Integrand size = 27, antiderivative size = 427 \[ \int \frac {(e x)^{5/2} \sqrt {a-b x^2}}{c+d x} \, dx=-\frac {2 \left (2 a-\frac {7 b c^2}{d^2}\right ) e^2 \sqrt {e x} \sqrt {a-b x^2}}{21 b d}-\frac {2 c e (e x)^{3/2} \sqrt {a-b x^2}}{5 d^2}+\frac {2 (e x)^{5/2} \sqrt {a-b x^2}}{7 d}+\frac {2 a^{3/4} c \left (5 b c^2-2 a d^2\right ) e^{5/2} \sqrt {1-\frac {b x^2}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )\right |-1\right )}{5 b^{3/4} d^4 \sqrt {a-b x^2}}-\frac {2 \sqrt [4]{a} \left (105 b^2 c^4+105 \sqrt {a} b^{3/2} c^3 d-70 a b c^2 d^2-42 a^{3/2} \sqrt {b} c d^3-10 a^2 d^4\right ) e^{5/2} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{105 b^{5/4} d^5 \sqrt {a-b x^2}}+\frac {2 \sqrt [4]{a} c^2 \left (b c^2-a d^2\right ) e^{5/2} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{\sqrt {b} c},\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} d^5 \sqrt {a-b x^2}} \] Output:
-2/21*(2*a-7*b*c^2/d^2)*e^2*(e*x)^(1/2)*(-b*x^2+a)^(1/2)/b/d-2/5*c*e*(e*x) ^(3/2)*(-b*x^2+a)^(1/2)/d^2+2/7*(e*x)^(5/2)*(-b*x^2+a)^(1/2)/d+2/5*a^(3/4) *c*(-2*a*d^2+5*b*c^2)*e^(5/2)*(1-b*x^2/a)^(1/2)*EllipticE(b^(1/4)*(e*x)^(1 /2)/a^(1/4)/e^(1/2),I)/b^(3/4)/d^4/(-b*x^2+a)^(1/2)-2/105*a^(1/4)*(105*b^2 *c^4+105*a^(1/2)*b^(3/2)*c^3*d-70*a*b*c^2*d^2-42*a^(3/2)*b^(1/2)*c*d^3-10* a^2*d^4)*e^(5/2)*(1-b*x^2/a)^(1/2)*EllipticF(b^(1/4)*(e*x)^(1/2)/a^(1/4)/e ^(1/2),I)/b^(5/4)/d^5/(-b*x^2+a)^(1/2)+2*a^(1/4)*c^2*(-a*d^2+b*c^2)*e^(5/2 )*(1-b*x^2/a)^(1/2)*EllipticPi(b^(1/4)*(e*x)^(1/2)/a^(1/4)/e^(1/2),-a^(1/2 )*d/b^(1/2)/c,I)/b^(1/4)/d^5/(-b*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 24.18 (sec) , antiderivative size = 403, normalized size of antiderivative = 0.94 \[ \int \frac {(e x)^{5/2} \sqrt {a-b x^2}}{c+d x} \, dx=\frac {(e x)^{5/2} \sqrt {a-b x^2} \left (2 d^2 x \left (-10 a d^2+b \left (35 c^2-21 c d x+15 d^2 x^2\right )\right )+\frac {-42 i \sqrt {a} \sqrt {b} c d \left (5 b c^2-2 a d^2\right ) \sqrt {1-\frac {a}{b x^2}} x^{3/2} E\left (\left .i \text {arcsinh}\left (\frac {\sqrt {-\frac {\sqrt {a}}{\sqrt {b}}}}{\sqrt {x}}\right )\right |-1\right )+2 i \sqrt {a} d \left (105 b^{3/2} c^3-35 \sqrt {a} b c^2 d-42 a \sqrt {b} c d^2+10 a^{3/2} d^3\right ) \sqrt {1-\frac {a}{b x^2}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-\frac {\sqrt {a}}{\sqrt {b}}}}{\sqrt {x}}\right ),-1\right )+42 c \left (\sqrt {-\frac {\sqrt {a}}{\sqrt {b}}} d \left (-5 b c^2+2 a d^2\right ) \left (a-b x^2\right )-5 i b c \left (b c^2-a d^2\right ) \sqrt {1-\frac {a}{b x^2}} x^{3/2} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} c}{\sqrt {a} d},i \text {arcsinh}\left (\frac {\sqrt {-\frac {\sqrt {a}}{\sqrt {b}}}}{\sqrt {x}}\right ),-1\right )\right )}{\sqrt {-\frac {\sqrt {a}}{\sqrt {b}}} \left (a-b x^2\right )}\right )}{105 b d^5 x^3} \] Input:
Integrate[((e*x)^(5/2)*Sqrt[a - b*x^2])/(c + d*x),x]
Output:
((e*x)^(5/2)*Sqrt[a - b*x^2]*(2*d^2*x*(-10*a*d^2 + b*(35*c^2 - 21*c*d*x + 15*d^2*x^2)) + ((-42*I)*Sqrt[a]*Sqrt[b]*c*d*(5*b*c^2 - 2*a*d^2)*Sqrt[1 - a /(b*x^2)]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt[-(Sqrt[a]/Sqrt[b])]/Sqrt[x]], - 1] + (2*I)*Sqrt[a]*d*(105*b^(3/2)*c^3 - 35*Sqrt[a]*b*c^2*d - 42*a*Sqrt[b]* c*d^2 + 10*a^(3/2)*d^3)*Sqrt[1 - a/(b*x^2)]*x^(3/2)*EllipticF[I*ArcSinh[Sq rt[-(Sqrt[a]/Sqrt[b])]/Sqrt[x]], -1] + 42*c*(Sqrt[-(Sqrt[a]/Sqrt[b])]*d*(- 5*b*c^2 + 2*a*d^2)*(a - b*x^2) - (5*I)*b*c*(b*c^2 - a*d^2)*Sqrt[1 - a/(b*x ^2)]*x^(3/2)*EllipticPi[-((Sqrt[b]*c)/(Sqrt[a]*d)), I*ArcSinh[Sqrt[-(Sqrt[ a]/Sqrt[b])]/Sqrt[x]], -1]))/(Sqrt[-(Sqrt[a]/Sqrt[b])]*(a - b*x^2))))/(105 *b*d^5*x^3)
Time = 2.99 (sec) , antiderivative size = 452, normalized size of antiderivative = 1.06, number of steps used = 21, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.741, Rules used = {616, 27, 1633, 25, 1543, 1542, 2427, 25, 2427, 25, 2427, 25, 27, 1513, 27, 765, 762, 1390, 1389, 327}
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.
| \(\displaystyle \int \frac {(e x)^{5/2} \sqrt {a-b x^2}}{c+d x} \, dx\) |
| \(\Big \downarrow \) 616 |
| \(\displaystyle \frac {2 \int \frac {e^4 x^3 \sqrt {a-b x^2}}{c e+d x e}d\sqrt {e x}}{e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \int \frac {e^3 x^3 \sqrt {a-b x^2}}{c e+d x e}d\sqrt {e x}\) |
| \(\Big \downarrow \) 1633 |
| \(\displaystyle 2 \left (\frac {\int -\frac {b d e^2 x^4-b c e^2 x^3+\frac {\left (b c^2-a d^2\right ) e^2 x^2}{d}-\frac {c \left (b c^2-a d^2\right ) e^2 x}{d^2}+\frac {c^2 \left (b c^2-a d^2\right ) e^2}{d^3}}{\sqrt {a-b x^2}}d\sqrt {e x}}{d^2}+\frac {c^3 e^3 \left (b c^2-a d^2\right ) \int \frac {1}{(c e+d x e) \sqrt {a-b x^2}}d\sqrt {e x}}{d^5}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 2 \left (\frac {c^3 e^3 \left (b c^2-a d^2\right ) \int \frac {1}{(c e+d x e) \sqrt {a-b x^2}}d\sqrt {e x}}{d^5}-\frac {\int \frac {b d e^2 x^4-b c e^2 x^3+\frac {\left (b c^2-a d^2\right ) e^2 x^2}{d}-\frac {c \left (b c^2-a d^2\right ) e^2 x}{d^2}+\frac {c^2 \left (b c^2-a d^2\right ) e^2}{d^3}}{\sqrt {a-b x^2}}d\sqrt {e x}}{d^2}\right )\) |
| \(\Big \downarrow \) 1543 |
| \(\displaystyle 2 \left (\frac {c^3 e^3 \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \int \frac {1}{(c e+d x e) \sqrt {1-\frac {b x^2}{a}}}d\sqrt {e x}}{d^5 \sqrt {a-b x^2}}-\frac {\int \frac {b d e^2 x^4-b c e^2 x^3+\frac {\left (b c^2-a d^2\right ) e^2 x^2}{d}-\frac {c \left (b c^2-a d^2\right ) e^2 x}{d^2}+\frac {c^2 \left (b c^2-a d^2\right ) e^2}{d^3}}{\sqrt {a-b x^2}}d\sqrt {e x}}{d^2}\right )\) |
| \(\Big \downarrow \) 1542 |
| \(\displaystyle 2 \left (\frac {\sqrt [4]{a} c^2 e^{5/2} \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{\sqrt {b} c},\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} d^5 \sqrt {a-b x^2}}-\frac {\int \frac {b d e^2 x^4-b c e^2 x^3+\frac {\left (b c^2-a d^2\right ) e^2 x^2}{d}-\frac {c \left (b c^2-a d^2\right ) e^2 x}{d^2}+\frac {c^2 \left (b c^2-a d^2\right ) e^2}{d^3}}{\sqrt {a-b x^2}}d\sqrt {e x}}{d^2}\right )\) |
| \(\Big \downarrow \) 2427 |
| \(\displaystyle 2 \left (\frac {\sqrt [4]{a} c^2 e^{5/2} \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{\sqrt {b} c},\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} d^5 \sqrt {a-b x^2}}-\frac {-\frac {e^2 \int -\frac {-7 b^2 c x^3+b \left (\frac {7 b c^2}{d}-2 a d\right ) x^2+7 b c \left (a-\frac {b c^2}{d^2}\right ) x+\frac {7 b c^2 \left (b c^2-a d^2\right )}{d^3}}{\sqrt {a-b x^2}}d\sqrt {e x}}{7 b}-\frac {1}{7} d (e x)^{5/2} \sqrt {a-b x^2}}{d^2}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 2 \left (\frac {\sqrt [4]{a} c^2 e^{5/2} \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{\sqrt {b} c},\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} d^5 \sqrt {a-b x^2}}-\frac {\frac {e^2 \int \frac {-7 b^2 c x^3+b \left (\frac {7 b c^2}{d}-2 a d\right ) x^2+7 b c \left (a-\frac {b c^2}{d^2}\right ) x+\frac {7 b c^2 \left (b c^2-a d^2\right )}{d^3}}{\sqrt {a-b x^2}}d\sqrt {e x}}{7 b}-\frac {1}{7} d (e x)^{5/2} \sqrt {a-b x^2}}{d^2}\right )\) |
| \(\Big \downarrow \) 2427 |
| \(\displaystyle 2 \left (\frac {\sqrt [4]{a} c^2 e^{5/2} \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{\sqrt {b} c},\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} d^5 \sqrt {a-b x^2}}-\frac {\frac {e^2 \left (\frac {7 b c (e x)^{3/2} \sqrt {a-b x^2}}{5 e}-\frac {e^2 \int -\frac {\frac {5 \left (\frac {7 b c^2}{d}-2 a d\right ) x^2 b^2}{e^2}+\frac {7 c \left (2 a-\frac {5 b c^2}{d^2}\right ) x b^2}{e^2}+\frac {35 c^2 \left (b c^2-a d^2\right ) b^2}{d^3 e^2}}{\sqrt {a-b x^2}}d\sqrt {e x}}{5 b}\right )}{7 b}-\frac {1}{7} d (e x)^{5/2} \sqrt {a-b x^2}}{d^2}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 2 \left (\frac {\sqrt [4]{a} c^2 e^{5/2} \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{\sqrt {b} c},\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} d^5 \sqrt {a-b x^2}}-\frac {\frac {e^2 \left (\frac {e^2 \int \frac {\frac {5 \left (\frac {7 b c^2}{d}-2 a d\right ) x^2 b^2}{e^2}+\frac {7 c \left (2 a-\frac {5 b c^2}{d^2}\right ) x b^2}{e^2}+\frac {35 c^2 \left (b c^2-a d^2\right ) b^2}{d^3 e^2}}{\sqrt {a-b x^2}}d\sqrt {e x}}{5 b}+\frac {7 b c (e x)^{3/2} \sqrt {a-b x^2}}{5 e}\right )}{7 b}-\frac {1}{7} d (e x)^{5/2} \sqrt {a-b x^2}}{d^2}\right )\) |
| \(\Big \downarrow \) 2427 |
| \(\displaystyle 2 \left (\frac {\sqrt [4]{a} c^2 e^{5/2} \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{\sqrt {b} c},\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} d^5 \sqrt {a-b x^2}}-\frac {\frac {e^2 \left (\frac {e^2 \left (-\frac {e^2 \int -\frac {b^2 \left (21 b c \left (2 a-\frac {5 b c^2}{d^2}\right ) e x d^3+5 \left (21 b^2 c^4-14 a b d^2 c^2-2 a^2 d^4\right ) e\right )}{d^3 e^5 \sqrt {a-b x^2}}d\sqrt {e x}}{3 b}-\frac {5 b \sqrt {e x} \sqrt {a-b x^2} \left (\frac {7 b c^2}{d}-2 a d\right )}{3 e^2}\right )}{5 b}+\frac {7 b c (e x)^{3/2} \sqrt {a-b x^2}}{5 e}\right )}{7 b}-\frac {1}{7} d (e x)^{5/2} \sqrt {a-b x^2}}{d^2}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 2 \left (\frac {\sqrt [4]{a} c^2 e^{5/2} \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{\sqrt {b} c},\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} d^5 \sqrt {a-b x^2}}-\frac {\frac {e^2 \left (\frac {e^2 \left (\frac {e^2 \int \frac {b^2 \left (5 \left (21 b^2 c^4-14 a b d^2 c^2-2 a^2 d^4\right ) e-21 b c d \left (5 b c^2-2 a d^2\right ) e x\right )}{d^3 e^5 \sqrt {a-b x^2}}d\sqrt {e x}}{3 b}-\frac {5 b \sqrt {e x} \sqrt {a-b x^2} \left (\frac {7 b c^2}{d}-2 a d\right )}{3 e^2}\right )}{5 b}+\frac {7 b c (e x)^{3/2} \sqrt {a-b x^2}}{5 e}\right )}{7 b}-\frac {1}{7} d (e x)^{5/2} \sqrt {a-b x^2}}{d^2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \left (\frac {\sqrt [4]{a} c^2 e^{5/2} \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{\sqrt {b} c},\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} d^5 \sqrt {a-b x^2}}-\frac {\frac {e^2 \left (\frac {e^2 \left (\frac {b \int \frac {5 \left (21 b^2 c^4-14 a b d^2 c^2-2 a^2 d^4\right ) e-21 b c d \left (5 b c^2-2 a d^2\right ) e x}{\sqrt {a-b x^2}}d\sqrt {e x}}{3 d^3 e^3}-\frac {5 b \sqrt {e x} \sqrt {a-b x^2} \left (\frac {7 b c^2}{d}-2 a d\right )}{3 e^2}\right )}{5 b}+\frac {7 b c (e x)^{3/2} \sqrt {a-b x^2}}{5 e}\right )}{7 b}-\frac {1}{7} d (e x)^{5/2} \sqrt {a-b x^2}}{d^2}\right )\) |
| \(\Big \downarrow \) 1513 |
| \(\displaystyle 2 \left (\frac {\sqrt [4]{a} c^2 e^{5/2} \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{\sqrt {b} c},\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} d^5 \sqrt {a-b x^2}}-\frac {\frac {e^2 \left (\frac {e^2 \left (\frac {b \left (e \left (-42 a^{3/2} \sqrt {b} c d^3-10 a^2 d^4+105 \sqrt {a} b^{3/2} c^3 d-70 a b c^2 d^2+105 b^2 c^4\right ) \int \frac {1}{\sqrt {a-b x^2}}d\sqrt {e x}-21 \sqrt {a} \sqrt {b} c d e \left (5 b c^2-2 a d^2\right ) \int \frac {\sqrt {b} x e+\sqrt {a} e}{\sqrt {a} e \sqrt {a-b x^2}}d\sqrt {e x}\right )}{3 d^3 e^3}-\frac {5 b \sqrt {e x} \sqrt {a-b x^2} \left (\frac {7 b c^2}{d}-2 a d\right )}{3 e^2}\right )}{5 b}+\frac {7 b c (e x)^{3/2} \sqrt {a-b x^2}}{5 e}\right )}{7 b}-\frac {1}{7} d (e x)^{5/2} \sqrt {a-b x^2}}{d^2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \left (\frac {\sqrt [4]{a} c^2 e^{5/2} \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{\sqrt {b} c},\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} d^5 \sqrt {a-b x^2}}-\frac {\frac {e^2 \left (\frac {e^2 \left (\frac {b \left (e \left (-42 a^{3/2} \sqrt {b} c d^3-10 a^2 d^4+105 \sqrt {a} b^{3/2} c^3 d-70 a b c^2 d^2+105 b^2 c^4\right ) \int \frac {1}{\sqrt {a-b x^2}}d\sqrt {e x}-21 \sqrt {b} c d \left (5 b c^2-2 a d^2\right ) \int \frac {\sqrt {b} x e+\sqrt {a} e}{\sqrt {a-b x^2}}d\sqrt {e x}\right )}{3 d^3 e^3}-\frac {5 b \sqrt {e x} \sqrt {a-b x^2} \left (\frac {7 b c^2}{d}-2 a d\right )}{3 e^2}\right )}{5 b}+\frac {7 b c (e x)^{3/2} \sqrt {a-b x^2}}{5 e}\right )}{7 b}-\frac {1}{7} d (e x)^{5/2} \sqrt {a-b x^2}}{d^2}\right )\) |
| \(\Big \downarrow \) 765 |
| \(\displaystyle 2 \left (\frac {\sqrt [4]{a} c^2 e^{5/2} \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{\sqrt {b} c},\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} d^5 \sqrt {a-b x^2}}-\frac {\frac {e^2 \left (\frac {e^2 \left (\frac {b \left (\frac {e \sqrt {1-\frac {b x^2}{a}} \left (-42 a^{3/2} \sqrt {b} c d^3-10 a^2 d^4+105 \sqrt {a} b^{3/2} c^3 d-70 a b c^2 d^2+105 b^2 c^4\right ) \int \frac {1}{\sqrt {1-\frac {b x^2}{a}}}d\sqrt {e x}}{\sqrt {a-b x^2}}-21 \sqrt {b} c d \left (5 b c^2-2 a d^2\right ) \int \frac {\sqrt {b} x e+\sqrt {a} e}{\sqrt {a-b x^2}}d\sqrt {e x}\right )}{3 d^3 e^3}-\frac {5 b \sqrt {e x} \sqrt {a-b x^2} \left (\frac {7 b c^2}{d}-2 a d\right )}{3 e^2}\right )}{5 b}+\frac {7 b c (e x)^{3/2} \sqrt {a-b x^2}}{5 e}\right )}{7 b}-\frac {1}{7} d (e x)^{5/2} \sqrt {a-b x^2}}{d^2}\right )\) |
| \(\Big \downarrow \) 762 |
| \(\displaystyle 2 \left (\frac {\sqrt [4]{a} c^2 e^{5/2} \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{\sqrt {b} c},\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} d^5 \sqrt {a-b x^2}}-\frac {\frac {e^2 \left (\frac {e^2 \left (\frac {b \left (\frac {\sqrt [4]{a} e^{3/2} \sqrt {1-\frac {b x^2}{a}} \left (-42 a^{3/2} \sqrt {b} c d^3-10 a^2 d^4+105 \sqrt {a} b^{3/2} c^3 d-70 a b c^2 d^2+105 b^2 c^4\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} \sqrt {a-b x^2}}-21 \sqrt {b} c d \left (5 b c^2-2 a d^2\right ) \int \frac {\sqrt {b} x e+\sqrt {a} e}{\sqrt {a-b x^2}}d\sqrt {e x}\right )}{3 d^3 e^3}-\frac {5 b \sqrt {e x} \sqrt {a-b x^2} \left (\frac {7 b c^2}{d}-2 a d\right )}{3 e^2}\right )}{5 b}+\frac {7 b c (e x)^{3/2} \sqrt {a-b x^2}}{5 e}\right )}{7 b}-\frac {1}{7} d (e x)^{5/2} \sqrt {a-b x^2}}{d^2}\right )\) |
| \(\Big \downarrow \) 1390 |
| \(\displaystyle 2 \left (\frac {\sqrt [4]{a} c^2 e^{5/2} \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{\sqrt {b} c},\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} d^5 \sqrt {a-b x^2}}-\frac {\frac {e^2 \left (\frac {e^2 \left (\frac {b \left (\frac {\sqrt [4]{a} e^{3/2} \sqrt {1-\frac {b x^2}{a}} \left (-42 a^{3/2} \sqrt {b} c d^3-10 a^2 d^4+105 \sqrt {a} b^{3/2} c^3 d-70 a b c^2 d^2+105 b^2 c^4\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} \sqrt {a-b x^2}}-\frac {21 \sqrt {b} c d \sqrt {1-\frac {b x^2}{a}} \left (5 b c^2-2 a d^2\right ) \int \frac {\sqrt {b} x e+\sqrt {a} e}{\sqrt {1-\frac {b x^2}{a}}}d\sqrt {e x}}{\sqrt {a-b x^2}}\right )}{3 d^3 e^3}-\frac {5 b \sqrt {e x} \sqrt {a-b x^2} \left (\frac {7 b c^2}{d}-2 a d\right )}{3 e^2}\right )}{5 b}+\frac {7 b c (e x)^{3/2} \sqrt {a-b x^2}}{5 e}\right )}{7 b}-\frac {1}{7} d (e x)^{5/2} \sqrt {a-b x^2}}{d^2}\right )\) |
| \(\Big \downarrow \) 1389 |
| \(\displaystyle 2 \left (\frac {\sqrt [4]{a} c^2 e^{5/2} \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{\sqrt {b} c},\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} d^5 \sqrt {a-b x^2}}-\frac {\frac {e^2 \left (\frac {e^2 \left (\frac {b \left (\frac {\sqrt [4]{a} e^{3/2} \sqrt {1-\frac {b x^2}{a}} \left (-42 a^{3/2} \sqrt {b} c d^3-10 a^2 d^4+105 \sqrt {a} b^{3/2} c^3 d-70 a b c^2 d^2+105 b^2 c^4\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} \sqrt {a-b x^2}}-\frac {21 \sqrt {a} \sqrt {b} c d e \sqrt {1-\frac {b x^2}{a}} \left (5 b c^2-2 a d^2\right ) \int \frac {\sqrt {\frac {\sqrt {b} x}{\sqrt {a}}+1}}{\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}d\sqrt {e x}}{\sqrt {a-b x^2}}\right )}{3 d^3 e^3}-\frac {5 b \sqrt {e x} \sqrt {a-b x^2} \left (\frac {7 b c^2}{d}-2 a d\right )}{3 e^2}\right )}{5 b}+\frac {7 b c (e x)^{3/2} \sqrt {a-b x^2}}{5 e}\right )}{7 b}-\frac {1}{7} d (e x)^{5/2} \sqrt {a-b x^2}}{d^2}\right )\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle 2 \left (\frac {\sqrt [4]{a} c^2 e^{5/2} \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{\sqrt {b} c},\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} d^5 \sqrt {a-b x^2}}-\frac {\frac {e^2 \left (\frac {e^2 \left (\frac {b \left (\frac {\sqrt [4]{a} e^{3/2} \sqrt {1-\frac {b x^2}{a}} \left (-42 a^{3/2} \sqrt {b} c d^3-10 a^2 d^4+105 \sqrt {a} b^{3/2} c^3 d-70 a b c^2 d^2+105 b^2 c^4\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} \sqrt {a-b x^2}}-\frac {21 a^{3/4} \sqrt [4]{b} c d e^{3/2} \sqrt {1-\frac {b x^2}{a}} \left (5 b c^2-2 a d^2\right ) E\left (\left .\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )\right |-1\right )}{\sqrt {a-b x^2}}\right )}{3 d^3 e^3}-\frac {5 b \sqrt {e x} \sqrt {a-b x^2} \left (\frac {7 b c^2}{d}-2 a d\right )}{3 e^2}\right )}{5 b}+\frac {7 b c (e x)^{3/2} \sqrt {a-b x^2}}{5 e}\right )}{7 b}-\frac {1}{7} d (e x)^{5/2} \sqrt {a-b x^2}}{d^2}\right )\) |
Input:
Int[((e*x)^(5/2)*Sqrt[a - b*x^2])/(c + d*x),x]
Output:
2*(-((-1/7*(d*(e*x)^(5/2)*Sqrt[a - b*x^2]) + (e^2*((7*b*c*(e*x)^(3/2)*Sqrt [a - b*x^2])/(5*e) + (e^2*((-5*b*((7*b*c^2)/d - 2*a*d)*Sqrt[e*x]*Sqrt[a - b*x^2])/(3*e^2) + (b*((-21*a^(3/4)*b^(1/4)*c*d*(5*b*c^2 - 2*a*d^2)*e^(3/2) *Sqrt[1 - (b*x^2)/a]*EllipticE[ArcSin[(b^(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt[e] )], -1])/Sqrt[a - b*x^2] + (a^(1/4)*(105*b^2*c^4 + 105*Sqrt[a]*b^(3/2)*c^3 *d - 70*a*b*c^2*d^2 - 42*a^(3/2)*Sqrt[b]*c*d^3 - 10*a^2*d^4)*e^(3/2)*Sqrt[ 1 - (b*x^2)/a]*EllipticF[ArcSin[(b^(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt[e])], -1 ])/(b^(1/4)*Sqrt[a - b*x^2])))/(3*d^3*e^3)))/(5*b)))/(7*b))/d^2) + (a^(1/4 )*c^2*(b*c^2 - a*d^2)*e^(5/2)*Sqrt[1 - (b*x^2)/a]*EllipticPi[-((Sqrt[a]*d) /(Sqrt[b]*c)), ArcSin[(b^(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt[e])], -1])/(b^(1/4 )*d^5*Sqrt[a - b*x^2]))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k/e Subst[Int[x^(k*(m + 1) - 1)*(c + d*(x^k/e))^n*(a + b*(x^(2*k)/e^2))^p, x], x, (e*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, p}, x] && ILtQ[n, 0] && FractionQ[m]
Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[(1/(Sqrt[a]*Rt[-b/a, 4]) )*EllipticF[ArcSin[Rt[-b/a, 4]*x], -1], x] /; FreeQ[{a, b}, x] && NegQ[b/a] && GtQ[a, 0]
Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[Sqrt[1 + b*(x^4/a)]/Sqrt [a + b*x^4] Int[1/Sqrt[1 + b*(x^4/a)], x], x] /; FreeQ[{a, b}, x] && NegQ [b/a] && !GtQ[a, 0]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Simp[d/Sq rt[a] Int[Sqrt[1 + e*(x^2/d)]/Sqrt[1 - e*(x^2/d)], x], x] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 + a*e^2, 0] && NegQ[c/a] && GtQ[a, 0]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Simp[Sqrt [1 + c*(x^4/a)]/Sqrt[a + c*x^4] Int[(d + e*x^2)/Sqrt[1 + c*(x^4/a)], x], x] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 + a*e^2, 0] && NegQ[c/a] && !GtQ [a, 0] && !(LtQ[a, 0] && GtQ[c, 0])
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[-c/a, 2]}, Simp[(d*q - e)/q Int[1/Sqrt[a + c*x^4], x], x] + Simp[e/q Int[(1 + q*x^2)/Sqrt[a + c*x^4], x], x]] /; FreeQ[{a, c, d, e}, x] && Neg Q[c/a] && NeQ[c*d^2 + a*e^2, 0]
Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[ {q = Rt[-c/a, 4]}, Simp[(1/(d*Sqrt[a]*q))*EllipticPi[-e/(d*q^2), ArcSin[q*x ], -1], x]] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] && GtQ[a, 0]
Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> Simp[ Sqrt[1 + c*(x^4/a)]/Sqrt[a + c*x^4] Int[1/((d + e*x^2)*Sqrt[1 + c*(x^4/a) ]), x], x] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] && !GtQ[a, 0]
Int[((x_)^(m_)*((a_) + (c_.)*(x_)^4)^(p_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(-d/e)^(m/2)*((c*d^2 + a*e^2)^(p + 1/2)/e^(2*p + 1)) Int[1/((d + e*x^2)*Sqrt[a + c*x^4]), x], x] + Simp[1/e^(2*p + 1) Int[(1/Sqrt[a + c*x ^4])*ExpandToSum[(e^(2*p + 1)*x^m*(a + c*x^4)^(p + 1/2) - (-d/e)^(m/2)*(c*d ^2 + a*e^2)^(p + 1/2))/(d + e*x^2), x], x], x] /; FreeQ[{a, c, d, e}, x] && IGtQ[p + 1/2, 0] && IGtQ[m/2, 0] && NeQ[c*d^2 - a*e^2, 0] && NegQ[c/a]
Int[(Pq_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{q = Expon[Pq, x ]}, With[{Pqq = Coeff[Pq, x, q]}, Simp[Pqq*x^(q - n + 1)*((a + b*x^n)^(p + 1)/(b*(q + n*p + 1))), x] + Simp[1/(b*(q + n*p + 1)) Int[ExpandToSum[b*(q + n*p + 1)*(Pq - Pqq*x^q) - a*Pqq*(q - n + 1)*x^(q - n), x]*(a + b*x^n)^p, x], x]] /; NeQ[q + n*p + 1, 0] && q - n >= 0 && (IntegerQ[2*p] || IntegerQ [p + (q + 1)/(2*n)])] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x] && IGtQ[n, 0]
Time = 1.91 (sec) , antiderivative size = 647, normalized size of antiderivative = 1.52
| method | result | size |
| elliptic | \(\frac {\sqrt {e x}\, \sqrt {\left (-b \,x^{2}+a \right ) e x}\, \left (\frac {2 e^{2} x^{2} \sqrt {-b e \,x^{3}+a e x}}{7 d}-\frac {2 c \,e^{2} x \sqrt {-b e \,x^{3}+a e x}}{5 d^{2}}-\frac {2 \left (\frac {\left (a \,d^{2}-b \,c^{2}\right ) e^{3}}{d^{3}}-\frac {5 e^{3} a}{7 d}\right ) \sqrt {-b e \,x^{3}+a e x}}{3 b e}+\frac {\left (\frac {c^{2} \left (a \,d^{2}-b \,c^{2}\right ) e^{3}}{d^{5}}+\frac {\left (\frac {\left (a \,d^{2}-b \,c^{2}\right ) e^{3}}{d^{3}}-\frac {5 e^{3} a}{7 d}\right ) a}{3 b}\right ) \sqrt {a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{b \sqrt {-b e \,x^{3}+a e x}}+\frac {\left (-\frac {c \left (a \,d^{2}-b \,c^{2}\right ) e^{3}}{d^{4}}+\frac {3 c \,e^{3} a}{5 d^{2}}\right ) \sqrt {a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}\, \left (-\frac {2 \sqrt {a b}\, \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{b \sqrt {-b e \,x^{3}+a e x}}-\frac {c^{3} \left (a \,d^{2}-b \,c^{2}\right ) e^{3} \sqrt {a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}, -\frac {\sqrt {a b}}{b \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right )}, \frac {\sqrt {2}}{2}\right )}{d^{6} b \sqrt {-b e \,x^{3}+a e x}\, \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right )}\right )}{e x \sqrt {-b \,x^{2}+a}}\) | \(647\) |
| risch | \(-\frac {2 \left (-15 b \,x^{2} d^{2}+21 b c d x +10 a \,d^{2}-35 b \,c^{2}\right ) \sqrt {-b \,x^{2}+a}\, x \,e^{3}}{105 b \,d^{3} \sqrt {e x}}-\frac {\left (\frac {\frac {21 d c \left (2 a \,d^{2}-5 b \,c^{2}\right ) \sqrt {a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}\, \left (-\frac {2 \sqrt {a b}\, \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{\sqrt {-b e \,x^{3}+a e x}}-\frac {10 a^{2} d^{4} \sqrt {a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{b \sqrt {-b e \,x^{3}+a e x}}+\frac {105 b \,c^{4} \sqrt {a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{\sqrt {-b e \,x^{3}+a e x}}-\frac {70 c^{2} d^{2} a \sqrt {a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{\sqrt {-b e \,x^{3}+a e x}}}{d^{2}}+\frac {105 c^{3} \left (a \,d^{2}-b \,c^{2}\right ) \sqrt {a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}, -\frac {\sqrt {a b}}{b \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right )}, \frac {\sqrt {2}}{2}\right )}{d^{3} \sqrt {-b e \,x^{3}+a e x}\, \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right )}\right ) e^{3} \sqrt {\left (-b \,x^{2}+a \right ) e x}}{105 b \,d^{3} \sqrt {e x}\, \sqrt {-b \,x^{2}+a}}\) | \(747\) |
| default | \(\text {Expression too large to display}\) | \(1307\) |
Input:
int((e*x)^(5/2)*(-b*x^2+a)^(1/2)/(d*x+c),x,method=_RETURNVERBOSE)
Output:
1/e/x*(e*x)^(1/2)/(-b*x^2+a)^(1/2)*((-b*x^2+a)*e*x)^(1/2)*(2/7/d*e^2*x^2*( -b*e*x^3+a*e*x)^(1/2)-2/5*c*e^2/d^2*x*(-b*e*x^3+a*e*x)^(1/2)-2/3*(1/d^3*(a *d^2-b*c^2)*e^3-5/7/d*e^3*a)/b/e*(-b*e*x^3+a*e*x)^(1/2)+(c^2*(a*d^2-b*c^2) *e^3/d^5+1/3*(1/d^3*(a*d^2-b*c^2)*e^3-5/7/d*e^3*a)/b*a)/b*(a*b)^(1/2)*((x+ 1/b*(a*b)^(1/2))*b/(a*b)^(1/2))^(1/2)*(-2*(x-1/b*(a*b)^(1/2))*b/(a*b)^(1/2 ))^(1/2)*(-b*x/(a*b)^(1/2))^(1/2)/(-b*e*x^3+a*e*x)^(1/2)*EllipticF(((x+1/b *(a*b)^(1/2))*b/(a*b)^(1/2))^(1/2),1/2*2^(1/2))+(-c/d^4*(a*d^2-b*c^2)*e^3+ 3/5*c*e^3/d^2*a)/b*(a*b)^(1/2)*((x+1/b*(a*b)^(1/2))*b/(a*b)^(1/2))^(1/2)*( -2*(x-1/b*(a*b)^(1/2))*b/(a*b)^(1/2))^(1/2)*(-b*x/(a*b)^(1/2))^(1/2)/(-b*e *x^3+a*e*x)^(1/2)*(-2/b*(a*b)^(1/2)*EllipticE(((x+1/b*(a*b)^(1/2))*b/(a*b) ^(1/2))^(1/2),1/2*2^(1/2))+1/b*(a*b)^(1/2)*EllipticF(((x+1/b*(a*b)^(1/2))* b/(a*b)^(1/2))^(1/2),1/2*2^(1/2)))-c^3*(a*d^2-b*c^2)*e^3/d^6/b*(a*b)^(1/2) *((x+1/b*(a*b)^(1/2))*b/(a*b)^(1/2))^(1/2)*(-2*(x-1/b*(a*b)^(1/2))*b/(a*b) ^(1/2))^(1/2)*(-b*x/(a*b)^(1/2))^(1/2)/(-b*e*x^3+a*e*x)^(1/2)/(c/d-1/b*(a* b)^(1/2))*EllipticPi(((x+1/b*(a*b)^(1/2))*b/(a*b)^(1/2))^(1/2),-1/b*(a*b)^ (1/2)/(c/d-1/b*(a*b)^(1/2)),1/2*2^(1/2)))
Timed out. \[ \int \frac {(e x)^{5/2} \sqrt {a-b x^2}}{c+d x} \, dx=\text {Timed out} \] Input:
integrate((e*x)^(5/2)*(-b*x^2+a)^(1/2)/(d*x+c),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {(e x)^{5/2} \sqrt {a-b x^2}}{c+d x} \, dx=\int \frac {\left (e x\right )^{\frac {5}{2}} \sqrt {a - b x^{2}}}{c + d x}\, dx \] Input:
integrate((e*x)**(5/2)*(-b*x**2+a)**(1/2)/(d*x+c),x)
Output:
Integral((e*x)**(5/2)*sqrt(a - b*x**2)/(c + d*x), x)
\[ \int \frac {(e x)^{5/2} \sqrt {a-b x^2}}{c+d x} \, dx=\int { \frac {\sqrt {-b x^{2} + a} \left (e x\right )^{\frac {5}{2}}}{d x + c} \,d x } \] Input:
integrate((e*x)^(5/2)*(-b*x^2+a)^(1/2)/(d*x+c),x, algorithm="maxima")
Output:
integrate(sqrt(-b*x^2 + a)*(e*x)^(5/2)/(d*x + c), x)
\[ \int \frac {(e x)^{5/2} \sqrt {a-b x^2}}{c+d x} \, dx=\int { \frac {\sqrt {-b x^{2} + a} \left (e x\right )^{\frac {5}{2}}}{d x + c} \,d x } \] Input:
integrate((e*x)^(5/2)*(-b*x^2+a)^(1/2)/(d*x+c),x, algorithm="giac")
Output:
integrate(sqrt(-b*x^2 + a)*(e*x)^(5/2)/(d*x + c), x)
Timed out. \[ \int \frac {(e x)^{5/2} \sqrt {a-b x^2}}{c+d x} \, dx=\int \frac {{\left (e\,x\right )}^{5/2}\,\sqrt {a-b\,x^2}}{c+d\,x} \,d x \] Input:
int(((e*x)^(5/2)*(a - b*x^2)^(1/2))/(c + d*x),x)
Output:
int(((e*x)^(5/2)*(a - b*x^2)^(1/2))/(c + d*x), x)
\[ \int \frac {(e x)^{5/2} \sqrt {a-b x^2}}{c+d x} \, dx=\frac {\sqrt {e}\, e^{2} \left (8 \sqrt {x}\, \sqrt {-b \,x^{2}+a}\, a d -42 \sqrt {x}\, \sqrt {-b \,x^{2}+a}\, b c x +30 \sqrt {x}\, \sqrt {-b \,x^{2}+a}\, b d \,x^{2}+42 \left (\int \frac {\sqrt {x}\, \sqrt {-b \,x^{2}+a}\, x^{2}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) a b \,d^{2}-105 \left (\int \frac {\sqrt {x}\, \sqrt {-b \,x^{2}+a}\, x^{2}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) b^{2} c^{2}-4 \left (\int \frac {\sqrt {x}\, \sqrt {-b \,x^{2}+a}}{-b d \,x^{4}-b c \,x^{3}+a d \,x^{2}+a c x}d x \right ) a^{2} c d -4 \left (\int \frac {\sqrt {x}\, \sqrt {-b \,x^{2}+a}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) a^{2} d^{2}+63 \left (\int \frac {\sqrt {x}\, \sqrt {-b \,x^{2}+a}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) a b \,c^{2}\right )}{105 b \,d^{2}} \] Input:
int((e*x)^(5/2)*(-b*x^2+a)^(1/2)/(d*x+c),x)
Output:
(sqrt(e)*e**2*(8*sqrt(x)*sqrt(a - b*x**2)*a*d - 42*sqrt(x)*sqrt(a - b*x**2 )*b*c*x + 30*sqrt(x)*sqrt(a - b*x**2)*b*d*x**2 + 42*int((sqrt(x)*sqrt(a - b*x**2)*x**2)/(a*c + a*d*x - b*c*x**2 - b*d*x**3),x)*a*b*d**2 - 105*int((s qrt(x)*sqrt(a - b*x**2)*x**2)/(a*c + a*d*x - b*c*x**2 - b*d*x**3),x)*b**2* c**2 - 4*int((sqrt(x)*sqrt(a - b*x**2))/(a*c*x + a*d*x**2 - b*c*x**3 - b*d *x**4),x)*a**2*c*d - 4*int((sqrt(x)*sqrt(a - b*x**2))/(a*c + a*d*x - b*c*x **2 - b*d*x**3),x)*a**2*d**2 + 63*int((sqrt(x)*sqrt(a - b*x**2))/(a*c + a* d*x - b*c*x**2 - b*d*x**3),x)*a*b*c**2))/(105*b*d**2)