Integrand size = 28, antiderivative size = 1063 \[ \int (d+e x)^3 \sqrt {f+g x} \sqrt {a+c x^2} \, dx=-\frac {4 \left (7 f g \left (33 c d^3 g+a e^2 (2 e f-33 d g)\right )+\frac {e \left (4 c f^2+5 a g^2\right ) \left (15 a e^2 g^2-c \left (16 e^2 f^2-66 d e f g+99 d^2 g^2\right )\right )}{c g}\right ) \sqrt {f+g x} \sqrt {a+c x^2}}{3465 c g^3}+\frac {2 (f+g x)^{3/2} \left (a e^2 g^2 (74 e f-231 d g)-c \left (64 e^3 f^3-264 d e^2 f^2 g+396 d^2 e f g^2-231 d^3 g^3\right )-5 e g \left (15 a e^2 g^2-c \left (16 e^2 f^2-66 d e f g+99 d^2 g^2\right )\right ) x\right ) \sqrt {a+c x^2}}{1155 c g^4}-\frac {2 e^2 (17 e f-33 d g) (f+g x)^{3/2} \left (a+c x^2\right )^{3/2}}{99 c g^2}+\frac {2 e^3 (f+g x)^{5/2} \left (a+c x^2\right )^{3/2}}{11 c g^2}+\frac {4 \left (\sqrt {c} f-\sqrt {-a} g\right ) \sqrt {\sqrt {c} f+\sqrt {-a} g} \left (3 a^2 e^2 g^4 (26 e f+231 d g)-c^2 f^2 \left (64 e^3 f^3-264 d e^2 f^2 g+396 d^2 e f g^2-231 d^3 g^3\right )-9 a c g^2 \left (6 e^3 f^3-33 d e^2 f^2 g+88 d^2 e f g^2+77 d^3 g^3\right )\right ) \sqrt {1-\frac {\sqrt {c} (f+g x)}{\sqrt {c} f-\sqrt {-a} g}} \sqrt {1-\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} E\left (\arcsin \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt {\sqrt {c} f+\sqrt {-a} g}}\right )|\frac {\sqrt {c} f+\sqrt {-a} g}{\sqrt {c} f-\sqrt {-a} g}\right )}{3465 c^{7/4} g^6 \sqrt {a+c x^2}}+\frac {4 \sqrt {\sqrt {c} f+\sqrt {-a} g} \left (75 a^3 e^3 g^5+3 (-a)^{5/2} \sqrt {c} e^2 g^4 (26 e f+231 d g)-9 a^2 c e g^3 \left (e^2 f^2+66 d e f g+55 d^2 g^2\right )-a c^2 f g \left (16 e^3 f^3-66 d e^2 f^2 g+99 d^2 e f g^2-924 d^3 g^3\right )-\sqrt {-a} c^{5/2} f^2 \left (64 e^3 f^3-264 d e^2 f^2 g+396 d^2 e f g^2-231 d^3 g^3\right )+9 (-a)^{3/2} c^{3/2} g^2 \left (6 e^3 f^3-33 d e^2 f^2 g+88 d^2 e f g^2+77 d^3 g^3\right )\right ) \sqrt {1-\frac {\sqrt {c} (f+g x)}{\sqrt {c} f-\sqrt {-a} g}} \sqrt {1-\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt {\sqrt {c} f+\sqrt {-a} g}}\right ),\frac {\sqrt {c} f+\sqrt {-a} g}{\sqrt {c} f-\sqrt {-a} g}\right )}{3465 c^{9/4} g^5 \sqrt {a+c x^2}} \] Output:
-4/3465*(7*f*g*(33*c*d^3*g+a*e^2*(-33*d*g+2*e*f))+e*(5*a*g^2+4*c*f^2)*(15* a*e^2*g^2-c*(99*d^2*g^2-66*d*e*f*g+16*e^2*f^2))/c/g)*(g*x+f)^(1/2)*(c*x^2+ a)^(1/2)/c/g^3+2/1155*(g*x+f)^(3/2)*(a*e^2*g^2*(-231*d*g+74*e*f)-c*(-231*d ^3*g^3+396*d^2*e*f*g^2-264*d*e^2*f^2*g+64*e^3*f^3)-5*e*g*(15*a*e^2*g^2-c*( 99*d^2*g^2-66*d*e*f*g+16*e^2*f^2))*x)*(c*x^2+a)^(1/2)/c/g^4-2/99*e^2*(-33* d*g+17*e*f)*(g*x+f)^(3/2)*(c*x^2+a)^(3/2)/c/g^2+2/11*e^3*(g*x+f)^(5/2)*(c* x^2+a)^(3/2)/c/g^2+4/3465*(c^(1/2)*f-(-a)^(1/2)*g)*(c^(1/2)*f+(-a)^(1/2)*g )^(1/2)*(3*a^2*e^2*g^4*(231*d*g+26*e*f)-c^2*f^2*(-231*d^3*g^3+396*d^2*e*f* g^2-264*d*e^2*f^2*g+64*e^3*f^3)-9*a*c*g^2*(77*d^3*g^3+88*d^2*e*f*g^2-33*d* e^2*f^2*g+6*e^3*f^3))*(1-c^(1/2)*(g*x+f)/(c^(1/2)*f-(-a)^(1/2)*g))^(1/2)*( 1-c^(1/2)*(g*x+f)/(c^(1/2)*f+(-a)^(1/2)*g))^(1/2)*EllipticE(c^(1/4)*(g*x+f )^(1/2)/(c^(1/2)*f+(-a)^(1/2)*g)^(1/2),((c^(1/2)*f+(-a)^(1/2)*g)/(c^(1/2)* f-(-a)^(1/2)*g))^(1/2))/c^(7/4)/g^6/(c*x^2+a)^(1/2)+4/3465*(c^(1/2)*f+(-a) ^(1/2)*g)^(1/2)*(75*a^3*e^3*g^5+3*(-a)^(5/2)*c^(1/2)*e^2*g^4*(231*d*g+26*e *f)-9*a^2*c*e*g^3*(55*d^2*g^2+66*d*e*f*g+e^2*f^2)-a*c^2*f*g*(-924*d^3*g^3+ 99*d^2*e*f*g^2-66*d*e^2*f^2*g+16*e^3*f^3)-(-a)^(1/2)*c^(5/2)*f^2*(-231*d^3 *g^3+396*d^2*e*f*g^2-264*d*e^2*f^2*g+64*e^3*f^3)+9*(-a)^(3/2)*c^(3/2)*g^2* (77*d^3*g^3+88*d^2*e*f*g^2-33*d*e^2*f^2*g+6*e^3*f^3))*(1-c^(1/2)*(g*x+f)/( c^(1/2)*f-(-a)^(1/2)*g))^(1/2)*(1-c^(1/2)*(g*x+f)/(c^(1/2)*f+(-a)^(1/2)*g) )^(1/2)*EllipticF(c^(1/4)*(g*x+f)^(1/2)/(c^(1/2)*f+(-a)^(1/2)*g)^(1/2),...
Result contains complex when optimal does not.
Time = 32.70 (sec) , antiderivative size = 1172, normalized size of antiderivative = 1.10 \[ \int (d+e x)^3 \sqrt {f+g x} \sqrt {a+c x^2} \, dx =\text {Too large to display} \] Input:
Integrate[(d + e*x)^3*Sqrt[f + g*x]*Sqrt[a + c*x^2],x]
Output:
Sqrt[f + g*x]*Sqrt[a + c*x^2]*((2*(-64*c^2*e^3*f^4 + 264*c^2*d*e^2*f^3*g - 396*c^2*d^2*e*f^2*g^2 - 46*a*c*e^3*f^2*g^2 + 231*c^2*d^3*f*g^3 + 264*a*c* d*e^2*f*g^3 + 990*a*c*d^2*e*g^4 - 150*a^2*e^3*g^4))/(3465*c^2*g^4) + (2*(4 8*c*e^3*f^3 - 198*c*d*e^2*f^2*g + 297*c*d^2*e*f*g^2 + 32*a*e^3*f*g^2 + 693 *c*d^3*g^3 + 462*a*d*e^2*g^3)*x)/(3465*c*g^3) + (2*e*(-8*c*e^2*f^2 + 33*c* d*e*f*g + 297*c*d^2*g^2 + 18*a*e^2*g^2)*x^2)/(693*c*g^2) + (2*e^2*(e*f + 3 3*d*g)*x^3)/(99*g) + (2*e^3*x^4)/11) - (4*(f + g*x)^(3/2)*(-(Sqrt[-f - (I* Sqrt[a]*g)/Sqrt[c]]*(-3*a^2*e^2*g^4*(26*e*f + 231*d*g) + c^2*f^2*(64*e^3*f ^3 - 264*d*e^2*f^2*g + 396*d^2*e*f*g^2 - 231*d^3*g^3) + 9*a*c*g^2*(6*e^3*f ^3 - 33*d*e^2*f^2*g + 88*d^2*e*f*g^2 + 77*d^3*g^3))*((a*g^2)/(f + g*x)^2 + c*(-1 + f/(f + g*x))^2)) + (I*Sqrt[c]*(Sqrt[c]*f + I*Sqrt[a]*g)*(-3*a^2*e ^2*g^4*(26*e*f + 231*d*g) + c^2*f^2*(64*e^3*f^3 - 264*d*e^2*f^2*g + 396*d^ 2*e*f*g^2 - 231*d^3*g^3) + 9*a*c*g^2*(6*e^3*f^3 - 33*d*e^2*f^2*g + 88*d^2* e*f*g^2 + 77*d^3*g^3))*Sqrt[1 - f/(f + g*x) - (I*Sqrt[a]*g)/(Sqrt[c]*(f + g*x))]*Sqrt[1 - f/(f + g*x) + (I*Sqrt[a]*g)/(Sqrt[c]*(f + g*x))]*EllipticE [I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I *Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)])/Sqrt[f + g*x] + (Sqrt[a]*g*(Sqrt[c ]*f + I*Sqrt[a]*g)*(-75*a^2*e^3*g^4 + (3*I)*a^(3/2)*Sqrt[c]*e^2*g^3*(e*f + 231*d*g) + 3*a*c*e*g^2*(2*e^2*f^2 - 33*d*e*f*g + 165*d^2*g^2) + c^2*f*(64 *e^3*f^3 - 264*d*e^2*f^2*g + 396*d^2*e*f*g^2 - 231*d^3*g^3) - (3*I)*Sqr...
Time = 3.65 (sec) , antiderivative size = 1311, normalized size of antiderivative = 1.23, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.536, Rules used = {722, 2185, 27, 2185, 27, 2185, 27, 2185, 25, 27, 599, 25, 1511, 1416, 1509}
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.
| \(\displaystyle \int \sqrt {a+c x^2} (d+e x)^3 \sqrt {f+g x} \, dx\) |
| \(\Big \downarrow \) 722 |
| \(\displaystyle \frac {\int \frac {(d+e x)^3 \left (c (e f-3 d g) x^2-2 (c d f-a e g) x+a (3 e f-d g)\right )}{\sqrt {f+g x} \sqrt {c x^2+a}}dx}{11 e}+\frac {2 \sqrt {a+c x^2} (d+e x)^4 \sqrt {f+g x}}{11 e}\) |
| \(\Big \downarrow \) 2185 |
| \(\displaystyle \frac {\frac {2 \int -\frac {-c e^2 g^4 \left (18 a e^2 g^2-c \left (29 e^2 f^2-96 d e g f+81 d^2 g^2\right )\right ) x^4-c e g^3 \left (2 a e^2 g^2 (10 e f+33 d g)-3 c \left (11 e^3 f^3-33 d e^2 g f^2+9 d^2 e g^2 f+27 d^3 g^3\right )\right ) x^3+3 c g^2 \left (a e^2 \left (7 e^2 f^2-48 d e g f-9 d^2 g^2\right ) g^2+c \left (5 e^4 f^4-15 d e^3 g f^3+15 d^3 e g^3 f+9 d^4 g^4\right )\right ) x^2+c g \left (3 a e \left (7 e^3 f^3-21 d e^2 g f^2-27 d^2 e g^2 f+3 d^3 g^3\right ) g^2+2 c \left (e^4 f^5-3 d e^3 g f^4+9 d^4 g^4 f\right )\right ) x+a c g^2 \left (7 e^4 f^4-21 d e^3 g f^3-27 d^3 e g^3 f+9 d^4 g^4\right )}{2 \sqrt {f+g x} \sqrt {c x^2+a}}dx}{9 c g^5}+\frac {2 e^3 \sqrt {a+c x^2} (f+g x)^{7/2} (e f-3 d g)}{9 g^4}}{11 e}+\frac {2 \sqrt {a+c x^2} (d+e x)^4 \sqrt {f+g x}}{11 e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {2 e^3 \sqrt {a+c x^2} (f+g x)^{7/2} (e f-3 d g)}{9 g^4}-\frac {\int \frac {-c e^2 g^4 \left (18 a e^2 g^2-c \left (29 e^2 f^2-96 d e g f+81 d^2 g^2\right )\right ) x^4-c e g^3 \left (2 a e^2 g^2 (10 e f+33 d g)-3 c \left (11 e^3 f^3-33 d e^2 g f^2+9 d^2 e g^2 f+27 d^3 g^3\right )\right ) x^3+3 c g^2 \left (a e^2 \left (7 e^2 f^2-48 d e g f-9 d^2 g^2\right ) g^2+c \left (5 e^4 f^4-15 d e^3 g f^3+15 d^3 e g^3 f+9 d^4 g^4\right )\right ) x^2+c g \left (3 a e \left (7 e^3 f^3-21 d e^2 g f^2-27 d^2 e g^2 f+3 d^3 g^3\right ) g^2+2 c \left (e^4 f^5-3 d e^3 g f^4+9 d^4 g^4 f\right )\right ) x+a c g^2 \left (7 e^4 f^4-21 d e^3 g f^3-27 d^3 e g^3 f+9 d^4 g^4\right )}{\sqrt {f+g x} \sqrt {c x^2+a}}dx}{9 c g^5}}{11 e}+\frac {2 \sqrt {a+c x^2} (d+e x)^4 \sqrt {f+g x}}{11 e}\) |
| \(\Big \downarrow \) 2185 |
| \(\displaystyle \frac {\frac {2 e^3 \sqrt {a+c x^2} (f+g x)^{7/2} (e f-3 d g)}{9 g^4}-\frac {\frac {2 \int \frac {c^2 e \left (2 a e^2 g^2 (74 e f-231 d g)-c \left (233 e^3 f^3-843 d e^2 g f^2+1107 d^2 e g^2 f-567 d^3 g^3\right )\right ) x^3 g^7+c \left (90 a^2 e^4 g^4+2 a c e^2 \left (100 e^2 f^2-264 d e g f-297 d^2 g^2\right ) g^2-c^2 \left (214 e^4 f^4-741 d e^3 g f^3+891 d^2 e^2 g^2 f^2-315 d^3 e g^3 f-189 d^4 g^4\right )\right ) x^2 g^6+3 a c \left (30 a e^4 f^2 g^2-c \left (32 e^4 f^4-111 d e^3 g f^3+135 d^2 e^2 g^2 f^2+63 d^3 e g^3 f-21 d^4 g^4\right )\right ) g^6+c \left (180 a^2 e^4 f g^4-a c e \left (107 e^3 f^3-519 d e^2 g f^2+1377 d^2 e g^2 f-63 d^3 g^3\right ) g^2-2 c^2 \left (22 e^4 f^5-75 d e^3 g f^4+81 d^2 e^2 g^2 f^3-63 d^4 g^4 f\right )\right ) x g^5}{2 \sqrt {f+g x} \sqrt {c x^2+a}}dx}{7 c g^4}-\frac {2}{7} e^2 g \sqrt {a+c x^2} (f+g x)^{5/2} \left (18 a e^2 g^2-c \left (81 d^2 g^2-96 d e f g+29 e^2 f^2\right )\right )}{9 c g^5}}{11 e}+\frac {2 \sqrt {a+c x^2} (d+e x)^4 \sqrt {f+g x}}{11 e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {2 e^3 \sqrt {a+c x^2} (f+g x)^{7/2} (e f-3 d g)}{9 g^4}-\frac {\frac {\int \frac {c^2 e \left (2 a e^2 g^2 (74 e f-231 d g)-c \left (233 e^3 f^3-843 d e^2 g f^2+1107 d^2 e g^2 f-567 d^3 g^3\right )\right ) x^3 g^7+c \left (90 a^2 e^4 g^4+2 a c e^2 \left (100 e^2 f^2-264 d e g f-297 d^2 g^2\right ) g^2-c^2 \left (214 e^4 f^4-741 d e^3 g f^3+891 d^2 e^2 g^2 f^2-315 d^3 e g^3 f-189 d^4 g^4\right )\right ) x^2 g^6+3 a c \left (30 a e^4 f^2 g^2-c \left (32 e^4 f^4-111 d e^3 g f^3+135 d^2 e^2 g^2 f^2+63 d^3 e g^3 f-21 d^4 g^4\right )\right ) g^6+c \left (180 a^2 e^4 f g^4-a c e \left (107 e^3 f^3-519 d e^2 g f^2+1377 d^2 e g^2 f-63 d^3 g^3\right ) g^2-2 c^2 \left (22 e^4 f^5-75 d e^3 g f^4+81 d^2 e^2 g^2 f^3-63 d^4 g^4 f\right )\right ) x g^5}{\sqrt {f+g x} \sqrt {c x^2+a}}dx}{7 c g^4}-\frac {2}{7} e^2 g \sqrt {a+c x^2} (f+g x)^{5/2} \left (18 a e^2 g^2-c \left (81 d^2 g^2-96 d e f g+29 e^2 f^2\right )\right )}{9 c g^5}}{11 e}+\frac {2 \sqrt {a+c x^2} (d+e x)^4 \sqrt {f+g x}}{11 e}\) |
| \(\Big \downarrow \) 2185 |
| \(\displaystyle \frac {\frac {2 e^3 \sqrt {a+c x^2} (f+g x)^{7/2} (e f-3 d g)}{9 g^4}-\frac {\frac {\frac {2 \int \frac {3 \left (c^2 \left (150 a^2 e^4 g^4-6 a c e^2 \left (2 e^2 f^2-33 d e g f+165 d^2 g^2\right ) g^2+c^2 \left (187 e^4 f^4-732 d e^3 g f^3+1098 d^2 e^2 g^2 f^2-798 d^3 e g^3 f+315 d^4 g^4\right )\right ) x^2 g^9+a c^2 \left (2 a f g^2 (e f+231 d g) e^3+c \left (73 e^4 f^4-288 d e^3 g f^3+432 d^2 e^2 g^2 f^2-882 d^3 e g^3 f+105 d^4 g^4\right )\right ) g^9+2 c^2 \left (a^2 e^3 (76 e f+231 d g) g^4-11 a c e \left (2 e^3 f^3-15 d e^2 g f^2+54 d^2 e g^2 f+21 d^3 g^3\right ) g^2+c^2 f \left (41 e^4 f^4-156 d e^3 g f^3+234 d^2 e^2 g^2 f^2-189 d^3 e g^3 f+105 d^4 g^4\right )\right ) x g^8\right )}{2 \sqrt {f+g x} \sqrt {c x^2+a}}dx}{5 c g^3}+\frac {2}{5} c e g^5 \sqrt {a+c x^2} (f+g x)^{3/2} \left (2 a e^2 g^2 (74 e f-231 d g)-c \left (-567 d^3 g^3+1107 d^2 e f g^2-843 d e^2 f^2 g+233 e^3 f^3\right )\right )}{7 c g^4}-\frac {2}{7} e^2 g \sqrt {a+c x^2} (f+g x)^{5/2} \left (18 a e^2 g^2-c \left (81 d^2 g^2-96 d e f g+29 e^2 f^2\right )\right )}{9 c g^5}}{11 e}+\frac {2 \sqrt {a+c x^2} (d+e x)^4 \sqrt {f+g x}}{11 e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {2 e^3 \sqrt {a+c x^2} (f+g x)^{7/2} (e f-3 d g)}{9 g^4}-\frac {\frac {\frac {3 \int \frac {c^2 \left (150 a^2 e^4 g^4-6 a c e^2 \left (2 e^2 f^2-33 d e g f+165 d^2 g^2\right ) g^2+c^2 \left (187 e^4 f^4-732 d e^3 g f^3+1098 d^2 e^2 g^2 f^2-798 d^3 e g^3 f+315 d^4 g^4\right )\right ) x^2 g^9+a c^2 \left (2 a f g^2 (e f+231 d g) e^3+c \left (73 e^4 f^4-288 d e^3 g f^3+432 d^2 e^2 g^2 f^2-882 d^3 e g^3 f+105 d^4 g^4\right )\right ) g^9+2 c^2 \left (a^2 e^3 (76 e f+231 d g) g^4-11 a c e \left (2 e^3 f^3-15 d e^2 g f^2+54 d^2 e g^2 f+21 d^3 g^3\right ) g^2+c^2 f \left (41 e^4 f^4-156 d e^3 g f^3+234 d^2 e^2 g^2 f^2-189 d^3 e g^3 f+105 d^4 g^4\right )\right ) x g^8}{\sqrt {f+g x} \sqrt {c x^2+a}}dx}{5 c g^3}+\frac {2}{5} c e g^5 \sqrt {a+c x^2} (f+g x)^{3/2} \left (2 a e^2 g^2 (74 e f-231 d g)-c \left (-567 d^3 g^3+1107 d^2 e f g^2-843 d e^2 f^2 g+233 e^3 f^3\right )\right )}{7 c g^4}-\frac {2}{7} e^2 g \sqrt {a+c x^2} (f+g x)^{5/2} \left (18 a e^2 g^2-c \left (81 d^2 g^2-96 d e f g+29 e^2 f^2\right )\right )}{9 c g^5}}{11 e}+\frac {2 \sqrt {a+c x^2} (d+e x)^4 \sqrt {f+g x}}{11 e}\) |
| \(\Big \downarrow \) 2185 |
| \(\displaystyle \frac {\frac {2 e^3 \sqrt {a+c x^2} (f+g x)^{7/2} (e f-3 d g)}{9 g^4}-\frac {\frac {\frac {3 \left (\frac {2 \int -\frac {c^2 e g^{10} \left (a g \left (75 a^2 e^3 g^4-9 a c e \left (e^2 f^2+66 d e g f+55 d^2 g^2\right ) g^2-c^2 f \left (16 e^3 f^3-66 d e^2 g f^2+99 d^2 e g^2 f-924 d^3 g^3\right )\right )-c \left (3 a^2 e^2 (26 e f+231 d g) g^4-9 a c \left (6 e^3 f^3-33 d e^2 g f^2+88 d^2 e g^2 f+77 d^3 g^3\right ) g^2-c^2 f^2 \left (64 e^3 f^3-264 d e^2 g f^2+396 d^2 e g^2 f-231 d^3 g^3\right )\right ) x\right )}{\sqrt {f+g x} \sqrt {c x^2+a}}dx}{3 c g^2}+\frac {2}{3} c g^8 \sqrt {a+c x^2} \sqrt {f+g x} \left (150 a^2 e^4 g^4-6 a c e^2 g^2 \left (165 d^2 g^2-33 d e f g+2 e^2 f^2\right )+c^2 \left (315 d^4 g^4-798 d^3 e f g^3+1098 d^2 e^2 f^2 g^2-732 d e^3 f^3 g+187 e^4 f^4\right )\right )\right )}{5 c g^3}+\frac {2}{5} c e g^5 \sqrt {a+c x^2} (f+g x)^{3/2} \left (2 a e^2 g^2 (74 e f-231 d g)-c \left (-567 d^3 g^3+1107 d^2 e f g^2-843 d e^2 f^2 g+233 e^3 f^3\right )\right )}{7 c g^4}-\frac {2}{7} e^2 g \sqrt {a+c x^2} (f+g x)^{5/2} \left (18 a e^2 g^2-c \left (81 d^2 g^2-96 d e f g+29 e^2 f^2\right )\right )}{9 c g^5}}{11 e}+\frac {2 \sqrt {a+c x^2} (d+e x)^4 \sqrt {f+g x}}{11 e}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {2 e^3 \sqrt {a+c x^2} (f+g x)^{7/2} (e f-3 d g)}{9 g^4}-\frac {\frac {\frac {3 \left (\frac {2}{3} c g^8 \sqrt {a+c x^2} \sqrt {f+g x} \left (150 a^2 e^4 g^4-6 a c e^2 g^2 \left (165 d^2 g^2-33 d e f g+2 e^2 f^2\right )+c^2 \left (315 d^4 g^4-798 d^3 e f g^3+1098 d^2 e^2 f^2 g^2-732 d e^3 f^3 g+187 e^4 f^4\right )\right )-\frac {2 \int \frac {c^2 e g^{10} \left (a g \left (75 a^2 e^3 g^4-9 a c e \left (e^2 f^2+66 d e g f+55 d^2 g^2\right ) g^2-c^2 f \left (16 e^3 f^3-66 d e^2 g f^2+99 d^2 e g^2 f-924 d^3 g^3\right )\right )-c \left (3 a^2 e^2 (26 e f+231 d g) g^4-9 a c \left (6 e^3 f^3-33 d e^2 g f^2+88 d^2 e g^2 f+77 d^3 g^3\right ) g^2-c^2 f^2 \left (64 e^3 f^3-264 d e^2 g f^2+396 d^2 e g^2 f-231 d^3 g^3\right )\right ) x\right )}{\sqrt {f+g x} \sqrt {c x^2+a}}dx}{3 c g^2}\right )}{5 c g^3}+\frac {2}{5} c e g^5 \sqrt {a+c x^2} (f+g x)^{3/2} \left (2 a e^2 g^2 (74 e f-231 d g)-c \left (-567 d^3 g^3+1107 d^2 e f g^2-843 d e^2 f^2 g+233 e^3 f^3\right )\right )}{7 c g^4}-\frac {2}{7} e^2 g \sqrt {a+c x^2} (f+g x)^{5/2} \left (18 a e^2 g^2-c \left (81 d^2 g^2-96 d e f g+29 e^2 f^2\right )\right )}{9 c g^5}}{11 e}+\frac {2 \sqrt {a+c x^2} (d+e x)^4 \sqrt {f+g x}}{11 e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {2 e^3 \sqrt {a+c x^2} (f+g x)^{7/2} (e f-3 d g)}{9 g^4}-\frac {\frac {\frac {3 \left (\frac {2}{3} c g^8 \sqrt {a+c x^2} \sqrt {f+g x} \left (150 a^2 e^4 g^4-6 a c e^2 g^2 \left (165 d^2 g^2-33 d e f g+2 e^2 f^2\right )+c^2 \left (315 d^4 g^4-798 d^3 e f g^3+1098 d^2 e^2 f^2 g^2-732 d e^3 f^3 g+187 e^4 f^4\right )\right )-\frac {2}{3} c e g^8 \int \frac {a g \left (75 a^2 e^3 g^4-9 a c e \left (e^2 f^2+66 d e g f+55 d^2 g^2\right ) g^2-c^2 f \left (16 e^3 f^3-66 d e^2 g f^2+99 d^2 e g^2 f-924 d^3 g^3\right )\right )-c \left (3 a^2 e^2 (26 e f+231 d g) g^4-9 a c \left (6 e^3 f^3-33 d e^2 g f^2+88 d^2 e g^2 f+77 d^3 g^3\right ) g^2-c^2 f^2 \left (64 e^3 f^3-264 d e^2 g f^2+396 d^2 e g^2 f-231 d^3 g^3\right )\right ) x}{\sqrt {f+g x} \sqrt {c x^2+a}}dx\right )}{5 c g^3}+\frac {2}{5} c e g^5 \sqrt {a+c x^2} (f+g x)^{3/2} \left (2 a e^2 g^2 (74 e f-231 d g)-c \left (-567 d^3 g^3+1107 d^2 e f g^2-843 d e^2 f^2 g+233 e^3 f^3\right )\right )}{7 c g^4}-\frac {2}{7} e^2 g \sqrt {a+c x^2} (f+g x)^{5/2} \left (18 a e^2 g^2-c \left (81 d^2 g^2-96 d e f g+29 e^2 f^2\right )\right )}{9 c g^5}}{11 e}+\frac {2 \sqrt {a+c x^2} (d+e x)^4 \sqrt {f+g x}}{11 e}\) |
| \(\Big \downarrow \) 599 |
| \(\displaystyle \frac {\frac {2 e^3 \sqrt {a+c x^2} (f+g x)^{7/2} (e f-3 d g)}{9 g^4}-\frac {\frac {\frac {3 \left (\frac {4}{3} c e g^6 \int -\frac {\left (c f^2+a g^2\right ) \left (75 a^2 e^3 g^4-3 a c e \left (2 e^2 f^2-33 d e g f+165 d^2 g^2\right ) g^2-c^2 f \left (64 e^3 f^3-264 d e^2 g f^2+396 d^2 e g^2 f-231 d^3 g^3\right )\right )-c \left (3 a^2 e^2 (26 e f+231 d g) g^4-9 a c \left (6 e^3 f^3-33 d e^2 g f^2+88 d^2 e g^2 f+77 d^3 g^3\right ) g^2-c^2 f^2 \left (64 e^3 f^3-264 d e^2 g f^2+396 d^2 e g^2 f-231 d^3 g^3\right )\right ) (f+g x)}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}+\frac {2}{3} c g^8 \sqrt {a+c x^2} \sqrt {f+g x} \left (150 a^2 e^4 g^4-6 a c e^2 g^2 \left (165 d^2 g^2-33 d e f g+2 e^2 f^2\right )+c^2 \left (315 d^4 g^4-798 d^3 e f g^3+1098 d^2 e^2 f^2 g^2-732 d e^3 f^3 g+187 e^4 f^4\right )\right )\right )}{5 c g^3}+\frac {2}{5} c e g^5 \sqrt {a+c x^2} (f+g x)^{3/2} \left (2 a e^2 g^2 (74 e f-231 d g)-c \left (-567 d^3 g^3+1107 d^2 e f g^2-843 d e^2 f^2 g+233 e^3 f^3\right )\right )}{7 c g^4}-\frac {2}{7} e^2 g \sqrt {a+c x^2} (f+g x)^{5/2} \left (18 a e^2 g^2-c \left (81 d^2 g^2-96 d e f g+29 e^2 f^2\right )\right )}{9 c g^5}}{11 e}+\frac {2 \sqrt {a+c x^2} (d+e x)^4 \sqrt {f+g x}}{11 e}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {2 e^3 \sqrt {a+c x^2} (f+g x)^{7/2} (e f-3 d g)}{9 g^4}-\frac {\frac {\frac {3 \left (\frac {2}{3} c g^8 \sqrt {a+c x^2} \sqrt {f+g x} \left (150 a^2 e^4 g^4-6 a c e^2 g^2 \left (165 d^2 g^2-33 d e f g+2 e^2 f^2\right )+c^2 \left (315 d^4 g^4-798 d^3 e f g^3+1098 d^2 e^2 f^2 g^2-732 d e^3 f^3 g+187 e^4 f^4\right )\right )-\frac {4}{3} c e g^6 \int \frac {\left (c f^2+a g^2\right ) \left (75 a^2 e^3 g^4-3 a c e \left (2 e^2 f^2-33 d e g f+165 d^2 g^2\right ) g^2-c^2 f \left (64 e^3 f^3-264 d e^2 g f^2+396 d^2 e g^2 f-231 d^3 g^3\right )\right )-c \left (3 a^2 e^2 (26 e f+231 d g) g^4-9 a c \left (6 e^3 f^3-33 d e^2 g f^2+88 d^2 e g^2 f+77 d^3 g^3\right ) g^2-c^2 f^2 \left (64 e^3 f^3-264 d e^2 g f^2+396 d^2 e g^2 f-231 d^3 g^3\right )\right ) (f+g x)}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}\right )}{5 c g^3}+\frac {2}{5} c e g^5 \sqrt {a+c x^2} (f+g x)^{3/2} \left (2 a e^2 g^2 (74 e f-231 d g)-c \left (-567 d^3 g^3+1107 d^2 e f g^2-843 d e^2 f^2 g+233 e^3 f^3\right )\right )}{7 c g^4}-\frac {2}{7} e^2 g \sqrt {a+c x^2} (f+g x)^{5/2} \left (18 a e^2 g^2-c \left (81 d^2 g^2-96 d e f g+29 e^2 f^2\right )\right )}{9 c g^5}}{11 e}+\frac {2 \sqrt {a+c x^2} (d+e x)^4 \sqrt {f+g x}}{11 e}\) |
| \(\Big \downarrow \) 1511 |
| \(\displaystyle \frac {2 \sqrt {f+g x} \sqrt {c x^2+a} (d+e x)^4}{11 e}+\frac {\frac {2 e^3 (e f-3 d g) (f+g x)^{7/2} \sqrt {c x^2+a}}{9 g^4}-\frac {\frac {\frac {2}{5} c e \left (2 a e^2 g^2 (74 e f-231 d g)-c \left (233 e^3 f^3-843 d e^2 g f^2+1107 d^2 e g^2 f-567 d^3 g^3\right )\right ) (f+g x)^{3/2} \sqrt {c x^2+a} g^5+\frac {3 \left (\frac {2}{3} c \left (150 a^2 e^4 g^4-6 a c e^2 \left (2 e^2 f^2-33 d e g f+165 d^2 g^2\right ) g^2+c^2 \left (187 e^4 f^4-732 d e^3 g f^3+1098 d^2 e^2 g^2 f^2-798 d^3 e g^3 f+315 d^4 g^4\right )\right ) \sqrt {f+g x} \sqrt {c x^2+a} g^8+\frac {4}{3} c e \left (-\sqrt {c f^2+a g^2} \left (\sqrt {c f^2+a g^2} \left (75 a^2 e^3 g^4-3 a c e \left (2 e^2 f^2-33 d e g f+165 d^2 g^2\right ) g^2-c^2 f \left (64 e^3 f^3-264 d e^2 g f^2+396 d^2 e g^2 f-231 d^3 g^3\right )\right )-\sqrt {c} \left (3 a^2 e^2 (26 e f+231 d g) g^4-9 a c \left (6 e^3 f^3-33 d e^2 g f^2+88 d^2 e g^2 f+77 d^3 g^3\right ) g^2-c^2 f^2 \left (64 e^3 f^3-264 d e^2 g f^2+396 d^2 e g^2 f-231 d^3 g^3\right )\right )\right ) \int \frac {1}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}-\sqrt {c} \sqrt {c f^2+a g^2} \left (3 a^2 e^2 (26 e f+231 d g) g^4-9 a c \left (6 e^3 f^3-33 d e^2 g f^2+88 d^2 e g^2 f+77 d^3 g^3\right ) g^2-c^2 f^2 \left (64 e^3 f^3-264 d e^2 g f^2+396 d^2 e g^2 f-231 d^3 g^3\right )\right ) \int \frac {1-\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}\right ) g^6\right )}{5 c g^3}}{7 c g^4}-\frac {2}{7} e^2 g \left (18 a e^2 g^2-c \left (29 e^2 f^2-96 d e g f+81 d^2 g^2\right )\right ) (f+g x)^{5/2} \sqrt {c x^2+a}}{9 c g^5}}{11 e}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle \frac {2 \sqrt {f+g x} \sqrt {c x^2+a} (d+e x)^4}{11 e}+\frac {\frac {2 e^3 (e f-3 d g) (f+g x)^{7/2} \sqrt {c x^2+a}}{9 g^4}-\frac {\frac {\frac {2}{5} c e \left (2 a e^2 g^2 (74 e f-231 d g)-c \left (233 e^3 f^3-843 d e^2 g f^2+1107 d^2 e g^2 f-567 d^3 g^3\right )\right ) (f+g x)^{3/2} \sqrt {c x^2+a} g^5+\frac {3 \left (\frac {2}{3} c \left (150 a^2 e^4 g^4-6 a c e^2 \left (2 e^2 f^2-33 d e g f+165 d^2 g^2\right ) g^2+c^2 \left (187 e^4 f^4-732 d e^3 g f^3+1098 d^2 e^2 g^2 f^2-798 d^3 e g^3 f+315 d^4 g^4\right )\right ) \sqrt {f+g x} \sqrt {c x^2+a} g^8+\frac {4}{3} c e \left (-\frac {\left (c f^2+a g^2\right )^{3/4} \left (\sqrt {c f^2+a g^2} \left (75 a^2 e^3 g^4-3 a c e \left (2 e^2 f^2-33 d e g f+165 d^2 g^2\right ) g^2-c^2 f \left (64 e^3 f^3-264 d e^2 g f^2+396 d^2 e g^2 f-231 d^3 g^3\right )\right )-\sqrt {c} \left (3 a^2 e^2 (26 e f+231 d g) g^4-9 a c \left (6 e^3 f^3-33 d e^2 g f^2+88 d^2 e g^2 f+77 d^3 g^3\right ) g^2-c^2 f^2 \left (64 e^3 f^3-264 d e^2 g f^2+396 d^2 e g^2 f-231 d^3 g^3\right )\right )\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right ) \sqrt {\frac {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}{\left (\frac {c f^2}{g^2}+a\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt [4]{c f^2+a g^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} f}{\sqrt {c f^2+a g^2}}+1\right )\right )}{2 \sqrt [4]{c} \sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}-\sqrt {c} \sqrt {c f^2+a g^2} \left (3 a^2 e^2 (26 e f+231 d g) g^4-9 a c \left (6 e^3 f^3-33 d e^2 g f^2+88 d^2 e g^2 f+77 d^3 g^3\right ) g^2-c^2 f^2 \left (64 e^3 f^3-264 d e^2 g f^2+396 d^2 e g^2 f-231 d^3 g^3\right )\right ) \int \frac {1-\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}\right ) g^6\right )}{5 c g^3}}{7 c g^4}-\frac {2}{7} e^2 g \left (18 a e^2 g^2-c \left (29 e^2 f^2-96 d e g f+81 d^2 g^2\right )\right ) (f+g x)^{5/2} \sqrt {c x^2+a}}{9 c g^5}}{11 e}\) |
| \(\Big \downarrow \) 1509 |
| \(\displaystyle \frac {2 \sqrt {f+g x} \sqrt {c x^2+a} (d+e x)^4}{11 e}+\frac {\frac {2 e^3 (e f-3 d g) (f+g x)^{7/2} \sqrt {c x^2+a}}{9 g^4}-\frac {\frac {\frac {2}{5} c e \left (2 a e^2 g^2 (74 e f-231 d g)-c \left (233 e^3 f^3-843 d e^2 g f^2+1107 d^2 e g^2 f-567 d^3 g^3\right )\right ) (f+g x)^{3/2} \sqrt {c x^2+a} g^5+\frac {3 \left (\frac {2}{3} c \left (150 a^2 e^4 g^4-6 a c e^2 \left (2 e^2 f^2-33 d e g f+165 d^2 g^2\right ) g^2+c^2 \left (187 e^4 f^4-732 d e^3 g f^3+1098 d^2 e^2 g^2 f^2-798 d^3 e g^3 f+315 d^4 g^4\right )\right ) \sqrt {f+g x} \sqrt {c x^2+a} g^8+\frac {4}{3} c e \left (-\sqrt {c} \sqrt {c f^2+a g^2} \left (3 a^2 e^2 (26 e f+231 d g) g^4-9 a c \left (6 e^3 f^3-33 d e^2 g f^2+88 d^2 e g^2 f+77 d^3 g^3\right ) g^2-c^2 f^2 \left (64 e^3 f^3-264 d e^2 g f^2+396 d^2 e g^2 f-231 d^3 g^3\right )\right ) \left (\frac {\sqrt [4]{c f^2+a g^2} \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right ) \sqrt {\frac {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}{\left (\frac {c f^2}{g^2}+a\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt [4]{c f^2+a g^2}}\right )|\frac {1}{2} \left (\frac {\sqrt {c} f}{\sqrt {c f^2+a g^2}}+1\right )\right )}{\sqrt [4]{c} \sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}-\frac {\sqrt {f+g x} \sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}{\left (\frac {c f^2}{g^2}+a\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right )}\right )-\frac {\left (c f^2+a g^2\right )^{3/4} \left (\sqrt {c f^2+a g^2} \left (75 a^2 e^3 g^4-3 a c e \left (2 e^2 f^2-33 d e g f+165 d^2 g^2\right ) g^2-c^2 f \left (64 e^3 f^3-264 d e^2 g f^2+396 d^2 e g^2 f-231 d^3 g^3\right )\right )-\sqrt {c} \left (3 a^2 e^2 (26 e f+231 d g) g^4-9 a c \left (6 e^3 f^3-33 d e^2 g f^2+88 d^2 e g^2 f+77 d^3 g^3\right ) g^2-c^2 f^2 \left (64 e^3 f^3-264 d e^2 g f^2+396 d^2 e g^2 f-231 d^3 g^3\right )\right )\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right ) \sqrt {\frac {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}{\left (\frac {c f^2}{g^2}+a\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt [4]{c f^2+a g^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} f}{\sqrt {c f^2+a g^2}}+1\right )\right )}{2 \sqrt [4]{c} \sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}\right ) g^6\right )}{5 c g^3}}{7 c g^4}-\frac {2}{7} e^2 g \left (18 a e^2 g^2-c \left (29 e^2 f^2-96 d e g f+81 d^2 g^2\right )\right ) (f+g x)^{5/2} \sqrt {c x^2+a}}{9 c g^5}}{11 e}\) |
Input:
Int[(d + e*x)^3*Sqrt[f + g*x]*Sqrt[a + c*x^2],x]
Output:
(2*(d + e*x)^4*Sqrt[f + g*x]*Sqrt[a + c*x^2])/(11*e) + ((2*e^3*(e*f - 3*d* g)*(f + g*x)^(7/2)*Sqrt[a + c*x^2])/(9*g^4) - ((-2*e^2*g*(18*a*e^2*g^2 - c *(29*e^2*f^2 - 96*d*e*f*g + 81*d^2*g^2))*(f + g*x)^(5/2)*Sqrt[a + c*x^2])/ 7 + ((2*c*e*g^5*(2*a*e^2*g^2*(74*e*f - 231*d*g) - c*(233*e^3*f^3 - 843*d*e ^2*f^2*g + 1107*d^2*e*f*g^2 - 567*d^3*g^3))*(f + g*x)^(3/2)*Sqrt[a + c*x^2 ])/5 + (3*((2*c*g^8*(150*a^2*e^4*g^4 - 6*a*c*e^2*g^2*(2*e^2*f^2 - 33*d*e*f *g + 165*d^2*g^2) + c^2*(187*e^4*f^4 - 732*d*e^3*f^3*g + 1098*d^2*e^2*f^2* g^2 - 798*d^3*e*f*g^3 + 315*d^4*g^4))*Sqrt[f + g*x]*Sqrt[a + c*x^2])/3 + ( 4*c*e*g^6*(-(Sqrt[c]*Sqrt[c*f^2 + a*g^2]*(3*a^2*e^2*g^4*(26*e*f + 231*d*g) - c^2*f^2*(64*e^3*f^3 - 264*d*e^2*f^2*g + 396*d^2*e*f*g^2 - 231*d^3*g^3) - 9*a*c*g^2*(6*e^3*f^3 - 33*d*e^2*f^2*g + 88*d^2*e*f*g^2 + 77*d^3*g^3))*(- ((Sqrt[f + g*x]*Sqrt[a + (c*f^2)/g^2 - (2*c*f*(f + g*x))/g^2 + (c*(f + g*x )^2)/g^2])/((a + (c*f^2)/g^2)*(1 + (Sqrt[c]*(f + g*x))/Sqrt[c*f^2 + a*g^2] ))) + ((c*f^2 + a*g^2)^(1/4)*(1 + (Sqrt[c]*(f + g*x))/Sqrt[c*f^2 + a*g^2]) *Sqrt[(a + (c*f^2)/g^2 - (2*c*f*(f + g*x))/g^2 + (c*(f + g*x)^2)/g^2)/((a + (c*f^2)/g^2)*(1 + (Sqrt[c]*(f + g*x))/Sqrt[c*f^2 + a*g^2])^2)]*EllipticE [2*ArcTan[(c^(1/4)*Sqrt[f + g*x])/(c*f^2 + a*g^2)^(1/4)], (1 + (Sqrt[c]*f) /Sqrt[c*f^2 + a*g^2])/2])/(c^(1/4)*Sqrt[a + (c*f^2)/g^2 - (2*c*f*(f + g*x) )/g^2 + (c*(f + g*x)^2)/g^2]))) - ((c*f^2 + a*g^2)^(3/4)*(Sqrt[c*f^2 + a*g ^2]*(75*a^2*e^3*g^4 - 3*a*c*e*g^2*(2*e^2*f^2 - 33*d*e*f*g + 165*d^2*g^2...
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((A_.) + (B_.)*(x_))/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2] ), x_Symbol] :> Simp[-2/d^2 Subst[Int[(B*c - A*d - B*x^2)/Sqrt[(b*c^2 + a *d^2)/d^2 - 2*b*c*(x^2/d^2) + b*(x^4/d^2)], x], x, Sqrt[c + d*x]], x] /; Fr eeQ[{a, b, c, d, A, B}, x] && PosQ[b/a]
Int[((d_.) + (e_.)*(x_))^(m_.)*Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_) + (c_.)*( x_)^2], x_Symbol] :> Simp[2*(d + e*x)^(m + 1)*Sqrt[f + g*x]*(Sqrt[a + c*x^2 ]/(e*(2*m + 5))), x] + Simp[1/(e*(2*m + 5)) Int[((d + e*x)^m/(Sqrt[f + g* x]*Sqrt[a + c*x^2]))*Simp[3*a*e*f - a*d*g - 2*(c*d*f - a*e*g)*x + (c*e*f - 3*c*d*g)*x^2, x], x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && IntegerQ[2*m ] && !LtQ[m, -1]
Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c /a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/ (2*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticF[2*ArcTan[q*x], 1/2 - b*(q^2/(4*c)) ], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo l] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + b*x^2 + c*x^4]/(a*(1 + q ^2*x^2))), x] + Simp[d*(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2* x^2)^2)]/(q*Sqrt[a + b*x^2 + c*x^4]))*EllipticE[2*ArcTan[q*x], 1/2 - b*(q^2 /(4*c))], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo l] :> With[{q = Rt[c/a, 2]}, Simp[(e + d*q)/q Int[1/Sqrt[a + b*x^2 + c*x^ 4], x], x] - Simp[e/q Int[(1 - q*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x] /; NeQ[e + d*q, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && Pos Q[c/a]
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]))
Time = 4.99 (sec) , antiderivative size = 1824, normalized size of antiderivative = 1.72
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(1824\) |
| risch | \(\text {Expression too large to display}\) | \(2562\) |
| default | \(\text {Expression too large to display}\) | \(6457\) |
Input:
int((e*x+d)^3*(g*x+f)^(1/2)*(c*x^2+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
((g*x+f)*(c*x^2+a))^(1/2)/(g*x+f)^(1/2)/(c*x^2+a)^(1/2)*(2/11*e^3*x^4*(c*g *x^3+c*f*x^2+a*g*x+a*f)^(1/2)+2/9*(3*c*d*e^2*g+1/11*f*e^3*c)/c/g*x^3*(c*g* x^3+c*f*x^2+a*g*x+a*f)^(1/2)+2/7*(2/11*a*e^3*g+3*c*d^2*e*g+3*c*d*e^2*f-8/9 /g*f*(3*c*d*e^2*g+1/11*f*e^3*c))/c/g*x^2*(c*g*x^3+c*f*x^2+a*g*x+a*f)^(1/2) +2/5*(3*a*d*e^2*g+3/11*a*e^3*f+c*d^3*g+3*c*d^2*e*f-6/7/g*f*(2/11*a*e^3*g+3 *c*d^2*e*g+3*c*d*e^2*f-8/9/g*f*(3*c*d*e^2*g+1/11*f*e^3*c))-7/9*a/c*(3*c*d* e^2*g+1/11*f*e^3*c))/c/g*x*(c*g*x^3+c*f*x^2+a*g*x+a*f)^(1/2)+2/3*(3*a*d^2* e*g+3*a*d*e^2*f+c*d^3*f-4/5/g*f*(3*a*d*e^2*g+3/11*a*e^3*f+c*d^3*g+3*c*d^2* e*f-6/7/g*f*(2/11*a*e^3*g+3*c*d^2*e*g+3*c*d*e^2*f-8/9/g*f*(3*c*d*e^2*g+1/1 1*f*e^3*c))-7/9*a/c*(3*c*d*e^2*g+1/11*f*e^3*c))-5/7*a/c*(2/11*a*e^3*g+3*c* d^2*e*g+3*c*d*e^2*f-8/9/g*f*(3*c*d*e^2*g+1/11*f*e^3*c))-2/3*a*f/c/g*(3*c*d *e^2*g+1/11*f*e^3*c))/c/g*(c*g*x^3+c*f*x^2+a*g*x+a*f)^(1/2)+2*(a*d^3*f-2/5 *a*f/c/g*(3*a*d*e^2*g+3/11*a*e^3*f+c*d^3*g+3*c*d^2*e*f-6/7/g*f*(2/11*a*e^3 *g+3*c*d^2*e*g+3*c*d*e^2*f-8/9/g*f*(3*c*d*e^2*g+1/11*f*e^3*c))-7/9*a/c*(3* c*d*e^2*g+1/11*f*e^3*c))-1/3*a/c*(3*a*d^2*e*g+3*a*d*e^2*f+c*d^3*f-4/5/g*f* (3*a*d*e^2*g+3/11*a*e^3*f+c*d^3*g+3*c*d^2*e*f-6/7/g*f*(2/11*a*e^3*g+3*c*d^ 2*e*g+3*c*d*e^2*f-8/9/g*f*(3*c*d*e^2*g+1/11*f*e^3*c))-7/9*a/c*(3*c*d*e^2*g +1/11*f*e^3*c))-5/7*a/c*(2/11*a*e^3*g+3*c*d^2*e*g+3*c*d*e^2*f-8/9/g*f*(3*c *d*e^2*g+1/11*f*e^3*c))-2/3*a*f/c/g*(3*c*d*e^2*g+1/11*f*e^3*c)))*(f/g-(-a* c)^(1/2)/c)*((x+f/g)/(f/g-(-a*c)^(1/2)/c))^(1/2)*((x-(-a*c)^(1/2)/c)/(-...
Time = 0.09 (sec) , antiderivative size = 765, normalized size of antiderivative = 0.72 \[ \int (d+e x)^3 \sqrt {f+g x} \sqrt {a+c x^2} \, dx =\text {Too large to display} \] Input:
integrate((e*x+d)^3*(g*x+f)^(1/2)*(c*x^2+a)^(1/2),x, algorithm="fricas")
Output:
-2/10395*(2*(64*c^3*e^3*f^6 - 264*c^3*d*e^2*f^5*g + 6*(66*c^3*d^2*e + 17*a *c^2*e^3)*f^4*g^2 - 33*(7*c^3*d^3 + 15*a*c^2*d*e^2)*f^3*g^3 + 3*(363*a*c^2 *d^2*e - 17*a^2*c*e^3)*f^2*g^4 - 99*(21*a*c^2*d^3 - 11*a^2*c*d*e^2)*f*g^5 + 45*(33*a^2*c*d^2*e - 5*a^3*e^3)*g^6)*sqrt(c*g)*weierstrassPInverse(4/3*( c*f^2 - 3*a*g^2)/(c*g^2), -8/27*(c*f^3 + 9*a*f*g^2)/(c*g^3), 1/3*(3*g*x + f)/g) + 6*(64*c^3*e^3*f^5*g - 264*c^3*d*e^2*f^4*g^2 + 18*(22*c^3*d^2*e + 3 *a*c^2*e^3)*f^3*g^3 - 33*(7*c^3*d^3 + 9*a*c^2*d*e^2)*f^2*g^4 + 6*(132*a*c^ 2*d^2*e - 13*a^2*c*e^3)*f*g^5 + 693*(a*c^2*d^3 - a^2*c*d*e^2)*g^6)*sqrt(c* g)*weierstrassZeta(4/3*(c*f^2 - 3*a*g^2)/(c*g^2), -8/27*(c*f^3 + 9*a*f*g^2 )/(c*g^3), weierstrassPInverse(4/3*(c*f^2 - 3*a*g^2)/(c*g^2), -8/27*(c*f^3 + 9*a*f*g^2)/(c*g^3), 1/3*(3*g*x + f)/g)) - 3*(315*c^3*e^3*g^6*x^4 - 64*c ^3*e^3*f^4*g^2 + 264*c^3*d*e^2*f^3*g^3 - 2*(198*c^3*d^2*e + 23*a*c^2*e^3)* f^2*g^4 + 33*(7*c^3*d^3 + 8*a*c^2*d*e^2)*f*g^5 + 30*(33*a*c^2*d^2*e - 5*a^ 2*c*e^3)*g^6 + 35*(c^3*e^3*f*g^5 + 33*c^3*d*e^2*g^6)*x^3 - 5*(8*c^3*e^3*f^ 2*g^4 - 33*c^3*d*e^2*f*g^5 - 9*(33*c^3*d^2*e + 2*a*c^2*e^3)*g^6)*x^2 + (48 *c^3*e^3*f^3*g^3 - 198*c^3*d*e^2*f^2*g^4 + (297*c^3*d^2*e + 32*a*c^2*e^3)* f*g^5 + 231*(3*c^3*d^3 + 2*a*c^2*d*e^2)*g^6)*x)*sqrt(c*x^2 + a)*sqrt(g*x + f))/(c^3*g^6)
\[ \int (d+e x)^3 \sqrt {f+g x} \sqrt {a+c x^2} \, dx=\int \sqrt {a + c x^{2}} \left (d + e x\right )^{3} \sqrt {f + g x}\, dx \] Input:
integrate((e*x+d)**3*(g*x+f)**(1/2)*(c*x**2+a)**(1/2),x)
Output:
Integral(sqrt(a + c*x**2)*(d + e*x)**3*sqrt(f + g*x), x)
\[ \int (d+e x)^3 \sqrt {f+g x} \sqrt {a+c x^2} \, dx=\int { \sqrt {c x^{2} + a} {\left (e x + d\right )}^{3} \sqrt {g x + f} \,d x } \] Input:
integrate((e*x+d)^3*(g*x+f)^(1/2)*(c*x^2+a)^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(c*x^2 + a)*(e*x + d)^3*sqrt(g*x + f), x)
\[ \int (d+e x)^3 \sqrt {f+g x} \sqrt {a+c x^2} \, dx=\int { \sqrt {c x^{2} + a} {\left (e x + d\right )}^{3} \sqrt {g x + f} \,d x } \] Input:
integrate((e*x+d)^3*(g*x+f)^(1/2)*(c*x^2+a)^(1/2),x, algorithm="giac")
Output:
integrate(sqrt(c*x^2 + a)*(e*x + d)^3*sqrt(g*x + f), x)
Timed out. \[ \int (d+e x)^3 \sqrt {f+g x} \sqrt {a+c x^2} \, dx=\int \sqrt {f+g\,x}\,\sqrt {c\,x^2+a}\,{\left (d+e\,x\right )}^3 \,d x \] Input:
int((f + g*x)^(1/2)*(a + c*x^2)^(1/2)*(d + e*x)^3,x)
Output:
int((f + g*x)^(1/2)*(a + c*x^2)^(1/2)*(d + e*x)^3, x)
\[ \int (d+e x)^3 \sqrt {f+g x} \sqrt {a+c x^2} \, dx=\text {too large to display} \] Input:
int((e*x+d)^3*(g*x+f)^(1/2)*(c*x^2+a)^(1/2),x)
Output:
( - 1386*sqrt(f + g*x)*sqrt(a + c*x**2)*a**2*d*e**2*g**4 - 456*sqrt(f + g* x)*sqrt(a + c*x**2)*a**2*e**3*f*g**3 + 1386*sqrt(f + g*x)*sqrt(a + c*x**2) *a*c*d**3*g**4 + 3564*sqrt(f + g*x)*sqrt(a + c*x**2)*a*c*d**2*e*f*g**3 - 6 6*sqrt(f + g*x)*sqrt(a + c*x**2)*a*c*d*e**2*f**2*g**2 + 924*sqrt(f + g*x)* sqrt(a + c*x**2)*a*c*d*e**2*f*g**3*x + 16*sqrt(f + g*x)*sqrt(a + c*x**2)*a *c*e**3*f**3*g + 64*sqrt(f + g*x)*sqrt(a + c*x**2)*a*c*e**3*f**2*g**2*x + 180*sqrt(f + g*x)*sqrt(a + c*x**2)*a*c*e**3*f*g**3*x**2 + 1386*sqrt(f + g* x)*sqrt(a + c*x**2)*c**2*d**3*f*g**3*x + 594*sqrt(f + g*x)*sqrt(a + c*x**2 )*c**2*d**2*e*f**2*g**2*x + 2970*sqrt(f + g*x)*sqrt(a + c*x**2)*c**2*d**2* e*f*g**3*x**2 - 396*sqrt(f + g*x)*sqrt(a + c*x**2)*c**2*d*e**2*f**3*g*x + 330*sqrt(f + g*x)*sqrt(a + c*x**2)*c**2*d*e**2*f**2*g**2*x**2 + 2310*sqrt( f + g*x)*sqrt(a + c*x**2)*c**2*d*e**2*f*g**3*x**3 + 96*sqrt(f + g*x)*sqrt( a + c*x**2)*c**2*e**3*f**4*x - 80*sqrt(f + g*x)*sqrt(a + c*x**2)*c**2*e**3 *f**3*g*x**2 + 70*sqrt(f + g*x)*sqrt(a + c*x**2)*c**2*e**3*f**2*g**2*x**3 + 630*sqrt(f + g*x)*sqrt(a + c*x**2)*c**2*e**3*f*g**3*x**4 + 2079*int((sqr t(f + g*x)*sqrt(a + c*x**2)*x**2)/(a*f + a*g*x + c*f*x**2 + c*g*x**3),x)*a **2*c*d*e**2*g**5 + 234*int((sqrt(f + g*x)*sqrt(a + c*x**2)*x**2)/(a*f + a *g*x + c*f*x**2 + c*g*x**3),x)*a**2*c*e**3*f*g**4 - 2079*int((sqrt(f + g*x )*sqrt(a + c*x**2)*x**2)/(a*f + a*g*x + c*f*x**2 + c*g*x**3),x)*a*c**2*d** 3*g**5 - 2376*int((sqrt(f + g*x)*sqrt(a + c*x**2)*x**2)/(a*f + a*g*x + ...