Integrand size = 25, antiderivative size = 505 \[ \int \frac {(c+d x)^{3/2}}{x^3 \left (a-b x^2\right )^{3/2}} \, dx=\frac {(c+d x)^{3/2}}{a x^2 \sqrt {a-b x^2}}-\frac {3 c \sqrt {c+d x} \sqrt {a-b x^2}}{2 a^2 x^2}-\frac {9 d \sqrt {c+d x} \sqrt {a-b x^2}}{4 a^2 x}+\frac {9 \sqrt {b} d \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{4 a^{3/2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {15 \sqrt {b} c d \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{4 a^{3/2} \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {3 \left (4 b c^2+a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{4 a^2 \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:
(d*x+c)^(3/2)/a/x^2/(-b*x^2+a)^(1/2)-3/2*c*(d*x+c)^(1/2)*(-b*x^2+a)^(1/2)/ a^2/x^2-9/4*d*(d*x+c)^(1/2)*(-b*x^2+a)^(1/2)/a^2/x+9/4*b^(1/2)*d*(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)-15/4*b^(1/2)*c*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 ))/a^(3/2)/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)-3/4*(a*d^2+4*b*c^2)*(b^(1/2)*(d* x+c)/(b^(1/2)*c+a^(1/2)*d))^(1/2)*((-b*x^2+a)/a)^(1/2)*EllipticPi(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^2/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 24.38 (sec) , antiderivative size = 889, normalized size of antiderivative = 1.76 \[ \int \frac {(c+d x)^{3/2}}{x^3 \left (a-b x^2\right )^{3/2}} \, dx=\frac {\sqrt {a-b x^2} \left (\frac {(c+d x) \left (-2 a c-5 a d x+6 b c x^2+9 b d x^3\right )}{a^2 x^2 \left (a-b x^2\right )}-\frac {3 \left (-3 b c^3 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}+3 a c d^2 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}+6 b c^2 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} (c+d x)-3 b c \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} (c+d x)^2+3 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 )-i \left (2 b c^2-3 \sqrt {a} \sqrt {b} c d+a d^2\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 )+4 i b c^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 )+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 )\right )}{a^2 c \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (-a+b x^2\right )}\right )}{4 \sqrt {c+d x}} \] Input:
Integrate[(c + d*x)^(3/2)/(x^3*(a - b*x^2)^(3/2)),x]
Output:
(Sqrt[a - b*x^2]*(((c + d*x)*(-2*a*c - 5*a*d*x + 6*b*c*x^2 + 9*b*d*x^3))/( a^2*x^2*(a - b*x^2)) - (3*(-3*b*c^3*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]] + 3*a*c *d^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]] + 6*b*c^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b ]]*(c + d*x) - 3*b*c*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(c + d*x)^2 + (3*I)*Sq rt[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)] - I*(2*b*c^2 - 3*Sqrt[a]*Sqrt[b]*c*d + 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)*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) ] + (4*I)*b*c^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)/(Sqr t[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)] + I*a*d^2*Sqrt[(d*(Sq rt[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*ArcS inh[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*c*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(-a + b* x^2))))/(4*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.
| \(\displaystyle \int \frac {(c+d x)^{3/2}}{x^3 \left (a-b x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {c^2}{x^3 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d^2}{x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {2 c d}{x^2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {(c+d x)^{3/2}}{x^3 \left (a-b x^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {c^2}{x^3 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d^2}{x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {2 c d}{x^2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {(c+d x)^{3/2}}{x^3 \left (a-b x^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {c^2}{x^3 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d^2}{x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {2 c d}{x^2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {(c+d x)^{3/2}}{x^3 \left (a-b x^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {c^2}{x^3 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d^2}{x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {2 c d}{x^2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {(c+d x)^{3/2}}{x^3 \left (a-b x^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {c^2}{x^3 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d^2}{x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {2 c d}{x^2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {(c+d x)^{3/2}}{x^3 \left (a-b x^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {c^2}{x^3 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d^2}{x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {2 c d}{x^2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {(c+d x)^{3/2}}{x^3 \left (a-b x^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {c^2}{x^3 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d^2}{x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {2 c d}{x^2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {(c+d x)^{3/2}}{x^3 \left (a-b x^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {c^2}{x^3 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d^2}{x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {2 c d}{x^2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {(c+d x)^{3/2}}{x^3 \left (a-b x^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {c^2}{x^3 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d^2}{x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {2 c d}{x^2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {(c+d x)^{3/2}}{x^3 \left (a-b x^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {c^2}{x^3 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d^2}{x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {2 c d}{x^2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {(c+d x)^{3/2}}{x^3 \left (a-b x^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {c^2}{x^3 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d^2}{x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {2 c d}{x^2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {(c+d x)^{3/2}}{x^3 \left (a-b x^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {c^2}{x^3 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d^2}{x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {2 c d}{x^2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {(c+d x)^{3/2}}{x^3 \left (a-b x^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {c^2}{x^3 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d^2}{x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {2 c d}{x^2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {(c+d x)^{3/2}}{x^3 \left (a-b x^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {c^2}{x^3 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d^2}{x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {2 c d}{x^2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {(c+d x)^{3/2}}{x^3 \left (a-b x^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {c^2}{x^3 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d^2}{x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {2 c d}{x^2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {(c+d x)^{3/2}}{x^3 \left (a-b x^2\right )^{3/2}}dx\) |
Input:
Int[(c + d*x)^(3/2)/(x^3*(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. \(908\) vs. \(2(406)=812\).
Time = 8.06 (sec) , antiderivative size = 909, normalized size of antiderivative = 1.80
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (d x +c \right ) \left (-b \,x^{2}+a \right )}\, \left (-\frac {c \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}{2 a^{2} x^{2}}-\frac {5 d \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}{4 a^{2} x}-\frac {2 \left (-b d x -b c \right ) \left (\frac {d x}{2 a^{2}}+\frac {c}{2 a^{2}}\right )}{\sqrt {\left (x^{2}-\frac {a}{b}\right ) \left (-b d x -b c \right )}}+\frac {3 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}}}\, \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 )}{2 a^{2} \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}-\frac {9 d^{2} b \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 )}{4 a^{2} \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}-\frac {3 \left (a \,d^{2}+4 b \,c^{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}}}\, d \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 )}{4 a^{2} \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}\, c}\right )}{\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}\) | \(909\) |
| risch | \(\text {Expression too large to display}\) | \(1111\) |
| default | \(\text {Expression too large to display}\) | \(1493\) |
Input:
int((d*x+c)^(3/2)/x^3/(-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)*(-1/2*c/a^2/x^2* (-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)-5/4*d/a^2*(-b*d*x^3-b*c*x^2+a*d*x+a*c)^ (1/2)/x-2*(-b*d*x-b*c)*(1/2*d/a^2*x+1/2*c/a^2)/((x^2-a/b)*(-b*d*x-b*c))^(1 /2)+3/2*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)*Ellipti cF(((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))-9/4/a^2*d^2*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)*((-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)))-3/4*(a*d^2+4*b*c^2)/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*EllipticPi(((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. \[ \int \frac {(c+d x)^{3/2}}{x^3 \left (a-b x^2\right )^{3/2}} \, dx=\text {Timed out} \] Input:
integrate((d*x+c)^(3/2)/x^3/(-b*x^2+a)^(3/2),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {(c+d x)^{3/2}}{x^3 \left (a-b x^2\right )^{3/2}} \, dx=\int \frac {\left (c + d x\right )^{\frac {3}{2}}}{x^{3} \left (a - b x^{2}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((d*x+c)**(3/2)/x**3/(-b*x**2+a)**(3/2),x)
Output:
Integral((c + d*x)**(3/2)/(x**3*(a - b*x**2)**(3/2)), x)
\[ \int \frac {(c+d x)^{3/2}}{x^3 \left (a-b x^2\right )^{3/2}} \, dx=\int { \frac {{\left (d x + c\right )}^{\frac {3}{2}}}{{\left (-b x^{2} + a\right )}^{\frac {3}{2}} x^{3}} \,d x } \] Input:
integrate((d*x+c)^(3/2)/x^3/(-b*x^2+a)^(3/2),x, algorithm="maxima")
Output:
integrate((d*x + c)^(3/2)/((-b*x^2 + a)^(3/2)*x^3), x)
\[ \int \frac {(c+d x)^{3/2}}{x^3 \left (a-b x^2\right )^{3/2}} \, dx=\int { \frac {{\left (d x + c\right )}^{\frac {3}{2}}}{{\left (-b x^{2} + a\right )}^{\frac {3}{2}} x^{3}} \,d x } \] Input:
integrate((d*x+c)^(3/2)/x^3/(-b*x^2+a)^(3/2),x, algorithm="giac")
Output:
integrate((d*x + c)^(3/2)/((-b*x^2 + a)^(3/2)*x^3), x)
Timed out. \[ \int \frac {(c+d x)^{3/2}}{x^3 \left (a-b x^2\right )^{3/2}} \, dx=\int \frac {{\left (c+d\,x\right )}^{3/2}}{x^3\,{\left (a-b\,x^2\right )}^{3/2}} \,d x \] Input:
int((c + d*x)^(3/2)/(x^3*(a - b*x^2)^(3/2)),x)
Output:
int((c + d*x)^(3/2)/(x^3*(a - b*x^2)^(3/2)), x)
\[ \int \frac {(c+d x)^{3/2}}{x^3 \left (a-b x^2\right )^{3/2}} \, dx=\int \frac {\left (d x +c \right )^{\frac {3}{2}}}{x^{3} \left (-b \,x^{2}+a \right )^{\frac {3}{2}}}d x \] Input:
int((d*x+c)^(3/2)/x^3/(-b*x^2+a)^(3/2),x)
Output:
int((d*x+c)^(3/2)/x^3/(-b*x^2+a)^(3/2),x)