3.1923 \(\int \frac{(a+b x) (a^2+2 a b x+b^2 x^2)^3}{d+e x} \, dx\)

Optimal. Leaf size=170 \[ \frac{b x (b d-a e)^6}{e^7}-\frac{(a+b x)^2 (b d-a e)^5}{2 e^6}+\frac{(a+b x)^3 (b d-a e)^4}{3 e^5}-\frac{(a+b x)^4 (b d-a e)^3}{4 e^4}+\frac{(a+b x)^5 (b d-a e)^2}{5 e^3}-\frac{(a+b x)^6 (b d-a e)}{6 e^2}-\frac{(b d-a e)^7 \log (d+e x)}{e^8}+\frac{(a+b x)^7}{7 e} \]

[Out]

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

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Rubi [A]  time = 0.0826123, antiderivative size = 170, 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} \[ \frac{b x (b d-a e)^6}{e^7}-\frac{(a+b x)^2 (b d-a e)^5}{2 e^6}+\frac{(a+b x)^3 (b d-a e)^4}{3 e^5}-\frac{(a+b x)^4 (b d-a e)^3}{4 e^4}+\frac{(a+b x)^5 (b d-a e)^2}{5 e^3}-\frac{(a+b x)^6 (b d-a e)}{6 e^2}-\frac{(b d-a e)^7 \log (d+e x)}{e^8}+\frac{(a+b x)^7}{7 e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.115576, size = 304, normalized size = 1.79 \[ \frac{b e x \left (147 a^2 b^4 e^2 \left (20 d^2 e^2 x^2-30 d^3 e x+60 d^4-15 d e^3 x^3+12 e^4 x^4\right )+1225 a^3 b^3 e^3 \left (6 d^2 e x-12 d^3-4 d e^2 x^2+3 e^3 x^3\right )+2450 a^4 b^2 e^4 \left (6 d^2-3 d e x+2 e^2 x^2\right )+4410 a^5 b e^5 (e x-2 d)+2940 a^6 e^6+49 a b^5 e \left (-20 d^3 e^2 x^2+15 d^2 e^3 x^3+30 d^4 e x-60 d^5-12 d e^4 x^4+10 e^5 x^5\right )+b^6 \left (140 d^4 e^2 x^2-105 d^3 e^3 x^3+84 d^2 e^4 x^4-210 d^5 e x+420 d^6-70 d e^5 x^5+60 e^6 x^6\right )\right )-420 (b d-a e)^7 \log (d+e x)}{420 e^8} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*e*x*(2940*a^6*e^6 + 4410*a^5*b*e^5*(-2*d + e*x) + 2450*a^4*b^2*e^4*(6*d^2 - 3*d*e*x + 2*e^2*x^2) + 1225*a^3
*b^3*e^3*(-12*d^3 + 6*d^2*e*x - 4*d*e^2*x^2 + 3*e^3*x^3) + 147*a^2*b^4*e^2*(60*d^4 - 30*d^3*e*x + 20*d^2*e^2*x
^2 - 15*d*e^3*x^3 + 12*e^4*x^4) + 49*a*b^5*e*(-60*d^5 + 30*d^4*e*x - 20*d^3*e^2*x^2 + 15*d^2*e^3*x^3 - 12*d*e^
4*x^4 + 10*e^5*x^5) + b^6*(420*d^6 - 210*d^5*e*x + 140*d^4*e^2*x^2 - 105*d^3*e^3*x^3 + 84*d^2*e^4*x^4 - 70*d*e
^5*x^5 + 60*e^6*x^6)) - 420*(b*d - a*e)^7*Log[d + e*x])/(420*e^8)

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Maple [B]  time = 0.006, size = 539, normalized size = 3.2 \begin{align*} 7\,{\frac{{b}^{5}{x}^{3}{a}^{2}{d}^{2}}{{e}^{3}}}-{\frac{35\,{b}^{4}{x}^{3}{a}^{3}d}{3\,{e}^{2}}}-7\,{\frac{a{d}^{5}{b}^{6}x}{{e}^{6}}}-{\frac{21\,{b}^{5}{x}^{2}{a}^{2}{d}^{3}}{2\,{e}^{4}}}-21\,{\frac{{a}^{5}d{b}^{2}x}{{e}^{2}}}+{\frac{7\,{b}^{6}{x}^{2}a{d}^{4}}{2\,{e}^{5}}}-{\frac{7\,{b}^{6}{x}^{3}a{d}^{3}}{3\,{e}^{4}}}+21\,{\frac{{a}^{2}{d}^{4}{b}^{5}x}{{e}^{5}}}+35\,{\frac{{a}^{4}{d}^{2}{b}^{3}x}{{e}^{3}}}-35\,{\frac{{a}^{3}{d}^{3}{b}^{4}x}{{e}^{4}}}-{\frac{35\,{b}^{3}{x}^{2}{a}^{4}d}{2\,{e}^{2}}}+{\frac{35\,{b}^{4}{x}^{2}{a}^{3}{d}^{2}}{2\,{e}^{3}}}+{\frac{7\,{b}^{6}{x}^{4}a{d}^{2}}{4\,{e}^{3}}}-7\,{\frac{\ln \left ( ex+d \right ){a}^{6}bd}{{e}^{2}}}+21\,{\frac{\ln \left ( ex+d \right ){a}^{5}{b}^{2}{d}^{2}}{{e}^{3}}}-35\,{\frac{\ln \left ( ex+d \right ){a}^{4}{b}^{3}{d}^{3}}{{e}^{4}}}+35\,{\frac{\ln \left ( ex+d \right ){a}^{3}{b}^{4}{d}^{4}}{{e}^{5}}}-{\frac{7\,{b}^{6}{x}^{5}ad}{5\,{e}^{2}}}-{\frac{21\,{b}^{5}{x}^{4}{a}^{2}d}{4\,{e}^{2}}}-21\,{\frac{\ln \left ( ex+d \right ){a}^{2}{b}^{5}{d}^{5}}{{e}^{6}}}+7\,{\frac{\ln \left ( ex+d \right ) a{b}^{6}{d}^{6}}{{e}^{7}}}+{\frac{{b}^{7}{x}^{3}{d}^{4}}{3\,{e}^{5}}}+{\frac{21\,{b}^{2}{x}^{2}{a}^{5}}{2\,e}}-{\frac{{b}^{7}{x}^{4}{d}^{3}}{4\,{e}^{4}}}+{\frac{35\,{b}^{3}{x}^{3}{a}^{4}}{3\,e}}+{\frac{21\,{b}^{5}{x}^{5}{a}^{2}}{5\,e}}+{\frac{{b}^{7}{x}^{5}{d}^{2}}{5\,{e}^{3}}}+{\frac{35\,{b}^{4}{x}^{4}{a}^{3}}{4\,e}}-{\frac{{b}^{7}{x}^{2}{d}^{5}}{2\,{e}^{6}}}-{\frac{\ln \left ( ex+d \right ){b}^{7}{d}^{7}}{{e}^{8}}}+{\frac{{b}^{7}{d}^{6}x}{{e}^{7}}}-{\frac{{b}^{7}{x}^{6}d}{6\,{e}^{2}}}+{\frac{7\,{b}^{6}{x}^{6}a}{6\,e}}+7\,{\frac{b{a}^{6}x}{e}}+{\frac{\ln \left ( ex+d \right ){a}^{7}}{e}}+{\frac{{b}^{7}{x}^{7}}{7\,e}} \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),x)

[Out]

7*b^5/e^3*x^3*a^2*d^2-35/3*b^4/e^2*x^3*a^3*d-7*b^6/e^6*a*d^5*x-21/2*b^5/e^4*x^2*a^2*d^3-21*b^2/e^2*a^5*d*x+7/2
*b^6/e^5*x^2*a*d^4-7/3*b^6/e^4*x^3*a*d^3+21*b^5/e^5*a^2*d^4*x+35*b^3/e^3*a^4*d^2*x-35*b^4/e^4*a^3*d^3*x-35/2*b
^3/e^2*x^2*a^4*d+35/2*b^4/e^3*x^2*a^3*d^2+7/4*b^6/e^3*x^4*a*d^2-7/e^2*ln(e*x+d)*a^6*b*d+21/e^3*ln(e*x+d)*a^5*b
^2*d^2-35/e^4*ln(e*x+d)*a^4*b^3*d^3+35/e^5*ln(e*x+d)*a^3*b^4*d^4-7/5*b^6/e^2*x^5*a*d-21/4*b^5/e^2*x^4*a^2*d-21
/e^6*ln(e*x+d)*a^2*b^5*d^5+7/e^7*ln(e*x+d)*a*b^6*d^6+1/3*b^7/e^5*x^3*d^4+21/2*b^2/e*x^2*a^5-1/4*b^7/e^4*x^4*d^
3+35/3*b^3/e*x^3*a^4+21/5*b^5/e*x^5*a^2+1/5*b^7/e^3*x^5*d^2+35/4*b^4/e*x^4*a^3-1/2*b^7/e^6*x^2*d^5-1/e^8*ln(e*
x+d)*b^7*d^7+b^7/e^7*d^6*x-1/6*b^7/e^2*x^6*d+7/6*b^6/e*x^6*a+7*b/e*a^6*x+1/e*ln(e*x+d)*a^7+1/7*b^7/e*x^7

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Maxima [B]  time = 0.977937, size = 618, normalized size = 3.64 \begin{align*} \frac{60 \, b^{7} e^{6} x^{7} - 70 \,{\left (b^{7} d e^{5} - 7 \, a b^{6} e^{6}\right )} x^{6} + 84 \,{\left (b^{7} d^{2} e^{4} - 7 \, a b^{6} d e^{5} + 21 \, a^{2} b^{5} e^{6}\right )} x^{5} - 105 \,{\left (b^{7} d^{3} e^{3} - 7 \, a b^{6} d^{2} e^{4} + 21 \, a^{2} b^{5} d e^{5} - 35 \, a^{3} b^{4} e^{6}\right )} x^{4} + 140 \,{\left (b^{7} d^{4} e^{2} - 7 \, a b^{6} d^{3} e^{3} + 21 \, a^{2} b^{5} d^{2} e^{4} - 35 \, a^{3} b^{4} d e^{5} + 35 \, a^{4} b^{3} e^{6}\right )} x^{3} - 210 \,{\left (b^{7} d^{5} e - 7 \, a b^{6} d^{4} e^{2} + 21 \, a^{2} b^{5} d^{3} e^{3} - 35 \, a^{3} b^{4} d^{2} e^{4} + 35 \, a^{4} b^{3} d e^{5} - 21 \, a^{5} b^{2} e^{6}\right )} x^{2} + 420 \,{\left (b^{7} d^{6} - 7 \, a b^{6} d^{5} e + 21 \, a^{2} b^{5} d^{4} e^{2} - 35 \, a^{3} b^{4} d^{3} e^{3} + 35 \, a^{4} b^{3} d^{2} e^{4} - 21 \, a^{5} b^{2} d e^{5} + 7 \, a^{6} b e^{6}\right )} x}{420 \, e^{7}} - \frac{{\left (b^{7} d^{7} - 7 \, a b^{6} d^{6} e + 21 \, a^{2} b^{5} d^{5} e^{2} - 35 \, a^{3} b^{4} d^{4} e^{3} + 35 \, a^{4} b^{3} d^{3} e^{4} - 21 \, a^{5} b^{2} d^{2} e^{5} + 7 \, a^{6} b d e^{6} - a^{7} e^{7}\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),x, algorithm="maxima")

[Out]

1/420*(60*b^7*e^6*x^7 - 70*(b^7*d*e^5 - 7*a*b^6*e^6)*x^6 + 84*(b^7*d^2*e^4 - 7*a*b^6*d*e^5 + 21*a^2*b^5*e^6)*x
^5 - 105*(b^7*d^3*e^3 - 7*a*b^6*d^2*e^4 + 21*a^2*b^5*d*e^5 - 35*a^3*b^4*e^6)*x^4 + 140*(b^7*d^4*e^2 - 7*a*b^6*
d^3*e^3 + 21*a^2*b^5*d^2*e^4 - 35*a^3*b^4*d*e^5 + 35*a^4*b^3*e^6)*x^3 - 210*(b^7*d^5*e - 7*a*b^6*d^4*e^2 + 21*
a^2*b^5*d^3*e^3 - 35*a^3*b^4*d^2*e^4 + 35*a^4*b^3*d*e^5 - 21*a^5*b^2*e^6)*x^2 + 420*(b^7*d^6 - 7*a*b^6*d^5*e +
21*a^2*b^5*d^4*e^2 - 35*a^3*b^4*d^3*e^3 + 35*a^4*b^3*d^2*e^4 - 21*a^5*b^2*d*e^5 + 7*a^6*b*e^6)*x)/e^7 - (b^7*
d^7 - 7*a*b^6*d^6*e + 21*a^2*b^5*d^5*e^2 - 35*a^3*b^4*d^4*e^3 + 35*a^4*b^3*d^3*e^4 - 21*a^5*b^2*d^2*e^5 + 7*a^
6*b*d*e^6 - a^7*e^7)*log(e*x + d)/e^8

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Fricas [B]  time = 1.49858, size = 952, normalized size = 5.6 \begin{align*} \frac{60 \, b^{7} e^{7} x^{7} - 70 \,{\left (b^{7} d e^{6} - 7 \, a b^{6} e^{7}\right )} x^{6} + 84 \,{\left (b^{7} d^{2} e^{5} - 7 \, a b^{6} d e^{6} + 21 \, a^{2} b^{5} e^{7}\right )} x^{5} - 105 \,{\left (b^{7} d^{3} e^{4} - 7 \, a b^{6} d^{2} e^{5} + 21 \, a^{2} b^{5} d e^{6} - 35 \, a^{3} b^{4} e^{7}\right )} x^{4} + 140 \,{\left (b^{7} d^{4} e^{3} - 7 \, a b^{6} d^{3} e^{4} + 21 \, a^{2} b^{5} d^{2} e^{5} - 35 \, a^{3} b^{4} d e^{6} + 35 \, a^{4} b^{3} e^{7}\right )} x^{3} - 210 \,{\left (b^{7} d^{5} e^{2} - 7 \, a b^{6} d^{4} e^{3} + 21 \, a^{2} b^{5} d^{3} e^{4} - 35 \, a^{3} b^{4} d^{2} e^{5} + 35 \, a^{4} b^{3} d e^{6} - 21 \, a^{5} b^{2} e^{7}\right )} x^{2} + 420 \,{\left (b^{7} d^{6} e - 7 \, 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} + 35 \, a^{4} b^{3} d^{2} e^{5} - 21 \, a^{5} b^{2} d e^{6} + 7 \, a^{6} b e^{7}\right )} x - 420 \,{\left (b^{7} d^{7} - 7 \, a b^{6} d^{6} e + 21 \, a^{2} b^{5} d^{5} e^{2} - 35 \, a^{3} b^{4} d^{4} e^{3} + 35 \, a^{4} b^{3} d^{3} e^{4} - 21 \, a^{5} b^{2} d^{2} e^{5} + 7 \, a^{6} b d e^{6} - a^{7} e^{7}\right )} \log \left (e x + d\right )}{420 \, 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),x, algorithm="fricas")

[Out]

1/420*(60*b^7*e^7*x^7 - 70*(b^7*d*e^6 - 7*a*b^6*e^7)*x^6 + 84*(b^7*d^2*e^5 - 7*a*b^6*d*e^6 + 21*a^2*b^5*e^7)*x
^5 - 105*(b^7*d^3*e^4 - 7*a*b^6*d^2*e^5 + 21*a^2*b^5*d*e^6 - 35*a^3*b^4*e^7)*x^4 + 140*(b^7*d^4*e^3 - 7*a*b^6*
d^3*e^4 + 21*a^2*b^5*d^2*e^5 - 35*a^3*b^4*d*e^6 + 35*a^4*b^3*e^7)*x^3 - 210*(b^7*d^5*e^2 - 7*a*b^6*d^4*e^3 + 2
1*a^2*b^5*d^3*e^4 - 35*a^3*b^4*d^2*e^5 + 35*a^4*b^3*d*e^6 - 21*a^5*b^2*e^7)*x^2 + 420*(b^7*d^6*e - 7*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 + 35*a^4*b^3*d^2*e^5 - 21*a^5*b^2*d*e^6 + 7*a^6*b*e^7)*x - 420*
(b^7*d^7 - 7*a*b^6*d^6*e + 21*a^2*b^5*d^5*e^2 - 35*a^3*b^4*d^4*e^3 + 35*a^4*b^3*d^3*e^4 - 21*a^5*b^2*d^2*e^5 +
7*a^6*b*d*e^6 - a^7*e^7)*log(e*x + d))/e^8

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Sympy [B]  time = 0.992577, size = 384, normalized size = 2.26 \begin{align*} \frac{b^{7} x^{7}}{7 e} + \frac{x^{6} \left (7 a b^{6} e - b^{7} d\right )}{6 e^{2}} + \frac{x^{5} \left (21 a^{2} b^{5} e^{2} - 7 a b^{6} d e + b^{7} d^{2}\right )}{5 e^{3}} + \frac{x^{4} \left (35 a^{3} b^{4} e^{3} - 21 a^{2} b^{5} d e^{2} + 7 a b^{6} d^{2} e - b^{7} d^{3}\right )}{4 e^{4}} + \frac{x^{3} \left (35 a^{4} b^{3} e^{4} - 35 a^{3} b^{4} d e^{3} + 21 a^{2} b^{5} d^{2} e^{2} - 7 a b^{6} d^{3} e + b^{7} d^{4}\right )}{3 e^{5}} + \frac{x^{2} \left (21 a^{5} b^{2} e^{5} - 35 a^{4} b^{3} d e^{4} + 35 a^{3} b^{4} d^{2} e^{3} - 21 a^{2} b^{5} d^{3} e^{2} + 7 a b^{6} d^{4} e - b^{7} d^{5}\right )}{2 e^{6}} + \frac{x \left (7 a^{6} b e^{6} - 21 a^{5} b^{2} d e^{5} + 35 a^{4} b^{3} d^{2} e^{4} - 35 a^{3} b^{4} d^{3} e^{3} + 21 a^{2} b^{5} d^{4} e^{2} - 7 a b^{6} d^{5} e + b^{7} d^{6}\right )}{e^{7}} + \frac{\left (a e - b d\right )^{7} \log{\left (d + e x \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),x)

[Out]

b**7*x**7/(7*e) + x**6*(7*a*b**6*e - b**7*d)/(6*e**2) + x**5*(21*a**2*b**5*e**2 - 7*a*b**6*d*e + b**7*d**2)/(5
*e**3) + x**4*(35*a**3*b**4*e**3 - 21*a**2*b**5*d*e**2 + 7*a*b**6*d**2*e - b**7*d**3)/(4*e**4) + x**3*(35*a**4
*b**3*e**4 - 35*a**3*b**4*d*e**3 + 21*a**2*b**5*d**2*e**2 - 7*a*b**6*d**3*e + b**7*d**4)/(3*e**5) + x**2*(21*a
**5*b**2*e**5 - 35*a**4*b**3*d*e**4 + 35*a**3*b**4*d**2*e**3 - 21*a**2*b**5*d**3*e**2 + 7*a*b**6*d**4*e - b**7
*d**5)/(2*e**6) + x*(7*a**6*b*e**6 - 21*a**5*b**2*d*e**5 + 35*a**4*b**3*d**2*e**4 - 35*a**3*b**4*d**3*e**3 + 2
1*a**2*b**5*d**4*e**2 - 7*a*b**6*d**5*e + b**7*d**6)/e**7 + (a*e - b*d)**7*log(d + e*x)/e**8

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Giac [B]  time = 1.08542, size = 633, normalized size = 3.72 \begin{align*} -{\left (b^{7} d^{7} - 7 \, a b^{6} d^{6} e + 21 \, a^{2} b^{5} d^{5} e^{2} - 35 \, a^{3} b^{4} d^{4} e^{3} + 35 \, a^{4} b^{3} d^{3} e^{4} - 21 \, a^{5} b^{2} d^{2} e^{5} + 7 \, a^{6} b d e^{6} - a^{7} e^{7}\right )} e^{\left (-8\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{1}{420} \,{\left (60 \, b^{7} x^{7} e^{6} - 70 \, b^{7} d x^{6} e^{5} + 84 \, b^{7} d^{2} x^{5} e^{4} - 105 \, b^{7} d^{3} x^{4} e^{3} + 140 \, b^{7} d^{4} x^{3} e^{2} - 210 \, b^{7} d^{5} x^{2} e + 420 \, b^{7} d^{6} x + 490 \, a b^{6} x^{6} e^{6} - 588 \, a b^{6} d x^{5} e^{5} + 735 \, a b^{6} d^{2} x^{4} e^{4} - 980 \, a b^{6} d^{3} x^{3} e^{3} + 1470 \, a b^{6} d^{4} x^{2} e^{2} - 2940 \, a b^{6} d^{5} x e + 1764 \, a^{2} b^{5} x^{5} e^{6} - 2205 \, a^{2} b^{5} d x^{4} e^{5} + 2940 \, a^{2} b^{5} d^{2} x^{3} e^{4} - 4410 \, a^{2} b^{5} d^{3} x^{2} e^{3} + 8820 \, a^{2} b^{5} d^{4} x e^{2} + 3675 \, a^{3} b^{4} x^{4} e^{6} - 4900 \, a^{3} b^{4} d x^{3} e^{5} + 7350 \, a^{3} b^{4} d^{2} x^{2} e^{4} - 14700 \, a^{3} b^{4} d^{3} x e^{3} + 4900 \, a^{4} b^{3} x^{3} e^{6} - 7350 \, a^{4} b^{3} d x^{2} e^{5} + 14700 \, a^{4} b^{3} d^{2} x e^{4} + 4410 \, a^{5} b^{2} x^{2} e^{6} - 8820 \, a^{5} b^{2} d x e^{5} + 2940 \, a^{6} b x e^{6}\right )} e^{\left (-7\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),x, algorithm="giac")

[Out]

-(b^7*d^7 - 7*a*b^6*d^6*e + 21*a^2*b^5*d^5*e^2 - 35*a^3*b^4*d^4*e^3 + 35*a^4*b^3*d^3*e^4 - 21*a^5*b^2*d^2*e^5
+ 7*a^6*b*d*e^6 - a^7*e^7)*e^(-8)*log(abs(x*e + d)) + 1/420*(60*b^7*x^7*e^6 - 70*b^7*d*x^6*e^5 + 84*b^7*d^2*x^
5*e^4 - 105*b^7*d^3*x^4*e^3 + 140*b^7*d^4*x^3*e^2 - 210*b^7*d^5*x^2*e + 420*b^7*d^6*x + 490*a*b^6*x^6*e^6 - 58
8*a*b^6*d*x^5*e^5 + 735*a*b^6*d^2*x^4*e^4 - 980*a*b^6*d^3*x^3*e^3 + 1470*a*b^6*d^4*x^2*e^2 - 2940*a*b^6*d^5*x*
e + 1764*a^2*b^5*x^5*e^6 - 2205*a^2*b^5*d*x^4*e^5 + 2940*a^2*b^5*d^2*x^3*e^4 - 4410*a^2*b^5*d^3*x^2*e^3 + 8820
*a^2*b^5*d^4*x*e^2 + 3675*a^3*b^4*x^4*e^6 - 4900*a^3*b^4*d*x^3*e^5 + 7350*a^3*b^4*d^2*x^2*e^4 - 14700*a^3*b^4*
d^3*x*e^3 + 4900*a^4*b^3*x^3*e^6 - 7350*a^4*b^3*d*x^2*e^5 + 14700*a^4*b^3*d^2*x*e^4 + 4410*a^5*b^2*x^2*e^6 - 8
820*a^5*b^2*d*x*e^5 + 2940*a^6*b*x*e^6)*e^(-7)