Integrand size = 27, antiderivative size = 645 \[ \int \frac {A+B x}{(d+e x)^{3/2} \left (a-c x^2\right )^{5/2}} \, dx=-\frac {2 (B d-A e)}{\left (c d^2-a e^2\right ) \sqrt {d+e x} \left (a-c x^2\right )^{3/2}}+\frac {\sqrt {d+e x} \left (a \left (7 B c d^2-8 A c d e+a B e^2\right )+c \left (A c d^2-8 a B d e+7 a A e^2\right ) x\right )}{3 a \left (c d^2-a e^2\right )^2 \left (a-c x^2\right )^{3/2}}-\frac {\sqrt {d+e x} \left (a e \left (A c d \left (c d^2-33 a e^2\right )+a B e \left (27 c d^2+5 a e^2\right )\right )-c \left (a B d e \left (3 c d^2+29 a e^2\right )+A \left (4 c^2 d^4-15 a c d^2 e^2-21 a^2 e^4\right )\right ) x\right )}{6 a^2 \left (c d^2-a e^2\right )^3 \sqrt {a-c x^2}}+\frac {\sqrt {c} \left (a B d e \left (3 c d^2+29 a e^2\right )+A \left (4 c^2 d^4-15 a c d^2 e^2-21 a^2 e^4\right )\right ) \sqrt {d+e x} \sqrt {1-\frac {c x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} e}{\sqrt {c} d+\sqrt {a} e}\right )}{6 a^{3/2} \left (c d^2-a e^2\right )^3 \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {a} e}} \sqrt {a-c x^2}}-\frac {\left (4 A c d \left (c d^2-3 a e^2\right )+a B e \left (3 c d^2+5 a e^2\right )\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {a} e}} \sqrt {1-\frac {c x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} e}{\sqrt {c} d+\sqrt {a} e}\right )}{6 a^{3/2} \sqrt {c} \left (c d^2-a e^2\right )^2 \sqrt {d+e x} \sqrt {a-c x^2}} \] Output:
(2*A*e-2*B*d)/(-a*e^2+c*d^2)/(e*x+d)^(1/2)/(-c*x^2+a)^(3/2)+1/3*(e*x+d)^(1 /2)*(a*(-8*A*c*d*e+B*a*e^2+7*B*c*d^2)+c*(7*A*a*e^2+A*c*d^2-8*B*a*d*e)*x)/a /(-a*e^2+c*d^2)^2/(-c*x^2+a)^(3/2)-1/6*(e*x+d)^(1/2)*(a*e*(A*c*d*(-33*a*e^ 2+c*d^2)+a*B*e*(5*a*e^2+27*c*d^2))-c*(a*B*d*e*(29*a*e^2+3*c*d^2)+A*(-21*a^ 2*e^4-15*a*c*d^2*e^2+4*c^2*d^4))*x)/a^2/(-a*e^2+c*d^2)^3/(-c*x^2+a)^(1/2)+ 1/6*c^(1/2)*(a*B*d*e*(29*a*e^2+3*c*d^2)+A*(-21*a^2*e^4-15*a*c*d^2*e^2+4*c^ 2*d^4))*(e*x+d)^(1/2)*(1-c*x^2/a)^(1/2)*EllipticE(1/2*(1-c^(1/2)*x/a^(1/2) )^(1/2)*2^(1/2),2^(1/2)*(a^(1/2)*e/(c^(1/2)*d+a^(1/2)*e))^(1/2))/a^(3/2)/( -a*e^2+c*d^2)^3/(c^(1/2)*(e*x+d)/(c^(1/2)*d+a^(1/2)*e))^(1/2)/(-c*x^2+a)^( 1/2)-1/6*(4*A*c*d*(-3*a*e^2+c*d^2)+a*B*e*(5*a*e^2+3*c*d^2))*(c^(1/2)*(e*x+ d)/(c^(1/2)*d+a^(1/2)*e))^(1/2)*(1-c*x^2/a)^(1/2)*EllipticF(1/2*(1-c^(1/2) *x/a^(1/2))^(1/2)*2^(1/2),2^(1/2)*(a^(1/2)*e/(c^(1/2)*d+a^(1/2)*e))^(1/2)) /a^(3/2)/c^(1/2)/(-a*e^2+c*d^2)^2/(e*x+d)^(1/2)/(-c*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 27.55 (sec) , antiderivative size = 789, normalized size of antiderivative = 1.22 \[ \int \frac {A+B x}{(d+e x)^{3/2} \left (a-c x^2\right )^{5/2}} \, dx=\frac {\sqrt {a-c x^2} \left (12 a^2 e^4 (-B d+A e)+e \left (a B d e \left (3 c d^2+29 a e^2\right )+A \left (4 c^2 d^4-15 a c d^2 e^2-21 a^2 e^4\right )\right )-\frac {2 a \left (-c d^2+a e^2\right ) (d+e x) \left (a^2 B e^2+A c^2 d^2 x+a c (B d (d-2 e x)+A e (-2 d+e x))\right )}{\left (a-c x^2\right )^2}-\frac {(d+e x) \left (-5 a^3 B e^4+4 A c^3 d^4 x+a c^2 d^2 e (3 B d x-A (d+15 e x))+a^2 c e^2 (3 A e (7 d-3 e x)+B d (-15 d+17 e x))\right )}{-a+c x^2}-\frac {i \sqrt {c} \left (\sqrt {c} d-\sqrt {a} e\right ) \left (a B d e \left (3 c d^2+29 a e^2\right )+A \left (4 c^2 d^4-15 a c d^2 e^2-21 a^2 e^4\right )\right ) \sqrt {\frac {e \left (\frac {\sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {\sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-d+\frac {\sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d+\sqrt {a} e}{\sqrt {c} d-\sqrt {a} e}\right )}{e \sqrt {-d+\frac {\sqrt {a} e}{\sqrt {c}}} \left (-a+c x^2\right )}-\frac {i \sqrt {a} \left (\sqrt {c} d-\sqrt {a} e\right ) \left (a B e \left (3 c d^2-24 \sqrt {a} \sqrt {c} d e+5 a e^2\right )+A \left (4 c^2 d^3+3 \sqrt {a} c^{3/2} d^2 e-12 a c d e^2+21 a^{3/2} \sqrt {c} e^3\right )\right ) \sqrt {\frac {e \left (\frac {\sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {\sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-d+\frac {\sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right ),\frac {\sqrt {c} d+\sqrt {a} e}{\sqrt {c} d-\sqrt {a} e}\right )}{\sqrt {-d+\frac {\sqrt {a} e}{\sqrt {c}}} \left (-a+c x^2\right )}\right )}{6 a^2 \left (c d^2-a e^2\right )^3 \sqrt {d+e x}} \] Input:
Integrate[(A + B*x)/((d + e*x)^(3/2)*(a - c*x^2)^(5/2)),x]
Output:
(Sqrt[a - c*x^2]*(12*a^2*e^4*(-(B*d) + A*e) + e*(a*B*d*e*(3*c*d^2 + 29*a*e ^2) + A*(4*c^2*d^4 - 15*a*c*d^2*e^2 - 21*a^2*e^4)) - (2*a*(-(c*d^2) + a*e^ 2)*(d + e*x)*(a^2*B*e^2 + A*c^2*d^2*x + a*c*(B*d*(d - 2*e*x) + A*e*(-2*d + e*x))))/(a - c*x^2)^2 - ((d + e*x)*(-5*a^3*B*e^4 + 4*A*c^3*d^4*x + a*c^2* d^2*e*(3*B*d*x - A*(d + 15*e*x)) + a^2*c*e^2*(3*A*e*(7*d - 3*e*x) + B*d*(- 15*d + 17*e*x))))/(-a + c*x^2) - (I*Sqrt[c]*(Sqrt[c]*d - Sqrt[a]*e)*(a*B*d *e*(3*c*d^2 + 29*a*e^2) + A*(4*c^2*d^4 - 15*a*c*d^2*e^2 - 21*a^2*e^4))*Sqr t[(e*(Sqrt[a]/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((Sqrt[a]*e)/Sqrt[c] - e*x)/ (d + e*x))]*(d + e*x)^(3/2)*EllipticE[I*ArcSinh[Sqrt[-d + (Sqrt[a]*e)/Sqrt [c]]/Sqrt[d + e*x]], (Sqrt[c]*d + Sqrt[a]*e)/(Sqrt[c]*d - Sqrt[a]*e)])/(e* Sqrt[-d + (Sqrt[a]*e)/Sqrt[c]]*(-a + c*x^2)) - (I*Sqrt[a]*(Sqrt[c]*d - Sqr t[a]*e)*(a*B*e*(3*c*d^2 - 24*Sqrt[a]*Sqrt[c]*d*e + 5*a*e^2) + A*(4*c^2*d^3 + 3*Sqrt[a]*c^(3/2)*d^2*e - 12*a*c*d*e^2 + 21*a^(3/2)*Sqrt[c]*e^3))*Sqrt[ (e*(Sqrt[a]/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*(d + e*x)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-d + (Sqrt[a]*e)/Sqrt[c ]]/Sqrt[d + e*x]], (Sqrt[c]*d + Sqrt[a]*e)/(Sqrt[c]*d - Sqrt[a]*e)])/(Sqrt [-d + (Sqrt[a]*e)/Sqrt[c]]*(-a + c*x^2))))/(6*a^2*(c*d^2 - a*e^2)^3*Sqrt[d + e*x])
Time = 1.05 (sec) , antiderivative size = 666, normalized size of antiderivative = 1.03, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.481, Rules used = {686, 27, 686, 27, 688, 27, 600, 509, 508, 327, 512, 511, 321}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {A+B x}{\left (a-c x^2\right )^{5/2} (d+e x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 686 |
| \(\displaystyle \frac {x (A c d-a B e)+a (B d-A e)}{3 a \left (a-c x^2\right )^{3/2} \sqrt {d+e x} \left (c d^2-a e^2\right )}-\frac {\int -\frac {c \left (4 A c d^2+3 a B e d-7 a A e^2+5 e (A c d-a B e) x\right )}{2 (d+e x)^{3/2} \left (a-c x^2\right )^{3/2}}dx}{3 a c \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {4 A c d^2+3 a B e d-7 a A e^2+5 e (A c d-a B e) x}{(d+e x)^{3/2} \left (a-c x^2\right )^{3/2}}dx}{6 a \left (c d^2-a e^2\right )}+\frac {x (A c d-a B e)+a (B d-A e)}{3 a \left (a-c x^2\right )^{3/2} \sqrt {d+e x} \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 686 |
| \(\displaystyle \frac {\frac {x \left (4 A c d \left (c d^2-3 a e^2\right )+a B e \left (5 a e^2+3 c d^2\right )\right )+a e \left (7 a A e^2-8 a B d e+A c d^2\right )}{a \sqrt {a-c x^2} \sqrt {d+e x} \left (c d^2-a e^2\right )}-\frac {\int -\frac {c e \left (3 a e \left (A c d^2-8 a B e d+7 a A e^2\right )+\left (4 A c d \left (c d^2-3 a e^2\right )+a B e \left (3 c d^2+5 a e^2\right )\right ) x\right )}{2 (d+e x)^{3/2} \sqrt {a-c x^2}}dx}{a c \left (c d^2-a e^2\right )}}{6 a \left (c d^2-a e^2\right )}+\frac {x (A c d-a B e)+a (B d-A e)}{3 a \left (a-c x^2\right )^{3/2} \sqrt {d+e x} \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {e \int \frac {3 a e \left (A c d^2-8 a B e d+7 a A e^2\right )+\left (4 A c d \left (c d^2-3 a e^2\right )+a B e \left (3 c d^2+5 a e^2\right )\right ) x}{(d+e x)^{3/2} \sqrt {a-c x^2}}dx}{2 a \left (c d^2-a e^2\right )}+\frac {x \left (4 A c d \left (c d^2-3 a e^2\right )+a B e \left (5 a e^2+3 c d^2\right )\right )+a e \left (7 a A e^2-8 a B d e+A c d^2\right )}{a \sqrt {a-c x^2} \sqrt {d+e x} \left (c d^2-a e^2\right )}}{6 a \left (c d^2-a e^2\right )}+\frac {x (A c d-a B e)+a (B d-A e)}{3 a \left (a-c x^2\right )^{3/2} \sqrt {d+e x} \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 688 |
| \(\displaystyle \frac {\frac {e \left (\frac {2 \int -\frac {a e \left (A c d \left (c d^2-33 a e^2\right )+a B e \left (27 c d^2+5 a e^2\right )\right )+c \left (a B d e \left (3 c d^2+29 a e^2\right )+A \left (4 c^2 d^4-15 a c e^2 d^2-21 a^2 e^4\right )\right ) x}{2 \sqrt {d+e x} \sqrt {a-c x^2}}dx}{c d^2-a e^2}-\frac {2 \sqrt {a-c x^2} \left (A \left (-21 a^2 e^4-15 a c d^2 e^2+4 c^2 d^4\right )+a B d e \left (29 a e^2+3 c d^2\right )\right )}{\sqrt {d+e x} \left (c d^2-a e^2\right )}\right )}{2 a \left (c d^2-a e^2\right )}+\frac {x \left (4 A c d \left (c d^2-3 a e^2\right )+a B e \left (5 a e^2+3 c d^2\right )\right )+a e \left (7 a A e^2-8 a B d e+A c d^2\right )}{a \sqrt {a-c x^2} \sqrt {d+e x} \left (c d^2-a e^2\right )}}{6 a \left (c d^2-a e^2\right )}+\frac {x (A c d-a B e)+a (B d-A e)}{3 a \left (a-c x^2\right )^{3/2} \sqrt {d+e x} \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {e \left (-\frac {\int \frac {a e \left (A c d \left (c d^2-33 a e^2\right )+a B e \left (27 c d^2+5 a e^2\right )\right )+c \left (a B d e \left (3 c d^2+29 a e^2\right )+A \left (4 c^2 d^4-15 a c e^2 d^2-21 a^2 e^4\right )\right ) x}{\sqrt {d+e x} \sqrt {a-c x^2}}dx}{c d^2-a e^2}-\frac {2 \sqrt {a-c x^2} \left (A \left (-21 a^2 e^4-15 a c d^2 e^2+4 c^2 d^4\right )+a B d e \left (29 a e^2+3 c d^2\right )\right )}{\sqrt {d+e x} \left (c d^2-a e^2\right )}\right )}{2 a \left (c d^2-a e^2\right )}+\frac {x \left (4 A c d \left (c d^2-3 a e^2\right )+a B e \left (5 a e^2+3 c d^2\right )\right )+a e \left (7 a A e^2-8 a B d e+A c d^2\right )}{a \sqrt {a-c x^2} \sqrt {d+e x} \left (c d^2-a e^2\right )}}{6 a \left (c d^2-a e^2\right )}+\frac {x (A c d-a B e)+a (B d-A e)}{3 a \left (a-c x^2\right )^{3/2} \sqrt {d+e x} \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 600 |
| \(\displaystyle \frac {\frac {e \left (-\frac {\frac {c \left (A \left (-21 a^2 e^4-15 a c d^2 e^2+4 c^2 d^4\right )+a B d e \left (29 a e^2+3 c d^2\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a-c x^2}}dx}{e}-\frac {\left (c d^2-a e^2\right ) \left (4 A c d \left (c d^2-3 a e^2\right )+a B e \left (5 a e^2+3 c d^2\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a-c x^2}}dx}{e}}{c d^2-a e^2}-\frac {2 \sqrt {a-c x^2} \left (A \left (-21 a^2 e^4-15 a c d^2 e^2+4 c^2 d^4\right )+a B d e \left (29 a e^2+3 c d^2\right )\right )}{\sqrt {d+e x} \left (c d^2-a e^2\right )}\right )}{2 a \left (c d^2-a e^2\right )}+\frac {x \left (4 A c d \left (c d^2-3 a e^2\right )+a B e \left (5 a e^2+3 c d^2\right )\right )+a e \left (7 a A e^2-8 a B d e+A c d^2\right )}{a \sqrt {a-c x^2} \sqrt {d+e x} \left (c d^2-a e^2\right )}}{6 a \left (c d^2-a e^2\right )}+\frac {x (A c d-a B e)+a (B d-A e)}{3 a \left (a-c x^2\right )^{3/2} \sqrt {d+e x} \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 509 |
| \(\displaystyle \frac {\frac {e \left (-\frac {\frac {c \sqrt {1-\frac {c x^2}{a}} \left (A \left (-21 a^2 e^4-15 a c d^2 e^2+4 c^2 d^4\right )+a B d e \left (29 a e^2+3 c d^2\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {1-\frac {c x^2}{a}}}dx}{e \sqrt {a-c x^2}}-\frac {\left (c d^2-a e^2\right ) \left (4 A c d \left (c d^2-3 a e^2\right )+a B e \left (5 a e^2+3 c d^2\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a-c x^2}}dx}{e}}{c d^2-a e^2}-\frac {2 \sqrt {a-c x^2} \left (A \left (-21 a^2 e^4-15 a c d^2 e^2+4 c^2 d^4\right )+a B d e \left (29 a e^2+3 c d^2\right )\right )}{\sqrt {d+e x} \left (c d^2-a e^2\right )}\right )}{2 a \left (c d^2-a e^2\right )}+\frac {x \left (4 A c d \left (c d^2-3 a e^2\right )+a B e \left (5 a e^2+3 c d^2\right )\right )+a e \left (7 a A e^2-8 a B d e+A c d^2\right )}{a \sqrt {a-c x^2} \sqrt {d+e x} \left (c d^2-a e^2\right )}}{6 a \left (c d^2-a e^2\right )}+\frac {x (A c d-a B e)+a (B d-A e)}{3 a \left (a-c x^2\right )^{3/2} \sqrt {d+e x} \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 508 |
| \(\displaystyle \frac {\frac {e \left (-\frac {-\frac {2 \sqrt {a} \sqrt {c} \sqrt {1-\frac {c x^2}{a}} \sqrt {d+e x} \left (A \left (-21 a^2 e^4-15 a c d^2 e^2+4 c^2 d^4\right )+a B d e \left (29 a e^2+3 c d^2\right )\right ) \int \frac {\sqrt {1-\frac {e \left (1-\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\frac {\sqrt {c} d}{\sqrt {a}}+e}}}{\sqrt {\frac {1}{2} \left (\frac {\sqrt {c} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}}{e \sqrt {a-c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}}}-\frac {\left (c d^2-a e^2\right ) \left (4 A c d \left (c d^2-3 a e^2\right )+a B e \left (5 a e^2+3 c d^2\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a-c x^2}}dx}{e}}{c d^2-a e^2}-\frac {2 \sqrt {a-c x^2} \left (A \left (-21 a^2 e^4-15 a c d^2 e^2+4 c^2 d^4\right )+a B d e \left (29 a e^2+3 c d^2\right )\right )}{\sqrt {d+e x} \left (c d^2-a e^2\right )}\right )}{2 a \left (c d^2-a e^2\right )}+\frac {x \left (4 A c d \left (c d^2-3 a e^2\right )+a B e \left (5 a e^2+3 c d^2\right )\right )+a e \left (7 a A e^2-8 a B d e+A c d^2\right )}{a \sqrt {a-c x^2} \sqrt {d+e x} \left (c d^2-a e^2\right )}}{6 a \left (c d^2-a e^2\right )}+\frac {x (A c d-a B e)+a (B d-A e)}{3 a \left (a-c x^2\right )^{3/2} \sqrt {d+e x} \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {\frac {e \left (-\frac {-\frac {\left (c d^2-a e^2\right ) \left (4 A c d \left (c d^2-3 a e^2\right )+a B e \left (5 a e^2+3 c d^2\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a-c x^2}}dx}{e}-\frac {2 \sqrt {a} \sqrt {c} \sqrt {1-\frac {c x^2}{a}} \sqrt {d+e x} \left (A \left (-21 a^2 e^4-15 a c d^2 e^2+4 c^2 d^4\right )+a B d e \left (29 a e^2+3 c d^2\right )\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {a}}+e}\right )}{e \sqrt {a-c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}}}}{c d^2-a e^2}-\frac {2 \sqrt {a-c x^2} \left (A \left (-21 a^2 e^4-15 a c d^2 e^2+4 c^2 d^4\right )+a B d e \left (29 a e^2+3 c d^2\right )\right )}{\sqrt {d+e x} \left (c d^2-a e^2\right )}\right )}{2 a \left (c d^2-a e^2\right )}+\frac {x \left (4 A c d \left (c d^2-3 a e^2\right )+a B e \left (5 a e^2+3 c d^2\right )\right )+a e \left (7 a A e^2-8 a B d e+A c d^2\right )}{a \sqrt {a-c x^2} \sqrt {d+e x} \left (c d^2-a e^2\right )}}{6 a \left (c d^2-a e^2\right )}+\frac {x (A c d-a B e)+a (B d-A e)}{3 a \left (a-c x^2\right )^{3/2} \sqrt {d+e x} \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 512 |
| \(\displaystyle \frac {\frac {e \left (-\frac {-\frac {\sqrt {1-\frac {c x^2}{a}} \left (c d^2-a e^2\right ) \left (4 A c d \left (c d^2-3 a e^2\right )+a B e \left (5 a e^2+3 c d^2\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {1-\frac {c x^2}{a}}}dx}{e \sqrt {a-c x^2}}-\frac {2 \sqrt {a} \sqrt {c} \sqrt {1-\frac {c x^2}{a}} \sqrt {d+e x} \left (A \left (-21 a^2 e^4-15 a c d^2 e^2+4 c^2 d^4\right )+a B d e \left (29 a e^2+3 c d^2\right )\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {a}}+e}\right )}{e \sqrt {a-c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}}}}{c d^2-a e^2}-\frac {2 \sqrt {a-c x^2} \left (A \left (-21 a^2 e^4-15 a c d^2 e^2+4 c^2 d^4\right )+a B d e \left (29 a e^2+3 c d^2\right )\right )}{\sqrt {d+e x} \left (c d^2-a e^2\right )}\right )}{2 a \left (c d^2-a e^2\right )}+\frac {x \left (4 A c d \left (c d^2-3 a e^2\right )+a B e \left (5 a e^2+3 c d^2\right )\right )+a e \left (7 a A e^2-8 a B d e+A c d^2\right )}{a \sqrt {a-c x^2} \sqrt {d+e x} \left (c d^2-a e^2\right )}}{6 a \left (c d^2-a e^2\right )}+\frac {x (A c d-a B e)+a (B d-A e)}{3 a \left (a-c x^2\right )^{3/2} \sqrt {d+e x} \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 511 |
| \(\displaystyle \frac {\frac {e \left (-\frac {\frac {2 \sqrt {a} \sqrt {1-\frac {c x^2}{a}} \left (c d^2-a e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}} \left (4 A c d \left (c d^2-3 a e^2\right )+a B e \left (5 a e^2+3 c d^2\right )\right ) \int \frac {1}{\sqrt {1-\frac {e \left (1-\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\frac {\sqrt {c} d}{\sqrt {a}}+e}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {c} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {c} e \sqrt {a-c x^2} \sqrt {d+e x}}-\frac {2 \sqrt {a} \sqrt {c} \sqrt {1-\frac {c x^2}{a}} \sqrt {d+e x} \left (A \left (-21 a^2 e^4-15 a c d^2 e^2+4 c^2 d^4\right )+a B d e \left (29 a e^2+3 c d^2\right )\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {a}}+e}\right )}{e \sqrt {a-c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}}}}{c d^2-a e^2}-\frac {2 \sqrt {a-c x^2} \left (A \left (-21 a^2 e^4-15 a c d^2 e^2+4 c^2 d^4\right )+a B d e \left (29 a e^2+3 c d^2\right )\right )}{\sqrt {d+e x} \left (c d^2-a e^2\right )}\right )}{2 a \left (c d^2-a e^2\right )}+\frac {x \left (4 A c d \left (c d^2-3 a e^2\right )+a B e \left (5 a e^2+3 c d^2\right )\right )+a e \left (7 a A e^2-8 a B d e+A c d^2\right )}{a \sqrt {a-c x^2} \sqrt {d+e x} \left (c d^2-a e^2\right )}}{6 a \left (c d^2-a e^2\right )}+\frac {x (A c d-a B e)+a (B d-A e)}{3 a \left (a-c x^2\right )^{3/2} \sqrt {d+e x} \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {\frac {e \left (-\frac {\frac {2 \sqrt {a} \sqrt {1-\frac {c x^2}{a}} \left (c d^2-a e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}} \left (4 A c d \left (c d^2-3 a e^2\right )+a B e \left (5 a e^2+3 c d^2\right )\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {a}}+e}\right )}{\sqrt {c} e \sqrt {a-c x^2} \sqrt {d+e x}}-\frac {2 \sqrt {a} \sqrt {c} \sqrt {1-\frac {c x^2}{a}} \sqrt {d+e x} \left (A \left (-21 a^2 e^4-15 a c d^2 e^2+4 c^2 d^4\right )+a B d e \left (29 a e^2+3 c d^2\right )\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {a}}+e}\right )}{e \sqrt {a-c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}}}}{c d^2-a e^2}-\frac {2 \sqrt {a-c x^2} \left (A \left (-21 a^2 e^4-15 a c d^2 e^2+4 c^2 d^4\right )+a B d e \left (29 a e^2+3 c d^2\right )\right )}{\sqrt {d+e x} \left (c d^2-a e^2\right )}\right )}{2 a \left (c d^2-a e^2\right )}+\frac {x \left (4 A c d \left (c d^2-3 a e^2\right )+a B e \left (5 a e^2+3 c d^2\right )\right )+a e \left (7 a A e^2-8 a B d e+A c d^2\right )}{a \sqrt {a-c x^2} \sqrt {d+e x} \left (c d^2-a e^2\right )}}{6 a \left (c d^2-a e^2\right )}+\frac {x (A c d-a B e)+a (B d-A e)}{3 a \left (a-c x^2\right )^{3/2} \sqrt {d+e x} \left (c d^2-a e^2\right )}\) |
Input:
Int[(A + B*x)/((d + e*x)^(3/2)*(a - c*x^2)^(5/2)),x]
Output:
(a*(B*d - A*e) + (A*c*d - a*B*e)*x)/(3*a*(c*d^2 - a*e^2)*Sqrt[d + e*x]*(a - c*x^2)^(3/2)) + ((a*e*(A*c*d^2 - 8*a*B*d*e + 7*a*A*e^2) + (4*A*c*d*(c*d^ 2 - 3*a*e^2) + a*B*e*(3*c*d^2 + 5*a*e^2))*x)/(a*(c*d^2 - a*e^2)*Sqrt[d + e *x]*Sqrt[a - c*x^2]) + (e*((-2*(a*B*d*e*(3*c*d^2 + 29*a*e^2) + A*(4*c^2*d^ 4 - 15*a*c*d^2*e^2 - 21*a^2*e^4))*Sqrt[a - c*x^2])/((c*d^2 - a*e^2)*Sqrt[d + e*x]) - ((-2*Sqrt[a]*Sqrt[c]*(a*B*d*e*(3*c*d^2 + 29*a*e^2) + A*(4*c^2*d ^4 - 15*a*c*d^2*e^2 - 21*a^2*e^4))*Sqrt[d + e*x]*Sqrt[1 - (c*x^2)/a]*Ellip ticE[ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[a]]/Sqrt[2]], (2*e)/((Sqrt[c]*d)/Sqr t[a] + e)])/(e*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[a]*e)]*Sqrt[a - c*x^2]) + (2*Sqrt[a]*(c*d^2 - a*e^2)*(4*A*c*d*(c*d^2 - 3*a*e^2) + a*B*e*(3 *c*d^2 + 5*a*e^2))*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[a]*e)]*Sqrt[ 1 - (c*x^2)/a]*EllipticF[ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[a]]/Sqrt[2]], (2 *e)/((Sqrt[c]*d)/Sqrt[a] + e)])/(Sqrt[c]*e*Sqrt[d + e*x]*Sqrt[a - c*x^2])) /(c*d^2 - a*e^2)))/(2*a*(c*d^2 - a*e^2)))/(6*a*(c*d^2 - a*e^2))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c /(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0] && !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ (Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) )], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
Int[Sqrt[(c_) + (d_.)*(x_)]/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> With[{q = Rt[-b/a, 2]}, Simp[-2*(Sqrt[c + d*x]/(Sqrt[a]*q*Sqrt[q*((c + d*x)/(d + c *q))])) Subst[Int[Sqrt[1 - 2*d*(x^2/(d + c*q))]/Sqrt[1 - x^2], x], x, Sqr t[(1 - q*x)/2]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[a, 0]
Int[Sqrt[(c_) + (d_.)*(x_)]/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[Sq rt[1 + b*(x^2/a)]/Sqrt[a + b*x^2] Int[Sqrt[c + d*x]/Sqrt[1 + b*(x^2/a)], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && !GtQ[a, 0]
Int[1/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Wit h[{q = Rt[-b/a, 2]}, Simp[-2*(Sqrt[q*((c + d*x)/(d + c*q))]/(Sqrt[a]*q*Sqrt [c + d*x])) Subst[Int[1/(Sqrt[1 - 2*d*(x^2/(d + c*q))]*Sqrt[1 - x^2]), x] , x, Sqrt[(1 - q*x)/2]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[ a, 0]
Int[1/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Sim p[Sqrt[1 + b*(x^2/a)]/Sqrt[a + b*x^2] Int[1/(Sqrt[c + d*x]*Sqrt[1 + b*(x^ 2/a)]), x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && !GtQ[a, 0]
Int[((A_.) + (B_.)*(x_))/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2] ), x_Symbol] :> Simp[B/d Int[Sqrt[c + d*x]/Sqrt[a + b*x^2], x], x] - Simp [(B*c - A*d)/d Int[1/(Sqrt[c + d*x]*Sqrt[a + b*x^2]), x], x] /; FreeQ[{a, b, c, d, A, B}, x] && NegQ[b/a]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))*(f*a*c*e - a*g*c*d + c*(c*d*f + a*e*g)*x)*((a + c*x^2)^(p + 1)/(2*a*c*(p + 1)*(c*d^2 + a*e^2))), x] + Simp[ 1/(2*a*c*(p + 1)*(c*d^2 + a*e^2)) Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Sim p[f*(c^2*d^2*(2*p + 3) + a*c*e^2*(m + 2*p + 3)) - a*c*d*e*g*m + c*e*(c*d*f + a*e*g)*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && LtQ [p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[(e*f - d*g)*(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1)/( (m + 1)*(c*d^2 + a*e^2))), x] + Simp[1/((m + 1)*(c*d^2 + a*e^2)) Int[(d + e*x)^(m + 1)*(a + c*x^2)^p*Simp[(c*d*f + a*e*g)*(m + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, p}, x] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Leaf count of result is larger than twice the leaf count of optimal. \(1194\) vs. \(2(571)=1142\).
Time = 10.76 (sec) , antiderivative size = 1195, normalized size of antiderivative = 1.85
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(1195\) |
| default | \(\text {Expression too large to display}\) | \(5787\) |
Input:
int((B*x+A)/(e*x+d)^(3/2)/(-c*x^2+a)^(5/2),x,method=_RETURNVERBOSE)
Output:
((e*x+d)*(-c*x^2+a))^(1/2)/(e*x+d)^(1/2)/(-c*x^2+a)^(1/2)*((1/3*(A*a*e^2+A *c*d^2-2*B*a*d*e)/c/a/(a*e^2-c*d^2)^2*x-1/3*(2*A*c*d*e-B*a*e^2-B*c*d^2)/(a *e^2-c*d^2)^2/c^2)*(-c*e*x^3-c*d*x^2+a*e*x+a*d)^(1/2)/(x^2-a/c)^2-2*(-c*e* x-c*d)*(1/12*(9*A*a^2*e^4+15*A*a*c*d^2*e^2-4*A*c^2*d^4-17*B*a^2*d*e^3-3*B* a*c*d^3*e)/(a*e^2-c*d^2)^3/a^2*x-1/12*e*(21*A*a*c*d*e^2-A*c^2*d^3-5*B*a^2* e^3-15*B*a*c*d^2*e)/(a*e^2-c*d^2)^3/a/c)/((x^2-a/c)*(-c*e*x-c*d))^(1/2)-2* (-c*e*x^2+a*e)*e^3/(a*e^2-c*d^2)^3*(A*e-B*d)/((x+d/e)*(-c*e*x^2+a*e))^(1/2 )+2*(-1/6/(a*e^2-c*d^2)^2*(12*A*a*c*d*e^2-4*A*c^2*d^3-5*B*a^2*e^3-3*B*a*c* d^2*e)/a^2+1/12*e^2*(21*A*a*c*d*e^2-A*c^2*d^3-5*B*a^2*e^3-15*B*a*c*d^2*e)/ (a*e^2-c*d^2)^3/a-1/6*c*d*(9*A*a^2*e^4+15*A*a*c*d^2*e^2-4*A*c^2*d^4-17*B*a ^2*d*e^3-3*B*a*c*d^3*e)/(a*e^2-c*d^2)^3/a^2-e^3*c*d*(A*e-B*d)/(a*e^2-c*d^2 )^3)*(d/e-1/c*(a*c)^(1/2))*((x+d/e)/(d/e-1/c*(a*c)^(1/2)))^(1/2)*((x-1/c*( a*c)^(1/2))/(-d/e-1/c*(a*c)^(1/2)))^(1/2)*((x+1/c*(a*c)^(1/2))/(-d/e+1/c*( a*c)^(1/2)))^(1/2)/(-c*e*x^3-c*d*x^2+a*e*x+a*d)^(1/2)*EllipticF(((x+d/e)/( d/e-1/c*(a*c)^(1/2)))^(1/2),((-d/e+1/c*(a*c)^(1/2))/(-d/e-1/c*(a*c)^(1/2)) )^(1/2))+2*(-1/12*c*e*(9*A*a^2*e^4+15*A*a*c*d^2*e^2-4*A*c^2*d^4-17*B*a^2*d *e^3-3*B*a*c*d^3*e)/(a*e^2-c*d^2)^3/a^2-c*e^4*(A*e-B*d)/(a*e^2-c*d^2)^3)*( d/e-1/c*(a*c)^(1/2))*((x+d/e)/(d/e-1/c*(a*c)^(1/2)))^(1/2)*((x-1/c*(a*c)^( 1/2))/(-d/e-1/c*(a*c)^(1/2)))^(1/2)*((x+1/c*(a*c)^(1/2))/(-d/e+1/c*(a*c)^( 1/2)))^(1/2)/(-c*e*x^3-c*d*x^2+a*e*x+a*d)^(1/2)*((-d/e-1/c*(a*c)^(1/2))...
Leaf count of result is larger than twice the leaf count of optimal. 1814 vs. \(2 (571) = 1142\).
Time = 0.24 (sec) , antiderivative size = 1814, normalized size of antiderivative = 2.81 \[ \int \frac {A+B x}{(d+e x)^{3/2} \left (a-c x^2\right )^{5/2}} \, dx=\text {Too large to display} \] Input:
integrate((B*x+A)/(e*x+d)^(3/2)/(-c*x^2+a)^(5/2),x, algorithm="fricas")
Output:
-1/18*((4*A*a^2*c^3*d^6 + 3*B*a^3*c^2*d^5*e - 18*A*a^3*c^2*d^4*e^2 - 52*B* a^4*c*d^3*e^3 + 78*A*a^4*c*d^2*e^4 - 15*B*a^5*d*e^5 + (4*A*c^5*d^5*e + 3*B *a*c^4*d^4*e^2 - 18*A*a*c^4*d^3*e^3 - 52*B*a^2*c^3*d^2*e^4 + 78*A*a^2*c^3* d*e^5 - 15*B*a^3*c^2*e^6)*x^5 + (4*A*c^5*d^6 + 3*B*a*c^4*d^5*e - 18*A*a*c^ 4*d^4*e^2 - 52*B*a^2*c^3*d^3*e^3 + 78*A*a^2*c^3*d^2*e^4 - 15*B*a^3*c^2*d*e ^5)*x^4 - 2*(4*A*a*c^4*d^5*e + 3*B*a^2*c^3*d^4*e^2 - 18*A*a^2*c^3*d^3*e^3 - 52*B*a^3*c^2*d^2*e^4 + 78*A*a^3*c^2*d*e^5 - 15*B*a^4*c*e^6)*x^3 - 2*(4*A *a*c^4*d^6 + 3*B*a^2*c^3*d^5*e - 18*A*a^2*c^3*d^4*e^2 - 52*B*a^3*c^2*d^3*e ^3 + 78*A*a^3*c^2*d^2*e^4 - 15*B*a^4*c*d*e^5)*x^2 + (4*A*a^2*c^3*d^5*e + 3 *B*a^3*c^2*d^4*e^2 - 18*A*a^3*c^2*d^3*e^3 - 52*B*a^4*c*d^2*e^4 + 78*A*a^4* c*d*e^5 - 15*B*a^5*e^6)*x)*sqrt(-c*e)*weierstrassPInverse(4/3*(c*d^2 + 3*a *e^2)/(c*e^2), -8/27*(c*d^3 - 9*a*d*e^2)/(c*e^3), 1/3*(3*e*x + d)/e) + 3*( 4*A*a^2*c^3*d^5*e + 3*B*a^3*c^2*d^4*e^2 - 15*A*a^3*c^2*d^3*e^3 + 29*B*a^4* c*d^2*e^4 - 21*A*a^4*c*d*e^5 + (4*A*c^5*d^4*e^2 + 3*B*a*c^4*d^3*e^3 - 15*A *a*c^4*d^2*e^4 + 29*B*a^2*c^3*d*e^5 - 21*A*a^2*c^3*e^6)*x^5 + (4*A*c^5*d^5 *e + 3*B*a*c^4*d^4*e^2 - 15*A*a*c^4*d^3*e^3 + 29*B*a^2*c^3*d^2*e^4 - 21*A* a^2*c^3*d*e^5)*x^4 - 2*(4*A*a*c^4*d^4*e^2 + 3*B*a^2*c^3*d^3*e^3 - 15*A*a^2 *c^3*d^2*e^4 + 29*B*a^3*c^2*d*e^5 - 21*A*a^3*c^2*e^6)*x^3 - 2*(4*A*a*c^4*d ^5*e + 3*B*a^2*c^3*d^4*e^2 - 15*A*a^2*c^3*d^3*e^3 + 29*B*a^3*c^2*d^2*e^4 - 21*A*a^3*c^2*d*e^5)*x^2 + (4*A*a^2*c^3*d^4*e^2 + 3*B*a^3*c^2*d^3*e^3 -...
Timed out. \[ \int \frac {A+B x}{(d+e x)^{3/2} \left (a-c x^2\right )^{5/2}} \, dx=\text {Timed out} \] Input:
integrate((B*x+A)/(e*x+d)**(3/2)/(-c*x**2+a)**(5/2),x)
Output:
Timed out
\[ \int \frac {A+B x}{(d+e x)^{3/2} \left (a-c x^2\right )^{5/2}} \, dx=\int { \frac {B x + A}{{\left (-c x^{2} + a\right )}^{\frac {5}{2}} {\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((B*x+A)/(e*x+d)^(3/2)/(-c*x^2+a)^(5/2),x, algorithm="maxima")
Output:
integrate((B*x + A)/((-c*x^2 + a)^(5/2)*(e*x + d)^(3/2)), x)
\[ \int \frac {A+B x}{(d+e x)^{3/2} \left (a-c x^2\right )^{5/2}} \, dx=\int { \frac {B x + A}{{\left (-c x^{2} + a\right )}^{\frac {5}{2}} {\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((B*x+A)/(e*x+d)^(3/2)/(-c*x^2+a)^(5/2),x, algorithm="giac")
Output:
integrate((B*x + A)/((-c*x^2 + a)^(5/2)*(e*x + d)^(3/2)), x)
Timed out. \[ \int \frac {A+B x}{(d+e x)^{3/2} \left (a-c x^2\right )^{5/2}} \, dx=\int \frac {A+B\,x}{{\left (a-c\,x^2\right )}^{5/2}\,{\left (d+e\,x\right )}^{3/2}} \,d x \] Input:
int((A + B*x)/((a - c*x^2)^(5/2)*(d + e*x)^(3/2)),x)
Output:
int((A + B*x)/((a - c*x^2)^(5/2)*(d + e*x)^(3/2)), x)
\[ \int \frac {A+B x}{(d+e x)^{3/2} \left (a-c x^2\right )^{5/2}} \, dx=\text {too large to display} \] Input:
int((B*x+A)/(e*x+d)^(3/2)/(-c*x^2+a)^(5/2),x)
Output:
(8*sqrt(a - c*x**2)*int(sqrt(d + e*x)/(sqrt(a - c*x**2)*a**2*d**4 - 2*sqrt (a - c*x**2)*a**2*d**2*e**2*x**2 + sqrt(a - c*x**2)*a**2*e**4*x**4 - 2*sqr t(a - c*x**2)*a*c*d**4*x**2 + 4*sqrt(a - c*x**2)*a*c*d**2*e**2*x**4 - 2*sq rt(a - c*x**2)*a*c*e**4*x**6 + sqrt(a - c*x**2)*c**2*d**4*x**4 - 2*sqrt(a - c*x**2)*c**2*d**2*e**2*x**6 + sqrt(a - c*x**2)*c**2*e**4*x**8),x)*a**2*b *d**4*e + 8*sqrt(a - c*x**2)*int(sqrt(d + e*x)/(sqrt(a - c*x**2)*a**2*d**4 - 2*sqrt(a - c*x**2)*a**2*d**2*e**2*x**2 + sqrt(a - c*x**2)*a**2*e**4*x** 4 - 2*sqrt(a - c*x**2)*a*c*d**4*x**2 + 4*sqrt(a - c*x**2)*a*c*d**2*e**2*x* *4 - 2*sqrt(a - c*x**2)*a*c*e**4*x**6 + sqrt(a - c*x**2)*c**2*d**4*x**4 - 2*sqrt(a - c*x**2)*c**2*d**2*e**2*x**6 + sqrt(a - c*x**2)*c**2*e**4*x**8), x)*a**2*b*d**3*e**2*x - 8*sqrt(a - c*x**2)*int(sqrt(d + e*x)/(sqrt(a - c*x **2)*a**2*d**4 - 2*sqrt(a - c*x**2)*a**2*d**2*e**2*x**2 + sqrt(a - c*x**2) *a**2*e**4*x**4 - 2*sqrt(a - c*x**2)*a*c*d**4*x**2 + 4*sqrt(a - c*x**2)*a* c*d**2*e**2*x**4 - 2*sqrt(a - c*x**2)*a*c*e**4*x**6 + sqrt(a - c*x**2)*c** 2*d**4*x**4 - 2*sqrt(a - c*x**2)*c**2*d**2*e**2*x**6 + sqrt(a - c*x**2)*c* *2*e**4*x**8),x)*a*b*c*d**4*e*x**2 - 8*sqrt(a - c*x**2)*int(sqrt(d + e*x)/ (sqrt(a - c*x**2)*a**2*d**4 - 2*sqrt(a - c*x**2)*a**2*d**2*e**2*x**2 + sqr t(a - c*x**2)*a**2*e**4*x**4 - 2*sqrt(a - c*x**2)*a*c*d**4*x**2 + 4*sqrt(a - c*x**2)*a*c*d**2*e**2*x**4 - 2*sqrt(a - c*x**2)*a*c*e**4*x**6 + sqrt(a - c*x**2)*c**2*d**4*x**4 - 2*sqrt(a - c*x**2)*c**2*d**2*e**2*x**6 + sqr...