Integrand size = 25, antiderivative size = 507 \[ \int \frac {(c+d x)^{5/2}}{x^2 \left (a-b x^2\right )^{3/2}} \, dx=\frac {(c+d x)^{5/2}}{a x \sqrt {a-b x^2}}+\frac {2 c d \sqrt {c+d x} \sqrt {a-b x^2}}{a^2}-\frac {2 c (c+d x)^{3/2} \sqrt {a-b x^2}}{a^2 x}+\frac {\left (2 b c^2+a d^2\right ) \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{a^{3/2} \sqrt {b} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {c \left (2 b c^2+3 a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{a^{3/2} \sqrt {b} \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {5 c^2 d \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{a \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:
(d*x+c)^(5/2)/a/x/(-b*x^2+a)^(1/2)+2*c*d*(d*x+c)^(1/2)*(-b*x^2+a)^(1/2)/a^ 2-2*c*(d*x+c)^(3/2)*(-b*x^2+a)^(1/2)/a^2/x+(a*d^2+2*b*c^2)*(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)/(b^(1/2)*(d*x+c) /(b^(1/2)*c+a^(1/2)*d))^(1/2)/(-b*x^2+a)^(1/2)-c*(3*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)-5*c^2*d*(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/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 24.02 (sec) , antiderivative size = 871, normalized size of antiderivative = 1.72 \[ \int \frac {(c+d x)^{5/2}}{x^2 \left (a-b x^2\right )^{3/2}} \, dx=\frac {\sqrt {a-b x^2} \left (\frac {(c+d x) \left (2 b c^2 x^2+a \left (-c^2+2 c d x+d^2 x^2\right )\right )}{a^2 x \left (a-b x^2\right )}-\frac {-2 b^2 c^4 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}+a b c^2 d^2 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}+a^2 d^4 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}+4 b^2 c^3 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} (c+d x)+2 a b c d^2 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} (c+d x)-2 b^2 c^2 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} (c+d x)^2-a b d^2 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} (c+d x)^2+i \sqrt {b} \left (2 b^{3/2} c^3-2 \sqrt {a} b c^2 d+a \sqrt {b} c d^2-a^{3/2} d^3\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right )|\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )+i \sqrt {a} \sqrt {b} d \left (2 b c^2-3 \sqrt {a} \sqrt {b} c d+a d^2\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right ),\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )+5 i a b c d^2 \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} \operatorname {EllipticPi}\left (\frac {\sqrt {b} c}{\sqrt {b} c-\sqrt {a} d},i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right ),\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )}{a^2 b d \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (-a+b x^2\right )}\right )}{\sqrt {c+d x}} \] Input:
Integrate[(c + d*x)^(5/2)/(x^2*(a - b*x^2)^(3/2)),x]
Output:
(Sqrt[a - b*x^2]*(((c + d*x)*(2*b*c^2*x^2 + a*(-c^2 + 2*c*d*x + d^2*x^2))) /(a^2*x*(a - b*x^2)) - (-2*b^2*c^4*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]] + a*b*c^ 2*d^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]] + a^2*d^4*Sqrt[-c + (Sqrt[a]*d)/Sqrt[ b]] + 4*b^2*c^3*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(c + d*x) + 2*a*b*c*d^2*Sqr t[-c + (Sqrt[a]*d)/Sqrt[b]]*(c + d*x) - 2*b^2*c^2*Sqrt[-c + (Sqrt[a]*d)/Sq rt[b]]*(c + d*x)^2 - a*b*d^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(c + d*x)^2 + I*Sqrt[b]*(2*b^(3/2)*c^3 - 2*Sqrt[a]*b*c^2*d + a*Sqrt[b]*c*d^2 - a^(3/2)*d ^3)*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*Sqrt[a]*Sqrt[b]*d*(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)] + (5*I)* a*b*c*d^2*Sqrt[(d*(Sqrt[a]/Sqrt[b] + x))/(c + d*x)]*Sqrt[-(((Sqrt[a]*d)/Sq rt[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]], (S qrt[b]*c + Sqrt[a]*d)/(Sqrt[b]*c - Sqrt[a]*d)])/(a^2*b*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.
| \(\displaystyle \int \frac {(c+d x)^{5/2}}{x^2 \left (a-b x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {c^3}{x^2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {3 c^2 d}{x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d^3 x}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {3 c d^2}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {(c+d x)^{5/2}}{x^2 \left (a-b x^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {c^3}{x^2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {3 c^2 d}{x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d^3 x}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {3 c d^2}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {(c+d x)^{5/2}}{x^2 \left (a-b x^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {c^3}{x^2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {3 c^2 d}{x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d^3 x}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {3 c d^2}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {(c+d x)^{5/2}}{x^2 \left (a-b x^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {c^3}{x^2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {3 c^2 d}{x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d^3 x}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {3 c d^2}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {(c+d x)^{5/2}}{x^2 \left (a-b x^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {c^3}{x^2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {3 c^2 d}{x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d^3 x}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {3 c d^2}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {(c+d x)^{5/2}}{x^2 \left (a-b x^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {c^3}{x^2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {3 c^2 d}{x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d^3 x}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {3 c d^2}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {(c+d x)^{5/2}}{x^2 \left (a-b x^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {c^3}{x^2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {3 c^2 d}{x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d^3 x}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {3 c d^2}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {(c+d x)^{5/2}}{x^2 \left (a-b x^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {c^3}{x^2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {3 c^2 d}{x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d^3 x}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {3 c d^2}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {(c+d x)^{5/2}}{x^2 \left (a-b x^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {c^3}{x^2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {3 c^2 d}{x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d^3 x}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {3 c d^2}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {(c+d x)^{5/2}}{x^2 \left (a-b x^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {c^3}{x^2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {3 c^2 d}{x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d^3 x}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {3 c d^2}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {(c+d x)^{5/2}}{x^2 \left (a-b x^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {c^3}{x^2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {3 c^2 d}{x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d^3 x}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {3 c d^2}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {(c+d x)^{5/2}}{x^2 \left (a-b x^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {c^3}{x^2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {3 c^2 d}{x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d^3 x}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {3 c d^2}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {(c+d x)^{5/2}}{x^2 \left (a-b x^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {c^3}{x^2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {3 c^2 d}{x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d^3 x}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {3 c d^2}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {(c+d x)^{5/2}}{x^2 \left (a-b x^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {c^3}{x^2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {3 c^2 d}{x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d^3 x}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {3 c d^2}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {(c+d x)^{5/2}}{x^2 \left (a-b x^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {c^3}{x^2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {3 c^2 d}{x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d^3 x}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {3 c d^2}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {(c+d x)^{5/2}}{x^2 \left (a-b x^2\right )^{3/2}}dx\) |
Input:
Int[(c + d*x)^(5/2)/(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. \(940\) vs. \(2(418)=836\).
Time = 8.26 (sec) , antiderivative size = 941, normalized size of antiderivative = 1.86
| 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 {\left (a \,d^{2}+b \,c^{2}\right ) x}{2 a^{2} b}+\frac {c d}{a b}\right )}{\sqrt {\left (x^{2}-\frac {a}{b}\right ) \left (-b d x -b c \right )}}-\frac {c^{2} \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}{a^{2} x}+\frac {2 \left (\frac {c \left (3 a \,d^{2}+b \,c^{2}\right )}{a^{2}}-\frac {d^{2} c}{a}-\frac {c \left (a \,d^{2}+b \,c^{2}\right )}{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 \left (-\frac {d \left (a \,d^{2}+b \,c^{2}\right )}{2 a^{2}}-\frac {b \,c^{2} d}{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}}}\, \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 )}{\sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}-\frac {5 c \,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}}\) | \(941\) |
| risch | \(-\frac {c^{2} \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{a^{2} x}-\frac {\left (\frac {c^{2} 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 )}{\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 {\left (a \,d^{2}+b \,c^{2}\right ) x}{2 a b}-\frac {c d}{b}\right )}{\sqrt {\left (x^{2}-\frac {a}{b}\right ) \left (-b d x -b c \right )}}+\frac {\left (-\frac {c \left (3 a \,d^{2}+b \,c^{2}\right )}{a}+c \,d^{2}+\frac {c \left (a \,d^{2}+b \,c^{2}\right )}{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 {\left (a \,d^{2}+b \,c^{2}\right ) 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 b \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}\right )+\frac {5 a \,c^{2} 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}}\) | \(1000\) |
| default | \(\text {Expression too large to display}\) | \(1444\) |
Input:
int((d*x+c)^(5/2)/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*(a*d^2+b*c^2)/a^2/b*x+c*d/a/b)/((x^2-a/b)*(-b*d*x-b*c))^(1/2)-c^2/a^ 2*(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)/x+2*(c*(3*a*d^2+b*c^2)/a^2-d^2*c/a-c* (a*d^2+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)*Ellipt icF(((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*(-1/2*d*(a*d^2+b*c^2)/a^2-1/2*b*c^2*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))*Ellipti cE(((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)))-5*c*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)*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)))
\[ \int \frac {(c+d x)^{5/2}}{x^2 \left (a-b x^2\right )^{3/2}} \, dx=\int { \frac {{\left (d x + c\right )}^{\frac {5}{2}}}{{\left (-b x^{2} + a\right )}^{\frac {3}{2}} x^{2}} \,d x } \] Input:
integrate((d*x+c)^(5/2)/x^2/(-b*x^2+a)^(3/2),x, algorithm="fricas")
Output:
integral((d^2*x^2 + 2*c*d*x + c^2)*sqrt(-b*x^2 + a)*sqrt(d*x + c)/(b^2*x^6 - 2*a*b*x^4 + a^2*x^2), x)
\[ \int \frac {(c+d x)^{5/2}}{x^2 \left (a-b x^2\right )^{3/2}} \, dx=\int \frac {\left (c + d x\right )^{\frac {5}{2}}}{x^{2} \left (a - b x^{2}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((d*x+c)**(5/2)/x**2/(-b*x**2+a)**(3/2),x)
Output:
Integral((c + d*x)**(5/2)/(x**2*(a - b*x**2)**(3/2)), x)
\[ \int \frac {(c+d x)^{5/2}}{x^2 \left (a-b x^2\right )^{3/2}} \, dx=\int { \frac {{\left (d x + c\right )}^{\frac {5}{2}}}{{\left (-b x^{2} + a\right )}^{\frac {3}{2}} x^{2}} \,d x } \] Input:
integrate((d*x+c)^(5/2)/x^2/(-b*x^2+a)^(3/2),x, algorithm="maxima")
Output:
integrate((d*x + c)^(5/2)/((-b*x^2 + a)^(3/2)*x^2), x)
\[ \int \frac {(c+d x)^{5/2}}{x^2 \left (a-b x^2\right )^{3/2}} \, dx=\int { \frac {{\left (d x + c\right )}^{\frac {5}{2}}}{{\left (-b x^{2} + a\right )}^{\frac {3}{2}} x^{2}} \,d x } \] Input:
integrate((d*x+c)^(5/2)/x^2/(-b*x^2+a)^(3/2),x, algorithm="giac")
Output:
integrate((d*x + c)^(5/2)/((-b*x^2 + a)^(3/2)*x^2), x)
Timed out. \[ \int \frac {(c+d x)^{5/2}}{x^2 \left (a-b x^2\right )^{3/2}} \, dx=\int \frac {{\left (c+d\,x\right )}^{5/2}}{x^2\,{\left (a-b\,x^2\right )}^{3/2}} \,d x \] Input:
int((c + d*x)^(5/2)/(x^2*(a - b*x^2)^(3/2)),x)
Output:
int((c + d*x)^(5/2)/(x^2*(a - b*x^2)^(3/2)), x)
\[ \int \frac {(c+d x)^{5/2}}{x^2 \left (a-b x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:
int((d*x+c)^(5/2)/x^2/(-b*x^2+a)^(3/2),x)
Output:
(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**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 - 10*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 - 10*sqrt(a - b*x**2)*int((sqrt(c + d*x)*x**2)/(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*d**4*x - 6*sqrt(a - b*x**2)*int((s qrt(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 - 9*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 - 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)*a*b*c**2*d**3*x - 3*sqr t(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 + 15*sqrt(a - b*x**2)*int((sqrt(c + d*x)*sqrt(a - ...