3.17.44 \(\int \frac {(a^2+2 a b x+b^2 x^2)^3}{(d+e x)^{7/2}} \, dx\) [1644]

Optimal. Leaf size=179 \[ -\frac {2 (b d-a e)^6}{5 e^7 (d+e x)^{5/2}}+\frac {4 b (b d-a e)^5}{e^7 (d+e x)^{3/2}}-\frac {30 b^2 (b d-a e)^4}{e^7 \sqrt {d+e x}}-\frac {40 b^3 (b d-a e)^3 \sqrt {d+e x}}{e^7}+\frac {10 b^4 (b d-a e)^2 (d+e x)^{3/2}}{e^7}-\frac {12 b^5 (b d-a e) (d+e x)^{5/2}}{5 e^7}+\frac {2 b^6 (d+e x)^{7/2}}{7 e^7} \]

[Out]

-2/5*(-a*e+b*d)^6/e^7/(e*x+d)^(5/2)+4*b*(-a*e+b*d)^5/e^7/(e*x+d)^(3/2)+10*b^4*(-a*e+b*d)^2*(e*x+d)^(3/2)/e^7-1
2/5*b^5*(-a*e+b*d)*(e*x+d)^(5/2)/e^7+2/7*b^6*(e*x+d)^(7/2)/e^7-30*b^2*(-a*e+b*d)^4/e^7/(e*x+d)^(1/2)-40*b^3*(-
a*e+b*d)^3*(e*x+d)^(1/2)/e^7

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Rubi [A]
time = 0.05, antiderivative size = 179, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {27, 45} \begin {gather*} -\frac {12 b^5 (d+e x)^{5/2} (b d-a e)}{5 e^7}+\frac {10 b^4 (d+e x)^{3/2} (b d-a e)^2}{e^7}-\frac {40 b^3 \sqrt {d+e x} (b d-a e)^3}{e^7}-\frac {30 b^2 (b d-a e)^4}{e^7 \sqrt {d+e x}}+\frac {4 b (b d-a e)^5}{e^7 (d+e x)^{3/2}}-\frac {2 (b d-a e)^6}{5 e^7 (d+e x)^{5/2}}+\frac {2 b^6 (d+e x)^{7/2}}{7 e^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x + b^2*x^2)^3/(d + e*x)^(7/2),x]

[Out]

(-2*(b*d - a*e)^6)/(5*e^7*(d + e*x)^(5/2)) + (4*b*(b*d - a*e)^5)/(e^7*(d + e*x)^(3/2)) - (30*b^2*(b*d - a*e)^4
)/(e^7*Sqrt[d + e*x]) - (40*b^3*(b*d - a*e)^3*Sqrt[d + e*x])/e^7 + (10*b^4*(b*d - a*e)^2*(d + e*x)^(3/2))/e^7
- (12*b^5*(b*d - a*e)*(d + e*x)^(5/2))/(5*e^7) + (2*b^6*(d + e*x)^(7/2))/(7*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 45

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)^{7/2}} \, dx &=\int \frac {(a+b x)^6}{(d+e x)^{7/2}} \, dx\\ &=\int \left (\frac {(-b d+a e)^6}{e^6 (d+e x)^{7/2}}-\frac {6 b (b d-a e)^5}{e^6 (d+e x)^{5/2}}+\frac {15 b^2 (b d-a e)^4}{e^6 (d+e x)^{3/2}}-\frac {20 b^3 (b d-a e)^3}{e^6 \sqrt {d+e x}}+\frac {15 b^4 (b d-a e)^2 \sqrt {d+e x}}{e^6}-\frac {6 b^5 (b d-a e) (d+e x)^{3/2}}{e^6}+\frac {b^6 (d+e x)^{5/2}}{e^6}\right ) \, dx\\ &=-\frac {2 (b d-a e)^6}{5 e^7 (d+e x)^{5/2}}+\frac {4 b (b d-a e)^5}{e^7 (d+e x)^{3/2}}-\frac {30 b^2 (b d-a e)^4}{e^7 \sqrt {d+e x}}-\frac {40 b^3 (b d-a e)^3 \sqrt {d+e x}}{e^7}+\frac {10 b^4 (b d-a e)^2 (d+e x)^{3/2}}{e^7}-\frac {12 b^5 (b d-a e) (d+e x)^{5/2}}{5 e^7}+\frac {2 b^6 (d+e x)^{7/2}}{7 e^7}\\ \end {align*}

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Mathematica [A]
time = 0.14, size = 291, normalized size = 1.63 \begin {gather*} -\frac {2 \left (7 a^6 e^6+14 a^5 b e^5 (2 d+5 e x)+35 a^4 b^2 e^4 \left (8 d^2+20 d e x+15 e^2 x^2\right )-140 a^3 b^3 e^3 \left (16 d^3+40 d^2 e x+30 d e^2 x^2+5 e^3 x^3\right )+35 a^2 b^4 e^2 \left (128 d^4+320 d^3 e x+240 d^2 e^2 x^2+40 d e^3 x^3-5 e^4 x^4\right )-14 a b^5 e \left (256 d^5+640 d^4 e x+480 d^3 e^2 x^2+80 d^2 e^3 x^3-10 d e^4 x^4+3 e^5 x^5\right )+b^6 \left (1024 d^6+2560 d^5 e x+1920 d^4 e^2 x^2+320 d^3 e^3 x^3-40 d^2 e^4 x^4+12 d e^5 x^5-5 e^6 x^6\right )\right )}{35 e^7 (d+e x)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + 2*a*b*x + b^2*x^2)^3/(d + e*x)^(7/2),x]

[Out]

(-2*(7*a^6*e^6 + 14*a^5*b*e^5*(2*d + 5*e*x) + 35*a^4*b^2*e^4*(8*d^2 + 20*d*e*x + 15*e^2*x^2) - 140*a^3*b^3*e^3
*(16*d^3 + 40*d^2*e*x + 30*d*e^2*x^2 + 5*e^3*x^3) + 35*a^2*b^4*e^2*(128*d^4 + 320*d^3*e*x + 240*d^2*e^2*x^2 +
40*d*e^3*x^3 - 5*e^4*x^4) - 14*a*b^5*e*(256*d^5 + 640*d^4*e*x + 480*d^3*e^2*x^2 + 80*d^2*e^3*x^3 - 10*d*e^4*x^
4 + 3*e^5*x^5) + b^6*(1024*d^6 + 2560*d^5*e*x + 1920*d^4*e^2*x^2 + 320*d^3*e^3*x^3 - 40*d^2*e^4*x^4 + 12*d*e^5
*x^5 - 5*e^6*x^6)))/(35*e^7*(d + e*x)^(5/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(383\) vs. \(2(159)=318\).
time = 0.78, size = 384, normalized size = 2.15

method	result	size
risch	\(\frac {2 b^{3} \left (5 b^{3} e^{3} x^{3}+42 a \,b^{2} e^{3} x^{2}-27 b^{3} d \,e^{2} x^{2}+175 a^{2} b \,e^{3} x -266 a \,b^{2} d \,e^{2} x +106 b^{3} d^{2} e x +700 e^{3} a^{3}-1925 a^{2} b d \,e^{2}+1792 a \,b^{2} d^{2} e -562 b^{3} d^{3}\right ) \sqrt {e x +d}}{35 e^{7}}-\frac {2 \left (75 b^{2} x^{2} e^{2}+10 a b \,e^{2} x +140 b^{2} d e x +a^{2} e^{2}+8 a b d e +66 b^{2} d^{2}\right ) \left (e^{4} a^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}{5 e^{7} \sqrt {e x +d}\, \left (e^{2} x^{2}+2 d x e +d^{2}\right )}\)	\(248\)
gosper	\(-\frac {2 \left (-5 b^{6} e^{6} x^{6}-42 a \,b^{5} e^{6} x^{5}+12 b^{6} d \,e^{5} x^{5}-175 a^{2} b^{4} e^{6} x^{4}+140 a \,b^{5} d \,e^{5} x^{4}-40 b^{6} d^{2} e^{4} x^{4}-700 a^{3} b^{3} e^{6} x^{3}+1400 a^{2} b^{4} d \,e^{5} x^{3}-1120 a \,b^{5} d^{2} e^{4} x^{3}+320 b^{6} d^{3} e^{3} x^{3}+525 a^{4} b^{2} e^{6} x^{2}-4200 a^{3} b^{3} d \,e^{5} x^{2}+8400 a^{2} b^{4} d^{2} e^{4} x^{2}-6720 a \,b^{5} d^{3} e^{3} x^{2}+1920 b^{6} d^{4} e^{2} x^{2}+70 a^{5} b \,e^{6} x +700 a^{4} b^{2} d \,e^{5} x -5600 a^{3} b^{3} d^{2} e^{4} x +11200 a^{2} b^{4} d^{3} e^{3} x -8960 a \,b^{5} d^{4} e^{2} x +2560 b^{6} d^{5} e x +7 a^{6} e^{6}+28 a^{5} b d \,e^{5}+280 b^{2} d^{2} e^{4} a^{4}-2240 a^{3} b^{3} d^{3} e^{3}+4480 d^{4} b^{4} e^{2} a^{2}-3584 a \,b^{5} d^{5} e +1024 b^{6} d^{6}\right )}{35 \left (e x +d \right )^{\frac {5}{2}} e^{7}}\)	\(377\)
trager	\(-\frac {2 \left (-5 b^{6} e^{6} x^{6}-42 a \,b^{5} e^{6} x^{5}+12 b^{6} d \,e^{5} x^{5}-175 a^{2} b^{4} e^{6} x^{4}+140 a \,b^{5} d \,e^{5} x^{4}-40 b^{6} d^{2} e^{4} x^{4}-700 a^{3} b^{3} e^{6} x^{3}+1400 a^{2} b^{4} d \,e^{5} x^{3}-1120 a \,b^{5} d^{2} e^{4} x^{3}+320 b^{6} d^{3} e^{3} x^{3}+525 a^{4} b^{2} e^{6} x^{2}-4200 a^{3} b^{3} d \,e^{5} x^{2}+8400 a^{2} b^{4} d^{2} e^{4} x^{2}-6720 a \,b^{5} d^{3} e^{3} x^{2}+1920 b^{6} d^{4} e^{2} x^{2}+70 a^{5} b \,e^{6} x +700 a^{4} b^{2} d \,e^{5} x -5600 a^{3} b^{3} d^{2} e^{4} x +11200 a^{2} b^{4} d^{3} e^{3} x -8960 a \,b^{5} d^{4} e^{2} x +2560 b^{6} d^{5} e x +7 a^{6} e^{6}+28 a^{5} b d \,e^{5}+280 b^{2} d^{2} e^{4} a^{4}-2240 a^{3} b^{3} d^{3} e^{3}+4480 d^{4} b^{4} e^{2} a^{2}-3584 a \,b^{5} d^{5} e +1024 b^{6} d^{6}\right )}{35 \left (e x +d \right )^{\frac {5}{2}} e^{7}}\)	\(377\)
derivativedivides	\(\frac {\frac {2 b^{6} \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {12 a \,b^{5} e \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {12 b^{6} d \left (e x +d \right )^{\frac {5}{2}}}{5}+10 a^{2} b^{4} e^{2} \left (e x +d \right )^{\frac {3}{2}}-20 a \,b^{5} d e \left (e x +d \right )^{\frac {3}{2}}+10 b^{6} d^{2} \left (e x +d \right )^{\frac {3}{2}}+40 a^{3} b^{3} e^{3} \sqrt {e x +d}-120 a^{2} b^{4} d \,e^{2} \sqrt {e x +d}+120 a \,b^{5} d^{2} e \sqrt {e x +d}-40 b^{6} d^{3} \sqrt {e x +d}-\frac {2 \left (a^{6} e^{6}-6 a^{5} b d \,e^{5}+15 b^{2} d^{2} e^{4} a^{4}-20 a^{3} b^{3} d^{3} e^{3}+15 d^{4} b^{4} e^{2} a^{2}-6 a \,b^{5} d^{5} e +b^{6} d^{6}\right )}{5 \left (e x +d \right )^{\frac {5}{2}}}-\frac {30 b^{2} \left (e^{4} a^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}{\sqrt {e x +d}}-\frac {4 b \left (a^{5} e^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}\right )}{\left (e x +d \right )^{\frac {3}{2}}}}{e^{7}}\)	\(384\)
default	\(\frac {\frac {2 b^{6} \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {12 a \,b^{5} e \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {12 b^{6} d \left (e x +d \right )^{\frac {5}{2}}}{5}+10 a^{2} b^{4} e^{2} \left (e x +d \right )^{\frac {3}{2}}-20 a \,b^{5} d e \left (e x +d \right )^{\frac {3}{2}}+10 b^{6} d^{2} \left (e x +d \right )^{\frac {3}{2}}+40 a^{3} b^{3} e^{3} \sqrt {e x +d}-120 a^{2} b^{4} d \,e^{2} \sqrt {e x +d}+120 a \,b^{5} d^{2} e \sqrt {e x +d}-40 b^{6} d^{3} \sqrt {e x +d}-\frac {2 \left (a^{6} e^{6}-6 a^{5} b d \,e^{5}+15 b^{2} d^{2} e^{4} a^{4}-20 a^{3} b^{3} d^{3} e^{3}+15 d^{4} b^{4} e^{2} a^{2}-6 a \,b^{5} d^{5} e +b^{6} d^{6}\right )}{5 \left (e x +d \right )^{\frac {5}{2}}}-\frac {30 b^{2} \left (e^{4} a^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}{\sqrt {e x +d}}-\frac {4 b \left (a^{5} e^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}\right )}{\left (e x +d \right )^{\frac {3}{2}}}}{e^{7}}\)	\(384\)

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)^(7/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 351 vs. \(2 (165) = 330\).
time = 0.27, size = 351, normalized size = 1.96 \begin {gather*} \frac {2}{35} \, {\left ({\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} b^{6} - 42 \, {\left (b^{6} d - a b^{5} e\right )} {\left (x e + d\right )}^{\frac {5}{2}} + 175 \, {\left (b^{6} d^{2} - 2 \, a b^{5} d e + a^{2} b^{4} e^{2}\right )} {\left (x e + d\right )}^{\frac {3}{2}} - 700 \, {\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 )} \sqrt {x e + d}\right )} e^{\left (-6\right )} - \frac {7 \, {\left (b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6} + 75 \, {\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 )} {\left (x e + d\right )}^{2} - 10 \, {\left (b^{6} d^{5} - 5 \, a b^{5} d^{4} e + 10 \, a^{2} b^{4} d^{3} e^{2} - 10 \, a^{3} b^{3} d^{2} e^{3} + 5 \, a^{4} b^{2} d e^{4} - a^{5} b e^{5}\right )} {\left (x e + d\right )}\right )} e^{\left (-6\right )}}{{\left (x e + d\right )}^{\frac {5}{2}}}\right )} e^{\left (-1\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)^(7/2),x, algorithm="maxima")

[Out]

2/35*((5*(x*e + d)^(7/2)*b^6 - 42*(b^6*d - a*b^5*e)*(x*e + d)^(5/2) + 175*(b^6*d^2 - 2*a*b^5*d*e + a^2*b^4*e^2
)*(x*e + d)^(3/2) - 700*(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)*sqrt(x*e + d))*e^(-6) - 7*(b
^6*d^6 - 6*a*b^5*d^5*e + 15*a^2*b^4*d^4*e^2 - 20*a^3*b^3*d^3*e^3 + 15*a^4*b^2*d^2*e^4 - 6*a^5*b*d*e^5 + a^6*e^
6 + 75*(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)*(x*e + d)^2 - 10*(b^6*d^5
- 5*a*b^5*d^4*e + 10*a^2*b^4*d^3*e^2 - 10*a^3*b^3*d^2*e^3 + 5*a^4*b^2*d*e^4 - a^5*b*e^5)*(x*e + d))*e^(-6)/(x
*e + d)^(5/2))*e^(-1)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 356 vs. \(2 (165) = 330\).
time = 3.10, size = 356, normalized size = 1.99 \begin {gather*} -\frac {2 \, {\left (1024 \, b^{6} d^{6} - {\left (5 \, b^{6} x^{6} + 42 \, a b^{5} x^{5} + 175 \, a^{2} b^{4} x^{4} + 700 \, a^{3} b^{3} x^{3} - 525 \, a^{4} b^{2} x^{2} - 70 \, a^{5} b x - 7 \, a^{6}\right )} e^{6} + 4 \, {\left (3 \, b^{6} d x^{5} + 35 \, a b^{5} d x^{4} + 350 \, a^{2} b^{4} d x^{3} - 1050 \, a^{3} b^{3} d x^{2} + 175 \, a^{4} b^{2} d x + 7 \, a^{5} b d\right )} e^{5} - 40 \, {\left (b^{6} d^{2} x^{4} + 28 \, a b^{5} d^{2} x^{3} - 210 \, a^{2} b^{4} d^{2} x^{2} + 140 \, a^{3} b^{3} d^{2} x - 7 \, a^{4} b^{2} d^{2}\right )} e^{4} + 320 \, {\left (b^{6} d^{3} x^{3} - 21 \, a b^{5} d^{3} x^{2} + 35 \, a^{2} b^{4} d^{3} x - 7 \, a^{3} b^{3} d^{3}\right )} e^{3} + 640 \, {\left (3 \, b^{6} d^{4} x^{2} - 14 \, a b^{5} d^{4} x + 7 \, a^{2} b^{4} d^{4}\right )} e^{2} + 512 \, {\left (5 \, b^{6} d^{5} x - 7 \, a b^{5} d^{5}\right )} e\right )} \sqrt {x e + d}}{35 \, {\left (x^{3} e^{10} + 3 \, d x^{2} e^{9} + 3 \, d^{2} x e^{8} + 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)^(7/2),x, algorithm="fricas")

[Out]

-2/35*(1024*b^6*d^6 - (5*b^6*x^6 + 42*a*b^5*x^5 + 175*a^2*b^4*x^4 + 700*a^3*b^3*x^3 - 525*a^4*b^2*x^2 - 70*a^5
*b*x - 7*a^6)*e^6 + 4*(3*b^6*d*x^5 + 35*a*b^5*d*x^4 + 350*a^2*b^4*d*x^3 - 1050*a^3*b^3*d*x^2 + 175*a^4*b^2*d*x
+ 7*a^5*b*d)*e^5 - 40*(b^6*d^2*x^4 + 28*a*b^5*d^2*x^3 - 210*a^2*b^4*d^2*x^2 + 140*a^3*b^3*d^2*x - 7*a^4*b^2*d
^2)*e^4 + 320*(b^6*d^3*x^3 - 21*a*b^5*d^3*x^2 + 35*a^2*b^4*d^3*x - 7*a^3*b^3*d^3)*e^3 + 640*(3*b^6*d^4*x^2 - 1
4*a*b^5*d^4*x + 7*a^2*b^4*d^4)*e^2 + 512*(5*b^6*d^5*x - 7*a*b^5*d^5)*e)*sqrt(x*e + d)/(x^3*e^10 + 3*d*x^2*e^9
+ 3*d^2*x*e^8 + d^3*e^7)

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Sympy [A]
time = 57.78, size = 221, normalized size = 1.23 \begin {gather*} \frac {2 b^{6} \left (d + e x\right )^{\frac {7}{2}}}{7 e^{7}} - \frac {30 b^{2} \left (a e - b d\right )^{4}}{e^{7} \sqrt {d + e x}} - \frac {4 b \left (a e - b d\right )^{5}}{e^{7} \left (d + e x\right )^{\frac {3}{2}}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \cdot \left (12 a b^{5} e - 12 b^{6} d\right )}{5 e^{7}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \cdot \left (30 a^{2} b^{4} e^{2} - 60 a b^{5} d e + 30 b^{6} d^{2}\right )}{3 e^{7}} + \frac {\sqrt {d + e x} \left (40 a^{3} b^{3} e^{3} - 120 a^{2} b^{4} d e^{2} + 120 a b^{5} d^{2} e - 40 b^{6} d^{3}\right )}{e^{7}} - \frac {2 \left (a e - b d\right )^{6}}{5 e^{7} \left (d + e x\right )^{\frac {5}{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)**(7/2),x)

[Out]

2*b**6*(d + e*x)**(7/2)/(7*e**7) - 30*b**2*(a*e - b*d)**4/(e**7*sqrt(d + e*x)) - 4*b*(a*e - b*d)**5/(e**7*(d +
e*x)**(3/2)) + (d + e*x)**(5/2)*(12*a*b**5*e - 12*b**6*d)/(5*e**7) + (d + e*x)**(3/2)*(30*a**2*b**4*e**2 - 60
*a*b**5*d*e + 30*b**6*d**2)/(3*e**7) + sqrt(d + e*x)*(40*a**3*b**3*e**3 - 120*a**2*b**4*d*e**2 + 120*a*b**5*d*
*2*e - 40*b**6*d**3)/e**7 - 2*(a*e - b*d)**6/(5*e**7*(d + e*x)**(5/2))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 458 vs. \(2 (165) = 330\).
time = 0.84, size = 458, normalized size = 2.56 \begin {gather*} \frac {2}{35} \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} b^{6} e^{42} - 42 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{6} d e^{42} + 175 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{6} d^{2} e^{42} - 700 \, \sqrt {x e + d} b^{6} d^{3} e^{42} + 42 \, {\left (x e + d\right )}^{\frac {5}{2}} a b^{5} e^{43} - 350 \, {\left (x e + d\right )}^{\frac {3}{2}} a b^{5} d e^{43} + 2100 \, \sqrt {x e + d} a b^{5} d^{2} e^{43} + 175 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{2} b^{4} e^{44} - 2100 \, \sqrt {x e + d} a^{2} b^{4} d e^{44} + 700 \, \sqrt {x e + d} a^{3} b^{3} e^{45}\right )} e^{\left (-49\right )} - \frac {2 \, {\left (75 \, {\left (x e + d\right )}^{2} b^{6} d^{4} - 10 \, {\left (x e + d\right )} b^{6} d^{5} + b^{6} d^{6} - 300 \, {\left (x e + d\right )}^{2} a b^{5} d^{3} e + 50 \, {\left (x e + d\right )} a b^{5} d^{4} e - 6 \, a b^{5} d^{5} e + 450 \, {\left (x e + d\right )}^{2} a^{2} b^{4} d^{2} e^{2} - 100 \, {\left (x e + d\right )} a^{2} b^{4} d^{3} e^{2} + 15 \, a^{2} b^{4} d^{4} e^{2} - 300 \, {\left (x e + d\right )}^{2} a^{3} b^{3} d e^{3} + 100 \, {\left (x e + d\right )} a^{3} b^{3} d^{2} e^{3} - 20 \, a^{3} b^{3} d^{3} e^{3} + 75 \, {\left (x e + d\right )}^{2} a^{4} b^{2} e^{4} - 50 \, {\left (x e + d\right )} a^{4} b^{2} d e^{4} + 15 \, a^{4} b^{2} d^{2} e^{4} + 10 \, {\left (x e + d\right )} a^{5} b e^{5} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}\right )} e^{\left (-7\right )}}{5 \, {\left (x e + d\right )}^{\frac {5}{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)^(7/2),x, algorithm="giac")

[Out]

2/35*(5*(x*e + d)^(7/2)*b^6*e^42 - 42*(x*e + d)^(5/2)*b^6*d*e^42 + 175*(x*e + d)^(3/2)*b^6*d^2*e^42 - 700*sqrt
(x*e + d)*b^6*d^3*e^42 + 42*(x*e + d)^(5/2)*a*b^5*e^43 - 350*(x*e + d)^(3/2)*a*b^5*d*e^43 + 2100*sqrt(x*e + d)
*a*b^5*d^2*e^43 + 175*(x*e + d)^(3/2)*a^2*b^4*e^44 - 2100*sqrt(x*e + d)*a^2*b^4*d*e^44 + 700*sqrt(x*e + d)*a^3
*b^3*e^45)*e^(-49) - 2/5*(75*(x*e + d)^2*b^6*d^4 - 10*(x*e + d)*b^6*d^5 + b^6*d^6 - 300*(x*e + d)^2*a*b^5*d^3*
e + 50*(x*e + d)*a*b^5*d^4*e - 6*a*b^5*d^5*e + 450*(x*e + d)^2*a^2*b^4*d^2*e^2 - 100*(x*e + d)*a^2*b^4*d^3*e^2
+ 15*a^2*b^4*d^4*e^2 - 300*(x*e + d)^2*a^3*b^3*d*e^3 + 100*(x*e + d)*a^3*b^3*d^2*e^3 - 20*a^3*b^3*d^3*e^3 + 7
5*(x*e + d)^2*a^4*b^2*e^4 - 50*(x*e + d)*a^4*b^2*d*e^4 + 15*a^4*b^2*d^2*e^4 + 10*(x*e + d)*a^5*b*e^5 - 6*a^5*b
*d*e^5 + a^6*e^6)*e^(-7)/(x*e + d)^(5/2)

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Mupad [B]
time = 0.55, size = 322, normalized size = 1.80 \begin {gather*} \frac {2\,b^6\,{\left (d+e\,x\right )}^{7/2}}{7\,e^7}-\frac {\left (12\,b^6\,d-12\,a\,b^5\,e\right )\,{\left (d+e\,x\right )}^{5/2}}{5\,e^7}-\frac {{\left (d+e\,x\right )}^2\,\left (30\,a^4\,b^2\,e^4-120\,a^3\,b^3\,d\,e^3+180\,a^2\,b^4\,d^2\,e^2-120\,a\,b^5\,d^3\,e+30\,b^6\,d^4\right )-\left (d+e\,x\right )\,\left (-4\,a^5\,b\,e^5+20\,a^4\,b^2\,d\,e^4-40\,a^3\,b^3\,d^2\,e^3+40\,a^2\,b^4\,d^3\,e^2-20\,a\,b^5\,d^4\,e+4\,b^6\,d^5\right )+\frac {2\,a^6\,e^6}{5}+\frac {2\,b^6\,d^6}{5}+6\,a^2\,b^4\,d^4\,e^2-8\,a^3\,b^3\,d^3\,e^3+6\,a^4\,b^2\,d^2\,e^4-\frac {12\,a\,b^5\,d^5\,e}{5}-\frac {12\,a^5\,b\,d\,e^5}{5}}{e^7\,{\left (d+e\,x\right )}^{5/2}}+\frac {40\,b^3\,{\left (a\,e-b\,d\right )}^3\,\sqrt {d+e\,x}}{e^7}+\frac {10\,b^4\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{3/2}}{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)^(7/2),x)

[Out]

(2*b^6*(d + e*x)^(7/2))/(7*e^7) - ((12*b^6*d - 12*a*b^5*e)*(d + e*x)^(5/2))/(5*e^7) - ((d + e*x)^2*(30*b^6*d^4
+ 30*a^4*b^2*e^4 - 120*a^3*b^3*d*e^3 + 180*a^2*b^4*d^2*e^2 - 120*a*b^5*d^3*e) - (d + e*x)*(4*b^6*d^5 - 4*a^5*
b*e^5 + 20*a^4*b^2*d*e^4 + 40*a^2*b^4*d^3*e^2 - 40*a^3*b^3*d^2*e^3 - 20*a*b^5*d^4*e) + (2*a^6*e^6)/5 + (2*b^6*
d^6)/5 + 6*a^2*b^4*d^4*e^2 - 8*a^3*b^3*d^3*e^3 + 6*a^4*b^2*d^2*e^4 - (12*a*b^5*d^5*e)/5 - (12*a^5*b*d*e^5)/5)/
(e^7*(d + e*x)^(5/2)) + (40*b^3*(a*e - b*d)^3*(d + e*x)^(1/2))/e^7 + (10*b^4*(a*e - b*d)^2*(d + e*x)^(3/2))/e^
7

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