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

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

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Rubi [A]  time = 0.26, antiderivative size = 186, 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} \begin {gather*} -\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} \end {gather*}

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*}

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Mathematica [B]  time = 0.11, size = 387, normalized size = 2.08 \begin {gather*} \frac {-60 a^7 e^7+420 a^6 b d e^6+1260 a^5 b^2 e^5 \left (-d^2+d e x+e^2 x^2\right )+1050 a^4 b^3 e^4 \left (2 d^3-4 d^2 e x-3 d e^2 x^2+e^3 x^3\right )+700 a^3 b^4 e^3 \left (-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\right )+105 a^2 b^5 e^2 \left (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\right )+42 a b^6 e \left (-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\right )+420 b (d+e x) (b d-a e)^6 \log (d+e x)+b^7 \left (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\right )}{60 e^8 (d+e x)} \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 0.44, size = 629, normalized size = 3.38 \begin {gather*} \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 {gather*}

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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giac [B]  time = 0.21, size = 542, normalized size = 2.91 \begin {gather*} \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 {gather*}

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)

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maple [B]  time = 0.06, size = 571, normalized size = 3.07 \begin {gather*} \frac {b^{7} x^{6}}{6 e^{2}}+\frac {7 a \,b^{6} x^{5}}{5 e^{2}}-\frac {2 b^{7} d \,x^{5}}{5 e^{3}}+\frac {21 a^{2} b^{5} x^{4}}{4 e^{2}}-\frac {7 a \,b^{6} d \,x^{4}}{2 e^{3}}+\frac {3 b^{7} d^{2} x^{4}}{4 e^{4}}+\frac {35 a^{3} b^{4} x^{3}}{3 e^{2}}-\frac {14 a^{2} b^{5} d \,x^{3}}{e^{3}}+\frac {7 a \,b^{6} d^{2} x^{3}}{e^{4}}-\frac {4 b^{7} d^{3} x^{3}}{3 e^{5}}+\frac {35 a^{4} b^{3} x^{2}}{2 e^{2}}-\frac {35 a^{3} b^{4} d \,x^{2}}{e^{3}}+\frac {63 a^{2} b^{5} d^{2} x^{2}}{2 e^{4}}-\frac {14 a \,b^{6} d^{3} x^{2}}{e^{5}}+\frac {5 b^{7} d^{4} x^{2}}{2 e^{6}}-\frac {a^{7}}{\left (e x +d \right ) e}+\frac {7 a^{6} b d}{\left (e x +d \right ) e^{2}}+\frac {7 a^{6} b \ln \left (e x +d \right )}{e^{2}}-\frac {21 a^{5} b^{2} d^{2}}{\left (e x +d \right ) e^{3}}-\frac {42 a^{5} b^{2} d \ln \left (e x +d \right )}{e^{3}}+\frac {21 a^{5} b^{2} x}{e^{2}}+\frac {35 a^{4} b^{3} d^{3}}{\left (e x +d \right ) e^{4}}+\frac {105 a^{4} b^{3} d^{2} \ln \left (e x +d \right )}{e^{4}}-\frac {70 a^{4} b^{3} d x}{e^{3}}-\frac {35 a^{3} b^{4} d^{4}}{\left (e x +d \right ) e^{5}}-\frac {140 a^{3} b^{4} d^{3} \ln \left (e x +d \right )}{e^{5}}+\frac {105 a^{3} b^{4} d^{2} x}{e^{4}}+\frac {21 a^{2} b^{5} d^{5}}{\left (e x +d \right ) e^{6}}+\frac {105 a^{2} b^{5} d^{4} \ln \left (e x +d \right )}{e^{6}}-\frac {84 a^{2} b^{5} d^{3} x}{e^{5}}-\frac {7 a \,b^{6} d^{6}}{\left (e x +d \right ) e^{7}}-\frac {42 a \,b^{6} d^{5} \ln \left (e x +d \right )}{e^{7}}+\frac {35 a \,b^{6} d^{4} x}{e^{6}}+\frac {b^{7} d^{7}}{\left (e x +d \right ) e^{8}}+\frac {7 b^{7} d^{6} \ln \left (e x +d \right )}{e^{8}}-\frac {6 b^{7} d^{5} x}{e^{7}} \end {gather*}

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

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maxima [B]  time = 0.79, size = 466, normalized size = 2.51 \begin {gather*} \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 {gather*}

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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mupad [B]  time = 2.00, size = 839, normalized size = 4.51 \begin {gather*} x^5\,\left (\frac {7\,a\,b^6}{5\,e^2}-\frac {2\,b^7\,d}{5\,e^3}\right )+x^2\,\left (\frac {d^2\,\left (\frac {2\,d\,\left (\frac {7\,a\,b^6}{e^2}-\frac {2\,b^7\,d}{e^3}\right )}{e}-\frac {21\,a^2\,b^5}{e^2}+\frac {b^7\,d^2}{e^4}\right )}{2\,e^2}-\frac {d\,\left (\frac {35\,a^3\,b^4}{e^2}+\frac {2\,d\,\left (\frac {2\,d\,\left (\frac {7\,a\,b^6}{e^2}-\frac {2\,b^7\,d}{e^3}\right )}{e}-\frac {21\,a^2\,b^5}{e^2}+\frac {b^7\,d^2}{e^4}\right )}{e}-\frac {d^2\,\left (\frac {7\,a\,b^6}{e^2}-\frac {2\,b^7\,d}{e^3}\right )}{e^2}\right )}{e}+\frac {35\,a^4\,b^3}{2\,e^2}\right )-x^4\,\left (\frac {d\,\left (\frac {7\,a\,b^6}{e^2}-\frac {2\,b^7\,d}{e^3}\right )}{2\,e}-\frac {21\,a^2\,b^5}{4\,e^2}+\frac {b^7\,d^2}{4\,e^4}\right )+x^3\,\left (\frac {35\,a^3\,b^4}{3\,e^2}+\frac {2\,d\,\left (\frac {2\,d\,\left (\frac {7\,a\,b^6}{e^2}-\frac {2\,b^7\,d}{e^3}\right )}{e}-\frac {21\,a^2\,b^5}{e^2}+\frac {b^7\,d^2}{e^4}\right )}{3\,e}-\frac {d^2\,\left (\frac {7\,a\,b^6}{e^2}-\frac {2\,b^7\,d}{e^3}\right )}{3\,e^2}\right )-x\,\left (\frac {d^2\,\left (\frac {35\,a^3\,b^4}{e^2}+\frac {2\,d\,\left (\frac {2\,d\,\left (\frac {7\,a\,b^6}{e^2}-\frac {2\,b^7\,d}{e^3}\right )}{e}-\frac {21\,a^2\,b^5}{e^2}+\frac {b^7\,d^2}{e^4}\right )}{e}-\frac {d^2\,\left (\frac {7\,a\,b^6}{e^2}-\frac {2\,b^7\,d}{e^3}\right )}{e^2}\right )}{e^2}-\frac {21\,a^5\,b^2}{e^2}+\frac {2\,d\,\left (\frac {d^2\,\left (\frac {2\,d\,\left (\frac {7\,a\,b^6}{e^2}-\frac {2\,b^7\,d}{e^3}\right )}{e}-\frac {21\,a^2\,b^5}{e^2}+\frac {b^7\,d^2}{e^4}\right )}{e^2}-\frac {2\,d\,\left (\frac {35\,a^3\,b^4}{e^2}+\frac {2\,d\,\left (\frac {2\,d\,\left (\frac {7\,a\,b^6}{e^2}-\frac {2\,b^7\,d}{e^3}\right )}{e}-\frac {21\,a^2\,b^5}{e^2}+\frac {b^7\,d^2}{e^4}\right )}{e}-\frac {d^2\,\left (\frac {7\,a\,b^6}{e^2}-\frac {2\,b^7\,d}{e^3}\right )}{e^2}\right )}{e}+\frac {35\,a^4\,b^3}{e^2}\right )}{e}\right )+\frac {\ln \left (d+e\,x\right )\,\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 )}{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}{e\,\left (x\,e^8+d\,e^7\right )}+\frac {b^7\,x^6}{6\,e^2} \end {gather*}

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)^2,x)

[Out]

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

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sympy [B]  time = 1.50, size = 428, normalized size = 2.30 \begin {gather*} \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}} + x^{5} \left (\frac {7 a b^{6}}{5 e^{2}} - \frac {2 b^{7} d}{5 e^{3}}\right ) + x^{4} \left (\frac {21 a^{2} b^{5}}{4 e^{2}} - \frac {7 a b^{6} d}{2 e^{3}} + \frac {3 b^{7} d^{2}}{4 e^{4}}\right ) + x^{3} \left (\frac {35 a^{3} b^{4}}{3 e^{2}} - \frac {14 a^{2} b^{5} d}{e^{3}} + \frac {7 a b^{6} d^{2}}{e^{4}} - \frac {4 b^{7} d^{3}}{3 e^{5}}\right ) + x^{2} \left (\frac {35 a^{4} b^{3}}{2 e^{2}} - \frac {35 a^{3} b^{4} d}{e^{3}} + \frac {63 a^{2} b^{5} d^{2}}{2 e^{4}} - \frac {14 a b^{6} d^{3}}{e^{5}} + \frac {5 b^{7} d^{4}}{2 e^{6}}\right ) + x \left (\frac {21 a^{5} b^{2}}{e^{2}} - \frac {70 a^{4} b^{3} d}{e^{3}} + \frac {105 a^{3} b^{4} d^{2}}{e^{4}} - \frac {84 a^{2} b^{5} d^{3}}{e^{5}} + \frac {35 a b^{6} d^{4}}{e^{6}} - \frac {6 b^{7} d^{5}}{e^{7}}\right ) + \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} \end {gather*}

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

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