\(\int \frac {(a^2+2 a b x+b^2 x^2)^3}{(d+e x)^{3/2}} \, dx\) [1642]

Optimal result

Integrand size = 28, antiderivative size = 179 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{3/2}} \, dx=-\frac {2 (b d-a e)^6}{e^7 \sqrt {d+e x}}-\frac {12 b (b d-a e)^5 \sqrt {d+e x}}{e^7}+\frac {10 b^2 (b d-a e)^4 (d+e x)^{3/2}}{e^7}-\frac {8 b^3 (b d-a e)^3 (d+e x)^{5/2}}{e^7}+\frac {30 b^4 (b d-a e)^2 (d+e x)^{7/2}}{7 e^7}-\frac {4 b^5 (b d-a e) (d+e x)^{9/2}}{3 e^7}+\frac {2 b^6 (d+e x)^{11/2}}{11 e^7} \]

[Out]

10*b^2*(-a*e+b*d)^4*(e*x+d)^(3/2)/e^7-8*b^3*(-a*e+b*d)^3*(e*x+d)^(5/2)/e^7+30/7*b^4*(-a*e+b*d)^2*(e*x+d)^(7/2)
/e^7-4/3*b^5*(-a*e+b*d)*(e*x+d)^(9/2)/e^7+2/11*b^6*(e*x+d)^(11/2)/e^7-2*(-a*e+b*d)^6/e^7/(e*x+d)^(1/2)-12*b*(-
a*e+b*d)^5*(e*x+d)^(1/2)/e^7

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {27, 45} \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{3/2}} \, dx=-\frac {4 b^5 (d+e x)^{9/2} (b d-a e)}{3 e^7}+\frac {30 b^4 (d+e x)^{7/2} (b d-a e)^2}{7 e^7}-\frac {8 b^3 (d+e x)^{5/2} (b d-a e)^3}{e^7}+\frac {10 b^2 (d+e x)^{3/2} (b d-a e)^4}{e^7}-\frac {12 b \sqrt {d+e x} (b d-a e)^5}{e^7}-\frac {2 (b d-a e)^6}{e^7 \sqrt {d+e x}}+\frac {2 b^6 (d+e x)^{11/2}}{11 e^7} \]

[In]

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

[Out]

(-2*(b*d - a*e)^6)/(e^7*Sqrt[d + e*x]) - (12*b*(b*d - a*e)^5*Sqrt[d + e*x])/e^7 + (10*b^2*(b*d - a*e)^4*(d + e
*x)^(3/2))/e^7 - (8*b^3*(b*d - a*e)^3*(d + e*x)^(5/2))/e^7 + (30*b^4*(b*d - a*e)^2*(d + e*x)^(7/2))/(7*e^7) -
(4*b^5*(b*d - a*e)*(d + e*x)^(9/2))/(3*e^7) + (2*b^6*(d + e*x)^(11/2))/(11*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*} \text {integral}& = \int \frac {(a+b x)^6}{(d+e x)^{3/2}} \, dx \\ & = \int \left (\frac {(-b d+a e)^6}{e^6 (d+e x)^{3/2}}-\frac {6 b (b d-a e)^5}{e^6 \sqrt {d+e x}}+\frac {15 b^2 (b d-a e)^4 \sqrt {d+e x}}{e^6}-\frac {20 b^3 (b d-a e)^3 (d+e x)^{3/2}}{e^6}+\frac {15 b^4 (b d-a e)^2 (d+e x)^{5/2}}{e^6}-\frac {6 b^5 (b d-a e) (d+e x)^{7/2}}{e^6}+\frac {b^6 (d+e x)^{9/2}}{e^6}\right ) \, dx \\ & = -\frac {2 (b d-a e)^6}{e^7 \sqrt {d+e x}}-\frac {12 b (b d-a e)^5 \sqrt {d+e x}}{e^7}+\frac {10 b^2 (b d-a e)^4 (d+e x)^{3/2}}{e^7}-\frac {8 b^3 (b d-a e)^3 (d+e x)^{5/2}}{e^7}+\frac {30 b^4 (b d-a e)^2 (d+e x)^{7/2}}{7 e^7}-\frac {4 b^5 (b d-a e) (d+e x)^{9/2}}{3 e^7}+\frac {2 b^6 (d+e x)^{11/2}}{11 e^7} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 288, normalized size of antiderivative = 1.61 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{3/2}} \, dx=\frac {2 \left (-231 a^6 e^6+1386 a^5 b e^5 (2 d+e x)+1155 a^4 b^2 e^4 \left (-8 d^2-4 d e x+e^2 x^2\right )+924 a^3 b^3 e^3 \left (16 d^3+8 d^2 e x-2 d e^2 x^2+e^3 x^3\right )+99 a^2 b^4 e^2 \left (-128 d^4-64 d^3 e x+16 d^2 e^2 x^2-8 d e^3 x^3+5 e^4 x^4\right )+22 a b^5 e \left (256 d^5+128 d^4 e x-32 d^3 e^2 x^2+16 d^2 e^3 x^3-10 d e^4 x^4+7 e^5 x^5\right )+b^6 \left (-1024 d^6-512 d^5 e x+128 d^4 e^2 x^2-64 d^3 e^3 x^3+40 d^2 e^4 x^4-28 d e^5 x^5+21 e^6 x^6\right )\right )}{231 e^7 \sqrt {d+e x}} \]

[In]

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

[Out]

(2*(-231*a^6*e^6 + 1386*a^5*b*e^5*(2*d + e*x) + 1155*a^4*b^2*e^4*(-8*d^2 - 4*d*e*x + e^2*x^2) + 924*a^3*b^3*e^
3*(16*d^3 + 8*d^2*e*x - 2*d*e^2*x^2 + e^3*x^3) + 99*a^2*b^4*e^2*(-128*d^4 - 64*d^3*e*x + 16*d^2*e^2*x^2 - 8*d*
e^3*x^3 + 5*e^4*x^4) + 22*a*b^5*e*(256*d^5 + 128*d^4*e*x - 32*d^3*e^2*x^2 + 16*d^2*e^3*x^3 - 10*d*e^4*x^4 + 7*
e^5*x^5) + b^6*(-1024*d^6 - 512*d^5*e*x + 128*d^4*e^2*x^2 - 64*d^3*e^3*x^3 + 40*d^2*e^4*x^4 - 28*d*e^5*x^5 + 2
1*e^6*x^6)))/(231*e^7*Sqrt[d + e*x])

Maple [A] (verified)

Time = 2.36 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.55

method	result	size
pseudoelliptic	\(\frac {\frac {2 \left (21 e^{6} x^{6}-28 d \,e^{5} x^{5}+40 d^{2} e^{4} x^{4}-64 x^{3} d^{3} e^{3}+128 d^{4} e^{2} x^{2}-512 d^{5} e x -1024 d^{6}\right ) b^{6}}{231}+\frac {1024 \left (\frac {7}{256} e^{5} x^{5}-\frac {5}{128} x^{4} d \,e^{4}+\frac {1}{16} d^{2} e^{3} x^{3}-\frac {1}{8} d^{3} e^{2} x^{2}+\frac {1}{2} d^{4} e x +d^{5}\right ) e a \,b^{5}}{21}-\frac {768 e^{2} \left (-\frac {5}{128} e^{4} x^{4}+\frac {1}{16} d \,e^{3} x^{3}-\frac {1}{8} d^{2} e^{2} x^{2}+\frac {1}{2} d^{3} e x +d^{4}\right ) a^{2} b^{4}}{7}+128 \left (\frac {1}{16} e^{3} x^{3}-\frac {1}{8} d \,e^{2} x^{2}+\frac {1}{2} d^{2} e x +d^{3}\right ) e^{3} a^{3} b^{3}-80 \left (-\frac {1}{8} x^{2} e^{2}+\frac {1}{2} d e x +d^{2}\right ) e^{4} a^{4} b^{2}+24 \left (\frac {e x}{2}+d \right ) e^{5} a^{5} b -2 a^{6} e^{6}}{\sqrt {e x +d}\, e^{7}}\)	\(278\)
risch	\(\frac {2 b \left (21 x^{5} e^{5} b^{5}+154 x^{4} a \,b^{4} e^{5}-49 x^{4} b^{5} d \,e^{4}+495 x^{3} a^{2} b^{3} e^{5}-374 x^{3} a \,b^{4} d \,e^{4}+89 x^{3} b^{5} d^{2} e^{3}+924 x^{2} a^{3} b^{2} e^{5}-1287 x^{2} a^{2} b^{3} d \,e^{4}+726 x^{2} a \,b^{4} d^{2} e^{3}-153 x^{2} b^{5} d^{3} e^{2}+1155 a^{4} b \,e^{5} x -2772 a^{3} b^{2} d \,e^{4} x +2871 x \,a^{2} b^{3} d^{2} e^{3}-1430 x a \,b^{4} d^{3} e^{2}+281 b^{5} d^{4} e x +1386 a^{5} e^{5}-5775 a^{4} b d \,e^{4}+10164 a^{3} b^{2} d^{2} e^{3}-9207 a^{2} b^{3} d^{3} e^{2}+4246 a \,b^{4} d^{4} e -793 b^{5} d^{5}\right ) \sqrt {e x +d}}{231 e^{7}}-\frac {2 \left (a^{6} e^{6}-6 a^{5} b d \,e^{5}+15 a^{4} b^{2} d^{2} e^{4}-20 a^{3} b^{3} d^{3} e^{3}+15 a^{2} b^{4} d^{4} e^{2}-6 a \,b^{5} d^{5} e +b^{6} d^{6}\right )}{e^{7} \sqrt {e x +d}}\)	\(364\)
gosper	\(-\frac {2 \left (-21 x^{6} b^{6} e^{6}-154 x^{5} a \,b^{5} e^{6}+28 x^{5} b^{6} d \,e^{5}-495 x^{4} a^{2} b^{4} e^{6}+220 x^{4} a \,b^{5} d \,e^{5}-40 x^{4} b^{6} d^{2} e^{4}-924 x^{3} a^{3} b^{3} e^{6}+792 x^{3} a^{2} b^{4} d \,e^{5}-352 x^{3} a \,b^{5} d^{2} e^{4}+64 x^{3} b^{6} d^{3} e^{3}-1155 x^{2} a^{4} b^{2} e^{6}+1848 x^{2} a^{3} b^{3} d \,e^{5}-1584 x^{2} a^{2} b^{4} d^{2} e^{4}+704 x^{2} a \,b^{5} d^{3} e^{3}-128 x^{2} b^{6} d^{4} e^{2}-1386 x \,a^{5} b \,e^{6}+4620 x \,a^{4} b^{2} d \,e^{5}-7392 x \,a^{3} b^{3} d^{2} e^{4}+6336 x \,a^{2} b^{4} d^{3} e^{3}-2816 x a \,b^{5} d^{4} e^{2}+512 x \,b^{6} d^{5} e +231 a^{6} e^{6}-2772 a^{5} b d \,e^{5}+9240 a^{4} b^{2} d^{2} e^{4}-14784 a^{3} b^{3} d^{3} e^{3}+12672 a^{2} b^{4} d^{4} e^{2}-5632 a \,b^{5} d^{5} e +1024 b^{6} d^{6}\right )}{231 \sqrt {e x +d}\, e^{7}}\)	\(377\)
trager	\(-\frac {2 \left (-21 x^{6} b^{6} e^{6}-154 x^{5} a \,b^{5} e^{6}+28 x^{5} b^{6} d \,e^{5}-495 x^{4} a^{2} b^{4} e^{6}+220 x^{4} a \,b^{5} d \,e^{5}-40 x^{4} b^{6} d^{2} e^{4}-924 x^{3} a^{3} b^{3} e^{6}+792 x^{3} a^{2} b^{4} d \,e^{5}-352 x^{3} a \,b^{5} d^{2} e^{4}+64 x^{3} b^{6} d^{3} e^{3}-1155 x^{2} a^{4} b^{2} e^{6}+1848 x^{2} a^{3} b^{3} d \,e^{5}-1584 x^{2} a^{2} b^{4} d^{2} e^{4}+704 x^{2} a \,b^{5} d^{3} e^{3}-128 x^{2} b^{6} d^{4} e^{2}-1386 x \,a^{5} b \,e^{6}+4620 x \,a^{4} b^{2} d \,e^{5}-7392 x \,a^{3} b^{3} d^{2} e^{4}+6336 x \,a^{2} b^{4} d^{3} e^{3}-2816 x a \,b^{5} d^{4} e^{2}+512 x \,b^{6} d^{5} e +231 a^{6} e^{6}-2772 a^{5} b d \,e^{5}+9240 a^{4} b^{2} d^{2} e^{4}-14784 a^{3} b^{3} d^{3} e^{3}+12672 a^{2} b^{4} d^{4} e^{2}-5632 a \,b^{5} d^{5} e +1024 b^{6} d^{6}\right )}{231 \sqrt {e x +d}\, e^{7}}\)	\(377\)
derivativedivides	\(\frac {\frac {2 b^{6} \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {4 a \,b^{5} e \left (e x +d \right )^{\frac {9}{2}}}{3}-\frac {4 b^{6} d \left (e x +d \right )^{\frac {9}{2}}}{3}+\frac {30 a^{2} b^{4} e^{2} \left (e x +d \right )^{\frac {7}{2}}}{7}-\frac {60 a \,b^{5} d e \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {30 b^{6} d^{2} \left (e x +d \right )^{\frac {7}{2}}}{7}+8 a^{3} b^{3} e^{3} \left (e x +d \right )^{\frac {5}{2}}-24 a^{2} b^{4} d \,e^{2} \left (e x +d \right )^{\frac {5}{2}}+24 a \,b^{5} d^{2} e \left (e x +d \right )^{\frac {5}{2}}-8 b^{6} d^{3} \left (e x +d \right )^{\frac {5}{2}}+10 a^{4} b^{2} e^{4} \left (e x +d \right )^{\frac {3}{2}}-40 a^{3} b^{3} d \,e^{3} \left (e x +d \right )^{\frac {3}{2}}+60 a^{2} b^{4} d^{2} e^{2} \left (e x +d \right )^{\frac {3}{2}}-40 a \,b^{5} d^{3} e \left (e x +d \right )^{\frac {3}{2}}+10 b^{6} d^{4} \left (e x +d \right )^{\frac {3}{2}}+12 a^{5} b \,e^{5} \sqrt {e x +d}-60 a^{4} b^{2} d \,e^{4} \sqrt {e x +d}+120 a^{3} b^{3} d^{2} e^{3} \sqrt {e x +d}-120 a^{2} b^{4} d^{3} e^{2} \sqrt {e x +d}+60 a \,b^{5} d^{4} e \sqrt {e x +d}-12 b^{6} d^{5} \sqrt {e x +d}-\frac {2 \left (a^{6} e^{6}-6 a^{5} b d \,e^{5}+15 a^{4} b^{2} d^{2} e^{4}-20 a^{3} b^{3} d^{3} e^{3}+15 a^{2} b^{4} d^{4} e^{2}-6 a \,b^{5} d^{5} e +b^{6} d^{6}\right )}{\sqrt {e x +d}}}{e^{7}}\)	\(448\)
default	\(\frac {\frac {2 b^{6} \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {4 a \,b^{5} e \left (e x +d \right )^{\frac {9}{2}}}{3}-\frac {4 b^{6} d \left (e x +d \right )^{\frac {9}{2}}}{3}+\frac {30 a^{2} b^{4} e^{2} \left (e x +d \right )^{\frac {7}{2}}}{7}-\frac {60 a \,b^{5} d e \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {30 b^{6} d^{2} \left (e x +d \right )^{\frac {7}{2}}}{7}+8 a^{3} b^{3} e^{3} \left (e x +d \right )^{\frac {5}{2}}-24 a^{2} b^{4} d \,e^{2} \left (e x +d \right )^{\frac {5}{2}}+24 a \,b^{5} d^{2} e \left (e x +d \right )^{\frac {5}{2}}-8 b^{6} d^{3} \left (e x +d \right )^{\frac {5}{2}}+10 a^{4} b^{2} e^{4} \left (e x +d \right )^{\frac {3}{2}}-40 a^{3} b^{3} d \,e^{3} \left (e x +d \right )^{\frac {3}{2}}+60 a^{2} b^{4} d^{2} e^{2} \left (e x +d \right )^{\frac {3}{2}}-40 a \,b^{5} d^{3} e \left (e x +d \right )^{\frac {3}{2}}+10 b^{6} d^{4} \left (e x +d \right )^{\frac {3}{2}}+12 a^{5} b \,e^{5} \sqrt {e x +d}-60 a^{4} b^{2} d \,e^{4} \sqrt {e x +d}+120 a^{3} b^{3} d^{2} e^{3} \sqrt {e x +d}-120 a^{2} b^{4} d^{3} e^{2} \sqrt {e x +d}+60 a \,b^{5} d^{4} e \sqrt {e x +d}-12 b^{6} d^{5} \sqrt {e x +d}-\frac {2 \left (a^{6} e^{6}-6 a^{5} b d \,e^{5}+15 a^{4} b^{2} d^{2} e^{4}-20 a^{3} b^{3} d^{3} e^{3}+15 a^{2} b^{4} d^{4} e^{2}-6 a \,b^{5} d^{5} e +b^{6} d^{6}\right )}{\sqrt {e x +d}}}{e^{7}}\)	\(448\)

[In]

int((b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^(3/2),x,method=_RETURNVERBOSE)

[Out]

2/231*((21*e^6*x^6-28*d*e^5*x^5+40*d^2*e^4*x^4-64*d^3*e^3*x^3+128*d^4*e^2*x^2-512*d^5*e*x-1024*d^6)*b^6+5632*(
7/256*e^5*x^5-5/128*x^4*d*e^4+1/16*d^2*e^3*x^3-1/8*d^3*e^2*x^2+1/2*d^4*e*x+d^5)*e*a*b^5-12672*e^2*(-5/128*e^4*
x^4+1/16*d*e^3*x^3-1/8*d^2*e^2*x^2+1/2*d^3*e*x+d^4)*a^2*b^4+14784*(1/16*e^3*x^3-1/8*d*e^2*x^2+1/2*d^2*e*x+d^3)
*e^3*a^3*b^3-9240*(-1/8*x^2*e^2+1/2*d*e*x+d^2)*e^4*a^4*b^2+2772*(1/2*e*x+d)*e^5*a^5*b-231*a^6*e^6)/(e*x+d)^(1/
2)/e^7

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 365 vs. \(2 (159) = 318\).

Time = 0.28 (sec) , antiderivative size = 365, normalized size of antiderivative = 2.04 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{3/2}} \, dx=\frac {2 \, {\left (21 \, b^{6} e^{6} x^{6} - 1024 \, b^{6} d^{6} + 5632 \, a b^{5} d^{5} e - 12672 \, a^{2} b^{4} d^{4} e^{2} + 14784 \, a^{3} b^{3} d^{3} e^{3} - 9240 \, a^{4} b^{2} d^{2} e^{4} + 2772 \, a^{5} b d e^{5} - 231 \, a^{6} e^{6} - 14 \, {\left (2 \, b^{6} d e^{5} - 11 \, a b^{5} e^{6}\right )} x^{5} + 5 \, {\left (8 \, b^{6} d^{2} e^{4} - 44 \, a b^{5} d e^{5} + 99 \, a^{2} b^{4} e^{6}\right )} x^{4} - 4 \, {\left (16 \, b^{6} d^{3} e^{3} - 88 \, a b^{5} d^{2} e^{4} + 198 \, a^{2} b^{4} d e^{5} - 231 \, a^{3} b^{3} e^{6}\right )} x^{3} + {\left (128 \, b^{6} d^{4} e^{2} - 704 \, a b^{5} d^{3} e^{3} + 1584 \, a^{2} b^{4} d^{2} e^{4} - 1848 \, a^{3} b^{3} d e^{5} + 1155 \, a^{4} b^{2} e^{6}\right )} x^{2} - 2 \, {\left (256 \, b^{6} d^{5} e - 1408 \, a b^{5} d^{4} e^{2} + 3168 \, a^{2} b^{4} d^{3} e^{3} - 3696 \, a^{3} b^{3} d^{2} e^{4} + 2310 \, a^{4} b^{2} d e^{5} - 693 \, a^{5} b e^{6}\right )} x\right )} \sqrt {e x + d}}{231 \, {\left (e^{8} x + d e^{7}\right )}} \]

[In]

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

[Out]

2/231*(21*b^6*e^6*x^6 - 1024*b^6*d^6 + 5632*a*b^5*d^5*e - 12672*a^2*b^4*d^4*e^2 + 14784*a^3*b^3*d^3*e^3 - 9240
*a^4*b^2*d^2*e^4 + 2772*a^5*b*d*e^5 - 231*a^6*e^6 - 14*(2*b^6*d*e^5 - 11*a*b^5*e^6)*x^5 + 5*(8*b^6*d^2*e^4 - 4
4*a*b^5*d*e^5 + 99*a^2*b^4*e^6)*x^4 - 4*(16*b^6*d^3*e^3 - 88*a*b^5*d^2*e^4 + 198*a^2*b^4*d*e^5 - 231*a^3*b^3*e
^6)*x^3 + (128*b^6*d^4*e^2 - 704*a*b^5*d^3*e^3 + 1584*a^2*b^4*d^2*e^4 - 1848*a^3*b^3*d*e^5 + 1155*a^4*b^2*e^6)
*x^2 - 2*(256*b^6*d^5*e - 1408*a*b^5*d^4*e^2 + 3168*a^2*b^4*d^3*e^3 - 3696*a^3*b^3*d^2*e^4 + 2310*a^4*b^2*d*e^
5 - 693*a^5*b*e^6)*x)*sqrt(e*x + d)/(e^8*x + d*e^7)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 416 vs. \(2 (167) = 334\).

Time = 10.17 (sec) , antiderivative size = 416, normalized size of antiderivative = 2.32 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{3/2}} \, dx=\begin {cases} \frac {2 \left (\frac {b^{6} \left (d + e x\right )^{\frac {11}{2}}}{11 e^{6}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \cdot \left (6 a b^{5} e - 6 b^{6} d\right )}{9 e^{6}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \cdot \left (15 a^{2} b^{4} e^{2} - 30 a b^{5} d e + 15 b^{6} d^{2}\right )}{7 e^{6}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \cdot \left (20 a^{3} b^{3} e^{3} - 60 a^{2} b^{4} d e^{2} + 60 a b^{5} d^{2} e - 20 b^{6} d^{3}\right )}{5 e^{6}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \cdot \left (15 a^{4} b^{2} e^{4} - 60 a^{3} b^{3} d e^{3} + 90 a^{2} b^{4} d^{2} e^{2} - 60 a b^{5} d^{3} e + 15 b^{6} d^{4}\right )}{3 e^{6}} + \frac {\sqrt {d + e x} \left (6 a^{5} b e^{5} - 30 a^{4} b^{2} d e^{4} + 60 a^{3} b^{3} d^{2} e^{3} - 60 a^{2} b^{4} d^{3} e^{2} + 30 a b^{5} d^{4} e - 6 b^{6} d^{5}\right )}{e^{6}} - \frac {\left (a e - b d\right )^{6}}{e^{6} \sqrt {d + e x}}\right )}{e} & \text {for}\: e \neq 0 \\\frac {a^{6} x + 3 a^{5} b x^{2} + 5 a^{4} b^{2} x^{3} + 5 a^{3} b^{3} x^{4} + 3 a^{2} b^{4} x^{5} + a b^{5} x^{6} + \frac {b^{6} x^{7}}{7}}{d^{\frac {3}{2}}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((2*(b**6*(d + e*x)**(11/2)/(11*e**6) + (d + e*x)**(9/2)*(6*a*b**5*e - 6*b**6*d)/(9*e**6) + (d + e*x)
**(7/2)*(15*a**2*b**4*e**2 - 30*a*b**5*d*e + 15*b**6*d**2)/(7*e**6) + (d + e*x)**(5/2)*(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)/(5*e**6) + (d + e*x)**(3/2)*(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)/(3*e**6) + sqrt(d + e*x)*(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)/e*
*6 - (a*e - b*d)**6/(e**6*sqrt(d + e*x)))/e, Ne(e, 0)), ((a**6*x + 3*a**5*b*x**2 + 5*a**4*b**2*x**3 + 5*a**3*b
**3*x**4 + 3*a**2*b**4*x**5 + a*b**5*x**6 + b**6*x**7/7)/d**(3/2), True))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 358 vs. \(2 (159) = 318\).

Time = 0.21 (sec) , antiderivative size = 358, normalized size of antiderivative = 2.00 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{3/2}} \, dx=\frac {2 \, {\left (\frac {21 \, {\left (e x + d\right )}^{\frac {11}{2}} b^{6} - 154 \, {\left (b^{6} d - a b^{5} e\right )} {\left (e x + d\right )}^{\frac {9}{2}} + 495 \, {\left (b^{6} d^{2} - 2 \, a b^{5} d e + a^{2} b^{4} e^{2}\right )} {\left (e x + d\right )}^{\frac {7}{2}} - 924 \, {\left (b^{6} d^{3} - 3 \, a b^{5} d^{2} e + 3 \, a^{2} b^{4} d e^{2} - a^{3} b^{3} e^{3}\right )} {\left (e x + d\right )}^{\frac {5}{2}} + 1155 \, {\left (b^{6} d^{4} - 4 \, a b^{5} d^{3} e + 6 \, a^{2} b^{4} d^{2} e^{2} - 4 \, a^{3} b^{3} d e^{3} + a^{4} b^{2} e^{4}\right )} {\left (e x + d\right )}^{\frac {3}{2}} - 1386 \, {\left (b^{6} d^{5} - 5 \, a b^{5} d^{4} e + 10 \, a^{2} b^{4} d^{3} e^{2} - 10 \, a^{3} b^{3} d^{2} e^{3} + 5 \, a^{4} b^{2} d e^{4} - a^{5} b e^{5}\right )} \sqrt {e x + d}}{e^{6}} - \frac {231 \, {\left (b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}\right )}}{\sqrt {e x + d} e^{6}}\right )}}{231 \, e} \]

[In]

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

[Out]

2/231*((21*(e*x + d)^(11/2)*b^6 - 154*(b^6*d - a*b^5*e)*(e*x + d)^(9/2) + 495*(b^6*d^2 - 2*a*b^5*d*e + a^2*b^4
*e^2)*(e*x + d)^(7/2) - 924*(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*x + d)^(5/2) + 1155*(
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*x + d)^(3/2) - 1386*(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)*sqrt(e*x + d))/e^6 - 2
31*(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)/(sqrt(e*x + d)*e^6))/e

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 479 vs. \(2 (159) = 318\).

Time = 0.28 (sec) , antiderivative size = 479, normalized size of antiderivative = 2.68 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{3/2}} \, dx=-\frac {2 \, {\left (b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}\right )}}{\sqrt {e x + d} e^{7}} + \frac {2 \, {\left (21 \, {\left (e x + d\right )}^{\frac {11}{2}} b^{6} e^{70} - 154 \, {\left (e x + d\right )}^{\frac {9}{2}} b^{6} d e^{70} + 495 \, {\left (e x + d\right )}^{\frac {7}{2}} b^{6} d^{2} e^{70} - 924 \, {\left (e x + d\right )}^{\frac {5}{2}} b^{6} d^{3} e^{70} + 1155 \, {\left (e x + d\right )}^{\frac {3}{2}} b^{6} d^{4} e^{70} - 1386 \, \sqrt {e x + d} b^{6} d^{5} e^{70} + 154 \, {\left (e x + d\right )}^{\frac {9}{2}} a b^{5} e^{71} - 990 \, {\left (e x + d\right )}^{\frac {7}{2}} a b^{5} d e^{71} + 2772 \, {\left (e x + d\right )}^{\frac {5}{2}} a b^{5} d^{2} e^{71} - 4620 \, {\left (e x + d\right )}^{\frac {3}{2}} a b^{5} d^{3} e^{71} + 6930 \, \sqrt {e x + d} a b^{5} d^{4} e^{71} + 495 \, {\left (e x + d\right )}^{\frac {7}{2}} a^{2} b^{4} e^{72} - 2772 \, {\left (e x + d\right )}^{\frac {5}{2}} a^{2} b^{4} d e^{72} + 6930 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{2} b^{4} d^{2} e^{72} - 13860 \, \sqrt {e x + d} a^{2} b^{4} d^{3} e^{72} + 924 \, {\left (e x + d\right )}^{\frac {5}{2}} a^{3} b^{3} e^{73} - 4620 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{3} b^{3} d e^{73} + 13860 \, \sqrt {e x + d} a^{3} b^{3} d^{2} e^{73} + 1155 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{4} b^{2} e^{74} - 6930 \, \sqrt {e x + d} a^{4} b^{2} d e^{74} + 1386 \, \sqrt {e x + d} a^{5} b e^{75}\right )}}{231 \, e^{77}} \]

[In]

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

[Out]

-2*(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)/(sqrt(e*x + d)*e^7) + 2/231*(21*(e*x + d)^(11/2)*b^6*e^70 - 154*(e*x + d)^(9/2)*b^6*d*e^70 + 495*(e*x
+ d)^(7/2)*b^6*d^2*e^70 - 924*(e*x + d)^(5/2)*b^6*d^3*e^70 + 1155*(e*x + d)^(3/2)*b^6*d^4*e^70 - 1386*sqrt(e*x
+ d)*b^6*d^5*e^70 + 154*(e*x + d)^(9/2)*a*b^5*e^71 - 990*(e*x + d)^(7/2)*a*b^5*d*e^71 + 2772*(e*x + d)^(5/2)*
a*b^5*d^2*e^71 - 4620*(e*x + d)^(3/2)*a*b^5*d^3*e^71 + 6930*sqrt(e*x + d)*a*b^5*d^4*e^71 + 495*(e*x + d)^(7/2)
*a^2*b^4*e^72 - 2772*(e*x + d)^(5/2)*a^2*b^4*d*e^72 + 6930*(e*x + d)^(3/2)*a^2*b^4*d^2*e^72 - 13860*sqrt(e*x +
d)*a^2*b^4*d^3*e^72 + 924*(e*x + d)^(5/2)*a^3*b^3*e^73 - 4620*(e*x + d)^(3/2)*a^3*b^3*d*e^73 + 13860*sqrt(e*x
+ d)*a^3*b^3*d^2*e^73 + 1155*(e*x + d)^(3/2)*a^4*b^2*e^74 - 6930*sqrt(e*x + d)*a^4*b^2*d*e^74 + 1386*sqrt(e*x
+ d)*a^5*b*e^75)/e^77

Mupad [B] (verification not implemented)

Time = 10.22 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.29 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{3/2}} \, dx=\frac {2\,b^6\,{\left (d+e\,x\right )}^{11/2}}{11\,e^7}-\frac {2\,a^6\,e^6-12\,a^5\,b\,d\,e^5+30\,a^4\,b^2\,d^2\,e^4-40\,a^3\,b^3\,d^3\,e^3+30\,a^2\,b^4\,d^4\,e^2-12\,a\,b^5\,d^5\,e+2\,b^6\,d^6}{e^7\,\sqrt {d+e\,x}}-\frac {\left (12\,b^6\,d-12\,a\,b^5\,e\right )\,{\left (d+e\,x\right )}^{9/2}}{9\,e^7}+\frac {10\,b^2\,{\left (a\,e-b\,d\right )}^4\,{\left (d+e\,x\right )}^{3/2}}{e^7}+\frac {8\,b^3\,{\left (a\,e-b\,d\right )}^3\,{\left (d+e\,x\right )}^{5/2}}{e^7}+\frac {30\,b^4\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{7/2}}{7\,e^7}+\frac {12\,b\,{\left (a\,e-b\,d\right )}^5\,\sqrt {d+e\,x}}{e^7} \]

[In]

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

[Out]

(2*b^6*(d + e*x)^(11/2))/(11*e^7) - (2*a^6*e^6 + 2*b^6*d^6 + 30*a^2*b^4*d^4*e^2 - 40*a^3*b^3*d^3*e^3 + 30*a^4*
b^2*d^2*e^4 - 12*a*b^5*d^5*e - 12*a^5*b*d*e^5)/(e^7*(d + e*x)^(1/2)) - ((12*b^6*d - 12*a*b^5*e)*(d + e*x)^(9/2
))/(9*e^7) + (10*b^2*(a*e - b*d)^4*(d + e*x)^(3/2))/e^7 + (8*b^3*(a*e - b*d)^3*(d + e*x)^(5/2))/e^7 + (30*b^4*
(a*e - b*d)^2*(d + e*x)^(7/2))/(7*e^7) + (12*b*(a*e - b*d)^5*(d + e*x)^(1/2))/e^7