Integrand size = 27, antiderivative size = 654 \[ \int \frac {(A+B x) \left (a-c x^2\right )^{3/2}}{(d+e x)^{9/2}} \, dx=\frac {8 c \left (32 B c d^3-4 A c d^2 e-33 a B d e^2+5 a A e^3\right ) \sqrt {a-c x^2}}{35 e^4 \left (c d^2-a e^2\right ) (d+e x)^{3/2}}-\frac {8 c \left (4 A c d e \left (c d^2-2 a e^2\right )-B \left (32 c^2 d^4-57 a c d^2 e^2+21 a^2 e^4\right )\right ) \sqrt {a-c x^2}}{35 e^4 \left (c d^2-a e^2\right )^2 \sqrt {d+e x}}-\frac {12 \left (7 a B e^2+4 c d (8 B d-A e)+5 c e (8 B d-A e) x\right ) \sqrt {a-c x^2}}{35 e^4 (d+e x)^{5/2}}+\frac {2 (8 B d-A e+7 B e x) \left (a-c x^2\right )^{3/2}}{7 e^2 (d+e x)^{7/2}}+\frac {8 \sqrt {a} c^{3/2} \left (4 A c d e \left (c d^2-2 a e^2\right )-B \left (32 c^2 d^4-57 a c d^2 e^2+21 a^2 e^4\right )\right ) \sqrt {d+e x} \sqrt {1-\frac {c x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} e}{\sqrt {c} d+\sqrt {a} e}\right )}{35 e^5 \left (c d^2-a e^2\right )^2 \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {a} e}} \sqrt {a-c x^2}}+\frac {8 \sqrt {a} c^{3/2} \left (32 B c d^3-4 A c d^2 e-33 a B d e^2+5 a A e^3\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {a} e}} \sqrt {1-\frac {c x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} e}{\sqrt {c} d+\sqrt {a} e}\right )}{35 e^5 \left (c d^2-a e^2\right ) \sqrt {d+e x} \sqrt {a-c x^2}} \] Output:
8/35*c*(5*A*a*e^3-4*A*c*d^2*e-33*B*a*d*e^2+32*B*c*d^3)*(-c*x^2+a)^(1/2)/e^ 4/(-a*e^2+c*d^2)/(e*x+d)^(3/2)-8/35*c*(4*A*c*d*e*(-2*a*e^2+c*d^2)-B*(21*a^ 2*e^4-57*a*c*d^2*e^2+32*c^2*d^4))*(-c*x^2+a)^(1/2)/e^4/(-a*e^2+c*d^2)^2/(e *x+d)^(1/2)-12/35*(7*B*a*e^2+4*c*d*(-A*e+8*B*d)+5*c*e*(-A*e+8*B*d)*x)*(-c* x^2+a)^(1/2)/e^4/(e*x+d)^(5/2)+2/7*(7*B*e*x-A*e+8*B*d)*(-c*x^2+a)^(3/2)/e^ 2/(e*x+d)^(7/2)+8/35*a^(1/2)*c^(3/2)*(4*A*c*d*e*(-2*a*e^2+c*d^2)-B*(21*a^2 *e^4-57*a*c*d^2*e^2+32*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))/e^5/(-a*e^2+c*d^2)^2/(c^(1/2)*(e*x+d)/(c^(1/2)*d+a^(1/2)*e ))^(1/2)/(-c*x^2+a)^(1/2)+8/35*a^(1/2)*c^(3/2)*(5*A*a*e^3-4*A*c*d^2*e-33*B *a*d*e^2+32*B*c*d^3)*(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))/e^5/(-a*e^2+c*d^2)/(e*x+d)^(1/2)/(-c*x ^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 28.34 (sec) , antiderivative size = 795, normalized size of antiderivative = 1.22 \[ \int \frac {(A+B x) \left (a-c x^2\right )^{3/2}}{(d+e x)^{9/2}} \, dx=\frac {2 \sqrt {a-c x^2} \left (4 c \left (4 A c d e \left (c d^2-2 a e^2\right )+B \left (-32 c^2 d^4+57 a c d^2 e^2-21 a^2 e^4\right )\right )+c \left (-16 A c d e \left (c d^2-2 a e^2\right )+B \left (93 c^2 d^4-158 a c d^2 e^2+49 a^2 e^4\right )\right )-\frac {5 (B d-A e) \left (c d^2-a e^2\right )^3}{(d+e x)^3}+\frac {\left (c d^2-a e^2\right )^2 \left (23 B c d^2-16 A c d e-7 a B e^2\right )}{(d+e x)^2}-\frac {c \left (c d^2-a e^2\right ) \left (47 B c d^3-19 A c d^2 e-43 a B d e^2+15 a A e^3\right )}{d+e x}+\frac {4 i c^{3/2} \left (\sqrt {c} d-\sqrt {a} e\right ) \left (-4 A c d e \left (c d^2-2 a e^2\right )+B \left (32 c^2 d^4-57 a c d^2 e^2+21 a^2 e^4\right )\right ) \sqrt {\frac {e \left (\frac {\sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {\sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-d+\frac {\sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d+\sqrt {a} e}{\sqrt {c} d-\sqrt {a} e}\right )}{e^2 \sqrt {-d+\frac {\sqrt {a} e}{\sqrt {c}}} \left (-a+c x^2\right )}-\frac {4 i \sqrt {a} c^{3/2} \left (\sqrt {c} d-\sqrt {a} e\right ) \left (A \sqrt {c} e \left (4 c d^2+3 \sqrt {a} \sqrt {c} d e-5 a e^2\right )+B \left (-32 c^{3/2} d^3-24 \sqrt {a} c d^2 e+33 a \sqrt {c} d e^2+21 a^{3/2} e^3\right )\right ) \sqrt {\frac {e \left (\frac {\sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {\sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-d+\frac {\sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right ),\frac {\sqrt {c} d+\sqrt {a} e}{\sqrt {c} d-\sqrt {a} e}\right )}{e \sqrt {-d+\frac {\sqrt {a} e}{\sqrt {c}}} \left (-a+c x^2\right )}\right )}{35 e^4 \left (c d^2-a e^2\right )^2 \sqrt {d+e x}} \] Input:
Integrate[((A + B*x)*(a - c*x^2)^(3/2))/(d + e*x)^(9/2),x]
Output:
(2*Sqrt[a - c*x^2]*(4*c*(4*A*c*d*e*(c*d^2 - 2*a*e^2) + B*(-32*c^2*d^4 + 57 *a*c*d^2*e^2 - 21*a^2*e^4)) + c*(-16*A*c*d*e*(c*d^2 - 2*a*e^2) + B*(93*c^2 *d^4 - 158*a*c*d^2*e^2 + 49*a^2*e^4)) - (5*(B*d - A*e)*(c*d^2 - a*e^2)^3)/ (d + e*x)^3 + ((c*d^2 - a*e^2)^2*(23*B*c*d^2 - 16*A*c*d*e - 7*a*B*e^2))/(d + e*x)^2 - (c*(c*d^2 - a*e^2)*(47*B*c*d^3 - 19*A*c*d^2*e - 43*a*B*d*e^2 + 15*a*A*e^3))/(d + e*x) + ((4*I)*c^(3/2)*(Sqrt[c]*d - Sqrt[a]*e)*(-4*A*c*d *e*(c*d^2 - 2*a*e^2) + B*(32*c^2*d^4 - 57*a*c*d^2*e^2 + 21*a^2*e^4))*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)*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^2* Sqrt[-d + (Sqrt[a]*e)/Sqrt[c]]*(-a + c*x^2)) - ((4*I)*Sqrt[a]*c^(3/2)*(Sqr t[c]*d - Sqrt[a]*e)*(A*Sqrt[c]*e*(4*c*d^2 + 3*Sqrt[a]*Sqrt[c]*d*e - 5*a*e^ 2) + B*(-32*c^(3/2)*d^3 - 24*Sqrt[a]*c*d^2*e + 33*a*Sqrt[c]*d*e^2 + 21*a^( 3/2)*e^3))*Sqrt[(e*(Sqrt[a]/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((Sqrt[a]*e)/S qrt[c] - e*x)/(d + e*x))]*(d + e*x)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-d + (S qrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d + Sqrt[a]*e)/(Sqrt[c]*d - Sq rt[a]*e)])/(e*Sqrt[-d + (Sqrt[a]*e)/Sqrt[c]]*(-a + c*x^2))))/(35*e^4*(c*d^ 2 - a*e^2)^2*Sqrt[d + e*x])
Time = 0.95 (sec) , antiderivative size = 650, normalized size of antiderivative = 0.99, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.481, Rules used = {680, 25, 27, 680, 25, 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.
| \(\displaystyle \int \frac {\left (a-c x^2\right )^{3/2} (A+B x)}{(d+e x)^{9/2}} \, dx\) |
| \(\Big \downarrow \) 680 |
| \(\displaystyle \frac {6 \int -\frac {c \left (5 a e (B d-A e)+\left (8 B c d^2-A c e d-7 a B e^2\right ) x\right ) \sqrt {a-c x^2}}{(d+e x)^{5/2}}dx}{35 e^2 \left (c d^2-a e^2\right )}-\frac {2 \left (a-c x^2\right )^{3/2} \left (e x \left (-7 a B e^2-6 A c d e+13 B c d^2\right )-5 a A e^3-2 a B d e^2-A c d^2 e+8 B c d^3\right )}{35 e^2 (d+e x)^{7/2} \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {6 \int \frac {c \left (5 a e (B d-A e)+\left (8 B c d^2-A c e d-7 a B e^2\right ) x\right ) \sqrt {a-c x^2}}{(d+e x)^{5/2}}dx}{35 e^2 \left (c d^2-a e^2\right )}-\frac {2 \left (a-c x^2\right )^{3/2} \left (e x \left (-7 a B e^2-6 A c d e+13 B c d^2\right )-5 a A e^3-2 a B d e^2-A c d^2 e+8 B c d^3\right )}{35 e^2 (d+e x)^{7/2} \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {6 c \int \frac {\left (5 a e (B d-A e)+\left (8 B c d^2-A c e d-7 a B e^2\right ) x\right ) \sqrt {a-c x^2}}{(d+e x)^{5/2}}dx}{35 e^2 \left (c d^2-a e^2\right )}-\frac {2 \left (a-c x^2\right )^{3/2} \left (e x \left (-7 a B e^2-6 A c d e+13 B c d^2\right )-5 a A e^3-2 a B d e^2-A c d^2 e+8 B c d^3\right )}{35 e^2 (d+e x)^{7/2} \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 680 |
| \(\displaystyle -\frac {6 c \left (\frac {2 \int -\frac {c \left (a e \left (8 B c d^3-A c e d^2-12 a B e^2 d+5 a A e^3\right )-\left (4 A c d e \left (c d^2-2 a e^2\right )-B \left (32 c^2 d^4-57 a c e^2 d^2+21 a^2 e^4\right )\right ) x\right )}{\sqrt {d+e x} \sqrt {a-c x^2}}dx}{3 e^2 \left (c d^2-a e^2\right )}+\frac {2 \sqrt {a-c x^2} \left (e x \left (A c d e \left (5 c d^2-13 a e^2\right )-B \left (21 a^2 e^4-69 a c d^2 e^2+40 c^2 d^4\right )\right )+A e \left (-5 a^2 e^4-7 a c d^2 e^2+4 c^2 d^4\right )-B \left (9 a^2 d e^4-49 a c d^3 e^2+32 c^2 d^5\right )\right )}{3 e^2 (d+e x)^{3/2} \left (c d^2-a e^2\right )}\right )}{35 e^2 \left (c d^2-a e^2\right )}-\frac {2 \left (a-c x^2\right )^{3/2} \left (e x \left (-7 a B e^2-6 A c d e+13 B c d^2\right )-5 a A e^3-2 a B d e^2-A c d^2 e+8 B c d^3\right )}{35 e^2 (d+e x)^{7/2} \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {6 c \left (\frac {2 \sqrt {a-c x^2} \left (e x \left (A c d e \left (5 c d^2-13 a e^2\right )-B \left (21 a^2 e^4-69 a c d^2 e^2+40 c^2 d^4\right )\right )+A e \left (-5 a^2 e^4-7 a c d^2 e^2+4 c^2 d^4\right )-B \left (9 a^2 d e^4-49 a c d^3 e^2+32 c^2 d^5\right )\right )}{3 e^2 (d+e x)^{3/2} \left (c d^2-a e^2\right )}-\frac {2 \int \frac {c \left (a e \left (8 B c d^3-A c e d^2-12 a B e^2 d+5 a A e^3\right )-\left (4 A c d e \left (c d^2-2 a e^2\right )-B \left (32 c^2 d^4-57 a c e^2 d^2+21 a^2 e^4\right )\right ) x\right )}{\sqrt {d+e x} \sqrt {a-c x^2}}dx}{3 e^2 \left (c d^2-a e^2\right )}\right )}{35 e^2 \left (c d^2-a e^2\right )}-\frac {2 \left (a-c x^2\right )^{3/2} \left (e x \left (-7 a B e^2-6 A c d e+13 B c d^2\right )-5 a A e^3-2 a B d e^2-A c d^2 e+8 B c d^3\right )}{35 e^2 (d+e x)^{7/2} \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {6 c \left (\frac {2 \sqrt {a-c x^2} \left (e x \left (A c d e \left (5 c d^2-13 a e^2\right )-B \left (21 a^2 e^4-69 a c d^2 e^2+40 c^2 d^4\right )\right )+A e \left (-5 a^2 e^4-7 a c d^2 e^2+4 c^2 d^4\right )-B \left (9 a^2 d e^4-49 a c d^3 e^2+32 c^2 d^5\right )\right )}{3 e^2 (d+e x)^{3/2} \left (c d^2-a e^2\right )}-\frac {2 c \int \frac {a e \left (8 B c d^3-A c e d^2-12 a B e^2 d+5 a A e^3\right )-\left (4 A c d e \left (c d^2-2 a e^2\right )-B \left (32 c^2 d^4-57 a c e^2 d^2+21 a^2 e^4\right )\right ) x}{\sqrt {d+e x} \sqrt {a-c x^2}}dx}{3 e^2 \left (c d^2-a e^2\right )}\right )}{35 e^2 \left (c d^2-a e^2\right )}-\frac {2 \left (a-c x^2\right )^{3/2} \left (e x \left (-7 a B e^2-6 A c d e+13 B c d^2\right )-5 a A e^3-2 a B d e^2-A c d^2 e+8 B c d^3\right )}{35 e^2 (d+e x)^{7/2} \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 600 |
| \(\displaystyle -\frac {6 c \left (\frac {2 \sqrt {a-c x^2} \left (e x \left (A c d e \left (5 c d^2-13 a e^2\right )-B \left (21 a^2 e^4-69 a c d^2 e^2+40 c^2 d^4\right )\right )+A e \left (-5 a^2 e^4-7 a c d^2 e^2+4 c^2 d^4\right )-B \left (9 a^2 d e^4-49 a c d^3 e^2+32 c^2 d^5\right )\right )}{3 e^2 (d+e x)^{3/2} \left (c d^2-a e^2\right )}-\frac {2 c \left (-\frac {\left (4 A c d e \left (c d^2-2 a e^2\right )-B \left (21 a^2 e^4-57 a c d^2 e^2+32 c^2 d^4\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a-c x^2}}dx}{e}-\frac {\left (c d^2-a e^2\right ) \left (5 a A e^3-33 a B d e^2-4 A c d^2 e+32 B c d^3\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a-c x^2}}dx}{e}\right )}{3 e^2 \left (c d^2-a e^2\right )}\right )}{35 e^2 \left (c d^2-a e^2\right )}-\frac {2 \left (a-c x^2\right )^{3/2} \left (e x \left (-7 a B e^2-6 A c d e+13 B c d^2\right )-5 a A e^3-2 a B d e^2-A c d^2 e+8 B c d^3\right )}{35 e^2 (d+e x)^{7/2} \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 509 |
| \(\displaystyle -\frac {6 c \left (\frac {2 \sqrt {a-c x^2} \left (e x \left (A c d e \left (5 c d^2-13 a e^2\right )-B \left (21 a^2 e^4-69 a c d^2 e^2+40 c^2 d^4\right )\right )+A e \left (-5 a^2 e^4-7 a c d^2 e^2+4 c^2 d^4\right )-B \left (9 a^2 d e^4-49 a c d^3 e^2+32 c^2 d^5\right )\right )}{3 e^2 (d+e x)^{3/2} \left (c d^2-a e^2\right )}-\frac {2 c \left (-\frac {\sqrt {1-\frac {c x^2}{a}} \left (4 A c d e \left (c d^2-2 a e^2\right )-B \left (21 a^2 e^4-57 a c d^2 e^2+32 c^2 d^4\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {1-\frac {c x^2}{a}}}dx}{e \sqrt {a-c x^2}}-\frac {\left (c d^2-a e^2\right ) \left (5 a A e^3-33 a B d e^2-4 A c d^2 e+32 B c d^3\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a-c x^2}}dx}{e}\right )}{3 e^2 \left (c d^2-a e^2\right )}\right )}{35 e^2 \left (c d^2-a e^2\right )}-\frac {2 \left (a-c x^2\right )^{3/2} \left (e x \left (-7 a B e^2-6 A c d e+13 B c d^2\right )-5 a A e^3-2 a B d e^2-A c d^2 e+8 B c d^3\right )}{35 e^2 (d+e x)^{7/2} \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 508 |
| \(\displaystyle -\frac {6 c \left (\frac {2 \sqrt {a-c x^2} \left (e x \left (A c d e \left (5 c d^2-13 a e^2\right )-B \left (21 a^2 e^4-69 a c d^2 e^2+40 c^2 d^4\right )\right )+A e \left (-5 a^2 e^4-7 a c d^2 e^2+4 c^2 d^4\right )-B \left (9 a^2 d e^4-49 a c d^3 e^2+32 c^2 d^5\right )\right )}{3 e^2 (d+e x)^{3/2} \left (c d^2-a e^2\right )}-\frac {2 c \left (\frac {2 \sqrt {a} \sqrt {1-\frac {c x^2}{a}} \sqrt {d+e x} \left (4 A c d e \left (c d^2-2 a e^2\right )-B \left (21 a^2 e^4-57 a c d^2 e^2+32 c^2 d^4\right )\right ) \int \frac {\sqrt {1-\frac {e \left (1-\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\frac {\sqrt {c} d}{\sqrt {a}}+e}}}{\sqrt {\frac {1}{2} \left (\frac {\sqrt {c} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {c} e \sqrt {a-c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}}}-\frac {\left (c d^2-a e^2\right ) \left (5 a A e^3-33 a B d e^2-4 A c d^2 e+32 B c d^3\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a-c x^2}}dx}{e}\right )}{3 e^2 \left (c d^2-a e^2\right )}\right )}{35 e^2 \left (c d^2-a e^2\right )}-\frac {2 \left (a-c x^2\right )^{3/2} \left (e x \left (-7 a B e^2-6 A c d e+13 B c d^2\right )-5 a A e^3-2 a B d e^2-A c d^2 e+8 B c d^3\right )}{35 e^2 (d+e x)^{7/2} \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle -\frac {6 c \left (\frac {2 \sqrt {a-c x^2} \left (e x \left (A c d e \left (5 c d^2-13 a e^2\right )-B \left (21 a^2 e^4-69 a c d^2 e^2+40 c^2 d^4\right )\right )+A e \left (-5 a^2 e^4-7 a c d^2 e^2+4 c^2 d^4\right )-B \left (9 a^2 d e^4-49 a c d^3 e^2+32 c^2 d^5\right )\right )}{3 e^2 (d+e x)^{3/2} \left (c d^2-a e^2\right )}-\frac {2 c \left (\frac {2 \sqrt {a} \sqrt {1-\frac {c x^2}{a}} \sqrt {d+e x} \left (4 A c d e \left (c d^2-2 a e^2\right )-B \left (21 a^2 e^4-57 a c d^2 e^2+32 c^2 d^4\right )\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {a}}+e}\right )}{\sqrt {c} e \sqrt {a-c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}}}-\frac {\left (c d^2-a e^2\right ) \left (5 a A e^3-33 a B d e^2-4 A c d^2 e+32 B c d^3\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a-c x^2}}dx}{e}\right )}{3 e^2 \left (c d^2-a e^2\right )}\right )}{35 e^2 \left (c d^2-a e^2\right )}-\frac {2 \left (a-c x^2\right )^{3/2} \left (e x \left (-7 a B e^2-6 A c d e+13 B c d^2\right )-5 a A e^3-2 a B d e^2-A c d^2 e+8 B c d^3\right )}{35 e^2 (d+e x)^{7/2} \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 512 |
| \(\displaystyle -\frac {6 c \left (\frac {2 \sqrt {a-c x^2} \left (e x \left (A c d e \left (5 c d^2-13 a e^2\right )-B \left (21 a^2 e^4-69 a c d^2 e^2+40 c^2 d^4\right )\right )+A e \left (-5 a^2 e^4-7 a c d^2 e^2+4 c^2 d^4\right )-B \left (9 a^2 d e^4-49 a c d^3 e^2+32 c^2 d^5\right )\right )}{3 e^2 (d+e x)^{3/2} \left (c d^2-a e^2\right )}-\frac {2 c \left (\frac {2 \sqrt {a} \sqrt {1-\frac {c x^2}{a}} \sqrt {d+e x} \left (4 A c d e \left (c d^2-2 a e^2\right )-B \left (21 a^2 e^4-57 a c d^2 e^2+32 c^2 d^4\right )\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {a}}+e}\right )}{\sqrt {c} e \sqrt {a-c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}}}-\frac {\sqrt {1-\frac {c x^2}{a}} \left (c d^2-a e^2\right ) \left (5 a A e^3-33 a B d e^2-4 A c d^2 e+32 B c d^3\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {1-\frac {c x^2}{a}}}dx}{e \sqrt {a-c x^2}}\right )}{3 e^2 \left (c d^2-a e^2\right )}\right )}{35 e^2 \left (c d^2-a e^2\right )}-\frac {2 \left (a-c x^2\right )^{3/2} \left (e x \left (-7 a B e^2-6 A c d e+13 B c d^2\right )-5 a A e^3-2 a B d e^2-A c d^2 e+8 B c d^3\right )}{35 e^2 (d+e x)^{7/2} \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 511 |
| \(\displaystyle -\frac {6 c \left (\frac {2 \sqrt {a-c x^2} \left (e x \left (A c d e \left (5 c d^2-13 a e^2\right )-B \left (21 a^2 e^4-69 a c d^2 e^2+40 c^2 d^4\right )\right )+A e \left (-5 a^2 e^4-7 a c d^2 e^2+4 c^2 d^4\right )-B \left (9 a^2 d e^4-49 a c d^3 e^2+32 c^2 d^5\right )\right )}{3 e^2 (d+e x)^{3/2} \left (c d^2-a e^2\right )}-\frac {2 c \left (\frac {2 \sqrt {a} \sqrt {1-\frac {c x^2}{a}} \left (c d^2-a e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}} \left (5 a A e^3-33 a B d e^2-4 A c d^2 e+32 B c d^3\right ) \int \frac {1}{\sqrt {1-\frac {e \left (1-\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\frac {\sqrt {c} d}{\sqrt {a}}+e}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {c} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {c} e \sqrt {a-c x^2} \sqrt {d+e x}}+\frac {2 \sqrt {a} \sqrt {1-\frac {c x^2}{a}} \sqrt {d+e x} \left (4 A c d e \left (c d^2-2 a e^2\right )-B \left (21 a^2 e^4-57 a c d^2 e^2+32 c^2 d^4\right )\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {a}}+e}\right )}{\sqrt {c} e \sqrt {a-c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}}}\right )}{3 e^2 \left (c d^2-a e^2\right )}\right )}{35 e^2 \left (c d^2-a e^2\right )}-\frac {2 \left (a-c x^2\right )^{3/2} \left (e x \left (-7 a B e^2-6 A c d e+13 B c d^2\right )-5 a A e^3-2 a B d e^2-A c d^2 e+8 B c d^3\right )}{35 e^2 (d+e x)^{7/2} \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle -\frac {6 c \left (\frac {2 \sqrt {a-c x^2} \left (e x \left (A c d e \left (5 c d^2-13 a e^2\right )-B \left (21 a^2 e^4-69 a c d^2 e^2+40 c^2 d^4\right )\right )+A e \left (-5 a^2 e^4-7 a c d^2 e^2+4 c^2 d^4\right )-B \left (9 a^2 d e^4-49 a c d^3 e^2+32 c^2 d^5\right )\right )}{3 e^2 (d+e x)^{3/2} \left (c d^2-a e^2\right )}-\frac {2 c \left (\frac {2 \sqrt {a} \sqrt {1-\frac {c x^2}{a}} \sqrt {d+e x} \left (4 A c d e \left (c d^2-2 a e^2\right )-B \left (21 a^2 e^4-57 a c d^2 e^2+32 c^2 d^4\right )\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {a}}+e}\right )}{\sqrt {c} e \sqrt {a-c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}}}+\frac {2 \sqrt {a} \sqrt {1-\frac {c x^2}{a}} \left (c d^2-a e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}} \left (5 a A e^3-33 a B d e^2-4 A c d^2 e+32 B c d^3\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {a}}+e}\right )}{\sqrt {c} e \sqrt {a-c x^2} \sqrt {d+e x}}\right )}{3 e^2 \left (c d^2-a e^2\right )}\right )}{35 e^2 \left (c d^2-a e^2\right )}-\frac {2 \left (a-c x^2\right )^{3/2} \left (e x \left (-7 a B e^2-6 A c d e+13 B c d^2\right )-5 a A e^3-2 a B d e^2-A c d^2 e+8 B c d^3\right )}{35 e^2 (d+e x)^{7/2} \left (c d^2-a e^2\right )}\) |
Input:
Int[((A + B*x)*(a - c*x^2)^(3/2))/(d + e*x)^(9/2),x]
Output:
(-2*(8*B*c*d^3 - A*c*d^2*e - 2*a*B*d*e^2 - 5*a*A*e^3 + e*(13*B*c*d^2 - 6*A *c*d*e - 7*a*B*e^2)*x)*(a - c*x^2)^(3/2))/(35*e^2*(c*d^2 - a*e^2)*(d + e*x )^(7/2)) - (6*c*((2*(A*e*(4*c^2*d^4 - 7*a*c*d^2*e^2 - 5*a^2*e^4) - B*(32*c ^2*d^5 - 49*a*c*d^3*e^2 + 9*a^2*d*e^4) + e*(A*c*d*e*(5*c*d^2 - 13*a*e^2) - B*(40*c^2*d^4 - 69*a*c*d^2*e^2 + 21*a^2*e^4))*x)*Sqrt[a - c*x^2])/(3*e^2* (c*d^2 - a*e^2)*(d + e*x)^(3/2)) - (2*c*((2*Sqrt[a]*(4*A*c*d*e*(c*d^2 - 2* a*e^2) - B*(32*c^2*d^4 - 57*a*c*d^2*e^2 + 21*a^2*e^4))*Sqrt[d + e*x]*Sqrt[ 1 - (c*x^2)/a]*EllipticE[ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[a]]/Sqrt[2]], (2 *e)/((Sqrt[c]*d)/Sqrt[a] + e)])/(Sqrt[c]*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)*(32*B*c*d ^3 - 4*A*c*d^2*e - 33*a*B*d*e^2 + 5*a*A*e^3)*Sqrt[(Sqrt[c]*(d + e*x))/(Sqr t[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])))/(3*e^2*(c*d^2 - a*e^2))))/(35*e^2*(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))*((a + c*x^2)^p/(e^2*(m + 1)*(m + 2)*(c*d^2 + a*e^2)))*((d*g - e*f*(m + 2))*(c*d^2 + a*e^2) - 2*c*d^2*p*(e* f - d*g) - e*(g*(m + 1)*(c*d^2 + a*e^2) + 2*c*d*p*(e*f - d*g))*x), x] - Sim p[p/(e^2*(m + 1)*(m + 2)*(c*d^2 + a*e^2)) Int[(d + e*x)^(m + 2)*(a + c*x^ 2)^(p - 1)*Simp[2*a*c*e*(e*f - d*g)*(m + 2) - c*(2*c*d*(d*g*(2*p + 1) - e*f *(m + 2*p + 2)) - 2*a*e^2*g*(m + 1))*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] && !ILtQ[m + 2*p + 3 , 0]
Time = 9.36 (sec) , antiderivative size = 1069, normalized size of antiderivative = 1.63
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(1069\) |
| default | \(\text {Expression too large to display}\) | \(11309\) |
Input:
int((B*x+A)*(-c*x^2+a)^(3/2)/(e*x+d)^(9/2),x,method=_RETURNVERBOSE)
Output:
((e*x+d)*(-c*x^2+a))^(1/2)/(e*x+d)^(1/2)/(-c*x^2+a)^(1/2)*(-2/7*(A*a*e^3-A *c*d^2*e-B*a*d*e^2+B*c*d^3)/e^8*(-c*e*x^3-c*d*x^2+a*e*x+a*d)^(1/2)/(x+d/e) ^4-2/35*(16*A*c*d*e+7*B*a*e^2-23*B*c*d^2)/e^7*(-c*e*x^3-c*d*x^2+a*e*x+a*d) ^(1/2)/(x+d/e)^3+2/35*c*(15*A*a*e^3-19*A*c*d^2*e-43*B*a*d*e^2+47*B*c*d^3)/ (a*e^2-c*d^2)/e^6*(-c*e*x^3-c*d*x^2+a*e*x+a*d)^(1/2)/(x+d/e)^2+2/35*(-c*e* x^2+a*e)/e^5/(a*e^2-c*d^2)^2*c*(32*A*a*c*d*e^3-16*A*c^2*d^3*e+49*B*a^2*e^4 -158*B*a*c*d^2*e^2+93*B*c^2*d^4)/((x+d/e)*(-c*e*x^2+a*e))^(1/2)+2*(c^2*(A* e-4*B*d)/e^5-1/35*c^2*(15*A*a*e^3-19*A*c*d^2*e-43*B*a*d*e^2+47*B*c*d^3)/(a *e^2-c*d^2)/e^5+1/35*c^2/e^5*d*(32*A*a*c*d*e^3-16*A*c^2*d^3*e+49*B*a^2*e^4 -158*B*a*c*d^2*e^2+93*B*c^2*d^4)/(a*e^2-c*d^2)^2)*(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*(B*c^2/e^4+1/35*c^ 2/e^4*(32*A*a*c*d*e^3-16*A*c^2*d^3*e+49*B*a^2*e^4-158*B*a*c*d^2*e^2+93*B*c ^2*d^4)/(a*e^2-c*d^2)^2)*(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))*EllipticE(((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))+1/c*(a*c)^(1/2)*El...
Leaf count of result is larger than twice the leaf count of optimal. 1392 vs. \(2 (572) = 1144\).
Time = 0.21 (sec) , antiderivative size = 1392, normalized size of antiderivative = 2.13 \[ \int \frac {(A+B x) \left (a-c x^2\right )^{3/2}}{(d+e x)^{9/2}} \, dx=\text {Too large to display} \] Input:
integrate((B*x+A)*(-c*x^2+a)^(3/2)/(e*x+d)^(9/2),x, algorithm="fricas")
Output:
2/105*(4*(32*B*c^3*d^9 - 4*A*c^3*d^8*e - 81*B*a*c^2*d^7*e^2 + 11*A*a*c^2*d ^6*e^3 + 57*B*a^2*c*d^5*e^4 - 15*A*a^2*c*d^4*e^5 + (32*B*c^3*d^5*e^4 - 4*A *c^3*d^4*e^5 - 81*B*a*c^2*d^3*e^6 + 11*A*a*c^2*d^2*e^7 + 57*B*a^2*c*d*e^8 - 15*A*a^2*c*e^9)*x^4 + 4*(32*B*c^3*d^6*e^3 - 4*A*c^3*d^5*e^4 - 81*B*a*c^2 *d^4*e^5 + 11*A*a*c^2*d^3*e^6 + 57*B*a^2*c*d^2*e^7 - 15*A*a^2*c*d*e^8)*x^3 + 6*(32*B*c^3*d^7*e^2 - 4*A*c^3*d^6*e^3 - 81*B*a*c^2*d^5*e^4 + 11*A*a*c^2 *d^4*e^5 + 57*B*a^2*c*d^3*e^6 - 15*A*a^2*c*d^2*e^7)*x^2 + 4*(32*B*c^3*d^8* e - 4*A*c^3*d^7*e^2 - 81*B*a*c^2*d^6*e^3 + 11*A*a*c^2*d^5*e^4 + 57*B*a^2*c *d^4*e^5 - 15*A*a^2*c*d^3*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 ) + 12*(32*B*c^3*d^8*e - 4*A*c^3*d^7*e^2 - 57*B*a*c^2*d^6*e^3 + 8*A*a*c^2* d^5*e^4 + 21*B*a^2*c*d^4*e^5 + (32*B*c^3*d^4*e^5 - 4*A*c^3*d^3*e^6 - 57*B* a*c^2*d^2*e^7 + 8*A*a*c^2*d*e^8 + 21*B*a^2*c*e^9)*x^4 + 4*(32*B*c^3*d^5*e^ 4 - 4*A*c^3*d^4*e^5 - 57*B*a*c^2*d^3*e^6 + 8*A*a*c^2*d^2*e^7 + 21*B*a^2*c* d*e^8)*x^3 + 6*(32*B*c^3*d^6*e^3 - 4*A*c^3*d^5*e^4 - 57*B*a*c^2*d^4*e^5 + 8*A*a*c^2*d^3*e^6 + 21*B*a^2*c*d^2*e^7)*x^2 + 4*(32*B*c^3*d^7*e^2 - 4*A*c^ 3*d^6*e^3 - 57*B*a*c^2*d^5*e^4 + 8*A*a*c^2*d^4*e^5 + 21*B*a^2*c*d^3*e^6)*x )*sqrt(-c*e)*weierstrassZeta(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), 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*(64*B*c^3*d^7*...
\[ \int \frac {(A+B x) \left (a-c x^2\right )^{3/2}}{(d+e x)^{9/2}} \, dx=\int \frac {\left (A + B x\right ) \left (a - c x^{2}\right )^{\frac {3}{2}}}{\left (d + e x\right )^{\frac {9}{2}}}\, dx \] Input:
integrate((B*x+A)*(-c*x**2+a)**(3/2)/(e*x+d)**(9/2),x)
Output:
Integral((A + B*x)*(a - c*x**2)**(3/2)/(d + e*x)**(9/2), x)
\[ \int \frac {(A+B x) \left (a-c x^2\right )^{3/2}}{(d+e x)^{9/2}} \, dx=\int { \frac {{\left (-c x^{2} + a\right )}^{\frac {3}{2}} {\left (B x + A\right )}}{{\left (e x + d\right )}^{\frac {9}{2}}} \,d x } \] Input:
integrate((B*x+A)*(-c*x^2+a)^(3/2)/(e*x+d)^(9/2),x, algorithm="maxima")
Output:
integrate((-c*x^2 + a)^(3/2)*(B*x + A)/(e*x + d)^(9/2), x)
\[ \int \frac {(A+B x) \left (a-c x^2\right )^{3/2}}{(d+e x)^{9/2}} \, dx=\int { \frac {{\left (-c x^{2} + a\right )}^{\frac {3}{2}} {\left (B x + A\right )}}{{\left (e x + d\right )}^{\frac {9}{2}}} \,d x } \] Input:
integrate((B*x+A)*(-c*x^2+a)^(3/2)/(e*x+d)^(9/2),x, algorithm="giac")
Output:
integrate((-c*x^2 + a)^(3/2)*(B*x + A)/(e*x + d)^(9/2), x)
Timed out. \[ \int \frac {(A+B x) \left (a-c x^2\right )^{3/2}}{(d+e x)^{9/2}} \, dx=\int \frac {{\left (a-c\,x^2\right )}^{3/2}\,\left (A+B\,x\right )}{{\left (d+e\,x\right )}^{9/2}} \,d x \] Input:
int(((a - c*x^2)^(3/2)*(A + B*x))/(d + e*x)^(9/2),x)
Output:
int(((a - c*x^2)^(3/2)*(A + B*x))/(d + e*x)^(9/2), x)
\[ \int \frac {(A+B x) \left (a-c x^2\right )^{3/2}}{(d+e x)^{9/2}} \, dx=\text {too large to display} \] Input:
int((B*x+A)*(-c*x^2+a)^(3/2)/(e*x+d)^(9/2),x)
Output:
(2*(2*sqrt(d + e*x)*sqrt(a - c*x**2)*a**2*b*e**3 - 3*sqrt(d + e*x)*sqrt(a - c*x**2)*a**2*c*d*e**2 + 24*sqrt(d + e*x)*sqrt(a - c*x**2)*a*b*c*d**2*e - sqrt(d + e*x)*sqrt(a - c*x**2)*a*b*c*d*e**2*x + 2*sqrt(d + e*x)*sqrt(a - c*x**2)*a*c**2*d**2*e*x + sqrt(d + e*x)*sqrt(a - c*x**2)*a*c**2*d*e**2*x** 2 - 16*sqrt(d + e*x)*sqrt(a - c*x**2)*b*c**2*d**3*x - 8*sqrt(d + e*x)*sqrt (a - c*x**2)*b*c**2*d**2*e*x**2 - sqrt(d + e*x)*sqrt(a - c*x**2)*b*c**2*d* e**2*x**3 - 5*int((sqrt(d + e*x)*sqrt(a - c*x**2)*x**2)/(a*d**5 + 5*a*d**4 *e*x + 10*a*d**3*e**2*x**2 + 10*a*d**2*e**3*x**3 + 5*a*d*e**4*x**4 + a*e** 5*x**5 - c*d**5*x**2 - 5*c*d**4*e*x**3 - 10*c*d**3*e**2*x**4 - 10*c*d**2*e **3*x**5 - 5*c*d*e**4*x**6 - c*e**5*x**7),x)*a**2*b*c*d**4*e**4 - 20*int(( sqrt(d + e*x)*sqrt(a - c*x**2)*x**2)/(a*d**5 + 5*a*d**4*e*x + 10*a*d**3*e* *2*x**2 + 10*a*d**2*e**3*x**3 + 5*a*d*e**4*x**4 + a*e**5*x**5 - c*d**5*x** 2 - 5*c*d**4*e*x**3 - 10*c*d**3*e**2*x**4 - 10*c*d**2*e**3*x**5 - 5*c*d*e* *4*x**6 - c*e**5*x**7),x)*a**2*b*c*d**3*e**5*x - 30*int((sqrt(d + e*x)*sqr t(a - c*x**2)*x**2)/(a*d**5 + 5*a*d**4*e*x + 10*a*d**3*e**2*x**2 + 10*a*d* *2*e**3*x**3 + 5*a*d*e**4*x**4 + a*e**5*x**5 - c*d**5*x**2 - 5*c*d**4*e*x* *3 - 10*c*d**3*e**2*x**4 - 10*c*d**2*e**3*x**5 - 5*c*d*e**4*x**6 - c*e**5* x**7),x)*a**2*b*c*d**2*e**6*x**2 - 20*int((sqrt(d + e*x)*sqrt(a - c*x**2)* x**2)/(a*d**5 + 5*a*d**4*e*x + 10*a*d**3*e**2*x**2 + 10*a*d**2*e**3*x**3 + 5*a*d*e**4*x**4 + a*e**5*x**5 - c*d**5*x**2 - 5*c*d**4*e*x**3 - 10*c*d...