Integrand size = 22, antiderivative size = 484 \[ \int \frac {1}{(c+d x)^{5/2} \left (a-b x^2\right )^{3/2}} \, dx=\frac {2 d}{3 \left (b c^2-a d^2\right ) (c+d x)^{3/2} \sqrt {a-b x^2}}+\frac {16 b c d}{3 \left (b c^2-a d^2\right )^2 \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {b \sqrt {c+d x} \left (a d \left (27 b c^2+5 a d^2\right )-b c \left (3 b c^2+29 a d^2\right ) x\right )}{3 a \left (b c^2-a d^2\right )^3 \sqrt {a-b x^2}}+\frac {b^{3/2} c \left (3 b c^2+29 a d^2\right ) \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{3 \sqrt {a} \left (b c^2-a d^2\right )^3 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {\sqrt {b} \left (3 b c^2+5 a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{3 \sqrt {a} \left (b c^2-a d^2\right )^2 \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:
2/3*d/(-a*d^2+b*c^2)/(d*x+c)^(3/2)/(-b*x^2+a)^(1/2)+16/3*b*c*d/(-a*d^2+b*c ^2)^2/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)-1/3*b*(d*x+c)^(1/2)*(a*d*(5*a*d^2+27* b*c^2)-b*c*(29*a*d^2+3*b*c^2)*x)/a/(-a*d^2+b*c^2)^3/(-b*x^2+a)^(1/2)+1/3*b ^(3/2)*c*(29*a*d^2+3*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^(1/2)/(-a*d^2+b*c^2)^3/(b^(1/2)*(d*x+c)/(b^(1/2)*c+a^(1/2)*d) )^(1/2)/(-b*x^2+a)^(1/2)-1/3*b^(1/2)*(5*a*d^2+3*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^(1 /2)/(-a*d^2+b*c^2)^2/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 2.18 (sec) , antiderivative size = 612, normalized size of antiderivative = 1.26 \[ \int \frac {1}{(c+d x)^{5/2} \left (a-b x^2\right )^{3/2}} \, dx=\frac {d \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (2 a^3 d^5+3 b^3 c^3 x (c+d x)^2-a^2 b d^3 \left (25 c^2+26 c d x+5 d^2 x^2\right )+a b^2 c d \left (-9 c^3-9 c^2 d x+31 c d^2 x^2+29 d^3 x^3\right )\right )+b (c+d x) \left (c d^2 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (3 b c^2+29 a d^2\right ) \left (a-b x^2\right )+i \sqrt {b} c \left (3 b^{3/2} c^3-3 \sqrt {a} b c^2 d+29 a \sqrt {b} c d^2-29 a^{3/2} d^3\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right )|\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )+i \sqrt {a} d \left (3 b^{3/2} c^3-27 \sqrt {a} b c^2 d+29 a \sqrt {b} c d^2-5 a^{3/2} d^3\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right ),\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )\right )}{3 a d \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (b c^2-a d^2\right )^3 (c+d x)^{3/2} \sqrt {a-b x^2}} \] Input:
Integrate[1/((c + d*x)^(5/2)*(a - b*x^2)^(3/2)),x]
Output:
(d*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(2*a^3*d^5 + 3*b^3*c^3*x*(c + d*x)^2 - a ^2*b*d^3*(25*c^2 + 26*c*d*x + 5*d^2*x^2) + a*b^2*c*d*(-9*c^3 - 9*c^2*d*x + 31*c*d^2*x^2 + 29*d^3*x^3)) + b*(c + d*x)*(c*d^2*Sqrt[-c + (Sqrt[a]*d)/Sq rt[b]]*(3*b*c^2 + 29*a*d^2)*(a - b*x^2) + I*Sqrt[b]*c*(3*b^(3/2)*c^3 - 3*S qrt[a]*b*c^2*d + 29*a*Sqrt[b]*c*d^2 - 29*a^(3/2)*d^3)*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)*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]*d*(3*b^(3 /2)*c^3 - 27*Sqrt[a]*b*c^2*d + 29*a*Sqrt[b]*c*d^2 - 5*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)*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*a*d *Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(b*c^2 - a*d^2)^3*(c + d*x)^(3/2)*Sqrt[a - b*x^2])
Time = 0.68 (sec) , antiderivative size = 509, normalized size of antiderivative = 1.05, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.591, Rules used = {496, 27, 688, 27, 688, 27, 600, 509, 508, 327, 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.
| \(\displaystyle \int \frac {1}{\left (a-b x^2\right )^{3/2} (c+d x)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 496 |
| \(\displaystyle \frac {\int -\frac {d (5 a d-3 b c x)}{2 (c+d x)^{5/2} \sqrt {a-b x^2}}dx}{a \left (b c^2-a d^2\right )}-\frac {a d-b c x}{a \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {d \int \frac {5 a d-3 b c x}{(c+d x)^{5/2} \sqrt {a-b x^2}}dx}{2 a \left (b c^2-a d^2\right )}-\frac {a d-b c x}{a \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 688 |
| \(\displaystyle -\frac {d \left (\frac {2 \int \frac {b \left (24 a c d-\left (3 b c^2+5 a d^2\right ) x\right )}{2 (c+d x)^{3/2} \sqrt {a-b x^2}}dx}{3 \left (b c^2-a d^2\right )}+\frac {2 \sqrt {a-b x^2} \left (5 a d^2+3 b c^2\right )}{3 (c+d x)^{3/2} \left (b c^2-a d^2\right )}\right )}{2 a \left (b c^2-a d^2\right )}-\frac {a d-b c x}{a \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {d \left (\frac {b \int \frac {24 a c d-\left (3 b c^2+5 a d^2\right ) x}{(c+d x)^{3/2} \sqrt {a-b x^2}}dx}{3 \left (b c^2-a d^2\right )}+\frac {2 \sqrt {a-b x^2} \left (5 a d^2+3 b c^2\right )}{3 (c+d x)^{3/2} \left (b c^2-a d^2\right )}\right )}{2 a \left (b c^2-a d^2\right )}-\frac {a d-b c x}{a \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 688 |
| \(\displaystyle -\frac {d \left (\frac {b \left (\frac {2 \int \frac {a d \left (27 b c^2+5 a d^2\right )+b c \left (3 b c^2+29 a d^2\right ) x}{2 \sqrt {c+d x} \sqrt {a-b x^2}}dx}{b c^2-a d^2}+\frac {2 c \sqrt {a-b x^2} \left (29 a d^2+3 b c^2\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}\right )}{3 \left (b c^2-a d^2\right )}+\frac {2 \sqrt {a-b x^2} \left (5 a d^2+3 b c^2\right )}{3 (c+d x)^{3/2} \left (b c^2-a d^2\right )}\right )}{2 a \left (b c^2-a d^2\right )}-\frac {a d-b c x}{a \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {d \left (\frac {b \left (\frac {\int \frac {a d \left (27 b c^2+5 a d^2\right )+b c \left (3 b c^2+29 a d^2\right ) x}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{b c^2-a d^2}+\frac {2 c \sqrt {a-b x^2} \left (29 a d^2+3 b c^2\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}\right )}{3 \left (b c^2-a d^2\right )}+\frac {2 \sqrt {a-b x^2} \left (5 a d^2+3 b c^2\right )}{3 (c+d x)^{3/2} \left (b c^2-a d^2\right )}\right )}{2 a \left (b c^2-a d^2\right )}-\frac {a d-b c x}{a \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 600 |
| \(\displaystyle -\frac {d \left (\frac {b \left (\frac {\frac {b c \left (29 a d^2+3 b c^2\right ) \int \frac {\sqrt {c+d x}}{\sqrt {a-b x^2}}dx}{d}-\frac {\left (b c^2-a d^2\right ) \left (5 a d^2+3 b c^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}}{b c^2-a d^2}+\frac {2 c \sqrt {a-b x^2} \left (29 a d^2+3 b c^2\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}\right )}{3 \left (b c^2-a d^2\right )}+\frac {2 \sqrt {a-b x^2} \left (5 a d^2+3 b c^2\right )}{3 (c+d x)^{3/2} \left (b c^2-a d^2\right )}\right )}{2 a \left (b c^2-a d^2\right )}-\frac {a d-b c x}{a \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 509 |
| \(\displaystyle -\frac {d \left (\frac {b \left (\frac {\frac {b c \sqrt {1-\frac {b x^2}{a}} \left (29 a d^2+3 b c^2\right ) \int \frac {\sqrt {c+d x}}{\sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}-\frac {\left (b c^2-a d^2\right ) \left (5 a d^2+3 b c^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}}{b c^2-a d^2}+\frac {2 c \sqrt {a-b x^2} \left (29 a d^2+3 b c^2\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}\right )}{3 \left (b c^2-a d^2\right )}+\frac {2 \sqrt {a-b x^2} \left (5 a d^2+3 b c^2\right )}{3 (c+d x)^{3/2} \left (b c^2-a d^2\right )}\right )}{2 a \left (b c^2-a d^2\right )}-\frac {a d-b c x}{a \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 508 |
| \(\displaystyle -\frac {d \left (\frac {b \left (\frac {-\frac {\left (b c^2-a d^2\right ) \left (5 a d^2+3 b c^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {2 \sqrt {a} \sqrt {b} c \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (29 a d^2+3 b c^2\right ) \int \frac {\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}}}{\sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{b c^2-a d^2}+\frac {2 c \sqrt {a-b x^2} \left (29 a d^2+3 b c^2\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}\right )}{3 \left (b c^2-a d^2\right )}+\frac {2 \sqrt {a-b x^2} \left (5 a d^2+3 b c^2\right )}{3 (c+d x)^{3/2} \left (b c^2-a d^2\right )}\right )}{2 a \left (b c^2-a d^2\right )}-\frac {a d-b c x}{a \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle -\frac {d \left (\frac {b \left (\frac {-\frac {\left (b c^2-a d^2\right ) \left (5 a d^2+3 b c^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {2 \sqrt {a} \sqrt {b} c \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (29 a d^2+3 b c^2\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{b c^2-a d^2}+\frac {2 c \sqrt {a-b x^2} \left (29 a d^2+3 b c^2\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}\right )}{3 \left (b c^2-a d^2\right )}+\frac {2 \sqrt {a-b x^2} \left (5 a d^2+3 b c^2\right )}{3 (c+d x)^{3/2} \left (b c^2-a d^2\right )}\right )}{2 a \left (b c^2-a d^2\right )}-\frac {a d-b c x}{a \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 512 |
| \(\displaystyle -\frac {d \left (\frac {b \left (\frac {-\frac {\sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \left (5 a d^2+3 b c^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}-\frac {2 \sqrt {a} \sqrt {b} c \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (29 a d^2+3 b c^2\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{b c^2-a d^2}+\frac {2 c \sqrt {a-b x^2} \left (29 a d^2+3 b c^2\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}\right )}{3 \left (b c^2-a d^2\right )}+\frac {2 \sqrt {a-b x^2} \left (5 a d^2+3 b c^2\right )}{3 (c+d x)^{3/2} \left (b c^2-a d^2\right )}\right )}{2 a \left (b c^2-a d^2\right )}-\frac {a d-b c x}{a \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 511 |
| \(\displaystyle -\frac {d \left (\frac {b \left (\frac {\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \left (5 a d^2+3 b c^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \int \frac {1}{\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {2 \sqrt {a} \sqrt {b} c \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (29 a d^2+3 b c^2\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{b c^2-a d^2}+\frac {2 c \sqrt {a-b x^2} \left (29 a d^2+3 b c^2\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}\right )}{3 \left (b c^2-a d^2\right )}+\frac {2 \sqrt {a-b x^2} \left (5 a d^2+3 b c^2\right )}{3 (c+d x)^{3/2} \left (b c^2-a d^2\right )}\right )}{2 a \left (b c^2-a d^2\right )}-\frac {a d-b c x}{a \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle -\frac {d \left (\frac {b \left (\frac {\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \left (5 a d^2+3 b c^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {2 \sqrt {a} \sqrt {b} c \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (29 a d^2+3 b c^2\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{b c^2-a d^2}+\frac {2 c \sqrt {a-b x^2} \left (29 a d^2+3 b c^2\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}\right )}{3 \left (b c^2-a d^2\right )}+\frac {2 \sqrt {a-b x^2} \left (5 a d^2+3 b c^2\right )}{3 (c+d x)^{3/2} \left (b c^2-a d^2\right )}\right )}{2 a \left (b c^2-a d^2\right )}-\frac {a d-b c x}{a \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )}\) |
Input:
Int[1/((c + d*x)^(5/2)*(a - b*x^2)^(3/2)),x]
Output:
-((a*d - b*c*x)/(a*(b*c^2 - a*d^2)*(c + d*x)^(3/2)*Sqrt[a - b*x^2])) - (d* ((2*(3*b*c^2 + 5*a*d^2)*Sqrt[a - b*x^2])/(3*(b*c^2 - a*d^2)*(c + d*x)^(3/2 )) + (b*((2*c*(3*b*c^2 + 29*a*d^2)*Sqrt[a - b*x^2])/((b*c^2 - a*d^2)*Sqrt[ c + d*x]) + ((-2*Sqrt[a]*Sqrt[b]*c*(3*b*c^2 + 29*a*d^2)*Sqrt[c + d*x]*Sqrt [1 - (b*x^2)/a]*EllipticE[ArcSin[Sqrt[1 - (Sqrt[b]*x)/Sqrt[a]]/Sqrt[2]], ( 2*d)/((Sqrt[b]*c)/Sqrt[a] + d)])/(d*Sqrt[(Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[a]*d)]*Sqrt[a - b*x^2]) + (2*Sqrt[a]*(b*c^2 - a*d^2)*(3*b*c^2 + 5*a*d ^2)*Sqrt[(Sqrt[b]*(c + d*x))/(Sqrt[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)])/(Sqrt[b]*d*Sqrt[c + d*x]*Sqrt[a - b*x^2]))/(b*c^2 - a*d^2 )))/(3*(b*c^2 - a*d^2))))/(2*a*(b*c^2 - a*d^2))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ (Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[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]
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ (-(a*d + b*c*x))*(c + d*x)^(n + 1)*((a + b*x^2)^(p + 1)/(2*a*(p + 1)*(b*c^2 + a*d^2))), x] + Simp[1/(2*a*(p + 1)*(b*c^2 + a*d^2)) Int[(c + d*x)^n*(a + b*x^2)^(p + 1)*Simp[b*c^2*(2*p + 3) + a*d^2*(n + 2*p + 3) + b*c*d*(n + 2 *p + 4)*x, x], x], x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[p, -1] && IntQuad raticQ[a, 0, b, c, d, n, p, x]
Int[Sqrt[(c_) + (d_.)*(x_)]/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> With[{q = Rt[-b/a, 2]}, Simp[-2*(Sqrt[c + d*x]/(Sqrt[a]*q*Sqrt[q*((c + d*x)/(d + c *q))])) Subst[Int[Sqrt[1 - 2*d*(x^2/(d + c*q))]/Sqrt[1 - x^2], x], x, Sqr t[(1 - q*x)/2]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[a, 0]
Int[Sqrt[(c_) + (d_.)*(x_)]/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[Sq rt[1 + b*(x^2/a)]/Sqrt[a + b*x^2] Int[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[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[((A_.) + (B_.)*(x_))/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2] ), x_Symbol] :> Simp[B/d Int[Sqrt[c + d*x]/Sqrt[a + b*x^2], x], x] - Simp [(B*c - A*d)/d Int[1/(Sqrt[c + d*x]*Sqrt[a + b*x^2]), x], x] /; FreeQ[{a, b, c, d, A, B}, x] && NegQ[b/a]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[(e*f - d*g)*(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1)/( (m + 1)*(c*d^2 + a*e^2))), x] + Simp[1/((m + 1)*(c*d^2 + a*e^2)) Int[(d + e*x)^(m + 1)*(a + c*x^2)^p*Simp[(c*d*f + a*e*g)*(m + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, p}, x] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Leaf count of result is larger than twice the leaf count of optimal. \(942\) vs. \(2(408)=816\).
Time = 8.15 (sec) , antiderivative size = 943, normalized size of antiderivative = 1.95
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (d x +c \right ) \left (-b \,x^{2}+a \right )}\, \left (-\frac {2 d \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}{3 \left (a \,d^{2}-b \,c^{2}\right )^{2} \left (x +\frac {c}{d}\right )^{2}}+\frac {20 \left (-b d \,x^{2}+a d \right ) d^{2} b c}{3 \left (a \,d^{2}-b \,c^{2}\right )^{3} \sqrt {\left (x +\frac {c}{d}\right ) \left (-b d \,x^{2}+a d \right )}}-\frac {2 \left (-b d x -b c \right ) \left (-\frac {b c \left (3 a \,d^{2}+b \,c^{2}\right ) x}{2 a \left (a \,d^{2}-b \,c^{2}\right )^{3}}+\frac {d \left (a \,d^{2}+3 b \,c^{2}\right )}{2 \left (a \,d^{2}-b \,c^{2}\right )^{3}}\right )}{\sqrt {\left (x^{2}-\frac {a}{b}\right ) \left (-b d x -b c \right )}}+\frac {2 \left (\frac {b \,d^{2}}{3 \left (a \,d^{2}-b \,c^{2}\right )^{2}}+\frac {10 b^{2} c^{2} d^{2}}{3 \left (a \,d^{2}-b \,c^{2}\right )^{3}}+\frac {b \left (a \,d^{2}+b \,c^{2}\right )}{\left (a \,d^{2}-b \,c^{2}\right )^{2} a}-\frac {b \,d^{2} \left (a \,d^{2}+3 b \,c^{2}\right )}{2 \left (a \,d^{2}-b \,c^{2}\right )^{3}}+\frac {b^{2} c^{2} \left (3 a \,d^{2}+b \,c^{2}\right )}{a \left (a \,d^{2}-b \,c^{2}\right )^{3}}\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 {10 b^{2} c \,d^{3}}{3 \left (a \,d^{2}-b \,c^{2}\right )^{3}}+\frac {b^{2} c d \left (3 a \,d^{2}+b \,c^{2}\right )}{2 a \left (a \,d^{2}-b \,c^{2}\right )^{3}}\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}}\right )}{\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}\) | \(943\) |
| default | \(\text {Expression too large to display}\) | \(2491\) |
Input:
int(1/(d*x+c)^(5/2)/(-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)*(-2/3*d/(a*d^2-b *c^2)^2*(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)/(x+c/d)^2+20/3*(-b*d*x^2+a*d)*d ^2/(a*d^2-b*c^2)^3*b*c/((x+c/d)*(-b*d*x^2+a*d))^(1/2)-2*(-b*d*x-b*c)*(-1/2 *b*c*(3*a*d^2+b*c^2)/a/(a*d^2-b*c^2)^3*x+1/2*d*(a*d^2+3*b*c^2)/(a*d^2-b*c^ 2)^3)/((x^2-a/b)*(-b*d*x-b*c))^(1/2)+2*(1/3*b*d^2/(a*d^2-b*c^2)^2+10/3*b^2 *c^2*d^2/(a*d^2-b*c^2)^3+b/(a*d^2-b*c^2)^2*(a*d^2+b*c^2)/a-1/2*b*d^2*(a*d^ 2+3*b*c^2)/(a*d^2-b*c^2)^3+b^2*c^2*(3*a*d^2+b*c^2)/a/(a*d^2-b*c^2)^3)*(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*(10/3*b^2*c*d^3/(a*d^2-b*c^2)^3+1/2*b^2*c*d*(3*a*d^2+b*c^2)/a/(a*d^2-b*c ^2)^3)*(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))))
Leaf count of result is larger than twice the leaf count of optimal. 925 vs. \(2 (408) = 816\).
Time = 0.13 (sec) , antiderivative size = 925, normalized size of antiderivative = 1.91 \[ \int \frac {1}{(c+d x)^{5/2} \left (a-b x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:
integrate(1/(d*x+c)^(5/2)/(-b*x^2+a)^(3/2),x, algorithm="fricas")
Output:
-1/9*((3*a*b^2*c^6 - 52*a^2*b*c^4*d^2 - 15*a^3*c^2*d^4 - (3*b^3*c^4*d^2 - 52*a*b^2*c^2*d^4 - 15*a^2*b*d^6)*x^4 - 2*(3*b^3*c^5*d - 52*a*b^2*c^3*d^3 - 15*a^2*b*c*d^5)*x^3 - (3*b^3*c^6 - 55*a*b^2*c^4*d^2 + 37*a^2*b*c^2*d^4 + 15*a^3*d^6)*x^2 + 2*(3*a*b^2*c^5*d - 52*a^2*b*c^3*d^3 - 15*a^3*c*d^5)*x)*s qrt(-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) + 3*(3*a*b^2*c^5*d + 29*a^2*b*c^3 *d^3 - (3*b^3*c^3*d^3 + 29*a*b^2*c*d^5)*x^4 - 2*(3*b^3*c^4*d^2 + 29*a*b^2* c^2*d^4)*x^3 - (3*b^3*c^5*d + 26*a*b^2*c^3*d^3 - 29*a^2*b*c*d^5)*x^2 + 2*( 3*a*b^2*c^4*d^2 + 29*a^2*b*c^2*d^4)*x)*sqrt(-b*d)*weierstrassZeta(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), weierstrassPInve rse(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)) + 3*(9*a*b^2*c^4*d^2 + 25*a^2*b*c^2*d^4 - 2*a^3*d^6 - (3*b ^3*c^3*d^3 + 29*a*b^2*c*d^5)*x^3 - (6*b^3*c^4*d^2 + 31*a*b^2*c^2*d^4 - 5*a ^2*b*d^6)*x^2 - (3*b^3*c^5*d - 9*a*b^2*c^3*d^3 - 26*a^2*b*c*d^5)*x)*sqrt(- b*x^2 + a)*sqrt(d*x + c))/(a^2*b^3*c^8*d - 3*a^3*b^2*c^6*d^3 + 3*a^4*b*c^4 *d^5 - a^5*c^2*d^7 - (a*b^4*c^6*d^3 - 3*a^2*b^3*c^4*d^5 + 3*a^3*b^2*c^2*d^ 7 - a^4*b*d^9)*x^4 - 2*(a*b^4*c^7*d^2 - 3*a^2*b^3*c^5*d^4 + 3*a^3*b^2*c^3* d^6 - a^4*b*c*d^8)*x^3 - (a*b^4*c^8*d - 4*a^2*b^3*c^6*d^3 + 6*a^3*b^2*c^4* d^5 - 4*a^4*b*c^2*d^7 + a^5*d^9)*x^2 + 2*(a^2*b^3*c^7*d^2 - 3*a^3*b^2*c^5* d^4 + 3*a^4*b*c^3*d^6 - a^5*c*d^8)*x)
\[ \int \frac {1}{(c+d x)^{5/2} \left (a-b x^2\right )^{3/2}} \, dx=\int \frac {1}{\left (a - b x^{2}\right )^{\frac {3}{2}} \left (c + d x\right )^{\frac {5}{2}}}\, dx \] Input:
integrate(1/(d*x+c)**(5/2)/(-b*x**2+a)**(3/2),x)
Output:
Integral(1/((a - b*x**2)**(3/2)*(c + d*x)**(5/2)), x)
\[ \int \frac {1}{(c+d x)^{5/2} \left (a-b x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-b x^{2} + a\right )}^{\frac {3}{2}} {\left (d x + c\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate(1/(d*x+c)^(5/2)/(-b*x^2+a)^(3/2),x, algorithm="maxima")
Output:
integrate(1/((-b*x^2 + a)^(3/2)*(d*x + c)^(5/2)), x)
\[ \int \frac {1}{(c+d x)^{5/2} \left (a-b x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-b x^{2} + a\right )}^{\frac {3}{2}} {\left (d x + c\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate(1/(d*x+c)^(5/2)/(-b*x^2+a)^(3/2),x, algorithm="giac")
Output:
integrate(1/((-b*x^2 + a)^(3/2)*(d*x + c)^(5/2)), x)
Timed out. \[ \int \frac {1}{(c+d x)^{5/2} \left (a-b x^2\right )^{3/2}} \, dx=\int \frac {1}{{\left (a-b\,x^2\right )}^{3/2}\,{\left (c+d\,x\right )}^{5/2}} \,d x \] Input:
int(1/((a - b*x^2)^(3/2)*(c + d*x)^(5/2)),x)
Output:
int(1/((a - b*x^2)^(3/2)*(c + d*x)^(5/2)), x)
\[ \int \frac {1}{(c+d x)^{5/2} \left (a-b x^2\right )^{3/2}} \, dx=\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{b^{2} d^{3} x^{7}+3 b^{2} c \,d^{2} x^{6}-2 a b \,d^{3} x^{5}+3 b^{2} c^{2} d \,x^{5}-6 a b c \,d^{2} x^{4}+b^{2} c^{3} x^{4}+a^{2} d^{3} x^{3}-6 a b \,c^{2} d \,x^{3}+3 a^{2} c \,d^{2} x^{2}-2 a b \,c^{3} x^{2}+3 a^{2} c^{2} d x +a^{2} c^{3}}d x \] Input:
int(1/(d*x+c)^(5/2)/(-b*x^2+a)^(3/2),x)
Output:
int((sqrt(c + d*x)*sqrt(a - b*x**2))/(a**2*c**3 + 3*a**2*c**2*d*x + 3*a**2 *c*d**2*x**2 + a**2*d**3*x**3 - 2*a*b*c**3*x**2 - 6*a*b*c**2*d*x**3 - 6*a* b*c*d**2*x**4 - 2*a*b*d**3*x**5 + b**2*c**3*x**4 + 3*b**2*c**2*d*x**5 + 3* b**2*c*d**2*x**6 + b**2*d**3*x**7),x)