3.16.71 \(\int \frac {(d+e x)^9}{(a d e+(c d^2+a e^2) x+c d e x^2)^3} \, dx\)

Optimal. Leaf size=221 \[ \frac {e^6 (a e+c d x)^4}{4 c^7 d^7}-\frac {6 e \left (c d^2-a e^2\right )^5}{c^7 d^7 (a e+c d x)}-\frac {\left (c d^2-a e^2\right )^6}{2 c^7 d^7 (a e+c d x)^2}+\frac {15 e^2 \left (c d^2-a e^2\right )^4 \log (a e+c d x)}{c^7 d^7}+\frac {2 e^5 \left (c d^2-a e^2\right ) (a e+c d x)^3}{c^7 d^7}+\frac {15 e^4 \left (c d^2-a e^2\right )^2 (a e+c d x)^2}{2 c^7 d^7}+\frac {20 e^3 x \left (c d^2-a e^2\right )^3}{c^6 d^6} \]

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Rubi [A]  time = 0.26, antiderivative size = 221, 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 {e^6 (a e+c d x)^4}{4 c^7 d^7}+\frac {2 e^5 \left (c d^2-a e^2\right ) (a e+c d x)^3}{c^7 d^7}+\frac {15 e^4 \left (c d^2-a e^2\right )^2 (a e+c d x)^2}{2 c^7 d^7}+\frac {20 e^3 x \left (c d^2-a e^2\right )^3}{c^6 d^6}-\frac {6 e \left (c d^2-a e^2\right )^5}{c^7 d^7 (a e+c d x)}-\frac {\left (c d^2-a e^2\right )^6}{2 c^7 d^7 (a e+c d x)^2}+\frac {15 e^2 \left (c d^2-a e^2\right )^4 \log (a e+c d x)}{c^7 d^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^9/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3,x]

[Out]

(20*e^3*(c*d^2 - a*e^2)^3*x)/(c^6*d^6) - (c*d^2 - a*e^2)^6/(2*c^7*d^7*(a*e + c*d*x)^2) - (6*e*(c*d^2 - a*e^2)^
5)/(c^7*d^7*(a*e + c*d*x)) + (15*e^4*(c*d^2 - a*e^2)^2*(a*e + c*d*x)^2)/(2*c^7*d^7) + (2*e^5*(c*d^2 - a*e^2)*(
a*e + c*d*x)^3)/(c^7*d^7) + (e^6*(a*e + c*d*x)^4)/(4*c^7*d^7) + (15*e^2*(c*d^2 - a*e^2)^4*Log[a*e + c*d*x])/(c
^7*d^7)

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 {(d+e x)^9}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx &=\int \frac {(d+e x)^6}{(a e+c d x)^3} \, dx\\ &=\int \left (\frac {20 \left (c d^2 e-a e^3\right )^3}{c^6 d^6}+\frac {\left (c d^2-a e^2\right )^6}{c^6 d^6 (a e+c d x)^3}+\frac {6 e \left (c d^2-a e^2\right )^5}{c^6 d^6 (a e+c d x)^2}+\frac {15 e^2 \left (c d^2-a e^2\right )^4}{c^6 d^6 (a e+c d x)}+\frac {15 e^4 \left (c d^2-a e^2\right )^2 (a e+c d x)}{c^6 d^6}+\frac {6 \left (c d^2 e^5-a e^7\right ) (a e+c d x)^2}{c^6 d^6}+\frac {e^6 (a e+c d x)^3}{c^6 d^6}\right ) \, dx\\ &=\frac {20 e^3 \left (c d^2-a e^2\right )^3 x}{c^6 d^6}-\frac {\left (c d^2-a e^2\right )^6}{2 c^7 d^7 (a e+c d x)^2}-\frac {6 e \left (c d^2-a e^2\right )^5}{c^7 d^7 (a e+c d x)}+\frac {15 e^4 \left (c d^2-a e^2\right )^2 (a e+c d x)^2}{2 c^7 d^7}+\frac {2 e^5 \left (c d^2-a e^2\right ) (a e+c d x)^3}{c^7 d^7}+\frac {e^6 (a e+c d x)^4}{4 c^7 d^7}+\frac {15 e^2 \left (c d^2-a e^2\right )^4 \log (a e+c d x)}{c^7 d^7}\\ \end {align*}

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Mathematica [A]  time = 0.13, size = 337, normalized size = 1.52 \begin {gather*} \frac {22 a^6 e^{12}-4 a^5 c d e^{10} (27 d+4 e x)+2 a^4 c^2 d^2 e^8 \left (105 d^2+12 d e x-34 e^2 x^2\right )-4 a^3 c^3 d^3 e^6 \left (50 d^3-15 d^2 e x-63 d e^2 x^2+5 e^3 x^3\right )+5 a^2 c^4 d^4 e^4 \left (18 d^4-32 d^3 e x-66 d^2 e^2 x^2+16 d e^3 x^3+e^4 x^4\right )-2 a c^5 d^5 e^2 \left (6 d^5-60 d^4 e x-80 d^3 e^2 x^2+60 d^2 e^3 x^3+10 d e^4 x^4+e^5 x^5\right )+60 e^2 \left (c d^2-a e^2\right )^4 (a e+c d x)^2 \log (a e+c d x)+c^6 d^6 \left (-2 d^6-24 d^5 e x+80 d^3 e^3 x^3+30 d^2 e^4 x^4+8 d e^5 x^5+e^6 x^6\right )}{4 c^7 d^7 (a e+c d x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^9/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3,x]

[Out]

(22*a^6*e^12 - 4*a^5*c*d*e^10*(27*d + 4*e*x) + 2*a^4*c^2*d^2*e^8*(105*d^2 + 12*d*e*x - 34*e^2*x^2) - 4*a^3*c^3
*d^3*e^6*(50*d^3 - 15*d^2*e*x - 63*d*e^2*x^2 + 5*e^3*x^3) + 5*a^2*c^4*d^4*e^4*(18*d^4 - 32*d^3*e*x - 66*d^2*e^
2*x^2 + 16*d*e^3*x^3 + e^4*x^4) - 2*a*c^5*d^5*e^2*(6*d^5 - 60*d^4*e*x - 80*d^3*e^2*x^2 + 60*d^2*e^3*x^3 + 10*d
*e^4*x^4 + e^5*x^5) + c^6*d^6*(-2*d^6 - 24*d^5*e*x + 80*d^3*e^3*x^3 + 30*d^2*e^4*x^4 + 8*d*e^5*x^5 + e^6*x^6)
+ 60*e^2*(c*d^2 - a*e^2)^4*(a*e + c*d*x)^2*Log[a*e + c*d*x])/(4*c^7*d^7*(a*e + c*d*x)^2)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(d+e x)^9}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(d + e*x)^9/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3,x]

[Out]

IntegrateAlgebraic[(d + e*x)^9/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3, x]

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fricas [B]  time = 0.40, size = 606, normalized size = 2.74 \begin {gather*} \frac {c^{6} d^{6} e^{6} x^{6} - 2 \, c^{6} d^{12} - 12 \, a c^{5} d^{10} e^{2} + 90 \, a^{2} c^{4} d^{8} e^{4} - 200 \, a^{3} c^{3} d^{6} e^{6} + 210 \, a^{4} c^{2} d^{4} e^{8} - 108 \, a^{5} c d^{2} e^{10} + 22 \, a^{6} e^{12} + 2 \, {\left (4 \, c^{6} d^{7} e^{5} - a c^{5} d^{5} e^{7}\right )} x^{5} + 5 \, {\left (6 \, c^{6} d^{8} e^{4} - 4 \, a c^{5} d^{6} e^{6} + a^{2} c^{4} d^{4} e^{8}\right )} x^{4} + 20 \, {\left (4 \, c^{6} d^{9} e^{3} - 6 \, a c^{5} d^{7} e^{5} + 4 \, a^{2} c^{4} d^{5} e^{7} - a^{3} c^{3} d^{3} e^{9}\right )} x^{3} + 2 \, {\left (80 \, a c^{5} d^{8} e^{4} - 165 \, a^{2} c^{4} d^{6} e^{6} + 126 \, a^{3} c^{3} d^{4} e^{8} - 34 \, a^{4} c^{2} d^{2} e^{10}\right )} x^{2} - 4 \, {\left (6 \, c^{6} d^{11} e - 30 \, a c^{5} d^{9} e^{3} + 40 \, a^{2} c^{4} d^{7} e^{5} - 15 \, a^{3} c^{3} d^{5} e^{7} - 6 \, a^{4} c^{2} d^{3} e^{9} + 4 \, a^{5} c d e^{11}\right )} x + 60 \, {\left (a^{2} c^{4} d^{8} e^{4} - 4 \, a^{3} c^{3} d^{6} e^{6} + 6 \, a^{4} c^{2} d^{4} e^{8} - 4 \, a^{5} c d^{2} e^{10} + a^{6} e^{12} + {\left (c^{6} d^{10} e^{2} - 4 \, a c^{5} d^{8} e^{4} + 6 \, a^{2} c^{4} d^{6} e^{6} - 4 \, a^{3} c^{3} d^{4} e^{8} + a^{4} c^{2} d^{2} e^{10}\right )} x^{2} + 2 \, {\left (a c^{5} d^{9} e^{3} - 4 \, a^{2} c^{4} d^{7} e^{5} + 6 \, a^{3} c^{3} d^{5} e^{7} - 4 \, a^{4} c^{2} d^{3} e^{9} + a^{5} c d e^{11}\right )} x\right )} \log \left (c d x + a e\right )}{4 \, {\left (c^{9} d^{9} x^{2} + 2 \, a c^{8} d^{8} e x + a^{2} c^{7} d^{7} e^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^9/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x, algorithm="fricas")

[Out]

1/4*(c^6*d^6*e^6*x^6 - 2*c^6*d^12 - 12*a*c^5*d^10*e^2 + 90*a^2*c^4*d^8*e^4 - 200*a^3*c^3*d^6*e^6 + 210*a^4*c^2
*d^4*e^8 - 108*a^5*c*d^2*e^10 + 22*a^6*e^12 + 2*(4*c^6*d^7*e^5 - a*c^5*d^5*e^7)*x^5 + 5*(6*c^6*d^8*e^4 - 4*a*c
^5*d^6*e^6 + a^2*c^4*d^4*e^8)*x^4 + 20*(4*c^6*d^9*e^3 - 6*a*c^5*d^7*e^5 + 4*a^2*c^4*d^5*e^7 - a^3*c^3*d^3*e^9)
*x^3 + 2*(80*a*c^5*d^8*e^4 - 165*a^2*c^4*d^6*e^6 + 126*a^3*c^3*d^4*e^8 - 34*a^4*c^2*d^2*e^10)*x^2 - 4*(6*c^6*d
^11*e - 30*a*c^5*d^9*e^3 + 40*a^2*c^4*d^7*e^5 - 15*a^3*c^3*d^5*e^7 - 6*a^4*c^2*d^3*e^9 + 4*a^5*c*d*e^11)*x + 6
0*(a^2*c^4*d^8*e^4 - 4*a^3*c^3*d^6*e^6 + 6*a^4*c^2*d^4*e^8 - 4*a^5*c*d^2*e^10 + a^6*e^12 + (c^6*d^10*e^2 - 4*a
*c^5*d^8*e^4 + 6*a^2*c^4*d^6*e^6 - 4*a^3*c^3*d^4*e^8 + a^4*c^2*d^2*e^10)*x^2 + 2*(a*c^5*d^9*e^3 - 4*a^2*c^4*d^
7*e^5 + 6*a^3*c^3*d^5*e^7 - 4*a^4*c^2*d^3*e^9 + a^5*c*d*e^11)*x)*log(c*d*x + a*e))/(c^9*d^9*x^2 + 2*a*c^8*d^8*
e*x + a^2*c^7*d^7*e^2)

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giac [B]  time = 15.08, size = 1068, normalized size = 4.83 \begin {gather*} \frac {15 \, {\left (c^{9} d^{18} e^{2} - 9 \, a c^{8} d^{16} e^{4} + 36 \, a^{2} c^{7} d^{14} e^{6} - 84 \, a^{3} c^{6} d^{12} e^{8} + 126 \, a^{4} c^{5} d^{10} e^{10} - 126 \, a^{5} c^{4} d^{8} e^{12} + 84 \, a^{6} c^{3} d^{6} e^{14} - 36 \, a^{7} c^{2} d^{4} e^{16} + 9 \, a^{8} c d^{2} e^{18} - a^{9} e^{20}\right )} \arctan \left (\frac {2 \, c d x e + c d^{2} + a e^{2}}{\sqrt {-c^{2} d^{4} + 2 \, a c d^{2} e^{2} - a^{2} e^{4}}}\right )}{{\left (c^{11} d^{15} - 4 \, a c^{10} d^{13} e^{2} + 6 \, a^{2} c^{9} d^{11} e^{4} - 4 \, a^{3} c^{8} d^{9} e^{6} + a^{4} c^{7} d^{7} e^{8}\right )} \sqrt {-c^{2} d^{4} + 2 \, a c d^{2} e^{2} - a^{2} e^{4}}} + \frac {15 \, {\left (c^{4} d^{8} e^{2} - 4 \, a c^{3} d^{6} e^{4} + 6 \, a^{2} c^{2} d^{4} e^{6} - 4 \, a^{3} c d^{2} e^{8} + a^{4} e^{10}\right )} \log \left (c d x^{2} e + c d^{2} x + a x e^{2} + a d e\right )}{2 \, c^{7} d^{7}} - \frac {c^{10} d^{22} + 2 \, a c^{9} d^{20} e^{2} - 63 \, a^{2} c^{8} d^{18} e^{4} + 312 \, a^{3} c^{7} d^{16} e^{6} - 798 \, a^{4} c^{6} d^{14} e^{8} + 1260 \, a^{5} c^{5} d^{12} e^{10} - 1302 \, a^{6} c^{4} d^{10} e^{12} + 888 \, a^{7} c^{3} d^{8} e^{14} - 387 \, a^{8} c^{2} d^{6} e^{16} + 98 \, a^{9} c d^{4} e^{18} - 11 \, a^{10} d^{2} e^{20} + 12 \, {\left (c^{10} d^{19} e^{3} - 9 \, a c^{9} d^{17} e^{5} + 36 \, a^{2} c^{8} d^{15} e^{7} - 84 \, a^{3} c^{7} d^{13} e^{9} + 126 \, a^{4} c^{6} d^{11} e^{11} - 126 \, a^{5} c^{5} d^{9} e^{13} + 84 \, a^{6} c^{4} d^{7} e^{15} - 36 \, a^{7} c^{3} d^{5} e^{17} + 9 \, a^{8} c^{2} d^{3} e^{19} - a^{9} c d e^{21}\right )} x^{3} + {\left (25 \, c^{10} d^{20} e^{2} - 214 \, a c^{9} d^{18} e^{4} + 801 \, a^{2} c^{8} d^{16} e^{6} - 1704 \, a^{3} c^{7} d^{14} e^{8} + 2226 \, a^{4} c^{6} d^{12} e^{10} - 1764 \, a^{5} c^{5} d^{10} e^{12} + 714 \, a^{6} c^{4} d^{8} e^{14} + 24 \, a^{7} c^{3} d^{6} e^{16} - 171 \, a^{8} c^{2} d^{4} e^{18} + 74 \, a^{9} c d^{2} e^{20} - 11 \, a^{10} e^{22}\right )} x^{2} + 2 \, {\left (7 \, c^{10} d^{21} e - 52 \, a c^{9} d^{19} e^{3} + 153 \, a^{2} c^{8} d^{17} e^{5} - 192 \, a^{3} c^{7} d^{15} e^{7} - 42 \, a^{4} c^{6} d^{13} e^{9} + 504 \, a^{5} c^{5} d^{11} e^{11} - 798 \, a^{6} c^{4} d^{9} e^{13} + 672 \, a^{7} c^{3} d^{7} e^{15} - 333 \, a^{8} c^{2} d^{5} e^{17} + 92 \, a^{9} c d^{3} e^{19} - 11 \, a^{10} d e^{21}\right )} x}{2 \, {\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )}^{2} {\left (c d x^{2} e + c d^{2} x + a x e^{2} + a d e\right )}^{2} c^{7} d^{7}} + \frac {{\left (c^{9} d^{9} x^{4} e^{18} + 8 \, c^{9} d^{10} x^{3} e^{17} + 30 \, c^{9} d^{11} x^{2} e^{16} + 80 \, c^{9} d^{12} x e^{15} - 4 \, a c^{8} d^{8} x^{3} e^{19} - 36 \, a c^{8} d^{9} x^{2} e^{18} - 180 \, a c^{8} d^{10} x e^{17} + 12 \, a^{2} c^{7} d^{7} x^{2} e^{20} + 144 \, a^{2} c^{7} d^{8} x e^{19} - 40 \, a^{3} c^{6} d^{6} x e^{21}\right )} e^{\left (-12\right )}}{4 \, c^{12} d^{12}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^9/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x, algorithm="giac")

[Out]

15*(c^9*d^18*e^2 - 9*a*c^8*d^16*e^4 + 36*a^2*c^7*d^14*e^6 - 84*a^3*c^6*d^12*e^8 + 126*a^4*c^5*d^10*e^10 - 126*
a^5*c^4*d^8*e^12 + 84*a^6*c^3*d^6*e^14 - 36*a^7*c^2*d^4*e^16 + 9*a^8*c*d^2*e^18 - a^9*e^20)*arctan((2*c*d*x*e
+ c*d^2 + a*e^2)/sqrt(-c^2*d^4 + 2*a*c*d^2*e^2 - a^2*e^4))/((c^11*d^15 - 4*a*c^10*d^13*e^2 + 6*a^2*c^9*d^11*e^
4 - 4*a^3*c^8*d^9*e^6 + a^4*c^7*d^7*e^8)*sqrt(-c^2*d^4 + 2*a*c*d^2*e^2 - a^2*e^4)) + 15/2*(c^4*d^8*e^2 - 4*a*c
^3*d^6*e^4 + 6*a^2*c^2*d^4*e^6 - 4*a^3*c*d^2*e^8 + a^4*e^10)*log(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e)/(c^7*d
^7) - 1/2*(c^10*d^22 + 2*a*c^9*d^20*e^2 - 63*a^2*c^8*d^18*e^4 + 312*a^3*c^7*d^16*e^6 - 798*a^4*c^6*d^14*e^8 +
1260*a^5*c^5*d^12*e^10 - 1302*a^6*c^4*d^10*e^12 + 888*a^7*c^3*d^8*e^14 - 387*a^8*c^2*d^6*e^16 + 98*a^9*c*d^4*e
^18 - 11*a^10*d^2*e^20 + 12*(c^10*d^19*e^3 - 9*a*c^9*d^17*e^5 + 36*a^2*c^8*d^15*e^7 - 84*a^3*c^7*d^13*e^9 + 12
6*a^4*c^6*d^11*e^11 - 126*a^5*c^5*d^9*e^13 + 84*a^6*c^4*d^7*e^15 - 36*a^7*c^3*d^5*e^17 + 9*a^8*c^2*d^3*e^19 -
a^9*c*d*e^21)*x^3 + (25*c^10*d^20*e^2 - 214*a*c^9*d^18*e^4 + 801*a^2*c^8*d^16*e^6 - 1704*a^3*c^7*d^14*e^8 + 22
26*a^4*c^6*d^12*e^10 - 1764*a^5*c^5*d^10*e^12 + 714*a^6*c^4*d^8*e^14 + 24*a^7*c^3*d^6*e^16 - 171*a^8*c^2*d^4*e
^18 + 74*a^9*c*d^2*e^20 - 11*a^10*e^22)*x^2 + 2*(7*c^10*d^21*e - 52*a*c^9*d^19*e^3 + 153*a^2*c^8*d^17*e^5 - 19
2*a^3*c^7*d^15*e^7 - 42*a^4*c^6*d^13*e^9 + 504*a^5*c^5*d^11*e^11 - 798*a^6*c^4*d^9*e^13 + 672*a^7*c^3*d^7*e^15
- 333*a^8*c^2*d^5*e^17 + 92*a^9*c*d^3*e^19 - 11*a^10*d*e^21)*x)/((c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)^2*(c*d*x
^2*e + c*d^2*x + a*x*e^2 + a*d*e)^2*c^7*d^7) + 1/4*(c^9*d^9*x^4*e^18 + 8*c^9*d^10*x^3*e^17 + 30*c^9*d^11*x^2*e
^16 + 80*c^9*d^12*x*e^15 - 4*a*c^8*d^8*x^3*e^19 - 36*a*c^8*d^9*x^2*e^18 - 180*a*c^8*d^10*x*e^17 + 12*a^2*c^7*d
^7*x^2*e^20 + 144*a^2*c^7*d^8*x*e^19 - 40*a^3*c^6*d^6*x*e^21)*e^(-12)/(c^12*d^12)

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maple [B]  time = 0.05, size = 544, normalized size = 2.46 \begin {gather*} -\frac {a^{6} e^{12}}{2 \left (c d x +a e \right )^{2} c^{7} d^{7}}+\frac {3 a^{5} e^{10}}{\left (c d x +a e \right )^{2} c^{6} d^{5}}-\frac {15 a^{4} e^{8}}{2 \left (c d x +a e \right )^{2} c^{5} d^{3}}+\frac {10 a^{3} e^{6}}{\left (c d x +a e \right )^{2} c^{4} d}-\frac {15 a^{2} d \,e^{4}}{2 \left (c d x +a e \right )^{2} c^{3}}+\frac {3 a \,d^{3} e^{2}}{\left (c d x +a e \right )^{2} c^{2}}-\frac {d^{5}}{2 \left (c d x +a e \right )^{2} c}+\frac {e^{6} x^{4}}{4 c^{3} d^{3}}-\frac {a \,e^{7} x^{3}}{c^{4} d^{4}}+\frac {2 e^{5} x^{3}}{c^{3} d^{2}}+\frac {6 a^{5} e^{11}}{\left (c d x +a e \right ) c^{7} d^{7}}-\frac {30 a^{4} e^{9}}{\left (c d x +a e \right ) c^{6} d^{5}}+\frac {60 a^{3} e^{7}}{\left (c d x +a e \right ) c^{5} d^{3}}-\frac {60 a^{2} e^{5}}{\left (c d x +a e \right ) c^{4} d}+\frac {3 a^{2} e^{8} x^{2}}{c^{5} d^{5}}+\frac {30 a d \,e^{3}}{\left (c d x +a e \right ) c^{3}}-\frac {9 a \,e^{6} x^{2}}{c^{4} d^{3}}-\frac {6 d^{3} e}{\left (c d x +a e \right ) c^{2}}+\frac {15 e^{4} x^{2}}{2 c^{3} d}+\frac {15 a^{4} e^{10} \ln \left (c d x +a e \right )}{c^{7} d^{7}}-\frac {60 a^{3} e^{8} \ln \left (c d x +a e \right )}{c^{6} d^{5}}-\frac {10 a^{3} e^{9} x}{c^{6} d^{6}}+\frac {90 a^{2} e^{6} \ln \left (c d x +a e \right )}{c^{5} d^{3}}+\frac {36 a^{2} e^{7} x}{c^{5} d^{4}}-\frac {60 a \,e^{4} \ln \left (c d x +a e \right )}{c^{4} d}-\frac {45 a \,e^{5} x}{c^{4} d^{2}}+\frac {15 d \,e^{2} \ln \left (c d x +a e \right )}{c^{3}}+\frac {20 e^{3} x}{c^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^9/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x)

[Out]

-e^7/c^4/d^4*x^3*a+3*e^8/c^5/d^5*x^2*a^2-9*e^6/c^4/d^3*x^2*a+6/d^7/c^7*e^11/(c*d*x+a*e)*a^5-30/d^5/c^6*e^9/(c*
d*x+a*e)*a^4+20*e^3/c^3*x-1/2/c*d^5/(c*d*x+a*e)^2+15/c^3*d*e^2*ln(c*d*x+a*e)-6*d^3/c^2*e/(c*d*x+a*e)+1/4*e^6/c
^3/d^3*x^4+2*e^5/c^3/d^2*x^3+15/2*e^4/c^3/d*x^2+15/c^7/d^7*e^10*ln(c*d*x+a*e)*a^4-60/c^6/d^5*e^8*ln(c*d*x+a*e)
*a^3+60/d^3/c^5*e^7/(c*d*x+a*e)*a^3-60/d/c^4*e^5/(c*d*x+a*e)*a^2+30*d/c^3*e^3/(c*d*x+a*e)*a+90/c^5/d^3*e^6*ln(
c*d*x+a*e)*a^2-60/c^4/d*e^4*ln(c*d*x+a*e)*a-10*e^9/c^6/d^6*a^3*x+36*e^7/c^5/d^4*a^2*x-45*e^5/c^4/d^2*a*x-1/2/c
^7/d^7/(c*d*x+a*e)^2*a^6*e^12+3/c^6/d^5/(c*d*x+a*e)^2*a^5*e^10-15/2/c^5/d^3/(c*d*x+a*e)^2*a^4*e^8+10/c^4/d/(c*
d*x+a*e)^2*a^3*e^6-15/2/c^3*d/(c*d*x+a*e)^2*a^2*e^4+3/c^2*d^3/(c*d*x+a*e)^2*a*e^2

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maxima [A]  time = 1.20, size = 408, normalized size = 1.85 \begin {gather*} -\frac {c^{6} d^{12} + 6 \, a c^{5} d^{10} e^{2} - 45 \, a^{2} c^{4} d^{8} e^{4} + 100 \, a^{3} c^{3} d^{6} e^{6} - 105 \, a^{4} c^{2} d^{4} e^{8} + 54 \, a^{5} c d^{2} e^{10} - 11 \, a^{6} e^{12} + 12 \, {\left (c^{6} d^{11} e - 5 \, a c^{5} d^{9} e^{3} + 10 \, a^{2} c^{4} d^{7} e^{5} - 10 \, a^{3} c^{3} d^{5} e^{7} + 5 \, a^{4} c^{2} d^{3} e^{9} - a^{5} c d e^{11}\right )} x}{2 \, {\left (c^{9} d^{9} x^{2} + 2 \, a c^{8} d^{8} e x + a^{2} c^{7} d^{7} e^{2}\right )}} + \frac {c^{3} d^{3} e^{6} x^{4} + 4 \, {\left (2 \, c^{3} d^{4} e^{5} - a c^{2} d^{2} e^{7}\right )} x^{3} + 6 \, {\left (5 \, c^{3} d^{5} e^{4} - 6 \, a c^{2} d^{3} e^{6} + 2 \, a^{2} c d e^{8}\right )} x^{2} + 4 \, {\left (20 \, c^{3} d^{6} e^{3} - 45 \, a c^{2} d^{4} e^{5} + 36 \, a^{2} c d^{2} e^{7} - 10 \, a^{3} e^{9}\right )} x}{4 \, c^{6} d^{6}} + \frac {15 \, {\left (c^{4} d^{8} e^{2} - 4 \, a c^{3} d^{6} e^{4} + 6 \, a^{2} c^{2} d^{4} e^{6} - 4 \, a^{3} c d^{2} e^{8} + a^{4} e^{10}\right )} \log \left (c d x + a e\right )}{c^{7} d^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^9/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x, algorithm="maxima")

[Out]

-1/2*(c^6*d^12 + 6*a*c^5*d^10*e^2 - 45*a^2*c^4*d^8*e^4 + 100*a^3*c^3*d^6*e^6 - 105*a^4*c^2*d^4*e^8 + 54*a^5*c*
d^2*e^10 - 11*a^6*e^12 + 12*(c^6*d^11*e - 5*a*c^5*d^9*e^3 + 10*a^2*c^4*d^7*e^5 - 10*a^3*c^3*d^5*e^7 + 5*a^4*c^
2*d^3*e^9 - a^5*c*d*e^11)*x)/(c^9*d^9*x^2 + 2*a*c^8*d^8*e*x + a^2*c^7*d^7*e^2) + 1/4*(c^3*d^3*e^6*x^4 + 4*(2*c
^3*d^4*e^5 - a*c^2*d^2*e^7)*x^3 + 6*(5*c^3*d^5*e^4 - 6*a*c^2*d^3*e^6 + 2*a^2*c*d*e^8)*x^2 + 4*(20*c^3*d^6*e^3
- 45*a*c^2*d^4*e^5 + 36*a^2*c*d^2*e^7 - 10*a^3*e^9)*x)/(c^6*d^6) + 15*(c^4*d^8*e^2 - 4*a*c^3*d^6*e^4 + 6*a^2*c
^2*d^4*e^6 - 4*a^3*c*d^2*e^8 + a^4*e^10)*log(c*d*x + a*e)/(c^7*d^7)

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mupad [B]  time = 0.13, size = 516, normalized size = 2.33 \begin {gather*} x^3\,\left (\frac {2\,e^5}{c^3\,d^2}-\frac {a\,e^7}{c^4\,d^4}\right )-x^2\,\left (\frac {3\,a^2\,e^8}{2\,c^5\,d^5}-\frac {15\,e^4}{2\,c^3\,d}+\frac {3\,a\,e\,\left (\frac {6\,e^5}{c^3\,d^2}-\frac {3\,a\,e^7}{c^4\,d^4}\right )}{2\,c\,d}\right )+\frac {x\,\left (6\,a^5\,e^{11}-30\,a^4\,c\,d^2\,e^9+60\,a^3\,c^2\,d^4\,e^7-60\,a^2\,c^3\,d^6\,e^5+30\,a\,c^4\,d^8\,e^3-6\,c^5\,d^{10}\,e\right )-\frac {-11\,a^6\,e^{12}+54\,a^5\,c\,d^2\,e^{10}-105\,a^4\,c^2\,d^4\,e^8+100\,a^3\,c^3\,d^6\,e^6-45\,a^2\,c^4\,d^8\,e^4+6\,a\,c^5\,d^{10}\,e^2+c^6\,d^{12}}{2\,c\,d}}{a^2\,c^6\,d^6\,e^2+2\,a\,c^7\,d^7\,e\,x+c^8\,d^8\,x^2}+x\,\left (\frac {20\,e^3}{c^3}-\frac {a^3\,e^9}{c^6\,d^6}-\frac {3\,a^2\,e^2\,\left (\frac {6\,e^5}{c^3\,d^2}-\frac {3\,a\,e^7}{c^4\,d^4}\right )}{c^2\,d^2}+\frac {3\,a\,e\,\left (\frac {3\,a^2\,e^8}{c^5\,d^5}-\frac {15\,e^4}{c^3\,d}+\frac {3\,a\,e\,\left (\frac {6\,e^5}{c^3\,d^2}-\frac {3\,a\,e^7}{c^4\,d^4}\right )}{c\,d}\right )}{c\,d}\right )+\frac {e^6\,x^4}{4\,c^3\,d^3}+\frac {\ln \left (a\,e+c\,d\,x\right )\,\left (15\,a^4\,e^{10}-60\,a^3\,c\,d^2\,e^8+90\,a^2\,c^2\,d^4\,e^6-60\,a\,c^3\,d^6\,e^4+15\,c^4\,d^8\,e^2\right )}{c^7\,d^7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^9/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^3,x)

[Out]

x^3*((2*e^5)/(c^3*d^2) - (a*e^7)/(c^4*d^4)) - x^2*((3*a^2*e^8)/(2*c^5*d^5) - (15*e^4)/(2*c^3*d) + (3*a*e*((6*e
^5)/(c^3*d^2) - (3*a*e^7)/(c^4*d^4)))/(2*c*d)) + (x*(6*a^5*e^11 - 6*c^5*d^10*e + 30*a*c^4*d^8*e^3 - 30*a^4*c*d
^2*e^9 - 60*a^2*c^3*d^6*e^5 + 60*a^3*c^2*d^4*e^7) - (c^6*d^12 - 11*a^6*e^12 + 6*a*c^5*d^10*e^2 + 54*a^5*c*d^2*
e^10 - 45*a^2*c^4*d^8*e^4 + 100*a^3*c^3*d^6*e^6 - 105*a^4*c^2*d^4*e^8)/(2*c*d))/(c^8*d^8*x^2 + a^2*c^6*d^6*e^2
+ 2*a*c^7*d^7*e*x) + x*((20*e^3)/c^3 - (a^3*e^9)/(c^6*d^6) - (3*a^2*e^2*((6*e^5)/(c^3*d^2) - (3*a*e^7)/(c^4*d
^4)))/(c^2*d^2) + (3*a*e*((3*a^2*e^8)/(c^5*d^5) - (15*e^4)/(c^3*d) + (3*a*e*((6*e^5)/(c^3*d^2) - (3*a*e^7)/(c^
4*d^4)))/(c*d)))/(c*d)) + (e^6*x^4)/(4*c^3*d^3) + (log(a*e + c*d*x)*(15*a^4*e^10 + 15*c^4*d^8*e^2 - 60*a*c^3*d
^6*e^4 - 60*a^3*c*d^2*e^8 + 90*a^2*c^2*d^4*e^6))/(c^7*d^7)

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sympy [A]  time = 7.53, size = 386, normalized size = 1.75 \begin {gather*} x^{3} \left (- \frac {a e^{7}}{c^{4} d^{4}} + \frac {2 e^{5}}{c^{3} d^{2}}\right ) + x^{2} \left (\frac {3 a^{2} e^{8}}{c^{5} d^{5}} - \frac {9 a e^{6}}{c^{4} d^{3}} + \frac {15 e^{4}}{2 c^{3} d}\right ) + x \left (- \frac {10 a^{3} e^{9}}{c^{6} d^{6}} + \frac {36 a^{2} e^{7}}{c^{5} d^{4}} - \frac {45 a e^{5}}{c^{4} d^{2}} + \frac {20 e^{3}}{c^{3}}\right ) + \frac {11 a^{6} e^{12} - 54 a^{5} c d^{2} e^{10} + 105 a^{4} c^{2} d^{4} e^{8} - 100 a^{3} c^{3} d^{6} e^{6} + 45 a^{2} c^{4} d^{8} e^{4} - 6 a c^{5} d^{10} e^{2} - c^{6} d^{12} + x \left (12 a^{5} c d e^{11} - 60 a^{4} c^{2} d^{3} e^{9} + 120 a^{3} c^{3} d^{5} e^{7} - 120 a^{2} c^{4} d^{7} e^{5} + 60 a c^{5} d^{9} e^{3} - 12 c^{6} d^{11} e\right )}{2 a^{2} c^{7} d^{7} e^{2} + 4 a c^{8} d^{8} e x + 2 c^{9} d^{9} x^{2}} + \frac {e^{6} x^{4}}{4 c^{3} d^{3}} + \frac {15 e^{2} \left (a e^{2} - c d^{2}\right )^{4} \log {\left (a e + c d x \right )}}{c^{7} d^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**9/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**3,x)

[Out]

x**3*(-a*e**7/(c**4*d**4) + 2*e**5/(c**3*d**2)) + x**2*(3*a**2*e**8/(c**5*d**5) - 9*a*e**6/(c**4*d**3) + 15*e*
*4/(2*c**3*d)) + x*(-10*a**3*e**9/(c**6*d**6) + 36*a**2*e**7/(c**5*d**4) - 45*a*e**5/(c**4*d**2) + 20*e**3/c**
3) + (11*a**6*e**12 - 54*a**5*c*d**2*e**10 + 105*a**4*c**2*d**4*e**8 - 100*a**3*c**3*d**6*e**6 + 45*a**2*c**4*
d**8*e**4 - 6*a*c**5*d**10*e**2 - c**6*d**12 + x*(12*a**5*c*d*e**11 - 60*a**4*c**2*d**3*e**9 + 120*a**3*c**3*d
**5*e**7 - 120*a**2*c**4*d**7*e**5 + 60*a*c**5*d**9*e**3 - 12*c**6*d**11*e))/(2*a**2*c**7*d**7*e**2 + 4*a*c**8
*d**8*e*x + 2*c**9*d**9*x**2) + e**6*x**4/(4*c**3*d**3) + 15*e**2*(a*e**2 - c*d**2)**4*log(a*e + c*d*x)/(c**7*
d**7)

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