3.1925 \(\int \frac{(a+b x) (a^2+2 a b x+b^2 x^2)^3}{(d+e x)^3} \, dx\)

Optimal. Leaf size=185 \[ -\frac{7 b^6 (d+e x)^4 (b d-a e)}{4 e^8}+\frac{7 b^5 (d+e x)^3 (b d-a e)^2}{e^8}-\frac{35 b^4 (d+e x)^2 (b d-a e)^3}{2 e^8}+\frac{35 b^3 x (b d-a e)^4}{e^7}-\frac{21 b^2 (b d-a e)^5 \log (d+e x)}{e^8}-\frac{7 b (b d-a e)^6}{e^8 (d+e x)}+\frac{(b d-a e)^7}{2 e^8 (d+e x)^2}+\frac{b^7 (d+e x)^5}{5 e^8} \]

[Out]

(35*b^3*(b*d - a*e)^4*x)/e^7 + (b*d - a*e)^7/(2*e^8*(d + e*x)^2) - (7*b*(b*d - a*e)^6)/(e^8*(d + e*x)) - (35*b
^4*(b*d - a*e)^3*(d + e*x)^2)/(2*e^8) + (7*b^5*(b*d - a*e)^2*(d + e*x)^3)/e^8 - (7*b^6*(b*d - a*e)*(d + e*x)^4
)/(4*e^8) + (b^7*(d + e*x)^5)/(5*e^8) - (21*b^2*(b*d - a*e)^5*Log[d + e*x])/e^8

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Rubi [A]  time = 0.22282, antiderivative size = 185, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {27, 43} \[ -\frac{7 b^6 (d+e x)^4 (b d-a e)}{4 e^8}+\frac{7 b^5 (d+e x)^3 (b d-a e)^2}{e^8}-\frac{35 b^4 (d+e x)^2 (b d-a e)^3}{2 e^8}+\frac{35 b^3 x (b d-a e)^4}{e^7}-\frac{21 b^2 (b d-a e)^5 \log (d+e x)}{e^8}-\frac{7 b (b d-a e)^6}{e^8 (d+e x)}+\frac{(b d-a e)^7}{2 e^8 (d+e x)^2}+\frac{b^7 (d+e x)^5}{5 e^8} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(35*b^3*(b*d - a*e)^4*x)/e^7 + (b*d - a*e)^7/(2*e^8*(d + e*x)^2) - (7*b*(b*d - a*e)^6)/(e^8*(d + e*x)) - (35*b
^4*(b*d - a*e)^3*(d + e*x)^2)/(2*e^8) + (7*b^5*(b*d - a*e)^2*(d + e*x)^3)/e^8 - (7*b^6*(b*d - a*e)*(d + e*x)^4
)/(4*e^8) + (b^7*(d + e*x)^5)/(5*e^8) - (21*b^2*(b*d - a*e)^5*Log[d + e*x])/e^8

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 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{(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^3} \, dx &=\int \frac{(a+b x)^7}{(d+e x)^3} \, dx\\ &=\int \left (\frac{35 b^3 (b d-a e)^4}{e^7}+\frac{(-b d+a e)^7}{e^7 (d+e x)^3}+\frac{7 b (b d-a e)^6}{e^7 (d+e x)^2}-\frac{21 b^2 (b d-a e)^5}{e^7 (d+e x)}-\frac{35 b^4 (b d-a e)^3 (d+e x)}{e^7}+\frac{21 b^5 (b d-a e)^2 (d+e x)^2}{e^7}-\frac{7 b^6 (b d-a e) (d+e x)^3}{e^7}+\frac{b^7 (d+e x)^4}{e^7}\right ) \, dx\\ &=\frac{35 b^3 (b d-a e)^4 x}{e^7}+\frac{(b d-a e)^7}{2 e^8 (d+e x)^2}-\frac{7 b (b d-a e)^6}{e^8 (d+e x)}-\frac{35 b^4 (b d-a e)^3 (d+e x)^2}{2 e^8}+\frac{7 b^5 (b d-a e)^2 (d+e x)^3}{e^8}-\frac{7 b^6 (b d-a e) (d+e x)^4}{4 e^8}+\frac{b^7 (d+e x)^5}{5 e^8}-\frac{21 b^2 (b d-a e)^5 \log (d+e x)}{e^8}\\ \end{align*}

Mathematica [B]  time = 0.12941, size = 388, normalized size = 2.1 \[ \frac{70 a^2 b^5 e^2 \left (63 d^3 e^2 x^2+20 d^2 e^3 x^3+6 d^4 e x-27 d^5-5 d e^4 x^4+2 e^5 x^5\right )+350 a^3 b^4 e^3 \left (-11 d^2 e^2 x^2+2 d^3 e x+7 d^4-4 d e^3 x^3+e^4 x^4\right )+350 a^4 b^3 e^4 \left (-4 d^2 e x-5 d^3+4 d e^2 x^2+2 e^3 x^3\right )+210 a^5 b^2 d e^5 (3 d+4 e x)-70 a^6 b e^6 (d+2 e x)-10 a^7 e^7+35 a b^6 e \left (-68 d^4 e^2 x^2-20 d^3 e^3 x^3+5 d^2 e^4 x^4-16 d^5 e x+22 d^6-2 d e^5 x^5+e^6 x^6\right )-420 b^2 (d+e x)^2 (b d-a e)^5 \log (d+e x)+b^7 \left (500 d^5 e^2 x^2+140 d^4 e^3 x^3-35 d^3 e^4 x^4+14 d^2 e^5 x^5+160 d^6 e x-130 d^7-7 d e^6 x^6+4 e^7 x^7\right )}{20 e^8 (d+e x)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-10*a^7*e^7 - 70*a^6*b*e^6*(d + 2*e*x) + 210*a^5*b^2*d*e^5*(3*d + 4*e*x) + 350*a^4*b^3*e^4*(-5*d^3 - 4*d^2*e*
x + 4*d*e^2*x^2 + 2*e^3*x^3) + 350*a^3*b^4*e^3*(7*d^4 + 2*d^3*e*x - 11*d^2*e^2*x^2 - 4*d*e^3*x^3 + e^4*x^4) +
70*a^2*b^5*e^2*(-27*d^5 + 6*d^4*e*x + 63*d^3*e^2*x^2 + 20*d^2*e^3*x^3 - 5*d*e^4*x^4 + 2*e^5*x^5) + 35*a*b^6*e*
(22*d^6 - 16*d^5*e*x - 68*d^4*e^2*x^2 - 20*d^3*e^3*x^3 + 5*d^2*e^4*x^4 - 2*d*e^5*x^5 + e^6*x^6) + b^7*(-130*d^
7 + 160*d^6*e*x + 500*d^5*e^2*x^2 + 140*d^4*e^3*x^3 - 35*d^3*e^4*x^4 + 14*d^2*e^5*x^5 - 7*d*e^6*x^6 + 4*e^7*x^
7) - 420*b^2*(b*d - a*e)^5*(d + e*x)^2*Log[d + e*x])/(20*e^8*(d + e*x)^2)

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Maple [B]  time = 0.013, size = 599, normalized size = 3.2 \begin{align*} 15\,{\frac{{b}^{7}{d}^{4}x}{{e}^{7}}}+{\frac{{b}^{7}{d}^{7}}{2\,{e}^{8} \left ( ex+d \right ) ^{2}}}+{\frac{7\,{b}^{6}{x}^{4}a}{4\,{e}^{3}}}-{\frac{3\,{b}^{7}{x}^{4}d}{4\,{e}^{4}}}+7\,{\frac{{b}^{5}{x}^{3}{a}^{2}}{{e}^{3}}}+2\,{\frac{{b}^{7}{x}^{3}{d}^{2}}{{e}^{5}}}+{\frac{35\,{b}^{4}{x}^{2}{a}^{3}}{2\,{e}^{3}}}-5\,{\frac{{b}^{7}{x}^{2}{d}^{3}}{{e}^{6}}}+35\,{\frac{{a}^{4}{b}^{3}x}{{e}^{3}}}+21\,{\frac{{b}^{2}\ln \left ( ex+d \right ){a}^{5}}{{e}^{3}}}-21\,{\frac{{b}^{7}\ln \left ( ex+d \right ){d}^{5}}{{e}^{8}}}-7\,{\frac{{a}^{6}b}{{e}^{2} \left ( ex+d \right ) }}-7\,{\frac{{b}^{7}{d}^{6}}{{e}^{8} \left ( ex+d \right ) }}+{\frac{{b}^{7}{x}^{5}}{5\,{e}^{3}}}-{\frac{{a}^{7}}{2\,e \left ( ex+d \right ) ^{2}}}-7\,{\frac{{b}^{6}{x}^{3}ad}{{e}^{4}}}-105\,{\frac{{a}^{3}d{b}^{4}x}{{e}^{4}}}-{\frac{63\,{b}^{5}{x}^{2}{a}^{2}d}{2\,{e}^{4}}}-{\frac{21\,{a}^{5}{d}^{2}{b}^{2}}{2\,{e}^{3} \left ( ex+d \right ) ^{2}}}+{\frac{35\,{a}^{4}{d}^{3}{b}^{3}}{2\,{e}^{4} \left ( ex+d \right ) ^{2}}}-{\frac{35\,{a}^{3}{d}^{4}{b}^{4}}{2\,{e}^{5} \left ( ex+d \right ) ^{2}}}+{\frac{21\,{a}^{2}{d}^{5}{b}^{5}}{2\,{e}^{6} \left ( ex+d \right ) ^{2}}}-{\frac{7\,a{d}^{6}{b}^{6}}{2\,{e}^{7} \left ( ex+d \right ) ^{2}}}+21\,{\frac{{b}^{6}{x}^{2}a{d}^{2}}{{e}^{5}}}+126\,{\frac{{a}^{2}{d}^{2}{b}^{5}x}{{e}^{5}}}-70\,{\frac{a{d}^{3}{b}^{6}x}{{e}^{6}}}+{\frac{7\,{a}^{6}db}{2\,{e}^{2} \left ( ex+d \right ) ^{2}}}-105\,{\frac{{b}^{3}\ln \left ( ex+d \right ){a}^{4}d}{{e}^{4}}}+210\,{\frac{{b}^{4}\ln \left ( ex+d \right ){a}^{3}{d}^{2}}{{e}^{5}}}-105\,{\frac{{a}^{2}{d}^{4}{b}^{5}}{{e}^{6} \left ( ex+d \right ) }}+42\,{\frac{{a}^{5}d{b}^{2}}{{e}^{3} \left ( ex+d \right ) }}-105\,{\frac{{a}^{4}{d}^{2}{b}^{3}}{{e}^{4} \left ( ex+d \right ) }}-210\,{\frac{{b}^{5}\ln \left ( ex+d \right ){a}^{2}{d}^{3}}{{e}^{6}}}+105\,{\frac{{b}^{6}\ln \left ( ex+d \right ) a{d}^{4}}{{e}^{7}}}+42\,{\frac{a{d}^{5}{b}^{6}}{{e}^{7} \left ( ex+d \right ) }}+140\,{\frac{{a}^{3}{d}^{3}{b}^{4}}{{e}^{5} \left ( ex+d \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

15*b^7/e^7*d^4*x+1/2/e^8/(e*x+d)^2*b^7*d^7+7/4*b^6/e^3*x^4*a-3/4*b^7/e^4*x^4*d+7*b^5/e^3*x^3*a^2+2*b^7/e^5*x^3
*d^2+35/2*b^4/e^3*x^2*a^3-5*b^7/e^6*x^2*d^3+35*b^3/e^3*a^4*x+21*b^2/e^3*ln(e*x+d)*a^5-21*b^7/e^8*ln(e*x+d)*d^5
-7*b/e^2/(e*x+d)*a^6-7*b^7/e^8/(e*x+d)*d^6+1/5*b^7/e^3*x^5-1/2/e/(e*x+d)^2*a^7-7*b^6/e^4*x^3*a*d-105*b^4/e^4*a
^3*d*x-63/2*b^5/e^4*x^2*a^2*d-21/2/e^3/(e*x+d)^2*d^2*a^5*b^2+35/2/e^4/(e*x+d)^2*a^4*b^3*d^3-35/2/e^5/(e*x+d)^2
*a^3*b^4*d^4+21/2/e^6/(e*x+d)^2*a^2*b^5*d^5-7/2/e^7/(e*x+d)^2*a*b^6*d^6+21*b^6/e^5*x^2*a*d^2+126*b^5/e^5*a^2*d
^2*x-70*b^6/e^6*a*d^3*x+7/2/e^2/(e*x+d)^2*d*a^6*b-105*b^3/e^4*ln(e*x+d)*a^4*d+210*b^4/e^5*ln(e*x+d)*a^3*d^2-10
5*b^5/e^6/(e*x+d)*a^2*d^4+42*b^2/e^3/(e*x+d)*a^5*d-105*b^3/e^4/(e*x+d)*a^4*d^2-210*b^5/e^6*ln(e*x+d)*a^2*d^3+1
05*b^6/e^7*ln(e*x+d)*a*d^4+42*b^6/e^7/(e*x+d)*a*d^5+140*b^4/e^5/(e*x+d)*a^3*d^3

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Maxima [B]  time = 1.01239, size = 639, normalized size = 3.45 \begin{align*} -\frac{13 \, b^{7} d^{7} - 77 \, a b^{6} d^{6} e + 189 \, a^{2} b^{5} d^{5} e^{2} - 245 \, a^{3} b^{4} d^{4} e^{3} + 175 \, a^{4} b^{3} d^{3} e^{4} - 63 \, a^{5} b^{2} d^{2} e^{5} + 7 \, a^{6} b d e^{6} + a^{7} e^{7} + 14 \,{\left (b^{7} d^{6} e - 6 \, a b^{6} d^{5} e^{2} + 15 \, a^{2} b^{5} d^{4} e^{3} - 20 \, a^{3} b^{4} d^{3} e^{4} + 15 \, a^{4} b^{3} d^{2} e^{5} - 6 \, a^{5} b^{2} d e^{6} + a^{6} b e^{7}\right )} x}{2 \,{\left (e^{10} x^{2} + 2 \, d e^{9} x + d^{2} e^{8}\right )}} + \frac{4 \, b^{7} e^{4} x^{5} - 5 \,{\left (3 \, b^{7} d e^{3} - 7 \, a b^{6} e^{4}\right )} x^{4} + 20 \,{\left (2 \, b^{7} d^{2} e^{2} - 7 \, a b^{6} d e^{3} + 7 \, a^{2} b^{5} e^{4}\right )} x^{3} - 10 \,{\left (10 \, b^{7} d^{3} e - 42 \, a b^{6} d^{2} e^{2} + 63 \, a^{2} b^{5} d e^{3} - 35 \, a^{3} b^{4} e^{4}\right )} x^{2} + 20 \,{\left (15 \, b^{7} d^{4} - 70 \, a b^{6} d^{3} e + 126 \, a^{2} b^{5} d^{2} e^{2} - 105 \, a^{3} b^{4} d e^{3} + 35 \, a^{4} b^{3} e^{4}\right )} x}{20 \, e^{7}} - \frac{21 \,{\left (b^{7} d^{5} - 5 \, a b^{6} d^{4} e + 10 \, a^{2} b^{5} d^{3} e^{2} - 10 \, a^{3} b^{4} d^{2} e^{3} + 5 \, a^{4} b^{3} d e^{4} - a^{5} b^{2} e^{5}\right )} \log \left (e x + d\right )}{e^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(13*b^7*d^7 - 77*a*b^6*d^6*e + 189*a^2*b^5*d^5*e^2 - 245*a^3*b^4*d^4*e^3 + 175*a^4*b^3*d^3*e^4 - 63*a^5*b
^2*d^2*e^5 + 7*a^6*b*d*e^6 + a^7*e^7 + 14*(b^7*d^6*e - 6*a*b^6*d^5*e^2 + 15*a^2*b^5*d^4*e^3 - 20*a^3*b^4*d^3*e
^4 + 15*a^4*b^3*d^2*e^5 - 6*a^5*b^2*d*e^6 + a^6*b*e^7)*x)/(e^10*x^2 + 2*d*e^9*x + d^2*e^8) + 1/20*(4*b^7*e^4*x
^5 - 5*(3*b^7*d*e^3 - 7*a*b^6*e^4)*x^4 + 20*(2*b^7*d^2*e^2 - 7*a*b^6*d*e^3 + 7*a^2*b^5*e^4)*x^3 - 10*(10*b^7*d
^3*e - 42*a*b^6*d^2*e^2 + 63*a^2*b^5*d*e^3 - 35*a^3*b^4*e^4)*x^2 + 20*(15*b^7*d^4 - 70*a*b^6*d^3*e + 126*a^2*b
^5*d^2*e^2 - 105*a^3*b^4*d*e^3 + 35*a^4*b^3*e^4)*x)/e^7 - 21*(b^7*d^5 - 5*a*b^6*d^4*e + 10*a^2*b^5*d^3*e^2 - 1
0*a^3*b^4*d^2*e^3 + 5*a^4*b^3*d*e^4 - a^5*b^2*e^5)*log(e*x + d)/e^8

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Fricas [B]  time = 1.52998, size = 1445, normalized size = 7.81 \begin{align*} \frac{4 \, b^{7} e^{7} x^{7} - 130 \, b^{7} d^{7} + 770 \, a b^{6} d^{6} e - 1890 \, a^{2} b^{5} d^{5} e^{2} + 2450 \, a^{3} b^{4} d^{4} e^{3} - 1750 \, a^{4} b^{3} d^{3} e^{4} + 630 \, a^{5} b^{2} d^{2} e^{5} - 70 \, a^{6} b d e^{6} - 10 \, a^{7} e^{7} - 7 \,{\left (b^{7} d e^{6} - 5 \, a b^{6} e^{7}\right )} x^{6} + 14 \,{\left (b^{7} d^{2} e^{5} - 5 \, a b^{6} d e^{6} + 10 \, a^{2} b^{5} e^{7}\right )} x^{5} - 35 \,{\left (b^{7} d^{3} e^{4} - 5 \, a b^{6} d^{2} e^{5} + 10 \, a^{2} b^{5} d e^{6} - 10 \, a^{3} b^{4} e^{7}\right )} x^{4} + 140 \,{\left (b^{7} d^{4} e^{3} - 5 \, a b^{6} d^{3} e^{4} + 10 \, a^{2} b^{5} d^{2} e^{5} - 10 \, a^{3} b^{4} d e^{6} + 5 \, a^{4} b^{3} e^{7}\right )} x^{3} + 10 \,{\left (50 \, b^{7} d^{5} e^{2} - 238 \, a b^{6} d^{4} e^{3} + 441 \, a^{2} b^{5} d^{3} e^{4} - 385 \, a^{3} b^{4} d^{2} e^{5} + 140 \, a^{4} b^{3} d e^{6}\right )} x^{2} + 20 \,{\left (8 \, b^{7} d^{6} e - 28 \, a b^{6} d^{5} e^{2} + 21 \, a^{2} b^{5} d^{4} e^{3} + 35 \, a^{3} b^{4} d^{3} e^{4} - 70 \, a^{4} b^{3} d^{2} e^{5} + 42 \, a^{5} b^{2} d e^{6} - 7 \, a^{6} b e^{7}\right )} x - 420 \,{\left (b^{7} d^{7} - 5 \, a b^{6} d^{6} e + 10 \, a^{2} b^{5} d^{5} e^{2} - 10 \, a^{3} b^{4} d^{4} e^{3} + 5 \, a^{4} b^{3} d^{3} e^{4} - a^{5} b^{2} d^{2} e^{5} +{\left (b^{7} d^{5} e^{2} - 5 \, a b^{6} d^{4} e^{3} + 10 \, a^{2} b^{5} d^{3} e^{4} - 10 \, a^{3} b^{4} d^{2} e^{5} + 5 \, a^{4} b^{3} d e^{6} - a^{5} b^{2} e^{7}\right )} x^{2} + 2 \,{\left (b^{7} d^{6} e - 5 \, a b^{6} d^{5} e^{2} + 10 \, a^{2} b^{5} d^{4} e^{3} - 10 \, a^{3} b^{4} d^{3} e^{4} + 5 \, a^{4} b^{3} d^{2} e^{5} - a^{5} b^{2} d e^{6}\right )} x\right )} \log \left (e x + d\right )}{20 \,{\left (e^{10} x^{2} + 2 \, d e^{9} x + d^{2} e^{8}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/20*(4*b^7*e^7*x^7 - 130*b^7*d^7 + 770*a*b^6*d^6*e - 1890*a^2*b^5*d^5*e^2 + 2450*a^3*b^4*d^4*e^3 - 1750*a^4*b
^3*d^3*e^4 + 630*a^5*b^2*d^2*e^5 - 70*a^6*b*d*e^6 - 10*a^7*e^7 - 7*(b^7*d*e^6 - 5*a*b^6*e^7)*x^6 + 14*(b^7*d^2
*e^5 - 5*a*b^6*d*e^6 + 10*a^2*b^5*e^7)*x^5 - 35*(b^7*d^3*e^4 - 5*a*b^6*d^2*e^5 + 10*a^2*b^5*d*e^6 - 10*a^3*b^4
*e^7)*x^4 + 140*(b^7*d^4*e^3 - 5*a*b^6*d^3*e^4 + 10*a^2*b^5*d^2*e^5 - 10*a^3*b^4*d*e^6 + 5*a^4*b^3*e^7)*x^3 +
10*(50*b^7*d^5*e^2 - 238*a*b^6*d^4*e^3 + 441*a^2*b^5*d^3*e^4 - 385*a^3*b^4*d^2*e^5 + 140*a^4*b^3*d*e^6)*x^2 +
20*(8*b^7*d^6*e - 28*a*b^6*d^5*e^2 + 21*a^2*b^5*d^4*e^3 + 35*a^3*b^4*d^3*e^4 - 70*a^4*b^3*d^2*e^5 + 42*a^5*b^2
*d*e^6 - 7*a^6*b*e^7)*x - 420*(b^7*d^7 - 5*a*b^6*d^6*e + 10*a^2*b^5*d^5*e^2 - 10*a^3*b^4*d^4*e^3 + 5*a^4*b^3*d
^3*e^4 - a^5*b^2*d^2*e^5 + (b^7*d^5*e^2 - 5*a*b^6*d^4*e^3 + 10*a^2*b^5*d^3*e^4 - 10*a^3*b^4*d^2*e^5 + 5*a^4*b^
3*d*e^6 - a^5*b^2*e^7)*x^2 + 2*(b^7*d^6*e - 5*a*b^6*d^5*e^2 + 10*a^2*b^5*d^4*e^3 - 10*a^3*b^4*d^3*e^4 + 5*a^4*
b^3*d^2*e^5 - a^5*b^2*d*e^6)*x)*log(e*x + d))/(e^10*x^2 + 2*d*e^9*x + d^2*e^8)

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Sympy [B]  time = 3.64868, size = 437, normalized size = 2.36 \begin{align*} \frac{b^{7} x^{5}}{5 e^{3}} + \frac{21 b^{2} \left (a e - b d\right )^{5} \log{\left (d + e x \right )}}{e^{8}} - \frac{a^{7} e^{7} + 7 a^{6} b d e^{6} - 63 a^{5} b^{2} d^{2} e^{5} + 175 a^{4} b^{3} d^{3} e^{4} - 245 a^{3} b^{4} d^{4} e^{3} + 189 a^{2} b^{5} d^{5} e^{2} - 77 a b^{6} d^{6} e + 13 b^{7} d^{7} + x \left (14 a^{6} b e^{7} - 84 a^{5} b^{2} d e^{6} + 210 a^{4} b^{3} d^{2} e^{5} - 280 a^{3} b^{4} d^{3} e^{4} + 210 a^{2} b^{5} d^{4} e^{3} - 84 a b^{6} d^{5} e^{2} + 14 b^{7} d^{6} e\right )}{2 d^{2} e^{8} + 4 d e^{9} x + 2 e^{10} x^{2}} + \frac{x^{4} \left (7 a b^{6} e - 3 b^{7} d\right )}{4 e^{4}} + \frac{x^{3} \left (7 a^{2} b^{5} e^{2} - 7 a b^{6} d e + 2 b^{7} d^{2}\right )}{e^{5}} + \frac{x^{2} \left (35 a^{3} b^{4} e^{3} - 63 a^{2} b^{5} d e^{2} + 42 a b^{6} d^{2} e - 10 b^{7} d^{3}\right )}{2 e^{6}} + \frac{x \left (35 a^{4} b^{3} e^{4} - 105 a^{3} b^{4} d e^{3} + 126 a^{2} b^{5} d^{2} e^{2} - 70 a b^{6} d^{3} e + 15 b^{7} d^{4}\right )}{e^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b**7*x**5/(5*e**3) + 21*b**2*(a*e - b*d)**5*log(d + e*x)/e**8 - (a**7*e**7 + 7*a**6*b*d*e**6 - 63*a**5*b**2*d*
*2*e**5 + 175*a**4*b**3*d**3*e**4 - 245*a**3*b**4*d**4*e**3 + 189*a**2*b**5*d**5*e**2 - 77*a*b**6*d**6*e + 13*
b**7*d**7 + x*(14*a**6*b*e**7 - 84*a**5*b**2*d*e**6 + 210*a**4*b**3*d**2*e**5 - 280*a**3*b**4*d**3*e**4 + 210*
a**2*b**5*d**4*e**3 - 84*a*b**6*d**5*e**2 + 14*b**7*d**6*e))/(2*d**2*e**8 + 4*d*e**9*x + 2*e**10*x**2) + x**4*
(7*a*b**6*e - 3*b**7*d)/(4*e**4) + x**3*(7*a**2*b**5*e**2 - 7*a*b**6*d*e + 2*b**7*d**2)/e**5 + x**2*(35*a**3*b
**4*e**3 - 63*a**2*b**5*d*e**2 + 42*a*b**6*d**2*e - 10*b**7*d**3)/(2*e**6) + x*(35*a**4*b**3*e**4 - 105*a**3*b
**4*d*e**3 + 126*a**2*b**5*d**2*e**2 - 70*a*b**6*d**3*e + 15*b**7*d**4)/e**7

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Giac [B]  time = 1.11125, size = 605, normalized size = 3.27 \begin{align*} -21 \,{\left (b^{7} d^{5} - 5 \, a b^{6} d^{4} e + 10 \, a^{2} b^{5} d^{3} e^{2} - 10 \, a^{3} b^{4} d^{2} e^{3} + 5 \, a^{4} b^{3} d e^{4} - a^{5} b^{2} e^{5}\right )} e^{\left (-8\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{1}{20} \,{\left (4 \, b^{7} x^{5} e^{12} - 15 \, b^{7} d x^{4} e^{11} + 40 \, b^{7} d^{2} x^{3} e^{10} - 100 \, b^{7} d^{3} x^{2} e^{9} + 300 \, b^{7} d^{4} x e^{8} + 35 \, a b^{6} x^{4} e^{12} - 140 \, a b^{6} d x^{3} e^{11} + 420 \, a b^{6} d^{2} x^{2} e^{10} - 1400 \, a b^{6} d^{3} x e^{9} + 140 \, a^{2} b^{5} x^{3} e^{12} - 630 \, a^{2} b^{5} d x^{2} e^{11} + 2520 \, a^{2} b^{5} d^{2} x e^{10} + 350 \, a^{3} b^{4} x^{2} e^{12} - 2100 \, a^{3} b^{4} d x e^{11} + 700 \, a^{4} b^{3} x e^{12}\right )} e^{\left (-15\right )} - \frac{{\left (13 \, b^{7} d^{7} - 77 \, a b^{6} d^{6} e + 189 \, a^{2} b^{5} d^{5} e^{2} - 245 \, a^{3} b^{4} d^{4} e^{3} + 175 \, a^{4} b^{3} d^{3} e^{4} - 63 \, a^{5} b^{2} d^{2} e^{5} + 7 \, a^{6} b d e^{6} + a^{7} e^{7} + 14 \,{\left (b^{7} d^{6} e - 6 \, a b^{6} d^{5} e^{2} + 15 \, a^{2} b^{5} d^{4} e^{3} - 20 \, a^{3} b^{4} d^{3} e^{4} + 15 \, a^{4} b^{3} d^{2} e^{5} - 6 \, a^{5} b^{2} d e^{6} + a^{6} b e^{7}\right )} x\right )} e^{\left (-8\right )}}{2 \,{\left (x e + d\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-21*(b^7*d^5 - 5*a*b^6*d^4*e + 10*a^2*b^5*d^3*e^2 - 10*a^3*b^4*d^2*e^3 + 5*a^4*b^3*d*e^4 - a^5*b^2*e^5)*e^(-8)
*log(abs(x*e + d)) + 1/20*(4*b^7*x^5*e^12 - 15*b^7*d*x^4*e^11 + 40*b^7*d^2*x^3*e^10 - 100*b^7*d^3*x^2*e^9 + 30
0*b^7*d^4*x*e^8 + 35*a*b^6*x^4*e^12 - 140*a*b^6*d*x^3*e^11 + 420*a*b^6*d^2*x^2*e^10 - 1400*a*b^6*d^3*x*e^9 + 1
40*a^2*b^5*x^3*e^12 - 630*a^2*b^5*d*x^2*e^11 + 2520*a^2*b^5*d^2*x*e^10 + 350*a^3*b^4*x^2*e^12 - 2100*a^3*b^4*d
*x*e^11 + 700*a^4*b^3*x*e^12)*e^(-15) - 1/2*(13*b^7*d^7 - 77*a*b^6*d^6*e + 189*a^2*b^5*d^5*e^2 - 245*a^3*b^4*d
^4*e^3 + 175*a^4*b^3*d^3*e^4 - 63*a^5*b^2*d^2*e^5 + 7*a^6*b*d*e^6 + a^7*e^7 + 14*(b^7*d^6*e - 6*a*b^6*d^5*e^2
+ 15*a^2*b^5*d^4*e^3 - 20*a^3*b^4*d^3*e^4 + 15*a^4*b^3*d^2*e^5 - 6*a^5*b^2*d*e^6 + a^6*b*e^7)*x)*e^(-8)/(x*e +
d)^2