Integrand size = 28, antiderivative size = 531 \[ \int \frac {(d+e x)^3 \sqrt {f+g x}}{\sqrt {a+c x^2}} \, dx=-\frac {2 e \left (25 a e^2 g^2+c \left (7 e^2 f^2+12 d e f g-90 d^2 g^2\right )\right ) \sqrt {f+g x} \sqrt {a+c x^2}}{105 c^2 g^2}+\frac {2 e (d+e x)^2 \sqrt {f+g x} \sqrt {a+c x^2}}{7 c}+\frac {2 e^2 (e f+11 d g) (f+g x)^{3/2} \sqrt {a+c x^2}}{35 c g^2}+\frac {2 \sqrt {-a} \left (a e^2 g^2 (19 e f+189 d g)-c \left (8 e^3 f^3-42 d e^2 f^2 g+105 d^2 e f g^2+105 d^3 g^3\right )\right ) \sqrt {f+g x} \sqrt {1+\frac {c x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{105 c^{3/2} g^3 \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {a+c x^2}}-\frac {2 \sqrt {-a} e \left (c f^2+a g^2\right ) \left (25 a e^2 g^2-c \left (8 e^2 f^2-42 d e f g+105 d^2 g^2\right )\right ) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {1+\frac {c x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{105 c^{5/2} g^3 \sqrt {f+g x} \sqrt {a+c x^2}} \]
2/35*e^2*(11*d*g+e*f)*(g*x+f)^(3/2)*(c*x^2+a)^(1/2)/c/g^2-2/105*e*(25*a*e^ 2*g^2+c*(-90*d^2*g^2+12*d*e*f*g+7*e^2*f^2))*(g*x+f)^(1/2)*(c*x^2+a)^(1/2)/ c^2/g^2+2/7*e*(e*x+d)^2*(g*x+f)^(1/2)*(c*x^2+a)^(1/2)/c+2/105*(a*e^2*g^2*( 189*d*g+19*e*f)-c*(105*d^3*g^3+105*d^2*e*f*g^2-42*d*e^2*f^2*g+8*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^3/(c*x^2+a)^(1/2)/((g*x+f)*c^(1/2)/(g*(-a)^(1/2)+f*c^(1/2)))^(1/2)-2/105 *e*(a*g^2+c*f^2)*(25*a*e^2*g^2-c*(105*d^2*g^2-42*d*e*f*g+8*e^2*f^2))*Ellip ticF(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^(5/2)/g^3/(g*x+f)^(1/2)/(c*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 25.52 (sec) , antiderivative size = 777, normalized size of antiderivative = 1.46 \[ \int \frac {(d+e x)^3 \sqrt {f+g x}}{\sqrt {a+c x^2}} \, dx=\frac {\sqrt {f+g x} \left (\frac {2 \left (a+c x^2\right ) \left (-25 a e^3 g^2+c e \left (105 d^2 g^2+21 d e g (f+3 g x)+e^2 \left (-4 f^2+3 f g x+15 g^2 x^2\right )\right )\right )}{c^2 g^2}-\frac {2 \left (g^2 \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}} \left (a e^2 g^2 (19 e f+189 d g)-c \left (8 e^3 f^3-42 d e^2 f^2 g+105 d^2 e f g^2+105 d^3 g^3\right )\right ) \left (a+c x^2\right )-\sqrt {c} \left (i a \sqrt {c} e^2 f g^2 (19 e f+189 d g)-a^{3/2} e^2 g^3 (19 e f+189 d g)-i c^{3/2} f \left (8 e^3 f^3-42 d e^2 f^2 g+105 d^2 e f g^2+105 d^3 g^3\right )+\sqrt {a} c g \left (8 e^3 f^3-42 d e^2 f^2 g+105 d^2 e f g^2+105 d^3 g^3\right )\right ) \sqrt {\frac {g \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{f+g x}} \sqrt {-\frac {\frac {i \sqrt {a} g}{\sqrt {c}}-g x}{f+g x}} (f+g x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}{\sqrt {f+g x}}\right )|\frac {\sqrt {c} f-i \sqrt {a} g}{\sqrt {c} f+i \sqrt {a} g}\right )-g \left (\sqrt {c} f+i \sqrt {a} g\right ) \left (105 i c^{3/2} d^3 g^2+25 a^{3/2} e^3 g^2+3 i a \sqrt {c} e^2 g (2 e f-63 d g)+\sqrt {a} c e \left (-8 e^2 f^2+42 d e f g-105 d^2 g^2\right )\right ) \sqrt {\frac {g \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{f+g x}} \sqrt {-\frac {\frac {i \sqrt {a} g}{\sqrt {c}}-g x}{f+g x}} (f+g x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}{\sqrt {f+g x}}\right ),\frac {\sqrt {c} f-i \sqrt {a} g}{\sqrt {c} f+i \sqrt {a} g}\right )\right )}{c^2 g^4 \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}} (f+g x)}\right )}{105 \sqrt {a+c x^2}} \]
(Sqrt[f + g*x]*((2*(a + c*x^2)*(-25*a*e^3*g^2 + c*e*(105*d^2*g^2 + 21*d*e* g*(f + 3*g*x) + e^2*(-4*f^2 + 3*f*g*x + 15*g^2*x^2))))/(c^2*g^2) - (2*(g^2 *Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(a*e^2*g^2*(19*e*f + 189*d*g) - c*(8*e^3 *f^3 - 42*d*e^2*f^2*g + 105*d^2*e*f*g^2 + 105*d^3*g^3))*(a + c*x^2) - Sqrt [c]*(I*a*Sqrt[c]*e^2*f*g^2*(19*e*f + 189*d*g) - a^(3/2)*e^2*g^3*(19*e*f + 189*d*g) - I*c^(3/2)*f*(8*e^3*f^3 - 42*d*e^2*f^2*g + 105*d^2*e*f*g^2 + 105 *d^3*g^3) + Sqrt[a]*c*g*(8*e^3*f^3 - 42*d*e^2*f^2*g + 105*d^2*e*f*g^2 + 10 5*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))]*(f + g*x)^(3/2)*EllipticE[I*ArcSinh[Sqrt[ -f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqr t[c]*f + I*Sqrt[a]*g)] - g*(Sqrt[c]*f + I*Sqrt[a]*g)*((105*I)*c^(3/2)*d^3* g^2 + 25*a^(3/2)*e^3*g^2 + (3*I)*a*Sqrt[c]*e^2*g*(2*e*f - 63*d*g) + Sqrt[a ]*c*e*(-8*e^2*f^2 + 42*d*e*f*g - 105*d^2*g^2))*Sqrt[(g*((I*Sqrt[a])/Sqrt[c ] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*(f + g *x)^(3/2)*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)]))/(c^2*g^4*Sqrt[ -f - (I*Sqrt[a]*g)/Sqrt[c]]*(f + g*x))))/(105*Sqrt[a + c*x^2])
Time = 1.80 (sec) , antiderivative size = 916, normalized size of antiderivative = 1.73, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.393, Rules used = {735, 25, 2185, 27, 2185, 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.
| \(\displaystyle \int \frac {(d+e x)^3 \sqrt {f+g x}}{\sqrt {a+c x^2}} \, dx\) |
| \(\Big \downarrow \) 735 |
| \(\displaystyle \frac {2 e \sqrt {a+c x^2} (d+e x)^2 \sqrt {f+g x}}{7 c}-\frac {\int -\frac {(d+e x) \left (7 c f d^2+c e (e f+11 d g) x^2-a e (4 e f+d g)-\left (5 a e^2 g-c d (12 e f+7 d g)\right ) x\right )}{\sqrt {f+g x} \sqrt {c x^2+a}}dx}{7 c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {(d+e x) \left (7 c f d^2+c e (e f+11 d g) x^2-a e (4 e f+d g)-\left (5 a e^2 g-c d (12 e f+7 d g)\right ) x\right )}{\sqrt {f+g x} \sqrt {c x^2+a}}dx}{7 c}+\frac {2 e \sqrt {a+c x^2} (d+e x)^2 \sqrt {f+g x}}{7 c}\) |
| \(\Big \downarrow \) 2185 |
| \(\displaystyle \frac {\frac {2 \int \frac {-c e \left (25 a e^2 g^2+c \left (7 e^2 f^2+12 d e g f-90 d^2 g^2\right )\right ) x^2 g^2+c \left (35 c d^3 f g-a e \left (3 e^2 f^2+53 d e g f+5 d^2 g^2\right )\right ) g^2-c \left (a e^2 (23 e f+63 d g) g^2+c \left (2 e^3 f^3+22 d e^2 g f^2-95 d^2 e g^2 f-35 d^3 g^3\right )\right ) x g}{2 \sqrt {f+g x} \sqrt {c x^2+a}}dx}{5 c g^3}+\frac {2 e^2 \sqrt {a+c x^2} (f+g x)^{3/2} (11 d g+e f)}{5 g^2}}{7 c}+\frac {2 e \sqrt {a+c x^2} (d+e x)^2 \sqrt {f+g x}}{7 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {-c e \left (25 a e^2 g^2+c \left (7 e^2 f^2+12 d e g f-90 d^2 g^2\right )\right ) x^2 g^2+c \left (35 c d^3 f g-a e \left (3 e^2 f^2+53 d e g f+5 d^2 g^2\right )\right ) g^2-c \left (a e^2 (23 e f+63 d g) g^2+c \left (2 e^3 f^3+22 d e^2 g f^2-95 d^2 e g^2 f-35 d^3 g^3\right )\right ) x g}{\sqrt {f+g x} \sqrt {c x^2+a}}dx}{5 c g^3}+\frac {2 e^2 \sqrt {a+c x^2} (f+g x)^{3/2} (11 d g+e f)}{5 g^2}}{7 c}+\frac {2 e \sqrt {a+c x^2} (d+e x)^2 \sqrt {f+g x}}{7 c}\) |
| \(\Big \downarrow \) 2185 |
| \(\displaystyle \frac {\frac {\frac {2 \int \frac {c g^3 \left (g \left (105 c^2 f g d^3+25 a^2 e^3 g^2-a c e \left (2 e^2 f^2+147 d e g f+105 d^2 g^2\right )\right )-c \left (a e^2 g^2 (19 e f+189 d g)-c \left (8 e^3 f^3-42 d e^2 g f^2+105 d^2 e g^2 f+105 d^3 g^3\right )\right ) x\right )}{2 \sqrt {f+g x} \sqrt {c x^2+a}}dx}{3 c g^2}-\frac {2}{3} e g \sqrt {a+c x^2} \sqrt {f+g x} \left (25 a e^2 g^2+c \left (-90 d^2 g^2+12 d e f g+7 e^2 f^2\right )\right )}{5 c g^3}+\frac {2 e^2 \sqrt {a+c x^2} (f+g x)^{3/2} (11 d g+e f)}{5 g^2}}{7 c}+\frac {2 e \sqrt {a+c x^2} (d+e x)^2 \sqrt {f+g x}}{7 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {1}{3} g \int \frac {g \left (105 c^2 f g d^3+25 a^2 e^3 g^2-a c e \left (2 e^2 f^2+147 d e g f+105 d^2 g^2\right )\right )-c \left (a e^2 g^2 (19 e f+189 d g)-c \left (8 e^3 f^3-42 d e^2 g f^2+105 d^2 e g^2 f+105 d^3 g^3\right )\right ) x}{\sqrt {f+g x} \sqrt {c x^2+a}}dx-\frac {2}{3} e g \sqrt {a+c x^2} \sqrt {f+g x} \left (25 a e^2 g^2+c \left (-90 d^2 g^2+12 d e f g+7 e^2 f^2\right )\right )}{5 c g^3}+\frac {2 e^2 \sqrt {a+c x^2} (f+g x)^{3/2} (11 d g+e f)}{5 g^2}}{7 c}+\frac {2 e \sqrt {a+c x^2} (d+e x)^2 \sqrt {f+g x}}{7 c}\) |
| \(\Big \downarrow \) 599 |
| \(\displaystyle \frac {\frac {-\frac {2 \int -\frac {e \left (c f^2+a g^2\right ) \left (25 a e^2 g^2-c \left (8 e^2 f^2-42 d e g f+105 d^2 g^2\right )\right )-c \left (a e^2 g^2 (19 e f+189 d g)-c \left (8 e^3 f^3-42 d e^2 g f^2+105 d^2 e g^2 f+105 d^3 g^3\right )\right ) (f+g x)}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{3 g}-\frac {2}{3} e g \sqrt {a+c x^2} \sqrt {f+g x} \left (25 a e^2 g^2+c \left (-90 d^2 g^2+12 d e f g+7 e^2 f^2\right )\right )}{5 c g^3}+\frac {2 e^2 \sqrt {a+c x^2} (f+g x)^{3/2} (11 d g+e f)}{5 g^2}}{7 c}+\frac {2 e \sqrt {a+c x^2} (d+e x)^2 \sqrt {f+g x}}{7 c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {2 \int \frac {e \left (c f^2+a g^2\right ) \left (25 a e^2 g^2-c \left (8 e^2 f^2-42 d e g f+105 d^2 g^2\right )\right )-c \left (a e^2 g^2 (19 e f+189 d g)-c \left (8 e^3 f^3-42 d e^2 g f^2+105 d^2 e g^2 f+105 d^3 g^3\right )\right ) (f+g x)}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{3 g}-\frac {2}{3} e g \sqrt {a+c x^2} \sqrt {f+g x} \left (25 a e^2 g^2+c \left (-90 d^2 g^2+12 d e f g+7 e^2 f^2\right )\right )}{5 c g^3}+\frac {2 e^2 \sqrt {a+c x^2} (f+g x)^{3/2} (11 d g+e f)}{5 g^2}}{7 c}+\frac {2 e \sqrt {a+c x^2} (d+e x)^2 \sqrt {f+g x}}{7 c}\) |
| \(\Big \downarrow \) 1511 |
| \(\displaystyle \frac {\frac {-\frac {2 \left (-\sqrt {a g^2+c f^2} \left (e \sqrt {a g^2+c f^2} \left (25 a e^2 g^2-c \left (105 d^2 g^2-42 d e f g+8 e^2 f^2\right )\right )-\sqrt {c} \left (a e^2 g^2 (189 d g+19 e f)-c \left (105 d^3 g^3+105 d^2 e f g^2-42 d e^2 f^2 g+8 e^3 f^3\right )\right )\right ) \int \frac {1}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}-\sqrt {c} \sqrt {a g^2+c f^2} \left (a e^2 g^2 (189 d g+19 e f)-c \left (105 d^3 g^3+105 d^2 e f g^2-42 d e^2 f^2 g+8 e^3 f^3\right )\right ) \int \frac {1-\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}\right )}{3 g}-\frac {2}{3} e g \sqrt {a+c x^2} \sqrt {f+g x} \left (25 a e^2 g^2+c \left (-90 d^2 g^2+12 d e f g+7 e^2 f^2\right )\right )}{5 c g^3}+\frac {2 e^2 \sqrt {a+c x^2} (f+g x)^{3/2} (11 d g+e f)}{5 g^2}}{7 c}+\frac {2 e \sqrt {a+c x^2} (d+e x)^2 \sqrt {f+g x}}{7 c}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle \frac {\frac {-\frac {2 \left (-\sqrt {c} \sqrt {a g^2+c f^2} \left (a e^2 g^2 (189 d g+19 e f)-c \left (105 d^3 g^3+105 d^2 e f g^2-42 d e^2 f^2 g+8 e^3 f^3\right )\right ) \int \frac {1-\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}-\frac {\left (a g^2+c f^2\right )^{3/4} \left (\frac {\sqrt {c} (f+g x)}{\sqrt {a g^2+c f^2}}+1\right ) \sqrt {\frac {a+\frac {c f^2}{g^2}-\frac {2 c f (f+g x)}{g^2}+\frac {c (f+g x)^2}{g^2}}{\left (a+\frac {c f^2}{g^2}\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {a g^2+c f^2}}+1\right )^2}} \left (e \sqrt {a g^2+c f^2} \left (25 a e^2 g^2-c \left (105 d^2 g^2-42 d e f g+8 e^2 f^2\right )\right )-\sqrt {c} \left (a e^2 g^2 (189 d g+19 e f)-c \left (105 d^3 g^3+105 d^2 e f g^2-42 d e^2 f^2 g+8 e^3 f^3\right )\right )\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt [4]{c f^2+a g^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} f}{\sqrt {c f^2+a g^2}}+1\right )\right )}{2 \sqrt [4]{c} \sqrt {a+\frac {c f^2}{g^2}-\frac {2 c f (f+g x)}{g^2}+\frac {c (f+g x)^2}{g^2}}}\right )}{3 g}-\frac {2}{3} e g \sqrt {a+c x^2} \sqrt {f+g x} \left (25 a e^2 g^2+c \left (-90 d^2 g^2+12 d e f g+7 e^2 f^2\right )\right )}{5 c g^3}+\frac {2 e^2 \sqrt {a+c x^2} (f+g x)^{3/2} (11 d g+e f)}{5 g^2}}{7 c}+\frac {2 e \sqrt {a+c x^2} (d+e x)^2 \sqrt {f+g x}}{7 c}\) |
| \(\Big \downarrow \) 1509 |
| \(\displaystyle \frac {2 e \sqrt {f+g x} \sqrt {c x^2+a} (d+e x)^2}{7 c}+\frac {\frac {2 (e f+11 d g) (f+g x)^{3/2} \sqrt {c x^2+a} e^2}{5 g^2}+\frac {-\frac {2}{3} e g \sqrt {f+g x} \sqrt {c x^2+a} \left (25 a e^2 g^2+c \left (7 e^2 f^2+12 d e g f-90 d^2 g^2\right )\right )-\frac {2 \left (-\sqrt {c} \sqrt {c f^2+a g^2} \left (a e^2 g^2 (19 e f+189 d g)-c \left (8 e^3 f^3-42 d e^2 g f^2+105 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 )-\frac {\left (c f^2+a g^2\right )^{3/4} \left (e \sqrt {c f^2+a g^2} \left (25 a e^2 g^2-c \left (8 e^2 f^2-42 d e g f+105 d^2 g^2\right )\right )-\sqrt {c} \left (a e^2 g^2 (19 e f+189 d g)-c \left (8 e^3 f^3-42 d e^2 g f^2+105 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 \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 )}{3 g}}{5 c g^3}}{7 c}\) |
(2*e*(d + e*x)^2*Sqrt[f + g*x]*Sqrt[a + c*x^2])/(7*c) + ((2*e^2*(e*f + 11* d*g)*(f + g*x)^(3/2)*Sqrt[a + c*x^2])/(5*g^2) + ((-2*e*g*(25*a*e^2*g^2 + c *(7*e^2*f^2 + 12*d*e*f*g - 90*d^2*g^2))*Sqrt[f + g*x]*Sqrt[a + c*x^2])/3 - (2*(-(Sqrt[c]*Sqrt[c*f^2 + a*g^2]*(a*e^2*g^2*(19*e*f + 189*d*g) - c*(8*e^ 3*f^3 - 42*d*e^2*f^2*g + 105*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))/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)*(e*Sqrt[c*f^2 + a*g^2]*(25*a*e^2 *g^2 - c*(8*e^2*f^2 - 42*d*e*f*g + 105*d^2*g^2)) - Sqrt[c]*(a*e^2*g^2*(19* e*f + 189*d*g) - c*(8*e^3*f^3 - 42*d*e^2*f^2*g + 105*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^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)]*EllipticF[2*ArcTan[(c^(1/4)*Sq rt[f + g*x])/(c*f^2 + a*g^2)^(1/4)], (1 + (Sqrt[c]*f)/Sqrt[c*f^2 + a*g^2]) /2])/(2*c^(1/4)*Sqrt[a + (c*f^2)/g^2 - (2*c*f*(f + g*x))/g^2 + (c*(f + ...
3.7.36.3.1 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[(f_.) + (g_.)*(x_)])/Sqrt[(a_) + (c_.)* (x_)^2], x_Symbol] :> Simp[2*e*(d + e*x)^(m - 1)*Sqrt[f + g*x]*(Sqrt[a + c* x^2]/(c*(2*m + 1))), x] - Simp[1/(c*(2*m + 1)) Int[((d + e*x)^(m - 2)/(Sq rt[f + g*x]*Sqrt[a + c*x^2]))*Simp[a*e*(d*g + 2*e*f*(m - 1)) - c*d^2*f*(2*m + 1) + (a*e^2*g*(2*m - 1) - c*d*(4*e*f*m + d*g*(2*m + 1)))*x - c*e*(e*f + d*g*(4*m - 1))*x^2, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && IntegerQ[ 2*m] && GtQ[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 = 3.37 (sec) , antiderivative size = 882, normalized size of antiderivative = 1.66
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (g x +f \right ) \left (c \,x^{2}+a \right )}\, \left (\frac {2 e^{3} x^{2} \sqrt {c g \,x^{3}+c f \,x^{2}+a g x +f a}}{7 c}+\frac {2 \left (3 d \,e^{2} g +\frac {1}{7} e^{3} f \right ) x \sqrt {c g \,x^{3}+c f \,x^{2}+a g x +f a}}{5 c g}+\frac {2 \left (3 d^{2} e g +3 d \,e^{2} f -\frac {4 f \left (3 d \,e^{2} g +\frac {1}{7} e^{3} f \right )}{5 g}-\frac {5 g a \,e^{3}}{7 c}\right ) \sqrt {c g \,x^{3}+c f \,x^{2}+a g x +f a}}{3 c g}+\frac {2 \left (d^{3} f -\frac {2 a f \left (3 d \,e^{2} g +\frac {1}{7} e^{3} f \right )}{5 c g}-\frac {a \left (3 d^{2} e g +3 d \,e^{2} f -\frac {4 f \left (3 d \,e^{2} g +\frac {1}{7} e^{3} f \right )}{5 g}-\frac {5 g a \,e^{3}}{7 c}\right )}{3 c}\right ) \left (\frac {f}{g}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}}\, F\left (\sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\right )}{\sqrt {c g \,x^{3}+c f \,x^{2}+a g x +f a}}+\frac {2 \left (d^{3} g +3 d^{2} e f -\frac {4 a f \,e^{3}}{7 c}-\frac {3 a \left (3 d \,e^{2} g +\frac {1}{7} e^{3} f \right )}{5 c}-\frac {2 f \left (3 d^{2} e g +3 d \,e^{2} f -\frac {4 f \left (3 d \,e^{2} g +\frac {1}{7} e^{3} f \right )}{5 g}-\frac {5 g a \,e^{3}}{7 c}\right )}{3 g}\right ) \left (\frac {f}{g}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}}\, \left (\left (-\frac {f}{g}-\frac {\sqrt {-a c}}{c}\right ) E\left (\sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\right )+\frac {\sqrt {-a c}\, F\left (\sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\right )}{c}\right )}{\sqrt {c g \,x^{3}+c f \,x^{2}+a g x +f a}}\right )}{\sqrt {g x +f}\, \sqrt {c \,x^{2}+a}}\) | \(882\) |
| risch | \(\text {Expression too large to display}\) | \(1610\) |
| default | \(\text {Expression too large to display}\) | \(3924\) |
((g*x+f)*(c*x^2+a))^(1/2)/(g*x+f)^(1/2)/(c*x^2+a)^(1/2)*(2/7*e^3/c*x^2*(c* g*x^3+c*f*x^2+a*g*x+a*f)^(1/2)+2/5*(3*d*e^2*g+1/7*e^3*f)/c/g*x*(c*g*x^3+c* f*x^2+a*g*x+a*f)^(1/2)+2/3*(3*d^2*e*g+3*d*e^2*f-4/5*f/g*(3*d*e^2*g+1/7*e^3 *f)-5/7/c*g*a*e^3)/c/g*(c*g*x^3+c*f*x^2+a*g*x+a*f)^(1/2)+2*(d^3*f-2/5*a/c* f/g*(3*d*e^2*g+1/7*e^3*f)-1/3/c*a*(3*d^2*e*g+3*d*e^2*f-4/5*f/g*(3*d*e^2*g+ 1/7*e^3*f)-5/7/c*g*a*e^3))*(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)*Ellip ticF(((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*(d^3*g+3*d^2*e*f-4/7*a/c*f*e^3-3/5/c*a*(3*d*e^2*g+1 /7*e^3*f)-2/3*f/g*(3*d^2*e*g+3*d*e^2*f-4/5*f/g*(3*d*e^2*g+1/7*e^3*f)-5/7/c *g*a*e^3))*(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+(-a*c)^(1/2)/c)/(- f/g-(-a*c)^(1/2)/c))^(1/2))+(-a*c)^(1/2)/c*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))))
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.11 (sec) , antiderivative size = 445, normalized size of antiderivative = 0.84 \[ \int \frac {(d+e x)^3 \sqrt {f+g x}}{\sqrt {a+c x^2}} \, dx=-\frac {2 \, {\left ({\left (8 \, c^{2} e^{3} f^{4} - 42 \, c^{2} d e^{2} f^{3} g + {\left (105 \, c^{2} d^{2} e - 13 \, a c e^{3}\right )} f^{2} g^{2} - 42 \, {\left (5 \, c^{2} d^{3} - 6 \, a c d e^{2}\right )} f g^{3} + 15 \, {\left (21 \, a c d^{2} e - 5 \, a^{2} e^{3}\right )} g^{4}\right )} \sqrt {c g} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c f^{2} - 3 \, a g^{2}\right )}}{3 \, c g^{2}}, -\frac {8 \, {\left (c f^{3} + 9 \, a f g^{2}\right )}}{27 \, c g^{3}}, \frac {3 \, g x + f}{3 \, g}\right ) + 3 \, {\left (8 \, c^{2} e^{3} f^{3} g - 42 \, c^{2} d e^{2} f^{2} g^{2} + {\left (105 \, c^{2} d^{2} e - 19 \, a c e^{3}\right )} f g^{3} + 21 \, {\left (5 \, c^{2} d^{3} - 9 \, a c d e^{2}\right )} g^{4}\right )} \sqrt {c g} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c f^{2} - 3 \, a g^{2}\right )}}{3 \, c g^{2}}, -\frac {8 \, {\left (c f^{3} + 9 \, a f g^{2}\right )}}{27 \, c g^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c f^{2} - 3 \, a g^{2}\right )}}{3 \, c g^{2}}, -\frac {8 \, {\left (c f^{3} + 9 \, a f g^{2}\right )}}{27 \, c g^{3}}, \frac {3 \, g x + f}{3 \, g}\right )\right ) - 3 \, {\left (15 \, c^{2} e^{3} g^{4} x^{2} - 4 \, c^{2} e^{3} f^{2} g^{2} + 21 \, c^{2} d e^{2} f g^{3} + 5 \, {\left (21 \, c^{2} d^{2} e - 5 \, a c e^{3}\right )} g^{4} + 3 \, {\left (c^{2} e^{3} f g^{3} + 21 \, c^{2} d e^{2} g^{4}\right )} x\right )} \sqrt {c x^{2} + a} \sqrt {g x + f}\right )}}{315 \, c^{3} g^{4}} \]
-2/315*((8*c^2*e^3*f^4 - 42*c^2*d*e^2*f^3*g + (105*c^2*d^2*e - 13*a*c*e^3) *f^2*g^2 - 42*(5*c^2*d^3 - 6*a*c*d*e^2)*f*g^3 + 15*(21*a*c*d^2*e - 5*a^2*e ^3)*g^4)*sqrt(c*g)*weierstrassPInverse(4/3*(c*f^2 - 3*a*g^2)/(c*g^2), -8/2 7*(c*f^3 + 9*a*f*g^2)/(c*g^3), 1/3*(3*g*x + f)/g) + 3*(8*c^2*e^3*f^3*g - 4 2*c^2*d*e^2*f^2*g^2 + (105*c^2*d^2*e - 19*a*c*e^3)*f*g^3 + 21*(5*c^2*d^3 - 9*a*c*d*e^2)*g^4)*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*( 15*c^2*e^3*g^4*x^2 - 4*c^2*e^3*f^2*g^2 + 21*c^2*d*e^2*f*g^3 + 5*(21*c^2*d^ 2*e - 5*a*c*e^3)*g^4 + 3*(c^2*e^3*f*g^3 + 21*c^2*d*e^2*g^4)*x)*sqrt(c*x^2 + a)*sqrt(g*x + f))/(c^3*g^4)
\[ \int \frac {(d+e x)^3 \sqrt {f+g x}}{\sqrt {a+c x^2}} \, dx=\int \frac {\left (d + e x\right )^{3} \sqrt {f + g x}}{\sqrt {a + c x^{2}}}\, dx \]
\[ \int \frac {(d+e x)^3 \sqrt {f+g x}}{\sqrt {a+c x^2}} \, dx=\int { \frac {{\left (e x + d\right )}^{3} \sqrt {g x + f}}{\sqrt {c x^{2} + a}} \,d x } \]
\[ \int \frac {(d+e x)^3 \sqrt {f+g x}}{\sqrt {a+c x^2}} \, dx=\int { \frac {{\left (e x + d\right )}^{3} \sqrt {g x + f}}{\sqrt {c x^{2} + a}} \,d x } \]
Timed out. \[ \int \frac {(d+e x)^3 \sqrt {f+g x}}{\sqrt {a+c x^2}} \, dx=\int \frac {\sqrt {f+g\,x}\,{\left (d+e\,x\right )}^3}{\sqrt {c\,x^2+a}} \,d x \]