\(\int \sqrt {b d+2 c d x} (a+b x+c x^2)^{3/2} \, dx\) [1345]

Optimal result

Integrand size = 28, antiderivative size = 281 \[ \int \sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^{3/2} \, dx=-\frac {\left (b^2-4 a c\right ) (b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}}{30 c^2 d}+\frac {(b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^{3/2}}{9 c d}+\frac {\left (b^2-4 a c\right )^{11/4} \sqrt {d} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\arcsin \left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{30 c^3 \sqrt {a+b x+c x^2}}-\frac {\left (b^2-4 a c\right )^{11/4} \sqrt {d} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ),-1\right )}{30 c^3 \sqrt {a+b x+c x^2}} \]

[Out]

1/9*(2*c*d*x+b*d)^(3/2)*(c*x^2+b*x+a)^(3/2)/c/d-1/30*(-4*a*c+b^2)*(2*c*d*x+b*d)^(3/2)*(c*x^2+b*x+a)^(1/2)/c^2/
d+1/30*(-4*a*c+b^2)^(11/4)*EllipticE((2*c*d*x+b*d)^(1/2)/(-4*a*c+b^2)^(1/4)/d^(1/2),I)*d^(1/2)*(-c*(c*x^2+b*x+
a)/(-4*a*c+b^2))^(1/2)/c^3/(c*x^2+b*x+a)^(1/2)-1/30*(-4*a*c+b^2)^(11/4)*EllipticF((2*c*d*x+b*d)^(1/2)/(-4*a*c+
b^2)^(1/4)/d^(1/2),I)*d^(1/2)*(-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(1/2)/c^3/(c*x^2+b*x+a)^(1/2)

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {699, 705, 704, 313, 227, 1213, 435} \[ \int \sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^{3/2} \, dx=-\frac {\sqrt {d} \left (b^2-4 a c\right )^{11/4} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ),-1\right )}{30 c^3 \sqrt {a+b x+c x^2}}+\frac {\sqrt {d} \left (b^2-4 a c\right )^{11/4} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\arcsin \left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{30 c^3 \sqrt {a+b x+c x^2}}-\frac {\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} (b d+2 c d x)^{3/2}}{30 c^2 d}+\frac {\left (a+b x+c x^2\right )^{3/2} (b d+2 c d x)^{3/2}}{9 c d} \]

[In]

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

[Out]

-1/30*((b^2 - 4*a*c)*(b*d + 2*c*d*x)^(3/2)*Sqrt[a + b*x + c*x^2])/(c^2*d) + ((b*d + 2*c*d*x)^(3/2)*(a + b*x +
c*x^2)^(3/2))/(9*c*d) + ((b^2 - 4*a*c)^(11/4)*Sqrt[d]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticE[A
rcSin[Sqrt[b*d + 2*c*d*x]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])], -1])/(30*c^3*Sqrt[a + b*x + c*x^2]) - ((b^2 - 4*a*c)
^(11/4)*Sqrt[d]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticF[ArcSin[Sqrt[b*d + 2*c*d*x]/((b^2 - 4*a*
c)^(1/4)*Sqrt[d])], -1])/(30*c^3*Sqrt[a + b*x + c*x^2])

Rule 227

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[EllipticF[ArcSin[Rt[-b, 4]*(x/Rt[a, 4])], -1]/(Rt[a, 4]*Rt[
-b, 4]), x] /; FreeQ[{a, b}, x] && NegQ[b/a] && GtQ[a, 0]

Rule 313

Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[-b/a, 2]}, Dist[-q^(-1), Int[1/Sqrt[a + b*x^4]
, x], x] + Dist[1/q, Int[(1 + q*x^2)/Sqrt[a + b*x^4], x], x]] /; FreeQ[{a, b}, x] && NegQ[b/a]

Rule 435

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*Ell
ipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0
]

Rule 699

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*((
a + b*x + c*x^2)^p/(e*(m + 2*p + 1))), x] - Dist[d*p*((b^2 - 4*a*c)/(b*e*(m + 2*p + 1))), Int[(d + e*x)^m*(a +
b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[2*c*d - b*e, 0] &&
NeQ[m + 2*p + 3, 0] && GtQ[p, 0] &&  !LtQ[m, -1] &&  !(IGtQ[(m - 1)/2, 0] && ( !IntegerQ[p] || LtQ[m, 2*p]))
&& RationalQ[m] && IntegerQ[2*p]

Rule 704

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

Rule 705

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

Rule 1213

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Dist[d/Sqrt[a], Int[Sqrt[1 + e*(x^2/d)]/Sqrt
[1 - e*(x^2/d)], x], x] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] && EqQ[c*d^2 + a*e^2, 0] && GtQ[a, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {(b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^{3/2}}{9 c d}-\frac {\left (b^2-4 a c\right ) \int \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2} \, dx}{6 c} \\ & = -\frac {\left (b^2-4 a c\right ) (b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}}{30 c^2 d}+\frac {(b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^{3/2}}{9 c d}+\frac {\left (b^2-4 a c\right )^2 \int \frac {\sqrt {b d+2 c d x}}{\sqrt {a+b x+c x^2}} \, dx}{60 c^2} \\ & = -\frac {\left (b^2-4 a c\right ) (b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}}{30 c^2 d}+\frac {(b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^{3/2}}{9 c d}+\frac {\left (\left (b^2-4 a c\right )^2 \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \int \frac {\sqrt {b d+2 c d x}}{\sqrt {-\frac {a c}{b^2-4 a c}-\frac {b c x}{b^2-4 a c}-\frac {c^2 x^2}{b^2-4 a c}}} \, dx}{60 c^2 \sqrt {a+b x+c x^2}} \\ & = -\frac {\left (b^2-4 a c\right ) (b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}}{30 c^2 d}+\frac {(b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^{3/2}}{9 c d}+\frac {\left (\left (b^2-4 a c\right )^2 \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {1-\frac {x^4}{\left (b^2-4 a c\right ) d^2}}} \, dx,x,\sqrt {b d+2 c d x}\right )}{30 c^3 d \sqrt {a+b x+c x^2}} \\ & = -\frac {\left (b^2-4 a c\right ) (b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}}{30 c^2 d}+\frac {(b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^{3/2}}{9 c d}-\frac {\left (\left (b^2-4 a c\right )^{5/2} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^4}{\left (b^2-4 a c\right ) d^2}}} \, dx,x,\sqrt {b d+2 c d x}\right )}{30 c^3 \sqrt {a+b x+c x^2}}+\frac {\left (\left (b^2-4 a c\right )^{5/2} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \text {Subst}\left (\int \frac {1+\frac {x^2}{\sqrt {b^2-4 a c} d}}{\sqrt {1-\frac {x^4}{\left (b^2-4 a c\right ) d^2}}} \, dx,x,\sqrt {b d+2 c d x}\right )}{30 c^3 \sqrt {a+b x+c x^2}} \\ & = -\frac {\left (b^2-4 a c\right ) (b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}}{30 c^2 d}+\frac {(b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^{3/2}}{9 c d}-\frac {\left (b^2-4 a c\right )^{11/4} \sqrt {d} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{30 c^3 \sqrt {a+b x+c x^2}}+\frac {\left (\left (b^2-4 a c\right )^{5/2} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {x^2}{\sqrt {b^2-4 a c} d}}}{\sqrt {1-\frac {x^2}{\sqrt {b^2-4 a c} d}}} \, dx,x,\sqrt {b d+2 c d x}\right )}{30 c^3 \sqrt {a+b x+c x^2}} \\ & = -\frac {\left (b^2-4 a c\right ) (b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}}{30 c^2 d}+\frac {(b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^{3/2}}{9 c d}+\frac {\left (b^2-4 a c\right )^{11/4} \sqrt {d} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{30 c^3 \sqrt {a+b x+c x^2}}-\frac {\left (b^2-4 a c\right )^{11/4} \sqrt {d} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{30 c^3 \sqrt {a+b x+c x^2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 10.05 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.35 \[ \int \sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^{3/2} \, dx=-\frac {\left (b^2-4 a c\right ) (d (b+2 c x))^{3/2} \sqrt {a+x (b+c x)} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {3}{4},\frac {7}{4},\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{24 c^2 d \sqrt {\frac {c (a+x (b+c x))}{-b^2+4 a c}}} \]

[In]

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

[Out]

-1/24*((b^2 - 4*a*c)*(d*(b + 2*c*x))^(3/2)*Sqrt[a + x*(b + c*x)]*Hypergeometric2F1[-3/2, 3/4, 7/4, (b + 2*c*x)
^2/(b^2 - 4*a*c)])/(c^2*d*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(699\) vs. \(2(237)=474\).

Time = 3.33 (sec) , antiderivative size = 700, normalized size of antiderivative = 2.49

method	result	size
default	\(\frac {\sqrt {d \left (2 c x +b \right )}\, \sqrt {c \,x^{2}+b x +a}\, \left (80 c^{6} x^{6}+240 b \,c^{5} x^{5}+192 \sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-b -2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, E\left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {2}\right ) a^{3} c^{3}-144 \sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-b -2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, E\left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {2}\right ) a^{2} b^{2} c^{2}+36 \sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-b -2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, E\left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {2}\right ) a \,b^{4} c -3 \sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-b -2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, E\left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {2}\right ) b^{6}+256 a \,c^{5} x^{4}+236 b^{2} c^{4} x^{4}+512 a b \,c^{4} x^{3}+72 x^{3} b^{3} c^{3}+176 a^{2} c^{4} x^{2}+296 a \,b^{2} c^{3} x^{2}-10 x^{2} b^{4} c^{2}+176 a^{2} b \,c^{3} x +40 x a \,b^{3} c^{2}-6 x \,b^{5} c +44 a^{2} b^{2} c^{2}-6 a \,b^{4} c \right )}{180 c^{3} \left (2 c^{2} x^{3}+3 c b \,x^{2}+2 a c x +b^{2} x +a b \right )}\)	\(700\)
elliptic	\(\text {Expression too large to display}\)	\(1516\)
risch	\(\text {Expression too large to display}\)	\(1782\)

[In]

int((2*c*d*x+b*d)^(1/2)*(c*x^2+b*x+a)^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/180*(d*(2*c*x+b))^(1/2)*(c*x^2+b*x+a)^(1/2)*(80*c^6*x^6+240*b*c^5*x^5+192*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*
a*c+b^2)^(1/2))^(1/2)*(-(2*c*x+b)/(-4*a*c+b^2)^(1/2))^(1/2)*((-b-2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))
^(1/2)*EllipticE(1/2*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),2^(1/2))*a^3*c^3-144*((b+
2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*(-(2*c*x+b)/(-4*a*c+b^2)^(1/2))^(1/2)*((-b-2*c*x+(-4*a*c+b
^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*EllipticE(1/2*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*2^(
1/2),2^(1/2))*a^2*b^2*c^2+36*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*(-(2*c*x+b)/(-4*a*c+b^2)^
(1/2))^(1/2)*((-b-2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*EllipticE(1/2*((b+2*c*x+(-4*a*c+b^2)^(1/
2))/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),2^(1/2))*a*b^4*c-3*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/
2)*(-(2*c*x+b)/(-4*a*c+b^2)^(1/2))^(1/2)*((-b-2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*EllipticE(1/
2*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),2^(1/2))*b^6+256*a*c^5*x^4+236*b^2*c^4*x^4+5
12*a*b*c^4*x^3+72*x^3*b^3*c^3+176*a^2*c^4*x^2+296*a*b^2*c^3*x^2-10*x^2*b^4*c^2+176*a^2*b*c^3*x+40*x*a*b^3*c^2-
6*x*b^5*c+44*a^2*b^2*c^2-6*a*b^4*c)/c^3/(2*c^2*x^3+3*b*c*x^2+2*a*c*x+b^2*x+a*b)

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.11 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.53 \[ \int \sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^{3/2} \, dx=-\frac {3 \, \sqrt {2} {\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt {c^{2} d} {\rm weierstrassZeta}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, {\rm weierstrassPInverse}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, \frac {2 \, c x + b}{2 \, c}\right )\right ) - {\left (20 \, c^{4} x^{3} + 30 \, b c^{3} x^{2} - 3 \, b^{3} c + 22 \, a b c^{2} + 4 \, {\left (b^{2} c^{2} + 11 \, a c^{3}\right )} x\right )} \sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a}}{90 \, c^{3}} \]

[In]

integrate((2*c*d*x+b*d)^(1/2)*(c*x^2+b*x+a)^(3/2),x, algorithm="fricas")

[Out]

-1/90*(3*sqrt(2)*(b^4 - 8*a*b^2*c + 16*a^2*c^2)*sqrt(c^2*d)*weierstrassZeta((b^2 - 4*a*c)/c^2, 0, weierstrassP
Inverse((b^2 - 4*a*c)/c^2, 0, 1/2*(2*c*x + b)/c)) - (20*c^4*x^3 + 30*b*c^3*x^2 - 3*b^3*c + 22*a*b*c^2 + 4*(b^2
*c^2 + 11*a*c^3)*x)*sqrt(2*c*d*x + b*d)*sqrt(c*x^2 + b*x + a))/c^3

Sympy [F]

\[ \int \sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^{3/2} \, dx=\int \sqrt {d \left (b + 2 c x\right )} \left (a + b x + c x^{2}\right )^{\frac {3}{2}}\, dx \]

[In]

integrate((2*c*d*x+b*d)**(1/2)*(c*x**2+b*x+a)**(3/2),x)

[Out]

Integral(sqrt(d*(b + 2*c*x))*(a + b*x + c*x**2)**(3/2), x)

Maxima [F]

\[ \int \sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^{3/2} \, dx=\int { \sqrt {2 \, c d x + b d} {\left (c x^{2} + b x + a\right )}^{\frac {3}{2}} \,d x } \]

[In]

integrate((2*c*d*x+b*d)^(1/2)*(c*x^2+b*x+a)^(3/2),x, algorithm="maxima")

[Out]

integrate(sqrt(2*c*d*x + b*d)*(c*x^2 + b*x + a)^(3/2), x)

Giac [F]

\[ \int \sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^{3/2} \, dx=\int { \sqrt {2 \, c d x + b d} {\left (c x^{2} + b x + a\right )}^{\frac {3}{2}} \,d x } \]

[In]

integrate((2*c*d*x+b*d)^(1/2)*(c*x^2+b*x+a)^(3/2),x, algorithm="giac")

[Out]

integrate(sqrt(2*c*d*x + b*d)*(c*x^2 + b*x + a)^(3/2), x)

Mupad [F(-1)]

Timed out. \[ \int \sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^{3/2} \, dx=\int \sqrt {b\,d+2\,c\,d\,x}\,{\left (c\,x^2+b\,x+a\right )}^{3/2} \,d x \]

[In]

int((b*d + 2*c*d*x)^(1/2)*(a + b*x + c*x^2)^(3/2),x)

[Out]

int((b*d + 2*c*d*x)^(1/2)*(a + b*x + c*x^2)^(3/2), x)