\(\int (d+e x)^{3/2} (a^2+2 a b x+b^2 x^2)^2 \, dx\) [1631]

Optimal result

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

[Out]

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

[In]

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

[Out]

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

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.19 \[ \int (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {2 (d+e x)^{5/2} \left (3003 a^4 e^4+1716 a^3 b e^3 (-2 d+5 e x)+286 a^2 b^2 e^2 \left (8 d^2-20 d e x+35 e^2 x^2\right )+52 a b^3 e \left (-16 d^3+40 d^2 e x-70 d e^2 x^2+105 e^3 x^3\right )+b^4 \left (128 d^4-320 d^3 e x+560 d^2 e^2 x^2-840 d e^3 x^3+1155 e^4 x^4\right )\right )}{15015 e^5} \]

[In]

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

[Out]

(2*(d + e*x)^(5/2)*(3003*a^4*e^4 + 1716*a^3*b*e^3*(-2*d + 5*e*x) + 286*a^2*b^2*e^2*(8*d^2 - 20*d*e*x + 35*e^2*
x^2) + 52*a*b^3*e*(-16*d^3 + 40*d^2*e*x - 70*d*e^2*x^2 + 105*e^3*x^3) + b^4*(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)))/(15015*e^5)

Maple [A] (verified)

Time = 2.24 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.11

method	result	size
pseudoelliptic	\(\frac {2 \left (\left (\frac {5}{13} b^{4} x^{4}+\frac {20}{11} a \,b^{3} x^{3}+\frac {10}{3} a^{2} b^{2} x^{2}+\frac {20}{7} a^{3} b x +a^{4}\right ) e^{4}-\frac {8 b \left (\frac {35}{143} b^{3} x^{3}+\frac {35}{33} a \,b^{2} x^{2}+\frac {5}{3} a^{2} b x +a^{3}\right ) d \,e^{3}}{7}+\frac {16 b^{2} d^{2} \left (\frac {35}{143} b^{2} x^{2}+\frac {10}{11} a b x +a^{2}\right ) e^{2}}{21}-\frac {64 b^{3} d^{3} \left (\frac {5 b x}{13}+a \right ) e}{231}+\frac {128 b^{4} d^{4}}{3003}\right ) \left (e x +d \right )^{\frac {5}{2}}}{5 e^{5}}\)	\(143\)
derivativedivides	\(\frac {\frac {2 b^{4} \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {4 \left (2 a e b -2 b^{2} d \right ) b^{2} \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a e b -2 b^{2} d \right )^{2}\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {4 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \left (2 a e b -2 b^{2} d \right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}}{e^{5}}\)	\(167\)
default	\(\frac {\frac {2 b^{4} \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {4 \left (2 a e b -2 b^{2} d \right ) b^{2} \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a e b -2 b^{2} d \right )^{2}\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {4 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \left (2 a e b -2 b^{2} d \right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}}{e^{5}}\)	\(167\)
gosper	\(\frac {2 \left (e x +d \right )^{\frac {5}{2}} \left (1155 b^{4} x^{4} e^{4}+5460 x^{3} a \,b^{3} e^{4}-840 x^{3} b^{4} d \,e^{3}+10010 x^{2} a^{2} b^{2} e^{4}-3640 x^{2} a \,b^{3} d \,e^{3}+560 x^{2} b^{4} d^{2} e^{2}+8580 x \,a^{3} b \,e^{4}-5720 x \,a^{2} b^{2} d \,e^{3}+2080 x a \,b^{3} d^{2} e^{2}-320 x \,b^{4} d^{3} e +3003 e^{4} a^{4}-3432 b \,e^{3} d \,a^{3}+2288 b^{2} e^{2} d^{2} a^{2}-832 a \,b^{3} d^{3} e +128 b^{4} d^{4}\right )}{15015 e^{5}}\)	\(186\)
trager	\(\frac {2 \left (1155 e^{6} b^{4} x^{6}+5460 a \,b^{3} e^{6} x^{5}+1470 b^{4} d \,e^{5} x^{5}+10010 a^{2} b^{2} e^{6} x^{4}+7280 a \,b^{3} d \,e^{5} x^{4}+35 b^{4} d^{2} e^{4} x^{4}+8580 a^{3} b \,e^{6} x^{3}+14300 a^{2} b^{2} d \,e^{5} x^{3}+260 a \,b^{3} d^{2} e^{4} x^{3}-40 b^{4} d^{3} e^{3} x^{3}+3003 a^{4} e^{6} x^{2}+13728 a^{3} b d \,e^{5} x^{2}+858 a^{2} b^{2} d^{2} e^{4} x^{2}-312 a \,b^{3} d^{3} e^{3} x^{2}+48 d^{4} e^{2} b^{4} x^{2}+6006 a^{4} d \,e^{5} x +1716 a^{3} b \,d^{2} e^{4} x -1144 a^{2} b^{2} d^{3} e^{3} x +416 a \,b^{3} d^{4} e^{2} x -64 b^{4} d^{5} e x +3003 a^{4} d^{2} e^{4}-3432 a^{3} b \,d^{3} e^{3}+2288 a^{2} b^{2} d^{4} e^{2}-832 a \,b^{3} d^{5} e +128 b^{4} d^{6}\right ) \sqrt {e x +d}}{15015 e^{5}}\)	\(332\)
risch	\(\frac {2 \left (1155 e^{6} b^{4} x^{6}+5460 a \,b^{3} e^{6} x^{5}+1470 b^{4} d \,e^{5} x^{5}+10010 a^{2} b^{2} e^{6} x^{4}+7280 a \,b^{3} d \,e^{5} x^{4}+35 b^{4} d^{2} e^{4} x^{4}+8580 a^{3} b \,e^{6} x^{3}+14300 a^{2} b^{2} d \,e^{5} x^{3}+260 a \,b^{3} d^{2} e^{4} x^{3}-40 b^{4} d^{3} e^{3} x^{3}+3003 a^{4} e^{6} x^{2}+13728 a^{3} b d \,e^{5} x^{2}+858 a^{2} b^{2} d^{2} e^{4} x^{2}-312 a \,b^{3} d^{3} e^{3} x^{2}+48 d^{4} e^{2} b^{4} x^{2}+6006 a^{4} d \,e^{5} x +1716 a^{3} b \,d^{2} e^{4} x -1144 a^{2} b^{2} d^{3} e^{3} x +416 a \,b^{3} d^{4} e^{2} x -64 b^{4} d^{5} e x +3003 a^{4} d^{2} e^{4}-3432 a^{3} b \,d^{3} e^{3}+2288 a^{2} b^{2} d^{4} e^{2}-832 a \,b^{3} d^{5} e +128 b^{4} d^{6}\right ) \sqrt {e x +d}}{15015 e^{5}}\)	\(332\)

[In]

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

[Out]

2/5*((5/13*b^4*x^4+20/11*a*b^3*x^3+10/3*a^2*b^2*x^2+20/7*a^3*b*x+a^4)*e^4-8/7*b*(35/143*b^3*x^3+35/33*a*b^2*x^
2+5/3*a^2*b*x+a^3)*d*e^3+16/21*b^2*d^2*(35/143*b^2*x^2+10/11*a*b*x+a^2)*e^2-64/231*b^3*d^3*(5/13*b*x+a)*e+128/
3003*b^4*d^4)*(e*x+d)^(5/2)/e^5

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 311 vs. \(2 (109) = 218\).

Time = 0.28 (sec) , antiderivative size = 311, normalized size of antiderivative = 2.41 \[ \int (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {2 \, {\left (1155 \, b^{4} e^{6} x^{6} + 128 \, b^{4} d^{6} - 832 \, a b^{3} d^{5} e + 2288 \, a^{2} b^{2} d^{4} e^{2} - 3432 \, a^{3} b d^{3} e^{3} + 3003 \, a^{4} d^{2} e^{4} + 210 \, {\left (7 \, b^{4} d e^{5} + 26 \, a b^{3} e^{6}\right )} x^{5} + 35 \, {\left (b^{4} d^{2} e^{4} + 208 \, a b^{3} d e^{5} + 286 \, a^{2} b^{2} e^{6}\right )} x^{4} - 20 \, {\left (2 \, b^{4} d^{3} e^{3} - 13 \, a b^{3} d^{2} e^{4} - 715 \, a^{2} b^{2} d e^{5} - 429 \, a^{3} b e^{6}\right )} x^{3} + 3 \, {\left (16 \, b^{4} d^{4} e^{2} - 104 \, a b^{3} d^{3} e^{3} + 286 \, a^{2} b^{2} d^{2} e^{4} + 4576 \, a^{3} b d e^{5} + 1001 \, a^{4} e^{6}\right )} x^{2} - 2 \, {\left (32 \, b^{4} d^{5} e - 208 \, a b^{3} d^{4} e^{2} + 572 \, a^{2} b^{2} d^{3} e^{3} - 858 \, a^{3} b d^{2} e^{4} - 3003 \, a^{4} d e^{5}\right )} x\right )} \sqrt {e x + d}}{15015 \, e^{5}} \]

[In]

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

[Out]

2/15015*(1155*b^4*e^6*x^6 + 128*b^4*d^6 - 832*a*b^3*d^5*e + 2288*a^2*b^2*d^4*e^2 - 3432*a^3*b*d^3*e^3 + 3003*a
^4*d^2*e^4 + 210*(7*b^4*d*e^5 + 26*a*b^3*e^6)*x^5 + 35*(b^4*d^2*e^4 + 208*a*b^3*d*e^5 + 286*a^2*b^2*e^6)*x^4 -
20*(2*b^4*d^3*e^3 - 13*a*b^3*d^2*e^4 - 715*a^2*b^2*d*e^5 - 429*a^3*b*e^6)*x^3 + 3*(16*b^4*d^4*e^2 - 104*a*b^3
*d^3*e^3 + 286*a^2*b^2*d^2*e^4 + 4576*a^3*b*d*e^5 + 1001*a^4*e^6)*x^2 - 2*(32*b^4*d^5*e - 208*a*b^3*d^4*e^2 +
572*a^2*b^2*d^3*e^3 - 858*a^3*b*d^2*e^4 - 3003*a^4*d*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 = 1.14 (sec) , antiderivative size = 272, normalized size of antiderivative = 2.11 \[ \int (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\begin {cases} \frac {2 \left (\frac {b^{4} \left (d + e x\right )^{\frac {13}{2}}}{13 e^{4}} + \frac {\left (d + e x\right )^{\frac {11}{2}} \cdot \left (4 a b^{3} e - 4 b^{4} d\right )}{11 e^{4}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \cdot \left (6 a^{2} b^{2} e^{2} - 12 a b^{3} d e + 6 b^{4} d^{2}\right )}{9 e^{4}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \cdot \left (4 a^{3} b e^{3} - 12 a^{2} b^{2} d e^{2} + 12 a b^{3} d^{2} e - 4 b^{4} d^{3}\right )}{7 e^{4}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \left (a^{4} e^{4} - 4 a^{3} b d e^{3} + 6 a^{2} b^{2} d^{2} e^{2} - 4 a b^{3} d^{3} e + b^{4} d^{4}\right )}{5 e^{4}}\right )}{e} & \text {for}\: e \neq 0 \\d^{\frac {3}{2}} \left (a^{4} x + 2 a^{3} b x^{2} + 2 a^{2} b^{2} x^{3} + a b^{3} x^{4} + \frac {b^{4} x^{5}}{5}\right ) & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((2*(b**4*(d + e*x)**(13/2)/(13*e**4) + (d + e*x)**(11/2)*(4*a*b**3*e - 4*b**4*d)/(11*e**4) + (d + e*
x)**(9/2)*(6*a**2*b**2*e**2 - 12*a*b**3*d*e + 6*b**4*d**2)/(9*e**4) + (d + e*x)**(7/2)*(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)/(7*e**4) + (d + e*x)**(5/2)*(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)/(5*e**4))/e, Ne(e, 0)), (d**(3/2)*(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 = 0.19 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.40 \[ \int (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {2 \, {\left (1155 \, {\left (e x + d\right )}^{\frac {13}{2}} b^{4} - 5460 \, {\left (b^{4} d - a b^{3} e\right )} {\left (e x + d\right )}^{\frac {11}{2}} + 10010 \, {\left (b^{4} d^{2} - 2 \, a b^{3} d e + a^{2} b^{2} e^{2}\right )} {\left (e x + d\right )}^{\frac {9}{2}} - 8580 \, {\left (b^{4} d^{3} - 3 \, a b^{3} d^{2} e + 3 \, a^{2} b^{2} d e^{2} - a^{3} b e^{3}\right )} {\left (e x + d\right )}^{\frac {7}{2}} + 3003 \, {\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )} {\left (e x + d\right )}^{\frac {5}{2}}\right )}}{15015 \, e^{5}} \]

[In]

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

[Out]

2/15015*(1155*(e*x + d)^(13/2)*b^4 - 5460*(b^4*d - a*b^3*e)*(e*x + d)^(11/2) + 10010*(b^4*d^2 - 2*a*b^3*d*e +
a^2*b^2*e^2)*(e*x + d)^(9/2) - 8580*(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)^(7/2) +
3003*(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)^(5/2))/e^5

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 807 vs. \(2 (109) = 218\).

Time = 0.32 (sec) , antiderivative size = 807, normalized size of antiderivative = 6.26 \[ \int (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\text {Too large to display} \]

[In]

integrate((e*x+d)^(3/2)*(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^2 + 30030*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*a^4*d + 60060*((e*x + d)^(3
/2) - 3*sqrt(e*x + d)*d)*a^3*b*d^2/e + 3003*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*
a^4 + 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^2/e^2 + 24024*(3*(e*x
+ d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*a^3*b*d/e + 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^2/e^3 + 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^2*b^2*d/e^2 + 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^3*b/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^2/e^4 + 1144*(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*b^3*d/e^3 + 858*(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/e^2 + 130*(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)*b^4*d/e^4 + 260*(
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^3/e^3 + 15*(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*sqr
t(e*x + d)*d^6)*b^4/e^4)/e

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.87 \[ \int (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {2\,b^4\,{\left (d+e\,x\right )}^{13/2}}{13\,e^5}-\frac {\left (8\,b^4\,d-8\,a\,b^3\,e\right )\,{\left (d+e\,x\right )}^{11/2}}{11\,e^5}+\frac {2\,{\left (a\,e-b\,d\right )}^4\,{\left (d+e\,x\right )}^{5/2}}{5\,e^5}+\frac {4\,b^2\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{9/2}}{3\,e^5}+\frac {8\,b\,{\left (a\,e-b\,d\right )}^3\,{\left (d+e\,x\right )}^{7/2}}{7\,e^5} \]

[In]

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

[Out]

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