\(\int \sqrt {d+e x} (a^2+2 a b x+b^2 x^2)^{5/2} \, dx\) [1694]

Optimal result

Integrand size = 30, antiderivative size = 318 \[ \int \sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=-\frac {2 (b d-a e)^5 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^6 (a+b x)}+\frac {2 b (b d-a e)^4 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (a+b x)}-\frac {20 b^2 (b d-a e)^3 (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^6 (a+b x)}+\frac {20 b^3 (b d-a e)^2 (d+e x)^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{9 e^6 (a+b x)}-\frac {10 b^4 (b d-a e) (d+e x)^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^6 (a+b x)}+\frac {2 b^5 (d+e x)^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}}{13 e^6 (a+b x)} \]

[Out]

-2/3*(-a*e+b*d)^5*(e*x+d)^(3/2)*((b*x+a)^2)^(1/2)/e^6/(b*x+a)+2*b*(-a*e+b*d)^4*(e*x+d)^(5/2)*((b*x+a)^2)^(1/2)
/e^6/(b*x+a)-20/7*b^2*(-a*e+b*d)^3*(e*x+d)^(7/2)*((b*x+a)^2)^(1/2)/e^6/(b*x+a)+20/9*b^3*(-a*e+b*d)^2*(e*x+d)^(
9/2)*((b*x+a)^2)^(1/2)/e^6/(b*x+a)-10/11*b^4*(-a*e+b*d)*(e*x+d)^(11/2)*((b*x+a)^2)^(1/2)/e^6/(b*x+a)+2/13*b^5*
(e*x+d)^(13/2)*((b*x+a)^2)^(1/2)/e^6/(b*x+a)

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 318, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {660, 45} \[ \int \sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=-\frac {20 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^3}{7 e^6 (a+b x)}+\frac {2 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^4}{e^6 (a+b x)}-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^5}{3 e^6 (a+b x)}+\frac {2 b^5 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{13/2}}{13 e^6 (a+b x)}-\frac {10 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (b d-a e)}{11 e^6 (a+b x)}+\frac {20 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)^2}{9 e^6 (a+b x)} \]

[In]

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

[Out]

(-2*(b*d - a*e)^5*(d + e*x)^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^6*(a + b*x)) + (2*b*(b*d - a*e)^4*(d + e
*x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^6*(a + b*x)) - (20*b^2*(b*d - a*e)^3*(d + e*x)^(7/2)*Sqrt[a^2 + 2*
a*b*x + b^2*x^2])/(7*e^6*(a + b*x)) + (20*b^3*(b*d - a*e)^2*(d + e*x)^(9/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*
e^6*(a + b*x)) - (10*b^4*(b*d - a*e)*(d + e*x)^(11/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*e^6*(a + b*x)) + (2*b
^5*(d + e*x)^(13/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(13*e^6*(a + b*x))

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])

Rule 660

Int[((d_.) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[(a + b*x + c*x^2)^Fra
cPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b,
c, d, e, m, p}, x] && EqQ[b^2 - 4*a*c, 0] &&  !IntegerQ[p] && NeQ[2*c*d - b*e, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (a b+b^2 x\right )^5 \sqrt {d+e x} \, dx}{b^4 \left (a b+b^2 x\right )} \\ & = \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (-\frac {b^5 (b d-a e)^5 \sqrt {d+e x}}{e^5}+\frac {5 b^6 (b d-a e)^4 (d+e x)^{3/2}}{e^5}-\frac {10 b^7 (b d-a e)^3 (d+e x)^{5/2}}{e^5}+\frac {10 b^8 (b d-a e)^2 (d+e x)^{7/2}}{e^5}-\frac {5 b^9 (b d-a e) (d+e x)^{9/2}}{e^5}+\frac {b^{10} (d+e x)^{11/2}}{e^5}\right ) \, dx}{b^4 \left (a b+b^2 x\right )} \\ & = -\frac {2 (b d-a e)^5 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^6 (a+b x)}+\frac {2 b (b d-a e)^4 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (a+b x)}-\frac {20 b^2 (b d-a e)^3 (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^6 (a+b x)}+\frac {20 b^3 (b d-a e)^2 (d+e x)^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{9 e^6 (a+b x)}-\frac {10 b^4 (b d-a e) (d+e x)^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^6 (a+b x)}+\frac {2 b^5 (d+e x)^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}}{13 e^6 (a+b x)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 235, normalized size of antiderivative = 0.74 \[ \int \sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {2 \sqrt {(a+b x)^2} (d+e x)^{3/2} \left (3003 a^5 e^5+3003 a^4 b e^4 (-2 d+3 e x)+858 a^3 b^2 e^3 \left (8 d^2-12 d e x+15 e^2 x^2\right )+286 a^2 b^3 e^2 \left (-16 d^3+24 d^2 e x-30 d e^2 x^2+35 e^3 x^3\right )+13 a b^4 e \left (128 d^4-192 d^3 e x+240 d^2 e^2 x^2-280 d e^3 x^3+315 e^4 x^4\right )+b^5 \left (-256 d^5+384 d^4 e x-480 d^3 e^2 x^2+560 d^2 e^3 x^3-630 d e^4 x^4+693 e^5 x^5\right )\right )}{9009 e^6 (a+b x)} \]

[In]

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

[Out]

(2*Sqrt[(a + b*x)^2]*(d + e*x)^(3/2)*(3003*a^5*e^5 + 3003*a^4*b*e^4*(-2*d + 3*e*x) + 858*a^3*b^2*e^3*(8*d^2 -
12*d*e*x + 15*e^2*x^2) + 286*a^2*b^3*e^2*(-16*d^3 + 24*d^2*e*x - 30*d*e^2*x^2 + 35*e^3*x^3) + 13*a*b^4*e*(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) + b^5*(-256*d^5 + 384*d^4*e*x - 480*d^3*e^2
*x^2 + 560*d^2*e^3*x^3 - 630*d*e^4*x^4 + 693*e^5*x^5)))/(9009*e^6*(a + b*x))

Maple [A] (verified)

Time = 2.21 (sec) , antiderivative size = 289, normalized size of antiderivative = 0.91

method	result	size
gosper	\(\frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (693 x^{5} e^{5} b^{5}+4095 x^{4} a \,b^{4} e^{5}-630 x^{4} b^{5} d \,e^{4}+10010 x^{3} a^{2} b^{3} e^{5}-3640 x^{3} a \,b^{4} d \,e^{4}+560 x^{3} b^{5} d^{2} e^{3}+12870 x^{2} a^{3} b^{2} e^{5}-8580 x^{2} a^{2} b^{3} d \,e^{4}+3120 x^{2} a \,b^{4} d^{2} e^{3}-480 x^{2} b^{5} d^{3} e^{2}+9009 a^{4} b \,e^{5} x -10296 a^{3} b^{2} d \,e^{4} x +6864 x \,a^{2} b^{3} d^{2} e^{3}-2496 x a \,b^{4} d^{3} e^{2}+384 b^{5} d^{4} e x +3003 a^{5} e^{5}-6006 a^{4} b d \,e^{4}+6864 a^{3} b^{2} d^{2} e^{3}-4576 a^{2} b^{3} d^{3} e^{2}+1664 a \,b^{4} d^{4} e -256 b^{5} d^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{9009 e^{6} \left (b x +a \right )^{5}}\)	\(289\)
default	\(\frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (693 x^{5} e^{5} b^{5}+4095 x^{4} a \,b^{4} e^{5}-630 x^{4} b^{5} d \,e^{4}+10010 x^{3} a^{2} b^{3} e^{5}-3640 x^{3} a \,b^{4} d \,e^{4}+560 x^{3} b^{5} d^{2} e^{3}+12870 x^{2} a^{3} b^{2} e^{5}-8580 x^{2} a^{2} b^{3} d \,e^{4}+3120 x^{2} a \,b^{4} d^{2} e^{3}-480 x^{2} b^{5} d^{3} e^{2}+9009 a^{4} b \,e^{5} x -10296 a^{3} b^{2} d \,e^{4} x +6864 x \,a^{2} b^{3} d^{2} e^{3}-2496 x a \,b^{4} d^{3} e^{2}+384 b^{5} d^{4} e x +3003 a^{5} e^{5}-6006 a^{4} b d \,e^{4}+6864 a^{3} b^{2} d^{2} e^{3}-4576 a^{2} b^{3} d^{3} e^{2}+1664 a \,b^{4} d^{4} e -256 b^{5} d^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{9009 e^{6} \left (b x +a \right )^{5}}\)	\(289\)
risch	\(\frac {2 \sqrt {\left (b x +a \right )^{2}}\, \left (693 b^{5} x^{6} e^{6}+4095 a \,b^{4} e^{6} x^{5}+63 b^{5} d \,e^{5} x^{5}+10010 a^{2} b^{3} e^{6} x^{4}+455 a \,b^{4} d \,e^{5} x^{4}-70 b^{5} d^{2} e^{4} x^{4}+12870 a^{3} b^{2} e^{6} x^{3}+1430 a^{2} b^{3} d \,e^{5} x^{3}-520 a \,b^{4} d^{2} e^{4} x^{3}+80 b^{5} d^{3} e^{3} x^{3}+9009 a^{4} b \,e^{6} x^{2}+2574 a^{3} b^{2} d \,e^{5} x^{2}-1716 a^{2} b^{3} d^{2} e^{4} x^{2}+624 a \,b^{4} d^{3} e^{3} x^{2}-96 b^{5} d^{4} e^{2} x^{2}+3003 a^{5} e^{6} x +3003 a^{4} b d \,e^{5} x -3432 a^{3} b^{2} d^{2} e^{4} x +2288 a^{2} b^{3} d^{3} e^{3} x -832 a \,b^{4} d^{4} e^{2} x +128 b^{5} d^{5} e x +3003 d \,a^{5} e^{5}-6006 a^{4} b \,d^{2} e^{4}+6864 a^{3} b^{2} d^{3} e^{3}-4576 a^{2} b^{3} d^{4} e^{2}+1664 a \,b^{4} d^{5} e -256 b^{5} d^{6}\right ) \sqrt {e x +d}}{9009 \left (b x +a \right ) e^{6}}\)	\(377\)

[In]

int((b^2*x^2+2*a*b*x+a^2)^(5/2)*(e*x+d)^(1/2),x,method=_RETURNVERBOSE)

[Out]

2/9009*(e*x+d)^(3/2)*(693*b^5*e^5*x^5+4095*a*b^4*e^5*x^4-630*b^5*d*e^4*x^4+10010*a^2*b^3*e^5*x^3-3640*a*b^4*d*
e^4*x^3+560*b^5*d^2*e^3*x^3+12870*a^3*b^2*e^5*x^2-8580*a^2*b^3*d*e^4*x^2+3120*a*b^4*d^2*e^3*x^2-480*b^5*d^3*e^
2*x^2+9009*a^4*b*e^5*x-10296*a^3*b^2*d*e^4*x+6864*a^2*b^3*d^2*e^3*x-2496*a*b^4*d^3*e^2*x+384*b^5*d^4*e*x+3003*
a^5*e^5-6006*a^4*b*d*e^4+6864*a^3*b^2*d^2*e^3-4576*a^2*b^3*d^3*e^2+1664*a*b^4*d^4*e-256*b^5*d^5)*((b*x+a)^2)^(
5/2)/e^6/(b*x+a)^5

Fricas [A] (verification not implemented)

none

Time = 0.58 (sec) , antiderivative size = 338, normalized size of antiderivative = 1.06 \[ \int \sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {2 \, {\left (693 \, b^{5} e^{6} x^{6} - 256 \, b^{5} d^{6} + 1664 \, a b^{4} d^{5} e - 4576 \, a^{2} b^{3} d^{4} e^{2} + 6864 \, a^{3} b^{2} d^{3} e^{3} - 6006 \, a^{4} b d^{2} e^{4} + 3003 \, a^{5} d e^{5} + 63 \, {\left (b^{5} d e^{5} + 65 \, a b^{4} e^{6}\right )} x^{5} - 35 \, {\left (2 \, b^{5} d^{2} e^{4} - 13 \, a b^{4} d e^{5} - 286 \, a^{2} b^{3} e^{6}\right )} x^{4} + 10 \, {\left (8 \, b^{5} d^{3} e^{3} - 52 \, a b^{4} d^{2} e^{4} + 143 \, a^{2} b^{3} d e^{5} + 1287 \, a^{3} b^{2} e^{6}\right )} x^{3} - 3 \, {\left (32 \, b^{5} d^{4} e^{2} - 208 \, a b^{4} d^{3} e^{3} + 572 \, a^{2} b^{3} d^{2} e^{4} - 858 \, a^{3} b^{2} d e^{5} - 3003 \, a^{4} b e^{6}\right )} x^{2} + {\left (128 \, b^{5} d^{5} e - 832 \, a b^{4} d^{4} e^{2} + 2288 \, a^{2} b^{3} d^{3} e^{3} - 3432 \, a^{3} b^{2} d^{2} e^{4} + 3003 \, a^{4} b d e^{5} + 3003 \, a^{5} e^{6}\right )} x\right )} \sqrt {e x + d}}{9009 \, e^{6}} \]

[In]

integrate((b^2*x^2+2*a*b*x+a^2)^(5/2)*(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

2/9009*(693*b^5*e^6*x^6 - 256*b^5*d^6 + 1664*a*b^4*d^5*e - 4576*a^2*b^3*d^4*e^2 + 6864*a^3*b^2*d^3*e^3 - 6006*
a^4*b*d^2*e^4 + 3003*a^5*d*e^5 + 63*(b^5*d*e^5 + 65*a*b^4*e^6)*x^5 - 35*(2*b^5*d^2*e^4 - 13*a*b^4*d*e^5 - 286*
a^2*b^3*e^6)*x^4 + 10*(8*b^5*d^3*e^3 - 52*a*b^4*d^2*e^4 + 143*a^2*b^3*d*e^5 + 1287*a^3*b^2*e^6)*x^3 - 3*(32*b^
5*d^4*e^2 - 208*a*b^4*d^3*e^3 + 572*a^2*b^3*d^2*e^4 - 858*a^3*b^2*d*e^5 - 3003*a^4*b*e^6)*x^2 + (128*b^5*d^5*e
- 832*a*b^4*d^4*e^2 + 2288*a^2*b^3*d^3*e^3 - 3432*a^3*b^2*d^2*e^4 + 3003*a^4*b*d*e^5 + 3003*a^5*e^6)*x)*sqrt(
e*x + d)/e^6

Sympy [F]

\[ \int \sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\int \sqrt {d + e x} \left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}\, dx \]

[In]

integrate((b**2*x**2+2*a*b*x+a**2)**(5/2)*(e*x+d)**(1/2),x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 338, normalized size of antiderivative = 1.06 \[ \int \sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {2 \, {\left (693 \, b^{5} e^{6} x^{6} - 256 \, b^{5} d^{6} + 1664 \, a b^{4} d^{5} e - 4576 \, a^{2} b^{3} d^{4} e^{2} + 6864 \, a^{3} b^{2} d^{3} e^{3} - 6006 \, a^{4} b d^{2} e^{4} + 3003 \, a^{5} d e^{5} + 63 \, {\left (b^{5} d e^{5} + 65 \, a b^{4} e^{6}\right )} x^{5} - 35 \, {\left (2 \, b^{5} d^{2} e^{4} - 13 \, a b^{4} d e^{5} - 286 \, a^{2} b^{3} e^{6}\right )} x^{4} + 10 \, {\left (8 \, b^{5} d^{3} e^{3} - 52 \, a b^{4} d^{2} e^{4} + 143 \, a^{2} b^{3} d e^{5} + 1287 \, a^{3} b^{2} e^{6}\right )} x^{3} - 3 \, {\left (32 \, b^{5} d^{4} e^{2} - 208 \, a b^{4} d^{3} e^{3} + 572 \, a^{2} b^{3} d^{2} e^{4} - 858 \, a^{3} b^{2} d e^{5} - 3003 \, a^{4} b e^{6}\right )} x^{2} + {\left (128 \, b^{5} d^{5} e - 832 \, a b^{4} d^{4} e^{2} + 2288 \, a^{2} b^{3} d^{3} e^{3} - 3432 \, a^{3} b^{2} d^{2} e^{4} + 3003 \, a^{4} b d e^{5} + 3003 \, a^{5} e^{6}\right )} x\right )} \sqrt {e x + d}}{9009 \, e^{6}} \]

[In]

integrate((b^2*x^2+2*a*b*x+a^2)^(5/2)*(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

2/9009*(693*b^5*e^6*x^6 - 256*b^5*d^6 + 1664*a*b^4*d^5*e - 4576*a^2*b^3*d^4*e^2 + 6864*a^3*b^2*d^3*e^3 - 6006*
a^4*b*d^2*e^4 + 3003*a^5*d*e^5 + 63*(b^5*d*e^5 + 65*a*b^4*e^6)*x^5 - 35*(2*b^5*d^2*e^4 - 13*a*b^4*d*e^5 - 286*
a^2*b^3*e^6)*x^4 + 10*(8*b^5*d^3*e^3 - 52*a*b^4*d^2*e^4 + 143*a^2*b^3*d*e^5 + 1287*a^3*b^2*e^6)*x^3 - 3*(32*b^
5*d^4*e^2 - 208*a*b^4*d^3*e^3 + 572*a^2*b^3*d^2*e^4 - 858*a^3*b^2*d*e^5 - 3003*a^4*b*e^6)*x^2 + (128*b^5*d^5*e
- 832*a*b^4*d^4*e^2 + 2288*a^2*b^3*d^3*e^3 - 3432*a^3*b^2*d^2*e^4 + 3003*a^4*b*d*e^5 + 3003*a^5*e^6)*x)*sqrt(
e*x + d)/e^6

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 713 vs. \(2 (230) = 460\).

Time = 0.34 (sec) , antiderivative size = 713, normalized size of antiderivative = 2.24 \[ \int \sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\text {Too large to display} \]

[In]

integrate((b^2*x^2+2*a*b*x+a^2)^(5/2)*(e*x+d)^(1/2),x, algorithm="giac")

[Out]

2/9009*(9009*sqrt(e*x + d)*a^5*d*sgn(b*x + a) + 3003*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*a^5*sgn(b*x + a) +
15015*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*a^4*b*d*sgn(b*x + a)/e + 6006*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3
/2)*d + 15*sqrt(e*x + d)*d^2)*a^3*b^2*d*sgn(b*x + a)/e^2 + 3003*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15
*sqrt(e*x + d)*d^2)*a^4*b*sgn(b*x + a)/e + 2574*(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^3*d*sgn(b*x + a)/e^3 + 2574*(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^2*sgn(b*x + a)/e^2 + 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)*a*b^4*d*sgn(b*x + a)/e^4
+ 286*(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*s
qrt(e*x + d)*d^4)*a^2*b^3*sgn(b*x + a)/e^3 + 13*(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^5*d*sgn(b*x + a)/e^5
+ 65*(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^4*sgn(b*x + a)/e^4 + 3*(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^5*sgn(b*x + a)/e^5)/e

Mupad [F(-1)]

Timed out. \[ \int \sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\int \sqrt {d+e\,x}\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2} \,d x \]

[In]

int((d + e*x)^(1/2)*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2),x)

[Out]

int((d + e*x)^(1/2)*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2), x)