3.12.96 \(\int \frac {(a^2+2 a b x+b^2 x^2)^3}{(d+e x)^5} \, dx\)

Optimal. Leaf size=155 \[ -\frac {b^5 x (5 b d-6 a e)}{e^6}+\frac {15 b^4 (b d-a e)^2 \log (d+e x)}{e^7}+\frac {20 b^3 (b d-a e)^3}{e^7 (d+e x)}-\frac {15 b^2 (b d-a e)^4}{2 e^7 (d+e x)^2}+\frac {2 b (b d-a e)^5}{e^7 (d+e x)^3}-\frac {(b d-a e)^6}{4 e^7 (d+e x)^4}+\frac {b^6 x^2}{2 e^5} \]

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Rubi [A]  time = 0.15, antiderivative size = 155, 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 {b^5 x (5 b d-6 a e)}{e^6}+\frac {20 b^3 (b d-a e)^3}{e^7 (d+e x)}-\frac {15 b^2 (b d-a e)^4}{2 e^7 (d+e x)^2}+\frac {15 b^4 (b d-a e)^2 \log (d+e x)}{e^7}+\frac {2 b (b d-a e)^5}{e^7 (d+e x)^3}-\frac {(b d-a e)^6}{4 e^7 (d+e x)^4}+\frac {b^6 x^2}{2 e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x + b^2*x^2)^3/(d + e*x)^5,x]

[Out]

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

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Mathematica [A]  time = 0.12, size = 301, normalized size = 1.94 \begin {gather*} -\frac {a^6 e^6+2 a^5 b e^5 (d+4 e x)+5 a^4 b^2 e^4 \left (d^2+4 d e x+6 e^2 x^2\right )+20 a^3 b^3 e^3 \left (d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right )-5 a^2 b^4 d e^2 \left (25 d^3+88 d^2 e x+108 d e^2 x^2+48 e^3 x^3\right )+2 a b^5 e \left (77 d^5+248 d^4 e x+252 d^3 e^2 x^2+48 d^2 e^3 x^3-48 d e^4 x^4-12 e^5 x^5\right )-60 b^4 (d+e x)^4 (b d-a e)^2 \log (d+e x)-\left (b^6 \left (57 d^6+168 d^5 e x+132 d^4 e^2 x^2-32 d^3 e^3 x^3-68 d^2 e^4 x^4-12 d e^5 x^5+2 e^6 x^6\right )\right )}{4 e^7 (d+e x)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + 2*a*b*x + b^2*x^2)^3/(d + e*x)^5,x]

[Out]

-1/4*(a^6*e^6 + 2*a^5*b*e^5*(d + 4*e*x) + 5*a^4*b^2*e^4*(d^2 + 4*d*e*x + 6*e^2*x^2) + 20*a^3*b^3*e^3*(d^3 + 4*
d^2*e*x + 6*d*e^2*x^2 + 4*e^3*x^3) - 5*a^2*b^4*d*e^2*(25*d^3 + 88*d^2*e*x + 108*d*e^2*x^2 + 48*e^3*x^3) + 2*a*
b^5*e*(77*d^5 + 248*d^4*e*x + 252*d^3*e^2*x^2 + 48*d^2*e^3*x^3 - 48*d*e^4*x^4 - 12*e^5*x^5) - b^6*(57*d^6 + 16
8*d^5*e*x + 132*d^4*e^2*x^2 - 32*d^3*e^3*x^3 - 68*d^2*e^4*x^4 - 12*d*e^5*x^5 + 2*e^6*x^6) - 60*b^4*(b*d - a*e)
^2*(d + e*x)^4*Log[d + e*x])/(e^7*(d + e*x)^4)

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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)^5} \, 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)^5,x]

[Out]

IntegrateAlgebraic[(a^2 + 2*a*b*x + b^2*x^2)^3/(d + e*x)^5, x]

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fricas [B]  time = 0.41, size = 571, normalized size = 3.68 \begin {gather*} \frac {2 \, b^{6} e^{6} x^{6} + 57 \, b^{6} d^{6} - 154 \, a b^{5} d^{5} e + 125 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} - 5 \, a^{4} b^{2} d^{2} e^{4} - 2 \, a^{5} b d e^{5} - a^{6} e^{6} - 12 \, {\left (b^{6} d e^{5} - 2 \, a b^{5} e^{6}\right )} x^{5} - 4 \, {\left (17 \, b^{6} d^{2} e^{4} - 24 \, a b^{5} d e^{5}\right )} x^{4} - 16 \, {\left (2 \, b^{6} d^{3} e^{3} + 6 \, a b^{5} d^{2} e^{4} - 15 \, a^{2} b^{4} d e^{5} + 5 \, a^{3} b^{3} e^{6}\right )} x^{3} + 6 \, {\left (22 \, b^{6} d^{4} e^{2} - 84 \, a b^{5} d^{3} e^{3} + 90 \, a^{2} b^{4} d^{2} e^{4} - 20 \, a^{3} b^{3} d e^{5} - 5 \, a^{4} b^{2} e^{6}\right )} x^{2} + 4 \, {\left (42 \, b^{6} d^{5} e - 124 \, a b^{5} d^{4} e^{2} + 110 \, a^{2} b^{4} d^{3} e^{3} - 20 \, a^{3} b^{3} d^{2} e^{4} - 5 \, a^{4} b^{2} d e^{5} - 2 \, a^{5} b e^{6}\right )} x + 60 \, {\left (b^{6} d^{6} - 2 \, a b^{5} d^{5} e + a^{2} b^{4} d^{4} e^{2} + {\left (b^{6} d^{2} e^{4} - 2 \, a b^{5} d e^{5} + a^{2} b^{4} e^{6}\right )} x^{4} + 4 \, {\left (b^{6} d^{3} e^{3} - 2 \, a b^{5} d^{2} e^{4} + a^{2} b^{4} d e^{5}\right )} x^{3} + 6 \, {\left (b^{6} d^{4} e^{2} - 2 \, a b^{5} d^{3} e^{3} + a^{2} b^{4} d^{2} e^{4}\right )} x^{2} + 4 \, {\left (b^{6} d^{5} e - 2 \, a b^{5} d^{4} e^{2} + a^{2} b^{4} d^{3} e^{3}\right )} x\right )} \log \left (e x + d\right )}{4 \, {\left (e^{11} x^{4} + 4 \, d e^{10} x^{3} + 6 \, d^{2} e^{9} x^{2} + 4 \, d^{3} e^{8} x + d^{4} 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)^5,x, algorithm="fricas")

[Out]

1/4*(2*b^6*e^6*x^6 + 57*b^6*d^6 - 154*a*b^5*d^5*e + 125*a^2*b^4*d^4*e^2 - 20*a^3*b^3*d^3*e^3 - 5*a^4*b^2*d^2*e
^4 - 2*a^5*b*d*e^5 - a^6*e^6 - 12*(b^6*d*e^5 - 2*a*b^5*e^6)*x^5 - 4*(17*b^6*d^2*e^4 - 24*a*b^5*d*e^5)*x^4 - 16
*(2*b^6*d^3*e^3 + 6*a*b^5*d^2*e^4 - 15*a^2*b^4*d*e^5 + 5*a^3*b^3*e^6)*x^3 + 6*(22*b^6*d^4*e^2 - 84*a*b^5*d^3*e
^3 + 90*a^2*b^4*d^2*e^4 - 20*a^3*b^3*d*e^5 - 5*a^4*b^2*e^6)*x^2 + 4*(42*b^6*d^5*e - 124*a*b^5*d^4*e^2 + 110*a^
2*b^4*d^3*e^3 - 20*a^3*b^3*d^2*e^4 - 5*a^4*b^2*d*e^5 - 2*a^5*b*e^6)*x + 60*(b^6*d^6 - 2*a*b^5*d^5*e + a^2*b^4*
d^4*e^2 + (b^6*d^2*e^4 - 2*a*b^5*d*e^5 + a^2*b^4*e^6)*x^4 + 4*(b^6*d^3*e^3 - 2*a*b^5*d^2*e^4 + a^2*b^4*d*e^5)*
x^3 + 6*(b^6*d^4*e^2 - 2*a*b^5*d^3*e^3 + a^2*b^4*d^2*e^4)*x^2 + 4*(b^6*d^5*e - 2*a*b^5*d^4*e^2 + a^2*b^4*d^3*e
^3)*x)*log(e*x + d))/(e^11*x^4 + 4*d*e^10*x^3 + 6*d^2*e^9*x^2 + 4*d^3*e^8*x + d^4*e^7)

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giac [B]  time = 0.18, size = 514, normalized size = 3.32 \begin {gather*} \frac {1}{2} \, {\left (b^{6} - \frac {12 \, {\left (b^{6} d e - a b^{5} e^{2}\right )} e^{\left (-1\right )}}{x e + d}\right )} {\left (x e + d\right )}^{2} e^{\left (-7\right )} - 15 \, {\left (b^{6} d^{2} - 2 \, a b^{5} d e + a^{2} b^{4} e^{2}\right )} e^{\left (-7\right )} \log \left (\frac {{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) + \frac {1}{4} \, {\left (\frac {80 \, b^{6} d^{3} e^{29}}{x e + d} - \frac {30 \, b^{6} d^{4} e^{29}}{{\left (x e + d\right )}^{2}} + \frac {8 \, b^{6} d^{5} e^{29}}{{\left (x e + d\right )}^{3}} - \frac {b^{6} d^{6} e^{29}}{{\left (x e + d\right )}^{4}} - \frac {240 \, a b^{5} d^{2} e^{30}}{x e + d} + \frac {120 \, a b^{5} d^{3} e^{30}}{{\left (x e + d\right )}^{2}} - \frac {40 \, a b^{5} d^{4} e^{30}}{{\left (x e + d\right )}^{3}} + \frac {6 \, a b^{5} d^{5} e^{30}}{{\left (x e + d\right )}^{4}} + \frac {240 \, a^{2} b^{4} d e^{31}}{x e + d} - \frac {180 \, a^{2} b^{4} d^{2} e^{31}}{{\left (x e + d\right )}^{2}} + \frac {80 \, a^{2} b^{4} d^{3} e^{31}}{{\left (x e + d\right )}^{3}} - \frac {15 \, a^{2} b^{4} d^{4} e^{31}}{{\left (x e + d\right )}^{4}} - \frac {80 \, a^{3} b^{3} e^{32}}{x e + d} + \frac {120 \, a^{3} b^{3} d e^{32}}{{\left (x e + d\right )}^{2}} - \frac {80 \, a^{3} b^{3} d^{2} e^{32}}{{\left (x e + d\right )}^{3}} + \frac {20 \, a^{3} b^{3} d^{3} e^{32}}{{\left (x e + d\right )}^{4}} - \frac {30 \, a^{4} b^{2} e^{33}}{{\left (x e + d\right )}^{2}} + \frac {40 \, a^{4} b^{2} d e^{33}}{{\left (x e + d\right )}^{3}} - \frac {15 \, a^{4} b^{2} d^{2} e^{33}}{{\left (x e + d\right )}^{4}} - \frac {8 \, a^{5} b e^{34}}{{\left (x e + d\right )}^{3}} + \frac {6 \, a^{5} b d e^{34}}{{\left (x e + d\right )}^{4}} - \frac {a^{6} e^{35}}{{\left (x e + d\right )}^{4}}\right )} e^{\left (-36\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)^5,x, algorithm="giac")

[Out]

1/2*(b^6 - 12*(b^6*d*e - a*b^5*e^2)*e^(-1)/(x*e + d))*(x*e + d)^2*e^(-7) - 15*(b^6*d^2 - 2*a*b^5*d*e + a^2*b^4
*e^2)*e^(-7)*log(abs(x*e + d)*e^(-1)/(x*e + d)^2) + 1/4*(80*b^6*d^3*e^29/(x*e + d) - 30*b^6*d^4*e^29/(x*e + d)
^2 + 8*b^6*d^5*e^29/(x*e + d)^3 - b^6*d^6*e^29/(x*e + d)^4 - 240*a*b^5*d^2*e^30/(x*e + d) + 120*a*b^5*d^3*e^30
/(x*e + d)^2 - 40*a*b^5*d^4*e^30/(x*e + d)^3 + 6*a*b^5*d^5*e^30/(x*e + d)^4 + 240*a^2*b^4*d*e^31/(x*e + d) - 1
80*a^2*b^4*d^2*e^31/(x*e + d)^2 + 80*a^2*b^4*d^3*e^31/(x*e + d)^3 - 15*a^2*b^4*d^4*e^31/(x*e + d)^4 - 80*a^3*b
^3*e^32/(x*e + d) + 120*a^3*b^3*d*e^32/(x*e + d)^2 - 80*a^3*b^3*d^2*e^32/(x*e + d)^3 + 20*a^3*b^3*d^3*e^32/(x*
e + d)^4 - 30*a^4*b^2*e^33/(x*e + d)^2 + 40*a^4*b^2*d*e^33/(x*e + d)^3 - 15*a^4*b^2*d^2*e^33/(x*e + d)^4 - 8*a
^5*b*e^34/(x*e + d)^3 + 6*a^5*b*d*e^34/(x*e + d)^4 - a^6*e^35/(x*e + d)^4)*e^(-36)

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maple [B]  time = 0.06, size = 498, normalized size = 3.21 \begin {gather*} -\frac {a^{6}}{4 \left (e x +d \right )^{4} e}+\frac {3 a^{5} b d}{2 \left (e x +d \right )^{4} e^{2}}-\frac {15 a^{4} b^{2} d^{2}}{4 \left (e x +d \right )^{4} e^{3}}+\frac {5 a^{3} b^{3} d^{3}}{\left (e x +d \right )^{4} e^{4}}-\frac {15 a^{2} b^{4} d^{4}}{4 \left (e x +d \right )^{4} e^{5}}+\frac {3 a \,b^{5} d^{5}}{2 \left (e x +d \right )^{4} e^{6}}-\frac {b^{6} d^{6}}{4 \left (e x +d \right )^{4} e^{7}}-\frac {2 a^{5} b}{\left (e x +d \right )^{3} e^{2}}+\frac {10 a^{4} b^{2} d}{\left (e x +d \right )^{3} e^{3}}-\frac {20 a^{3} b^{3} d^{2}}{\left (e x +d \right )^{3} e^{4}}+\frac {20 a^{2} b^{4} d^{3}}{\left (e x +d \right )^{3} e^{5}}-\frac {10 a \,b^{5} d^{4}}{\left (e x +d \right )^{3} e^{6}}+\frac {2 b^{6} d^{5}}{\left (e x +d \right )^{3} e^{7}}-\frac {15 a^{4} b^{2}}{2 \left (e x +d \right )^{2} e^{3}}+\frac {30 a^{3} b^{3} d}{\left (e x +d \right )^{2} e^{4}}-\frac {45 a^{2} b^{4} d^{2}}{\left (e x +d \right )^{2} e^{5}}+\frac {30 a \,b^{5} d^{3}}{\left (e x +d \right )^{2} e^{6}}-\frac {15 b^{6} d^{4}}{2 \left (e x +d \right )^{2} e^{7}}+\frac {b^{6} x^{2}}{2 e^{5}}-\frac {20 a^{3} b^{3}}{\left (e x +d \right ) e^{4}}+\frac {60 a^{2} b^{4} d}{\left (e x +d \right ) e^{5}}+\frac {15 a^{2} b^{4} \ln \left (e x +d \right )}{e^{5}}-\frac {60 a \,b^{5} d^{2}}{\left (e x +d \right ) e^{6}}-\frac {30 a \,b^{5} d \ln \left (e x +d \right )}{e^{6}}+\frac {6 a \,b^{5} x}{e^{5}}+\frac {20 b^{6} d^{3}}{\left (e x +d \right ) e^{7}}+\frac {15 b^{6} d^{2} \ln \left (e x +d \right )}{e^{7}}-\frac {5 b^{6} d 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)^5,x)

[Out]

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

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maxima [B]  time = 1.52, size = 387, normalized size = 2.50 \begin {gather*} \frac {57 \, b^{6} d^{6} - 154 \, a b^{5} d^{5} e + 125 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} - 5 \, a^{4} b^{2} d^{2} e^{4} - 2 \, a^{5} b d e^{5} - a^{6} e^{6} + 80 \, {\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} + 30 \, {\left (7 \, b^{6} d^{4} e^{2} - 20 \, a b^{5} d^{3} e^{3} + 18 \, 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} + 4 \, {\left (47 \, b^{6} d^{5} e - 130 \, a b^{5} d^{4} e^{2} + 110 \, a^{2} b^{4} d^{3} e^{3} - 20 \, a^{3} b^{3} d^{2} e^{4} - 5 \, a^{4} b^{2} d e^{5} - 2 \, a^{5} b e^{6}\right )} x}{4 \, {\left (e^{11} x^{4} + 4 \, d e^{10} x^{3} + 6 \, d^{2} e^{9} x^{2} + 4 \, d^{3} e^{8} x + d^{4} e^{7}\right )}} + \frac {b^{6} e x^{2} - 2 \, {\left (5 \, b^{6} d - 6 \, a b^{5} e\right )} x}{2 \, e^{6}} + \frac {15 \, {\left (b^{6} d^{2} - 2 \, a b^{5} d e + a^{2} b^{4} e^{2}\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)^5,x, algorithm="maxima")

[Out]

1/4*(57*b^6*d^6 - 154*a*b^5*d^5*e + 125*a^2*b^4*d^4*e^2 - 20*a^3*b^3*d^3*e^3 - 5*a^4*b^2*d^2*e^4 - 2*a^5*b*d*e
^5 - a^6*e^6 + 80*(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 + 30*(7*b^6*d^4*e^2 - 20
*a*b^5*d^3*e^3 + 18*a^2*b^4*d^2*e^4 - 4*a^3*b^3*d*e^5 - a^4*b^2*e^6)*x^2 + 4*(47*b^6*d^5*e - 130*a*b^5*d^4*e^2
+ 110*a^2*b^4*d^3*e^3 - 20*a^3*b^3*d^2*e^4 - 5*a^4*b^2*d*e^5 - 2*a^5*b*e^6)*x)/(e^11*x^4 + 4*d*e^10*x^3 + 6*d
^2*e^9*x^2 + 4*d^3*e^8*x + d^4*e^7) + 1/2*(b^6*e*x^2 - 2*(5*b^6*d - 6*a*b^5*e)*x)/e^6 + 15*(b^6*d^2 - 2*a*b^5*
d*e + a^2*b^4*e^2)*log(e*x + d)/e^7

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mupad [B]  time = 0.15, size = 387, normalized size = 2.50 \begin {gather*} x\,\left (\frac {6\,a\,b^5}{e^5}-\frac {5\,b^6\,d}{e^6}\right )-\frac {x^2\,\left (\frac {15\,a^4\,b^2\,e^5}{2}+30\,a^3\,b^3\,d\,e^4-135\,a^2\,b^4\,d^2\,e^3+150\,a\,b^5\,d^3\,e^2-\frac {105\,b^6\,d^4\,e}{2}\right )+\frac {a^6\,e^6+2\,a^5\,b\,d\,e^5+5\,a^4\,b^2\,d^2\,e^4+20\,a^3\,b^3\,d^3\,e^3-125\,a^2\,b^4\,d^4\,e^2+154\,a\,b^5\,d^5\,e-57\,b^6\,d^6}{4\,e}+x\,\left (2\,a^5\,b\,e^5+5\,a^4\,b^2\,d\,e^4+20\,a^3\,b^3\,d^2\,e^3-110\,a^2\,b^4\,d^3\,e^2+130\,a\,b^5\,d^4\,e-47\,b^6\,d^5\right )+x^3\,\left (20\,a^3\,b^3\,e^5-60\,a^2\,b^4\,d\,e^4+60\,a\,b^5\,d^2\,e^3-20\,b^6\,d^3\,e^2\right )}{d^4\,e^6+4\,d^3\,e^7\,x+6\,d^2\,e^8\,x^2+4\,d\,e^9\,x^3+e^{10}\,x^4}+\frac {b^6\,x^2}{2\,e^5}+\frac {\ln \left (d+e\,x\right )\,\left (15\,a^2\,b^4\,e^2-30\,a\,b^5\,d\,e+15\,b^6\,d^2\right )}{e^7} \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)^5,x)

[Out]

x*((6*a*b^5)/e^5 - (5*b^6*d)/e^6) - (x^2*((15*a^4*b^2*e^5)/2 - (105*b^6*d^4*e)/2 + 150*a*b^5*d^3*e^2 + 30*a^3*
b^3*d*e^4 - 135*a^2*b^4*d^2*e^3) + (a^6*e^6 - 57*b^6*d^6 - 125*a^2*b^4*d^4*e^2 + 20*a^3*b^3*d^3*e^3 + 5*a^4*b^
2*d^2*e^4 + 154*a*b^5*d^5*e + 2*a^5*b*d*e^5)/(4*e) + x*(2*a^5*b*e^5 - 47*b^6*d^5 + 5*a^4*b^2*d*e^4 - 110*a^2*b
^4*d^3*e^2 + 20*a^3*b^3*d^2*e^3 + 130*a*b^5*d^4*e) + x^3*(20*a^3*b^3*e^5 - 20*b^6*d^3*e^2 + 60*a*b^5*d^2*e^3 -
60*a^2*b^4*d*e^4))/(d^4*e^6 + e^10*x^4 + 4*d^3*e^7*x + 4*d*e^9*x^3 + 6*d^2*e^8*x^2) + (b^6*x^2)/(2*e^5) + (lo
g(d + e*x)*(15*b^6*d^2 + 15*a^2*b^4*e^2 - 30*a*b^5*d*e))/e^7

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sympy [B]  time = 10.02, size = 394, normalized size = 2.54 \begin {gather*} \frac {b^{6} x^{2}}{2 e^{5}} + \frac {15 b^{4} \left (a e - b d\right )^{2} \log {\left (d + e x \right )}}{e^{7}} + x \left (\frac {6 a b^{5}}{e^{5}} - \frac {5 b^{6} d}{e^{6}}\right ) + \frac {- a^{6} e^{6} - 2 a^{5} b d e^{5} - 5 a^{4} b^{2} d^{2} e^{4} - 20 a^{3} b^{3} d^{3} e^{3} + 125 a^{2} b^{4} d^{4} e^{2} - 154 a b^{5} d^{5} e + 57 b^{6} d^{6} + x^{3} \left (- 80 a^{3} b^{3} e^{6} + 240 a^{2} b^{4} d e^{5} - 240 a b^{5} d^{2} e^{4} + 80 b^{6} d^{3} e^{3}\right ) + x^{2} \left (- 30 a^{4} b^{2} e^{6} - 120 a^{3} b^{3} d e^{5} + 540 a^{2} b^{4} d^{2} e^{4} - 600 a b^{5} d^{3} e^{3} + 210 b^{6} d^{4} e^{2}\right ) + x \left (- 8 a^{5} b e^{6} - 20 a^{4} b^{2} d e^{5} - 80 a^{3} b^{3} d^{2} e^{4} + 440 a^{2} b^{4} d^{3} e^{3} - 520 a b^{5} d^{4} e^{2} + 188 b^{6} d^{5} e\right )}{4 d^{4} e^{7} + 16 d^{3} e^{8} x + 24 d^{2} e^{9} x^{2} + 16 d e^{10} x^{3} + 4 e^{11} x^{4}} \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)**5,x)

[Out]

b**6*x**2/(2*e**5) + 15*b**4*(a*e - b*d)**2*log(d + e*x)/e**7 + x*(6*a*b**5/e**5 - 5*b**6*d/e**6) + (-a**6*e**
6 - 2*a**5*b*d*e**5 - 5*a**4*b**2*d**2*e**4 - 20*a**3*b**3*d**3*e**3 + 125*a**2*b**4*d**4*e**2 - 154*a*b**5*d*
*5*e + 57*b**6*d**6 + x**3*(-80*a**3*b**3*e**6 + 240*a**2*b**4*d*e**5 - 240*a*b**5*d**2*e**4 + 80*b**6*d**3*e*
*3) + x**2*(-30*a**4*b**2*e**6 - 120*a**3*b**3*d*e**5 + 540*a**2*b**4*d**2*e**4 - 600*a*b**5*d**3*e**3 + 210*b
**6*d**4*e**2) + x*(-8*a**5*b*e**6 - 20*a**4*b**2*d*e**5 - 80*a**3*b**3*d**2*e**4 + 440*a**2*b**4*d**3*e**3 -
520*a*b**5*d**4*e**2 + 188*b**6*d**5*e))/(4*d**4*e**7 + 16*d**3*e**8*x + 24*d**2*e**9*x**2 + 16*d*e**10*x**3 +
4*e**11*x**4)

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