\(\int (b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2} \, dx\) [1328]

Optimal result

Integrand size = 28, antiderivative size = 180 \[ \int (b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2} \, dx=-\frac {2 \left (b^2-4 a c\right ) d \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}}{21 c}+\frac {(b d+2 c d x)^{5/2} \sqrt {a+b x+c x^2}}{7 c d}-\frac {\left (b^2-4 a c\right )^{9/4} d^{3/2} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ),-1\right )}{21 c^2 \sqrt {a+b x+c x^2}} \]

[Out]

1/7*(2*c*d*x+b*d)^(5/2)*(c*x^2+b*x+a)^(1/2)/c/d-2/21*(-4*a*c+b^2)*d*(2*c*d*x+b*d)^(1/2)*(c*x^2+b*x+a)^(1/2)/c-
1/21*(-4*a*c+b^2)^(9/4)*d^(3/2)*EllipticF((2*c*d*x+b*d)^(1/2)/(-4*a*c+b^2)^(1/4)/d^(1/2),I)*(-c*(c*x^2+b*x+a)/
(-4*a*c+b^2))^(1/2)/c^2/(c*x^2+b*x+a)^(1/2)

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {699, 706, 705, 703, 227} \[ \int (b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2} \, dx=-\frac {d^{3/2} \left (b^2-4 a c\right )^{9/4} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ),-1\right )}{21 c^2 \sqrt {a+b x+c x^2}}-\frac {2 d \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \sqrt {b d+2 c d x}}{21 c}+\frac {\sqrt {a+b x+c x^2} (b d+2 c d x)^{5/2}}{7 c d} \]

[In]

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

[Out]

(-2*(b^2 - 4*a*c)*d*Sqrt[b*d + 2*c*d*x]*Sqrt[a + b*x + c*x^2])/(21*c) + ((b*d + 2*c*d*x)^(5/2)*Sqrt[a + b*x +
c*x^2])/(7*c*d) - ((b^2 - 4*a*c)^(9/4)*d^(3/2)*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticF[ArcSin[S
qrt[b*d + 2*c*d*x]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])], -1])/(21*c^2*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 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 703

Int[1/(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[1/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 706

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

Rubi steps \begin{align*} \text {integral}& = \frac {(b d+2 c d x)^{5/2} \sqrt {a+b x+c x^2}}{7 c d}-\frac {\left (b^2-4 a c\right ) \int \frac {(b d+2 c d x)^{3/2}}{\sqrt {a+b x+c x^2}} \, dx}{14 c} \\ & = -\frac {2 \left (b^2-4 a c\right ) d \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}}{21 c}+\frac {(b d+2 c d x)^{5/2} \sqrt {a+b x+c x^2}}{7 c d}-\frac {\left (\left (b^2-4 a c\right )^2 d^2\right ) \int \frac {1}{\sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}} \, dx}{42 c} \\ & = -\frac {2 \left (b^2-4 a c\right ) d \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}}{21 c}+\frac {(b d+2 c d x)^{5/2} \sqrt {a+b x+c x^2}}{7 c d}-\frac {\left (\left (b^2-4 a c\right )^2 d^2 \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \int \frac {1}{\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}{42 c \sqrt {a+b x+c x^2}} \\ & = -\frac {2 \left (b^2-4 a c\right ) d \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}}{21 c}+\frac {(b d+2 c d x)^{5/2} \sqrt {a+b x+c x^2}}{7 c d}-\frac {\left (\left (b^2-4 a c\right )^2 d \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 )}{21 c^2 \sqrt {a+b x+c x^2}} \\ & = -\frac {2 \left (b^2-4 a c\right ) d \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}}{21 c}+\frac {(b d+2 c d x)^{5/2} \sqrt {a+b x+c x^2}}{7 c d}-\frac {\left (b^2-4 a c\right )^{9/4} d^{3/2} \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 )}{21 c^2 \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 = 10.09 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.61 \[ \int (b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2} \, dx=\frac {1}{14} d \sqrt {d (b+2 c x)} \sqrt {a+x (b+c x)} \left (8 (a+x (b+c x))+\frac {\left (b^2-4 a c\right ) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{4},\frac {5}{4},\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{c \sqrt {\frac {c (a+x (b+c x))}{-b^2+4 a c}}}\right ) \]

[In]

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

[Out]

(d*Sqrt[d*(b + 2*c*x)]*Sqrt[a + x*(b + c*x)]*(8*(a + x*(b + c*x)) + ((b^2 - 4*a*c)*Hypergeometric2F1[-1/2, 1/4
, 5/4, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(c*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)])))/14

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(489\) vs. \(2(152)=304\).

Time = 2.74 (sec) , antiderivative size = 490, normalized size of antiderivative = 2.72

method	result	size
risch	\(\frac {\left (12 c^{2} x^{2}+12 b c x +8 a c +b^{2}\right ) \left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}\, d^{2}}{21 c \sqrt {d \left (2 c x +b \right )}}-\frac {\left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) \left (\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {\frac {x +\frac {b}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b}{2 c}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, F\left (\sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}, \sqrt {\frac {-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b}{2 c}}}\right ) d^{2} \sqrt {d \left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )}}{21 c \sqrt {2 c^{2} d \,x^{3}+3 b c d \,x^{2}+2 a d x c +b^{2} d x +a b d}\, \sqrt {d \left (2 c x +b \right )}\, \sqrt {c \,x^{2}+b x +a}}\)	\(490\)
default	\(-\frac {\sqrt {d \left (2 c x +b \right )}\, \sqrt {c \,x^{2}+b x +a}\, d \left (-48 c^{5} x^{5}+16 \sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-b -2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, F\left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {2}\right ) \sqrt {-4 a c +b^{2}}\, a^{2} c^{2}-8 \sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-b -2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, F\left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {2}\right ) \sqrt {-4 a c +b^{2}}\, a \,b^{2} c +\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-b -2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, F\left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {2}\right ) \sqrt {-4 a c +b^{2}}\, b^{4}-120 b \,x^{4} c^{4}-80 a \,c^{4} x^{3}-100 b^{2} c^{3} x^{3}-120 a b \,c^{3} x^{2}-30 x^{2} b^{3} c^{2}-32 a^{2} c^{3} x -44 a \,c^{2} b^{2} x -2 c x \,b^{4}-16 a^{2} b \,c^{2}-2 a \,b^{3} c \right )}{42 c^{2} \left (2 c^{2} x^{3}+3 c b \,x^{2}+2 a c x +b^{2} x +a b \right )}\)	\(564\)
elliptic	\(\text {Expression too large to display}\)	\(1268\)

[In]

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

[Out]

1/21/c*(12*c^2*x^2+12*b*c*x+8*a*c+b^2)*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)*d^2/(d*(2*c*x+b))^(1/2)-1/21/c*(16*a^2*c^
2-8*a*b^2*c+b^4)*(1/2/c*(-b+(-4*a*c+b^2)^(1/2))+1/2*(b+(-4*a*c+b^2)^(1/2))/c)*((x+1/2*(b+(-4*a*c+b^2)^(1/2))/c
)/(1/2/c*(-b+(-4*a*c+b^2)^(1/2))+1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2)*((x+1/2/c*b)/(-1/2*(b+(-4*a*c+b^2)^(1/2)
)/c+1/2/c*b))^(1/2)*((x-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))/(-1/2*(b+(-4*a*c+b^2)^(1/2))/c-1/2/c*(-b+(-4*a*c+b^2)^(
1/2))))^(1/2)/(2*c^2*d*x^3+3*b*c*d*x^2+2*a*c*d*x+b^2*d*x+a*b*d)^(1/2)*EllipticF(((x+1/2*(b+(-4*a*c+b^2)^(1/2))
/c)/(1/2/c*(-b+(-4*a*c+b^2)^(1/2))+1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2),((-1/2*(b+(-4*a*c+b^2)^(1/2))/c-1/2/c*
(-b+(-4*a*c+b^2)^(1/2)))/(-1/2*(b+(-4*a*c+b^2)^(1/2))/c+1/2/c*b))^(1/2))*d^2*(d*(2*c*x+b)*(c*x^2+b*x+a))^(1/2)
/(d*(2*c*x+b))^(1/2)/(c*x^2+b*x+a)^(1/2)

Fricas [C] (verification not implemented)

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

Time = 0.09 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.67 \[ \int (b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2} \, dx=-\frac {\sqrt {2} {\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt {c^{2} d} d {\rm weierstrassPInverse}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, \frac {2 \, c x + b}{2 \, c}\right ) - 2 \, {\left (12 \, c^{4} d x^{2} + 12 \, b c^{3} d x + {\left (b^{2} c^{2} + 8 \, a c^{3}\right )} d\right )} \sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a}}{42 \, c^{3}} \]

[In]

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

[Out]

-1/42*(sqrt(2)*(b^4 - 8*a*b^2*c + 16*a^2*c^2)*sqrt(c^2*d)*d*weierstrassPInverse((b^2 - 4*a*c)/c^2, 0, 1/2*(2*c
*x + b)/c) - 2*(12*c^4*d*x^2 + 12*b*c^3*d*x + (b^2*c^2 + 8*a*c^3)*d)*sqrt(2*c*d*x + b*d)*sqrt(c*x^2 + b*x + a)
)/c^3

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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