Integrand size = 25, antiderivative size = 495 \[ \int \frac {(c+d x)^{5/2}}{x^3 \sqrt {a-b x^2}} \, dx=-\frac {c^2 \sqrt {c+d x} \sqrt {a-b x^2}}{2 a x^2}-\frac {9 c d \sqrt {c+d x} \sqrt {a-b x^2}}{4 a x}+\frac {9 \sqrt {b} c d \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{4 \sqrt {a} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {d \left (11 b c^2+8 a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{4 \sqrt {a} \sqrt {b} \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {c \left (4 b c^2+15 a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{4 a \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:
-1/2*c^2*(d*x+c)^(1/2)*(-b*x^2+a)^(1/2)/a/x^2-9/4*c*d*(d*x+c)^(1/2)*(-b*x^ 2+a)^(1/2)/a/x+9/4*b^(1/2)*c*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^(1/2)/(b^(1/2)*(d*x+c)/(b^(1/2)*c+a^(1/2)*d))^(1/2)/(-b*x^ 2+a)^(1/2)-1/4*d*(8*a*d^2+11*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)/b^(1/2)/(d*x+c) ^(1/2)/(-b*x^2+a)^(1/2)-1/4*c*(15*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/(d *x+c)^(1/2)/(-b*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 23.72 (sec) , antiderivative size = 863, normalized size of antiderivative = 1.74 \[ \int \frac {(c+d x)^{5/2}}{x^3 \sqrt {a-b x^2}} \, dx=\frac {\sqrt {a-b x^2} \left (-\frac {c (c+d x) (2 c+9 d x)}{a x^2}-\frac {-9 b c^3 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}+9 a c d^2 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}+18 b c^2 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} (c+d x)-9 b c \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} (c+d x)^2+9 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^2-9 \sqrt {a} \sqrt {b} c d+7 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 )+15 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 \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)^(5/2)/(x^3*Sqrt[a - b*x^2]),x]
Output:
(Sqrt[a - b*x^2]*(-((c*(c + d*x)*(2*c + 9*d*x))/(a*x^2)) - (-9*b*c^3*Sqrt[ -c + (Sqrt[a]*d)/Sqrt[b]] + 9*a*c*d^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]] + 18* b*c^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(c + d*x) - 9*b*c*Sqrt[-c + (Sqrt[a]* d)/Sqrt[b]]*(c + d*x)^2 + (9*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)] - I*(2*b* c^2 - 9*Sqrt[a]*Sqrt[b]*c*d + 7*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)*Elli pticF[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)/(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)] + (15*I)*a*d^2*Sqrt[(d*(Sqrt[a]/Sqrt[b] + x))/(c + d*x)]*Sq rt[-(((Sqrt[a]*d)/Sqrt[b] - d*x)/(c + d*x))]*(c + d*x)^(3/2)*EllipticPi[(S qrt[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*Sqr t[-c + (Sqrt[a]*d)/Sqrt[b]]*(-a + b*x^2))))/(4*Sqrt[c + d*x])
Time = 1.54 (sec) , antiderivative size = 544, normalized size of antiderivative = 1.10, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.640, Rules used = {636, 2352, 25, 2351, 600, 509, 508, 327, 512, 511, 321, 633, 632, 186, 413, 412}
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.
| \(\displaystyle \int \frac {(c+d x)^{5/2}}{x^3 \sqrt {a-b x^2}} \, dx\) |
| \(\Big \downarrow \) 636 |
| \(\displaystyle \frac {\int \frac {9 a d c^2+2 \left (b c^2+6 a d^2\right ) x c+d \left (b c^2+4 a d^2\right ) x^2}{x^2 \sqrt {c+d x} \sqrt {a-b x^2}}dx}{4 a}-\frac {c^2 \sqrt {a-b x^2} \sqrt {c+d x}}{2 a x^2}\) |
| \(\Big \downarrow \) 2352 |
| \(\displaystyle \frac {-\frac {\int -\frac {-9 a b d^2 x^2 c^2+a \left (4 b c^2+15 a d^2\right ) c^2+2 a d \left (b c^2+4 a d^2\right ) x c}{x \sqrt {c+d x} \sqrt {a-b x^2}}dx}{2 a c}-\frac {9 c d \sqrt {a-b x^2} \sqrt {c+d x}}{x}}{4 a}-\frac {c^2 \sqrt {a-b x^2} \sqrt {c+d x}}{2 a x^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {-9 a b d^2 x^2 c^2+a \left (4 b c^2+15 a d^2\right ) c^2+2 a d \left (b c^2+4 a d^2\right ) x c}{x \sqrt {c+d x} \sqrt {a-b x^2}}dx}{2 a c}-\frac {9 c d \sqrt {a-b x^2} \sqrt {c+d x}}{x}}{4 a}-\frac {c^2 \sqrt {a-b x^2} \sqrt {c+d x}}{2 a x^2}\) |
| \(\Big \downarrow \) 2351 |
| \(\displaystyle \frac {\frac {\int \frac {2 a b d c^3-9 a b d^2 x c^2+8 a^2 d^3 c}{\sqrt {c+d x} \sqrt {a-b x^2}}dx+a c^2 \left (15 a d^2+4 b c^2\right ) \int \frac {1}{x \sqrt {c+d x} \sqrt {a-b x^2}}dx}{2 a c}-\frac {9 c d \sqrt {a-b x^2} \sqrt {c+d x}}{x}}{4 a}-\frac {c^2 \sqrt {a-b x^2} \sqrt {c+d x}}{2 a x^2}\) |
| \(\Big \downarrow \) 600 |
| \(\displaystyle \frac {\frac {a c^2 \left (15 a d^2+4 b c^2\right ) \int \frac {1}{x \sqrt {c+d x} \sqrt {a-b x^2}}dx+a c d \left (8 a d^2+11 b c^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx-9 a b c^2 d \int \frac {\sqrt {c+d x}}{\sqrt {a-b x^2}}dx}{2 a c}-\frac {9 c d \sqrt {a-b x^2} \sqrt {c+d x}}{x}}{4 a}-\frac {c^2 \sqrt {a-b x^2} \sqrt {c+d x}}{2 a x^2}\) |
| \(\Big \downarrow \) 509 |
| \(\displaystyle \frac {\frac {a c^2 \left (15 a d^2+4 b c^2\right ) \int \frac {1}{x \sqrt {c+d x} \sqrt {a-b x^2}}dx+a c d \left (8 a d^2+11 b c^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx-\frac {9 a b c^2 d \sqrt {1-\frac {b x^2}{a}} \int \frac {\sqrt {c+d x}}{\sqrt {1-\frac {b x^2}{a}}}dx}{\sqrt {a-b x^2}}}{2 a c}-\frac {9 c d \sqrt {a-b x^2} \sqrt {c+d x}}{x}}{4 a}-\frac {c^2 \sqrt {a-b x^2} \sqrt {c+d x}}{2 a x^2}\) |
| \(\Big \downarrow \) 508 |
| \(\displaystyle \frac {\frac {\frac {18 a^{3/2} \sqrt {b} c^2 d \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \int \frac {\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}}}{\sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}+a c^2 \left (15 a d^2+4 b c^2\right ) \int \frac {1}{x \sqrt {c+d x} \sqrt {a-b x^2}}dx+a c d \left (8 a d^2+11 b c^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{2 a c}-\frac {9 c d \sqrt {a-b x^2} \sqrt {c+d x}}{x}}{4 a}-\frac {c^2 \sqrt {a-b x^2} \sqrt {c+d x}}{2 a x^2}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {\frac {a c^2 \left (15 a d^2+4 b c^2\right ) \int \frac {1}{x \sqrt {c+d x} \sqrt {a-b x^2}}dx+a c d \left (8 a d^2+11 b c^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx+\frac {18 a^{3/2} \sqrt {b} c^2 d \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{2 a c}-\frac {9 c d \sqrt {a-b x^2} \sqrt {c+d x}}{x}}{4 a}-\frac {c^2 \sqrt {a-b x^2} \sqrt {c+d x}}{2 a x^2}\) |
| \(\Big \downarrow \) 512 |
| \(\displaystyle \frac {\frac {a c^2 \left (15 a d^2+4 b c^2\right ) \int \frac {1}{x \sqrt {c+d x} \sqrt {a-b x^2}}dx+\frac {a c d \sqrt {1-\frac {b x^2}{a}} \left (8 a d^2+11 b c^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}}}dx}{\sqrt {a-b x^2}}+\frac {18 a^{3/2} \sqrt {b} c^2 d \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{2 a c}-\frac {9 c d \sqrt {a-b x^2} \sqrt {c+d x}}{x}}{4 a}-\frac {c^2 \sqrt {a-b x^2} \sqrt {c+d x}}{2 a x^2}\) |
| \(\Big \downarrow \) 511 |
| \(\displaystyle \frac {\frac {-\frac {2 a^{3/2} c d \sqrt {1-\frac {b x^2}{a}} \left (8 a d^2+11 b c^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \int \frac {1}{\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} \sqrt {a-b x^2} \sqrt {c+d x}}+a c^2 \left (15 a d^2+4 b c^2\right ) \int \frac {1}{x \sqrt {c+d x} \sqrt {a-b x^2}}dx+\frac {18 a^{3/2} \sqrt {b} c^2 d \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{2 a c}-\frac {9 c d \sqrt {a-b x^2} \sqrt {c+d x}}{x}}{4 a}-\frac {c^2 \sqrt {a-b x^2} \sqrt {c+d x}}{2 a x^2}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {\frac {a c^2 \left (15 a d^2+4 b c^2\right ) \int \frac {1}{x \sqrt {c+d x} \sqrt {a-b x^2}}dx-\frac {2 a^{3/2} c d \sqrt {1-\frac {b x^2}{a}} \left (8 a d^2+11 b c^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} \sqrt {a-b x^2} \sqrt {c+d x}}+\frac {18 a^{3/2} \sqrt {b} c^2 d \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{2 a c}-\frac {9 c d \sqrt {a-b x^2} \sqrt {c+d x}}{x}}{4 a}-\frac {c^2 \sqrt {a-b x^2} \sqrt {c+d x}}{2 a x^2}\) |
| \(\Big \downarrow \) 633 |
| \(\displaystyle \frac {\frac {\frac {a c^2 \sqrt {1-\frac {b x^2}{a}} \left (15 a d^2+4 b c^2\right ) \int \frac {1}{x \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}}}dx}{\sqrt {a-b x^2}}-\frac {2 a^{3/2} c d \sqrt {1-\frac {b x^2}{a}} \left (8 a d^2+11 b c^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} \sqrt {a-b x^2} \sqrt {c+d x}}+\frac {18 a^{3/2} \sqrt {b} c^2 d \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{2 a c}-\frac {9 c d \sqrt {a-b x^2} \sqrt {c+d x}}{x}}{4 a}-\frac {c^2 \sqrt {a-b x^2} \sqrt {c+d x}}{2 a x^2}\) |
| \(\Big \downarrow \) 632 |
| \(\displaystyle \frac {\frac {\frac {a c^2 \sqrt {1-\frac {b x^2}{a}} \left (15 a d^2+4 b c^2\right ) \int \frac {1}{x \sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}} \sqrt {\frac {\sqrt {b} x}{\sqrt {a}}+1} \sqrt {c+d x}}dx}{\sqrt {a-b x^2}}-\frac {2 a^{3/2} c d \sqrt {1-\frac {b x^2}{a}} \left (8 a d^2+11 b c^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} \sqrt {a-b x^2} \sqrt {c+d x}}+\frac {18 a^{3/2} \sqrt {b} c^2 d \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{2 a c}-\frac {9 c d \sqrt {a-b x^2} \sqrt {c+d x}}{x}}{4 a}-\frac {c^2 \sqrt {a-b x^2} \sqrt {c+d x}}{2 a x^2}\) |
| \(\Big \downarrow \) 186 |
| \(\displaystyle \frac {\frac {-\frac {2 a c^2 \sqrt {1-\frac {b x^2}{a}} \left (15 a d^2+4 b c^2\right ) \int \frac {\sqrt {a}}{\sqrt {b} x \sqrt {\frac {\sqrt {b} x}{\sqrt {a}}+1} \sqrt {c+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}}}d\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {a-b x^2}}-\frac {2 a^{3/2} c d \sqrt {1-\frac {b x^2}{a}} \left (8 a d^2+11 b c^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} \sqrt {a-b x^2} \sqrt {c+d x}}+\frac {18 a^{3/2} \sqrt {b} c^2 d \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{2 a c}-\frac {9 c d \sqrt {a-b x^2} \sqrt {c+d x}}{x}}{4 a}-\frac {c^2 \sqrt {a-b x^2} \sqrt {c+d x}}{2 a x^2}\) |
| \(\Big \downarrow \) 413 |
| \(\displaystyle \frac {\frac {-\frac {2 a c^2 \sqrt {1-\frac {b x^2}{a}} \left (15 a d^2+4 b c^2\right ) \sqrt {1-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d+\sqrt {b} c}} \int \frac {\sqrt {a}}{\sqrt {b} x \sqrt {\frac {\sqrt {b} x}{\sqrt {a}}+1} \sqrt {1-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b} c+\sqrt {a} d}}}d\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {a-b x^2} \sqrt {-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {\sqrt {a} d}{\sqrt {b}}+c}}-\frac {2 a^{3/2} c d \sqrt {1-\frac {b x^2}{a}} \left (8 a d^2+11 b c^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} \sqrt {a-b x^2} \sqrt {c+d x}}+\frac {18 a^{3/2} \sqrt {b} c^2 d \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{2 a c}-\frac {9 c d \sqrt {a-b x^2} \sqrt {c+d x}}{x}}{4 a}-\frac {c^2 \sqrt {a-b x^2} \sqrt {c+d x}}{2 a x^2}\) |
| \(\Big \downarrow \) 412 |
| \(\displaystyle \frac {\frac {-\frac {2 a^{3/2} c d \sqrt {1-\frac {b x^2}{a}} \left (8 a d^2+11 b c^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} \sqrt {a-b x^2} \sqrt {c+d x}}+\frac {18 a^{3/2} \sqrt {b} c^2 d \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}-\frac {2 a c^2 \sqrt {1-\frac {b x^2}{a}} \left (15 a d^2+4 b c^2\right ) \sqrt {1-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{\sqrt {a-b x^2} \sqrt {-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {\sqrt {a} d}{\sqrt {b}}+c}}}{2 a c}-\frac {9 c d \sqrt {a-b x^2} \sqrt {c+d x}}{x}}{4 a}-\frac {c^2 \sqrt {a-b x^2} \sqrt {c+d x}}{2 a x^2}\) |
Input:
Int[(c + d*x)^(5/2)/(x^3*Sqrt[a - b*x^2]),x]
Output:
-1/2*(c^2*Sqrt[c + d*x]*Sqrt[a - b*x^2])/(a*x^2) + ((-9*c*d*Sqrt[c + d*x]* Sqrt[a - b*x^2])/x + ((18*a^(3/2)*Sqrt[b]*c^2*d*Sqrt[c + d*x]*Sqrt[1 - (b* x^2)/a]*EllipticE[ArcSin[Sqrt[1 - (Sqrt[b]*x)/Sqrt[a]]/Sqrt[2]], (2*d)/((S qrt[b]*c)/Sqrt[a] + d)])/(Sqrt[(Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[a]*d) ]*Sqrt[a - b*x^2]) - (2*a^(3/2)*c*d*(11*b*c^2 + 8*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)])/(Sqr t[b]*Sqrt[c + d*x]*Sqrt[a - b*x^2]) - (2*a*c^2*(4*b*c^2 + 15*a*d^2)*Sqrt[1 - (b*x^2)/a]*Sqrt[1 - (Sqrt[a]*d*(1 - (Sqrt[b]*x)/Sqrt[a]))/(Sqrt[b]*c + Sqrt[a]*d)]*EllipticPi[2, ArcSin[Sqrt[1 - (Sqrt[b]*x)/Sqrt[a]]/Sqrt[2]], ( 2*Sqrt[a]*d)/(Sqrt[b]*c + Sqrt[a]*d)])/(Sqrt[a - b*x^2]*Sqrt[c + (Sqrt[a]* d)/Sqrt[b] - (Sqrt[a]*d*(1 - (Sqrt[b]*x)/Sqrt[a]))/Sqrt[b]]))/(2*a*c))/(4* a)
Int[1/(((a_.) + (b_.)*(x_))*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_ )]*Sqrt[(g_.) + (h_.)*(x_)]), x_] :> Simp[-2 Subst[Int[1/(Simp[b*c - a*d - b*x^2, x]*Sqrt[Simp[(d*e - c*f)/d + f*(x^2/d), x]]*Sqrt[Simp[(d*g - c*h)/ d + h*(x^2/d), x]]), x], x, Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && GtQ[(d*e - c*f)/d, 0]
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[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x _)^2]), x_Symbol] :> Simp[(1/(a*Sqrt[c]*Sqrt[e]*Rt[-d/c, 2]))*EllipticPi[b* (c/(a*d)), ArcSin[Rt[-d/c, 2]*x], c*(f/(d*e))], x] /; FreeQ[{a, b, c, d, e, f}, x] && !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] && !( !GtQ[f/e, 0] && S implerSqrtQ[-f/e, -d/c])
Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x _)^2]), x_Symbol] :> Simp[Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2] Int[1/((a + b*x^2)*Sqrt[1 + (d/c)*x^2]*Sqrt[e + f*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && !GtQ[c, 0]
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[1/((x_)*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] : > With[{q = Rt[-b/a, 2]}, Simp[1/Sqrt[a] Int[1/(x*Sqrt[c + d*x]*Sqrt[1 - q*x]*Sqrt[1 + q*x]), x], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[ a, 0]
Int[1/((x_)*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] : > Simp[Sqrt[1 + b*(x^2/a)]/Sqrt[a + b*x^2] Int[1/(x*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[(((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_))/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[c^(n - 1/2)*(e*x)^(m + 1)*Sqrt[c + d*x]*(Sqrt[a + b*x^2] /(a*e*(m + 1))), x] - Simp[1/(2*a*e*(m + 1)) Int[((e*x)^(m + 1)/(Sqrt[c + d*x]*Sqrt[a + b*x^2]))*ExpandToSum[(2*a*c^(n + 1/2)*(m + 1) + a*c^(n - 1/2 )*d*(2*m + 3)*x + 2*b*c^(n + 1/2)*(m + 2)*x^2 + b*c^(n - 1/2)*d*(2*m + 5)*x ^3 - 2*a*(m + 1)*(c + d*x)^(n + 1/2))/x, x], x], x] /; FreeQ[{a, b, c, d, e }, x] && IGtQ[n + 3/2, 0] && LtQ[m, -1] && IntegerQ[2*m]
Int[((Px_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_.))/(x_), x_S ymbol] :> Int[PolynomialQuotient[Px, x, x]*(c + d*x)^n*(a + b*x^2)^p, x] + Simp[PolynomialRemainder[Px, x, x] Int[(c + d*x)^n*((a + b*x^2)^p/x), x], x] /; FreeQ[{a, b, c, d, n, p}, x] && PolynomialQ[Px, x]
Int[((Px_)*((e_.)*(x_))^(m_))/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x _)^2]), x_Symbol] :> With[{Px0 = Coefficient[Px, x, 0]}, Simp[Px0*(e*x)^(m + 1)*Sqrt[c + d*x]*(Sqrt[a + b*x^2]/(a*c*e*(m + 1))), x] + Simp[1/(2*a*c*e* (m + 1)) Int[((e*x)^(m + 1)/(Sqrt[c + d*x]*Sqrt[a + b*x^2]))*ExpandToSum[ 2*a*c*(m + 1)*((Px - Px0)/x) - Px0*(a*d*(2*m + 3) + 2*b*c*(m + 2)*x + b*d*( 2*m + 5)*x^2), x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && PolynomialQ[Px, x] && LtQ[m, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(819\) vs. \(2(400)=800\).
Time = 4.28 (sec) , antiderivative size = 820, normalized size of antiderivative = 1.66
| method | result | size |
| risch | \(-\frac {c \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, \left (9 d x +2 c \right )}{4 a \,x^{2}}+\frac {\left (\frac {8 a \,d^{3} \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 {2 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}}}\, \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 )}{\sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}-\frac {c \left (15 a \,d^{2}+4 b \,c^{2}\right ) \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}}-\frac {9 c \,d^{2} \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}}\right ) \sqrt {\left (d x +c \right ) \left (-b \,x^{2}+a \right )}}{8 a \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}\) | \(820\) |
| elliptic | \(\frac {\sqrt {\left (d x +c \right ) \left (-b \,x^{2}+a \right )}\, \left (-\frac {c^{2} \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}{2 a \,x^{2}}-\frac {9 c d \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}{4 a x}+\frac {2 \left (d^{3}+\frac {d b \,c^{2}}{4 a}\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 {9 b 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}}}\, \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 \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}-\frac {\left (15 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 \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}\right )}{\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}\) | \(870\) |
| default | \(\text {Expression too large to display}\) | \(1636\) |
Input:
int((d*x+c)^(5/2)/x^3/(-b*x^2+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/4*c*(d*x+c)^(1/2)*(-b*x^2+a)^(1/2)*(9*d*x+2*c)/a/x^2+1/8/a*(8*a*d^3/b*( a*b)^(1/2)*2^(1/2)*((x+1/b*(a*b)^(1/2))*b/(a*b)^(1/2))^(1/2)*((x+c/d)/(c/d -1/b*(a*b)^(1/2)))^(1/2)*(-2*(x-1/b*(a*b)^(1/2))*b/(a*b)^(1/2))^(1/2)/(-b* d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)*EllipticF(1/2*2^(1/2)*((x+1/b*(a*b)^(1/2))* b/(a*b)^(1/2))^(1/2),(-2/b*(a*b)^(1/2)/(c/d-1/b*(a*b)^(1/2)))^(1/2))+2*c^2 *d*(a*b)^(1/2)*2^(1/2)*((x+1/b*(a*b)^(1/2))*b/(a*b)^(1/2))^(1/2)*((x+c/d)/ (c/d-1/b*(a*b)^(1/2)))^(1/2)*(-2*(x-1/b*(a*b)^(1/2))*b/(a*b)^(1/2))^(1/2)/ (-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)*EllipticF(1/2*2^(1/2)*((x+1/b*(a*b)^(1/ 2))*b/(a*b)^(1/2))^(1/2),(-2/b*(a*b)^(1/2)/(c/d-1/b*(a*b)^(1/2)))^(1/2))-c *(15*a*d^2+4*b*c^2)*2^(1/2)*((x+1/b*(a*b)^(1/2))*b/(a*b)^(1/2))^(1/2)*((x+ c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2)*(-2*(x-1/b*(a*b)^(1/2))*b/(a*b)^(1/2))^( 1/2)/(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)*EllipticPi(1/2*2^(1/2)*((x+1/b*(a* b)^(1/2))*b/(a*b)^(1/2))^(1/2),2,(-2/b*(a*b)^(1/2)/(c/d-1/b*(a*b)^(1/2)))^ (1/2))-9*c*d^2*(a*b)^(1/2)*2^(1/2)*((x+1/b*(a*b)^(1/2))*b/(a*b)^(1/2))^(1/ 2)*((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2)*(-2*(x-1/b*(a*b)^(1/2))*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))*Elli pticE(1/2*2^(1/2)*((x+1/b*(a*b)^(1/2))*b/(a*b)^(1/2))^(1/2),(-2/b*(a*b)^(1 /2)/(c/d-1/b*(a*b)^(1/2)))^(1/2))-c/d*EllipticF(1/2*2^(1/2)*((x+1/b*(a*b)^ (1/2))*b/(a*b)^(1/2))^(1/2),(-2/b*(a*b)^(1/2)/(c/d-1/b*(a*b)^(1/2)))^(1/2) )))*((d*x+c)*(-b*x^2+a))^(1/2)/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)
\[ \int \frac {(c+d x)^{5/2}}{x^3 \sqrt {a-b x^2}} \, dx=\int { \frac {{\left (d x + c\right )}^{\frac {5}{2}}}{\sqrt {-b x^{2} + a} x^{3}} \,d x } \] Input:
integrate((d*x+c)^(5/2)/x^3/(-b*x^2+a)^(1/2),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*x^5 - a*x^3), x)
\[ \int \frac {(c+d x)^{5/2}}{x^3 \sqrt {a-b x^2}} \, dx=\int \frac {\left (c + d x\right )^{\frac {5}{2}}}{x^{3} \sqrt {a - b x^{2}}}\, dx \] Input:
integrate((d*x+c)**(5/2)/x**3/(-b*x**2+a)**(1/2),x)
Output:
Integral((c + d*x)**(5/2)/(x**3*sqrt(a - b*x**2)), x)
\[ \int \frac {(c+d x)^{5/2}}{x^3 \sqrt {a-b x^2}} \, dx=\int { \frac {{\left (d x + c\right )}^{\frac {5}{2}}}{\sqrt {-b x^{2} + a} x^{3}} \,d x } \] Input:
integrate((d*x+c)^(5/2)/x^3/(-b*x^2+a)^(1/2),x, algorithm="maxima")
Output:
integrate((d*x + c)^(5/2)/(sqrt(-b*x^2 + a)*x^3), x)
\[ \int \frac {(c+d x)^{5/2}}{x^3 \sqrt {a-b x^2}} \, dx=\int { \frac {{\left (d x + c\right )}^{\frac {5}{2}}}{\sqrt {-b x^{2} + a} x^{3}} \,d x } \] Input:
integrate((d*x+c)^(5/2)/x^3/(-b*x^2+a)^(1/2),x, algorithm="giac")
Output:
integrate((d*x + c)^(5/2)/(sqrt(-b*x^2 + a)*x^3), x)
Timed out. \[ \int \frac {(c+d x)^{5/2}}{x^3 \sqrt {a-b x^2}} \, dx=\int \frac {{\left (c+d\,x\right )}^{5/2}}{x^3\,\sqrt {a-b\,x^2}} \,d x \] Input:
int((c + d*x)^(5/2)/(x^3*(a - b*x^2)^(1/2)),x)
Output:
int((c + d*x)^(5/2)/(x^3*(a - b*x^2)^(1/2)), x)
\[ \int \frac {(c+d x)^{5/2}}{x^3 \sqrt {a-b x^2}} \, dx=\frac {-18 \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, a \,d^{3} x -2 \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, b \,c^{3}-12 \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, b \,c^{2} d x -9 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, x}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) a b \,d^{4} x^{2}-6 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, x}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) b^{2} c^{2} d^{2} x^{2}-18 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{-b d \,x^{5}-b c \,x^{4}+a d \,x^{3}+a c \,x^{2}}d x \right ) a^{2} c \,d^{3} x^{2}-3 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{-b d \,x^{5}-b c \,x^{4}+a d \,x^{3}+a c \,x^{2}}d x \right ) a b \,c^{3} d \,x^{2}-9 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{-b d \,x^{4}-b c \,x^{3}+a d \,x^{2}+a c x}d x \right ) a^{2} d^{4} x^{2}+6 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{-b d \,x^{4}-b c \,x^{3}+a d \,x^{2}+a c x}d x \right ) a b \,c^{2} d^{2} x^{2}+2 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{-b d \,x^{4}-b c \,x^{3}+a d \,x^{2}+a c x}d x \right ) b^{2} c^{4} x^{2}+4 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) a b c \,d^{3} x^{2}+\left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) b^{2} c^{3} d \,x^{2}}{4 a b c \,x^{2}} \] Input:
int((d*x+c)^(5/2)/x^3/(-b*x^2+a)^(1/2),x)
Output:
( - 18*sqrt(c + d*x)*sqrt(a - b*x**2)*a*d**3*x - 2*sqrt(c + d*x)*sqrt(a - b*x**2)*b*c**3 - 12*sqrt(c + d*x)*sqrt(a - b*x**2)*b*c**2*d*x - 9*int((sqr t(c + d*x)*sqrt(a - b*x**2)*x)/(a*c + a*d*x - b*c*x**2 - b*d*x**3),x)*a*b* d**4*x**2 - 6*int((sqrt(c + d*x)*sqrt(a - b*x**2)*x)/(a*c + a*d*x - b*c*x* *2 - b*d*x**3),x)*b**2*c**2*d**2*x**2 - 18*int((sqrt(c + d*x)*sqrt(a - b*x **2))/(a*c*x**2 + a*d*x**3 - b*c*x**4 - b*d*x**5),x)*a**2*c*d**3*x**2 - 3* int((sqrt(c + d*x)*sqrt(a - b*x**2))/(a*c*x**2 + a*d*x**3 - b*c*x**4 - b*d *x**5),x)*a*b*c**3*d*x**2 - 9*int((sqrt(c + d*x)*sqrt(a - b*x**2))/(a*c*x + a*d*x**2 - b*c*x**3 - b*d*x**4),x)*a**2*d**4*x**2 + 6*int((sqrt(c + d*x) *sqrt(a - b*x**2))/(a*c*x + a*d*x**2 - b*c*x**3 - b*d*x**4),x)*a*b*c**2*d* *2*x**2 + 2*int((sqrt(c + d*x)*sqrt(a - b*x**2))/(a*c*x + a*d*x**2 - b*c*x **3 - b*d*x**4),x)*b**2*c**4*x**2 + 4*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*b*c*d**3*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)*b**2*c**3*d*x* *2)/(4*a*b*c*x**2)