gather*} -\frac {\frac {20\,a^3\,e^6+10\,a^2\,c\,d^2\,e^4+4\,a\,c^2\,d^4\,e^2+c^3\,d^6}{140\,e^4}+\frac {c^3\,d^3\,x^3}{4\,e}+\frac {c\,d\,x\,\left (10\,a^2\,e^4+4\,a\,c\,d^2\,e^2+c^2\,d^4\right )}{20\,e^3}+\frac {3\,c^2\,d^2\,x^2\,\left (c\,d^2+4\,a\,e^2\right )}{20\,e^2}}{d^7+7\,d^6\,e\,x+21\,d^5\,e^2\,x^2+35\,d^4\,e^3\,x^3+35\,d^3\,e^4\,x^4+21\,d^2\,e^5\,x^5+7\,d\,e^6\,x^6+e^7\,x^7} \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)^11,x)

[Out]

-((20*a^3*e^6 + c^3*d^6 + 4*a*c^2*d^4*e^2 + 10*a^2*c*d^2*e^4)/(140*e^4) + (c^3*d^3*x^3)/(4*e) + (c*d*x*(10*a^2
*e^4 + c^2*d^4 + 4*a*c*d^2*e^2))/(20*e^3) + (3*c^2*d^2*x^2*(4*a*e^2 + c*d^2))/(20*e^2))/(d^7 + e^7*x^7 + 7*d*e
^6*x^6 + 21*d^5*e^2*x^2 + 35*d^4*e^3*x^3 + 35*d^3*e^4*x^4 + 21*d^2*e^5*x^5 + 7*d^6*e*x)

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