\(\int \frac {\sqrt {a x+b x^2+c x^3}}{\sqrt {d x}} \, dx\) [5]

Optimal result

Integrand size = 26, antiderivative size = 107 \[ \int \frac {\sqrt {a x+b x^2+c x^3}}{\sqrt {d x}} \, dx=\frac {(b+2 c x) \sqrt {a x+b x^2+c x^3}}{4 c \sqrt {d x}}-\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {\sqrt {d x} (b+2 c x)}{2 \sqrt {c} \sqrt {d} \sqrt {a x+b x^2+c x^3}}\right )}{8 c^{3/2} \sqrt {d}} \] Output:

1/4*(2*c*x+b)*(c*x^3+b*x^2+a*x)^(1/2)/c/(d*x)^(1/2)-1/8*(-4*a*c+b^2)*arcta 
nh(1/2*(d*x)^(1/2)*(2*c*x+b)/c^(1/2)/d^(1/2)/(c*x^3+b*x^2+a*x)^(1/2))/c^(3 
/2)/d^(1/2)
 

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.91 \[ \int \frac {\sqrt {a x+b x^2+c x^3}}{\sqrt {d x}} \, dx=\frac {\sqrt {x (a+x (b+c x))} \left (\sqrt {c} (b+2 c x)+\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a}-\sqrt {a+x (b+c x)}}\right )}{\sqrt {a+x (b+c x)}}\right )}{4 c^{3/2} \sqrt {d x}} \] Input:

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

Output:

(Sqrt[x*(a + x*(b + c*x))]*(Sqrt[c]*(b + 2*c*x) + ((b^2 - 4*a*c)*ArcTanh[( 
Sqrt[c]*x)/(Sqrt[a] - Sqrt[a + x*(b + c*x)])])/Sqrt[a + x*(b + c*x)]))/(4* 
c^(3/2)*Sqrt[d*x])
 

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.07, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {2467, 30, 1087, 1092, 219}

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.

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

\(\Big \downarrow \) 2467

\(\displaystyle \frac {\sqrt {a x+b x^2+c x^3} \int \frac {\sqrt {x} \sqrt {c x^2+b x+a}}{\sqrt {d x}}dx}{\sqrt {x} \sqrt {a+b x+c x^2}}\)

\(\Big \downarrow \) 30

\(\displaystyle \frac {\sqrt {a x+b x^2+c x^3} \int \sqrt {c x^2+b x+a}dx}{\sqrt {d x} \sqrt {a+b x+c x^2}}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {\sqrt {a x+b x^2+c x^3} \left (\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{4 c}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{8 c}\right )}{\sqrt {d x} \sqrt {a+b x+c x^2}}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {\sqrt {a x+b x^2+c x^3} \left (\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{4 c}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{4 c}\right )}{\sqrt {d x} \sqrt {a+b x+c x^2}}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\sqrt {a x+b x^2+c x^3} \left (\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{4 c}-\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 c^{3/2}}\right )}{\sqrt {d x} \sqrt {a+b x+c x^2}}\)

Input:

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

Output:

(Sqrt[a*x + b*x^2 + c*x^3]*(((b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(4*c) - (( 
b^2 - 4*a*c)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(8*c^ 
(3/2))))/(Sqrt[d*x]*Sqrt[a + b*x + c*x^2])
 

Defintions of rubi rules used

rule 30

Int[(u_.)*((a_.)*(x_))^(m_.)*((b_.)*(x_)^(i_.))^(p_), x_Symbol] :> Simp[b^I 
ntPart[p]*((b*x^i)^FracPart[p]/(a^(i*IntPart[p])*(a*x)^(i*FracPart[p]))) 
Int[u*(a*x)^(m + i*p), x], x] /; FreeQ[{a, b, i, m, p}, x] && IntegerQ[i] & 
&  !IntegerQ[p]
 

rule 219

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] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 1087

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] - Simp[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] && 
GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[3*p])
 

rule 1092

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

rule 2467

Int[(Fx_.)*(Px_)^(p_), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[Px^F 
racPart[p]/(x^(r*FracPart[p])*ExpandToSum[Px/x^r, x]^FracPart[p])   Int[x^( 
p*r)*ExpandToSum[Px/x^r, x]^p*Fx, x], x] /; IGtQ[r, 0]] /; FreeQ[p, x] && P 
olyQ[Px, x] &&  !IntegerQ[p] &&  !MonomialQ[Px, x] &&  !PolyQ[Fx, x]
 

Maple [A] (verified)

Time = 0.11 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.21

method	result	size
risch	\(\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right ) x}{4 c \sqrt {d x}\, \sqrt {x \left (c \,x^{2}+b x +a \right )}}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {1}{2} b d +x c d}{\sqrt {c d}}+\sqrt {c d \,x^{2}+b d x +a d}\right ) \sqrt {d \left (c \,x^{2}+b x +a \right )}\, x}{8 c \sqrt {c d}\, \sqrt {d x}\, \sqrt {x \left (c \,x^{2}+b x +a \right )}}\)	\(130\)
default	\(\frac {\sqrt {x \left (c \,x^{2}+b x +a \right )}\, \left (4 \ln \left (\frac {2 x c d +2 \sqrt {d \left (c \,x^{2}+b x +a \right )}\, \sqrt {c d}+b d}{2 \sqrt {c d}}\right ) a c d -\ln \left (\frac {2 x c d +2 \sqrt {d \left (c \,x^{2}+b x +a \right )}\, \sqrt {c d}+b d}{2 \sqrt {c d}}\right ) b^{2} d +4 \sqrt {c d}\, \sqrt {d \left (c \,x^{2}+b x +a \right )}\, c x +2 \sqrt {c d}\, \sqrt {d \left (c \,x^{2}+b x +a \right )}\, b \right )}{8 \sqrt {d x}\, \sqrt {d \left (c \,x^{2}+b x +a \right )}\, c \sqrt {c d}}\)	\(177\)

Input:

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

Output:

1/4*(2*c*x+b)*(c*x^2+b*x+a)/c/(d*x)^(1/2)/(x*(c*x^2+b*x+a))^(1/2)*x+1/8*(4 
*a*c-b^2)/c*ln((1/2*b*d+x*c*d)/(c*d)^(1/2)+(c*d*x^2+b*d*x+a*d)^(1/2))/(c*d 
)^(1/2)*(d*(c*x^2+b*x+a))^(1/2)/(d*x)^(1/2)/(x*(c*x^2+b*x+a))^(1/2)*x
 

Fricas [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 249, normalized size of antiderivative = 2.33 \[ \int \frac {\sqrt {a x+b x^2+c x^3}}{\sqrt {d x}} \, dx=\left [-\frac {{\left (b^{2} - 4 \, a c\right )} \sqrt {c d} x \log \left (\frac {8 \, c^{2} d x^{3} + 8 \, b c d x^{2} + {\left (b^{2} + 4 \, a c\right )} d x + 4 \, \sqrt {c x^{3} + b x^{2} + a x} \sqrt {c d} {\left (2 \, c x + b\right )} \sqrt {d x}}{x}\right ) - 4 \, \sqrt {c x^{3} + b x^{2} + a x} {\left (2 \, c^{2} x + b c\right )} \sqrt {d x}}{16 \, c^{2} d x}, \frac {{\left (b^{2} - 4 \, a c\right )} \sqrt {-c d} x \arctan \left (\frac {\sqrt {c x^{3} + b x^{2} + a x} \sqrt {-c d} {\left (2 \, c x + b\right )} \sqrt {d x}}{2 \, {\left (c^{2} d x^{3} + b c d x^{2} + a c d x\right )}}\right ) + 2 \, \sqrt {c x^{3} + b x^{2} + a x} {\left (2 \, c^{2} x + b c\right )} \sqrt {d x}}{8 \, c^{2} d x}\right ] \] Input:

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

Output:

[-1/16*((b^2 - 4*a*c)*sqrt(c*d)*x*log((8*c^2*d*x^3 + 8*b*c*d*x^2 + (b^2 + 
4*a*c)*d*x + 4*sqrt(c*x^3 + b*x^2 + a*x)*sqrt(c*d)*(2*c*x + b)*sqrt(d*x))/ 
x) - 4*sqrt(c*x^3 + b*x^2 + a*x)*(2*c^2*x + b*c)*sqrt(d*x))/(c^2*d*x), 1/8 
*((b^2 - 4*a*c)*sqrt(-c*d)*x*arctan(1/2*sqrt(c*x^3 + b*x^2 + a*x)*sqrt(-c* 
d)*(2*c*x + b)*sqrt(d*x)/(c^2*d*x^3 + b*c*d*x^2 + a*c*d*x)) + 2*sqrt(c*x^3 
 + b*x^2 + a*x)*(2*c^2*x + b*c)*sqrt(d*x))/(c^2*d*x)]
 

Sympy [F]

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

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

Output:

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

Maxima [F]

\[ \int \frac {\sqrt {a x+b x^2+c x^3}}{\sqrt {d x}} \, dx=\int { \frac {\sqrt {c x^{3} + b x^{2} + a x}}{\sqrt {d x}} \,d x } \] Input:

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 205 vs. \(2 (85) = 170\).

Time = 0.18 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.92 \[ \int \frac {\sqrt {a x+b x^2+c x^3}}{\sqrt {d x}} \, dx=\frac {{\left (2 \, \sqrt {c d^{3} x^{2} + b d^{3} x + a d^{3}} {\left (2 \, d x + \frac {b d}{c}\right )} + \frac {{\left (b^{2} d^{3} - 4 \, a c d^{3}\right )} \log \left ({\left | -b d^{2} - 2 \, {\left (\sqrt {c d} d x - \sqrt {c d^{3} x^{2} + b d^{3} x + a d^{3}}\right )} \sqrt {c d} \right |}\right )}{\sqrt {c d} c} - \frac {b^{2} d^{3} \log \left ({\left | -b d^{2} + 2 \, \sqrt {a d^{3}} \sqrt {c d} \right |}\right ) - 4 \, a c d^{3} \log \left ({\left | -b d^{2} + 2 \, \sqrt {a d^{3}} \sqrt {c d} \right |}\right ) + 2 \, \sqrt {a d^{3}} \sqrt {c d} b d}{\sqrt {c d} c}\right )} {\left | d \right |}^{2}}{8 \, d^{5}} \] Input:

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

Output:

1/8*(2*sqrt(c*d^3*x^2 + b*d^3*x + a*d^3)*(2*d*x + b*d/c) + (b^2*d^3 - 4*a* 
c*d^3)*log(abs(-b*d^2 - 2*(sqrt(c*d)*d*x - sqrt(c*d^3*x^2 + b*d^3*x + a*d^ 
3))*sqrt(c*d)))/(sqrt(c*d)*c) - (b^2*d^3*log(abs(-b*d^2 + 2*sqrt(a*d^3)*sq 
rt(c*d))) - 4*a*c*d^3*log(abs(-b*d^2 + 2*sqrt(a*d^3)*sqrt(c*d))) + 2*sqrt( 
a*d^3)*sqrt(c*d)*b*d)/(sqrt(c*d)*c))*abs(d)^2/d^5
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.20 \[ \int \frac {\sqrt {a x+b x^2+c x^3}}{\sqrt {d x}} \, dx=\frac {\sqrt {d}\, \left (2 \sqrt {c \,x^{2}+b x +a}\, b c +4 \sqrt {c \,x^{2}+b x +a}\, c^{2} x +4 \sqrt {c}\, \mathrm {log}\left (\frac {2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) a c -\sqrt {c}\, \mathrm {log}\left (\frac {2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) b^{2}\right )}{8 c^{2} d} \] Input:

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

Output:

(sqrt(d)*(2*sqrt(a + b*x + c*x**2)*b*c + 4*sqrt(a + b*x + c*x**2)*c**2*x + 
 4*sqrt(c)*log((2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)/sqrt(4*a*c - 
 b**2))*a*c - sqrt(c)*log((2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)/s 
qrt(4*a*c - b**2))*b**2))/(8*c**2*d)