\(\int \frac {(d x)^{3/2}}{(a+b x+c x^2)^{3/2}} \, dx\) [334]

Optimal result

Integrand size = 22, antiderivative size = 462 \[ \int \frac {(d x)^{3/2}}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {2 d \sqrt {d x} (2 a+b x)}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}-\frac {b \sqrt {-b+\sqrt {b^2-4 a c}} \left (b+\sqrt {b^2-4 a c}\right ) d^{3/2} \sqrt {1+\frac {2 c x}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x}{b+\sqrt {b^2-4 a c}}} E\left (\arcsin \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d x}}{\sqrt {-b+\sqrt {b^2-4 a c}} \sqrt {d}}\right )|\frac {b-\sqrt {b^2-4 a c}}{b+\sqrt {b^2-4 a c}}\right )}{\sqrt {2} c^{3/2} \left (b^2-4 a c\right ) \sqrt {a+x (b+c x)}}+\frac {\sqrt {-b+\sqrt {b^2-4 a c}} \left (b^2-4 a c+b \sqrt {b^2-4 a c}\right ) d^{3/2} \sqrt {1+\frac {2 c x}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x}{b+\sqrt {b^2-4 a c}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d x}}{\sqrt {-b+\sqrt {b^2-4 a c}} \sqrt {d}}\right ),\frac {b-\sqrt {b^2-4 a c}}{b+\sqrt {b^2-4 a c}}\right )}{\sqrt {2} c^{3/2} \left (b^2-4 a c\right ) \sqrt {a+x (b+c x)}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 12.22 (sec) , antiderivative size = 458, normalized size of antiderivative = 0.99 \[ \int \frac {(d x)^{3/2}}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {d^2 \left (-4 \sqrt {\frac {a}{b+\sqrt {b^2-4 a c}}} \left (b^2 x+a (b-2 c x)\right )+i b \left (-b+\sqrt {b^2-4 a c}\right ) \sqrt {1+\frac {2 a}{\left (b+\sqrt {b^2-4 a c}\right ) x}} x^{3/2} \sqrt {\frac {4 a+2 b x-2 \sqrt {b^2-4 a c} x}{b x-\sqrt {b^2-4 a c} x}} E\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {a}{b+\sqrt {b^2-4 a c}}}}{\sqrt {x}}\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )-i \left (-b^2+4 a c+b \sqrt {b^2-4 a c}\right ) \sqrt {1+\frac {2 a}{\left (b+\sqrt {b^2-4 a c}\right ) x}} x^{3/2} \sqrt {\frac {4 a+2 b x-2 \sqrt {b^2-4 a c} x}{b x-\sqrt {b^2-4 a c} x}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {a}{b+\sqrt {b^2-4 a c}}}}{\sqrt {x}}\right ),\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )\right )}{2 c \left (b^2-4 a c\right ) \sqrt {\frac {a}{b+\sqrt {b^2-4 a c}}} \sqrt {d x} \sqrt {a+x (b+c x)}} \] Input:

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

Output:

(d^2*(-4*Sqrt[a/(b + Sqrt[b^2 - 4*a*c])]*(b^2*x + a*(b - 2*c*x)) + I*b*(-b 
 + Sqrt[b^2 - 4*a*c])*Sqrt[1 + (2*a)/((b + Sqrt[b^2 - 4*a*c])*x)]*x^(3/2)* 
Sqrt[(4*a + 2*b*x - 2*Sqrt[b^2 - 4*a*c]*x)/(b*x - Sqrt[b^2 - 4*a*c]*x)]*El 
lipticE[I*ArcSinh[(Sqrt[2]*Sqrt[a/(b + Sqrt[b^2 - 4*a*c])])/Sqrt[x]], (b + 
 Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])] - I*(-b^2 + 4*a*c + b*Sqrt[b^ 
2 - 4*a*c])*Sqrt[1 + (2*a)/((b + Sqrt[b^2 - 4*a*c])*x)]*x^(3/2)*Sqrt[(4*a 
+ 2*b*x - 2*Sqrt[b^2 - 4*a*c]*x)/(b*x - Sqrt[b^2 - 4*a*c]*x)]*EllipticF[I* 
ArcSinh[(Sqrt[2]*Sqrt[a/(b + Sqrt[b^2 - 4*a*c])])/Sqrt[x]], (b + Sqrt[b^2 
- 4*a*c])/(b - Sqrt[b^2 - 4*a*c])]))/(2*c*(b^2 - 4*a*c)*Sqrt[a/(b + Sqrt[b 
^2 - 4*a*c])]*Sqrt[d*x]*Sqrt[a + x*(b + c*x)])
 

Rubi [A] (verified)

Time = 0.84 (sec) , antiderivative size = 347, normalized size of antiderivative = 0.75, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {1164, 27, 1241, 1240, 1511, 27, 1416, 1509}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 1164

\(\displaystyle \frac {2 d \sqrt {d x} (2 a+b x)}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}-\frac {2 \int \frac {d^2 (2 a+b x)}{2 \sqrt {d x} \sqrt {c x^2+b x+a}}dx}{b^2-4 a c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 d \sqrt {d x} (2 a+b x)}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}-\frac {d^2 \int \frac {2 a+b x}{\sqrt {d x} \sqrt {c x^2+b x+a}}dx}{b^2-4 a c}\)

\(\Big \downarrow \) 1241

\(\displaystyle \frac {2 d \sqrt {d x} (2 a+b x)}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}-\frac {d^2 \sqrt {x} \int \frac {2 a+b x}{\sqrt {x} \sqrt {c x^2+b x+a}}dx}{\sqrt {d x} \left (b^2-4 a c\right )}\)

\(\Big \downarrow \) 1240

\(\displaystyle \frac {2 d \sqrt {d x} (2 a+b x)}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}-\frac {2 d^2 \sqrt {x} \int \frac {2 a+b x}{\sqrt {c x^2+b x+a}}d\sqrt {x}}{\sqrt {d x} \left (b^2-4 a c\right )}\)

\(\Big \downarrow \) 1511

\(\displaystyle \frac {2 d \sqrt {d x} (2 a+b x)}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}-\frac {2 d^2 \sqrt {x} \left (\sqrt {a} \left (2 \sqrt {a}+\frac {b}{\sqrt {c}}\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}d\sqrt {x}-\frac {\sqrt {a} b \int \frac {\sqrt {a}-\sqrt {c} x}{\sqrt {a} \sqrt {c x^2+b x+a}}d\sqrt {x}}{\sqrt {c}}\right )}{\sqrt {d x} \left (b^2-4 a c\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 d \sqrt {d x} (2 a+b x)}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}-\frac {2 d^2 \sqrt {x} \left (\sqrt {a} \left (2 \sqrt {a}+\frac {b}{\sqrt {c}}\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}d\sqrt {x}-\frac {b \int \frac {\sqrt {a}-\sqrt {c} x}{\sqrt {c x^2+b x+a}}d\sqrt {x}}{\sqrt {c}}\right )}{\sqrt {d x} \left (b^2-4 a c\right )}\)

\(\Big \downarrow \) 1416

\(\displaystyle \frac {2 d \sqrt {d x} (2 a+b x)}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}-\frac {2 d^2 \sqrt {x} \left (\frac {\sqrt [4]{a} \left (2 \sqrt {a}+\frac {b}{\sqrt {c}}\right ) \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+b x+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 \sqrt [4]{c} \sqrt {a+b x+c x^2}}-\frac {b \int \frac {\sqrt {a}-\sqrt {c} x}{\sqrt {c x^2+b x+a}}d\sqrt {x}}{\sqrt {c}}\right )}{\sqrt {d x} \left (b^2-4 a c\right )}\)

\(\Big \downarrow \) 1509

\(\displaystyle \frac {2 d \sqrt {d x} (2 a+b x)}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}-\frac {2 d^2 \sqrt {x} \left (\frac {\sqrt [4]{a} \left (2 \sqrt {a}+\frac {b}{\sqrt {c}}\right ) \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+b x+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 \sqrt [4]{c} \sqrt {a+b x+c x^2}}-\frac {b \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+b x+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{\sqrt [4]{c} \sqrt {a+b x+c x^2}}-\frac {\sqrt {x} \sqrt {a+b x+c x^2}}{\sqrt {a}+\sqrt {c} x}\right )}{\sqrt {c}}\right )}{\sqrt {d x} \left (b^2-4 a c\right )}\)

Input:

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

Output:

(2*d*Sqrt[d*x]*(2*a + b*x))/((b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2]) - (2*d^2 
*Sqrt[x]*(-((b*(-((Sqrt[x]*Sqrt[a + b*x + c*x^2])/(Sqrt[a] + Sqrt[c]*x)) + 
 (a^(1/4)*(Sqrt[a] + Sqrt[c]*x)*Sqrt[(a + b*x + c*x^2)/(Sqrt[a] + Sqrt[c]* 
x)^2]*EllipticE[2*ArcTan[(c^(1/4)*Sqrt[x])/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[ 
c]))/4])/(c^(1/4)*Sqrt[a + b*x + c*x^2])))/Sqrt[c]) + (a^(1/4)*(2*Sqrt[a] 
+ b/Sqrt[c])*(Sqrt[a] + Sqrt[c]*x)*Sqrt[(a + b*x + c*x^2)/(Sqrt[a] + Sqrt[ 
c]*x)^2]*EllipticF[2*ArcTan[(c^(1/4)*Sqrt[x])/a^(1/4)], (2 - b/(Sqrt[a]*Sq 
rt[c]))/4])/(2*c^(1/4)*Sqrt[a + b*x + c*x^2])))/((b^2 - 4*a*c)*Sqrt[d*x])
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 1164

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[(d + e*x)^(m - 1)*(d*b - 2*a*e + (2*c*d - b*e)*x)*((a + b*x 
+ c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] + Simp[1/((p + 1)*(b^2 - 4*a* 
c))   Int[(d + e*x)^(m - 2)*Simp[e*(2*a*e*(m - 1) + b*d*(2*p - m + 4)) - 2* 
c*d^2*(2*p + 3) + e*(b*e - 2*d*c)*(m + 2*p + 2)*x, x]*(a + b*x + c*x^2)^(p 
+ 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && LtQ[p, -1] && GtQ[m, 1] && Int 
QuadraticQ[a, b, c, d, e, m, p, x]
 

rule 1240

Int[((f_) + (g_.)*(x_))/(Sqrt[x_]*Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2]), 
x_Symbol] :> Simp[2   Subst[Int[(f + g*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x, 
 Sqrt[x]], x] /; FreeQ[{a, b, c, f, g}, x]
 

rule 1241

Int[((f_) + (g_.)*(x_))/(Sqrt[(e_)*(x_)]*Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_ 
)^2]), x_Symbol] :> Simp[Sqrt[x]/Sqrt[e*x]   Int[(f + g*x)/(Sqrt[x]*Sqrt[a 
+ b*x + c*x^2]), x], x] /; FreeQ[{a, b, c, e, f, g}, x]
 

rule 1416

Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c 
/a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/ 
(2*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticF[2*ArcTan[q*x], 1/2 - b*(q^2/(4*c)) 
], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
 

rule 1509

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo 
l] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + b*x^2 + c*x^4]/(a*(1 + q 
^2*x^2))), x] + Simp[d*(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2* 
x^2)^2)]/(q*Sqrt[a + b*x^2 + c*x^4]))*EllipticE[2*ArcTan[q*x], 1/2 - b*(q^2 
/(4*c))], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 
- 4*a*c, 0] && PosQ[c/a]
 

rule 1511

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo 
l] :> With[{q = Rt[c/a, 2]}, Simp[(e + d*q)/q   Int[1/Sqrt[a + b*x^2 + c*x^ 
4], x], x] - Simp[e/q   Int[(1 - q*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x] /; 
NeQ[e + d*q, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && Pos 
Q[c/a]
 

Maple [A] (verified)

Time = 1.44 (sec) , antiderivative size = 686, normalized size of antiderivative = 1.48

method	result	size
default	\(\frac {d \sqrt {d x}\, \left (2 \sqrt {-4 a c +b^{2}}\, \sqrt {\frac {2 c x +\sqrt {-4 a c +b^{2}}+b}{b +\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-2 c x +\sqrt {-4 a c +b^{2}}-b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {c x}{b +\sqrt {-4 a c +b^{2}}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {2 c x +\sqrt {-4 a c +b^{2}}+b}{b +\sqrt {-4 a c +b^{2}}}}, \frac {\sqrt {2}\, \sqrt {\frac {b +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}}{2}\right ) a c +4 \sqrt {\frac {2 c x +\sqrt {-4 a c +b^{2}}+b}{b +\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-2 c x +\sqrt {-4 a c +b^{2}}-b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {c x}{b +\sqrt {-4 a c +b^{2}}}}\, \operatorname {EllipticE}\left (\sqrt {\frac {2 c x +\sqrt {-4 a c +b^{2}}+b}{b +\sqrt {-4 a c +b^{2}}}}, \frac {\sqrt {2}\, \sqrt {\frac {b +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}}{2}\right ) a b c -\sqrt {\frac {2 c x +\sqrt {-4 a c +b^{2}}+b}{b +\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-2 c x +\sqrt {-4 a c +b^{2}}-b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {c x}{b +\sqrt {-4 a c +b^{2}}}}\, \operatorname {EllipticE}\left (\sqrt {\frac {2 c x +\sqrt {-4 a c +b^{2}}+b}{b +\sqrt {-4 a c +b^{2}}}}, \frac {\sqrt {2}\, \sqrt {\frac {b +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}}{2}\right ) b^{3}-\sqrt {-4 a c +b^{2}}\, \sqrt {\frac {2 c x +\sqrt {-4 a c +b^{2}}+b}{b +\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-2 c x +\sqrt {-4 a c +b^{2}}-b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {c x}{b +\sqrt {-4 a c +b^{2}}}}\, \operatorname {EllipticE}\left (\sqrt {\frac {2 c x +\sqrt {-4 a c +b^{2}}+b}{b +\sqrt {-4 a c +b^{2}}}}, \frac {\sqrt {2}\, \sqrt {\frac {b +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}}{2}\right ) b^{2}-2 b \,c^{2} x^{2}-4 a \,c^{2} x \right )}{x \sqrt {c \,x^{2}+b x +a}\, c^{2} \left (4 a c -b^{2}\right )}\)	\(686\)
elliptic	\(\frac {\sqrt {d x}\, \sqrt {d x \left (c \,x^{2}+b x +a \right )}\, \left (-\frac {2 c d x \left (\frac {b d x}{c \left (4 a c -b^{2}\right )}+\frac {2 a d}{c \left (4 a c -b^{2}\right )}\right )}{\sqrt {\left (\frac {a}{c}+\frac {b x}{c}+x^{2}\right ) c d x}}+\frac {2 d^{2} a \left (b +\sqrt {-4 a c +b^{2}}\right ) \sqrt {2}\, \sqrt {\frac {\left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) c}{b +\sqrt {-4 a c +b^{2}}}}\, \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 {2 c x}{b +\sqrt {-4 a c +b^{2}}}}\, \operatorname {EllipticF}\left (\sqrt {2}\, \sqrt {\frac {\left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) c}{b +\sqrt {-4 a c +b^{2}}}}, \frac {\sqrt {-\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right )}{c \left (-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}}{2}\right )}{\left (4 a c -b^{2}\right ) c \sqrt {c d \,x^{3}+b d \,x^{2}+a d x}}+\frac {b \,d^{2} \left (b +\sqrt {-4 a c +b^{2}}\right ) \sqrt {2}\, \sqrt {\frac {\left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) c}{b +\sqrt {-4 a c +b^{2}}}}\, \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 {2 c x}{b +\sqrt {-4 a c +b^{2}}}}\, \left (\left (-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \operatorname {EllipticE}\left (\sqrt {2}\, \sqrt {\frac {\left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) c}{b +\sqrt {-4 a c +b^{2}}}}, \frac {\sqrt {-\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right )}{c \left (-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}}{2}\right )+\frac {\left (-b +\sqrt {-4 a c +b^{2}}\right ) \operatorname {EllipticF}\left (\sqrt {2}\, \sqrt {\frac {\left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) c}{b +\sqrt {-4 a c +b^{2}}}}, \frac {\sqrt {-\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right )}{c \left (-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}}{2}\right )}{2 c}\right )}{\left (4 a c -b^{2}\right ) c \sqrt {c d \,x^{3}+b d \,x^{2}+a d x}}\right )}{d x \sqrt {c \,x^{2}+b x +a}}\)	\(819\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 282, normalized size of antiderivative = 0.61 \[ \int \frac {(d x)^{3/2}}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {2 \, {\left ({\left ({\left (b^{2} c - 6 \, a c^{2}\right )} d x^{2} + {\left (b^{3} - 6 \, a b c\right )} d x + {\left (a b^{2} - 6 \, a^{2} c\right )} d\right )} \sqrt {c d} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} - 3 \, a c\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, b^{3} - 9 \, a b c\right )}}{27 \, c^{3}}, \frac {3 \, c x + b}{3 \, c}\right ) + 3 \, {\left (b c^{2} d x^{2} + b^{2} c d x + a b c d\right )} \sqrt {c d} {\rm weierstrassZeta}\left (\frac {4 \, {\left (b^{2} - 3 \, a c\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, b^{3} - 9 \, a b c\right )}}{27 \, c^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} - 3 \, a c\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, b^{3} - 9 \, a b c\right )}}{27 \, c^{3}}, \frac {3 \, c x + b}{3 \, c}\right )\right ) + 3 \, {\left (b c^{2} d x + 2 \, a c^{2} d\right )} \sqrt {c x^{2} + b x + a} \sqrt {d x}\right )}}{3 \, {\left (a b^{2} c^{2} - 4 \, a^{2} c^{3} + {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} x^{2} + {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} x\right )}} \] Input:

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

Output:

2/3*(((b^2*c - 6*a*c^2)*d*x^2 + (b^3 - 6*a*b*c)*d*x + (a*b^2 - 6*a^2*c)*d) 
*sqrt(c*d)*weierstrassPInverse(4/3*(b^2 - 3*a*c)/c^2, -4/27*(2*b^3 - 9*a*b 
*c)/c^3, 1/3*(3*c*x + b)/c) + 3*(b*c^2*d*x^2 + b^2*c*d*x + a*b*c*d)*sqrt(c 
*d)*weierstrassZeta(4/3*(b^2 - 3*a*c)/c^2, -4/27*(2*b^3 - 9*a*b*c)/c^3, we 
ierstrassPInverse(4/3*(b^2 - 3*a*c)/c^2, -4/27*(2*b^3 - 9*a*b*c)/c^3, 1/3* 
(3*c*x + b)/c)) + 3*(b*c^2*d*x + 2*a*c^2*d)*sqrt(c*x^2 + b*x + a)*sqrt(d*x 
))/(a*b^2*c^2 - 4*a^2*c^3 + (b^2*c^3 - 4*a*c^4)*x^2 + (b^3*c^2 - 4*a*b*c^3 
)*x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {(d x)^{3/2}}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {\sqrt {d}\, d \left (-10 \sqrt {x}\, \sqrt {c \,x^{2}+b x +a}\, a +8 \sqrt {x}\, \sqrt {c \,x^{2}+b x +a}\, b x -2 \sqrt {x}\, \sqrt {c \,x^{2}+b x +a}\, c \,x^{2}+3 \left (\int \frac {\sqrt {x}\, \sqrt {c \,x^{2}+b x +a}\, x^{3}}{c^{2} x^{4}+2 b c \,x^{3}+2 a c \,x^{2}+b^{2} x^{2}+2 a b x +a^{2}}d x \right ) a \,c^{2}+3 \left (\int \frac {\sqrt {x}\, \sqrt {c \,x^{2}+b x +a}\, x^{3}}{c^{2} x^{4}+2 b c \,x^{3}+2 a c \,x^{2}+b^{2} x^{2}+2 a b x +a^{2}}d x \right ) b \,c^{2} x +3 \left (\int \frac {\sqrt {x}\, \sqrt {c \,x^{2}+b x +a}\, x^{3}}{c^{2} x^{4}+2 b c \,x^{3}+2 a c \,x^{2}+b^{2} x^{2}+2 a b x +a^{2}}d x \right ) c^{3} x^{2}-12 \left (\int \frac {\sqrt {x}\, \sqrt {c \,x^{2}+b x +a}}{c^{2} x^{4}+2 b c \,x^{3}+2 a c \,x^{2}+b^{2} x^{2}+2 a b x +a^{2}}d x \right ) a^{2} b -12 \left (\int \frac {\sqrt {x}\, \sqrt {c \,x^{2}+b x +a}}{c^{2} x^{4}+2 b c \,x^{3}+2 a c \,x^{2}+b^{2} x^{2}+2 a b x +a^{2}}d x \right ) a \,b^{2} x -12 \left (\int \frac {\sqrt {x}\, \sqrt {c \,x^{2}+b x +a}}{c^{2} x^{4}+2 b c \,x^{3}+2 a c \,x^{2}+b^{2} x^{2}+2 a b x +a^{2}}d x \right ) a b c \,x^{2}+5 \left (\int \frac {\sqrt {x}\, \sqrt {c \,x^{2}+b x +a}}{c^{2} x^{5}+2 b c \,x^{4}+2 a c \,x^{3}+b^{2} x^{3}+2 a b \,x^{2}+a^{2} x}d x \right ) a^{3}+5 \left (\int \frac {\sqrt {x}\, \sqrt {c \,x^{2}+b x +a}}{c^{2} x^{5}+2 b c \,x^{4}+2 a c \,x^{3}+b^{2} x^{3}+2 a b \,x^{2}+a^{2} x}d x \right ) a^{2} b x +5 \left (\int \frac {\sqrt {x}\, \sqrt {c \,x^{2}+b x +a}}{c^{2} x^{5}+2 b c \,x^{4}+2 a c \,x^{3}+b^{2} x^{3}+2 a b \,x^{2}+a^{2} x}d x \right ) a^{2} c \,x^{2}\right )}{8 b^{2} \left (c \,x^{2}+b x +a \right )} \] Input:

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

Output:

(sqrt(d)*d*( - 10*sqrt(x)*sqrt(a + b*x + c*x**2)*a + 8*sqrt(x)*sqrt(a + b* 
x + c*x**2)*b*x - 2*sqrt(x)*sqrt(a + b*x + c*x**2)*c*x**2 + 3*int((sqrt(x) 
*sqrt(a + b*x + c*x**2)*x**3)/(a**2 + 2*a*b*x + 2*a*c*x**2 + b**2*x**2 + 2 
*b*c*x**3 + c**2*x**4),x)*a*c**2 + 3*int((sqrt(x)*sqrt(a + b*x + c*x**2)*x 
**3)/(a**2 + 2*a*b*x + 2*a*c*x**2 + b**2*x**2 + 2*b*c*x**3 + c**2*x**4),x) 
*b*c**2*x + 3*int((sqrt(x)*sqrt(a + b*x + c*x**2)*x**3)/(a**2 + 2*a*b*x + 
2*a*c*x**2 + b**2*x**2 + 2*b*c*x**3 + c**2*x**4),x)*c**3*x**2 - 12*int((sq 
rt(x)*sqrt(a + b*x + c*x**2))/(a**2 + 2*a*b*x + 2*a*c*x**2 + b**2*x**2 + 2 
*b*c*x**3 + c**2*x**4),x)*a**2*b - 12*int((sqrt(x)*sqrt(a + b*x + c*x**2)) 
/(a**2 + 2*a*b*x + 2*a*c*x**2 + b**2*x**2 + 2*b*c*x**3 + c**2*x**4),x)*a*b 
**2*x - 12*int((sqrt(x)*sqrt(a + b*x + c*x**2))/(a**2 + 2*a*b*x + 2*a*c*x* 
*2 + b**2*x**2 + 2*b*c*x**3 + c**2*x**4),x)*a*b*c*x**2 + 5*int((sqrt(x)*sq 
rt(a + b*x + c*x**2))/(a**2*x + 2*a*b*x**2 + 2*a*c*x**3 + b**2*x**3 + 2*b* 
c*x**4 + c**2*x**5),x)*a**3 + 5*int((sqrt(x)*sqrt(a + b*x + c*x**2))/(a**2 
*x + 2*a*b*x**2 + 2*a*c*x**3 + b**2*x**3 + 2*b*c*x**4 + c**2*x**5),x)*a**2 
*b*x + 5*int((sqrt(x)*sqrt(a + b*x + c*x**2))/(a**2*x + 2*a*b*x**2 + 2*a*c 
*x**3 + b**2*x**3 + 2*b*c*x**4 + c**2*x**5),x)*a**2*c*x**2))/(8*b**2*(a + 
b*x + c*x**2))