3.1.46 \(\int x^2 \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+e x+d x^2} \, dx\) [46]

Optimal. Leaf size=317 \[ -\frac {\left (2 a d \left (4 c d-5 e^2\right )-b \left (12 c d e-7 e^3\right )\right ) (e+2 d x) \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+e x+d x^2}}{128 d^4 (a+b x)}+\frac {b x^2 \sqrt {a^2+2 a b x+b^2 x^2} \left (c+e x+d x^2\right )^{3/2}}{5 d (a+b x)}-\frac {\left (32 b c d+50 a d e-35 b e^2-6 d (10 a d-7 b e) x\right ) \sqrt {a^2+2 a b x+b^2 x^2} \left (c+e x+d x^2\right )^{3/2}}{240 d^3 (a+b x)}-\frac {\left (4 c d-e^2\right ) \left (8 a c d^2-12 b c d e-10 a d e^2+7 b e^3\right ) \sqrt {a^2+2 a b x+b^2 x^2} \tanh ^{-1}\left (\frac {e+2 d x}{2 \sqrt {d} \sqrt {c+e x+d x^2}}\right )}{256 d^{9/2} (a+b x)} \]

[Out]

1/5*b*x^2*(d*x^2+e*x+c)^(3/2)*((b*x+a)^2)^(1/2)/d/(b*x+a)-1/240*(32*b*c*d+50*a*d*e-35*b*e^2-6*d*(10*a*d-7*b*e)
*x)*(d*x^2+e*x+c)^(3/2)*((b*x+a)^2)^(1/2)/d^3/(b*x+a)-1/256*(4*c*d-e^2)*(8*a*c*d^2-10*a*d*e^2-12*b*c*d*e+7*b*e
^3)*arctanh(1/2*(2*d*x+e)/d^(1/2)/(d*x^2+e*x+c)^(1/2))*((b*x+a)^2)^(1/2)/d^(9/2)/(b*x+a)-1/128*(2*a*d*(4*c*d-5
*e^2)-b*(12*c*d*e-7*e^3))*(2*d*x+e)*((b*x+a)^2)^(1/2)*(d*x^2+e*x+c)^(1/2)/d^4/(b*x+a)

________________________________________________________________________________________

Rubi [A]
time = 0.19, antiderivative size = 317, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {1014, 846, 793, 626, 635, 212} \begin {gather*} -\frac {\sqrt {a^2+2 a b x+b^2 x^2} (2 d x+e) \sqrt {c+d x^2+e x} \left (2 a d \left (4 c d-5 e^2\right )-b \left (12 c d e-7 e^3\right )\right )}{128 d^4 (a+b x)}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (c+d x^2+e x\right )^{3/2} \left (-6 d x (10 a d-7 b e)+50 a d e+32 b c d-35 b e^2\right )}{240 d^3 (a+b x)}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (4 c d-e^2\right ) \left (8 a c d^2-10 a d e^2-12 b c d e+7 b e^3\right ) \tanh ^{-1}\left (\frac {2 d x+e}{2 \sqrt {d} \sqrt {c+d x^2+e x}}\right )}{256 d^{9/2} (a+b x)}+\frac {b x^2 \sqrt {a^2+2 a b x+b^2 x^2} \left (c+d x^2+e x\right )^{3/2}}{5 d (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/128*((2*a*d*(4*c*d - 5*e^2) - b*(12*c*d*e - 7*e^3))*(e + 2*d*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*Sqrt[c + e*x
+ d*x^2])/(d^4*(a + b*x)) + (b*x^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(c + e*x + d*x^2)^(3/2))/(5*d*(a + b*x)) - ((
32*b*c*d + 50*a*d*e - 35*b*e^2 - 6*d*(10*a*d - 7*b*e)*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(c + e*x + d*x^2)^(3/2)
)/(240*d^3*(a + b*x)) - ((4*c*d - e^2)*(8*a*c*d^2 - 12*b*c*d*e - 10*a*d*e^2 + 7*b*e^3)*Sqrt[a^2 + 2*a*b*x + b^
2*x^2]*ArcTanh[(e + 2*d*x)/(2*Sqrt[d]*Sqrt[c + e*x + d*x^2])])/(256*d^(9/2)*(a + b*x))

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 626

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x)*((a + b*x + c*x^2)^p/(2*c*(2*p + 1
))), x] - Dist[p*((b^2 - 4*a*c)/(2*c*(2*p + 1))), Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x]
&& NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && IntegerQ[4*p]

Rule 635

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 793

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(b
*e*g*(p + 2) - c*(e*f + d*g)*(2*p + 3) - 2*c*e*g*(p + 1)*x))*((a + b*x + c*x^2)^(p + 1)/(2*c^2*(p + 1)*(2*p +
3))), x] + Dist[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p + 3))/(2*c^2*(2*p + 3)), Int[(
a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b^2 - 4*a*c, 0] &&  !LeQ[p, -1]

Rule 846

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[g*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 2))), x] + Dist[1/(c*(m + 2*p + 2)), Int[(d + e*x)^(m
- 1)*(a + b*x + c*x^2)^p*Simp[m*(c*d*f - a*e*g) + d*(2*c*f - b*g)*(p + 1) + (m*(c*e*f + c*d*g - b*e*g) + e*(p
+ 1)*(2*c*f - b*g))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 -
b*d*e + a*e^2, 0] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
&&  !(IGtQ[m, 0] && EqQ[f, 0])

Rule 1014

Int[((g_.) + (h_.)*(x_))^(m_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_) + (e_.)*(x_) + (f_.)*(x_)^2)^(q_)
, x_Symbol] :> Dist[(a + b*x + c*x^2)^FracPart[p]/((4*c)^IntPart[p]*(b + 2*c*x)^(2*FracPart[p])), Int[(g + h*x
)^m*(b + 2*c*x)^(2*p)*(d + e*x + f*x^2)^q, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, p, q}, x] && EqQ[b^2 -
4*a*c, 0]

Rubi steps

\begin {align*} \int x^2 \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+e x+d x^2} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int x^2 \left (2 a b+2 b^2 x\right ) \sqrt {c+e x+d x^2} \, dx}{2 a b+2 b^2 x}\\ &=\frac {b x^2 \sqrt {a^2+2 a b x+b^2 x^2} \left (c+e x+d x^2\right )^{3/2}}{5 d (a+b x)}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int x \left (-4 b^2 c+b (10 a d-7 b e) x\right ) \sqrt {c+e x+d x^2} \, dx}{5 d \left (2 a b+2 b^2 x\right )}\\ &=\frac {b x^2 \sqrt {a^2+2 a b x+b^2 x^2} \left (c+e x+d x^2\right )^{3/2}}{5 d (a+b x)}-\frac {\left (32 b c d+50 a d e-35 b e^2-6 d (10 a d-7 b e) x\right ) \sqrt {a^2+2 a b x+b^2 x^2} \left (c+e x+d x^2\right )^{3/2}}{240 d^3 (a+b x)}+\frac {\left (\left (16 b^2 c d e-2 b c d (10 a d-7 b e)+\frac {5}{2} b e^2 (10 a d-7 b e)\right ) \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \sqrt {c+e x+d x^2} \, dx}{40 d^3 \left (2 a b+2 b^2 x\right )}\\ &=-\frac {\left (8 a c d^2-12 b c d e-10 a d e^2+7 b e^3\right ) (e+2 d x) \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+e x+d x^2}}{128 d^4 (a+b x)}+\frac {b x^2 \sqrt {a^2+2 a b x+b^2 x^2} \left (c+e x+d x^2\right )^{3/2}}{5 d (a+b x)}-\frac {\left (32 b c d+50 a d e-35 b e^2-6 d (10 a d-7 b e) x\right ) \sqrt {a^2+2 a b x+b^2 x^2} \left (c+e x+d x^2\right )^{3/2}}{240 d^3 (a+b x)}+\frac {\left (\left (4 c d-e^2\right ) \left (16 b^2 c d e-2 b c d (10 a d-7 b e)+\frac {5}{2} b e^2 (10 a d-7 b e)\right ) \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \frac {1}{\sqrt {c+e x+d x^2}} \, dx}{320 d^4 \left (2 a b+2 b^2 x\right )}\\ &=-\frac {\left (8 a c d^2-12 b c d e-10 a d e^2+7 b e^3\right ) (e+2 d x) \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+e x+d x^2}}{128 d^4 (a+b x)}+\frac {b x^2 \sqrt {a^2+2 a b x+b^2 x^2} \left (c+e x+d x^2\right )^{3/2}}{5 d (a+b x)}-\frac {\left (32 b c d+50 a d e-35 b e^2-6 d (10 a d-7 b e) x\right ) \sqrt {a^2+2 a b x+b^2 x^2} \left (c+e x+d x^2\right )^{3/2}}{240 d^3 (a+b x)}+\frac {\left (\left (4 c d-e^2\right ) \left (16 b^2 c d e-2 b c d (10 a d-7 b e)+\frac {5}{2} b e^2 (10 a d-7 b e)\right ) \sqrt {a^2+2 a b x+b^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{4 d-x^2} \, dx,x,\frac {e+2 d x}{\sqrt {c+e x+d x^2}}\right )}{160 d^4 \left (2 a b+2 b^2 x\right )}\\ &=-\frac {\left (8 a c d^2-12 b c d e-10 a d e^2+7 b e^3\right ) (e+2 d x) \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+e x+d x^2}}{128 d^4 (a+b x)}+\frac {b x^2 \sqrt {a^2+2 a b x+b^2 x^2} \left (c+e x+d x^2\right )^{3/2}}{5 d (a+b x)}-\frac {\left (32 b c d+50 a d e-35 b e^2-6 d (10 a d-7 b e) x\right ) \sqrt {a^2+2 a b x+b^2 x^2} \left (c+e x+d x^2\right )^{3/2}}{240 d^3 (a+b x)}-\frac {\left (4 c d-e^2\right ) \left (8 a c d^2-12 b c d e-10 a d e^2+7 b e^3\right ) \sqrt {a^2+2 a b x+b^2 x^2} \tanh ^{-1}\left (\frac {e+2 d x}{2 \sqrt {d} \sqrt {c+e x+d x^2}}\right )}{256 d^{9/2} (a+b x)}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.63, size = 236, normalized size = 0.74 \begin {gather*} \frac {\sqrt {(a+b x)^2} \left (2 \sqrt {d} \sqrt {c+x (e+d x)} \left (10 a d \left (15 e^3-10 d e^2 x+8 d^2 e x^2+48 d^3 x^3+4 c d (-13 e+6 d x)\right )+b \left (-256 c^2 d^2-105 e^4+70 d e^3 x-56 d^2 e^2 x^2+48 d^3 e x^3+384 d^4 x^4+4 c d \left (115 e^2-58 d e x+32 d^2 x^2\right )\right )\right )+15 \left (4 c d-e^2\right ) \left (2 a d \left (4 c d-5 e^2\right )+b \left (-12 c d e+7 e^3\right )\right ) \log \left (e+2 d x-2 \sqrt {d} \sqrt {c+x (e+d x)}\right )\right )}{3840 d^{9/2} (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[(a + b*x)^2]*(2*Sqrt[d]*Sqrt[c + x*(e + d*x)]*(10*a*d*(15*e^3 - 10*d*e^2*x + 8*d^2*e*x^2 + 48*d^3*x^3 +
4*c*d*(-13*e + 6*d*x)) + b*(-256*c^2*d^2 - 105*e^4 + 70*d*e^3*x - 56*d^2*e^2*x^2 + 48*d^3*e*x^3 + 384*d^4*x^4
+ 4*c*d*(115*e^2 - 58*d*e*x + 32*d^2*x^2))) + 15*(4*c*d - e^2)*(2*a*d*(4*c*d - 5*e^2) + b*(-12*c*d*e + 7*e^3))
*Log[e + 2*d*x - 2*Sqrt[d]*Sqrt[c + x*(e + d*x)]]))/(3840*d^(9/2)*(a + b*x))

________________________________________________________________________________________

Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.16, size = 530, normalized size = 1.67

method	result	size
risch	\(\frac {\left (384 b \,x^{4} d^{4}+480 a \,d^{4} x^{3}+48 b \,d^{3} e \,x^{3}+80 a \,d^{3} e \,x^{2}+128 b c \,d^{3} x^{2}-56 b \,d^{2} e^{2} x^{2}+240 a c \,d^{3} x -100 a \,d^{2} e^{2} x -232 b c \,d^{2} e x +70 b d \,e^{3} x -520 a c \,d^{2} e +150 a d \,e^{3}-256 b \,c^{2} d^{2}+460 b c d \,e^{2}-105 b \,e^{4}\right ) \sqrt {d \,x^{2}+e x +c}\, \sqrt {\left (b x +a \right )^{2}}}{1920 d^{4} \left (b x +a \right )}+\frac {\left (-\frac {\ln \left (\frac {\frac {e}{2}+d x}{\sqrt {d}}+\sqrt {d \,x^{2}+e x +c}\right ) a \,c^{2}}{8 d^{\frac {3}{2}}}+\frac {3 \ln \left (\frac {\frac {e}{2}+d x}{\sqrt {d}}+\sqrt {d \,x^{2}+e x +c}\right ) a c \,e^{2}}{16 d^{\frac {5}{2}}}-\frac {5 \ln \left (\frac {\frac {e}{2}+d x}{\sqrt {d}}+\sqrt {d \,x^{2}+e x +c}\right ) a \,e^{4}}{128 d^{\frac {7}{2}}}+\frac {3 \ln \left (\frac {\frac {e}{2}+d x}{\sqrt {d}}+\sqrt {d \,x^{2}+e x +c}\right ) b \,c^{2} e}{16 d^{\frac {5}{2}}}-\frac {5 \ln \left (\frac {\frac {e}{2}+d x}{\sqrt {d}}+\sqrt {d \,x^{2}+e x +c}\right ) b c \,e^{3}}{32 d^{\frac {7}{2}}}+\frac {7 \ln \left (\frac {\frac {e}{2}+d x}{\sqrt {d}}+\sqrt {d \,x^{2}+e x +c}\right ) b \,e^{5}}{256 d^{\frac {9}{2}}}\right ) \sqrt {\left (b x +a \right )^{2}}}{b x +a}\)	\(394\)
default	\(-\frac {\mathrm {csgn}\left (b x +a \right ) \left (-768 d^{\frac {9}{2}} \left (d \,x^{2}+e x +c \right )^{\frac {3}{2}} b \,x^{2}-960 d^{\frac {9}{2}} \left (d \,x^{2}+e x +c \right )^{\frac {3}{2}} a x +672 d^{\frac {7}{2}} \left (d \,x^{2}+e x +c \right )^{\frac {3}{2}} b e x +800 d^{\frac {7}{2}} \left (d \,x^{2}+e x +c \right )^{\frac {3}{2}} a e +512 d^{\frac {7}{2}} \left (d \,x^{2}+e x +c \right )^{\frac {3}{2}} b c -560 d^{\frac {5}{2}} \left (d \,x^{2}+e x +c \right )^{\frac {3}{2}} b \,e^{2}+480 d^{\frac {9}{2}} \sqrt {d \,x^{2}+e x +c}\, a c x -600 d^{\frac {7}{2}} \sqrt {d \,x^{2}+e x +c}\, a \,e^{2} x -720 d^{\frac {7}{2}} \sqrt {d \,x^{2}+e x +c}\, b c e x +420 d^{\frac {5}{2}} \sqrt {d \,x^{2}+e x +c}\, b \,e^{3} x +240 d^{\frac {7}{2}} \sqrt {d \,x^{2}+e x +c}\, a c e -300 d^{\frac {5}{2}} \sqrt {d \,x^{2}+e x +c}\, a \,e^{3}-360 d^{\frac {5}{2}} \sqrt {d \,x^{2}+e x +c}\, b c \,e^{2}+210 d^{\frac {3}{2}} \sqrt {d \,x^{2}+e x +c}\, b \,e^{4}+480 \ln \left (\frac {2 \sqrt {d \,x^{2}+e x +c}\, \sqrt {d}+2 d x +e}{2 \sqrt {d}}\right ) a \,c^{2} d^{4}-720 \ln \left (\frac {2 \sqrt {d \,x^{2}+e x +c}\, \sqrt {d}+2 d x +e}{2 \sqrt {d}}\right ) a c \,d^{3} e^{2}+150 \ln \left (\frac {2 \sqrt {d \,x^{2}+e x +c}\, \sqrt {d}+2 d x +e}{2 \sqrt {d}}\right ) a \,d^{2} e^{4}-720 \ln \left (\frac {2 \sqrt {d \,x^{2}+e x +c}\, \sqrt {d}+2 d x +e}{2 \sqrt {d}}\right ) b \,c^{2} d^{3} e +600 \ln \left (\frac {2 \sqrt {d \,x^{2}+e x +c}\, \sqrt {d}+2 d x +e}{2 \sqrt {d}}\right ) b c \,d^{2} e^{3}-105 \ln \left (\frac {2 \sqrt {d \,x^{2}+e x +c}\, \sqrt {d}+2 d x +e}{2 \sqrt {d}}\right ) b d \,e^{5}\right )}{3840 d^{\frac {11}{2}}}\)	\(530\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3840*csgn(b*x+a)*(-768*d^(9/2)*(d*x^2+e*x+c)^(3/2)*b*x^2-960*d^(9/2)*(d*x^2+e*x+c)^(3/2)*a*x+672*d^(7/2)*(d
*x^2+e*x+c)^(3/2)*b*e*x+800*d^(7/2)*(d*x^2+e*x+c)^(3/2)*a*e+512*d^(7/2)*(d*x^2+e*x+c)^(3/2)*b*c-560*d^(5/2)*(d
*x^2+e*x+c)^(3/2)*b*e^2+480*d^(9/2)*(d*x^2+e*x+c)^(1/2)*a*c*x-600*d^(7/2)*(d*x^2+e*x+c)^(1/2)*a*e^2*x-720*d^(7
/2)*(d*x^2+e*x+c)^(1/2)*b*c*e*x+420*d^(5/2)*(d*x^2+e*x+c)^(1/2)*b*e^3*x+240*d^(7/2)*(d*x^2+e*x+c)^(1/2)*a*c*e-
300*d^(5/2)*(d*x^2+e*x+c)^(1/2)*a*e^3-360*d^(5/2)*(d*x^2+e*x+c)^(1/2)*b*c*e^2+210*d^(3/2)*(d*x^2+e*x+c)^(1/2)*
b*e^4+480*ln(1/2*(2*(d*x^2+e*x+c)^(1/2)*d^(1/2)+2*d*x+e)/d^(1/2))*a*c^2*d^4-720*ln(1/2*(2*(d*x^2+e*x+c)^(1/2)*
d^(1/2)+2*d*x+e)/d^(1/2))*a*c*d^3*e^2+150*ln(1/2*(2*(d*x^2+e*x+c)^(1/2)*d^(1/2)+2*d*x+e)/d^(1/2))*a*d^2*e^4-72
0*ln(1/2*(2*(d*x^2+e*x+c)^(1/2)*d^(1/2)+2*d*x+e)/d^(1/2))*b*c^2*d^3*e+600*ln(1/2*(2*(d*x^2+e*x+c)^(1/2)*d^(1/2
)+2*d*x+e)/d^(1/2))*b*c*d^2*e^3-105*ln(1/2*(2*(d*x^2+e*x+c)^(1/2)*d^(1/2)+2*d*x+e)/d^(1/2))*b*d*e^5)/d^(11/2)

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(d*x^2 + x*e + c)*sqrt((b*x + a)^2)*x^2, x)

________________________________________________________________________________________

Fricas [A]
time = 0.39, size = 516, normalized size = 1.63 \begin {gather*} \left [-\frac {15 \, {\left (32 \, a c^{2} d^{3} - 48 \, b c^{2} d^{2} e - 48 \, a c d^{2} e^{2} + 40 \, b c d e^{3} + 10 \, a d e^{4} - 7 \, b e^{5}\right )} \sqrt {d} \log \left (8 \, d^{2} x^{2} + 8 \, d x e + 4 \, \sqrt {d x^{2} + x e + c} {\left (2 \, d x + e\right )} \sqrt {d} + 4 \, c d + e^{2}\right ) - 4 \, {\left (384 \, b d^{5} x^{4} + 480 \, a d^{5} x^{3} + 128 \, b c d^{4} x^{2} + 240 \, a c d^{4} x - 256 \, b c^{2} d^{3} - 105 \, b d e^{4} + 10 \, {\left (7 \, b d^{2} x + 15 \, a d^{2}\right )} e^{3} - 4 \, {\left (14 \, b d^{3} x^{2} + 25 \, a d^{3} x - 115 \, b c d^{2}\right )} e^{2} + 8 \, {\left (6 \, b d^{4} x^{3} + 10 \, a d^{4} x^{2} - 29 \, b c d^{3} x - 65 \, a c d^{3}\right )} e\right )} \sqrt {d x^{2} + x e + c}}{7680 \, d^{5}}, \frac {15 \, {\left (32 \, a c^{2} d^{3} - 48 \, b c^{2} d^{2} e - 48 \, a c d^{2} e^{2} + 40 \, b c d e^{3} + 10 \, a d e^{4} - 7 \, b e^{5}\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {d x^{2} + x e + c} {\left (2 \, d x + e\right )} \sqrt {-d}}{2 \, {\left (d^{2} x^{2} + d x e + c d\right )}}\right ) + 2 \, {\left (384 \, b d^{5} x^{4} + 480 \, a d^{5} x^{3} + 128 \, b c d^{4} x^{2} + 240 \, a c d^{4} x - 256 \, b c^{2} d^{3} - 105 \, b d e^{4} + 10 \, {\left (7 \, b d^{2} x + 15 \, a d^{2}\right )} e^{3} - 4 \, {\left (14 \, b d^{3} x^{2} + 25 \, a d^{3} x - 115 \, b c d^{2}\right )} e^{2} + 8 \, {\left (6 \, b d^{4} x^{3} + 10 \, a d^{4} x^{2} - 29 \, b c d^{3} x - 65 \, a c d^{3}\right )} e\right )} \sqrt {d x^{2} + x e + c}}{3840 \, d^{5}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/7680*(15*(32*a*c^2*d^3 - 48*b*c^2*d^2*e - 48*a*c*d^2*e^2 + 40*b*c*d*e^3 + 10*a*d*e^4 - 7*b*e^5)*sqrt(d)*lo
g(8*d^2*x^2 + 8*d*x*e + 4*sqrt(d*x^2 + x*e + c)*(2*d*x + e)*sqrt(d) + 4*c*d + e^2) - 4*(384*b*d^5*x^4 + 480*a*
d^5*x^3 + 128*b*c*d^4*x^2 + 240*a*c*d^4*x - 256*b*c^2*d^3 - 105*b*d*e^4 + 10*(7*b*d^2*x + 15*a*d^2)*e^3 - 4*(1
4*b*d^3*x^2 + 25*a*d^3*x - 115*b*c*d^2)*e^2 + 8*(6*b*d^4*x^3 + 10*a*d^4*x^2 - 29*b*c*d^3*x - 65*a*c*d^3)*e)*sq
rt(d*x^2 + x*e + c))/d^5, 1/3840*(15*(32*a*c^2*d^3 - 48*b*c^2*d^2*e - 48*a*c*d^2*e^2 + 40*b*c*d*e^3 + 10*a*d*e
^4 - 7*b*e^5)*sqrt(-d)*arctan(1/2*sqrt(d*x^2 + x*e + c)*(2*d*x + e)*sqrt(-d)/(d^2*x^2 + d*x*e + c*d)) + 2*(384
*b*d^5*x^4 + 480*a*d^5*x^3 + 128*b*c*d^4*x^2 + 240*a*c*d^4*x - 256*b*c^2*d^3 - 105*b*d*e^4 + 10*(7*b*d^2*x + 1
5*a*d^2)*e^3 - 4*(14*b*d^3*x^2 + 25*a*d^3*x - 115*b*c*d^2)*e^2 + 8*(6*b*d^4*x^3 + 10*a*d^4*x^2 - 29*b*c*d^3*x
- 65*a*c*d^3)*e)*sqrt(d*x^2 + x*e + c))/d^5]

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{2} \sqrt {c + d x^{2} + e x} \sqrt {\left (a + b x\right )^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]
time = 4.87, size = 368, normalized size = 1.16 \begin {gather*} \frac {1}{1920} \, \sqrt {d x^{2} + x e + c} {\left (2 \, {\left (4 \, {\left (6 \, {\left (8 \, b x \mathrm {sgn}\left (b x + a\right ) + \frac {10 \, a d^{4} \mathrm {sgn}\left (b x + a\right ) + b d^{3} e \mathrm {sgn}\left (b x + a\right )}{d^{4}}\right )} x + \frac {16 \, b c d^{3} \mathrm {sgn}\left (b x + a\right ) + 10 \, a d^{3} e \mathrm {sgn}\left (b x + a\right ) - 7 \, b d^{2} e^{2} \mathrm {sgn}\left (b x + a\right )}{d^{4}}\right )} x + \frac {120 \, a c d^{3} \mathrm {sgn}\left (b x + a\right ) - 116 \, b c d^{2} e \mathrm {sgn}\left (b x + a\right ) - 50 \, a d^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) + 35 \, b d e^{3} \mathrm {sgn}\left (b x + a\right )}{d^{4}}\right )} x - \frac {256 \, b c^{2} d^{2} \mathrm {sgn}\left (b x + a\right ) + 520 \, a c d^{2} e \mathrm {sgn}\left (b x + a\right ) - 460 \, b c d e^{2} \mathrm {sgn}\left (b x + a\right ) - 150 \, a d e^{3} \mathrm {sgn}\left (b x + a\right ) + 105 \, b e^{4} \mathrm {sgn}\left (b x + a\right )}{d^{4}}\right )} + \frac {{\left (32 \, a c^{2} d^{3} \mathrm {sgn}\left (b x + a\right ) - 48 \, b c^{2} d^{2} e \mathrm {sgn}\left (b x + a\right ) - 48 \, a c d^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) + 40 \, b c d e^{3} \mathrm {sgn}\left (b x + a\right ) + 10 \, a d e^{4} \mathrm {sgn}\left (b x + a\right ) - 7 \, b e^{5} \mathrm {sgn}\left (b x + a\right )\right )} \log \left ({\left | -2 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + x e + c}\right )} \sqrt {d} - e \right |}\right )}{256 \, d^{\frac {9}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1920*sqrt(d*x^2 + x*e + c)*(2*(4*(6*(8*b*x*sgn(b*x + a) + (10*a*d^4*sgn(b*x + a) + b*d^3*e*sgn(b*x + a))/d^4
)*x + (16*b*c*d^3*sgn(b*x + a) + 10*a*d^3*e*sgn(b*x + a) - 7*b*d^2*e^2*sgn(b*x + a))/d^4)*x + (120*a*c*d^3*sgn
(b*x + a) - 116*b*c*d^2*e*sgn(b*x + a) - 50*a*d^2*e^2*sgn(b*x + a) + 35*b*d*e^3*sgn(b*x + a))/d^4)*x - (256*b*
c^2*d^2*sgn(b*x + a) + 520*a*c*d^2*e*sgn(b*x + a) - 460*b*c*d*e^2*sgn(b*x + a) - 150*a*d*e^3*sgn(b*x + a) + 10
5*b*e^4*sgn(b*x + a))/d^4) + 1/256*(32*a*c^2*d^3*sgn(b*x + a) - 48*b*c^2*d^2*e*sgn(b*x + a) - 48*a*c*d^2*e^2*s
gn(b*x + a) + 40*b*c*d*e^3*sgn(b*x + a) + 10*a*d*e^4*sgn(b*x + a) - 7*b*e^5*sgn(b*x + a))*log(abs(-2*(sqrt(d)*
x - sqrt(d*x^2 + x*e + c))*sqrt(d) - e))/d^(9/2)

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^2\,\sqrt {{\left (a+b\,x\right )}^2}\,\sqrt {d\,x^2+e\,x+c} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________