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

Optimal. Leaf size=158 \[ -\frac {2 b^5 (d+e x)^3 (b d-a e)}{e^7}+\frac {15 b^4 (d+e x)^2 (b d-a e)^2}{2 e^7}-\frac {20 b^3 x (b d-a e)^3}{e^6}+\frac {15 b^2 (b d-a e)^4 \log (d+e x)}{e^7}+\frac {6 b (b d-a e)^5}{e^7 (d+e x)}-\frac {(b d-a e)^6}{2 e^7 (d+e x)^2}+\frac {b^6 (d+e x)^4}{4 e^7} \]

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Rubi [A]  time = 0.18, antiderivative size = 158, 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 {2 b^5 (d+e x)^3 (b d-a e)}{e^7}+\frac {15 b^4 (d+e x)^2 (b d-a e)^2}{2 e^7}-\frac {20 b^3 x (b d-a e)^3}{e^6}+\frac {15 b^2 (b d-a e)^4 \log (d+e x)}{e^7}+\frac {6 b (b d-a e)^5}{e^7 (d+e x)}-\frac {(b d-a e)^6}{2 e^7 (d+e x)^2}+\frac {b^6 (d+e x)^4}{4 e^7} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]  time = 0.11, size = 303, normalized size = 1.92 \begin {gather*} \frac {-2 a^6 e^6-12 a^5 b e^5 (d+2 e x)+30 a^4 b^2 d e^4 (3 d+4 e x)+40 a^3 b^3 e^3 \left (-5 d^3-4 d^2 e x+4 d e^2 x^2+2 e^3 x^3\right )+30 a^2 b^4 e^2 \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 )+4 a b^5 e \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 )+60 b^2 (d+e x)^2 (b d-a e)^4 \log (d+e x)+b^6 \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 )}{4 e^7 (d+e x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*a^6*e^6 - 12*a^5*b*e^5*(d + 2*e*x) + 30*a^4*b^2*d*e^4*(3*d + 4*e*x) + 40*a^3*b^3*e^3*(-5*d^3 - 4*d^2*e*x +
4*d*e^2*x^2 + 2*e^3*x^3) + 30*a^2*b^4*e^2*(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) + 4*a*
b^5*e*(-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) + b^6*(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) + 60*b^2*(b*d - a*e)^4*(d + e*
x)^2*Log[d + e*x])/(4*e^7*(d + e*x)^2)

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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)^3} \, 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)^3,x]

[Out]

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

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fricas [B]  time = 0.41, size = 548, normalized size = 3.47 \begin {gather*} \frac {b^{6} e^{6} x^{6} + 22 \, b^{6} d^{6} - 108 \, a b^{5} d^{5} e + 210 \, a^{2} b^{4} d^{4} e^{2} - 200 \, a^{3} b^{3} d^{3} e^{3} + 90 \, a^{4} b^{2} d^{2} e^{4} - 12 \, a^{5} b d e^{5} - 2 \, a^{6} e^{6} - 2 \, {\left (b^{6} d e^{5} - 4 \, a b^{5} e^{6}\right )} x^{5} + 5 \, {\left (b^{6} d^{2} e^{4} - 4 \, a b^{5} d e^{5} + 6 \, a^{2} b^{4} e^{6}\right )} x^{4} - 20 \, {\left (b^{6} d^{3} e^{3} - 4 \, a b^{5} d^{2} e^{4} + 6 \, a^{2} b^{4} d e^{5} - 4 \, a^{3} b^{3} e^{6}\right )} x^{3} - 2 \, {\left (34 \, b^{6} d^{4} e^{2} - 126 \, a b^{5} d^{3} e^{3} + 165 \, a^{2} b^{4} d^{2} e^{4} - 80 \, a^{3} b^{3} d e^{5}\right )} x^{2} - 4 \, {\left (4 \, b^{6} d^{5} e - 6 \, a b^{5} d^{4} e^{2} - 15 \, a^{2} b^{4} d^{3} e^{3} + 40 \, a^{3} b^{3} d^{2} e^{4} - 30 \, a^{4} b^{2} d e^{5} + 6 \, a^{5} b e^{6}\right )} x + 60 \, {\left (b^{6} d^{6} - 4 \, a b^{5} d^{5} e + 6 \, a^{2} b^{4} d^{4} e^{2} - 4 \, a^{3} b^{3} d^{3} e^{3} + a^{4} b^{2} d^{2} e^{4} + {\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} + 2 \, {\left (b^{6} d^{5} e - 4 \, a b^{5} d^{4} e^{2} + 6 \, a^{2} b^{4} d^{3} e^{3} - 4 \, a^{3} b^{3} d^{2} e^{4} + a^{4} b^{2} d e^{5}\right )} x\right )} \log \left (e x + d\right )}{4 \, {\left (e^{9} x^{2} + 2 \, d e^{8} x + d^{2} 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)^3,x, algorithm="fricas")

[Out]

1/4*(b^6*e^6*x^6 + 22*b^6*d^6 - 108*a*b^5*d^5*e + 210*a^2*b^4*d^4*e^2 - 200*a^3*b^3*d^3*e^3 + 90*a^4*b^2*d^2*e
^4 - 12*a^5*b*d*e^5 - 2*a^6*e^6 - 2*(b^6*d*e^5 - 4*a*b^5*e^6)*x^5 + 5*(b^6*d^2*e^4 - 4*a*b^5*d*e^5 + 6*a^2*b^4
*e^6)*x^4 - 20*(b^6*d^3*e^3 - 4*a*b^5*d^2*e^4 + 6*a^2*b^4*d*e^5 - 4*a^3*b^3*e^6)*x^3 - 2*(34*b^6*d^4*e^2 - 126
*a*b^5*d^3*e^3 + 165*a^2*b^4*d^2*e^4 - 80*a^3*b^3*d*e^5)*x^2 - 4*(4*b^6*d^5*e - 6*a*b^5*d^4*e^2 - 15*a^2*b^4*d
^3*e^3 + 40*a^3*b^3*d^2*e^4 - 30*a^4*b^2*d*e^5 + 6*a^5*b*e^6)*x + 60*(b^6*d^6 - 4*a*b^5*d^5*e + 6*a^2*b^4*d^4*
e^2 - 4*a^3*b^3*d^3*e^3 + a^4*b^2*d^2*e^4 + (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 + 2*(b^6*d^5*e - 4*a*b^5*d^4*e^2 + 6*a^2*b^4*d^3*e^3 - 4*a^3*b^3*d^2*e^4 + a^4*b^2*d*e^5
)*x)*log(e*x + d))/(e^9*x^2 + 2*d*e^8*x + d^2*e^7)

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giac [B]  time = 0.19, size = 341, normalized size = 2.16 \begin {gather*} 15 \, {\left (b^{6} d^{4} - 4 \, a b^{5} d^{3} e + 6 \, a^{2} b^{4} d^{2} e^{2} - 4 \, a^{3} b^{3} d e^{3} + a^{4} b^{2} e^{4}\right )} e^{\left (-7\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {1}{4} \, {\left (b^{6} x^{4} e^{9} - 4 \, b^{6} d x^{3} e^{8} + 12 \, b^{6} d^{2} x^{2} e^{7} - 40 \, b^{6} d^{3} x e^{6} + 8 \, a b^{5} x^{3} e^{9} - 36 \, a b^{5} d x^{2} e^{8} + 144 \, a b^{5} d^{2} x e^{7} + 30 \, a^{2} b^{4} x^{2} e^{9} - 180 \, a^{2} b^{4} d x e^{8} + 80 \, a^{3} b^{3} x e^{9}\right )} e^{\left (-12\right )} + \frac {{\left (11 \, b^{6} d^{6} - 54 \, a b^{5} d^{5} e + 105 \, a^{2} b^{4} d^{4} e^{2} - 100 \, a^{3} b^{3} d^{3} e^{3} + 45 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} - a^{6} e^{6} + 12 \, {\left (b^{6} d^{5} e - 5 \, a b^{5} d^{4} e^{2} + 10 \, a^{2} b^{4} d^{3} e^{3} - 10 \, 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 )}}{2 \, {\left (x e + d\right )}^{2}} \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)^3,x, algorithm="giac")

[Out]

15*(b^6*d^4 - 4*a*b^5*d^3*e + 6*a^2*b^4*d^2*e^2 - 4*a^3*b^3*d*e^3 + a^4*b^2*e^4)*e^(-7)*log(abs(x*e + d)) + 1/
4*(b^6*x^4*e^9 - 4*b^6*d*x^3*e^8 + 12*b^6*d^2*x^2*e^7 - 40*b^6*d^3*x*e^6 + 8*a*b^5*x^3*e^9 - 36*a*b^5*d*x^2*e^
8 + 144*a*b^5*d^2*x*e^7 + 30*a^2*b^4*x^2*e^9 - 180*a^2*b^4*d*x*e^8 + 80*a^3*b^3*x*e^9)*e^(-12) + 1/2*(11*b^6*d
^6 - 54*a*b^5*d^5*e + 105*a^2*b^4*d^4*e^2 - 100*a^3*b^3*d^3*e^3 + 45*a^4*b^2*d^2*e^4 - 6*a^5*b*d*e^5 - a^6*e^6
+ 12*(b^6*d^5*e - 5*a*b^5*d^4*e^2 + 10*a^2*b^4*d^3*e^3 - 10*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)^2

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maple [B]  time = 0.06, size = 464, normalized size = 2.94 \begin {gather*} \frac {b^{6} x^{4}}{4 e^{3}}+\frac {2 a \,b^{5} x^{3}}{e^{3}}-\frac {b^{6} d \,x^{3}}{e^{4}}-\frac {a^{6}}{2 \left (e x +d \right )^{2} e}+\frac {3 a^{5} b d}{\left (e x +d \right )^{2} e^{2}}-\frac {15 a^{4} b^{2} d^{2}}{2 \left (e x +d \right )^{2} e^{3}}+\frac {10 a^{3} b^{3} d^{3}}{\left (e x +d \right )^{2} e^{4}}-\frac {15 a^{2} b^{4} d^{4}}{2 \left (e x +d \right )^{2} e^{5}}+\frac {15 a^{2} b^{4} x^{2}}{2 e^{3}}+\frac {3 a \,b^{5} d^{5}}{\left (e x +d \right )^{2} e^{6}}-\frac {9 a \,b^{5} d \,x^{2}}{e^{4}}-\frac {b^{6} d^{6}}{2 \left (e x +d \right )^{2} e^{7}}+\frac {3 b^{6} d^{2} x^{2}}{e^{5}}-\frac {6 a^{5} b}{\left (e x +d \right ) e^{2}}+\frac {30 a^{4} b^{2} d}{\left (e x +d \right ) e^{3}}+\frac {15 a^{4} b^{2} \ln \left (e x +d \right )}{e^{3}}-\frac {60 a^{3} b^{3} d^{2}}{\left (e x +d \right ) e^{4}}-\frac {60 a^{3} b^{3} d \ln \left (e x +d \right )}{e^{4}}+\frac {20 a^{3} b^{3} x}{e^{3}}+\frac {60 a^{2} b^{4} d^{3}}{\left (e x +d \right ) e^{5}}+\frac {90 a^{2} b^{4} d^{2} \ln \left (e x +d \right )}{e^{5}}-\frac {45 a^{2} b^{4} d x}{e^{4}}-\frac {30 a \,b^{5} d^{4}}{\left (e x +d \right ) e^{6}}-\frac {60 a \,b^{5} d^{3} \ln \left (e x +d \right )}{e^{6}}+\frac {36 a \,b^{5} d^{2} x}{e^{5}}+\frac {6 b^{6} d^{5}}{\left (e x +d \right ) e^{7}}+\frac {15 b^{6} d^{4} \ln \left (e x +d \right )}{e^{7}}-\frac {10 b^{6} d^{3} 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)^3,x)

[Out]

-b^6/e^4*x^3*d+15/2*b^4/e^3*x^2*a^2+3*b^6/e^5*x^2*d^2+20*b^3/e^3*a^3*x-10*b^6/e^6*d^3*x+15*b^2/e^3*ln(e*x+d)*a
^4+15*b^6/e^7*ln(e*x+d)*d^4-6*b/e^2/(e*x+d)*a^5+6*b^6/e^7/(e*x+d)*d^5-1/2/e^7/(e*x+d)^2*d^6*b^6+2*b^5/e^3*x^3*
a-15/2/e^5/(e*x+d)^2*d^4*a^2*b^4+30*b^2/e^3/(e*x+d)*a^4*d-60*b^3/e^4/(e*x+d)*a^3*d^2+60*b^4/e^5/(e*x+d)*a^2*d^
3-30*b^5/e^6/(e*x+d)*a*d^4+3/e^2/(e*x+d)^2*d*a^5*b-15/2/e^3/(e*x+d)^2*d^2*a^4*b^2+10/e^4/(e*x+d)^2*d^3*a^3*b^3
+36*b^5/e^5*a*d^2*x-60*b^3/e^4*ln(e*x+d)*d*a^3+90*b^4/e^5*ln(e*x+d)*a^2*d^2-60*b^5/e^6*ln(e*x+d)*a*d^3-9*b^5/e
^4*x^2*a*d-45*b^4/e^4*a^2*d*x+3/e^6/(e*x+d)^2*d^5*a*b^5+1/4*b^6/e^3*x^4-1/2/e/(e*x+d)^2*a^6

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maxima [B]  time = 1.51, size = 364, normalized size = 2.30 \begin {gather*} \frac {11 \, b^{6} d^{6} - 54 \, a b^{5} d^{5} e + 105 \, a^{2} b^{4} d^{4} e^{2} - 100 \, a^{3} b^{3} d^{3} e^{3} + 45 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} - a^{6} e^{6} + 12 \, {\left (b^{6} d^{5} e - 5 \, a b^{5} d^{4} e^{2} + 10 \, a^{2} b^{4} d^{3} e^{3} - 10 \, a^{3} b^{3} d^{2} e^{4} + 5 \, a^{4} b^{2} d e^{5} - a^{5} b e^{6}\right )} x}{2 \, {\left (e^{9} x^{2} + 2 \, d e^{8} x + d^{2} e^{7}\right )}} + \frac {b^{6} e^{3} x^{4} - 4 \, {\left (b^{6} d e^{2} - 2 \, a b^{5} e^{3}\right )} x^{3} + 6 \, {\left (2 \, b^{6} d^{2} e - 6 \, a b^{5} d e^{2} + 5 \, a^{2} b^{4} e^{3}\right )} x^{2} - 4 \, {\left (10 \, b^{6} d^{3} - 36 \, a b^{5} d^{2} e + 45 \, a^{2} b^{4} d e^{2} - 20 \, a^{3} b^{3} e^{3}\right )} x}{4 \, e^{6}} + \frac {15 \, {\left (b^{6} d^{4} - 4 \, a b^{5} d^{3} e + 6 \, a^{2} b^{4} d^{2} e^{2} - 4 \, a^{3} b^{3} d e^{3} + a^{4} b^{2} e^{4}\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)^3,x, algorithm="maxima")

[Out]

1/2*(11*b^6*d^6 - 54*a*b^5*d^5*e + 105*a^2*b^4*d^4*e^2 - 100*a^3*b^3*d^3*e^3 + 45*a^4*b^2*d^2*e^4 - 6*a^5*b*d*
e^5 - a^6*e^6 + 12*(b^6*d^5*e - 5*a*b^5*d^4*e^2 + 10*a^2*b^4*d^3*e^3 - 10*a^3*b^3*d^2*e^4 + 5*a^4*b^2*d*e^5 -
a^5*b*e^6)*x)/(e^9*x^2 + 2*d*e^8*x + d^2*e^7) + 1/4*(b^6*e^3*x^4 - 4*(b^6*d*e^2 - 2*a*b^5*e^3)*x^3 + 6*(2*b^6*
d^2*e - 6*a*b^5*d*e^2 + 5*a^2*b^4*e^3)*x^2 - 4*(10*b^6*d^3 - 36*a*b^5*d^2*e + 45*a^2*b^4*d*e^2 - 20*a^3*b^3*e^
3)*x)/e^6 + 15*(b^6*d^4 - 4*a*b^5*d^3*e + 6*a^2*b^4*d^2*e^2 - 4*a^3*b^3*d*e^3 + a^4*b^2*e^4)*log(e*x + d)/e^7

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mupad [B]  time = 0.57, size = 441, normalized size = 2.79 \begin {gather*} x\,\left (\frac {20\,a^3\,b^3}{e^3}-\frac {b^6\,d^3}{e^6}+\frac {3\,d\,\left (\frac {3\,d\,\left (\frac {6\,a\,b^5}{e^3}-\frac {3\,b^6\,d}{e^4}\right )}{e}-\frac {15\,a^2\,b^4}{e^3}+\frac {3\,b^6\,d^2}{e^5}\right )}{e}-\frac {3\,d^2\,\left (\frac {6\,a\,b^5}{e^3}-\frac {3\,b^6\,d}{e^4}\right )}{e^2}\right )-\frac {\frac {a^6\,e^6+6\,a^5\,b\,d\,e^5-45\,a^4\,b^2\,d^2\,e^4+100\,a^3\,b^3\,d^3\,e^3-105\,a^2\,b^4\,d^4\,e^2+54\,a\,b^5\,d^5\,e-11\,b^6\,d^6}{2\,e}-x\,\left (-6\,a^5\,b\,e^5+30\,a^4\,b^2\,d\,e^4-60\,a^3\,b^3\,d^2\,e^3+60\,a^2\,b^4\,d^3\,e^2-30\,a\,b^5\,d^4\,e+6\,b^6\,d^5\right )}{d^2\,e^6+2\,d\,e^7\,x+e^8\,x^2}+x^3\,\left (\frac {2\,a\,b^5}{e^3}-\frac {b^6\,d}{e^4}\right )-x^2\,\left (\frac {3\,d\,\left (\frac {6\,a\,b^5}{e^3}-\frac {3\,b^6\,d}{e^4}\right )}{2\,e}-\frac {15\,a^2\,b^4}{2\,e^3}+\frac {3\,b^6\,d^2}{2\,e^5}\right )+\frac {\ln \left (d+e\,x\right )\,\left (15\,a^4\,b^2\,e^4-60\,a^3\,b^3\,d\,e^3+90\,a^2\,b^4\,d^2\,e^2-60\,a\,b^5\,d^3\,e+15\,b^6\,d^4\right )}{e^7}+\frac {b^6\,x^4}{4\,e^3} \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)^3,x)

[Out]

x*((20*a^3*b^3)/e^3 - (b^6*d^3)/e^6 + (3*d*((3*d*((6*a*b^5)/e^3 - (3*b^6*d)/e^4))/e - (15*a^2*b^4)/e^3 + (3*b^
6*d^2)/e^5))/e - (3*d^2*((6*a*b^5)/e^3 - (3*b^6*d)/e^4))/e^2) - ((a^6*e^6 - 11*b^6*d^6 - 105*a^2*b^4*d^4*e^2 +
100*a^3*b^3*d^3*e^3 - 45*a^4*b^2*d^2*e^4 + 54*a*b^5*d^5*e + 6*a^5*b*d*e^5)/(2*e) - x*(6*b^6*d^5 - 6*a^5*b*e^5
+ 30*a^4*b^2*d*e^4 + 60*a^2*b^4*d^3*e^2 - 60*a^3*b^3*d^2*e^3 - 30*a*b^5*d^4*e))/(d^2*e^6 + e^8*x^2 + 2*d*e^7*
x) + x^3*((2*a*b^5)/e^3 - (b^6*d)/e^4) - x^2*((3*d*((6*a*b^5)/e^3 - (3*b^6*d)/e^4))/(2*e) - (15*a^2*b^4)/(2*e^
3) + (3*b^6*d^2)/(2*e^5)) + (log(d + e*x)*(15*b^6*d^4 + 15*a^4*b^2*e^4 - 60*a^3*b^3*d*e^3 + 90*a^2*b^4*d^2*e^2
- 60*a*b^5*d^3*e))/e^7 + (b^6*x^4)/(4*e^3)

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sympy [B]  time = 2.30, size = 340, normalized size = 2.15 \begin {gather*} \frac {b^{6} x^{4}}{4 e^{3}} + \frac {15 b^{2} \left (a e - b d\right )^{4} \log {\left (d + e x \right )}}{e^{7}} + x^{3} \left (\frac {2 a b^{5}}{e^{3}} - \frac {b^{6} d}{e^{4}}\right ) + x^{2} \left (\frac {15 a^{2} b^{4}}{2 e^{3}} - \frac {9 a b^{5} d}{e^{4}} + \frac {3 b^{6} d^{2}}{e^{5}}\right ) + x \left (\frac {20 a^{3} b^{3}}{e^{3}} - \frac {45 a^{2} b^{4} d}{e^{4}} + \frac {36 a b^{5} d^{2}}{e^{5}} - \frac {10 b^{6} d^{3}}{e^{6}}\right ) + \frac {- a^{6} e^{6} - 6 a^{5} b d e^{5} + 45 a^{4} b^{2} d^{2} e^{4} - 100 a^{3} b^{3} d^{3} e^{3} + 105 a^{2} b^{4} d^{4} e^{2} - 54 a b^{5} d^{5} e + 11 b^{6} d^{6} + x \left (- 12 a^{5} b e^{6} + 60 a^{4} b^{2} d e^{5} - 120 a^{3} b^{3} d^{2} e^{4} + 120 a^{2} b^{4} d^{3} e^{3} - 60 a b^{5} d^{4} e^{2} + 12 b^{6} d^{5} e\right )}{2 d^{2} e^{7} + 4 d e^{8} x + 2 e^{9} x^{2}} \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)**3,x)

[Out]

b**6*x**4/(4*e**3) + 15*b**2*(a*e - b*d)**4*log(d + e*x)/e**7 + x**3*(2*a*b**5/e**3 - b**6*d/e**4) + x**2*(15*
a**2*b**4/(2*e**3) - 9*a*b**5*d/e**4 + 3*b**6*d**2/e**5) + x*(20*a**3*b**3/e**3 - 45*a**2*b**4*d/e**4 + 36*a*b
**5*d**2/e**5 - 10*b**6*d**3/e**6) + (-a**6*e**6 - 6*a**5*b*d*e**5 + 45*a**4*b**2*d**2*e**4 - 100*a**3*b**3*d*
*3*e**3 + 105*a**2*b**4*d**4*e**2 - 54*a*b**5*d**5*e + 11*b**6*d**6 + x*(-12*a**5*b*e**6 + 60*a**4*b**2*d*e**5
- 120*a**3*b**3*d**2*e**4 + 120*a**2*b**4*d**3*e**3 - 60*a*b**5*d**4*e**2 + 12*b**6*d**5*e))/(2*d**2*e**7 + 4
*d*e**8*x + 2*e**9*x**2)

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