\(\int (d+e x)^{3/2} (a^2+2 a b x+b^2 x^2)^3 \, dx\) [1639]

Optimal result

Integrand size = 28, antiderivative size = 187 \[ \int (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {2 (b d-a e)^6 (d+e x)^{5/2}}{5 e^7}-\frac {12 b (b d-a e)^5 (d+e x)^{7/2}}{7 e^7}+\frac {10 b^2 (b d-a e)^4 (d+e x)^{9/2}}{3 e^7}-\frac {40 b^3 (b d-a e)^3 (d+e x)^{11/2}}{11 e^7}+\frac {30 b^4 (b d-a e)^2 (d+e x)^{13/2}}{13 e^7}-\frac {4 b^5 (b d-a e) (d+e x)^{15/2}}{5 e^7}+\frac {2 b^6 (d+e x)^{17/2}}{17 e^7} \]

[Out]

2/5*(-a*e+b*d)^6*(e*x+d)^(5/2)/e^7-12/7*b*(-a*e+b*d)^5*(e*x+d)^(7/2)/e^7+10/3*b^2*(-a*e+b*d)^4*(e*x+d)^(9/2)/e
^7-40/11*b^3*(-a*e+b*d)^3*(e*x+d)^(11/2)/e^7+30/13*b^4*(-a*e+b*d)^2*(e*x+d)^(13/2)/e^7-4/5*b^5*(-a*e+b*d)*(e*x
+d)^(15/2)/e^7+2/17*b^6*(e*x+d)^(17/2)/e^7

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 187, 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} \[ \int (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=-\frac {4 b^5 (d+e x)^{15/2} (b d-a e)}{5 e^7}+\frac {30 b^4 (d+e x)^{13/2} (b d-a e)^2}{13 e^7}-\frac {40 b^3 (d+e x)^{11/2} (b d-a e)^3}{11 e^7}+\frac {10 b^2 (d+e x)^{9/2} (b d-a e)^4}{3 e^7}-\frac {12 b (d+e x)^{7/2} (b d-a e)^5}{7 e^7}+\frac {2 (d+e x)^{5/2} (b d-a e)^6}{5 e^7}+\frac {2 b^6 (d+e x)^{17/2}}{17 e^7} \]

[In]

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

[Out]

(2*(b*d - a*e)^6*(d + e*x)^(5/2))/(5*e^7) - (12*b*(b*d - a*e)^5*(d + e*x)^(7/2))/(7*e^7) + (10*b^2*(b*d - a*e)
^4*(d + e*x)^(9/2))/(3*e^7) - (40*b^3*(b*d - a*e)^3*(d + e*x)^(11/2))/(11*e^7) + (30*b^4*(b*d - a*e)^2*(d + e*
x)^(13/2))/(13*e^7) - (4*b^5*(b*d - a*e)*(d + e*x)^(15/2))/(5*e^7) + (2*b^6*(d + e*x)^(17/2))/(17*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 (a+b x)^6 (d+e x)^{3/2} \, dx \\ & = \int \left (\frac {(-b d+a e)^6 (d+e x)^{3/2}}{e^6}-\frac {6 b (b d-a e)^5 (d+e x)^{5/2}}{e^6}+\frac {15 b^2 (b d-a e)^4 (d+e x)^{7/2}}{e^6}-\frac {20 b^3 (b d-a e)^3 (d+e x)^{9/2}}{e^6}+\frac {15 b^4 (b d-a e)^2 (d+e x)^{11/2}}{e^6}-\frac {6 b^5 (b d-a e) (d+e x)^{13/2}}{e^6}+\frac {b^6 (d+e x)^{15/2}}{e^6}\right ) \, dx \\ & = \frac {2 (b d-a e)^6 (d+e x)^{5/2}}{5 e^7}-\frac {12 b (b d-a e)^5 (d+e x)^{7/2}}{7 e^7}+\frac {10 b^2 (b d-a e)^4 (d+e x)^{9/2}}{3 e^7}-\frac {40 b^3 (b d-a e)^3 (d+e x)^{11/2}}{11 e^7}+\frac {30 b^4 (b d-a e)^2 (d+e x)^{13/2}}{13 e^7}-\frac {4 b^5 (b d-a e) (d+e x)^{15/2}}{5 e^7}+\frac {2 b^6 (d+e x)^{17/2}}{17 e^7} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.56 \[ \int (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {2 (d+e x)^{5/2} \left (51051 a^6 e^6+43758 a^5 b e^5 (-2 d+5 e x)+12155 a^4 b^2 e^4 \left (8 d^2-20 d e x+35 e^2 x^2\right )+4420 a^3 b^3 e^3 \left (-16 d^3+40 d^2 e x-70 d e^2 x^2+105 e^3 x^3\right )+255 a^2 b^4 e^2 \left (128 d^4-320 d^3 e x+560 d^2 e^2 x^2-840 d e^3 x^3+1155 e^4 x^4\right )+34 a b^5 e \left (-256 d^5+640 d^4 e x-1120 d^3 e^2 x^2+1680 d^2 e^3 x^3-2310 d e^4 x^4+3003 e^5 x^5\right )+b^6 \left (1024 d^6-2560 d^5 e x+4480 d^4 e^2 x^2-6720 d^3 e^3 x^3+9240 d^2 e^4 x^4-12012 d e^5 x^5+15015 e^6 x^6\right )\right )}{255255 e^7} \]

[In]

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

[Out]

(2*(d + e*x)^(5/2)*(51051*a^6*e^6 + 43758*a^5*b*e^5*(-2*d + 5*e*x) + 12155*a^4*b^2*e^4*(8*d^2 - 20*d*e*x + 35*
e^2*x^2) + 4420*a^3*b^3*e^3*(-16*d^3 + 40*d^2*e*x - 70*d*e^2*x^2 + 105*e^3*x^3) + 255*a^2*b^4*e^2*(128*d^4 - 3
20*d^3*e*x + 560*d^2*e^2*x^2 - 840*d*e^3*x^3 + 1155*e^4*x^4) + 34*a*b^5*e*(-256*d^5 + 640*d^4*e*x - 1120*d^3*e
^2*x^2 + 1680*d^2*e^3*x^3 - 2310*d*e^4*x^4 + 3003*e^5*x^5) + b^6*(1024*d^6 - 2560*d^5*e*x + 4480*d^4*e^2*x^2 -
6720*d^3*e^3*x^3 + 9240*d^2*e^4*x^4 - 12012*d*e^5*x^5 + 15015*e^6*x^6)))/(255255*e^7)

Maple [A] (verified)

Time = 2.31 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.48

method	result	size
pseudoelliptic	\(\frac {2 \left (e x +d \right )^{\frac {5}{2}} \left (\left (\frac {5}{17} b^{6} x^{6}+a^{6}+2 a \,x^{5} b^{5}+\frac {75}{13} a^{2} x^{4} b^{4}+\frac {100}{11} a^{3} x^{3} b^{3}+\frac {25}{3} a^{4} x^{2} b^{2}+\frac {30}{7} a^{5} x b \right ) e^{6}-\frac {12 b \left (\frac {7}{51} b^{5} x^{5}+\frac {35}{39} a \,b^{4} x^{4}+\frac {350}{143} a^{2} b^{3} x^{3}+\frac {350}{99} a^{3} b^{2} x^{2}+\frac {25}{9} a^{4} b x +a^{5}\right ) d \,e^{5}}{7}+\frac {40 b^{2} \left (\frac {21}{221} b^{4} x^{4}+\frac {84}{143} a \,b^{3} x^{3}+\frac {210}{143} a^{2} b^{2} x^{2}+\frac {20}{11} a^{3} b x +a^{4}\right ) d^{2} e^{4}}{21}-\frac {320 \left (\frac {21}{221} b^{3} x^{3}+\frac {7}{13} a \,b^{2} x^{2}+\frac {15}{13} a^{2} b x +a^{3}\right ) b^{3} d^{3} e^{3}}{231}+\frac {640 \left (\frac {7}{51} b^{2} x^{2}+\frac {2}{3} a b x +a^{2}\right ) b^{4} d^{4} e^{2}}{1001}-\frac {512 \left (\frac {5 b x}{17}+a \right ) b^{5} d^{5} e}{3003}+\frac {1024 b^{6} d^{6}}{51051}\right )}{5 e^{7}}\)	\(276\)
gosper	\(\frac {2 \left (e x +d \right )^{\frac {5}{2}} \left (15015 x^{6} b^{6} e^{6}+102102 x^{5} a \,b^{5} e^{6}-12012 x^{5} b^{6} d \,e^{5}+294525 x^{4} a^{2} b^{4} e^{6}-78540 x^{4} a \,b^{5} d \,e^{5}+9240 x^{4} b^{6} d^{2} e^{4}+464100 x^{3} a^{3} b^{3} e^{6}-214200 x^{3} a^{2} b^{4} d \,e^{5}+57120 x^{3} a \,b^{5} d^{2} e^{4}-6720 x^{3} b^{6} d^{3} e^{3}+425425 x^{2} a^{4} b^{2} e^{6}-309400 x^{2} a^{3} b^{3} d \,e^{5}+142800 x^{2} a^{2} b^{4} d^{2} e^{4}-38080 x^{2} a \,b^{5} d^{3} e^{3}+4480 x^{2} b^{6} d^{4} e^{2}+218790 x \,a^{5} b \,e^{6}-243100 x \,a^{4} b^{2} d \,e^{5}+176800 x \,a^{3} b^{3} d^{2} e^{4}-81600 x \,a^{2} b^{4} d^{3} e^{3}+21760 x a \,b^{5} d^{4} e^{2}-2560 x \,b^{6} d^{5} e +51051 a^{6} e^{6}-87516 a^{5} b d \,e^{5}+97240 a^{4} b^{2} d^{2} e^{4}-70720 a^{3} b^{3} d^{3} e^{3}+32640 a^{2} b^{4} d^{4} e^{2}-8704 a \,b^{5} d^{5} e +1024 b^{6} d^{6}\right )}{255255 e^{7}}\)	\(377\)
derivativedivides	\(\frac {\frac {2 b^{6} \left (e x +d \right )^{\frac {17}{2}}}{17}+\frac {2 \left (2 a e b -2 b^{2} d \right ) b^{4} \left (e x +d \right )^{\frac {15}{2}}}{5}+\frac {2 \left (\left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) b^{4}+2 \left (2 a e b -2 b^{2} d \right )^{2} b^{2}+b^{2} \left (2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a e b -2 b^{2} d \right )^{2}\right )\right ) \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (4 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \left (2 a e b -2 b^{2} d \right ) b^{2}+\left (2 a e b -2 b^{2} d \right ) \left (2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a e b -2 b^{2} d \right )^{2}\right )\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (\left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \left (2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a e b -2 b^{2} d \right )^{2}\right )+2 \left (2 a e b -2 b^{2} d \right )^{2} \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )+b^{2} \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )^{2}\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {6 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )^{2} \left (2 a e b -2 b^{2} d \right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )^{3} \left (e x +d \right )^{\frac {5}{2}}}{5}}{e^{7}}\)	\(457\)
default	\(\frac {\frac {2 b^{6} \left (e x +d \right )^{\frac {17}{2}}}{17}+\frac {2 \left (2 a e b -2 b^{2} d \right ) b^{4} \left (e x +d \right )^{\frac {15}{2}}}{5}+\frac {2 \left (\left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) b^{4}+2 \left (2 a e b -2 b^{2} d \right )^{2} b^{2}+b^{2} \left (2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a e b -2 b^{2} d \right )^{2}\right )\right ) \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (4 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \left (2 a e b -2 b^{2} d \right ) b^{2}+\left (2 a e b -2 b^{2} d \right ) \left (2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a e b -2 b^{2} d \right )^{2}\right )\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (\left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \left (2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a e b -2 b^{2} d \right )^{2}\right )+2 \left (2 a e b -2 b^{2} d \right )^{2} \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )+b^{2} \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )^{2}\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {6 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )^{2} \left (2 a e b -2 b^{2} d \right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )^{3} \left (e x +d \right )^{\frac {5}{2}}}{5}}{e^{7}}\)	\(457\)
trager	\(\frac {2 \left (15015 b^{6} e^{8} x^{8}+102102 a \,b^{5} e^{8} x^{7}+18018 b^{6} d \,e^{7} x^{7}+294525 a^{2} b^{4} e^{8} x^{6}+125664 a \,b^{5} d \,e^{7} x^{6}+231 b^{6} d^{2} e^{6} x^{6}+464100 a^{3} b^{3} e^{8} x^{5}+374850 a^{2} b^{4} d \,e^{7} x^{5}+2142 a \,b^{5} d^{2} e^{6} x^{5}-252 b^{6} d^{3} e^{5} x^{5}+425425 a^{4} b^{2} e^{8} x^{4}+618800 a^{3} b^{3} d \,e^{7} x^{4}+8925 a^{2} b^{4} d^{2} e^{6} x^{4}-2380 a \,b^{5} d^{3} e^{5} x^{4}+280 b^{6} d^{4} e^{4} x^{4}+218790 a^{5} b \,e^{8} x^{3}+607750 a^{4} b^{2} d \,e^{7} x^{3}+22100 a^{3} b^{3} d^{2} e^{6} x^{3}-10200 a^{2} b^{4} d^{3} e^{5} x^{3}+2720 a \,b^{5} d^{4} e^{4} x^{3}-320 b^{6} d^{5} e^{3} x^{3}+51051 a^{6} e^{8} x^{2}+350064 a^{5} b d \,e^{7} x^{2}+36465 a^{4} b^{2} d^{2} e^{6} x^{2}-26520 a^{3} b^{3} d^{3} e^{5} x^{2}+12240 a^{2} b^{4} d^{4} e^{4} x^{2}-3264 a \,b^{5} d^{5} e^{3} x^{2}+384 b^{6} d^{6} e^{2} x^{2}+102102 a^{6} d \,e^{7} x +43758 a^{5} b \,d^{2} e^{6} x -48620 a^{4} b^{2} d^{3} e^{5} x +35360 a^{3} b^{3} d^{4} e^{4} x -16320 a^{2} b^{4} d^{5} e^{3} x +4352 a \,b^{5} d^{6} e^{2} x -512 b^{6} d^{7} e x +51051 a^{6} d^{2} e^{6}-87516 a^{5} b \,d^{3} e^{5}+97240 a^{4} b^{2} d^{4} e^{4}-70720 a^{3} b^{3} d^{5} e^{3}+32640 a^{2} b^{4} d^{6} e^{2}-8704 a \,b^{5} d^{7} e +1024 b^{6} d^{8}\right ) \sqrt {e x +d}}{255255 e^{7}}\)	\(591\)
risch	\(\frac {2 \left (15015 b^{6} e^{8} x^{8}+102102 a \,b^{5} e^{8} x^{7}+18018 b^{6} d \,e^{7} x^{7}+294525 a^{2} b^{4} e^{8} x^{6}+125664 a \,b^{5} d \,e^{7} x^{6}+231 b^{6} d^{2} e^{6} x^{6}+464100 a^{3} b^{3} e^{8} x^{5}+374850 a^{2} b^{4} d \,e^{7} x^{5}+2142 a \,b^{5} d^{2} e^{6} x^{5}-252 b^{6} d^{3} e^{5} x^{5}+425425 a^{4} b^{2} e^{8} x^{4}+618800 a^{3} b^{3} d \,e^{7} x^{4}+8925 a^{2} b^{4} d^{2} e^{6} x^{4}-2380 a \,b^{5} d^{3} e^{5} x^{4}+280 b^{6} d^{4} e^{4} x^{4}+218790 a^{5} b \,e^{8} x^{3}+607750 a^{4} b^{2} d \,e^{7} x^{3}+22100 a^{3} b^{3} d^{2} e^{6} x^{3}-10200 a^{2} b^{4} d^{3} e^{5} x^{3}+2720 a \,b^{5} d^{4} e^{4} x^{3}-320 b^{6} d^{5} e^{3} x^{3}+51051 a^{6} e^{8} x^{2}+350064 a^{5} b d \,e^{7} x^{2}+36465 a^{4} b^{2} d^{2} e^{6} x^{2}-26520 a^{3} b^{3} d^{3} e^{5} x^{2}+12240 a^{2} b^{4} d^{4} e^{4} x^{2}-3264 a \,b^{5} d^{5} e^{3} x^{2}+384 b^{6} d^{6} e^{2} x^{2}+102102 a^{6} d \,e^{7} x +43758 a^{5} b \,d^{2} e^{6} x -48620 a^{4} b^{2} d^{3} e^{5} x +35360 a^{3} b^{3} d^{4} e^{4} x -16320 a^{2} b^{4} d^{5} e^{3} x +4352 a \,b^{5} d^{6} e^{2} x -512 b^{6} d^{7} e x +51051 a^{6} d^{2} e^{6}-87516 a^{5} b \,d^{3} e^{5}+97240 a^{4} b^{2} d^{4} e^{4}-70720 a^{3} b^{3} d^{5} e^{3}+32640 a^{2} b^{4} d^{6} e^{2}-8704 a \,b^{5} d^{7} e +1024 b^{6} d^{8}\right ) \sqrt {e x +d}}{255255 e^{7}}\)	\(591\)

[In]

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

[Out]

2/5*(e*x+d)^(5/2)*((5/17*b^6*x^6+a^6+2*a*x^5*b^5+75/13*a^2*x^4*b^4+100/11*a^3*x^3*b^3+25/3*a^4*x^2*b^2+30/7*a^
5*x*b)*e^6-12/7*b*(7/51*b^5*x^5+35/39*a*b^4*x^4+350/143*a^2*b^3*x^3+350/99*a^3*b^2*x^2+25/9*a^4*b*x+a^5)*d*e^5
+40/21*b^2*(21/221*b^4*x^4+84/143*a*b^3*x^3+210/143*a^2*b^2*x^2+20/11*a^3*b*x+a^4)*d^2*e^4-320/231*(21/221*b^3
*x^3+7/13*a*b^2*x^2+15/13*a^2*b*x+a^3)*b^3*d^3*e^3+640/1001*(7/51*b^2*x^2+2/3*a*b*x+a^2)*b^4*d^4*e^2-512/3003*
(5/17*b*x+a)*b^5*d^5*e+1024/51051*b^6*d^6)/e^7

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 541 vs. \(2 (159) = 318\).

Time = 0.39 (sec) , antiderivative size = 541, normalized size of antiderivative = 2.89 \[ \int (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {2 \, {\left (15015 \, b^{6} e^{8} x^{8} + 1024 \, b^{6} d^{8} - 8704 \, a b^{5} d^{7} e + 32640 \, a^{2} b^{4} d^{6} e^{2} - 70720 \, a^{3} b^{3} d^{5} e^{3} + 97240 \, a^{4} b^{2} d^{4} e^{4} - 87516 \, a^{5} b d^{3} e^{5} + 51051 \, a^{6} d^{2} e^{6} + 6006 \, {\left (3 \, b^{6} d e^{7} + 17 \, a b^{5} e^{8}\right )} x^{7} + 231 \, {\left (b^{6} d^{2} e^{6} + 544 \, a b^{5} d e^{7} + 1275 \, a^{2} b^{4} e^{8}\right )} x^{6} - 42 \, {\left (6 \, b^{6} d^{3} e^{5} - 51 \, a b^{5} d^{2} e^{6} - 8925 \, a^{2} b^{4} d e^{7} - 11050 \, a^{3} b^{3} e^{8}\right )} x^{5} + 35 \, {\left (8 \, b^{6} d^{4} e^{4} - 68 \, a b^{5} d^{3} e^{5} + 255 \, a^{2} b^{4} d^{2} e^{6} + 17680 \, a^{3} b^{3} d e^{7} + 12155 \, a^{4} b^{2} e^{8}\right )} x^{4} - 10 \, {\left (32 \, b^{6} d^{5} e^{3} - 272 \, a b^{5} d^{4} e^{4} + 1020 \, a^{2} b^{4} d^{3} e^{5} - 2210 \, a^{3} b^{3} d^{2} e^{6} - 60775 \, a^{4} b^{2} d e^{7} - 21879 \, a^{5} b e^{8}\right )} x^{3} + 3 \, {\left (128 \, b^{6} d^{6} e^{2} - 1088 \, a b^{5} d^{5} e^{3} + 4080 \, a^{2} b^{4} d^{4} e^{4} - 8840 \, a^{3} b^{3} d^{3} e^{5} + 12155 \, a^{4} b^{2} d^{2} e^{6} + 116688 \, a^{5} b d e^{7} + 17017 \, a^{6} e^{8}\right )} x^{2} - 2 \, {\left (256 \, b^{6} d^{7} e - 2176 \, a b^{5} d^{6} e^{2} + 8160 \, a^{2} b^{4} d^{5} e^{3} - 17680 \, a^{3} b^{3} d^{4} e^{4} + 24310 \, a^{4} b^{2} d^{3} e^{5} - 21879 \, a^{5} b d^{2} e^{6} - 51051 \, a^{6} d e^{7}\right )} x\right )} \sqrt {e x + d}}{255255 \, e^{7}} \]

[In]

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

[Out]

2/255255*(15015*b^6*e^8*x^8 + 1024*b^6*d^8 - 8704*a*b^5*d^7*e + 32640*a^2*b^4*d^6*e^2 - 70720*a^3*b^3*d^5*e^3
+ 97240*a^4*b^2*d^4*e^4 - 87516*a^5*b*d^3*e^5 + 51051*a^6*d^2*e^6 + 6006*(3*b^6*d*e^7 + 17*a*b^5*e^8)*x^7 + 23
1*(b^6*d^2*e^6 + 544*a*b^5*d*e^7 + 1275*a^2*b^4*e^8)*x^6 - 42*(6*b^6*d^3*e^5 - 51*a*b^5*d^2*e^6 - 8925*a^2*b^4
*d*e^7 - 11050*a^3*b^3*e^8)*x^5 + 35*(8*b^6*d^4*e^4 - 68*a*b^5*d^3*e^5 + 255*a^2*b^4*d^2*e^6 + 17680*a^3*b^3*d
*e^7 + 12155*a^4*b^2*e^8)*x^4 - 10*(32*b^6*d^5*e^3 - 272*a*b^5*d^4*e^4 + 1020*a^2*b^4*d^3*e^5 - 2210*a^3*b^3*d
^2*e^6 - 60775*a^4*b^2*d*e^7 - 21879*a^5*b*e^8)*x^3 + 3*(128*b^6*d^6*e^2 - 1088*a*b^5*d^5*e^3 + 4080*a^2*b^4*d
^4*e^4 - 8840*a^3*b^3*d^3*e^5 + 12155*a^4*b^2*d^2*e^6 + 116688*a^5*b*d*e^7 + 17017*a^6*e^8)*x^2 - 2*(256*b^6*d
^7*e - 2176*a*b^5*d^6*e^2 + 8160*a^2*b^4*d^5*e^3 - 17680*a^3*b^3*d^4*e^4 + 24310*a^4*b^2*d^3*e^5 - 21879*a^5*b
*d^2*e^6 - 51051*a^6*d*e^7)*x)*sqrt(e*x + d)/e^7

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 495 vs. \(2 (173) = 346\).

Time = 1.37 (sec) , antiderivative size = 495, normalized size of antiderivative = 2.65 \[ \int (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\begin {cases} \frac {2 \left (\frac {b^{6} \left (d + e x\right )^{\frac {17}{2}}}{17 e^{6}} + \frac {\left (d + e x\right )^{\frac {15}{2}} \cdot \left (6 a b^{5} e - 6 b^{6} d\right )}{15 e^{6}} + \frac {\left (d + e x\right )^{\frac {13}{2}} \cdot \left (15 a^{2} b^{4} e^{2} - 30 a b^{5} d e + 15 b^{6} d^{2}\right )}{13 e^{6}} + \frac {\left (d + e x\right )^{\frac {11}{2}} \cdot \left (20 a^{3} b^{3} e^{3} - 60 a^{2} b^{4} d e^{2} + 60 a b^{5} d^{2} e - 20 b^{6} d^{3}\right )}{11 e^{6}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \cdot \left (15 a^{4} b^{2} e^{4} - 60 a^{3} b^{3} d e^{3} + 90 a^{2} b^{4} d^{2} e^{2} - 60 a b^{5} d^{3} e + 15 b^{6} d^{4}\right )}{9 e^{6}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \cdot \left (6 a^{5} b e^{5} - 30 a^{4} b^{2} d e^{4} + 60 a^{3} b^{3} d^{2} e^{3} - 60 a^{2} b^{4} d^{3} e^{2} + 30 a b^{5} d^{4} e - 6 b^{6} d^{5}\right )}{7 e^{6}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \left (a^{6} e^{6} - 6 a^{5} b d e^{5} + 15 a^{4} b^{2} d^{2} e^{4} - 20 a^{3} b^{3} d^{3} e^{3} + 15 a^{2} b^{4} d^{4} e^{2} - 6 a b^{5} d^{5} e + b^{6} d^{6}\right )}{5 e^{6}}\right )}{e} & \text {for}\: e \neq 0 \\d^{\frac {3}{2}} \left (a^{6} x + 3 a^{5} b x^{2} + 5 a^{4} b^{2} x^{3} + 5 a^{3} b^{3} x^{4} + 3 a^{2} b^{4} x^{5} + a b^{5} x^{6} + \frac {b^{6} x^{7}}{7}\right ) & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((2*(b**6*(d + e*x)**(17/2)/(17*e**6) + (d + e*x)**(15/2)*(6*a*b**5*e - 6*b**6*d)/(15*e**6) + (d + e*
x)**(13/2)*(15*a**2*b**4*e**2 - 30*a*b**5*d*e + 15*b**6*d**2)/(13*e**6) + (d + e*x)**(11/2)*(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)/(11*e**6) + (d + e*x)**(9/2)*(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)/(9*e**6) + (d + e*x)**(7/2)*(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)/(7*e**6) + (d + e*x)**(5/2)*(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)/(5*e**6))/e, Ne(e, 0)), (d**(3/2)*(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), True))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 350 vs. \(2 (159) = 318\).

Time = 0.20 (sec) , antiderivative size = 350, normalized size of antiderivative = 1.87 \[ \int (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {2 \, {\left (15015 \, {\left (e x + d\right )}^{\frac {17}{2}} b^{6} - 102102 \, {\left (b^{6} d - a b^{5} e\right )} {\left (e x + d\right )}^{\frac {15}{2}} + 294525 \, {\left (b^{6} d^{2} - 2 \, a b^{5} d e + a^{2} b^{4} e^{2}\right )} {\left (e x + d\right )}^{\frac {13}{2}} - 464100 \, {\left (b^{6} d^{3} - 3 \, a b^{5} d^{2} e + 3 \, a^{2} b^{4} d e^{2} - a^{3} b^{3} e^{3}\right )} {\left (e x + d\right )}^{\frac {11}{2}} + 425425 \, {\left (b^{6} d^{4} - 4 \, a b^{5} d^{3} e + 6 \, a^{2} b^{4} d^{2} e^{2} - 4 \, a^{3} b^{3} d e^{3} + a^{4} b^{2} e^{4}\right )} {\left (e x + d\right )}^{\frac {9}{2}} - 218790 \, {\left (b^{6} d^{5} - 5 \, a b^{5} d^{4} e + 10 \, a^{2} b^{4} d^{3} e^{2} - 10 \, a^{3} b^{3} d^{2} e^{3} + 5 \, a^{4} b^{2} d e^{4} - a^{5} b e^{5}\right )} {\left (e x + d\right )}^{\frac {7}{2}} + 51051 \, {\left (b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}\right )} {\left (e x + d\right )}^{\frac {5}{2}}\right )}}{255255 \, e^{7}} \]

[In]

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

[Out]

2/255255*(15015*(e*x + d)^(17/2)*b^6 - 102102*(b^6*d - a*b^5*e)*(e*x + d)^(15/2) + 294525*(b^6*d^2 - 2*a*b^5*d
*e + a^2*b^4*e^2)*(e*x + d)^(13/2) - 464100*(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
)^(11/2) + 425425*(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)^(9/2
) - 218790*(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)*(
e*x + d)^(7/2) + 51051*(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)*(e*x + d)^(5/2))/e^7

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1397 vs. \(2 (159) = 318\).

Time = 0.29 (sec) , antiderivative size = 1397, normalized size of antiderivative = 7.47 \[ \int (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\text {Too large to display} \]

[In]

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

[Out]

2/765765*(765765*sqrt(e*x + d)*a^6*d^2 + 510510*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*a^6*d + 1531530*((e*x +
d)^(3/2) - 3*sqrt(e*x + d)*d)*a^5*b*d^2/e + 51051*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)
*d^2)*a^6 + 765765*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*a^4*b^2*d^2/e^2 + 612612*
(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*a^5*b*d/e + 437580*(5*(e*x + d)^(7/2) - 21*(
e*x + d)^(5/2)*d + 35*(e*x + d)^(3/2)*d^2 - 35*sqrt(e*x + d)*d^3)*a^3*b^3*d^2/e^3 + 656370*(5*(e*x + d)^(7/2)
- 21*(e*x + d)^(5/2)*d + 35*(e*x + d)^(3/2)*d^2 - 35*sqrt(e*x + d)*d^3)*a^4*b^2*d/e^2 + 131274*(5*(e*x + d)^(7
/2) - 21*(e*x + d)^(5/2)*d + 35*(e*x + d)^(3/2)*d^2 - 35*sqrt(e*x + d)*d^3)*a^5*b/e + 36465*(35*(e*x + d)^(9/2
) - 180*(e*x + d)^(7/2)*d + 378*(e*x + d)^(5/2)*d^2 - 420*(e*x + d)^(3/2)*d^3 + 315*sqrt(e*x + d)*d^4)*a^2*b^4
*d^2/e^4 + 97240*(35*(e*x + d)^(9/2) - 180*(e*x + d)^(7/2)*d + 378*(e*x + d)^(5/2)*d^2 - 420*(e*x + d)^(3/2)*d
^3 + 315*sqrt(e*x + d)*d^4)*a^3*b^3*d/e^3 + 36465*(35*(e*x + d)^(9/2) - 180*(e*x + d)^(7/2)*d + 378*(e*x + d)^
(5/2)*d^2 - 420*(e*x + d)^(3/2)*d^3 + 315*sqrt(e*x + d)*d^4)*a^4*b^2/e^2 + 6630*(63*(e*x + d)^(11/2) - 385*(e*
x + d)^(9/2)*d + 990*(e*x + d)^(7/2)*d^2 - 1386*(e*x + d)^(5/2)*d^3 + 1155*(e*x + d)^(3/2)*d^4 - 693*sqrt(e*x
+ d)*d^5)*a*b^5*d^2/e^5 + 33150*(63*(e*x + d)^(11/2) - 385*(e*x + d)^(9/2)*d + 990*(e*x + d)^(7/2)*d^2 - 1386*
(e*x + d)^(5/2)*d^3 + 1155*(e*x + d)^(3/2)*d^4 - 693*sqrt(e*x + d)*d^5)*a^2*b^4*d/e^4 + 22100*(63*(e*x + d)^(1
1/2) - 385*(e*x + d)^(9/2)*d + 990*(e*x + d)^(7/2)*d^2 - 1386*(e*x + d)^(5/2)*d^3 + 1155*(e*x + d)^(3/2)*d^4 -
693*sqrt(e*x + d)*d^5)*a^3*b^3/e^3 + 255*(231*(e*x + d)^(13/2) - 1638*(e*x + d)^(11/2)*d + 5005*(e*x + d)^(9/
2)*d^2 - 8580*(e*x + d)^(7/2)*d^3 + 9009*(e*x + d)^(5/2)*d^4 - 6006*(e*x + d)^(3/2)*d^5 + 3003*sqrt(e*x + d)*d
^6)*b^6*d^2/e^6 + 3060*(231*(e*x + d)^(13/2) - 1638*(e*x + d)^(11/2)*d + 5005*(e*x + d)^(9/2)*d^2 - 8580*(e*x
+ d)^(7/2)*d^3 + 9009*(e*x + d)^(5/2)*d^4 - 6006*(e*x + d)^(3/2)*d^5 + 3003*sqrt(e*x + d)*d^6)*a*b^5*d/e^5 + 3
825*(231*(e*x + d)^(13/2) - 1638*(e*x + d)^(11/2)*d + 5005*(e*x + d)^(9/2)*d^2 - 8580*(e*x + d)^(7/2)*d^3 + 90
09*(e*x + d)^(5/2)*d^4 - 6006*(e*x + d)^(3/2)*d^5 + 3003*sqrt(e*x + d)*d^6)*a^2*b^4/e^4 + 238*(429*(e*x + d)^(
15/2) - 3465*(e*x + d)^(13/2)*d + 12285*(e*x + d)^(11/2)*d^2 - 25025*(e*x + d)^(9/2)*d^3 + 32175*(e*x + d)^(7/
2)*d^4 - 27027*(e*x + d)^(5/2)*d^5 + 15015*(e*x + d)^(3/2)*d^6 - 6435*sqrt(e*x + d)*d^7)*b^6*d/e^6 + 714*(429*
(e*x + d)^(15/2) - 3465*(e*x + d)^(13/2)*d + 12285*(e*x + d)^(11/2)*d^2 - 25025*(e*x + d)^(9/2)*d^3 + 32175*(e
*x + d)^(7/2)*d^4 - 27027*(e*x + d)^(5/2)*d^5 + 15015*(e*x + d)^(3/2)*d^6 - 6435*sqrt(e*x + d)*d^7)*a*b^5/e^5
+ 7*(6435*(e*x + d)^(17/2) - 58344*(e*x + d)^(15/2)*d + 235620*(e*x + d)^(13/2)*d^2 - 556920*(e*x + d)^(11/2)*
d^3 + 850850*(e*x + d)^(9/2)*d^4 - 875160*(e*x + d)^(7/2)*d^5 + 612612*(e*x + d)^(5/2)*d^6 - 291720*(e*x + d)^
(3/2)*d^7 + 109395*sqrt(e*x + d)*d^8)*b^6/e^6)/e

Mupad [B] (verification not implemented)

Time = 9.53 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.87 \[ \int (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {2\,b^6\,{\left (d+e\,x\right )}^{17/2}}{17\,e^7}-\frac {\left (12\,b^6\,d-12\,a\,b^5\,e\right )\,{\left (d+e\,x\right )}^{15/2}}{15\,e^7}+\frac {2\,{\left (a\,e-b\,d\right )}^6\,{\left (d+e\,x\right )}^{5/2}}{5\,e^7}+\frac {10\,b^2\,{\left (a\,e-b\,d\right )}^4\,{\left (d+e\,x\right )}^{9/2}}{3\,e^7}+\frac {40\,b^3\,{\left (a\,e-b\,d\right )}^3\,{\left (d+e\,x\right )}^{11/2}}{11\,e^7}+\frac {30\,b^4\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{13/2}}{13\,e^7}+\frac {12\,b\,{\left (a\,e-b\,d\right )}^5\,{\left (d+e\,x\right )}^{7/2}}{7\,e^7} \]

[In]

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

[Out]

(2*b^6*(d + e*x)^(17/2))/(17*e^7) - ((12*b^6*d - 12*a*b^5*e)*(d + e*x)^(15/2))/(15*e^7) + (2*(a*e - b*d)^6*(d
+ e*x)^(5/2))/(5*e^7) + (10*b^2*(a*e - b*d)^4*(d + e*x)^(9/2))/(3*e^7) + (40*b^3*(a*e - b*d)^3*(d + e*x)^(11/2
))/(11*e^7) + (30*b^4*(a*e - b*d)^2*(d + e*x)^(13/2))/(13*e^7) + (12*b*(a*e - b*d)^5*(d + e*x)^(7/2))/(7*e^7)