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

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Rubi [A]  time = 0.22, antiderivative size = 185, normalized size of antiderivative = 1.00, 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*}

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Mathematica [B]  time = 0.12, size = 388, normalized size = 2.10 \[ \frac {-10 a^7 e^7-70 a^6 b e^6 (d+2 e x)+210 a^5 b^2 d e^5 (3 d+4 e x)+350 a^4 b^3 e^4 \left (-5 d^3-4 d^2 e x+4 d e^2 x^2+2 e^3 x^3\right )+350 a^3 b^4 e^3 \left (7 d^4+2 d^3 e x-11 d^2 e^2 x^2-4 d e^3 x^3+e^4 x^4\right )+70 a^2 b^5 e^2 \left (-27 d^5+6 d^4 e x+63 d^3 e^2 x^2+20 d^2 e^3 x^3-5 d e^4 x^4+2 e^5 x^5\right )+35 a b^6 e \left (22 d^6-16 d^5 e x-68 d^4 e^2 x^2-20 d^3 e^3 x^3+5 d^2 e^4 x^4-2 d e^5 x^5+e^6 x^6\right )-420 b^2 (d+e x)^2 (b d-a e)^5 \log (d+e x)+b^7 \left (-130 d^7+160 d^6 e x+500 d^5 e^2 x^2+140 d^4 e^3 x^3-35 d^3 e^4 x^4+14 d^2 e^5 x^5-7 d e^6 x^6+4 e^7 x^7\right )}{20 e^8 (d+e x)^2} \]

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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fricas [B]  time = 1.84, size = 701, normalized size = 3.79 \[ \frac {4 \, b^{7} e^{7} x^{7} - 130 \, b^{7} d^{7} + 770 \, a b^{6} d^{6} e - 1890 \, a^{2} b^{5} d^{5} e^{2} + 2450 \, a^{3} b^{4} d^{4} e^{3} - 1750 \, a^{4} b^{3} d^{3} e^{4} + 630 \, a^{5} b^{2} d^{2} e^{5} - 70 \, a^{6} b d e^{6} - 10 \, a^{7} e^{7} - 7 \, {\left (b^{7} d e^{6} - 5 \, a b^{6} e^{7}\right )} x^{6} + 14 \, {\left (b^{7} d^{2} e^{5} - 5 \, a b^{6} d e^{6} + 10 \, a^{2} b^{5} e^{7}\right )} x^{5} - 35 \, {\left (b^{7} d^{3} e^{4} - 5 \, a b^{6} d^{2} e^{5} + 10 \, a^{2} b^{5} d e^{6} - 10 \, a^{3} b^{4} e^{7}\right )} x^{4} + 140 \, {\left (b^{7} d^{4} e^{3} - 5 \, a b^{6} d^{3} e^{4} + 10 \, a^{2} b^{5} d^{2} e^{5} - 10 \, a^{3} b^{4} d e^{6} + 5 \, a^{4} b^{3} e^{7}\right )} x^{3} + 10 \, {\left (50 \, b^{7} d^{5} e^{2} - 238 \, a b^{6} d^{4} e^{3} + 441 \, a^{2} b^{5} d^{3} e^{4} - 385 \, a^{3} b^{4} d^{2} e^{5} + 140 \, a^{4} b^{3} d e^{6}\right )} x^{2} + 20 \, {\left (8 \, b^{7} d^{6} e - 28 \, a b^{6} d^{5} e^{2} + 21 \, a^{2} b^{5} d^{4} e^{3} + 35 \, a^{3} b^{4} d^{3} e^{4} - 70 \, a^{4} b^{3} d^{2} e^{5} + 42 \, a^{5} b^{2} d e^{6} - 7 \, a^{6} b e^{7}\right )} x - 420 \, {\left (b^{7} d^{7} - 5 \, a b^{6} d^{6} e + 10 \, a^{2} b^{5} d^{5} e^{2} - 10 \, a^{3} b^{4} d^{4} e^{3} + 5 \, a^{4} b^{3} d^{3} e^{4} - a^{5} b^{2} d^{2} e^{5} + {\left (b^{7} d^{5} e^{2} - 5 \, a b^{6} d^{4} e^{3} + 10 \, a^{2} b^{5} d^{3} e^{4} - 10 \, a^{3} b^{4} d^{2} e^{5} + 5 \, a^{4} b^{3} d e^{6} - a^{5} b^{2} e^{7}\right )} x^{2} + 2 \, {\left (b^{7} d^{6} e - 5 \, a b^{6} d^{5} e^{2} + 10 \, a^{2} b^{5} d^{4} e^{3} - 10 \, a^{3} b^{4} d^{3} e^{4} + 5 \, a^{4} b^{3} d^{2} e^{5} - a^{5} b^{2} d e^{6}\right )} x\right )} \log \left (e x + d\right )}{20 \, {\left (e^{10} x^{2} + 2 \, d e^{9} x + d^{2} e^{8}\right )}} \]

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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giac [B]  time = 0.21, size = 448, normalized size = 2.42 \[ -21 \, {\left (b^{7} d^{5} - 5 \, a b^{6} d^{4} e + 10 \, a^{2} b^{5} d^{3} e^{2} - 10 \, a^{3} b^{4} d^{2} e^{3} + 5 \, a^{4} b^{3} d e^{4} - a^{5} b^{2} e^{5}\right )} e^{\left (-8\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {1}{20} \, {\left (4 \, b^{7} x^{5} e^{12} - 15 \, b^{7} d x^{4} e^{11} + 40 \, b^{7} d^{2} x^{3} e^{10} - 100 \, b^{7} d^{3} x^{2} e^{9} + 300 \, b^{7} d^{4} x e^{8} + 35 \, a b^{6} x^{4} e^{12} - 140 \, a b^{6} d x^{3} e^{11} + 420 \, a b^{6} d^{2} x^{2} e^{10} - 1400 \, a b^{6} d^{3} x e^{9} + 140 \, a^{2} b^{5} x^{3} e^{12} - 630 \, a^{2} b^{5} d x^{2} e^{11} + 2520 \, a^{2} b^{5} d^{2} x e^{10} + 350 \, a^{3} b^{4} x^{2} e^{12} - 2100 \, a^{3} b^{4} d x e^{11} + 700 \, a^{4} b^{3} x e^{12}\right )} e^{\left (-15\right )} - \frac {{\left (13 \, b^{7} d^{7} - 77 \, a b^{6} d^{6} e + 189 \, a^{2} b^{5} d^{5} e^{2} - 245 \, a^{3} b^{4} d^{4} e^{3} + 175 \, a^{4} b^{3} d^{3} e^{4} - 63 \, a^{5} b^{2} d^{2} e^{5} + 7 \, a^{6} b d e^{6} + a^{7} e^{7} + 14 \, {\left (b^{7} d^{6} e - 6 \, a b^{6} d^{5} e^{2} + 15 \, a^{2} b^{5} d^{4} e^{3} - 20 \, a^{3} b^{4} d^{3} e^{4} + 15 \, a^{4} b^{3} d^{2} e^{5} - 6 \, a^{5} b^{2} d e^{6} + a^{6} b e^{7}\right )} x\right )} e^{\left (-8\right )}}{2 \, {\left (x e + d\right )}^{2}} \]

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

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maple [B]  time = 0.06, size = 599, normalized size = 3.24 \[ \frac {b^{7} x^{5}}{5 e^{3}}+\frac {7 a \,b^{6} x^{4}}{4 e^{3}}-\frac {3 b^{7} d \,x^{4}}{4 e^{4}}+\frac {7 a^{2} b^{5} x^{3}}{e^{3}}-\frac {7 a \,b^{6} d \,x^{3}}{e^{4}}+\frac {2 b^{7} d^{2} x^{3}}{e^{5}}-\frac {a^{7}}{2 \left (e x +d \right )^{2} e}+\frac {7 a^{6} b d}{2 \left (e x +d \right )^{2} e^{2}}-\frac {21 a^{5} b^{2} d^{2}}{2 \left (e x +d \right )^{2} e^{3}}+\frac {35 a^{4} b^{3} d^{3}}{2 \left (e x +d \right )^{2} e^{4}}-\frac {35 a^{3} b^{4} d^{4}}{2 \left (e x +d \right )^{2} e^{5}}+\frac {35 a^{3} b^{4} x^{2}}{2 e^{3}}+\frac {21 a^{2} b^{5} d^{5}}{2 \left (e x +d \right )^{2} e^{6}}-\frac {63 a^{2} b^{5} d \,x^{2}}{2 e^{4}}-\frac {7 a \,b^{6} d^{6}}{2 \left (e x +d \right )^{2} e^{7}}+\frac {21 a \,b^{6} d^{2} x^{2}}{e^{5}}+\frac {b^{7} d^{7}}{2 \left (e x +d \right )^{2} e^{8}}-\frac {5 b^{7} d^{3} x^{2}}{e^{6}}-\frac {7 a^{6} b}{\left (e x +d \right ) e^{2}}+\frac {42 a^{5} b^{2} d}{\left (e x +d \right ) e^{3}}+\frac {21 a^{5} b^{2} \ln \left (e x +d \right )}{e^{3}}-\frac {105 a^{4} b^{3} d^{2}}{\left (e x +d \right ) e^{4}}-\frac {105 a^{4} b^{3} d \ln \left (e x +d \right )}{e^{4}}+\frac {35 a^{4} b^{3} x}{e^{3}}+\frac {140 a^{3} b^{4} d^{3}}{\left (e x +d \right ) e^{5}}+\frac {210 a^{3} b^{4} d^{2} \ln \left (e x +d \right )}{e^{5}}-\frac {105 a^{3} b^{4} d x}{e^{4}}-\frac {105 a^{2} b^{5} d^{4}}{\left (e x +d \right ) e^{6}}-\frac {210 a^{2} b^{5} d^{3} \ln \left (e x +d \right )}{e^{6}}+\frac {126 a^{2} b^{5} d^{2} x}{e^{5}}+\frac {42 a \,b^{6} d^{5}}{\left (e x +d \right ) e^{7}}+\frac {105 a \,b^{6} d^{4} \ln \left (e x +d \right )}{e^{7}}-\frac {70 a \,b^{6} d^{3} x}{e^{6}}-\frac {7 b^{7} d^{6}}{\left (e x +d \right ) e^{8}}-\frac {21 b^{7} d^{5} \ln \left (e x +d \right )}{e^{8}}+\frac {15 b^{7} d^{4} x}{e^{7}} \]

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]

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

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maxima [B]  time = 0.67, size = 473, normalized size = 2.56 \[ -\frac {13 \, b^{7} d^{7} - 77 \, a b^{6} d^{6} e + 189 \, a^{2} b^{5} d^{5} e^{2} - 245 \, a^{3} b^{4} d^{4} e^{3} + 175 \, a^{4} b^{3} d^{3} e^{4} - 63 \, a^{5} b^{2} d^{2} e^{5} + 7 \, a^{6} b d e^{6} + a^{7} e^{7} + 14 \, {\left (b^{7} d^{6} e - 6 \, a b^{6} d^{5} e^{2} + 15 \, a^{2} b^{5} d^{4} e^{3} - 20 \, a^{3} b^{4} d^{3} e^{4} + 15 \, a^{4} b^{3} d^{2} e^{5} - 6 \, a^{5} b^{2} d e^{6} + a^{6} b e^{7}\right )} x}{2 \, {\left (e^{10} x^{2} + 2 \, d e^{9} x + d^{2} e^{8}\right )}} + \frac {4 \, b^{7} e^{4} x^{5} - 5 \, {\left (3 \, b^{7} d e^{3} - 7 \, a b^{6} e^{4}\right )} x^{4} + 20 \, {\left (2 \, b^{7} d^{2} e^{2} - 7 \, a b^{6} d e^{3} + 7 \, a^{2} b^{5} e^{4}\right )} x^{3} - 10 \, {\left (10 \, b^{7} d^{3} e - 42 \, a b^{6} d^{2} e^{2} + 63 \, a^{2} b^{5} d e^{3} - 35 \, a^{3} b^{4} e^{4}\right )} x^{2} + 20 \, {\left (15 \, b^{7} d^{4} - 70 \, a b^{6} d^{3} e + 126 \, a^{2} b^{5} d^{2} e^{2} - 105 \, a^{3} b^{4} d e^{3} + 35 \, a^{4} b^{3} e^{4}\right )} x}{20 \, e^{7}} - \frac {21 \, {\left (b^{7} d^{5} - 5 \, a b^{6} d^{4} e + 10 \, a^{2} b^{5} d^{3} e^{2} - 10 \, a^{3} b^{4} d^{2} e^{3} + 5 \, a^{4} b^{3} d e^{4} - a^{5} b^{2} e^{5}\right )} \log \left (e x + d\right )}{e^{8}} \]

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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mupad [B]  time = 2.02, size = 690, normalized size = 3.73 \[ x^4\,\left (\frac {7\,a\,b^6}{4\,e^3}-\frac {3\,b^7\,d}{4\,e^4}\right )-x^3\,\left (\frac {d\,\left (\frac {7\,a\,b^6}{e^3}-\frac {3\,b^7\,d}{e^4}\right )}{e}-\frac {7\,a^2\,b^5}{e^3}+\frac {b^7\,d^2}{e^5}\right )+x^2\,\left (\frac {35\,a^3\,b^4}{2\,e^3}-\frac {b^7\,d^3}{2\,e^6}+\frac {3\,d\,\left (\frac {3\,d\,\left (\frac {7\,a\,b^6}{e^3}-\frac {3\,b^7\,d}{e^4}\right )}{e}-\frac {21\,a^2\,b^5}{e^3}+\frac {3\,b^7\,d^2}{e^5}\right )}{2\,e}-\frac {3\,d^2\,\left (\frac {7\,a\,b^6}{e^3}-\frac {3\,b^7\,d}{e^4}\right )}{2\,e^2}\right )+x\,\left (\frac {3\,d^2\,\left (\frac {3\,d\,\left (\frac {7\,a\,b^6}{e^3}-\frac {3\,b^7\,d}{e^4}\right )}{e}-\frac {21\,a^2\,b^5}{e^3}+\frac {3\,b^7\,d^2}{e^5}\right )}{e^2}+\frac {35\,a^4\,b^3}{e^3}-\frac {3\,d\,\left (\frac {35\,a^3\,b^4}{e^3}-\frac {b^7\,d^3}{e^6}+\frac {3\,d\,\left (\frac {3\,d\,\left (\frac {7\,a\,b^6}{e^3}-\frac {3\,b^7\,d}{e^4}\right )}{e}-\frac {21\,a^2\,b^5}{e^3}+\frac {3\,b^7\,d^2}{e^5}\right )}{e}-\frac {3\,d^2\,\left (\frac {7\,a\,b^6}{e^3}-\frac {3\,b^7\,d}{e^4}\right )}{e^2}\right )}{e}-\frac {d^3\,\left (\frac {7\,a\,b^6}{e^3}-\frac {3\,b^7\,d}{e^4}\right )}{e^3}\right )-\frac {\frac {a^7\,e^7+7\,a^6\,b\,d\,e^6-63\,a^5\,b^2\,d^2\,e^5+175\,a^4\,b^3\,d^3\,e^4-245\,a^3\,b^4\,d^4\,e^3+189\,a^2\,b^5\,d^5\,e^2-77\,a\,b^6\,d^6\,e+13\,b^7\,d^7}{2\,e}+x\,\left (7\,a^6\,b\,e^6-42\,a^5\,b^2\,d\,e^5+105\,a^4\,b^3\,d^2\,e^4-140\,a^3\,b^4\,d^3\,e^3+105\,a^2\,b^5\,d^4\,e^2-42\,a\,b^6\,d^5\,e+7\,b^7\,d^6\right )}{d^2\,e^7+2\,d\,e^8\,x+e^9\,x^2}-\frac {\ln \left (d+e\,x\right )\,\left (-21\,a^5\,b^2\,e^5+105\,a^4\,b^3\,d\,e^4-210\,a^3\,b^4\,d^2\,e^3+210\,a^2\,b^5\,d^3\,e^2-105\,a\,b^6\,d^4\,e+21\,b^7\,d^5\right )}{e^8}+\frac {b^7\,x^5}{5\,e^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^4*((7*a*b^6)/(4*e^3) - (3*b^7*d)/(4*e^4)) - x^3*((d*((7*a*b^6)/e^3 - (3*b^7*d)/e^4))/e - (7*a^2*b^5)/e^3 + (
b^7*d^2)/e^5) + x^2*((35*a^3*b^4)/(2*e^3) - (b^7*d^3)/(2*e^6) + (3*d*((3*d*((7*a*b^6)/e^3 - (3*b^7*d)/e^4))/e
- (21*a^2*b^5)/e^3 + (3*b^7*d^2)/e^5))/(2*e) - (3*d^2*((7*a*b^6)/e^3 - (3*b^7*d)/e^4))/(2*e^2)) + x*((3*d^2*((
3*d*((7*a*b^6)/e^3 - (3*b^7*d)/e^4))/e - (21*a^2*b^5)/e^3 + (3*b^7*d^2)/e^5))/e^2 + (35*a^4*b^3)/e^3 - (3*d*((
35*a^3*b^4)/e^3 - (b^7*d^3)/e^6 + (3*d*((3*d*((7*a*b^6)/e^3 - (3*b^7*d)/e^4))/e - (21*a^2*b^5)/e^3 + (3*b^7*d^
2)/e^5))/e - (3*d^2*((7*a*b^6)/e^3 - (3*b^7*d)/e^4))/e^2))/e - (d^3*((7*a*b^6)/e^3 - (3*b^7*d)/e^4))/e^3) - ((
a^7*e^7 + 13*b^7*d^7 + 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 -
77*a*b^6*d^6*e + 7*a^6*b*d*e^6)/(2*e) + x*(7*b^7*d^6 + 7*a^6*b*e^6 - 42*a^5*b^2*d*e^5 + 105*a^2*b^5*d^4*e^2 -
140*a^3*b^4*d^3*e^3 + 105*a^4*b^3*d^2*e^4 - 42*a*b^6*d^5*e))/(d^2*e^7 + e^9*x^2 + 2*d*e^8*x) - (log(d + e*x)*(
21*b^7*d^5 - 21*a^5*b^2*e^5 + 105*a^4*b^3*d*e^4 + 210*a^2*b^5*d^3*e^2 - 210*a^3*b^4*d^2*e^3 - 105*a*b^6*d^4*e)
)/e^8 + (b^7*x^5)/(5*e^3)

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sympy [B]  time = 3.02, size = 447, normalized size = 2.42 \[ \frac {b^{7} x^{5}}{5 e^{3}} + \frac {21 b^{2} \left (a e - b d\right )^{5} \log {\left (d + e x \right )}}{e^{8}} + x^{4} \left (\frac {7 a b^{6}}{4 e^{3}} - \frac {3 b^{7} d}{4 e^{4}}\right ) + x^{3} \left (\frac {7 a^{2} b^{5}}{e^{3}} - \frac {7 a b^{6} d}{e^{4}} + \frac {2 b^{7} d^{2}}{e^{5}}\right ) + x^{2} \left (\frac {35 a^{3} b^{4}}{2 e^{3}} - \frac {63 a^{2} b^{5} d}{2 e^{4}} + \frac {21 a b^{6} d^{2}}{e^{5}} - \frac {5 b^{7} d^{3}}{e^{6}}\right ) + x \left (\frac {35 a^{4} b^{3}}{e^{3}} - \frac {105 a^{3} b^{4} d}{e^{4}} + \frac {126 a^{2} b^{5} d^{2}}{e^{5}} - \frac {70 a b^{6} d^{3}}{e^{6}} + \frac {15 b^{7} d^{4}}{e^{7}}\right ) + \frac {- a^{7} e^{7} - 7 a^{6} b d e^{6} + 63 a^{5} b^{2} d^{2} e^{5} - 175 a^{4} b^{3} d^{3} e^{4} + 245 a^{3} b^{4} d^{4} e^{3} - 189 a^{2} b^{5} d^{5} e^{2} + 77 a b^{6} d^{6} e - 13 b^{7} d^{7} + x \left (- 14 a^{6} b e^{7} + 84 a^{5} b^{2} d e^{6} - 210 a^{4} b^{3} d^{2} e^{5} + 280 a^{3} b^{4} d^{3} e^{4} - 210 a^{2} b^{5} d^{4} e^{3} + 84 a b^{6} d^{5} e^{2} - 14 b^{7} d^{6} e\right )}{2 d^{2} e^{8} + 4 d e^{9} x + 2 e^{10} x^{2}} \]

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 + x**4*(7*a*b**6/(4*e**3) - 3*b**7*d/(4*e**4)) +
x**3*(7*a**2*b**5/e**3 - 7*a*b**6*d/e**4 + 2*b**7*d**2/e**5) + x**2*(35*a**3*b**4/(2*e**3) - 63*a**2*b**5*d/(
2*e**4) + 21*a*b**6*d**2/e**5 - 5*b**7*d**3/e**6) + x*(35*a**4*b**3/e**3 - 105*a**3*b**4*d/e**4 + 126*a**2*b**
5*d**2/e**5 - 70*a*b**6*d**3/e**6 + 15*b**7*d**4/e**7) + (-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)

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