Integrand size = 25, antiderivative size = 510 \[ \int \frac {(c+d x)^{5/2} \sqrt {a-b x^2}}{x^3} \, dx=\frac {29}{12} d^2 \sqrt {c+d x} \sqrt {a-b x^2}-\frac {5 d (c+d x)^{3/2} \sqrt {a-b x^2}}{4 x}-\frac {(c+d x)^{5/2} \sqrt {a-b x^2}}{2 x^2}+\frac {83 \sqrt {a} \sqrt {b} c d \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{12 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {\sqrt {a} d \left (17 b c^2+16 a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{12 \sqrt {b} \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {c \left (4 b c^2-15 a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{4 \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:
29/12*d^2*(d*x+c)^(1/2)*(-b*x^2+a)^(1/2)-5/4*d*(d*x+c)^(3/2)*(-b*x^2+a)^(1 /2)/x-1/2*(d*x+c)^(5/2)*(-b*x^2+a)^(1/2)/x^2+83/12*a^(1/2)*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))/(b^(1/2)*(d*x+c)/(b^ (1/2)*c+a^(1/2)*d))^(1/2)/(-b*x^2+a)^(1/2)-1/12*a^(1/2)*d*(16*a*d^2+17*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)*Ellipt icF(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))/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))/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 23.66 (sec) , antiderivative size = 866, normalized size of antiderivative = 1.70 \[ \int \frac {(c+d x)^{5/2} \sqrt {a-b x^2}}{x^3} \, dx=\frac {\sqrt {a-b x^2} \left (\frac {(c+d x) \left (-6 c^2-27 c d x+8 d^2 x^2\right )}{x^2}-\frac {-83 b c^3 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}+83 a c d^2 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}+166 b c^2 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} (c+d x)-83 b c \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} (c+d x)^2+83 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 (54 b c^2-83 \sqrt {a} \sqrt {b} c d+29 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 )-12 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 )+45 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 )}{\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (-a+b x^2\right )}\right )}{12 \sqrt {c+d x}} \] Input:
Integrate[((c + d*x)^(5/2)*Sqrt[a - b*x^2])/x^3,x]
Output:
(Sqrt[a - b*x^2]*(((c + d*x)*(-6*c^2 - 27*c*d*x + 8*d^2*x^2))/x^2 - (-83*b *c^3*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]] + 83*a*c*d^2*Sqrt[-c + (Sqrt[a]*d)/Sqr t[b]] + 166*b*c^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(c + d*x) - 83*b*c*Sqrt[- c + (Sqrt[a]*d)/Sqrt[b]]*(c + d*x)^2 + (83*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*(54*b*c^2 - 83*Sqrt[a]*Sqrt[b]*c*d + 29*a*d^2)*Sqrt[(d*(Sqrt[a]/S qrt[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)] - (12*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 + Sqr t[a]*d)/(Sqrt[b]*c - Sqrt[a]*d)] + (45*I)*a*d^2*Sqrt[(d*(Sqrt[a]/Sqrt[b] + x))/(c + d*x)]*Sqrt[-(((Sqrt[a]*d)/Sqrt[b] - d*x)/(c + d*x))]*(c + d*x)^( 3/2)*EllipticPi[(Sqrt[b]*c)/(Sqrt[b]*c - Sqrt[a]*d), I*ArcSinh[Sqrt[-c + ( Sqrt[a]*d)/Sqrt[b]]/Sqrt[c + d*x]], (Sqrt[b]*c + Sqrt[a]*d)/(Sqrt[b]*c - S qrt[a]*d)])/(Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(-a + b*x^2))))/(12*Sqrt[c + d *x])
Time = 1.84 (sec) , antiderivative size = 571, normalized size of antiderivative = 1.12, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.680, Rules used = {628, 2352, 2351, 633, 632, 186, 413, 412, 2185, 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.
| \(\displaystyle \int \frac {\sqrt {a-b x^2} (c+d x)^{5/2}}{x^3} \, dx\) |
| \(\Big \downarrow \) 628 |
| \(\displaystyle \frac {1}{4} \int \frac {-4 b d^3 x^4-12 b c d^2 x^3-d \left (11 b c^2-4 a d^2\right ) x^2-2 c \left (b c^2-6 a d^2\right ) x+9 a c^2 d}{x^2 \sqrt {c+d x} \sqrt {a-b x^2}}dx-\frac {c^2 \sqrt {a-b x^2} \sqrt {c+d x}}{2 x^2}\) |
| \(\Big \downarrow \) 2352 |
| \(\displaystyle \frac {1}{4} \left (-\frac {\int \frac {8 a b c d^3 x^3+33 a b c^2 d^2 x^2+2 a c d \left (11 b c^2-4 a d^2\right ) x+a c^2 \left (4 b c^2-15 a d^2\right )}{x \sqrt {c+d x} \sqrt {a-b x^2}}dx}{2 a c}-\frac {9 c d \sqrt {a-b x^2} \sqrt {c+d x}}{x}\right )-\frac {c^2 \sqrt {a-b x^2} \sqrt {c+d x}}{2 x^2}\) |
| \(\Big \downarrow \) 2351 |
| \(\displaystyle \frac {1}{4} \left (-\frac {\int \frac {22 a b d c^3+33 a b d^2 x c^2-8 a^2 d^3 c+8 a b d^3 x^2 c}{\sqrt {c+d x} \sqrt {a-b x^2}}dx+a c^2 \left (4 b c^2-15 a d^2\right ) \int \frac {1}{x \sqrt {c+d x} \sqrt {a-b x^2}}dx}{2 a c}-\frac {9 c d \sqrt {a-b x^2} \sqrt {c+d x}}{x}\right )-\frac {c^2 \sqrt {a-b x^2} \sqrt {c+d x}}{2 x^2}\) |
| \(\Big \downarrow \) 633 |
| \(\displaystyle \frac {1}{4} \left (-\frac {\int \frac {22 a b d c^3+33 a b d^2 x c^2-8 a^2 d^3 c+8 a b d^3 x^2 c}{\sqrt {c+d x} \sqrt {a-b x^2}}dx+\frac {a c^2 \sqrt {1-\frac {b x^2}{a}} \left (4 b c^2-15 a d^2\right ) \int \frac {1}{x \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}}}dx}{\sqrt {a-b x^2}}}{2 a c}-\frac {9 c d \sqrt {a-b x^2} \sqrt {c+d x}}{x}\right )-\frac {c^2 \sqrt {a-b x^2} \sqrt {c+d x}}{2 x^2}\) |
| \(\Big \downarrow \) 632 |
| \(\displaystyle \frac {1}{4} \left (-\frac {\int \frac {22 a b d c^3+33 a b d^2 x c^2-8 a^2 d^3 c+8 a b d^3 x^2 c}{\sqrt {c+d x} \sqrt {a-b x^2}}dx+\frac {a c^2 \sqrt {1-\frac {b x^2}{a}} \left (4 b c^2-15 a d^2\right ) \int \frac {1}{x \sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}} \sqrt {\frac {\sqrt {b} x}{\sqrt {a}}+1} \sqrt {c+d x}}dx}{\sqrt {a-b x^2}}}{2 a c}-\frac {9 c d \sqrt {a-b x^2} \sqrt {c+d x}}{x}\right )-\frac {c^2 \sqrt {a-b x^2} \sqrt {c+d x}}{2 x^2}\) |
| \(\Big \downarrow \) 186 |
| \(\displaystyle \frac {1}{4} \left (-\frac {\int \frac {22 a b d c^3+33 a b d^2 x c^2-8 a^2 d^3 c+8 a b d^3 x^2 c}{\sqrt {c+d x} \sqrt {a-b x^2}}dx-\frac {2 a c^2 \sqrt {1-\frac {b x^2}{a}} \left (4 b c^2-15 a d^2\right ) \int \frac {\sqrt {a}}{\sqrt {b} x \sqrt {\frac {\sqrt {b} x}{\sqrt {a}}+1} \sqrt {c+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}}}d\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {a-b x^2}}}{2 a c}-\frac {9 c d \sqrt {a-b x^2} \sqrt {c+d x}}{x}\right )-\frac {c^2 \sqrt {a-b x^2} \sqrt {c+d x}}{2 x^2}\) |
| \(\Big \downarrow \) 413 |
| \(\displaystyle \frac {1}{4} \left (-\frac {\int \frac {22 a b d c^3+33 a b d^2 x c^2-8 a^2 d^3 c+8 a b d^3 x^2 c}{\sqrt {c+d x} \sqrt {a-b x^2}}dx-\frac {2 a c^2 \sqrt {1-\frac {b x^2}{a}} \left (4 b c^2-15 a d^2\right ) \sqrt {1-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d+\sqrt {b} c}} \int \frac {\sqrt {a}}{\sqrt {b} x \sqrt {\frac {\sqrt {b} x}{\sqrt {a}}+1} \sqrt {1-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b} c+\sqrt {a} d}}}d\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {a-b x^2} \sqrt {-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {\sqrt {a} d}{\sqrt {b}}+c}}}{2 a c}-\frac {9 c d \sqrt {a-b x^2} \sqrt {c+d x}}{x}\right )-\frac {c^2 \sqrt {a-b x^2} \sqrt {c+d x}}{2 x^2}\) |
| \(\Big \downarrow \) 412 |
| \(\displaystyle \frac {1}{4} \left (-\frac {\int \frac {22 a b d c^3+33 a b d^2 x c^2-8 a^2 d^3 c+8 a b d^3 x^2 c}{\sqrt {c+d x} \sqrt {a-b x^2}}dx-\frac {2 a c^2 \sqrt {1-\frac {b x^2}{a}} \left (4 b c^2-15 a d^2\right ) \sqrt {1-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{\sqrt {a-b x^2} \sqrt {-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {\sqrt {a} d}{\sqrt {b}}+c}}}{2 a c}-\frac {9 c d \sqrt {a-b x^2} \sqrt {c+d x}}{x}\right )-\frac {c^2 \sqrt {a-b x^2} \sqrt {c+d x}}{2 x^2}\) |
| \(\Big \downarrow \) 2185 |
| \(\displaystyle \frac {1}{4} \left (-\frac {-\frac {2 \int -\frac {a b c d^3 \left (2 \left (33 b c^2-8 a d^2\right )+83 b c d x\right )}{2 \sqrt {c+d x} \sqrt {a-b x^2}}dx}{3 b d^2}-\frac {2 a c^2 \sqrt {1-\frac {b x^2}{a}} \left (4 b c^2-15 a d^2\right ) \sqrt {1-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{\sqrt {a-b x^2} \sqrt {-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {\sqrt {a} d}{\sqrt {b}}+c}}-\frac {16}{3} a c d^2 \sqrt {a-b x^2} \sqrt {c+d x}}{2 a c}-\frac {9 c d \sqrt {a-b x^2} \sqrt {c+d x}}{x}\right )-\frac {c^2 \sqrt {a-b x^2} \sqrt {c+d x}}{2 x^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{4} \left (-\frac {\frac {1}{3} a c d \int \frac {2 \left (33 b c^2-8 a d^2\right )+83 b c d x}{\sqrt {c+d x} \sqrt {a-b x^2}}dx-\frac {2 a c^2 \sqrt {1-\frac {b x^2}{a}} \left (4 b c^2-15 a d^2\right ) \sqrt {1-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{\sqrt {a-b x^2} \sqrt {-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {\sqrt {a} d}{\sqrt {b}}+c}}-\frac {16}{3} a c d^2 \sqrt {a-b x^2} \sqrt {c+d x}}{2 a c}-\frac {9 c d \sqrt {a-b x^2} \sqrt {c+d x}}{x}\right )-\frac {c^2 \sqrt {a-b x^2} \sqrt {c+d x}}{2 x^2}\) |
| \(\Big \downarrow \) 600 |
| \(\displaystyle \frac {1}{4} \left (-\frac {\frac {1}{3} a c d \left (83 b c \int \frac {\sqrt {c+d x}}{\sqrt {a-b x^2}}dx-\left (16 a d^2+17 b c^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx\right )-\frac {2 a c^2 \sqrt {1-\frac {b x^2}{a}} \left (4 b c^2-15 a d^2\right ) \sqrt {1-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{\sqrt {a-b x^2} \sqrt {-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {\sqrt {a} d}{\sqrt {b}}+c}}-\frac {16}{3} a c d^2 \sqrt {a-b x^2} \sqrt {c+d x}}{2 a c}-\frac {9 c d \sqrt {a-b x^2} \sqrt {c+d x}}{x}\right )-\frac {c^2 \sqrt {a-b x^2} \sqrt {c+d x}}{2 x^2}\) |
| \(\Big \downarrow \) 509 |
| \(\displaystyle \frac {1}{4} \left (-\frac {\frac {1}{3} a c d \left (\frac {83 b c \sqrt {1-\frac {b x^2}{a}} \int \frac {\sqrt {c+d x}}{\sqrt {1-\frac {b x^2}{a}}}dx}{\sqrt {a-b x^2}}-\left (16 a d^2+17 b c^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx\right )-\frac {2 a c^2 \sqrt {1-\frac {b x^2}{a}} \left (4 b c^2-15 a d^2\right ) \sqrt {1-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{\sqrt {a-b x^2} \sqrt {-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {\sqrt {a} d}{\sqrt {b}}+c}}-\frac {16}{3} a c d^2 \sqrt {a-b x^2} \sqrt {c+d x}}{2 a c}-\frac {9 c d \sqrt {a-b x^2} \sqrt {c+d x}}{x}\right )-\frac {c^2 \sqrt {a-b x^2} \sqrt {c+d x}}{2 x^2}\) |
| \(\Big \downarrow \) 508 |
| \(\displaystyle \frac {1}{4} \left (-\frac {\frac {1}{3} a c d \left (-\left (\left (16 a d^2+17 b c^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx\right )-\frac {166 \sqrt {a} \sqrt {b} c \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \int \frac {\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}}}{\sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )-\frac {2 a c^2 \sqrt {1-\frac {b x^2}{a}} \left (4 b c^2-15 a d^2\right ) \sqrt {1-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{\sqrt {a-b x^2} \sqrt {-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {\sqrt {a} d}{\sqrt {b}}+c}}-\frac {16}{3} a c d^2 \sqrt {a-b x^2} \sqrt {c+d x}}{2 a c}-\frac {9 c d \sqrt {a-b x^2} \sqrt {c+d x}}{x}\right )-\frac {c^2 \sqrt {a-b x^2} \sqrt {c+d x}}{2 x^2}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {1}{4} \left (-\frac {\frac {1}{3} a c d \left (-\left (16 a d^2+17 b c^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx-\frac {166 \sqrt {a} \sqrt {b} c \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )-\frac {2 a c^2 \sqrt {1-\frac {b x^2}{a}} \left (4 b c^2-15 a d^2\right ) \sqrt {1-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{\sqrt {a-b x^2} \sqrt {-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {\sqrt {a} d}{\sqrt {b}}+c}}-\frac {16}{3} a c d^2 \sqrt {a-b x^2} \sqrt {c+d x}}{2 a c}-\frac {9 c d \sqrt {a-b x^2} \sqrt {c+d x}}{x}\right )-\frac {c^2 \sqrt {a-b x^2} \sqrt {c+d x}}{2 x^2}\) |
| \(\Big \downarrow \) 512 |
| \(\displaystyle \frac {1}{4} \left (-\frac {\frac {1}{3} a c d \left (-\frac {\sqrt {1-\frac {b x^2}{a}} \left (16 a d^2+17 b c^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}}}dx}{\sqrt {a-b x^2}}-\frac {166 \sqrt {a} \sqrt {b} c \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )-\frac {2 a c^2 \sqrt {1-\frac {b x^2}{a}} \left (4 b c^2-15 a d^2\right ) \sqrt {1-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{\sqrt {a-b x^2} \sqrt {-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {\sqrt {a} d}{\sqrt {b}}+c}}-\frac {16}{3} a c d^2 \sqrt {a-b x^2} \sqrt {c+d x}}{2 a c}-\frac {9 c d \sqrt {a-b x^2} \sqrt {c+d x}}{x}\right )-\frac {c^2 \sqrt {a-b x^2} \sqrt {c+d x}}{2 x^2}\) |
| \(\Big \downarrow \) 511 |
| \(\displaystyle \frac {1}{4} \left (-\frac {\frac {1}{3} a c d \left (\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \left (16 a d^2+17 b c^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \int \frac {1}{\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {166 \sqrt {a} \sqrt {b} c \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )-\frac {2 a c^2 \sqrt {1-\frac {b x^2}{a}} \left (4 b c^2-15 a d^2\right ) \sqrt {1-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{\sqrt {a-b x^2} \sqrt {-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {\sqrt {a} d}{\sqrt {b}}+c}}-\frac {16}{3} a c d^2 \sqrt {a-b x^2} \sqrt {c+d x}}{2 a c}-\frac {9 c d \sqrt {a-b x^2} \sqrt {c+d x}}{x}\right )-\frac {c^2 \sqrt {a-b x^2} \sqrt {c+d x}}{2 x^2}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {1}{4} \left (-\frac {-\frac {2 a c^2 \sqrt {1-\frac {b x^2}{a}} \left (4 b c^2-15 a d^2\right ) \sqrt {1-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{\sqrt {a-b x^2} \sqrt {-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {\sqrt {a} d}{\sqrt {b}}+c}}+\frac {1}{3} a c d \left (\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \left (16 a d^2+17 b c^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {166 \sqrt {a} \sqrt {b} c \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )-\frac {16}{3} a c d^2 \sqrt {a-b x^2} \sqrt {c+d x}}{2 a c}-\frac {9 c d \sqrt {a-b x^2} \sqrt {c+d x}}{x}\right )-\frac {c^2 \sqrt {a-b x^2} \sqrt {c+d x}}{2 x^2}\) |
Input:
Int[((c + d*x)^(5/2)*Sqrt[a - b*x^2])/x^3,x]
Output:
-1/2*(c^2*Sqrt[c + d*x]*Sqrt[a - b*x^2])/x^2 + ((-9*c*d*Sqrt[c + d*x]*Sqrt [a - b*x^2])/x - ((-16*a*c*d^2*Sqrt[c + d*x]*Sqrt[a - b*x^2])/3 + (a*c*d*( (-166*Sqrt[a]*Sqrt[b]*c*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)]) /(Sqrt[(Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[a]*d)]*Sqrt[a - b*x^2]) + (2* Sqrt[a]*(17*b*c^2 + 16*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]]/S qrt[2]], (2*d)/((Sqrt[b]*c)/Sqrt[a] + d)])/(Sqrt[b]*Sqrt[c + d*x]*Sqrt[a - b*x^2])))/3 - (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
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[((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]/( e*(m + 1))), x] - Simp[1/(2*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) - 2*b*(m + 1)*x^2*(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[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[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] : > With[{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[f*(d + e*x) ^(m + q - 1)*((a + b*x^2)^(p + 1)/(b*e^(q - 1)*(m + q + 2*p + 1))), x] + Si mp[1/(b*e^q*(m + q + 2*p + 1)) Int[(d + e*x)^m*(a + b*x^2)^p*ExpandToSum[ b*e^q*(m + q + 2*p + 1)*Pq - b*f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x )^(q - 2)*(a*e^2*(m + q - 1) - b*d^2*(m + q + 2*p + 1) - 2*b*d*e*(m + q + p )*x), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, d , e, m, p}, x] && PolyQ[Pq, x] && NeQ[b*d^2 + a*e^2, 0] && !(EqQ[d, 0] && True) && !(IGtQ[m, 0] && RationalQ[a, b, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))
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. \(886\) vs. \(2(409)=818\).
Time = 3.29 (sec) , antiderivative size = 887, normalized size of antiderivative = 1.74
| method | result | size |
| 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 x^{2}}-\frac {9 c d \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}{4 x}+\frac {2 d^{2} \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}{3}+\frac {2 \left (\frac {2}{3} a \,d^{3}-\frac {11}{4} b \,c^{2} d \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 {83 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 )}{12 \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 \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}\right )}{\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}\) | \(887\) |
| risch | \(\text {Expression too large to display}\) | \(1262\) |
| default | \(\text {Expression too large to display}\) | \(1654\) |
Input:
int((d*x+c)^(5/2)*(-b*x^2+a)^(1/2)/x^3,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^2/x^2*(- b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)-9/4*c*d/x*(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1 /2)+2/3*d^2*(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)+2*(2/3*a*d^3-11/4*b*c^2*d)* (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))-83/12*b*c*d^2*(c/d-1/b*(a*b)^(1/2))*((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1 /2)*((x-1/b*(a*b)^(1/2))/(-c/d-1/b*(a*b)^(1/2)))^(1/2)*((x+1/b*(a*b)^(1/2) )/(-c/d+1/b*(a*b)^(1/2)))^(1/2)/(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)*((-c/d- 1/b*(a*b)^(1/2))*EllipticE(((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2),((-c/d+1/ b*(a*b)^(1/2))/(-c/d-1/b*(a*b)^(1/2)))^(1/2))+1/b*(a*b)^(1/2)*EllipticF((( x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2),((-c/d+1/b*(a*b)^(1/2))/(-c/d-1/b*(a*b )^(1/2)))^(1/2)))-1/4*(15*a*d^2-4*b*c^2)*(c/d-1/b*(a*b)^(1/2))*((x+c/d)/(c /d-1/b*(a*b)^(1/2)))^(1/2)*((x-1/b*(a*b)^(1/2))/(-c/d-1/b*(a*b)^(1/2)))^(1 /2)*((x+1/b*(a*b)^(1/2))/(-c/d+1/b*(a*b)^(1/2)))^(1/2)/(-b*d*x^3-b*c*x^2+a *d*x+a*c)^(1/2)*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. \[ \int \frac {(c+d x)^{5/2} \sqrt {a-b x^2}}{x^3} \, dx=\text {Timed out} \] Input:
integrate((d*x+c)^(5/2)*(-b*x^2+a)^(1/2)/x^3,x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {(c+d x)^{5/2} \sqrt {a-b x^2}}{x^3} \, dx=\int \frac {\sqrt {a - b x^{2}} \left (c + d x\right )^{\frac {5}{2}}}{x^{3}}\, dx \] Input:
integrate((d*x+c)**(5/2)*(-b*x**2+a)**(1/2)/x**3,x)
Output:
Integral(sqrt(a - b*x**2)*(c + d*x)**(5/2)/x**3, x)
\[ \int \frac {(c+d x)^{5/2} \sqrt {a-b x^2}}{x^3} \, dx=\int { \frac {\sqrt {-b x^{2} + a} {\left (d x + c\right )}^{\frac {5}{2}}}{x^{3}} \,d x } \] Input:
integrate((d*x+c)^(5/2)*(-b*x^2+a)^(1/2)/x^3,x, algorithm="maxima")
Output:
integrate(sqrt(-b*x^2 + a)*(d*x + c)^(5/2)/x^3, x)
\[ \int \frac {(c+d x)^{5/2} \sqrt {a-b x^2}}{x^3} \, dx=\int { \frac {\sqrt {-b x^{2} + a} {\left (d x + c\right )}^{\frac {5}{2}}}{x^{3}} \,d x } \] Input:
integrate((d*x+c)^(5/2)*(-b*x^2+a)^(1/2)/x^3,x, algorithm="giac")
Output:
integrate(sqrt(-b*x^2 + a)*(d*x + c)^(5/2)/x^3, x)
Timed out. \[ \int \frac {(c+d x)^{5/2} \sqrt {a-b x^2}}{x^3} \, dx=\int \frac {\sqrt {a-b\,x^2}\,{\left (c+d\,x\right )}^{5/2}}{x^3} \,d x \] Input:
int(((a - b*x^2)^(1/2)*(c + d*x)^(5/2))/x^3,x)
Output:
int(((a - b*x^2)^(1/2)*(c + d*x)^(5/2))/x^3, x)
\[ \int \frac {(c+d x)^{5/2} \sqrt {a-b x^2}}{x^3} \, dx=\int \frac {\left (d x +c \right )^{\frac {5}{2}} \sqrt {-b \,x^{2}+a}}{x^{3}}d x \] Input:
int((d*x+c)^(5/2)*(-b*x^2+a)^(1/2)/x^3,x)
Output:
int((d*x+c)^(5/2)*(-b*x^2+a)^(1/2)/x^3,x)