Integrand size = 25, antiderivative size = 465 \[ \int \frac {(c+d x)^{3/2}}{x^2 \left (a-b x^2\right )^{3/2}} \, dx=\frac {(c+d x)^{3/2}}{a x \sqrt {a-b x^2}}-\frac {2 c \sqrt {c+d x} \sqrt {a-b x^2}}{a^2 x}+\frac {2 \sqrt {b} c \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{a^{3/2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {\left (2 b c^2+a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{a^{3/2} \sqrt {b} \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {3 c d \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{a \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:
(d*x+c)^(3/2)/a/x/(-b*x^2+a)^(1/2)-2*c*(d*x+c)^(1/2)*(-b*x^2+a)^(1/2)/a^2/ x+2*b^(1/2)*c*(d*x+c)^(1/2)*(1-b*x^2/a)^(1/2)*EllipticE(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))/a^( 3/2)/(b^(1/2)*(d*x+c)/(b^(1/2)*c+a^(1/2)*d))^(1/2)/(-b*x^2+a)^(1/2)-(a*d^2 +2*b*c^2)*(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))/a^(3/2)/b^(1/2)/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)-3* c*d*(b^(1/2)*(d*x+c)/(b^(1/2)*c+a^(1/2)*d))^(1/2)*((-b*x^2+a)/a)^(1/2)*Ell ipticPi(1/2*(1-b^(1/2)*x/a^(1/2))^(1/2)*2^(1/2),2,2^(1/2)*(a^(1/2)*d/(b^(1 /2)*c+a^(1/2)*d))^(1/2))/a/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 23.89 (sec) , antiderivative size = 706, normalized size of antiderivative = 1.52 \[ \int \frac {(c+d x)^{3/2}}{x^2 \left (a-b x^2\right )^{3/2}} \, dx=\frac {\sqrt {a-b x^2} \left (\frac {(c+d x) \left (-a c+a d x+2 b c x^2\right )}{a^2 x \left (a-b x^2\right )}-\frac {-2 b c^3 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}+2 a c d^2 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}+4 b c^2 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} (c+d x)-2 b c \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} (c+d x)^2+2 i \sqrt {b} c \left (\sqrt {b} c-\sqrt {a} d\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right )|\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )+2 i \sqrt {a} d \left (\sqrt {b} c-\sqrt {a} d\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right ),\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )+3 i a d^2 \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} \operatorname {EllipticPi}\left (\frac {\sqrt {b} c}{\sqrt {b} c-\sqrt {a} d},i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right ),\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )}{a^2 d \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (-a+b x^2\right )}\right )}{\sqrt {c+d x}} \] Input:
Integrate[(c + d*x)^(3/2)/(x^2*(a - b*x^2)^(3/2)),x]
Output:
(Sqrt[a - b*x^2]*(((c + d*x)*(-(a*c) + a*d*x + 2*b*c*x^2))/(a^2*x*(a - b*x ^2)) - (-2*b*c^3*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]] + 2*a*c*d^2*Sqrt[-c + (Sqr t[a]*d)/Sqrt[b]] + 4*b*c^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(c + d*x) - 2*b* c*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(c + d*x)^2 + (2*I)*Sqrt[b]*c*(Sqrt[b]*c - Sqrt[a]*d)*Sqrt[(d*(Sqrt[a]/Sqrt[b] + x))/(c + d*x)]*Sqrt[-(((Sqrt[a]*d) /Sqrt[b] - d*x)/(c + d*x))]*(c + d*x)^(3/2)*EllipticE[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)] + (2*I)*Sqrt[a]*d*(Sqrt[b]*c - Sqrt[a]*d)*Sqrt[(d*(Sqrt[a]/Sqr t[b] + x))/(c + d*x)]*Sqrt[-(((Sqrt[a]*d)/Sqrt[b] - d*x)/(c + d*x))]*(c + d*x)^(3/2)*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)] + (3*I)*a*d^2*Sqrt[(d *(Sqrt[a]/Sqrt[b] + x))/(c + d*x)]*Sqrt[-(((Sqrt[a]*d)/Sqrt[b] - d*x)/(c + d*x))]*(c + d*x)^(3/2)*EllipticPi[(Sqrt[b]*c)/(Sqrt[b]*c - Sqrt[a]*d), 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)])/(a^2*d*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(-a + b*x^2))))/Sqrt[c + d*x]
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.
| \(\displaystyle \int \frac {(c+d x)^{3/2}}{x^2 \left (a-b x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {c^2}{x^2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d^2}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {2 c d}{x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {(c+d x)^{3/2}}{x^2 \left (a-b x^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {c^2}{x^2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d^2}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {2 c d}{x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {(c+d x)^{3/2}}{x^2 \left (a-b x^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {c^2}{x^2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d^2}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {2 c d}{x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {(c+d x)^{3/2}}{x^2 \left (a-b x^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {c^2}{x^2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d^2}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {2 c d}{x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {(c+d x)^{3/2}}{x^2 \left (a-b x^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {c^2}{x^2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d^2}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {2 c d}{x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {(c+d x)^{3/2}}{x^2 \left (a-b x^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {c^2}{x^2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d^2}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {2 c d}{x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {(c+d x)^{3/2}}{x^2 \left (a-b x^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {c^2}{x^2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d^2}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {2 c d}{x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {(c+d x)^{3/2}}{x^2 \left (a-b x^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {c^2}{x^2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d^2}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {2 c d}{x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {(c+d x)^{3/2}}{x^2 \left (a-b x^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {c^2}{x^2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d^2}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {2 c d}{x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {(c+d x)^{3/2}}{x^2 \left (a-b x^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {c^2}{x^2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d^2}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {2 c d}{x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {(c+d x)^{3/2}}{x^2 \left (a-b x^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {c^2}{x^2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d^2}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {2 c d}{x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {(c+d x)^{3/2}}{x^2 \left (a-b x^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {c^2}{x^2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d^2}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {2 c d}{x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {(c+d x)^{3/2}}{x^2 \left (a-b x^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {c^2}{x^2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d^2}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {2 c d}{x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {(c+d x)^{3/2}}{x^2 \left (a-b x^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {c^2}{x^2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d^2}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {2 c d}{x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {(c+d x)^{3/2}}{x^2 \left (a-b x^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {c^2}{x^2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d^2}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {2 c d}{x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {(c+d x)^{3/2}}{x^2 \left (a-b x^2\right )^{3/2}}dx\) |
Input:
Int[(c + d*x)^(3/2)/(x^2*(a - b*x^2)^(3/2)),x]
Output:
$Aborted
Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo l] :> Int[ExpandIntegrand[(a + b*x^2)^p/Sqrt[c + d*x], x^m*(c + d*x)^(n + 1 /2), x], x] /; FreeQ[{a, b, c, d, m}, x] && IntegerQ[p + 1/2] && IntegerQ[n + 1/2] && IntegerQ[m]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Leaf count of result is larger than twice the leaf count of optimal. \(891\) vs. \(2(380)=760\).
Time = 7.00 (sec) , antiderivative size = 892, normalized size of antiderivative = 1.92
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (d x +c \right ) \left (-b \,x^{2}+a \right )}\, \left (-\frac {2 \left (-b d x -b c \right ) \left (\frac {c x}{2 a^{2}}+\frac {d}{2 b a}\right )}{\sqrt {\left (x^{2}-\frac {a}{b}\right ) \left (-b d x -b c \right )}}-\frac {c \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}{a^{2} x}+\frac {2 \left (\frac {a \,d^{2}+b \,c^{2}}{a^{2}}-\frac {d^{2}}{2 a}-\frac {b \,c^{2}}{a^{2}}\right ) \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 )}{\sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}-\frac {2 b c 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}}}\, \left (\left (-\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \operatorname {EllipticE}\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 )+\frac {\sqrt {a 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}\right )}{a^{2} \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}-\frac {3 d^{2} \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 {EllipticPi}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, -\frac {\left (-\frac {c}{d}+\frac {\sqrt {a b}}{b}\right ) d}{c}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )}{a \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}\right )}{\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}\) | \(892\) |
| risch | \(-\frac {c \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{a^{2} x}-\frac {\left (\frac {d c \sqrt {a b}\, \sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \left (\left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \operatorname {EllipticE}\left (\frac {\sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}}{2}, \sqrt {-\frac {2 \sqrt {a b}}{b \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right )}}\right )-\frac {c \operatorname {EllipticF}\left (\frac {\sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}}{2}, \sqrt {-\frac {2 \sqrt {a b}}{b \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right )}}\right )}{d}\right )}{\sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}+2 a \left (-\frac {2 \left (-b d x -b c \right ) \left (-\frac {c x}{2 a}-\frac {d}{2 b}\right )}{\sqrt {\left (x^{2}-\frac {a}{b}\right ) \left (-b d x -b c \right )}}+\frac {\left (-\frac {a \,d^{2}+b \,c^{2}}{a}+\frac {d^{2}}{2}+\frac {c^{2} b}{a}\right ) \sqrt {a b}\, \sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}}{2}, \sqrt {-\frac {2 \sqrt {a b}}{b \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right )}}\right )}{b \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}+\frac {c d \sqrt {a b}\, \sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \left (\left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \operatorname {EllipticE}\left (\frac {\sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}}{2}, \sqrt {-\frac {2 \sqrt {a b}}{b \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right )}}\right )-\frac {c \operatorname {EllipticF}\left (\frac {\sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}}{2}, \sqrt {-\frac {2 \sqrt {a b}}{b \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right )}}\right )}{d}\right )}{2 a \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}\right )+\frac {3 a c d \sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \operatorname {EllipticPi}\left (\frac {\sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}}{2}, 2, \sqrt {-\frac {2 \sqrt {a b}}{b \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right )}}\right )}{\sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}\right ) \sqrt {\left (d x +c \right ) \left (-b \,x^{2}+a \right )}}{2 a^{2} \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}\) | \(957\) |
| default | \(\text {Expression too large to display}\) | \(1132\) |
Input:
int((d*x+c)^(3/2)/x^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)*(-2*(-b*d*x-b*c) *(1/2*c/a^2*x+1/2*d/b/a)/((x^2-a/b)*(-b*d*x-b*c))^(1/2)-c/a^2*(-b*d*x^3-b* c*x^2+a*d*x+a*c)^(1/2)/x+2*((a*d^2+b*c^2)/a^2-1/2/a*d^2-1/a^2*b*c^2)*(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))-2 *b*c*d/a^2*(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)*((-c/d-1/b*(a* b)^(1/2))*EllipticE(((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))+1/b*(a*b)^(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)))-3*d^2/a*(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)*Ellipt icPi(((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2),-(-c/d+1/b*(a*b)^(1/2))/c*d,((- c/d+1/b*(a*b)^(1/2))/(-c/d-1/b*(a*b)^(1/2)))^(1/2)))
Timed out. \[ \int \frac {(c+d x)^{3/2}}{x^2 \left (a-b x^2\right )^{3/2}} \, dx=\text {Timed out} \] Input:
integrate((d*x+c)^(3/2)/x^2/(-b*x^2+a)^(3/2),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {(c+d x)^{3/2}}{x^2 \left (a-b x^2\right )^{3/2}} \, dx=\int \frac {\left (c + d x\right )^{\frac {3}{2}}}{x^{2} \left (a - b x^{2}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((d*x+c)**(3/2)/x**2/(-b*x**2+a)**(3/2),x)
Output:
Integral((c + d*x)**(3/2)/(x**2*(a - b*x**2)**(3/2)), x)
\[ \int \frac {(c+d x)^{3/2}}{x^2 \left (a-b x^2\right )^{3/2}} \, dx=\int { \frac {{\left (d x + c\right )}^{\frac {3}{2}}}{{\left (-b x^{2} + a\right )}^{\frac {3}{2}} x^{2}} \,d x } \] Input:
integrate((d*x+c)^(3/2)/x^2/(-b*x^2+a)^(3/2),x, algorithm="maxima")
Output:
integrate((d*x + c)^(3/2)/((-b*x^2 + a)^(3/2)*x^2), x)
\[ \int \frac {(c+d x)^{3/2}}{x^2 \left (a-b x^2\right )^{3/2}} \, dx=\int { \frac {{\left (d x + c\right )}^{\frac {3}{2}}}{{\left (-b x^{2} + a\right )}^{\frac {3}{2}} x^{2}} \,d x } \] Input:
integrate((d*x+c)^(3/2)/x^2/(-b*x^2+a)^(3/2),x, algorithm="giac")
Output:
integrate((d*x + c)^(3/2)/((-b*x^2 + a)^(3/2)*x^2), x)
Timed out. \[ \int \frac {(c+d x)^{3/2}}{x^2 \left (a-b x^2\right )^{3/2}} \, dx=\int \frac {{\left (c+d\,x\right )}^{3/2}}{x^2\,{\left (a-b\,x^2\right )}^{3/2}} \,d x \] Input:
int((c + d*x)^(3/2)/(x^2*(a - b*x^2)^(3/2)),x)
Output:
int((c + d*x)^(3/2)/(x^2*(a - b*x^2)^(3/2)), x)
\[ \int \frac {(c+d x)^{3/2}}{x^2 \left (a-b x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:
int((d*x+c)^(3/2)/x^2/(-b*x^2+a)^(3/2),x)
Output:
(2*sqrt(a - b*x**2)*int(sqrt(c + d*x)/(sqrt(a - b*x**2)*a*c**2 - sqrt(a - b*x**2)*a*d**2*x**2 - sqrt(a - b*x**2)*b*c**2*x**2 + sqrt(a - b*x**2)*b*d* *2*x**4),x)*a**2*c*d**4*x - 8*sqrt(a - b*x**2)*int(sqrt(c + d*x)/(sqrt(a - b*x**2)*a*c**2 - sqrt(a - b*x**2)*a*d**2*x**2 - sqrt(a - b*x**2)*b*c**2*x **2 + sqrt(a - b*x**2)*b*d**2*x**4),x)*a*b*c**3*d**2*x - 3*sqrt(a - b*x**2 )*int((sqrt(c + d*x)*x**3)/(sqrt(a - b*x**2)*a*c**2 - sqrt(a - b*x**2)*a*d **2*x**2 - sqrt(a - b*x**2)*b*c**2*x**2 + sqrt(a - b*x**2)*b*d**2*x**4),x) *a*b*d**5*x - 3*sqrt(a - b*x**2)*int((sqrt(c + d*x)*sqrt(a - b*x**2)*x**2) /(a**2*c**2 - a**2*d**2*x**2 - 2*a*b*c**2*x**2 + 2*a*b*d**2*x**4 + b**2*c* *2*x**4 - b**2*d**2*x**6),x)*a*b*c*d**4*x - 6*sqrt(a - b*x**2)*int((sqrt(c + d*x)*sqrt(a - b*x**2)*x**2)/(a**2*c**2 - a**2*d**2*x**2 - 2*a*b*c**2*x* *2 + 2*a*b*d**2*x**4 + b**2*c**2*x**4 - b**2*d**2*x**6),x)*b**2*c**3*d**2* x - sqrt(a - b*x**2)*int((sqrt(c + d*x)*sqrt(a - b*x**2)*x)/(a**2*c**2 - a **2*d**2*x**2 - 2*a*b*c**2*x**2 + 2*a*b*d**2*x**4 + b**2*c**2*x**4 - b**2* d**2*x**6),x)*a*b*c**2*d**3*x - 2*sqrt(a - b*x**2)*int((sqrt(c + d*x)*sqrt (a - b*x**2)*x)/(a**2*c**2 - a**2*d**2*x**2 - 2*a*b*c**2*x**2 + 2*a*b*d**2 *x**4 + b**2*c**2*x**4 - b**2*d**2*x**6),x)*b**2*c**4*d*x + 6*sqrt(a - b*x **2)*int((sqrt(c + d*x)*sqrt(a - b*x**2))/(a**2*c**2*x - a**2*d**2*x**3 - 2*a*b*c**2*x**3 + 2*a*b*d**2*x**5 + b**2*c**2*x**5 - b**2*d**2*x**7),x)*a* b*c**4*d*x + sqrt(a - b*x**2)*int((sqrt(c + d*x)*sqrt(a - b*x**2))/(a**...