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

[Out]

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

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Rubi [A]  time = 0.08, antiderivative size = 170, 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} \[ \frac {b x (b d-a e)^6}{e^7}-\frac {(a+b x)^2 (b d-a e)^5}{2 e^6}+\frac {(a+b x)^3 (b d-a e)^4}{3 e^5}-\frac {(a+b x)^4 (b d-a e)^3}{4 e^4}+\frac {(a+b x)^5 (b d-a e)^2}{5 e^3}-\frac {(a+b x)^6 (b d-a e)}{6 e^2}-\frac {(b d-a e)^7 \log (d+e x)}{e^8}+\frac {(a+b x)^7}{7 e} \]

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*}

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Mathematica [A]  time = 0.11, size = 304, normalized size = 1.79 \[ \frac {b e x \left (2940 a^6 e^6+4410 a^5 b e^5 (e x-2 d)+2450 a^4 b^2 e^4 \left (6 d^2-3 d e x+2 e^2 x^2\right )+1225 a^3 b^3 e^3 \left (-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right )+147 a^2 b^4 e^2 \left (60 d^4-30 d^3 e x+20 d^2 e^2 x^2-15 d e^3 x^3+12 e^4 x^4\right )+49 a b^5 e \left (-60 d^5+30 d^4 e x-20 d^3 e^2 x^2+15 d^2 e^3 x^3-12 d e^4 x^4+10 e^5 x^5\right )+b^6 \left (420 d^6-210 d^5 e x+140 d^4 e^2 x^2-105 d^3 e^3 x^3+84 d^2 e^4 x^4-70 d e^5 x^5+60 e^6 x^6\right )\right )-420 (b d-a e)^7 \log (d+e x)}{420 e^8} \]

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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fricas [B]  time = 1.35, size = 459, normalized size = 2.70 \[ \frac {60 \, b^{7} e^{7} x^{7} - 70 \, {\left (b^{7} d e^{6} - 7 \, a b^{6} e^{7}\right )} x^{6} + 84 \, {\left (b^{7} d^{2} e^{5} - 7 \, a b^{6} d e^{6} + 21 \, a^{2} b^{5} e^{7}\right )} x^{5} - 105 \, {\left (b^{7} d^{3} e^{4} - 7 \, a b^{6} d^{2} e^{5} + 21 \, a^{2} b^{5} d e^{6} - 35 \, a^{3} b^{4} e^{7}\right )} x^{4} + 140 \, {\left (b^{7} d^{4} e^{3} - 7 \, a b^{6} d^{3} e^{4} + 21 \, a^{2} b^{5} d^{2} e^{5} - 35 \, a^{3} b^{4} d e^{6} + 35 \, a^{4} b^{3} e^{7}\right )} x^{3} - 210 \, {\left (b^{7} d^{5} e^{2} - 7 \, a b^{6} d^{4} e^{3} + 21 \, a^{2} b^{5} d^{3} e^{4} - 35 \, a^{3} b^{4} d^{2} e^{5} + 35 \, a^{4} b^{3} d e^{6} - 21 \, a^{5} b^{2} e^{7}\right )} x^{2} + 420 \, {\left (b^{7} d^{6} e - 7 \, a b^{6} d^{5} e^{2} + 21 \, a^{2} b^{5} d^{4} e^{3} - 35 \, a^{3} b^{4} d^{3} e^{4} + 35 \, a^{4} b^{3} d^{2} e^{5} - 21 \, a^{5} b^{2} d e^{6} + 7 \, a^{6} b e^{7}\right )} x - 420 \, {\left (b^{7} d^{7} - 7 \, a b^{6} d^{6} e + 21 \, a^{2} b^{5} d^{5} e^{2} - 35 \, a^{3} b^{4} d^{4} e^{3} + 35 \, a^{4} b^{3} d^{3} e^{4} - 21 \, a^{5} b^{2} d^{2} e^{5} + 7 \, a^{6} b d e^{6} - a^{7} e^{7}\right )} \log \left (e x + d\right )}{420 \, e^{8}} \]

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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giac [B]  time = 0.17, size = 469, normalized size = 2.76 \[ -{\left (b^{7} d^{7} - 7 \, a b^{6} d^{6} e + 21 \, a^{2} b^{5} d^{5} e^{2} - 35 \, a^{3} b^{4} d^{4} e^{3} + 35 \, a^{4} b^{3} d^{3} e^{4} - 21 \, a^{5} b^{2} d^{2} e^{5} + 7 \, a^{6} b d e^{6} - a^{7} e^{7}\right )} e^{\left (-8\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {1}{420} \, {\left (60 \, b^{7} x^{7} e^{6} - 70 \, b^{7} d x^{6} e^{5} + 84 \, b^{7} d^{2} x^{5} e^{4} - 105 \, b^{7} d^{3} x^{4} e^{3} + 140 \, b^{7} d^{4} x^{3} e^{2} - 210 \, b^{7} d^{5} x^{2} e + 420 \, b^{7} d^{6} x + 490 \, a b^{6} x^{6} e^{6} - 588 \, a b^{6} d x^{5} e^{5} + 735 \, a b^{6} d^{2} x^{4} e^{4} - 980 \, a b^{6} d^{3} x^{3} e^{3} + 1470 \, a b^{6} d^{4} x^{2} e^{2} - 2940 \, a b^{6} d^{5} x e + 1764 \, a^{2} b^{5} x^{5} e^{6} - 2205 \, a^{2} b^{5} d x^{4} e^{5} + 2940 \, a^{2} b^{5} d^{2} x^{3} e^{4} - 4410 \, a^{2} b^{5} d^{3} x^{2} e^{3} + 8820 \, a^{2} b^{5} d^{4} x e^{2} + 3675 \, a^{3} b^{4} x^{4} e^{6} - 4900 \, a^{3} b^{4} d x^{3} e^{5} + 7350 \, a^{3} b^{4} d^{2} x^{2} e^{4} - 14700 \, a^{3} b^{4} d^{3} x e^{3} + 4900 \, a^{4} b^{3} x^{3} e^{6} - 7350 \, a^{4} b^{3} d x^{2} e^{5} + 14700 \, a^{4} b^{3} d^{2} x e^{4} + 4410 \, a^{5} b^{2} x^{2} e^{6} - 8820 \, a^{5} b^{2} d x e^{5} + 2940 \, a^{6} b x e^{6}\right )} e^{\left (-7\right )} \]

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)

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maple [B]  time = 0.05, size = 539, normalized size = 3.17 \[ \frac {b^{7} x^{7}}{7 e}+\frac {7 a \,b^{6} x^{6}}{6 e}-\frac {b^{7} d \,x^{6}}{6 e^{2}}+\frac {21 a^{2} b^{5} x^{5}}{5 e}-\frac {7 a \,b^{6} d \,x^{5}}{5 e^{2}}+\frac {b^{7} d^{2} x^{5}}{5 e^{3}}+\frac {35 a^{3} b^{4} x^{4}}{4 e}-\frac {21 a^{2} b^{5} d \,x^{4}}{4 e^{2}}+\frac {7 a \,b^{6} d^{2} x^{4}}{4 e^{3}}-\frac {b^{7} d^{3} x^{4}}{4 e^{4}}+\frac {35 a^{4} b^{3} x^{3}}{3 e}-\frac {35 a^{3} b^{4} d \,x^{3}}{3 e^{2}}+\frac {7 a^{2} b^{5} d^{2} x^{3}}{e^{3}}-\frac {7 a \,b^{6} d^{3} x^{3}}{3 e^{4}}+\frac {b^{7} d^{4} x^{3}}{3 e^{5}}+\frac {21 a^{5} b^{2} x^{2}}{2 e}-\frac {35 a^{4} b^{3} d \,x^{2}}{2 e^{2}}+\frac {35 a^{3} b^{4} d^{2} x^{2}}{2 e^{3}}-\frac {21 a^{2} b^{5} d^{3} x^{2}}{2 e^{4}}+\frac {7 a \,b^{6} d^{4} x^{2}}{2 e^{5}}-\frac {b^{7} d^{5} x^{2}}{2 e^{6}}+\frac {a^{7} \ln \left (e x +d \right )}{e}-\frac {7 a^{6} b d \ln \left (e x +d \right )}{e^{2}}+\frac {7 a^{6} b x}{e}+\frac {21 a^{5} b^{2} d^{2} \ln \left (e x +d \right )}{e^{3}}-\frac {21 a^{5} b^{2} d x}{e^{2}}-\frac {35 a^{4} b^{3} d^{3} \ln \left (e x +d \right )}{e^{4}}+\frac {35 a^{4} b^{3} d^{2} x}{e^{3}}+\frac {35 a^{3} b^{4} d^{4} \ln \left (e x +d \right )}{e^{5}}-\frac {35 a^{3} b^{4} d^{3} x}{e^{4}}-\frac {21 a^{2} b^{5} d^{5} \ln \left (e x +d \right )}{e^{6}}+\frac {21 a^{2} b^{5} d^{4} x}{e^{5}}+\frac {7 a \,b^{6} d^{6} \ln \left (e x +d \right )}{e^{7}}-\frac {7 a \,b^{6} d^{5} x}{e^{6}}-\frac {b^{7} d^{7} \ln \left (e x +d \right )}{e^{8}}+\frac {b^{7} d^{6} x}{e^{7}} \]

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),x)

[Out]

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

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maxima [B]  time = 0.65, size = 458, normalized size = 2.69 \[ \frac {60 \, b^{7} e^{6} x^{7} - 70 \, {\left (b^{7} d e^{5} - 7 \, a b^{6} e^{6}\right )} x^{6} + 84 \, {\left (b^{7} d^{2} e^{4} - 7 \, a b^{6} d e^{5} + 21 \, a^{2} b^{5} e^{6}\right )} x^{5} - 105 \, {\left (b^{7} d^{3} e^{3} - 7 \, a b^{6} d^{2} e^{4} + 21 \, a^{2} b^{5} d e^{5} - 35 \, a^{3} b^{4} e^{6}\right )} x^{4} + 140 \, {\left (b^{7} d^{4} e^{2} - 7 \, a b^{6} d^{3} e^{3} + 21 \, a^{2} b^{5} d^{2} e^{4} - 35 \, a^{3} b^{4} d e^{5} + 35 \, a^{4} b^{3} e^{6}\right )} x^{3} - 210 \, {\left (b^{7} d^{5} e - 7 \, a b^{6} d^{4} e^{2} + 21 \, a^{2} b^{5} d^{3} e^{3} - 35 \, a^{3} b^{4} d^{2} e^{4} + 35 \, a^{4} b^{3} d e^{5} - 21 \, a^{5} b^{2} e^{6}\right )} x^{2} + 420 \, {\left (b^{7} d^{6} - 7 \, a b^{6} d^{5} e + 21 \, a^{2} b^{5} d^{4} e^{2} - 35 \, a^{3} b^{4} d^{3} e^{3} + 35 \, a^{4} b^{3} d^{2} e^{4} - 21 \, a^{5} b^{2} d e^{5} + 7 \, a^{6} b e^{6}\right )} x}{420 \, e^{7}} - \frac {{\left (b^{7} d^{7} - 7 \, a b^{6} d^{6} e + 21 \, a^{2} b^{5} d^{5} e^{2} - 35 \, a^{3} b^{4} d^{4} e^{3} + 35 \, a^{4} b^{3} d^{3} e^{4} - 21 \, a^{5} b^{2} d^{2} e^{5} + 7 \, a^{6} b d e^{6} - a^{7} e^{7}\right )} \log \left (e x + d\right )}{e^{8}} \]

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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mupad [B]  time = 0.07, size = 510, normalized size = 3.00 \[ x^6\,\left (\frac {7\,a\,b^6}{6\,e}-\frac {b^7\,d}{6\,e^2}\right )+x\,\left (\frac {7\,a^6\,b}{e}-\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {7\,a\,b^6}{e}-\frac {b^7\,d}{e^2}\right )}{e}-\frac {21\,a^2\,b^5}{e}\right )}{e}+\frac {35\,a^3\,b^4}{e}\right )}{e}-\frac {35\,a^4\,b^3}{e}\right )}{e}+\frac {21\,a^5\,b^2}{e}\right )}{e}\right )+x^4\,\left (\frac {d\,\left (\frac {d\,\left (\frac {7\,a\,b^6}{e}-\frac {b^7\,d}{e^2}\right )}{e}-\frac {21\,a^2\,b^5}{e}\right )}{4\,e}+\frac {35\,a^3\,b^4}{4\,e}\right )+x^2\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {7\,a\,b^6}{e}-\frac {b^7\,d}{e^2}\right )}{e}-\frac {21\,a^2\,b^5}{e}\right )}{e}+\frac {35\,a^3\,b^4}{e}\right )}{e}-\frac {35\,a^4\,b^3}{e}\right )}{2\,e}+\frac {21\,a^5\,b^2}{2\,e}\right )-x^5\,\left (\frac {d\,\left (\frac {7\,a\,b^6}{e}-\frac {b^7\,d}{e^2}\right )}{5\,e}-\frac {21\,a^2\,b^5}{5\,e}\right )-x^3\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {7\,a\,b^6}{e}-\frac {b^7\,d}{e^2}\right )}{e}-\frac {21\,a^2\,b^5}{e}\right )}{e}+\frac {35\,a^3\,b^4}{e}\right )}{3\,e}-\frac {35\,a^4\,b^3}{3\,e}\right )+\frac {\ln \left (d+e\,x\right )\,\left (a^7\,e^7-7\,a^6\,b\,d\,e^6+21\,a^5\,b^2\,d^2\,e^5-35\,a^4\,b^3\,d^3\,e^4+35\,a^3\,b^4\,d^4\,e^3-21\,a^2\,b^5\,d^5\,e^2+7\,a\,b^6\,d^6\,e-b^7\,d^7\right )}{e^8}+\frac {b^7\,x^7}{7\,e} \]

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),x)

[Out]

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

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sympy [B]  time = 0.83, size = 408, normalized size = 2.40 \[ \frac {b^{7} x^{7}}{7 e} + x^{6} \left (\frac {7 a b^{6}}{6 e} - \frac {b^{7} d}{6 e^{2}}\right ) + x^{5} \left (\frac {21 a^{2} b^{5}}{5 e} - \frac {7 a b^{6} d}{5 e^{2}} + \frac {b^{7} d^{2}}{5 e^{3}}\right ) + x^{4} \left (\frac {35 a^{3} b^{4}}{4 e} - \frac {21 a^{2} b^{5} d}{4 e^{2}} + \frac {7 a b^{6} d^{2}}{4 e^{3}} - \frac {b^{7} d^{3}}{4 e^{4}}\right ) + x^{3} \left (\frac {35 a^{4} b^{3}}{3 e} - \frac {35 a^{3} b^{4} d}{3 e^{2}} + \frac {7 a^{2} b^{5} d^{2}}{e^{3}} - \frac {7 a b^{6} d^{3}}{3 e^{4}} + \frac {b^{7} d^{4}}{3 e^{5}}\right ) + x^{2} \left (\frac {21 a^{5} b^{2}}{2 e} - \frac {35 a^{4} b^{3} d}{2 e^{2}} + \frac {35 a^{3} b^{4} d^{2}}{2 e^{3}} - \frac {21 a^{2} b^{5} d^{3}}{2 e^{4}} + \frac {7 a b^{6} d^{4}}{2 e^{5}} - \frac {b^{7} d^{5}}{2 e^{6}}\right ) + x \left (\frac {7 a^{6} b}{e} - \frac {21 a^{5} b^{2} d}{e^{2}} + \frac {35 a^{4} b^{3} d^{2}}{e^{3}} - \frac {35 a^{3} b^{4} d^{3}}{e^{4}} + \frac {21 a^{2} b^{5} d^{4}}{e^{5}} - \frac {7 a b^{6} d^{5}}{e^{6}} + \frac {b^{7} d^{6}}{e^{7}}\right ) + \frac {\left (a e - b d\right )^{7} \log {\left (d + e x \right )}}{e^{8}} \]

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),x)

[Out]

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

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