Integrand size = 28, antiderivative size = 666 \[ \int \frac {(d+e x)^3 \sqrt {a+c x^2}}{\sqrt {f+g x}} \, dx=-\frac {4 \left (9 a e^2 g^2 (2 e f-5 d g)+c \left (76 e^3 f^3-204 d e^2 f^2 g+168 d^2 e f g^2-35 d^3 g^3\right )\right ) \sqrt {f+g x} \sqrt {a+c x^2}}{315 c g^4}+\frac {2 (d+e x)^3 \sqrt {f+g x} \sqrt {a+c x^2}}{9 g}+\frac {4 e \left (7 a e^2 g^2+c \left (64 e^2 f^2-111 d e f g+42 d^2 g^2\right )\right ) (f+g x)^{3/2} \sqrt {a+c x^2}}{315 c g^4}-\frac {4 e^2 (4 e f-3 d g) (f+g x)^{5/2} \sqrt {a+c x^2}}{63 g^4}+\frac {4 \sqrt {-a} \left (21 a^2 e^3 g^4-3 a c e g^2 \left (10 e^2 f^2-39 d e f g+63 d^2 g^2\right )-c^2 f \left (64 e^3 f^3-216 d e^2 f^2 g+252 d^2 e f g^2-105 d^3 g^3\right )\right ) \sqrt {f+g x} \sqrt {1+\frac {c x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{315 c^{3/2} g^5 \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {a+c x^2}}-\frac {4 \sqrt {-a} \left (c f^2+a g^2\right ) \left (9 a e^2 g^2 (2 e f-5 d g)-c \left (64 e^3 f^3-216 d e^2 f^2 g+252 d^2 e f g^2-105 d^3 g^3\right )\right ) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {1+\frac {c x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{315 c^{3/2} g^5 \sqrt {f+g x} \sqrt {a+c x^2}} \]
4/315*e*(7*a*e^2*g^2+c*(42*d^2*g^2-111*d*e*f*g+64*e^2*f^2))*(g*x+f)^(3/2)* (c*x^2+a)^(1/2)/c/g^4-4/63*e^2*(-3*d*g+4*e*f)*(g*x+f)^(5/2)*(c*x^2+a)^(1/2 )/g^4-4/315*(9*a*e^2*g^2*(-5*d*g+2*e*f)+c*(-35*d^3*g^3+168*d^2*e*f*g^2-204 *d*e^2*f^2*g+76*e^3*f^3))*(g*x+f)^(1/2)*(c*x^2+a)^(1/2)/c/g^4+2/9*(e*x+d)^ 3*(g*x+f)^(1/2)*(c*x^2+a)^(1/2)/g+4/315*(21*a^2*e^3*g^4-3*a*c*e*g^2*(63*d^ 2*g^2-39*d*e*f*g+10*e^2*f^2)-c^2*f*(-105*d^3*g^3+252*d^2*e*f*g^2-216*d*e^2 *f^2*g+64*e^3*f^3))*EllipticE(1/2*(1-x*c^(1/2)/(-a)^(1/2))^(1/2)*2^(1/2),( -2*a*g/(-a*g+f*(-a)^(1/2)*c^(1/2)))^(1/2))*(-a)^(1/2)*(g*x+f)^(1/2)*(1+c*x ^2/a)^(1/2)/c^(3/2)/g^5/(c*x^2+a)^(1/2)/((g*x+f)*c^(1/2)/(g*(-a)^(1/2)+f*c ^(1/2)))^(1/2)-4/315*(a*g^2+c*f^2)*(9*a*e^2*g^2*(-5*d*g+2*e*f)-c*(-105*d^3 *g^3+252*d^2*e*f*g^2-216*d*e^2*f^2*g+64*e^3*f^3))*EllipticF(1/2*(1-x*c^(1/ 2)/(-a)^(1/2))^(1/2)*2^(1/2),(-2*a*g/(-a*g+f*(-a)^(1/2)*c^(1/2)))^(1/2))*( -a)^(1/2)*(1+c*x^2/a)^(1/2)*((g*x+f)*c^(1/2)/(g*(-a)^(1/2)+f*c^(1/2)))^(1/ 2)/c^(3/2)/g^5/(g*x+f)^(1/2)/(c*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 27.66 (sec) , antiderivative size = 872, normalized size of antiderivative = 1.31 \[ \int \frac {(d+e x)^3 \sqrt {a+c x^2}}{\sqrt {f+g x}} \, dx=\frac {\sqrt {f+g x} \left (\frac {2 \left (a+c x^2\right ) \left (2 a e^2 g^2 (-11 e f+45 d g+7 e g x)+c \left (105 d^3 g^3+63 d^2 e g^2 (-4 f+3 g x)+27 d e^2 g \left (8 f^2-6 f g x+5 g^2 x^2\right )+e^3 \left (-64 f^3+48 f^2 g x-40 f g^2 x^2+35 g^3 x^3\right )\right )\right )}{c g^4}+\frac {4 (f+g x) \left (\frac {g^2 \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}} \left (-21 a^2 e^3 g^4+3 a c e g^2 \left (10 e^2 f^2-39 d e f g+63 d^2 g^2\right )+c^2 f \left (64 e^3 f^3-216 d e^2 f^2 g+252 d^2 e f g^2-105 d^3 g^3\right )\right ) \left (a+c x^2\right )}{(f+g x)^2}+\frac {i \sqrt {c} \left (\sqrt {c} f+i \sqrt {a} g\right ) \left (21 a^2 e^3 g^4-3 a c e g^2 \left (10 e^2 f^2-39 d e f g+63 d^2 g^2\right )+c^2 f \left (-64 e^3 f^3+216 d e^2 f^2 g-252 d^2 e f g^2+105 d^3 g^3\right )\right ) \sqrt {\frac {g \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{f+g x}} \sqrt {-\frac {\frac {i \sqrt {a} g}{\sqrt {c}}-g x}{f+g x}} E\left (i \text {arcsinh}\left (\frac {\sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}{\sqrt {f+g x}}\right )|\frac {\sqrt {c} f-i \sqrt {a} g}{\sqrt {c} f+i \sqrt {a} g}\right )}{\sqrt {f+g x}}+\frac {\sqrt {a} \sqrt {c} g \left (\sqrt {c} f+i \sqrt {a} g\right ) \left (-21 i a^{3/2} e^3 g^3+9 a \sqrt {c} e^2 g^2 (2 e f-5 d g)+3 i \sqrt {a} c e g \left (16 e^2 f^2-54 d e f g+63 d^2 g^2\right )+c^{3/2} \left (-64 e^3 f^3+216 d e^2 f^2 g-252 d^2 e f g^2+105 d^3 g^3\right )\right ) \sqrt {\frac {g \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{f+g x}} \sqrt {-\frac {\frac {i \sqrt {a} g}{\sqrt {c}}-g x}{f+g x}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}{\sqrt {f+g x}}\right ),\frac {\sqrt {c} f-i \sqrt {a} g}{\sqrt {c} f+i \sqrt {a} g}\right )}{\sqrt {f+g x}}\right )}{c^2 g^6 \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}\right )}{315 \sqrt {a+c x^2}} \]
(Sqrt[f + g*x]*((2*(a + c*x^2)*(2*a*e^2*g^2*(-11*e*f + 45*d*g + 7*e*g*x) + c*(105*d^3*g^3 + 63*d^2*e*g^2*(-4*f + 3*g*x) + 27*d*e^2*g*(8*f^2 - 6*f*g* x + 5*g^2*x^2) + e^3*(-64*f^3 + 48*f^2*g*x - 40*f*g^2*x^2 + 35*g^3*x^3)))) /(c*g^4) + (4*(f + g*x)*((g^2*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(-21*a^2*e^ 3*g^4 + 3*a*c*e*g^2*(10*e^2*f^2 - 39*d*e*f*g + 63*d^2*g^2) + c^2*f*(64*e^3 *f^3 - 216*d*e^2*f^2*g + 252*d^2*e*f*g^2 - 105*d^3*g^3))*(a + c*x^2))/(f + g*x)^2 + (I*Sqrt[c]*(Sqrt[c]*f + I*Sqrt[a]*g)*(21*a^2*e^3*g^4 - 3*a*c*e*g ^2*(10*e^2*f^2 - 39*d*e*f*g + 63*d^2*g^2) + c^2*f*(-64*e^3*f^3 + 216*d*e^2 *f^2*g - 252*d^2*e*f*g^2 + 105*d^3*g^3))*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x) )/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*EllipticE[I* ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sq rt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)])/Sqrt[f + g*x] + (Sqrt[a]*Sqrt[c]*g*(S qrt[c]*f + I*Sqrt[a]*g)*((-21*I)*a^(3/2)*e^3*g^3 + 9*a*Sqrt[c]*e^2*g^2*(2* e*f - 5*d*g) + (3*I)*Sqrt[a]*c*e*g*(16*e^2*f^2 - 54*d*e*f*g + 63*d^2*g^2) + c^(3/2)*(-64*e^3*f^3 + 216*d*e^2*f^2*g - 252*d^2*e*f*g^2 + 105*d^3*g^3)) *Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[ c] - g*x)/(f + g*x))]*EllipticF[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)])/Sqr t[f + g*x]))/(c^2*g^6*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]])))/(315*Sqrt[a + c* x^2])
Time = 2.70 (sec) , antiderivative size = 1085, normalized size of antiderivative = 1.63, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {723, 27, 2185, 27, 2185, 27, 2185, 27, 599, 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.
| \(\displaystyle \int \frac {\sqrt {a+c x^2} (d+e x)^3}{\sqrt {f+g x}} \, dx\) |
| \(\Big \downarrow \) 723 |
| \(\displaystyle \frac {2 \sqrt {a+c x^2} (d+e x)^3 \sqrt {f+g x}}{9 g}-\frac {\int \frac {2 (d+e x)^2 \left (c (4 e f-3 d g) x^2+(c d f-a e g) x+a (3 e f-4 d g)\right )}{\sqrt {f+g x} \sqrt {c x^2+a}}dx}{9 g}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \sqrt {a+c x^2} (d+e x)^3 \sqrt {f+g x}}{9 g}-\frac {2 \int \frac {(d+e x)^2 \left (c (4 e f-3 d g) x^2+(c d f-a e g) x+a (3 e f-4 d g)\right )}{\sqrt {f+g x} \sqrt {c x^2+a}}dx}{9 g}\) |
| \(\Big \downarrow \) 2185 |
| \(\displaystyle \frac {2 \sqrt {a+c x^2} (d+e x)^3 \sqrt {f+g x}}{9 g}-\frac {2 \left (\frac {2 \int -\frac {c e g^3 \left (7 a e^2 g^2+c \left (64 e^2 f^2-111 d e g f+42 d^2 g^2\right )\right ) x^3-c g^2 \left (a e^2 g^2 (e f-27 d g)-c \left (44 e^3 f^3-33 d e^2 g f^2-42 d^2 e g^2 f+21 d^3 g^3\right )\right ) x^2+c g \left (a e \left (40 e^2 f^2-72 d e g f+63 d^2 g^2\right ) g^2+c \left (8 e^3 f^4-6 d e^2 g f^3-7 d^3 g^3 f\right )\right ) x+a c g^2 \left (20 e^3 f^3-15 d e^2 g f^2-21 d^2 e g^2 f+28 d^3 g^3\right )}{2 \sqrt {f+g x} \sqrt {c x^2+a}}dx}{7 c g^4}+\frac {2 e^2 \sqrt {a+c x^2} (f+g x)^{5/2} (4 e f-3 d g)}{7 g^3}\right )}{9 g}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \sqrt {a+c x^2} (d+e x)^3 \sqrt {f+g x}}{9 g}-\frac {2 \left (\frac {2 e^2 \sqrt {a+c x^2} (f+g x)^{5/2} (4 e f-3 d g)}{7 g^3}-\frac {\int \frac {c e g^3 \left (7 a e^2 g^2+c \left (64 e^2 f^2-111 d e g f+42 d^2 g^2\right )\right ) x^3-c g^2 \left (a e^2 g^2 (e f-27 d g)-c \left (44 e^3 f^3-33 d e^2 g f^2-42 d^2 e g^2 f+21 d^3 g^3\right )\right ) x^2+c g \left (a e \left (40 e^2 f^2-72 d e g f+63 d^2 g^2\right ) g^2+c \left (8 e^3 f^4-6 d e^2 g f^3-7 d^3 g^3 f\right )\right ) x+a c g^2 \left (20 e^3 f^3-15 d e^2 g f^2-21 d^2 e g^2 f+28 d^3 g^3\right )}{\sqrt {f+g x} \sqrt {c x^2+a}}dx}{7 c g^4}\right )}{9 g}\) |
| \(\Big \downarrow \) 2185 |
| \(\displaystyle \frac {2 \sqrt {a+c x^2} (d+e x)^3 \sqrt {f+g x}}{9 g}-\frac {2 \left (\frac {2 e^2 \sqrt {a+c x^2} (f+g x)^{5/2} (4 e f-3 d g)}{7 g^3}-\frac {\frac {2 \int -\frac {3 c^2 \left (9 a e^2 (2 e f-5 d g) g^2+c \left (76 e^3 f^3-204 d e^2 g f^2+168 d^2 e g^2 f-35 d^3 g^3\right )\right ) x^2 g^5+a c \left (21 a f g^2 e^3+c \left (92 e^3 f^3-258 d e^2 g f^2+231 d^2 e g^2 f-140 d^3 g^3\right )\right ) g^5+c \left (21 a^2 e^3 g^4+3 a c e \left (2 e^2 f^2+9 d e g f-63 d^2 g^2\right ) g^2+c^2 f \left (88 e^3 f^3-192 d e^2 g f^2+84 d^2 e g^2 f+35 d^3 g^3\right )\right ) x g^4}{2 \sqrt {f+g x} \sqrt {c x^2+a}}dx}{5 c g^3}+\frac {2}{5} e g \sqrt {a+c x^2} (f+g x)^{3/2} \left (7 a e^2 g^2+c \left (42 d^2 g^2-111 d e f g+64 e^2 f^2\right )\right )}{7 c g^4}\right )}{9 g}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \sqrt {a+c x^2} (d+e x)^3 \sqrt {f+g x}}{9 g}-\frac {2 \left (\frac {2 e^2 \sqrt {a+c x^2} (f+g x)^{5/2} (4 e f-3 d g)}{7 g^3}-\frac {\frac {2}{5} e g \sqrt {a+c x^2} (f+g x)^{3/2} \left (7 a e^2 g^2+c \left (42 d^2 g^2-111 d e f g+64 e^2 f^2\right )\right )-\frac {\int \frac {3 c^2 \left (9 a e^2 (2 e f-5 d g) g^2+c \left (76 e^3 f^3-204 d e^2 g f^2+168 d^2 e g^2 f-35 d^3 g^3\right )\right ) x^2 g^5+a c \left (21 a f g^2 e^3+c \left (92 e^3 f^3-258 d e^2 g f^2+231 d^2 e g^2 f-140 d^3 g^3\right )\right ) g^5+c \left (21 a^2 e^3 g^4+3 a c e \left (2 e^2 f^2+9 d e g f-63 d^2 g^2\right ) g^2+c^2 f \left (88 e^3 f^3-192 d e^2 g f^2+84 d^2 e g^2 f+35 d^3 g^3\right )\right ) x g^4}{\sqrt {f+g x} \sqrt {c x^2+a}}dx}{5 c g^3}}{7 c g^4}\right )}{9 g}\) |
| \(\Big \downarrow \) 2185 |
| \(\displaystyle \frac {2 \sqrt {a+c x^2} (d+e x)^3 \sqrt {f+g x}}{9 g}-\frac {2 \left (\frac {2 e^2 \sqrt {a+c x^2} (f+g x)^{5/2} (4 e f-3 d g)}{7 g^3}-\frac {\frac {2}{5} e g \sqrt {a+c x^2} (f+g x)^{3/2} \left (7 a e^2 g^2+c \left (42 d^2 g^2-111 d e f g+64 e^2 f^2\right )\right )-\frac {\frac {2 \int \frac {3 c^2 g^6 \left (a g \left (3 a e^2 (e f+15 d g) g^2+c \left (16 e^3 f^3-54 d e^2 g f^2+63 d^2 e g^2 f-105 d^3 g^3\right )\right )+\left (21 a^2 e^3 g^4-3 a c e \left (10 e^2 f^2-39 d e g f+63 d^2 g^2\right ) g^2-c^2 f \left (64 e^3 f^3-216 d e^2 g f^2+252 d^2 e g^2 f-105 d^3 g^3\right )\right ) x\right )}{2 \sqrt {f+g x} \sqrt {c x^2+a}}dx}{3 c g^2}+2 c g^4 \sqrt {a+c x^2} \sqrt {f+g x} \left (9 a e^2 g^2 (2 e f-5 d g)+c \left (-35 d^3 g^3+168 d^2 e f g^2-204 d e^2 f^2 g+76 e^3 f^3\right )\right )}{5 c g^3}}{7 c g^4}\right )}{9 g}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \sqrt {a+c x^2} (d+e x)^3 \sqrt {f+g x}}{9 g}-\frac {2 \left (\frac {2 e^2 \sqrt {a+c x^2} (f+g x)^{5/2} (4 e f-3 d g)}{7 g^3}-\frac {\frac {2}{5} e g \sqrt {a+c x^2} (f+g x)^{3/2} \left (7 a e^2 g^2+c \left (42 d^2 g^2-111 d e f g+64 e^2 f^2\right )\right )-\frac {c g^4 \int \frac {a g \left (3 a e^2 (e f+15 d g) g^2+c \left (16 e^3 f^3-54 d e^2 g f^2+63 d^2 e g^2 f-105 d^3 g^3\right )\right )+\left (21 a^2 e^3 g^4-3 a c e \left (10 e^2 f^2-39 d e g f+63 d^2 g^2\right ) g^2-c^2 f \left (64 e^3 f^3-216 d e^2 g f^2+252 d^2 e g^2 f-105 d^3 g^3\right )\right ) x}{\sqrt {f+g x} \sqrt {c x^2+a}}dx+2 c g^4 \sqrt {a+c x^2} \sqrt {f+g x} \left (9 a e^2 g^2 (2 e f-5 d g)+c \left (-35 d^3 g^3+168 d^2 e f g^2-204 d e^2 f^2 g+76 e^3 f^3\right )\right )}{5 c g^3}}{7 c g^4}\right )}{9 g}\) |
| \(\Big \downarrow \) 599 |
| \(\displaystyle \frac {2 \sqrt {a+c x^2} (d+e x)^3 \sqrt {f+g x}}{9 g}-\frac {2 \left (\frac {2 e^2 \sqrt {a+c x^2} (f+g x)^{5/2} (4 e f-3 d g)}{7 g^3}-\frac {\frac {2}{5} e g \sqrt {a+c x^2} (f+g x)^{3/2} \left (7 a e^2 g^2+c \left (42 d^2 g^2-111 d e f g+64 e^2 f^2\right )\right )-\frac {2 c g^4 \sqrt {a+c x^2} \sqrt {f+g x} \left (9 a e^2 g^2 (2 e f-5 d g)+c \left (-35 d^3 g^3+168 d^2 e f g^2-204 d e^2 f^2 g+76 e^3 f^3\right )\right )-2 c g^2 \int \frac {\left (c f^2+a g^2\right ) \left (9 a e^2 g^2 (2 e f-5 d g)-c \left (64 e^3 f^3-216 d e^2 g f^2+252 d^2 e g^2 f-105 d^3 g^3\right )\right )-\left (21 a^2 e^3 g^4-3 a c e \left (10 e^2 f^2-39 d e g f+63 d^2 g^2\right ) g^2-c^2 f \left (64 e^3 f^3-216 d e^2 g f^2+252 d^2 e g^2 f-105 d^3 g^3\right )\right ) (f+g x)}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{5 c g^3}}{7 c g^4}\right )}{9 g}\) |
| \(\Big \downarrow \) 1511 |
| \(\displaystyle \frac {2 \sqrt {a+c x^2} (d+e x)^3 \sqrt {f+g x}}{9 g}-\frac {2 \left (\frac {2 e^2 \sqrt {a+c x^2} (f+g x)^{5/2} (4 e f-3 d g)}{7 g^3}-\frac {\frac {2}{5} e g \sqrt {a+c x^2} (f+g x)^{3/2} \left (7 a e^2 g^2+c \left (42 d^2 g^2-111 d e f g+64 e^2 f^2\right )\right )-\frac {2 c g^4 \sqrt {a+c x^2} \sqrt {f+g x} \left (9 a e^2 g^2 (2 e f-5 d g)+c \left (-35 d^3 g^3+168 d^2 e f g^2-204 d e^2 f^2 g+76 e^3 f^3\right )\right )-2 c g^2 \left (\frac {\sqrt {a g^2+c f^2} \left (21 a^2 e^3 g^4-3 a c e g^2 \left (63 d^2 g^2-39 d e f g+10 e^2 f^2\right )-c^2 f \left (-105 d^3 g^3+252 d^2 e f g^2-216 d e^2 f^2 g+64 e^3 f^3\right )\right ) \int \frac {1-\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{\sqrt {c}}-\frac {\sqrt {a g^2+c f^2} \left (21 a^2 e^3 g^4-3 a c e g^2 \left (63 d^2 g^2-39 d e f g+10 e^2 f^2\right )-\sqrt {c} \sqrt {a g^2+c f^2} \left (9 a e^2 g^2 (2 e f-5 d g)-c \left (-105 d^3 g^3+252 d^2 e f g^2-216 d e^2 f^2 g+64 e^3 f^3\right )\right )-c^2 f \left (-105 d^3 g^3+252 d^2 e f g^2-216 d e^2 f^2 g+64 e^3 f^3\right )\right ) \int \frac {1}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{\sqrt {c}}\right )}{5 c g^3}}{7 c g^4}\right )}{9 g}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle \frac {2 (d+e x)^3 \sqrt {f+g x} \sqrt {c x^2+a}}{9 g}-\frac {2 \left (\frac {2 e^2 (4 e f-3 d g) (f+g x)^{5/2} \sqrt {c x^2+a}}{7 g^3}-\frac {\frac {2}{5} e g \left (7 a e^2 g^2+c \left (64 e^2 f^2-111 d e g f+42 d^2 g^2\right )\right ) (f+g x)^{3/2} \sqrt {c x^2+a}-\frac {2 c g^4 \left (9 a e^2 (2 e f-5 d g) g^2+c \left (76 e^3 f^3-204 d e^2 g f^2+168 d^2 e g^2 f-35 d^3 g^3\right )\right ) \sqrt {f+g x} \sqrt {c x^2+a}-2 c g^2 \left (\frac {\sqrt {c f^2+a g^2} \left (21 a^2 e^3 g^4-3 a c e \left (10 e^2 f^2-39 d e g f+63 d^2 g^2\right ) g^2-c^2 f \left (64 e^3 f^3-216 d e^2 g f^2+252 d^2 e g^2 f-105 d^3 g^3\right )\right ) \int \frac {1-\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{\sqrt {c}}-\frac {\left (c f^2+a g^2\right )^{3/4} \left (21 a^2 e^3 g^4-3 a c e \left (10 e^2 f^2-39 d e g f+63 d^2 g^2\right ) g^2-c^2 f \left (64 e^3 f^3-216 d e^2 g f^2+252 d^2 e g^2 f-105 d^3 g^3\right )-\sqrt {c} \sqrt {c f^2+a g^2} \left (9 a e^2 g^2 (2 e f-5 d g)-c \left (64 e^3 f^3-216 d e^2 g f^2+252 d^2 e g^2 f-105 d^3 g^3\right )\right )\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right ) \sqrt {\frac {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}{\left (\frac {c f^2}{g^2}+a\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt [4]{c f^2+a g^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} f}{\sqrt {c f^2+a g^2}}+1\right )\right )}{2 c^{3/4} \sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}\right )}{5 c g^3}}{7 c g^4}\right )}{9 g}\) |
| \(\Big \downarrow \) 1509 |
| \(\displaystyle \frac {2 (d+e x)^3 \sqrt {f+g x} \sqrt {c x^2+a}}{9 g}-\frac {2 \left (\frac {2 e^2 (4 e f-3 d g) (f+g x)^{5/2} \sqrt {c x^2+a}}{7 g^3}-\frac {\frac {2}{5} e g \left (7 a e^2 g^2+c \left (64 e^2 f^2-111 d e g f+42 d^2 g^2\right )\right ) (f+g x)^{3/2} \sqrt {c x^2+a}-\frac {2 c g^4 \left (9 a e^2 (2 e f-5 d g) g^2+c \left (76 e^3 f^3-204 d e^2 g f^2+168 d^2 e g^2 f-35 d^3 g^3\right )\right ) \sqrt {f+g x} \sqrt {c x^2+a}-2 c g^2 \left (\frac {\sqrt {c f^2+a g^2} \left (21 a^2 e^3 g^4-3 a c e \left (10 e^2 f^2-39 d e g f+63 d^2 g^2\right ) g^2-c^2 f \left (64 e^3 f^3-216 d e^2 g f^2+252 d^2 e g^2 f-105 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 )}{\sqrt {c}}-\frac {\left (c f^2+a g^2\right )^{3/4} \left (21 a^2 e^3 g^4-3 a c e \left (10 e^2 f^2-39 d e g f+63 d^2 g^2\right ) g^2-c^2 f \left (64 e^3 f^3-216 d e^2 g f^2+252 d^2 e g^2 f-105 d^3 g^3\right )-\sqrt {c} \sqrt {c f^2+a g^2} \left (9 a e^2 g^2 (2 e f-5 d g)-c \left (64 e^3 f^3-216 d e^2 g f^2+252 d^2 e g^2 f-105 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 c^{3/4} \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 )}{5 c g^3}}{7 c g^4}\right )}{9 g}\) |
(2*(d + e*x)^3*Sqrt[f + g*x]*Sqrt[a + c*x^2])/(9*g) - (2*((2*e^2*(4*e*f - 3*d*g)*(f + g*x)^(5/2)*Sqrt[a + c*x^2])/(7*g^3) - ((2*e*g*(7*a*e^2*g^2 + c *(64*e^2*f^2 - 111*d*e*f*g + 42*d^2*g^2))*(f + g*x)^(3/2)*Sqrt[a + c*x^2]) /5 - (2*c*g^4*(9*a*e^2*g^2*(2*e*f - 5*d*g) + c*(76*e^3*f^3 - 204*d*e^2*f^2 *g + 168*d^2*e*f*g^2 - 35*d^3*g^3))*Sqrt[f + g*x]*Sqrt[a + c*x^2] - 2*c*g^ 2*((Sqrt[c*f^2 + a*g^2]*(21*a^2*e^3*g^4 - 3*a*c*e*g^2*(10*e^2*f^2 - 39*d*e *f*g + 63*d^2*g^2) - c^2*f*(64*e^3*f^3 - 216*d*e^2*f^2*g + 252*d^2*e*f*g^2 - 105*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))/S qrt[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])))/Sqrt[c] - ((c*f^2 + a*g ^2)^(3/4)*(21*a^2*e^3*g^4 - 3*a*c*e*g^2*(10*e^2*f^2 - 39*d*e*f*g + 63*d^2* g^2) - c^2*f*(64*e^3*f^3 - 216*d*e^2*f^2*g + 252*d^2*e*f*g^2 - 105*d^3*g^3 ) - Sqrt[c]*Sqrt[c*f^2 + a*g^2]*(9*a*e^2*g^2*(2*e*f - 5*d*g) - c*(64*e^3*f ^3 - 216*d*e^2*f^2*g + 252*d^2*e*f*g^2 - 105*d^3*g^3)))*(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...
3.7.29.3.1 Defintions of rubi rules used
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[(a_) + (c_.)*(x_)^2])/Sqrt[(f_.) + (g_ .)*(x_)], x_Symbol] :> Simp[2*(d + e*x)^m*Sqrt[f + g*x]*(Sqrt[a + c*x^2]/(g *(2*m + 3))), x] - Simp[1/(g*(2*m + 3)) Int[((d + e*x)^(m - 1)/(Sqrt[f + g*x]*Sqrt[a + c*x^2]))*Simp[2*a*(e*f*m - d*g*(m + 1)) + (2*c*d*f - 2*a*e*g) *x - (2*c*(d*g*m - e*f*(m + 1)))*x^2, x], x], x] /; FreeQ[{a, c, d, e, f, g }, x] && IntegerQ[2*m] && GtQ[m, 0]
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 = 2.44 (sec) , antiderivative size = 1156, normalized size of antiderivative = 1.74
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(1156\) |
| risch | \(\text {Expression too large to display}\) | \(1937\) |
| default | \(\text {Expression too large to display}\) | \(5079\) |
((g*x+f)*(c*x^2+a))^(1/2)/(g*x+f)^(1/2)/(c*x^2+a)^(1/2)*(2/9*e^3/g*x^3*(c* g*x^3+c*f*x^2+a*g*x+a*f)^(1/2)+2/7*(3*c*d*e^2-8/9*e^3/g*c*f)/c/g*x^2*(c*g* x^3+c*f*x^2+a*g*x+a*f)^(1/2)+2/5*(2/9*a*e^3+3*c*d^2*e-6/7*(3*c*d*e^2-8/9*e ^3/g*c*f)/g*f)/c/g*x*(c*g*x^3+c*f*x^2+a*g*x+a*f)^(1/2)+2/3*(3*a*d*e^2+c*d^ 3-2/3*e^3/g*f*a-5/7*(3*c*d*e^2-8/9*e^3/g*c*f)/c*a-4/5*(2/9*a*e^3+3*c*d^2*e -6/7*(3*c*d*e^2-8/9*e^3/g*c*f)/g*f)/g*f)/c/g*(c*g*x^3+c*f*x^2+a*g*x+a*f)^( 1/2)+2*(a*d^3-2/5*(2/9*a*e^3+3*c*d^2*e-6/7*(3*c*d*e^2-8/9*e^3/g*c*f)/g*f)/ c/g*f*a-1/3*(3*a*d*e^2+c*d^3-2/3*e^3/g*f*a-5/7*(3*c*d*e^2-8/9*e^3/g*c*f)/c *a-4/5*(2/9*a*e^3+3*c*d^2*e-6/7*(3*c*d*e^2-8/9*e^3/g*c*f)/g*f)/g*f)/c*a)*( 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)/(-f/g-(-a*c)^(1/2)/c))^(1/2)*((x+(-a*c)^(1/2)/c)/(-f/g+(-a*c)^(1/2)/c)) ^(1/2)/(c*g*x^3+c*f*x^2+a*g*x+a*f)^(1/2)*EllipticF(((x+f/g)/(f/g-(-a*c)^(1 /2)/c))^(1/2),((-f/g+(-a*c)^(1/2)/c)/(-f/g-(-a*c)^(1/2)/c))^(1/2))+2*(3*a* d^2*e-4/7*(3*c*d*e^2-8/9*e^3/g*c*f)/c/g*f*a-3/5*(2/9*a*e^3+3*c*d^2*e-6/7*( 3*c*d*e^2-8/9*e^3/g*c*f)/g*f)/c*a-2/3*(3*a*d*e^2+c*d^3-2/3*e^3/g*f*a-5/7*( 3*c*d*e^2-8/9*e^3/g*c*f)/c*a-4/5*(2/9*a*e^3+3*c*d^2*e-6/7*(3*c*d*e^2-8/9*e ^3/g*c*f)/g*f)/g*f)/g*f)*(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)/(-f/g-(-a*c)^(1/2)/c))^(1/2)*((x+(-a*c)^(1/2) /c)/(-f/g+(-a*c)^(1/2)/c))^(1/2)/(c*g*x^3+c*f*x^2+a*g*x+a*f)^(1/2)*((-f/g- (-a*c)^(1/2)/c)*EllipticE(((x+f/g)/(f/g-(-a*c)^(1/2)/c))^(1/2),((-f/g+(...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.18 (sec) , antiderivative size = 578, normalized size of antiderivative = 0.87 \[ \int \frac {(d+e x)^3 \sqrt {a+c x^2}}{\sqrt {f+g x}} \, dx=-\frac {2 \, {\left (2 \, {\left (64 \, c^{2} e^{3} f^{5} - 216 \, c^{2} d e^{2} f^{4} g + 6 \, {\left (42 \, c^{2} d^{2} e + 13 \, a c e^{3}\right )} f^{3} g^{2} - 3 \, {\left (35 \, c^{2} d^{3} + 93 \, a c d e^{2}\right )} f^{2} g^{3} + 6 \, {\left (63 \, a c d^{2} e - 2 \, a^{2} e^{3}\right )} f g^{4} - 45 \, {\left (7 \, a c d^{3} - 3 \, a^{2} d e^{2}\right )} g^{5}\right )} \sqrt {c g} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c f^{2} - 3 \, a g^{2}\right )}}{3 \, c g^{2}}, -\frac {8 \, {\left (c f^{3} + 9 \, a f g^{2}\right )}}{27 \, c g^{3}}, \frac {3 \, g x + f}{3 \, g}\right ) + 6 \, {\left (64 \, c^{2} e^{3} f^{4} g - 216 \, c^{2} d e^{2} f^{3} g^{2} + 6 \, {\left (42 \, c^{2} d^{2} e + 5 \, a c e^{3}\right )} f^{2} g^{3} - 3 \, {\left (35 \, c^{2} d^{3} + 39 \, a c d e^{2}\right )} f g^{4} + 21 \, {\left (9 \, a c d^{2} e - a^{2} e^{3}\right )} g^{5}\right )} \sqrt {c g} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c f^{2} - 3 \, a g^{2}\right )}}{3 \, c g^{2}}, -\frac {8 \, {\left (c f^{3} + 9 \, a f g^{2}\right )}}{27 \, c g^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c f^{2} - 3 \, a g^{2}\right )}}{3 \, c g^{2}}, -\frac {8 \, {\left (c f^{3} + 9 \, a f g^{2}\right )}}{27 \, c g^{3}}, \frac {3 \, g x + f}{3 \, g}\right )\right ) - 3 \, {\left (35 \, c^{2} e^{3} g^{5} x^{3} - 64 \, c^{2} e^{3} f^{3} g^{2} + 216 \, c^{2} d e^{2} f^{2} g^{3} - 2 \, {\left (126 \, c^{2} d^{2} e + 11 \, a c e^{3}\right )} f g^{4} + 15 \, {\left (7 \, c^{2} d^{3} + 6 \, a c d e^{2}\right )} g^{5} - 5 \, {\left (8 \, c^{2} e^{3} f g^{4} - 27 \, c^{2} d e^{2} g^{5}\right )} x^{2} + {\left (48 \, c^{2} e^{3} f^{2} g^{3} - 162 \, c^{2} d e^{2} f g^{4} + 7 \, {\left (27 \, c^{2} d^{2} e + 2 \, a c e^{3}\right )} g^{5}\right )} x\right )} \sqrt {c x^{2} + a} \sqrt {g x + f}\right )}}{945 \, c^{2} g^{6}} \]
-2/945*(2*(64*c^2*e^3*f^5 - 216*c^2*d*e^2*f^4*g + 6*(42*c^2*d^2*e + 13*a*c *e^3)*f^3*g^2 - 3*(35*c^2*d^3 + 93*a*c*d*e^2)*f^2*g^3 + 6*(63*a*c*d^2*e - 2*a^2*e^3)*f*g^4 - 45*(7*a*c*d^3 - 3*a^2*d*e^2)*g^5)*sqrt(c*g)*weierstrass PInverse(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^2*e^3*f^4*g - 216*c^2*d*e^2*f^3*g^2 + 6*(42* c^2*d^2*e + 5*a*c*e^3)*f^2*g^3 - 3*(35*c^2*d^3 + 39*a*c*d*e^2)*f*g^4 + 21* (9*a*c*d^2*e - a^2*e^3)*g^5)*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*(35*c^2*e^3*g^5*x^3 - 64*c^2*e^3*f^3*g^2 + 216*c^2*d*e^2*f^2*g^3 - 2*(126*c^2*d^2*e + 11*a*c*e^3)*f*g^4 + 15*(7*c^2*d^3 + 6*a*c*d*e^2)*g^5 - 5*(8*c^2*e^3*f*g^4 - 27*c^2*d*e^2*g^5)*x^2 + (48*c^2*e^3*f^2*g^3 - 162*c ^2*d*e^2*f*g^4 + 7*(27*c^2*d^2*e + 2*a*c*e^3)*g^5)*x)*sqrt(c*x^2 + a)*sqrt (g*x + f))/(c^2*g^6)
\[ \int \frac {(d+e x)^3 \sqrt {a+c x^2}}{\sqrt {f+g x}} \, dx=\int \frac {\sqrt {a + c x^{2}} \left (d + e x\right )^{3}}{\sqrt {f + g x}}\, dx \]
\[ \int \frac {(d+e x)^3 \sqrt {a+c x^2}}{\sqrt {f+g x}} \, dx=\int { \frac {\sqrt {c x^{2} + a} {\left (e x + d\right )}^{3}}{\sqrt {g x + f}} \,d x } \]
\[ \int \frac {(d+e x)^3 \sqrt {a+c x^2}}{\sqrt {f+g x}} \, dx=\int { \frac {\sqrt {c x^{2} + a} {\left (e x + d\right )}^{3}}{\sqrt {g x + f}} \,d x } \]
Timed out. \[ \int \frac {(d+e x)^3 \sqrt {a+c x^2}}{\sqrt {f+g x}} \, dx=\int \frac {\sqrt {c\,x^2+a}\,{\left (d+e\,x\right )}^3}{\sqrt {f+g\,x}} \,d x \]