Integrand size = 23, antiderivative size = 158 \[ \int \frac {x \sqrt {c+d x}}{\left (a-b x^2\right )^{3/2}} \, dx=\frac {\sqrt {c+d x}}{b \sqrt {a-b x^2}}+\frac {\sqrt {a} d \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{b^{3/2} \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:
(d*x+c)^(1/2)/b/(-b*x^2+a)^(1/2)+a^(1/2)*d*(b^(1/2)*(d*x+c)/(b^(1/2)*c+a^( 1/2)*d))^(1/2)*(1-b*x^2/a)^(1/2)*EllipticF(1/2*(1-b^(1/2)*x/a^(1/2))^(1/2) *2^(1/2),2^(1/2)*(a^(1/2)*d/(b^(1/2)*c+a^(1/2)*d))^(1/2))/b^(3/2)/(d*x+c)^ (1/2)/(-b*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 21.87 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.15 \[ \int \frac {x \sqrt {c+d x}}{\left (a-b x^2\right )^{3/2}} \, dx=\frac {\sqrt {c+d x}-\frac {i \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right ),\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )}{\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}}{b \sqrt {a-b x^2}} \] Input:
Integrate[(x*Sqrt[c + d*x])/(a - b*x^2)^(3/2),x]
Output:
(Sqrt[c + d*x] - (I*Sqrt[(d*(Sqrt[a]/Sqrt[b] + x))/(c + d*x)]*Sqrt[-(((Sqr t[a]*d)/Sqrt[b] - d*x)/(c + d*x))]*(c + d*x)*EllipticF[I*ArcSinh[Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]/Sqrt[c + d*x]], (Sqrt[b]*c + Sqrt[a]*d)/(Sqrt[b]*c - Sqrt[a]*d)])/Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]])/(b*Sqrt[a - b*x^2])
Time = 0.28 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.96, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {592, 512, 511, 321}
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.
| \(\displaystyle \int \frac {x \sqrt {c+d x}}{\left (a-b x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 592 |
| \(\displaystyle \frac {\sqrt {c+d x}}{b \sqrt {a-b x^2}}-\frac {d \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{2 b}\) |
| \(\Big \downarrow \) 512 |
| \(\displaystyle \frac {\sqrt {c+d x}}{b \sqrt {a-b x^2}}-\frac {d \sqrt {1-\frac {b x^2}{a}} \int \frac {1}{\sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}}}dx}{2 b \sqrt {a-b x^2}}\) |
| \(\Big \downarrow \) 511 |
| \(\displaystyle \frac {\sqrt {a} d \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \int \frac {1}{\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{b^{3/2} \sqrt {a-b x^2} \sqrt {c+d x}}+\frac {\sqrt {c+d x}}{b \sqrt {a-b x^2}}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {\sqrt {a} d \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{b^{3/2} \sqrt {a-b x^2} \sqrt {c+d x}}+\frac {\sqrt {c+d x}}{b \sqrt {a-b x^2}}\) |
Input:
Int[(x*Sqrt[c + d*x])/(a - b*x^2)^(3/2),x]
Output:
Sqrt[c + d*x]/(b*Sqrt[a - b*x^2]) + (Sqrt[a]*d*Sqrt[(Sqrt[b]*(c + d*x))/(S qrt[b]*c + Sqrt[a]*d)]*Sqrt[1 - (b*x^2)/a]*EllipticF[ArcSin[Sqrt[1 - (Sqrt [b]*x)/Sqrt[a]]/Sqrt[2]], (2*d)/((Sqrt[b]*c)/Sqrt[a] + d)])/(b^(3/2)*Sqrt[ c + d*x]*Sqrt[a - b*x^2])
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[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] && !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
Int[1/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Wit h[{q = Rt[-b/a, 2]}, Simp[-2*(Sqrt[q*((c + d*x)/(d + c*q))]/(Sqrt[a]*q*Sqrt [c + d*x])) Subst[Int[1/(Sqrt[1 - 2*d*(x^2/(d + c*q))]*Sqrt[1 - x^2]), x] , x, Sqrt[(1 - q*x)/2]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[ a, 0]
Int[1/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Sim p[Sqrt[1 + b*(x^2/a)]/Sqrt[a + b*x^2] Int[1/(Sqrt[c + d*x]*Sqrt[1 + b*(x^ 2/a)]), x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && !GtQ[a, 0]
Int[(x_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c + d*x)^n*((a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] - Simp[d*(n/(2*b* (p + 1))) Int[(c + d*x)^(n - 1)*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d}, x] && LtQ[p, -1] && GtQ[n, 0] && (IntegerQ[n] || IntegerQ[p] || I ntegersQ[2*n, 2*p])
Leaf count of result is larger than twice the leaf count of optimal. \(280\) vs. \(2(127)=254\).
Time = 0.79 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.78
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (d x +c \right ) \left (-b \,x^{2}+a \right )}\, \left (-\frac {-b d x -b c}{b^{2} \sqrt {\left (x^{2}-\frac {a}{b}\right ) \left (-b d x -b c \right )}}-\frac {d \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x -\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x +\frac {\sqrt {a b}}{b}}{-\frac {c}{d}+\frac {\sqrt {a b}}{b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )}{b \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}\right )}{\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}\) | \(281\) |
| default | \(\frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, \left (\sqrt {a b}\, \sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {a b}-b c}}\, \sqrt {\frac {\left (-b x +\sqrt {a b}\right ) d}{d \sqrt {a b}+b c}}\, \sqrt {\frac {\left (b x +\sqrt {a b}\right ) d}{d \sqrt {a b}-b c}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {a b}-b c}}, \sqrt {-\frac {d \sqrt {a b}-b c}{d \sqrt {a b}+b c}}\right ) d -\sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {a b}-b c}}\, \sqrt {\frac {\left (-b x +\sqrt {a b}\right ) d}{d \sqrt {a b}+b c}}\, \sqrt {\frac {\left (b x +\sqrt {a b}\right ) d}{d \sqrt {a b}-b c}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {a b}-b c}}, \sqrt {-\frac {d \sqrt {a b}-b c}{d \sqrt {a b}+b c}}\right ) b c +b d x +b c \right )}{\left (-b d \,x^{3}-b c \,x^{2}+a d x +a c \right ) b^{2}}\) | \(329\) |
Input:
int(x*(d*x+c)^(1/2)/(-b*x^2+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
((d*x+c)*(-b*x^2+a))^(1/2)/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)*(-(-b*d*x-b*c)/b ^2/((x^2-a/b)*(-b*d*x-b*c))^(1/2)-d/b*(c/d-1/b*(a*b)^(1/2))*((x+c/d)/(c/d- 1/b*(a*b)^(1/2)))^(1/2)*((x-1/b*(a*b)^(1/2))/(-c/d-1/b*(a*b)^(1/2)))^(1/2) *((x+1/b*(a*b)^(1/2))/(-c/d+1/b*(a*b)^(1/2)))^(1/2)/(-b*d*x^3-b*c*x^2+a*d* x+a*c)^(1/2)*EllipticF(((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2),((-c/d+1/b*(a *b)^(1/2))/(-c/d-1/b*(a*b)^(1/2)))^(1/2)))
Time = 0.09 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.68 \[ \int \frac {x \sqrt {c+d x}}{\left (a-b x^2\right )^{3/2}} \, dx=\frac {{\left (b x^{2} - a\right )} \sqrt {-b d} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, \frac {3 \, d x + c}{3 \, d}\right ) - \sqrt {-b x^{2} + a} \sqrt {d x + c} b}{b^{3} x^{2} - a b^{2}} \] Input:
integrate(x*(d*x+c)^(1/2)/(-b*x^2+a)^(3/2),x, algorithm="fricas")
Output:
((b*x^2 - a)*sqrt(-b*d)*weierstrassPInverse(4/3*(b*c^2 + 3*a*d^2)/(b*d^2), -8/27*(b*c^3 - 9*a*c*d^2)/(b*d^3), 1/3*(3*d*x + c)/d) - sqrt(-b*x^2 + a)* sqrt(d*x + c)*b)/(b^3*x^2 - a*b^2)
\[ \int \frac {x \sqrt {c+d x}}{\left (a-b x^2\right )^{3/2}} \, dx=\int \frac {x \sqrt {c + d x}}{\left (a - b x^{2}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(x*(d*x+c)**(1/2)/(-b*x**2+a)**(3/2),x)
Output:
Integral(x*sqrt(c + d*x)/(a - b*x**2)**(3/2), x)
\[ \int \frac {x \sqrt {c+d x}}{\left (a-b x^2\right )^{3/2}} \, dx=\int { \frac {\sqrt {d x + c} x}{{\left (-b x^{2} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(x*(d*x+c)^(1/2)/(-b*x^2+a)^(3/2),x, algorithm="maxima")
Output:
integrate(sqrt(d*x + c)*x/(-b*x^2 + a)^(3/2), x)
\[ \int \frac {x \sqrt {c+d x}}{\left (a-b x^2\right )^{3/2}} \, dx=\int { \frac {\sqrt {d x + c} x}{{\left (-b x^{2} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(x*(d*x+c)^(1/2)/(-b*x^2+a)^(3/2),x, algorithm="giac")
Output:
integrate(sqrt(d*x + c)*x/(-b*x^2 + a)^(3/2), x)
Timed out. \[ \int \frac {x \sqrt {c+d x}}{\left (a-b x^2\right )^{3/2}} \, dx=\int \frac {x\,\sqrt {c+d\,x}}{{\left (a-b\,x^2\right )}^{3/2}} \,d x \] Input:
int((x*(c + d*x)^(1/2))/(a - b*x^2)^(3/2),x)
Output:
int((x*(c + d*x)^(1/2))/(a - b*x^2)^(3/2), x)
\[ \int \frac {x \sqrt {c+d x}}{\left (a-b x^2\right )^{3/2}} \, dx=\frac {2 \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}-\left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) a d +\left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) b d \,x^{2}}{2 b \left (-b \,x^{2}+a \right )} \] Input:
int(x*(d*x+c)^(1/2)/(-b*x^2+a)^(3/2),x)
Output:
(2*sqrt(c + d*x)*sqrt(a - b*x**2) - int((sqrt(c + d*x)*sqrt(a - b*x**2))/( a*c + a*d*x - b*c*x**2 - b*d*x**3),x)*a*d + int((sqrt(c + d*x)*sqrt(a - b* x**2))/(a*c + a*d*x - b*c*x**2 - b*d*x**3),x)*b*d*x**2)/(2*b*(a - b*x**2))