3.1924 \(\int \frac{(a+b x) (a^2+2 a b x+b^2 x^2)^3}{(d+e x)^2} \, dx\)

Optimal. Leaf size=186 \[ -\frac{7 b^6 (d+e x)^5 (b d-a e)}{5 e^8}+\frac{21 b^5 (d+e x)^4 (b d-a e)^2}{4 e^8}-\frac{35 b^4 (d+e x)^3 (b d-a e)^3}{3 e^8}+\frac{35 b^3 (d+e x)^2 (b d-a e)^4}{2 e^8}-\frac{21 b^2 x (b d-a e)^5}{e^7}+\frac{(b d-a e)^7}{e^8 (d+e x)}+\frac{7 b (b d-a e)^6 \log (d+e x)}{e^8}+\frac{b^7 (d+e x)^6}{6 e^8} \]

[Out]

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

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Rubi [A]  time = 0.257907, antiderivative size = 186, 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} \[ -\frac{7 b^6 (d+e x)^5 (b d-a e)}{5 e^8}+\frac{21 b^5 (d+e x)^4 (b d-a e)^2}{4 e^8}-\frac{35 b^4 (d+e x)^3 (b d-a e)^3}{3 e^8}+\frac{35 b^3 (d+e x)^2 (b d-a e)^4}{2 e^8}-\frac{21 b^2 x (b d-a e)^5}{e^7}+\frac{(b d-a e)^7}{e^8 (d+e x)}+\frac{7 b (b d-a e)^6 \log (d+e x)}{e^8}+\frac{b^7 (d+e x)^6}{6 e^8} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-21*b^2*(b*d - a*e)^5*x)/e^7 + (b*d - a*e)^7/(e^8*(d + e*x)) + (35*b^3*(b*d - a*e)^4*(d + e*x)^2)/(2*e^8) - (
35*b^4*(b*d - a*e)^3*(d + e*x)^3)/(3*e^8) + (21*b^5*(b*d - a*e)^2*(d + e*x)^4)/(4*e^8) - (7*b^6*(b*d - a*e)*(d
+ e*x)^5)/(5*e^8) + (b^7*(d + e*x)^6)/(6*e^8) + (7*b*(b*d - a*e)^6*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)^2} \, dx &=\int \frac{(a+b x)^7}{(d+e x)^2} \, dx\\ &=\int \left (-\frac{21 b^2 (b d-a e)^5}{e^7}+\frac{(-b d+a e)^7}{e^7 (d+e x)^2}+\frac{7 b (b d-a e)^6}{e^7 (d+e x)}+\frac{35 b^3 (b d-a e)^4 (d+e x)}{e^7}-\frac{35 b^4 (b d-a e)^3 (d+e x)^2}{e^7}+\frac{21 b^5 (b d-a e)^2 (d+e x)^3}{e^7}-\frac{7 b^6 (b d-a e) (d+e x)^4}{e^7}+\frac{b^7 (d+e x)^5}{e^7}\right ) \, dx\\ &=-\frac{21 b^2 (b d-a e)^5 x}{e^7}+\frac{(b d-a e)^7}{e^8 (d+e x)}+\frac{35 b^3 (b d-a e)^4 (d+e x)^2}{2 e^8}-\frac{35 b^4 (b d-a e)^3 (d+e x)^3}{3 e^8}+\frac{21 b^5 (b d-a e)^2 (d+e x)^4}{4 e^8}-\frac{7 b^6 (b d-a e) (d+e x)^5}{5 e^8}+\frac{b^7 (d+e x)^6}{6 e^8}+\frac{7 b (b d-a e)^6 \log (d+e x)}{e^8}\\ \end{align*}

Mathematica [B]  time = 0.11972, size = 387, normalized size = 2.08 \[ \frac{105 a^2 b^5 e^2 \left (-30 d^3 e^2 x^2+10 d^2 e^3 x^3-48 d^4 e x+12 d^5-5 d e^4 x^4+3 e^5 x^5\right )+700 a^3 b^4 e^3 \left (6 d^2 e^2 x^2+9 d^3 e x-3 d^4-2 d e^3 x^3+e^4 x^4\right )+1050 a^4 b^3 e^4 \left (-4 d^2 e x+2 d^3-3 d e^2 x^2+e^3 x^3\right )+1260 a^5 b^2 e^5 \left (-d^2+d e x+e^2 x^2\right )+420 a^6 b d e^6-60 a^7 e^7+42 a b^6 e \left (30 d^4 e^2 x^2-10 d^3 e^3 x^3+5 d^2 e^4 x^4+50 d^5 e x-10 d^6-3 d e^5 x^5+2 e^6 x^6\right )+420 b (d+e x) (b d-a e)^6 \log (d+e x)+b^7 \left (-210 d^5 e^2 x^2+70 d^4 e^3 x^3-35 d^3 e^4 x^4+21 d^2 e^5 x^5-360 d^6 e x+60 d^7-14 d e^6 x^6+10 e^7 x^7\right )}{60 e^8 (d+e x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(420*a^6*b*d*e^6 - 60*a^7*e^7 + 1260*a^5*b^2*e^5*(-d^2 + d*e*x + e^2*x^2) + 1050*a^4*b^3*e^4*(2*d^3 - 4*d^2*e*
x - 3*d*e^2*x^2 + e^3*x^3) + 700*a^3*b^4*e^3*(-3*d^4 + 9*d^3*e*x + 6*d^2*e^2*x^2 - 2*d*e^3*x^3 + e^4*x^4) + 10
5*a^2*b^5*e^2*(12*d^5 - 48*d^4*e*x - 30*d^3*e^2*x^2 + 10*d^2*e^3*x^3 - 5*d*e^4*x^4 + 3*e^5*x^5) + 42*a*b^6*e*(
-10*d^6 + 50*d^5*e*x + 30*d^4*e^2*x^2 - 10*d^3*e^3*x^3 + 5*d^2*e^4*x^4 - 3*d*e^5*x^5 + 2*e^6*x^6) + b^7*(60*d^
7 - 360*d^6*e*x - 210*d^5*e^2*x^2 + 70*d^4*e^3*x^3 - 35*d^3*e^4*x^4 + 21*d^2*e^5*x^5 - 14*d*e^6*x^6 + 10*e^7*x
^7) + 420*b*(b*d - a*e)^6*(d + e*x)*Log[d + e*x])/(60*e^8*(d + e*x))

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Maple [B]  time = 0.013, size = 571, normalized size = 3.1 \begin{align*}{\frac{7\,{b}^{6}{x}^{5}a}{5\,{e}^{2}}}-{\frac{2\,{b}^{7}{x}^{5}d}{5\,{e}^{3}}}+{\frac{21\,{b}^{5}{x}^{4}{a}^{2}}{4\,{e}^{2}}}+{\frac{3\,{b}^{7}{x}^{4}{d}^{2}}{4\,{e}^{4}}}+{\frac{35\,{b}^{4}{x}^{3}{a}^{3}}{3\,{e}^{2}}}-{\frac{4\,{b}^{7}{x}^{3}{d}^{3}}{3\,{e}^{5}}}+{\frac{35\,{b}^{3}{x}^{2}{a}^{4}}{2\,{e}^{2}}}+{\frac{5\,{b}^{7}{x}^{2}{d}^{4}}{2\,{e}^{6}}}+21\,{\frac{{a}^{5}{b}^{2}x}{{e}^{2}}}-6\,{\frac{{b}^{7}{d}^{5}x}{{e}^{7}}}+7\,{\frac{b\ln \left ( ex+d \right ){a}^{6}}{{e}^{2}}}+7\,{\frac{{b}^{7}\ln \left ( ex+d \right ){d}^{6}}{{e}^{8}}}+{\frac{{b}^{7}{d}^{7}}{{e}^{8} \left ( ex+d \right ) }}-{\frac{{a}^{7}}{e \left ( ex+d \right ) }}+{\frac{{b}^{7}{x}^{6}}{6\,{e}^{2}}}-84\,{\frac{{a}^{2}{d}^{3}{b}^{5}x}{{e}^{5}}}+35\,{\frac{a{d}^{4}{b}^{6}x}{{e}^{6}}}+7\,{\frac{{a}^{6}db}{{e}^{2} \left ( ex+d \right ) }}-21\,{\frac{{a}^{5}{d}^{2}{b}^{2}}{{e}^{3} \left ( ex+d \right ) }}+105\,{\frac{{b}^{3}\ln \left ( ex+d \right ){a}^{4}{d}^{2}}{{e}^{4}}}-140\,{\frac{{b}^{4}\ln \left ( ex+d \right ){a}^{3}{d}^{3}}{{e}^{5}}}+105\,{\frac{{b}^{5}\ln \left ( ex+d \right ){a}^{2}{d}^{4}}{{e}^{6}}}-42\,{\frac{{b}^{6}\ln \left ( ex+d \right ) a{d}^{5}}{{e}^{7}}}+21\,{\frac{{a}^{2}{d}^{5}{b}^{5}}{{e}^{6} \left ( ex+d \right ) }}-7\,{\frac{a{d}^{6}{b}^{6}}{{e}^{7} \left ( ex+d \right ) }}-42\,{\frac{{b}^{2}\ln \left ( ex+d \right ){a}^{5}d}{{e}^{3}}}+35\,{\frac{{a}^{4}{d}^{3}{b}^{3}}{{e}^{4} \left ( ex+d \right ) }}-35\,{\frac{{a}^{3}{d}^{4}{b}^{4}}{{e}^{5} \left ( ex+d \right ) }}-{\frac{7\,{b}^{6}{x}^{4}ad}{2\,{e}^{3}}}+{\frac{63\,{b}^{5}{x}^{2}{a}^{2}{d}^{2}}{2\,{e}^{4}}}-14\,{\frac{{b}^{5}{x}^{3}{a}^{2}d}{{e}^{3}}}-14\,{\frac{{b}^{6}{x}^{2}a{d}^{3}}{{e}^{5}}}-70\,{\frac{{a}^{4}d{b}^{3}x}{{e}^{3}}}+105\,{\frac{{a}^{3}{d}^{2}{b}^{4}x}{{e}^{4}}}+7\,{\frac{{b}^{6}{x}^{3}a{d}^{2}}{{e}^{4}}}-35\,{\frac{{b}^{4}{x}^{2}{a}^{3}d}{{e}^{3}}} \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)^2,x)

[Out]

7/5*b^6/e^2*x^5*a-2/5*b^7/e^3*x^5*d+21/4*b^5/e^2*x^4*a^2+3/4*b^7/e^4*x^4*d^2+35/3*b^4/e^2*x^3*a^3-4/3*b^7/e^5*
x^3*d^3+35/2*b^3/e^2*x^2*a^4+5/2*b^7/e^6*x^2*d^4+21*b^2/e^2*a^5*x-6*b^7/e^7*d^5*x+7*b/e^2*ln(e*x+d)*a^6+7*b^7/
e^8*ln(e*x+d)*d^6+1/e^8/(e*x+d)*b^7*d^7-1/e/(e*x+d)*a^7+1/6*b^7/e^2*x^6-84*b^5/e^5*a^2*d^3*x+35*b^6/e^6*a*d^4*
x+7/e^2/(e*x+d)*d*a^6*b-21/e^3/(e*x+d)*a^5*b^2*d^2+105*b^3/e^4*ln(e*x+d)*a^4*d^2-140*b^4/e^5*ln(e*x+d)*a^3*d^3
+105*b^5/e^6*ln(e*x+d)*a^2*d^4-42*b^6/e^7*ln(e*x+d)*a*d^5+21/e^6/(e*x+d)*a^2*b^5*d^5-7/e^7/(e*x+d)*a*b^6*d^6-4
2*b^2/e^3*ln(e*x+d)*a^5*d+35/e^4/(e*x+d)*a^4*b^3*d^3-35/e^5/(e*x+d)*a^3*b^4*d^4-7/2*b^6/e^3*x^4*a*d+63/2*b^5/e
^4*x^2*a^2*d^2-14*b^5/e^3*x^3*a^2*d-14*b^6/e^5*x^2*a*d^3-70*b^3/e^3*a^4*d*x+105*b^4/e^4*a^3*d^2*x+7*b^6/e^4*x^
3*a*d^2-35*b^4/e^3*x^2*a^3*d

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Maxima [B]  time = 0.972245, size = 629, normalized size = 3.38 \begin{align*} \frac{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^{9} x + d e^{8}} + \frac{10 \, b^{7} e^{5} x^{6} - 12 \,{\left (2 \, b^{7} d e^{4} - 7 \, a b^{6} e^{5}\right )} x^{5} + 15 \,{\left (3 \, b^{7} d^{2} e^{3} - 14 \, a b^{6} d e^{4} + 21 \, a^{2} b^{5} e^{5}\right )} x^{4} - 20 \,{\left (4 \, b^{7} d^{3} e^{2} - 21 \, a b^{6} d^{2} e^{3} + 42 \, a^{2} b^{5} d e^{4} - 35 \, a^{3} b^{4} e^{5}\right )} x^{3} + 30 \,{\left (5 \, b^{7} d^{4} e - 28 \, a b^{6} d^{3} e^{2} + 63 \, a^{2} b^{5} d^{2} e^{3} - 70 \, a^{3} b^{4} d e^{4} + 35 \, a^{4} b^{3} e^{5}\right )} x^{2} - 60 \,{\left (6 \, b^{7} d^{5} - 35 \, a b^{6} d^{4} e + 84 \, a^{2} b^{5} d^{3} e^{2} - 105 \, a^{3} b^{4} d^{2} e^{3} + 70 \, a^{4} b^{3} d e^{4} - 21 \, a^{5} b^{2} e^{5}\right )} x}{60 \, e^{7}} + \frac{7 \,{\left (b^{7} d^{6} - 6 \, a b^{6} d^{5} e + 15 \, a^{2} b^{5} d^{4} e^{2} - 20 \, a^{3} b^{4} d^{3} e^{3} + 15 \, a^{4} b^{3} d^{2} e^{4} - 6 \, a^{5} b^{2} d e^{5} + a^{6} b e^{6}\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)^2,x, algorithm="maxima")

[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^9*x + d*e^8) + 1/60*(10*b^7*e^5*x^6 - 12*(2*b^7*d*e^4 - 7*a*b^6*e^5)*x^5 + 15*(3*
b^7*d^2*e^3 - 14*a*b^6*d*e^4 + 21*a^2*b^5*e^5)*x^4 - 20*(4*b^7*d^3*e^2 - 21*a*b^6*d^2*e^3 + 42*a^2*b^5*d*e^4 -
35*a^3*b^4*e^5)*x^3 + 30*(5*b^7*d^4*e - 28*a*b^6*d^3*e^2 + 63*a^2*b^5*d^2*e^3 - 70*a^3*b^4*d*e^4 + 35*a^4*b^3
*e^5)*x^2 - 60*(6*b^7*d^5 - 35*a*b^6*d^4*e + 84*a^2*b^5*d^3*e^2 - 105*a^3*b^4*d^2*e^3 + 70*a^4*b^3*d*e^4 - 21*
a^5*b^2*e^5)*x)/e^7 + 7*(b^7*d^6 - 6*a*b^6*d^5*e + 15*a^2*b^5*d^4*e^2 - 20*a^3*b^4*d^3*e^3 + 15*a^4*b^3*d^2*e^
4 - 6*a^5*b^2*d*e^5 + a^6*b*e^6)*log(e*x + d)/e^8

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Fricas [B]  time = 1.56819, size = 1310, normalized size = 7.04 \begin{align*} \frac{10 \, b^{7} e^{7} x^{7} + 60 \, b^{7} d^{7} - 420 \, a b^{6} d^{6} e + 1260 \, a^{2} b^{5} d^{5} e^{2} - 2100 \, a^{3} b^{4} d^{4} e^{3} + 2100 \, a^{4} b^{3} d^{3} e^{4} - 1260 \, a^{5} b^{2} d^{2} e^{5} + 420 \, a^{6} b d e^{6} - 60 \, a^{7} e^{7} - 14 \,{\left (b^{7} d e^{6} - 6 \, a b^{6} e^{7}\right )} x^{6} + 21 \,{\left (b^{7} d^{2} e^{5} - 6 \, a b^{6} d e^{6} + 15 \, a^{2} b^{5} e^{7}\right )} x^{5} - 35 \,{\left (b^{7} d^{3} e^{4} - 6 \, a b^{6} d^{2} e^{5} + 15 \, a^{2} b^{5} d e^{6} - 20 \, a^{3} b^{4} e^{7}\right )} x^{4} + 70 \,{\left (b^{7} d^{4} e^{3} - 6 \, a b^{6} d^{3} e^{4} + 15 \, a^{2} b^{5} d^{2} e^{5} - 20 \, a^{3} b^{4} d e^{6} + 15 \, a^{4} b^{3} e^{7}\right )} x^{3} - 210 \,{\left (b^{7} d^{5} e^{2} - 6 \, a b^{6} d^{4} e^{3} + 15 \, a^{2} b^{5} d^{3} e^{4} - 20 \, a^{3} b^{4} d^{2} e^{5} + 15 \, a^{4} b^{3} d e^{6} - 6 \, a^{5} b^{2} e^{7}\right )} x^{2} - 60 \,{\left (6 \, b^{7} d^{6} e - 35 \, a b^{6} d^{5} e^{2} + 84 \, a^{2} b^{5} d^{4} e^{3} - 105 \, a^{3} b^{4} d^{3} e^{4} + 70 \, a^{4} b^{3} d^{2} e^{5} - 21 \, a^{5} b^{2} d e^{6}\right )} x + 420 \,{\left (b^{7} d^{7} - 6 \, a b^{6} d^{6} e + 15 \, a^{2} b^{5} d^{5} e^{2} - 20 \, a^{3} b^{4} d^{4} e^{3} + 15 \, a^{4} b^{3} d^{3} e^{4} - 6 \, a^{5} b^{2} d^{2} e^{5} + a^{6} b d e^{6} +{\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 )} \log \left (e x + d\right )}{60 \,{\left (e^{9} x + d 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)^2,x, algorithm="fricas")

[Out]

1/60*(10*b^7*e^7*x^7 + 60*b^7*d^7 - 420*a*b^6*d^6*e + 1260*a^2*b^5*d^5*e^2 - 2100*a^3*b^4*d^4*e^3 + 2100*a^4*b
^3*d^3*e^4 - 1260*a^5*b^2*d^2*e^5 + 420*a^6*b*d*e^6 - 60*a^7*e^7 - 14*(b^7*d*e^6 - 6*a*b^6*e^7)*x^6 + 21*(b^7*
d^2*e^5 - 6*a*b^6*d*e^6 + 15*a^2*b^5*e^7)*x^5 - 35*(b^7*d^3*e^4 - 6*a*b^6*d^2*e^5 + 15*a^2*b^5*d*e^6 - 20*a^3*
b^4*e^7)*x^4 + 70*(b^7*d^4*e^3 - 6*a*b^6*d^3*e^4 + 15*a^2*b^5*d^2*e^5 - 20*a^3*b^4*d*e^6 + 15*a^4*b^3*e^7)*x^3
- 210*(b^7*d^5*e^2 - 6*a*b^6*d^4*e^3 + 15*a^2*b^5*d^3*e^4 - 20*a^3*b^4*d^2*e^5 + 15*a^4*b^3*d*e^6 - 6*a^5*b^2
*e^7)*x^2 - 60*(6*b^7*d^6*e - 35*a*b^6*d^5*e^2 + 84*a^2*b^5*d^4*e^3 - 105*a^3*b^4*d^3*e^4 + 70*a^4*b^3*d^2*e^5
- 21*a^5*b^2*d*e^6)*x + 420*(b^7*d^7 - 6*a*b^6*d^6*e + 15*a^2*b^5*d^5*e^2 - 20*a^3*b^4*d^4*e^3 + 15*a^4*b^3*d
^3*e^4 - 6*a^5*b^2*d^2*e^5 + a^6*b*d*e^6 + (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)*log(e*x + d))/(e^9*x + d*e^8)

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Sympy [B]  time = 1.79035, size = 410, normalized size = 2.2 \begin{align*} \frac{b^{7} x^{6}}{6 e^{2}} + \frac{7 b \left (a e - b d\right )^{6} \log{\left (d + e x \right )}}{e^{8}} - \frac{a^{7} e^{7} - 7 a^{6} b d e^{6} + 21 a^{5} b^{2} d^{2} e^{5} - 35 a^{4} b^{3} d^{3} e^{4} + 35 a^{3} b^{4} d^{4} e^{3} - 21 a^{2} b^{5} d^{5} e^{2} + 7 a b^{6} d^{6} e - b^{7} d^{7}}{d e^{8} + e^{9} x} + \frac{x^{5} \left (7 a b^{6} e - 2 b^{7} d\right )}{5 e^{3}} + \frac{x^{4} \left (21 a^{2} b^{5} e^{2} - 14 a b^{6} d e + 3 b^{7} d^{2}\right )}{4 e^{4}} + \frac{x^{3} \left (35 a^{3} b^{4} e^{3} - 42 a^{2} b^{5} d e^{2} + 21 a b^{6} d^{2} e - 4 b^{7} d^{3}\right )}{3 e^{5}} + \frac{x^{2} \left (35 a^{4} b^{3} e^{4} - 70 a^{3} b^{4} d e^{3} + 63 a^{2} b^{5} d^{2} e^{2} - 28 a b^{6} d^{3} e + 5 b^{7} d^{4}\right )}{2 e^{6}} + \frac{x \left (21 a^{5} b^{2} e^{5} - 70 a^{4} b^{3} d e^{4} + 105 a^{3} b^{4} d^{2} e^{3} - 84 a^{2} b^{5} d^{3} e^{2} + 35 a b^{6} d^{4} e - 6 b^{7} d^{5}\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)**2,x)

[Out]

b**7*x**6/(6*e**2) + 7*b*(a*e - b*d)**6*log(d + e*x)/e**8 - (a**7*e**7 - 7*a**6*b*d*e**6 + 21*a**5*b**2*d**2*e
**5 - 35*a**4*b**3*d**3*e**4 + 35*a**3*b**4*d**4*e**3 - 21*a**2*b**5*d**5*e**2 + 7*a*b**6*d**6*e - b**7*d**7)/
(d*e**8 + e**9*x) + x**5*(7*a*b**6*e - 2*b**7*d)/(5*e**3) + x**4*(21*a**2*b**5*e**2 - 14*a*b**6*d*e + 3*b**7*d
**2)/(4*e**4) + x**3*(35*a**3*b**4*e**3 - 42*a**2*b**5*d*e**2 + 21*a*b**6*d**2*e - 4*b**7*d**3)/(3*e**5) + x**
2*(35*a**4*b**3*e**4 - 70*a**3*b**4*d*e**3 + 63*a**2*b**5*d**2*e**2 - 28*a*b**6*d**3*e + 5*b**7*d**4)/(2*e**6)
+ x*(21*a**5*b**2*e**5 - 70*a**4*b**3*d*e**4 + 105*a**3*b**4*d**2*e**3 - 84*a**2*b**5*d**3*e**2 + 35*a*b**6*d
**4*e - 6*b**7*d**5)/e**7

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Giac [B]  time = 1.1109, size = 732, normalized size = 3.94 \begin{align*} \frac{1}{60} \,{\left (10 \, b^{7} - \frac{84 \,{\left (b^{7} d e - a b^{6} e^{2}\right )} e^{\left (-1\right )}}{x e + d} + \frac{315 \,{\left (b^{7} d^{2} e^{2} - 2 \, a b^{6} d e^{3} + a^{2} b^{5} e^{4}\right )} e^{\left (-2\right )}}{{\left (x e + d\right )}^{2}} - \frac{700 \,{\left (b^{7} d^{3} e^{3} - 3 \, a b^{6} d^{2} e^{4} + 3 \, a^{2} b^{5} d e^{5} - a^{3} b^{4} e^{6}\right )} e^{\left (-3\right )}}{{\left (x e + d\right )}^{3}} + \frac{1050 \,{\left (b^{7} d^{4} e^{4} - 4 \, a b^{6} d^{3} e^{5} + 6 \, a^{2} b^{5} d^{2} e^{6} - 4 \, a^{3} b^{4} d e^{7} + a^{4} b^{3} e^{8}\right )} e^{\left (-4\right )}}{{\left (x e + d\right )}^{4}} - \frac{1260 \,{\left (b^{7} d^{5} e^{5} - 5 \, a b^{6} d^{4} e^{6} + 10 \, a^{2} b^{5} d^{3} e^{7} - 10 \, a^{3} b^{4} d^{2} e^{8} + 5 \, a^{4} b^{3} d e^{9} - a^{5} b^{2} e^{10}\right )} e^{\left (-5\right )}}{{\left (x e + d\right )}^{5}}\right )}{\left (x e + d\right )}^{6} e^{\left (-8\right )} - 7 \,{\left (b^{7} d^{6} - 6 \, a b^{6} d^{5} e + 15 \, a^{2} b^{5} d^{4} e^{2} - 20 \, a^{3} b^{4} d^{3} e^{3} + 15 \, a^{4} b^{3} d^{2} e^{4} - 6 \, a^{5} b^{2} d e^{5} + a^{6} b e^{6}\right )} e^{\left (-8\right )} \log \left (\frac{{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) +{\left (\frac{b^{7} d^{7} e^{6}}{x e + d} - \frac{7 \, a b^{6} d^{6} e^{7}}{x e + d} + \frac{21 \, a^{2} b^{5} d^{5} e^{8}}{x e + d} - \frac{35 \, a^{3} b^{4} d^{4} e^{9}}{x e + d} + \frac{35 \, a^{4} b^{3} d^{3} e^{10}}{x e + d} - \frac{21 \, a^{5} b^{2} d^{2} e^{11}}{x e + d} + \frac{7 \, a^{6} b d e^{12}}{x e + d} - \frac{a^{7} e^{13}}{x e + d}\right )} e^{\left (-14\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)^2,x, algorithm="giac")

[Out]

1/60*(10*b^7 - 84*(b^7*d*e - a*b^6*e^2)*e^(-1)/(x*e + d) + 315*(b^7*d^2*e^2 - 2*a*b^6*d*e^3 + a^2*b^5*e^4)*e^(
-2)/(x*e + d)^2 - 700*(b^7*d^3*e^3 - 3*a*b^6*d^2*e^4 + 3*a^2*b^5*d*e^5 - a^3*b^4*e^6)*e^(-3)/(x*e + d)^3 + 105
0*(b^7*d^4*e^4 - 4*a*b^6*d^3*e^5 + 6*a^2*b^5*d^2*e^6 - 4*a^3*b^4*d*e^7 + a^4*b^3*e^8)*e^(-4)/(x*e + d)^4 - 126
0*(b^7*d^5*e^5 - 5*a*b^6*d^4*e^6 + 10*a^2*b^5*d^3*e^7 - 10*a^3*b^4*d^2*e^8 + 5*a^4*b^3*d*e^9 - a^5*b^2*e^10)*e
^(-5)/(x*e + d)^5)*(x*e + d)^6*e^(-8) - 7*(b^7*d^6 - 6*a*b^6*d^5*e + 15*a^2*b^5*d^4*e^2 - 20*a^3*b^4*d^3*e^3 +
15*a^4*b^3*d^2*e^4 - 6*a^5*b^2*d*e^5 + a^6*b*e^6)*e^(-8)*log(abs(x*e + d)*e^(-1)/(x*e + d)^2) + (b^7*d^7*e^6/
(x*e + d) - 7*a*b^6*d^6*e^7/(x*e + d) + 21*a^2*b^5*d^5*e^8/(x*e + d) - 35*a^3*b^4*d^4*e^9/(x*e + d) + 35*a^4*b
^3*d^3*e^10/(x*e + d) - 21*a^5*b^2*d^2*e^11/(x*e + d) + 7*a^6*b*d*e^12/(x*e + d) - a^7*e^13/(x*e + d))*e^(-14)