3.12.95 \(\int \frac {(a^2+2 a b x+b^2 x^2)^3}{(d+e x)^4} \, dx\)

Optimal. Leaf size=156 \[ -\frac {3 b^5 (d+e x)^2 (b d-a e)}{e^7}+\frac {15 b^4 x (b d-a e)^2}{e^6}-\frac {20 b^3 (b d-a e)^3 \log (d+e x)}{e^7}-\frac {15 b^2 (b d-a e)^4}{e^7 (d+e x)}+\frac {3 b (b d-a e)^5}{e^7 (d+e x)^2}-\frac {(b d-a e)^6}{3 e^7 (d+e x)^3}+\frac {b^6 (d+e x)^3}{3 e^7} \]

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Rubi [A]  time = 0.16, antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {27, 43} \begin {gather*} -\frac {3 b^5 (d+e x)^2 (b d-a e)}{e^7}+\frac {15 b^4 x (b d-a e)^2}{e^6}-\frac {15 b^2 (b d-a e)^4}{e^7 (d+e x)}-\frac {20 b^3 (b d-a e)^3 \log (d+e x)}{e^7}+\frac {3 b (b d-a e)^5}{e^7 (d+e x)^2}-\frac {(b d-a e)^6}{3 e^7 (d+e x)^3}+\frac {b^6 (d+e x)^3}{3 e^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x + b^2*x^2)^3/(d + e*x)^4,x]

[Out]

(15*b^4*(b*d - a*e)^2*x)/e^6 - (b*d - a*e)^6/(3*e^7*(d + e*x)^3) + (3*b*(b*d - a*e)^5)/(e^7*(d + e*x)^2) - (15
*b^2*(b*d - a*e)^4)/(e^7*(d + e*x)) - (3*b^5*(b*d - a*e)*(d + e*x)^2)/e^7 + (b^6*(d + e*x)^3)/(3*e^7) - (20*b^
3*(b*d - a*e)^3*Log[d + e*x])/e^7

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 {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^4} \, dx &=\int \frac {(a+b x)^6}{(d+e x)^4} \, dx\\ &=\int \left (\frac {15 b^4 (b d-a e)^2}{e^6}+\frac {(-b d+a e)^6}{e^6 (d+e x)^4}-\frac {6 b (b d-a e)^5}{e^6 (d+e x)^3}+\frac {15 b^2 (b d-a e)^4}{e^6 (d+e x)^2}-\frac {20 b^3 (b d-a e)^3}{e^6 (d+e x)}-\frac {6 b^5 (b d-a e) (d+e x)}{e^6}+\frac {b^6 (d+e x)^2}{e^6}\right ) \, dx\\ &=\frac {15 b^4 (b d-a e)^2 x}{e^6}-\frac {(b d-a e)^6}{3 e^7 (d+e x)^3}+\frac {3 b (b d-a e)^5}{e^7 (d+e x)^2}-\frac {15 b^2 (b d-a e)^4}{e^7 (d+e x)}-\frac {3 b^5 (b d-a e) (d+e x)^2}{e^7}+\frac {b^6 (d+e x)^3}{3 e^7}-\frac {20 b^3 (b d-a e)^3 \log (d+e x)}{e^7}\\ \end {align*}

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Mathematica [A]  time = 0.11, size = 302, normalized size = 1.94 \begin {gather*} \frac {-a^6 e^6-3 a^5 b e^5 (d+3 e x)-15 a^4 b^2 e^4 \left (d^2+3 d e x+3 e^2 x^2\right )+10 a^3 b^3 d e^3 \left (11 d^2+27 d e x+18 e^2 x^2\right )+15 a^2 b^4 e^2 \left (-13 d^4-27 d^3 e x-9 d^2 e^2 x^2+9 d e^3 x^3+3 e^4 x^4\right )+3 a b^5 e \left (47 d^5+81 d^4 e x-9 d^3 e^2 x^2-63 d^2 e^3 x^3-15 d e^4 x^4+3 e^5 x^5\right )-60 b^3 (d+e x)^3 (b d-a e)^3 \log (d+e x)+b^6 \left (-37 d^6-51 d^5 e x+39 d^4 e^2 x^2+73 d^3 e^3 x^3+15 d^2 e^4 x^4-3 d e^5 x^5+e^6 x^6\right )}{3 e^7 (d+e x)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + 2*a*b*x + b^2*x^2)^3/(d + e*x)^4,x]

[Out]

(-(a^6*e^6) - 3*a^5*b*e^5*(d + 3*e*x) - 15*a^4*b^2*e^4*(d^2 + 3*d*e*x + 3*e^2*x^2) + 10*a^3*b^3*d*e^3*(11*d^2
+ 27*d*e*x + 18*e^2*x^2) + 15*a^2*b^4*e^2*(-13*d^4 - 27*d^3*e*x - 9*d^2*e^2*x^2 + 9*d*e^3*x^3 + 3*e^4*x^4) + 3
*a*b^5*e*(47*d^5 + 81*d^4*e*x - 9*d^3*e^2*x^2 - 63*d^2*e^3*x^3 - 15*d*e^4*x^4 + 3*e^5*x^5) + b^6*(-37*d^6 - 51
*d^5*e*x + 39*d^4*e^2*x^2 + 73*d^3*e^3*x^3 + 15*d^2*e^4*x^4 - 3*d*e^5*x^5 + e^6*x^6) - 60*b^3*(b*d - a*e)^3*(d
+ e*x)^3*Log[d + e*x])/(3*e^7*(d + e*x)^3)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a^2 + 2*a*b*x + b^2*x^2)^3/(d + e*x)^4,x]

[Out]

IntegrateAlgebraic[(a^2 + 2*a*b*x + b^2*x^2)^3/(d + e*x)^4, x]

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fricas [B]  time = 0.40, size = 576, normalized size = 3.69 \begin {gather*} \frac {b^{6} e^{6} x^{6} - 37 \, b^{6} d^{6} + 141 \, a b^{5} d^{5} e - 195 \, a^{2} b^{4} d^{4} e^{2} + 110 \, a^{3} b^{3} d^{3} e^{3} - 15 \, a^{4} b^{2} d^{2} e^{4} - 3 \, a^{5} b d e^{5} - a^{6} e^{6} - 3 \, {\left (b^{6} d e^{5} - 3 \, a b^{5} e^{6}\right )} x^{5} + 15 \, {\left (b^{6} d^{2} e^{4} - 3 \, a b^{5} d e^{5} + 3 \, a^{2} b^{4} e^{6}\right )} x^{4} + {\left (73 \, b^{6} d^{3} e^{3} - 189 \, a b^{5} d^{2} e^{4} + 135 \, a^{2} b^{4} d e^{5}\right )} x^{3} + 3 \, {\left (13 \, b^{6} d^{4} e^{2} - 9 \, a b^{5} d^{3} e^{3} - 45 \, a^{2} b^{4} d^{2} e^{4} + 60 \, a^{3} b^{3} d e^{5} - 15 \, a^{4} b^{2} e^{6}\right )} x^{2} - 3 \, {\left (17 \, b^{6} d^{5} e - 81 \, a b^{5} d^{4} e^{2} + 135 \, a^{2} b^{4} d^{3} e^{3} - 90 \, a^{3} b^{3} d^{2} e^{4} + 15 \, a^{4} b^{2} d e^{5} + 3 \, a^{5} b e^{6}\right )} x - 60 \, {\left (b^{6} d^{6} - 3 \, a b^{5} d^{5} e + 3 \, a^{2} b^{4} d^{4} e^{2} - a^{3} b^{3} d^{3} e^{3} + {\left (b^{6} d^{3} e^{3} - 3 \, a b^{5} d^{2} e^{4} + 3 \, a^{2} b^{4} d e^{5} - a^{3} b^{3} e^{6}\right )} x^{3} + 3 \, {\left (b^{6} d^{4} e^{2} - 3 \, a b^{5} d^{3} e^{3} + 3 \, a^{2} b^{4} d^{2} e^{4} - a^{3} b^{3} d e^{5}\right )} x^{2} + 3 \, {\left (b^{6} d^{5} e - 3 \, a b^{5} d^{4} e^{2} + 3 \, a^{2} b^{4} d^{3} e^{3} - a^{3} b^{3} d^{2} e^{4}\right )} x\right )} \log \left (e x + d\right )}{3 \, {\left (e^{10} x^{3} + 3 \, d e^{9} x^{2} + 3 \, d^{2} e^{8} x + d^{3} e^{7}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^4,x, algorithm="fricas")

[Out]

1/3*(b^6*e^6*x^6 - 37*b^6*d^6 + 141*a*b^5*d^5*e - 195*a^2*b^4*d^4*e^2 + 110*a^3*b^3*d^3*e^3 - 15*a^4*b^2*d^2*e
^4 - 3*a^5*b*d*e^5 - a^6*e^6 - 3*(b^6*d*e^5 - 3*a*b^5*e^6)*x^5 + 15*(b^6*d^2*e^4 - 3*a*b^5*d*e^5 + 3*a^2*b^4*e
^6)*x^4 + (73*b^6*d^3*e^3 - 189*a*b^5*d^2*e^4 + 135*a^2*b^4*d*e^5)*x^3 + 3*(13*b^6*d^4*e^2 - 9*a*b^5*d^3*e^3 -
45*a^2*b^4*d^2*e^4 + 60*a^3*b^3*d*e^5 - 15*a^4*b^2*e^6)*x^2 - 3*(17*b^6*d^5*e - 81*a*b^5*d^4*e^2 + 135*a^2*b^
4*d^3*e^3 - 90*a^3*b^3*d^2*e^4 + 15*a^4*b^2*d*e^5 + 3*a^5*b*e^6)*x - 60*(b^6*d^6 - 3*a*b^5*d^5*e + 3*a^2*b^4*d
^4*e^2 - a^3*b^3*d^3*e^3 + (b^6*d^3*e^3 - 3*a*b^5*d^2*e^4 + 3*a^2*b^4*d*e^5 - a^3*b^3*e^6)*x^3 + 3*(b^6*d^4*e^
2 - 3*a*b^5*d^3*e^3 + 3*a^2*b^4*d^2*e^4 - a^3*b^3*d*e^5)*x^2 + 3*(b^6*d^5*e - 3*a*b^5*d^4*e^2 + 3*a^2*b^4*d^3*
e^3 - a^3*b^3*d^2*e^4)*x)*log(e*x + d))/(e^10*x^3 + 3*d*e^9*x^2 + 3*d^2*e^8*x + d^3*e^7)

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giac [B]  time = 0.21, size = 335, normalized size = 2.15 \begin {gather*} -20 \, {\left (b^{6} d^{3} - 3 \, a b^{5} d^{2} e + 3 \, a^{2} b^{4} d e^{2} - a^{3} b^{3} e^{3}\right )} e^{\left (-7\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {1}{3} \, {\left (b^{6} x^{3} e^{8} - 6 \, b^{6} d x^{2} e^{7} + 30 \, b^{6} d^{2} x e^{6} + 9 \, a b^{5} x^{2} e^{8} - 72 \, a b^{5} d x e^{7} + 45 \, a^{2} b^{4} x e^{8}\right )} e^{\left (-12\right )} - \frac {{\left (37 \, b^{6} d^{6} - 141 \, a b^{5} d^{5} e + 195 \, a^{2} b^{4} d^{4} e^{2} - 110 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} + 3 \, a^{5} b d e^{5} + a^{6} e^{6} + 45 \, {\left (b^{6} d^{4} e^{2} - 4 \, a b^{5} d^{3} e^{3} + 6 \, a^{2} b^{4} d^{2} e^{4} - 4 \, a^{3} b^{3} d e^{5} + a^{4} b^{2} e^{6}\right )} x^{2} + 9 \, {\left (9 \, b^{6} d^{5} e - 35 \, a b^{5} d^{4} e^{2} + 50 \, a^{2} b^{4} d^{3} e^{3} - 30 \, a^{3} b^{3} d^{2} e^{4} + 5 \, a^{4} b^{2} d e^{5} + a^{5} b e^{6}\right )} x\right )} e^{\left (-7\right )}}{3 \, {\left (x e + d\right )}^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^4,x, algorithm="giac")

[Out]

-20*(b^6*d^3 - 3*a*b^5*d^2*e + 3*a^2*b^4*d*e^2 - a^3*b^3*e^3)*e^(-7)*log(abs(x*e + d)) + 1/3*(b^6*x^3*e^8 - 6*
b^6*d*x^2*e^7 + 30*b^6*d^2*x*e^6 + 9*a*b^5*x^2*e^8 - 72*a*b^5*d*x*e^7 + 45*a^2*b^4*x*e^8)*e^(-12) - 1/3*(37*b^
6*d^6 - 141*a*b^5*d^5*e + 195*a^2*b^4*d^4*e^2 - 110*a^3*b^3*d^3*e^3 + 15*a^4*b^2*d^2*e^4 + 3*a^5*b*d*e^5 + a^6
*e^6 + 45*(b^6*d^4*e^2 - 4*a*b^5*d^3*e^3 + 6*a^2*b^4*d^2*e^4 - 4*a^3*b^3*d*e^5 + a^4*b^2*e^6)*x^2 + 9*(9*b^6*d
^5*e - 35*a*b^5*d^4*e^2 + 50*a^2*b^4*d^3*e^3 - 30*a^3*b^3*d^2*e^4 + 5*a^4*b^2*d*e^5 + a^5*b*e^6)*x)*e^(-7)/(x*
e + d)^3

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maple [B]  time = 0.05, size = 483, normalized size = 3.10 \begin {gather*} -\frac {a^{6}}{3 \left (e x +d \right )^{3} e}+\frac {2 a^{5} b d}{\left (e x +d \right )^{3} e^{2}}-\frac {5 a^{4} b^{2} d^{2}}{\left (e x +d \right )^{3} e^{3}}+\frac {20 a^{3} b^{3} d^{3}}{3 \left (e x +d \right )^{3} e^{4}}-\frac {5 a^{2} b^{4} d^{4}}{\left (e x +d \right )^{3} e^{5}}+\frac {2 a \,b^{5} d^{5}}{\left (e x +d \right )^{3} e^{6}}-\frac {b^{6} d^{6}}{3 \left (e x +d \right )^{3} e^{7}}+\frac {b^{6} x^{3}}{3 e^{4}}-\frac {3 a^{5} b}{\left (e x +d \right )^{2} e^{2}}+\frac {15 a^{4} b^{2} d}{\left (e x +d \right )^{2} e^{3}}-\frac {30 a^{3} b^{3} d^{2}}{\left (e x +d \right )^{2} e^{4}}+\frac {30 a^{2} b^{4} d^{3}}{\left (e x +d \right )^{2} e^{5}}-\frac {15 a \,b^{5} d^{4}}{\left (e x +d \right )^{2} e^{6}}+\frac {3 a \,b^{5} x^{2}}{e^{4}}+\frac {3 b^{6} d^{5}}{\left (e x +d \right )^{2} e^{7}}-\frac {2 b^{6} d \,x^{2}}{e^{5}}-\frac {15 a^{4} b^{2}}{\left (e x +d \right ) e^{3}}+\frac {60 a^{3} b^{3} d}{\left (e x +d \right ) e^{4}}+\frac {20 a^{3} b^{3} \ln \left (e x +d \right )}{e^{4}}-\frac {90 a^{2} b^{4} d^{2}}{\left (e x +d \right ) e^{5}}-\frac {60 a^{2} b^{4} d \ln \left (e x +d \right )}{e^{5}}+\frac {15 a^{2} b^{4} x}{e^{4}}+\frac {60 a \,b^{5} d^{3}}{\left (e x +d \right ) e^{6}}+\frac {60 a \,b^{5} d^{2} \ln \left (e x +d \right )}{e^{6}}-\frac {24 a \,b^{5} d x}{e^{5}}-\frac {15 b^{6} d^{4}}{\left (e x +d \right ) e^{7}}-\frac {20 b^{6} d^{3} \ln \left (e x +d \right )}{e^{7}}+\frac {10 b^{6} d^{2} x}{e^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^4,x)

[Out]

20*b^3/e^4*ln(e*x+d)*a^3-20*b^6/e^7*ln(e*x+d)*d^3-15*b^6/e^7/(e*x+d)*d^4-3*b/e^2/(e*x+d)^2*a^5+3*b^6/e^7/(e*x+
d)^2*d^5-1/3/e^7/(e*x+d)^3*d^6*b^6+3*b^5/e^4*x^2*a-2*b^6/e^5*x^2*d+15*b^4/e^4*a^2*x+10*b^6/e^6*d^2*x-15*b^2/e^
3/(e*x+d)*a^4+15*b^2/e^3/(e*x+d)^2*a^4*d-30*b^3/e^4/(e*x+d)^2*a^3*d^2+30*b^4/e^5/(e*x+d)^2*a^2*d^3-15*b^5/e^6/
(e*x+d)^2*a*d^4+2/e^2/(e*x+d)^3*d*a^5*b-5/e^3/(e*x+d)^3*d^2*a^4*b^2+20/3/e^4/(e*x+d)^3*d^3*a^3*b^3-5/e^5/(e*x+
d)^3*d^4*a^2*b^4+2/e^6/(e*x+d)^3*d^5*a*b^5+60*b^3/e^4/(e*x+d)*d*a^3-90*b^4/e^5/(e*x+d)*a^2*d^2+60*b^5/e^6/(e*x
+d)*a*d^3-24*b^5/e^5*a*d*x-60*b^4/e^5*ln(e*x+d)*a^2*d-1/3/e/(e*x+d)^3*a^6+1/3*b^6/e^4*x^3+60*b^5/e^6*ln(e*x+d)
*a*d^2

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maxima [B]  time = 1.62, size = 374, normalized size = 2.40 \begin {gather*} -\frac {37 \, b^{6} d^{6} - 141 \, a b^{5} d^{5} e + 195 \, a^{2} b^{4} d^{4} e^{2} - 110 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} + 3 \, a^{5} b d e^{5} + a^{6} e^{6} + 45 \, {\left (b^{6} d^{4} e^{2} - 4 \, a b^{5} d^{3} e^{3} + 6 \, a^{2} b^{4} d^{2} e^{4} - 4 \, a^{3} b^{3} d e^{5} + a^{4} b^{2} e^{6}\right )} x^{2} + 9 \, {\left (9 \, b^{6} d^{5} e - 35 \, a b^{5} d^{4} e^{2} + 50 \, a^{2} b^{4} d^{3} e^{3} - 30 \, a^{3} b^{3} d^{2} e^{4} + 5 \, a^{4} b^{2} d e^{5} + a^{5} b e^{6}\right )} x}{3 \, {\left (e^{10} x^{3} + 3 \, d e^{9} x^{2} + 3 \, d^{2} e^{8} x + d^{3} e^{7}\right )}} + \frac {b^{6} e^{2} x^{3} - 3 \, {\left (2 \, b^{6} d e - 3 \, a b^{5} e^{2}\right )} x^{2} + 3 \, {\left (10 \, b^{6} d^{2} - 24 \, a b^{5} d e + 15 \, a^{2} b^{4} e^{2}\right )} x}{3 \, e^{6}} - \frac {20 \, {\left (b^{6} d^{3} - 3 \, a b^{5} d^{2} e + 3 \, a^{2} b^{4} d e^{2} - a^{3} b^{3} e^{3}\right )} \log \left (e x + d\right )}{e^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^4,x, algorithm="maxima")

[Out]

-1/3*(37*b^6*d^6 - 141*a*b^5*d^5*e + 195*a^2*b^4*d^4*e^2 - 110*a^3*b^3*d^3*e^3 + 15*a^4*b^2*d^2*e^4 + 3*a^5*b*
d*e^5 + a^6*e^6 + 45*(b^6*d^4*e^2 - 4*a*b^5*d^3*e^3 + 6*a^2*b^4*d^2*e^4 - 4*a^3*b^3*d*e^5 + a^4*b^2*e^6)*x^2 +
9*(9*b^6*d^5*e - 35*a*b^5*d^4*e^2 + 50*a^2*b^4*d^3*e^3 - 30*a^3*b^3*d^2*e^4 + 5*a^4*b^2*d*e^5 + a^5*b*e^6)*x)
/(e^10*x^3 + 3*d*e^9*x^2 + 3*d^2*e^8*x + d^3*e^7) + 1/3*(b^6*e^2*x^3 - 3*(2*b^6*d*e - 3*a*b^5*e^2)*x^2 + 3*(10
*b^6*d^2 - 24*a*b^5*d*e + 15*a^2*b^4*e^2)*x)/e^6 - 20*(b^6*d^3 - 3*a*b^5*d^2*e + 3*a^2*b^4*d*e^2 - a^3*b^3*e^3
)*log(e*x + d)/e^7

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mupad [B]  time = 0.58, size = 393, normalized size = 2.52 \begin {gather*} x^2\,\left (\frac {3\,a\,b^5}{e^4}-\frac {2\,b^6\,d}{e^5}\right )-\frac {x^2\,\left (15\,a^4\,b^2\,e^5-60\,a^3\,b^3\,d\,e^4+90\,a^2\,b^4\,d^2\,e^3-60\,a\,b^5\,d^3\,e^2+15\,b^6\,d^4\,e\right )+\frac {a^6\,e^6+3\,a^5\,b\,d\,e^5+15\,a^4\,b^2\,d^2\,e^4-110\,a^3\,b^3\,d^3\,e^3+195\,a^2\,b^4\,d^4\,e^2-141\,a\,b^5\,d^5\,e+37\,b^6\,d^6}{3\,e}+x\,\left (3\,a^5\,b\,e^5+15\,a^4\,b^2\,d\,e^4-90\,a^3\,b^3\,d^2\,e^3+150\,a^2\,b^4\,d^3\,e^2-105\,a\,b^5\,d^4\,e+27\,b^6\,d^5\right )}{d^3\,e^6+3\,d^2\,e^7\,x+3\,d\,e^8\,x^2+e^9\,x^3}-x\,\left (\frac {4\,d\,\left (\frac {6\,a\,b^5}{e^4}-\frac {4\,b^6\,d}{e^5}\right )}{e}-\frac {15\,a^2\,b^4}{e^4}+\frac {6\,b^6\,d^2}{e^6}\right )-\frac {\ln \left (d+e\,x\right )\,\left (-20\,a^3\,b^3\,e^3+60\,a^2\,b^4\,d\,e^2-60\,a\,b^5\,d^2\,e+20\,b^6\,d^3\right )}{e^7}+\frac {b^6\,x^3}{3\,e^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a^2 + b^2*x^2 + 2*a*b*x)^3/(d + e*x)^4,x)

[Out]

x^2*((3*a*b^5)/e^4 - (2*b^6*d)/e^5) - (x^2*(15*b^6*d^4*e + 15*a^4*b^2*e^5 - 60*a*b^5*d^3*e^2 - 60*a^3*b^3*d*e^
4 + 90*a^2*b^4*d^2*e^3) + (a^6*e^6 + 37*b^6*d^6 + 195*a^2*b^4*d^4*e^2 - 110*a^3*b^3*d^3*e^3 + 15*a^4*b^2*d^2*e
^4 - 141*a*b^5*d^5*e + 3*a^5*b*d*e^5)/(3*e) + x*(27*b^6*d^5 + 3*a^5*b*e^5 + 15*a^4*b^2*d*e^4 + 150*a^2*b^4*d^3
*e^2 - 90*a^3*b^3*d^2*e^3 - 105*a*b^5*d^4*e))/(d^3*e^6 + e^9*x^3 + 3*d^2*e^7*x + 3*d*e^8*x^2) - x*((4*d*((6*a*
b^5)/e^4 - (4*b^6*d)/e^5))/e - (15*a^2*b^4)/e^4 + (6*b^6*d^2)/e^6) - (log(d + e*x)*(20*b^6*d^3 - 20*a^3*b^3*e^
3 + 60*a^2*b^4*d*e^2 - 60*a*b^5*d^2*e))/e^7 + (b^6*x^3)/(3*e^4)

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sympy [B]  time = 4.35, size = 367, normalized size = 2.35 \begin {gather*} \frac {b^{6} x^{3}}{3 e^{4}} + \frac {20 b^{3} \left (a e - b d\right )^{3} \log {\left (d + e x \right )}}{e^{7}} + x^{2} \left (\frac {3 a b^{5}}{e^{4}} - \frac {2 b^{6} d}{e^{5}}\right ) + x \left (\frac {15 a^{2} b^{4}}{e^{4}} - \frac {24 a b^{5} d}{e^{5}} + \frac {10 b^{6} d^{2}}{e^{6}}\right ) + \frac {- a^{6} e^{6} - 3 a^{5} b d e^{5} - 15 a^{4} b^{2} d^{2} e^{4} + 110 a^{3} b^{3} d^{3} e^{3} - 195 a^{2} b^{4} d^{4} e^{2} + 141 a b^{5} d^{5} e - 37 b^{6} d^{6} + x^{2} \left (- 45 a^{4} b^{2} e^{6} + 180 a^{3} b^{3} d e^{5} - 270 a^{2} b^{4} d^{2} e^{4} + 180 a b^{5} d^{3} e^{3} - 45 b^{6} d^{4} e^{2}\right ) + x \left (- 9 a^{5} b e^{6} - 45 a^{4} b^{2} d e^{5} + 270 a^{3} b^{3} d^{2} e^{4} - 450 a^{2} b^{4} d^{3} e^{3} + 315 a b^{5} d^{4} e^{2} - 81 b^{6} d^{5} e\right )}{3 d^{3} e^{7} + 9 d^{2} e^{8} x + 9 d e^{9} x^{2} + 3 e^{10} x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b**2*x**2+2*a*b*x+a**2)**3/(e*x+d)**4,x)

[Out]

b**6*x**3/(3*e**4) + 20*b**3*(a*e - b*d)**3*log(d + e*x)/e**7 + x**2*(3*a*b**5/e**4 - 2*b**6*d/e**5) + x*(15*a
**2*b**4/e**4 - 24*a*b**5*d/e**5 + 10*b**6*d**2/e**6) + (-a**6*e**6 - 3*a**5*b*d*e**5 - 15*a**4*b**2*d**2*e**4
+ 110*a**3*b**3*d**3*e**3 - 195*a**2*b**4*d**4*e**2 + 141*a*b**5*d**5*e - 37*b**6*d**6 + x**2*(-45*a**4*b**2*
e**6 + 180*a**3*b**3*d*e**5 - 270*a**2*b**4*d**2*e**4 + 180*a*b**5*d**3*e**3 - 45*b**6*d**4*e**2) + x*(-9*a**5
*b*e**6 - 45*a**4*b**2*d*e**5 + 270*a**3*b**3*d**2*e**4 - 450*a**2*b**4*d**3*e**3 + 315*a*b**5*d**4*e**2 - 81*
b**6*d**5*e))/(3*d**3*e**7 + 9*d**2*e**8*x + 9*d*e**9*x**2 + 3*e**10*x**3)

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