\(\int \sqrt {d+e x} (a^2+2 a b x+b^2 x^2)^2 \, dx\) [1632]

Optimal result

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

[Out]

2/3*(-a*e+b*d)^4*(e*x+d)^(3/2)/e^5-8/5*b*(-a*e+b*d)^3*(e*x+d)^(5/2)/e^5+12/7*b^2*(-a*e+b*d)^2*(e*x+d)^(7/2)/e^
5-8/9*b^3*(-a*e+b*d)*(e*x+d)^(9/2)/e^5+2/11*b^4*(e*x+d)^(11/2)/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} \[ \int \sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=-\frac {8 b^3 (d+e x)^{9/2} (b d-a e)}{9 e^5}+\frac {12 b^2 (d+e x)^{7/2} (b d-a e)^2}{7 e^5}-\frac {8 b (d+e x)^{5/2} (b d-a e)^3}{5 e^5}+\frac {2 (d+e x)^{3/2} (b d-a e)^4}{3 e^5}+\frac {2 b^4 (d+e x)^{11/2}}{11 e^5} \]

[In]

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

[Out]

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

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.19 \[ \int \sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {2 (d+e x)^{3/2} \left (1155 a^4 e^4+924 a^3 b e^3 (-2 d+3 e x)+198 a^2 b^2 e^2 \left (8 d^2-12 d e x+15 e^2 x^2\right )+44 a b^3 e \left (-16 d^3+24 d^2 e x-30 d e^2 x^2+35 e^3 x^3\right )+b^4 \left (128 d^4-192 d^3 e x+240 d^2 e^2 x^2-280 d e^3 x^3+315 e^4 x^4\right )\right )}{3465 e^5} \]

[In]

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

[Out]

(2*(d + e*x)^(3/2)*(1155*a^4*e^4 + 924*a^3*b*e^3*(-2*d + 3*e*x) + 198*a^2*b^2*e^2*(8*d^2 - 12*d*e*x + 15*e^2*x
^2) + 44*a*b^3*e*(-16*d^3 + 24*d^2*e*x - 30*d*e^2*x^2 + 35*e^3*x^3) + b^4*(128*d^4 - 192*d^3*e*x + 240*d^2*e^2
*x^2 - 280*d*e^3*x^3 + 315*e^4*x^4)))/(3465*e^5)

Maple [A] (verified)

Time = 2.28 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.11

method	result	size
pseudoelliptic	\(\frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (\left (\frac {3}{11} b^{4} x^{4}+\frac {4}{3} a \,b^{3} x^{3}+\frac {18}{7} a^{2} b^{2} x^{2}+\frac {12}{5} a^{3} b x +a^{4}\right ) e^{4}-\frac {8 b \left (\frac {5}{33} b^{3} x^{3}+\frac {5}{7} a \,b^{2} x^{2}+\frac {9}{7} a^{2} b x +a^{3}\right ) d \,e^{3}}{5}+\frac {48 \left (\frac {5}{33} b^{2} x^{2}+\frac {2}{3} a b x +a^{2}\right ) b^{2} d^{2} e^{2}}{35}-\frac {64 b^{3} d^{3} \left (\frac {3 b x}{11}+a \right ) e}{105}+\frac {128 b^{4} d^{4}}{1155}\right )}{3 e^{5}}\)	\(143\)
derivativedivides	\(\frac {\frac {2 b^{4} \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {4 \left (2 a e b -2 b^{2} d \right ) b^{2} \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a e b -2 b^{2} d \right )^{2}\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {4 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \left (2 a e b -2 b^{2} d \right ) \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}}{e^{5}}\)	\(167\)
default	\(\frac {\frac {2 b^{4} \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {4 \left (2 a e b -2 b^{2} d \right ) b^{2} \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a e b -2 b^{2} d \right )^{2}\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {4 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \left (2 a e b -2 b^{2} d \right ) \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}}{e^{5}}\)	\(167\)
gosper	\(\frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (315 b^{4} x^{4} e^{4}+1540 x^{3} a \,b^{3} e^{4}-280 x^{3} b^{4} d \,e^{3}+2970 x^{2} a^{2} b^{2} e^{4}-1320 x^{2} a \,b^{3} d \,e^{3}+240 x^{2} b^{4} d^{2} e^{2}+2772 x \,a^{3} b \,e^{4}-2376 x \,a^{2} b^{2} d \,e^{3}+1056 x a \,b^{3} d^{2} e^{2}-192 x \,b^{4} d^{3} e +1155 e^{4} a^{4}-1848 b \,e^{3} d \,a^{3}+1584 b^{2} e^{2} d^{2} a^{2}-704 a \,b^{3} d^{3} e +128 b^{4} d^{4}\right )}{3465 e^{5}}\)	\(186\)
trager	\(\frac {2 \left (315 b^{4} e^{5} x^{5}+1540 a \,b^{3} e^{5} x^{4}+35 b^{4} d \,e^{4} x^{4}+2970 a^{2} b^{2} e^{5} x^{3}+220 a \,b^{3} d \,e^{4} x^{3}-40 d^{2} e^{3} b^{4} x^{3}+2772 a^{3} b \,e^{5} x^{2}+594 a^{2} b^{2} d \,e^{4} x^{2}-264 a \,b^{3} d^{2} e^{3} x^{2}+48 b^{4} d^{3} e^{2} x^{2}+1155 e^{5} a^{4} x +924 a^{3} b d \,e^{4} x -792 a^{2} b^{2} d^{2} e^{3} x +352 a \,b^{3} d^{3} e^{2} x -64 b^{4} d^{4} e x +1155 d \,e^{4} a^{4}-1848 a^{3} b \,d^{2} e^{3}+1584 a^{2} b^{2} d^{3} e^{2}-704 a \,b^{3} d^{4} e +128 b^{4} d^{5}\right ) \sqrt {e x +d}}{3465 e^{5}}\)	\(257\)
risch	\(\frac {2 \left (315 b^{4} e^{5} x^{5}+1540 a \,b^{3} e^{5} x^{4}+35 b^{4} d \,e^{4} x^{4}+2970 a^{2} b^{2} e^{5} x^{3}+220 a \,b^{3} d \,e^{4} x^{3}-40 d^{2} e^{3} b^{4} x^{3}+2772 a^{3} b \,e^{5} x^{2}+594 a^{2} b^{2} d \,e^{4} x^{2}-264 a \,b^{3} d^{2} e^{3} x^{2}+48 b^{4} d^{3} e^{2} x^{2}+1155 e^{5} a^{4} x +924 a^{3} b d \,e^{4} x -792 a^{2} b^{2} d^{2} e^{3} x +352 a \,b^{3} d^{3} e^{2} x -64 b^{4} d^{4} e x +1155 d \,e^{4} a^{4}-1848 a^{3} b \,d^{2} e^{3}+1584 a^{2} b^{2} d^{3} e^{2}-704 a \,b^{3} d^{4} e +128 b^{4} d^{5}\right ) \sqrt {e x +d}}{3465 e^{5}}\)	\(257\)

[In]

int((b^2*x^2+2*a*b*x+a^2)^2*(e*x+d)^(1/2),x,method=_RETURNVERBOSE)

[Out]

2/3*(e*x+d)^(3/2)*((3/11*b^4*x^4+4/3*a*b^3*x^3+18/7*a^2*b^2*x^2+12/5*a^3*b*x+a^4)*e^4-8/5*b*(5/33*b^3*x^3+5/7*
a*b^2*x^2+9/7*a^2*b*x+a^3)*d*e^3+48/35*(5/33*b^2*x^2+2/3*a*b*x+a^2)*b^2*d^2*e^2-64/105*b^3*d^3*(3/11*b*x+a)*e+
128/1155*b^4*d^4)/e^5

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 245 vs. \(2 (109) = 218\).

Time = 0.26 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.90 \[ \int \sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {2 \, {\left (315 \, b^{4} e^{5} x^{5} + 128 \, b^{4} d^{5} - 704 \, a b^{3} d^{4} e + 1584 \, a^{2} b^{2} d^{3} e^{2} - 1848 \, a^{3} b d^{2} e^{3} + 1155 \, a^{4} d e^{4} + 35 \, {\left (b^{4} d e^{4} + 44 \, a b^{3} e^{5}\right )} x^{4} - 10 \, {\left (4 \, b^{4} d^{2} e^{3} - 22 \, a b^{3} d e^{4} - 297 \, a^{2} b^{2} e^{5}\right )} x^{3} + 6 \, {\left (8 \, b^{4} d^{3} e^{2} - 44 \, a b^{3} d^{2} e^{3} + 99 \, a^{2} b^{2} d e^{4} + 462 \, a^{3} b e^{5}\right )} x^{2} - {\left (64 \, b^{4} d^{4} e - 352 \, a b^{3} d^{3} e^{2} + 792 \, a^{2} b^{2} d^{2} e^{3} - 924 \, a^{3} b d e^{4} - 1155 \, a^{4} e^{5}\right )} x\right )} \sqrt {e x + d}}{3465 \, e^{5}} \]

[In]

integrate((b^2*x^2+2*a*b*x+a^2)^2*(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

2/3465*(315*b^4*e^5*x^5 + 128*b^4*d^5 - 704*a*b^3*d^4*e + 1584*a^2*b^2*d^3*e^2 - 1848*a^3*b*d^2*e^3 + 1155*a^4
*d*e^4 + 35*(b^4*d*e^4 + 44*a*b^3*e^5)*x^4 - 10*(4*b^4*d^2*e^3 - 22*a*b^3*d*e^4 - 297*a^2*b^2*e^5)*x^3 + 6*(8*
b^4*d^3*e^2 - 44*a*b^3*d^2*e^3 + 99*a^2*b^2*d*e^4 + 462*a^3*b*e^5)*x^2 - (64*b^4*d^4*e - 352*a*b^3*d^3*e^2 + 7
92*a^2*b^2*d^2*e^3 - 924*a^3*b*d*e^4 - 1155*a^4*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 = 1.03 (sec) , antiderivative size = 272, normalized size of antiderivative = 2.11 \[ \int \sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\begin {cases} \frac {2 \left (\frac {b^{4} \left (d + e x\right )^{\frac {11}{2}}}{11 e^{4}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \cdot \left (4 a b^{3} e - 4 b^{4} d\right )}{9 e^{4}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \cdot \left (6 a^{2} b^{2} e^{2} - 12 a b^{3} d e + 6 b^{4} d^{2}\right )}{7 e^{4}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \cdot \left (4 a^{3} b e^{3} - 12 a^{2} b^{2} d e^{2} + 12 a b^{3} d^{2} e - 4 b^{4} d^{3}\right )}{5 e^{4}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \left (a^{4} e^{4} - 4 a^{3} b d e^{3} + 6 a^{2} b^{2} d^{2} e^{2} - 4 a b^{3} d^{3} e + b^{4} d^{4}\right )}{3 e^{4}}\right )}{e} & \text {for}\: e \neq 0 \\\sqrt {d} \left (a^{4} x + 2 a^{3} b x^{2} + 2 a^{2} b^{2} x^{3} + a b^{3} x^{4} + \frac {b^{4} x^{5}}{5}\right ) & \text {otherwise} \end {cases} \]

[In]

integrate((b**2*x**2+2*a*b*x+a**2)**2*(e*x+d)**(1/2),x)

[Out]

Piecewise((2*(b**4*(d + e*x)**(11/2)/(11*e**4) + (d + e*x)**(9/2)*(4*a*b**3*e - 4*b**4*d)/(9*e**4) + (d + e*x)
**(7/2)*(6*a**2*b**2*e**2 - 12*a*b**3*d*e + 6*b**4*d**2)/(7*e**4) + (d + e*x)**(5/2)*(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)/(5*e**4) + (d + e*x)**(3/2)*(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)/(3*e**4))/e, Ne(e, 0)), (sqrt(d)*(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 \[ \int \sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {2 \, {\left (315 \, {\left (e x + d\right )}^{\frac {11}{2}} b^{4} - 1540 \, {\left (b^{4} d - a b^{3} e\right )} {\left (e x + d\right )}^{\frac {9}{2}} + 2970 \, {\left (b^{4} d^{2} - 2 \, a b^{3} d e + a^{2} b^{2} e^{2}\right )} {\left (e x + d\right )}^{\frac {7}{2}} - 2772 \, {\left (b^{4} d^{3} - 3 \, a b^{3} d^{2} e + 3 \, a^{2} b^{2} d e^{2} - a^{3} b e^{3}\right )} {\left (e x + d\right )}^{\frac {5}{2}} + 1155 \, {\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )} {\left (e x + d\right )}^{\frac {3}{2}}\right )}}{3465 \, e^{5}} \]

[In]

integrate((b^2*x^2+2*a*b*x+a^2)^2*(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

2/3465*(315*(e*x + d)^(11/2)*b^4 - 1540*(b^4*d - a*b^3*e)*(e*x + d)^(9/2) + 2970*(b^4*d^2 - 2*a*b^3*d*e + a^2*
b^2*e^2)*(e*x + d)^(7/2) - 2772*(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)^(5/2) + 1155
*(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)^(3/2))/e^5

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 470 vs. \(2 (109) = 218\).

Time = 0.29 (sec) , antiderivative size = 470, normalized size of antiderivative = 3.64 \[ \int \sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {2 \, {\left (3465 \, \sqrt {e x + d} a^{4} d + 1155 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {e x + d} d\right )} a^{4} + \frac {4620 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {e x + d} d\right )} a^{3} b d}{e} + \frac {1386 \, {\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} - 10 \, {\left (e x + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {e x + d} d^{2}\right )} a^{2} b^{2} d}{e^{2}} + \frac {924 \, {\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} - 10 \, {\left (e x + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {e x + d} d^{2}\right )} a^{3} b}{e} + \frac {396 \, {\left (5 \, {\left (e x + d\right )}^{\frac {7}{2}} - 21 \, {\left (e x + d\right )}^{\frac {5}{2}} d + 35 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {e x + d} d^{3}\right )} a b^{3} d}{e^{3}} + \frac {594 \, {\left (5 \, {\left (e x + d\right )}^{\frac {7}{2}} - 21 \, {\left (e x + d\right )}^{\frac {5}{2}} d + 35 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {e x + d} d^{3}\right )} a^{2} b^{2}}{e^{2}} + \frac {11 \, {\left (35 \, {\left (e x + d\right )}^{\frac {9}{2}} - 180 \, {\left (e x + d\right )}^{\frac {7}{2}} d + 378 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {e x + d} d^{4}\right )} b^{4} d}{e^{4}} + \frac {44 \, {\left (35 \, {\left (e x + d\right )}^{\frac {9}{2}} - 180 \, {\left (e x + d\right )}^{\frac {7}{2}} d + 378 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {e x + d} d^{4}\right )} a b^{3}}{e^{3}} + \frac {5 \, {\left (63 \, {\left (e x + d\right )}^{\frac {11}{2}} - 385 \, {\left (e x + d\right )}^{\frac {9}{2}} d + 990 \, {\left (e x + d\right )}^{\frac {7}{2}} d^{2} - 1386 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{3} + 1155 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{4} - 693 \, \sqrt {e x + d} d^{5}\right )} b^{4}}{e^{4}}\right )}}{3465 \, e} \]

[In]

integrate((b^2*x^2+2*a*b*x+a^2)^2*(e*x+d)^(1/2),x, algorithm="giac")

[Out]

2/3465*(3465*sqrt(e*x + d)*a^4*d + 1155*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*a^4 + 4620*((e*x + d)^(3/2) - 3*
sqrt(e*x + d)*d)*a^3*b*d/e + 1386*(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/
e^2 + 924*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*a^3*b/e + 396*(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/e^3 + 594*(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/e^2 + 11*(35*(e*x + d)^(9/2) - 1
80*(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/e^4 +
44*(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/e^3 + 5*(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/e^4)/e

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.87 \[ \int \sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {2\,b^4\,{\left (d+e\,x\right )}^{11/2}}{11\,e^5}-\frac {\left (8\,b^4\,d-8\,a\,b^3\,e\right )\,{\left (d+e\,x\right )}^{9/2}}{9\,e^5}+\frac {2\,{\left (a\,e-b\,d\right )}^4\,{\left (d+e\,x\right )}^{3/2}}{3\,e^5}+\frac {12\,b^2\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{7/2}}{7\,e^5}+\frac {8\,b\,{\left (a\,e-b\,d\right )}^3\,{\left (d+e\,x\right )}^{5/2}}{5\,e^5} \]

[In]

int((d + e*x)^(1/2)*(a^2 + b^2*x^2 + 2*a*b*x)^2,x)

[Out]

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