\(\int (d+e x)^{5/2} (a^2+2 a b x+b^2 x^2)^2 \, dx\) [1630]

Optimal result

Integrand size = 28, antiderivative size = 129 \[ \int (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {2 (b d-a e)^4 (d+e x)^{7/2}}{7 e^5}-\frac {8 b (b d-a e)^3 (d+e x)^{9/2}}{9 e^5}+\frac {12 b^2 (b d-a e)^2 (d+e x)^{11/2}}{11 e^5}-\frac {8 b^3 (b d-a e) (d+e x)^{13/2}}{13 e^5}+\frac {2 b^4 (d+e x)^{15/2}}{15 e^5} \]

[Out]

2/7*(-a*e+b*d)^4*(e*x+d)^(7/2)/e^5-8/9*b*(-a*e+b*d)^3*(e*x+d)^(9/2)/e^5+12/11*b^2*(-a*e+b*d)^2*(e*x+d)^(11/2)/
e^5-8/13*b^3*(-a*e+b*d)*(e*x+d)^(13/2)/e^5+2/15*b^4*(e*x+d)^(15/2)/e^5

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 129, 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} \[ \int (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=-\frac {8 b^3 (d+e x)^{13/2} (b d-a e)}{13 e^5}+\frac {12 b^2 (d+e x)^{11/2} (b d-a e)^2}{11 e^5}-\frac {8 b (d+e x)^{9/2} (b d-a e)^3}{9 e^5}+\frac {2 (d+e x)^{7/2} (b d-a e)^4}{7 e^5}+\frac {2 b^4 (d+e x)^{15/2}}{15 e^5} \]

[In]

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

[Out]

(2*(b*d - a*e)^4*(d + e*x)^(7/2))/(7*e^5) - (8*b*(b*d - a*e)^3*(d + e*x)^(9/2))/(9*e^5) + (12*b^2*(b*d - a*e)^
2*(d + e*x)^(11/2))/(11*e^5) - (8*b^3*(b*d - a*e)*(d + e*x)^(13/2))/(13*e^5) + (2*b^4*(d + e*x)^(15/2))/(15*e^
5)

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)^4 (d+e x)^{5/2} \, dx \\ & = \int \left (\frac {(-b d+a e)^4 (d+e x)^{5/2}}{e^4}-\frac {4 b (b d-a e)^3 (d+e x)^{7/2}}{e^4}+\frac {6 b^2 (b d-a e)^2 (d+e x)^{9/2}}{e^4}-\frac {4 b^3 (b d-a e) (d+e x)^{11/2}}{e^4}+\frac {b^4 (d+e x)^{13/2}}{e^4}\right ) \, dx \\ & = \frac {2 (b d-a e)^4 (d+e x)^{7/2}}{7 e^5}-\frac {8 b (b d-a e)^3 (d+e x)^{9/2}}{9 e^5}+\frac {12 b^2 (b d-a e)^2 (d+e x)^{11/2}}{11 e^5}-\frac {8 b^3 (b d-a e) (d+e x)^{13/2}}{13 e^5}+\frac {2 b^4 (d+e x)^{15/2}}{15 e^5} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.19 \[ \int (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {2 (d+e x)^{7/2} \left (6435 a^4 e^4+2860 a^3 b e^3 (-2 d+7 e x)+390 a^2 b^2 e^2 \left (8 d^2-28 d e x+63 e^2 x^2\right )+60 a b^3 e \left (-16 d^3+56 d^2 e x-126 d e^2 x^2+231 e^3 x^3\right )+b^4 \left (128 d^4-448 d^3 e x+1008 d^2 e^2 x^2-1848 d e^3 x^3+3003 e^4 x^4\right )\right )}{45045 e^5} \]

[In]

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

[Out]

(2*(d + e*x)^(7/2)*(6435*a^4*e^4 + 2860*a^3*b*e^3*(-2*d + 7*e*x) + 390*a^2*b^2*e^2*(8*d^2 - 28*d*e*x + 63*e^2*
x^2) + 60*a*b^3*e*(-16*d^3 + 56*d^2*e*x - 126*d*e^2*x^2 + 231*e^3*x^3) + b^4*(128*d^4 - 448*d^3*e*x + 1008*d^2
*e^2*x^2 - 1848*d*e^3*x^3 + 3003*e^4*x^4)))/(45045*e^5)

Maple [A] (verified)

Time = 2.27 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.12

method	result	size
pseudoelliptic	\(\frac {2 \left (e x +d \right )^{\frac {7}{2}} \left (\left (\frac {7}{15} e^{4} x^{4}-\frac {56}{195} d \,e^{3} x^{3}+\frac {112}{715} d^{2} e^{2} x^{2}-\frac {448}{6435} d^{3} e x +\frac {128}{6435} d^{4}\right ) b^{4}-\frac {64 e a \left (-\frac {231}{16} e^{3} x^{3}+\frac {63}{8} d \,e^{2} x^{2}-\frac {7}{2} d^{2} e x +d^{3}\right ) b^{3}}{429}+\frac {16 \left (\frac {63}{8} x^{2} e^{2}-\frac {7}{2} d e x +d^{2}\right ) e^{2} a^{2} b^{2}}{33}-\frac {8 \left (-\frac {7 e x}{2}+d \right ) e^{3} a^{3} b}{9}+e^{4} a^{4}\right )}{7 e^{5}}\)	\(144\)
derivativedivides	\(\frac {\frac {2 b^{4} \left (e x +d \right )^{\frac {15}{2}}}{15}+\frac {4 \left (2 a e b -2 b^{2} d \right ) b^{2} \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a e b -2 b^{2} d \right )^{2}\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {4 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \left (2 a e b -2 b^{2} d \right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )^{2} \left (e x +d \right )^{\frac {7}{2}}}{7}}{e^{5}}\)	\(167\)
default	\(\frac {\frac {2 b^{4} \left (e x +d \right )^{\frac {15}{2}}}{15}+\frac {4 \left (2 a e b -2 b^{2} d \right ) b^{2} \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a e b -2 b^{2} d \right )^{2}\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {4 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \left (2 a e b -2 b^{2} d \right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )^{2} \left (e x +d \right )^{\frac {7}{2}}}{7}}{e^{5}}\)	\(167\)
gosper	\(\frac {2 \left (e x +d \right )^{\frac {7}{2}} \left (3003 b^{4} x^{4} e^{4}+13860 x^{3} a \,b^{3} e^{4}-1848 x^{3} b^{4} d \,e^{3}+24570 x^{2} a^{2} b^{2} e^{4}-7560 x^{2} a \,b^{3} d \,e^{3}+1008 x^{2} b^{4} d^{2} e^{2}+20020 x \,a^{3} b \,e^{4}-10920 x \,a^{2} b^{2} d \,e^{3}+3360 x a \,b^{3} d^{2} e^{2}-448 x \,b^{4} d^{3} e +6435 e^{4} a^{4}-5720 b \,e^{3} d \,a^{3}+3120 b^{2} e^{2} d^{2} a^{2}-960 a \,b^{3} d^{3} e +128 b^{4} d^{4}\right )}{45045 e^{5}}\)	\(186\)
trager	\(\frac {2 \left (3003 b^{4} e^{7} x^{7}+13860 a \,b^{3} e^{7} x^{6}+7161 b^{4} d \,e^{6} x^{6}+24570 a^{2} b^{2} e^{7} x^{5}+34020 a \,b^{3} d \,e^{6} x^{5}+4473 b^{4} d^{2} e^{5} x^{5}+20020 a^{3} b \,e^{7} x^{4}+62790 a^{2} b^{2} d \,e^{6} x^{4}+22260 a \,b^{3} d^{2} e^{5} x^{4}+35 b^{4} d^{3} e^{4} x^{4}+6435 e^{7} a^{4} x^{3}+54340 a^{3} b d \,e^{6} x^{3}+44070 a^{2} b^{2} d^{2} e^{5} x^{3}+300 a \,b^{3} d^{3} e^{4} x^{3}-40 b^{4} d^{4} e^{3} x^{3}+19305 d \,e^{6} a^{4} x^{2}+42900 a^{3} b \,d^{2} e^{5} x^{2}+1170 a^{2} b^{2} d^{3} e^{4} x^{2}-360 a \,b^{3} d^{4} e^{3} x^{2}+48 b^{4} d^{5} e^{2} x^{2}+19305 d^{2} e^{5} a^{4} x +2860 a^{3} b \,d^{3} e^{4} x -1560 a^{2} b^{2} d^{4} e^{3} x +480 a \,b^{3} d^{5} e^{2} x -64 b^{4} d^{6} e x +6435 d^{3} e^{4} a^{4}-5720 a^{3} b \,d^{4} e^{3}+3120 a^{2} b^{2} d^{5} e^{2}-960 a \,b^{3} d^{6} e +128 b^{4} d^{7}\right ) \sqrt {e x +d}}{45045 e^{5}}\)	\(407\)
risch	\(\frac {2 \left (3003 b^{4} e^{7} x^{7}+13860 a \,b^{3} e^{7} x^{6}+7161 b^{4} d \,e^{6} x^{6}+24570 a^{2} b^{2} e^{7} x^{5}+34020 a \,b^{3} d \,e^{6} x^{5}+4473 b^{4} d^{2} e^{5} x^{5}+20020 a^{3} b \,e^{7} x^{4}+62790 a^{2} b^{2} d \,e^{6} x^{4}+22260 a \,b^{3} d^{2} e^{5} x^{4}+35 b^{4} d^{3} e^{4} x^{4}+6435 e^{7} a^{4} x^{3}+54340 a^{3} b d \,e^{6} x^{3}+44070 a^{2} b^{2} d^{2} e^{5} x^{3}+300 a \,b^{3} d^{3} e^{4} x^{3}-40 b^{4} d^{4} e^{3} x^{3}+19305 d \,e^{6} a^{4} x^{2}+42900 a^{3} b \,d^{2} e^{5} x^{2}+1170 a^{2} b^{2} d^{3} e^{4} x^{2}-360 a \,b^{3} d^{4} e^{3} x^{2}+48 b^{4} d^{5} e^{2} x^{2}+19305 d^{2} e^{5} a^{4} x +2860 a^{3} b \,d^{3} e^{4} x -1560 a^{2} b^{2} d^{4} e^{3} x +480 a \,b^{3} d^{5} e^{2} x -64 b^{4} d^{6} e x +6435 d^{3} e^{4} a^{4}-5720 a^{3} b \,d^{4} e^{3}+3120 a^{2} b^{2} d^{5} e^{2}-960 a \,b^{3} d^{6} e +128 b^{4} d^{7}\right ) \sqrt {e x +d}}{45045 e^{5}}\)	\(407\)

[In]

int((e*x+d)^(5/2)*(b^2*x^2+2*a*b*x+a^2)^2,x,method=_RETURNVERBOSE)

[Out]

2/7*(e*x+d)^(7/2)*((7/15*e^4*x^4-56/195*d*e^3*x^3+112/715*d^2*e^2*x^2-448/6435*d^3*e*x+128/6435*d^4)*b^4-64/42
9*e*a*(-231/16*e^3*x^3+63/8*d*e^2*x^2-7/2*d^2*e*x+d^3)*b^3+16/33*(63/8*x^2*e^2-7/2*d*e*x+d^2)*e^2*a^2*b^2-8/9*
(-7/2*e*x+d)*e^3*a^3*b+e^4*a^4)/e^5

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 377 vs. \(2 (109) = 218\).

Time = 0.27 (sec) , antiderivative size = 377, normalized size of antiderivative = 2.92 \[ \int (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {2 \, {\left (3003 \, b^{4} e^{7} x^{7} + 128 \, b^{4} d^{7} - 960 \, a b^{3} d^{6} e + 3120 \, a^{2} b^{2} d^{5} e^{2} - 5720 \, a^{3} b d^{4} e^{3} + 6435 \, a^{4} d^{3} e^{4} + 231 \, {\left (31 \, b^{4} d e^{6} + 60 \, a b^{3} e^{7}\right )} x^{6} + 63 \, {\left (71 \, b^{4} d^{2} e^{5} + 540 \, a b^{3} d e^{6} + 390 \, a^{2} b^{2} e^{7}\right )} x^{5} + 35 \, {\left (b^{4} d^{3} e^{4} + 636 \, a b^{3} d^{2} e^{5} + 1794 \, a^{2} b^{2} d e^{6} + 572 \, a^{3} b e^{7}\right )} x^{4} - 5 \, {\left (8 \, b^{4} d^{4} e^{3} - 60 \, a b^{3} d^{3} e^{4} - 8814 \, a^{2} b^{2} d^{2} e^{5} - 10868 \, a^{3} b d e^{6} - 1287 \, a^{4} e^{7}\right )} x^{3} + 3 \, {\left (16 \, b^{4} d^{5} e^{2} - 120 \, a b^{3} d^{4} e^{3} + 390 \, a^{2} b^{2} d^{3} e^{4} + 14300 \, a^{3} b d^{2} e^{5} + 6435 \, a^{4} d e^{6}\right )} x^{2} - {\left (64 \, b^{4} d^{6} e - 480 \, a b^{3} d^{5} e^{2} + 1560 \, a^{2} b^{2} d^{4} e^{3} - 2860 \, a^{3} b d^{3} e^{4} - 19305 \, a^{4} d^{2} e^{5}\right )} x\right )} \sqrt {e x + d}}{45045 \, e^{5}} \]

[In]

integrate((e*x+d)^(5/2)*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="fricas")

[Out]

2/45045*(3003*b^4*e^7*x^7 + 128*b^4*d^7 - 960*a*b^3*d^6*e + 3120*a^2*b^2*d^5*e^2 - 5720*a^3*b*d^4*e^3 + 6435*a
^4*d^3*e^4 + 231*(31*b^4*d*e^6 + 60*a*b^3*e^7)*x^6 + 63*(71*b^4*d^2*e^5 + 540*a*b^3*d*e^6 + 390*a^2*b^2*e^7)*x
^5 + 35*(b^4*d^3*e^4 + 636*a*b^3*d^2*e^5 + 1794*a^2*b^2*d*e^6 + 572*a^3*b*e^7)*x^4 - 5*(8*b^4*d^4*e^3 - 60*a*b
^3*d^3*e^4 - 8814*a^2*b^2*d^2*e^5 - 10868*a^3*b*d*e^6 - 1287*a^4*e^7)*x^3 + 3*(16*b^4*d^5*e^2 - 120*a*b^3*d^4*
e^3 + 390*a^2*b^2*d^3*e^4 + 14300*a^3*b*d^2*e^5 + 6435*a^4*d*e^6)*x^2 - (64*b^4*d^6*e - 480*a*b^3*d^5*e^2 + 15
60*a^2*b^2*d^4*e^3 - 2860*a^3*b*d^3*e^4 - 19305*a^4*d^2*e^5)*x)*sqrt(e*x + d)/e^5

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 272 vs. \(2 (119) = 238\).

Time = 1.27 (sec) , antiderivative size = 272, normalized size of antiderivative = 2.11 \[ \int (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\begin {cases} \frac {2 \left (\frac {b^{4} \left (d + e x\right )^{\frac {15}{2}}}{15 e^{4}} + \frac {\left (d + e x\right )^{\frac {13}{2}} \cdot \left (4 a b^{3} e - 4 b^{4} d\right )}{13 e^{4}} + \frac {\left (d + e x\right )^{\frac {11}{2}} \cdot \left (6 a^{2} b^{2} e^{2} - 12 a b^{3} d e + 6 b^{4} d^{2}\right )}{11 e^{4}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \cdot \left (4 a^{3} b e^{3} - 12 a^{2} b^{2} d e^{2} + 12 a b^{3} d^{2} e - 4 b^{4} d^{3}\right )}{9 e^{4}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \left (a^{4} e^{4} - 4 a^{3} b d e^{3} + 6 a^{2} b^{2} d^{2} e^{2} - 4 a b^{3} d^{3} e + b^{4} d^{4}\right )}{7 e^{4}}\right )}{e} & \text {for}\: e \neq 0 \\d^{\frac {5}{2}} \left (a^{4} x + 2 a^{3} b x^{2} + 2 a^{2} b^{2} x^{3} + a b^{3} x^{4} + \frac {b^{4} x^{5}}{5}\right ) & \text {otherwise} \end {cases} \]

[In]

integrate((e*x+d)**(5/2)*(b**2*x**2+2*a*b*x+a**2)**2,x)

[Out]

Piecewise((2*(b**4*(d + e*x)**(15/2)/(15*e**4) + (d + e*x)**(13/2)*(4*a*b**3*e - 4*b**4*d)/(13*e**4) + (d + e*
x)**(11/2)*(6*a**2*b**2*e**2 - 12*a*b**3*d*e + 6*b**4*d**2)/(11*e**4) + (d + e*x)**(9/2)*(4*a**3*b*e**3 - 12*a
**2*b**2*d*e**2 + 12*a*b**3*d**2*e - 4*b**4*d**3)/(9*e**4) + (d + e*x)**(7/2)*(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)/(7*e**4))/e, Ne(e, 0)), (d**(5/2)*(a**4*x + 2*a**3*b*x**2
+ 2*a**2*b**2*x**3 + a*b**3*x**4 + b**4*x**5/5), True))

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.40 \[ \int (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {2 \, {\left (3003 \, {\left (e x + d\right )}^{\frac {15}{2}} b^{4} - 13860 \, {\left (b^{4} d - a b^{3} e\right )} {\left (e x + d\right )}^{\frac {13}{2}} + 24570 \, {\left (b^{4} d^{2} - 2 \, a b^{3} d e + a^{2} b^{2} e^{2}\right )} {\left (e x + d\right )}^{\frac {11}{2}} - 20020 \, {\left (b^{4} d^{3} - 3 \, a b^{3} d^{2} e + 3 \, a^{2} b^{2} d e^{2} - a^{3} b e^{3}\right )} {\left (e x + d\right )}^{\frac {9}{2}} + 6435 \, {\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )} {\left (e x + d\right )}^{\frac {7}{2}}\right )}}{45045 \, e^{5}} \]

[In]

integrate((e*x+d)^(5/2)*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="maxima")

[Out]

2/45045*(3003*(e*x + d)^(15/2)*b^4 - 13860*(b^4*d - a*b^3*e)*(e*x + d)^(13/2) + 24570*(b^4*d^2 - 2*a*b^3*d*e +
a^2*b^2*e^2)*(e*x + d)^(11/2) - 20020*(b^4*d^3 - 3*a*b^3*d^2*e + 3*a^2*b^2*d*e^2 - a^3*b*e^3)*(e*x + d)^(9/2)
+ 6435*(b^4*d^4 - 4*a*b^3*d^3*e + 6*a^2*b^2*d^2*e^2 - 4*a^3*b*d*e^3 + a^4*e^4)*(e*x + d)^(7/2))/e^5

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1204 vs. \(2 (109) = 218\).

Time = 0.31 (sec) , antiderivative size = 1204, normalized size of antiderivative = 9.33 \[ \int (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\text {Too large to display} \]

[In]

integrate((e*x+d)^(5/2)*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="giac")

[Out]

2/45045*(45045*sqrt(e*x + d)*a^4*d^3 + 45045*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*a^4*d^2 + 60060*((e*x + d)^
(3/2) - 3*sqrt(e*x + d)*d)*a^3*b*d^3/e + 9009*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2
)*a^4*d + 18018*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*a^2*b^2*d^3/e^2 + 36036*(3*(
e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*a^3*b*d^2/e + 1287*(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 + 5148*(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*b^3*d^3/e^3 + 23166*(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^2*b^2*d^2/e^2 + 15444*(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*d/e + 143*(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)*b^4*d^3/e^4 + 1716*(3
5*(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*b^3*d^2/e^3 + 2574*(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^2*d/e^2 + 572*(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/e + 195*(63*(e*x + d)^(11/2) - 3
85*(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*sqr
t(e*x + d)*d^5)*b^4*d^2/e^4 + 780*(63*(e*x + d)^(11/2) - 385*(e*x + d)^(9/2)*d + 990*(e*x + d)^(7/2)*d^2 - 138
6*(e*x + d)^(5/2)*d^3 + 1155*(e*x + d)^(3/2)*d^4 - 693*sqrt(e*x + d)*d^5)*a*b^3*d/e^3 + 390*(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 - 6
93*sqrt(e*x + d)*d^5)*a^2*b^2/e^2 + 45*(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^4*d/e^4 + 60*(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^3/e^3 + 7*(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^4/e^4)/e

Mupad [B] (verification not implemented)

Time = 9.51 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.87 \[ \int (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {2\,b^4\,{\left (d+e\,x\right )}^{15/2}}{15\,e^5}-\frac {\left (8\,b^4\,d-8\,a\,b^3\,e\right )\,{\left (d+e\,x\right )}^{13/2}}{13\,e^5}+\frac {2\,{\left (a\,e-b\,d\right )}^4\,{\left (d+e\,x\right )}^{7/2}}{7\,e^5}+\frac {12\,b^2\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{11/2}}{11\,e^5}+\frac {8\,b\,{\left (a\,e-b\,d\right )}^3\,{\left (d+e\,x\right )}^{9/2}}{9\,e^5} \]

[In]

int((d + e*x)^(5/2)*(a^2 + b^2*x^2 + 2*a*b*x)^2,x)

[Out]

(2*b^4*(d + e*x)^(15/2))/(15*e^5) - ((8*b^4*d - 8*a*b^3*e)*(d + e*x)^(13/2))/(13*e^5) + (2*(a*e - b*d)^4*(d +
e*x)^(7/2))/(7*e^5) + (12*b^2*(a*e - b*d)^2*(d + e*x)^(11/2))/(11*e^5) + (8*b*(a*e - b*d)^3*(d + e*x)^(9/2))/(
9*e^5)