Integrand size = 28, antiderivative size = 457 \[ \int \frac {(d+e x)^3}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx=-\frac {8 e^2 (e f-3 d g) \sqrt {f+g x} \sqrt {a+c x^2}}{15 c g^2}+\frac {2 e^2 (d+e x) \sqrt {f+g x} \sqrt {a+c x^2}}{5 c g}+\frac {2 \sqrt {-a} e \left (9 a e^2 g^2-c \left (8 e^2 f^2-30 d e f g+45 d^2 g^2\right )\right ) \sqrt {f+g x} \sqrt {1+\frac {c x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{15 c^{3/2} g^3 \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {a+c x^2}}-\frac {2 \sqrt {-a} \left (a e^2 g^2 (7 e f-15 d g)-c \left (8 e^3 f^3-30 d e^2 f^2 g+45 d^2 e f g^2-15 d^3 g^3\right )\right ) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {1+\frac {c x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{15 c^{3/2} g^3 \sqrt {f+g x} \sqrt {a+c x^2}} \]
-8/15*e^2*(-3*d*g+e*f)*(g*x+f)^(1/2)*(c*x^2+a)^(1/2)/c/g^2+2/5*e^2*(e*x+d) *(g*x+f)^(1/2)*(c*x^2+a)^(1/2)/c/g+2/15*e*(9*a*e^2*g^2-c*(45*d^2*g^2-30*d* e*f*g+8*e^2*f^2))*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)-2/15*(a*e^2*g^2*(-15*d*g+7*e*f)-c*(-15*d^3*g^3+45*d^2*e*f*g^2 -30*d*e^2*f^2*g+8*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^3/(g*x+f) ^(1/2)/(c*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 24.72 (sec) , antiderivative size = 619, normalized size of antiderivative = 1.35 \[ \int \frac {(d+e x)^3}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx=\frac {\sqrt {f+g x} \left (\frac {2 e^2 (-4 e f+15 d g+3 e g x) \left (a+c x^2\right )}{c g^2}+\frac {2 (f+g x) \left (\frac {e g^2 \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}} \left (-9 a e^2 g^2+c \left (8 e^2 f^2-30 d e f g+45 d^2 g^2\right )\right ) \left (a+c x^2\right )}{(f+g x)^2}+\frac {\sqrt {c} e \left (-i \sqrt {c} f+\sqrt {a} g\right ) \left (-9 a e^2 g^2+c \left (8 e^2 f^2-30 d e f g+45 d^2 g^2\right )\right ) \sqrt {\frac {g \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{f+g x}} \sqrt {-\frac {\frac {i \sqrt {a} g}{\sqrt {c}}-g x}{f+g x}} E\left (i \text {arcsinh}\left (\frac {\sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}{\sqrt {f+g x}}\right )|\frac {\sqrt {c} f-i \sqrt {a} g}{\sqrt {c} f+i \sqrt {a} g}\right )}{\sqrt {f+g x}}+\frac {\sqrt {c} g \left (15 i c^{3/2} d^3 g^2+9 a^{3/2} e^3 g^2-i a \sqrt {c} e^2 g (2 e f+15 d g)+\sqrt {a} c e \left (-8 e^2 f^2+30 d e f g-45 d^2 g^2\right )\right ) \sqrt {\frac {g \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{f+g x}} \sqrt {-\frac {\frac {i \sqrt {a} g}{\sqrt {c}}-g x}{f+g x}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}{\sqrt {f+g x}}\right ),\frac {\sqrt {c} f-i \sqrt {a} g}{\sqrt {c} f+i \sqrt {a} g}\right )}{\sqrt {f+g x}}\right )}{c^2 g^4 \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}\right )}{15 \sqrt {a+c x^2}} \]
(Sqrt[f + g*x]*((2*e^2*(-4*e*f + 15*d*g + 3*e*g*x)*(a + c*x^2))/(c*g^2) + (2*(f + g*x)*((e*g^2*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(-9*a*e^2*g^2 + c*(8 *e^2*f^2 - 30*d*e*f*g + 45*d^2*g^2))*(a + c*x^2))/(f + g*x)^2 + (Sqrt[c]*e *((-I)*Sqrt[c]*f + Sqrt[a]*g)*(-9*a*e^2*g^2 + c*(8*e^2*f^2 - 30*d*e*f*g + 45*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))]*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[c]*g*((15*I)*c^(3/2)*d^3*g^2 + 9*a^(3/2)*e^ 3*g^2 - I*a*Sqrt[c]*e^2*g*(2*e*f + 15*d*g) + Sqrt[a]*c*e*(-8*e^2*f^2 + 30* d*e*f*g - 45*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))]*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)])/Sqrt[f + g*x]))/(c^2*g^4*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c ]])))/(15*Sqrt[a + c*x^2])
Time = 1.31 (sec) , antiderivative size = 831, normalized size of antiderivative = 1.82, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.321, Rules used = {728, 25, 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.
| \(\displaystyle \int \frac {(d+e x)^3}{\sqrt {a+c x^2} \sqrt {f+g x}} \, dx\) |
| \(\Big \downarrow \) 728 |
| \(\displaystyle \frac {2 e^2 \sqrt {a+c x^2} (d+e x) \sqrt {f+g x}}{5 c g}-\frac {\int -\frac {5 c g d^3-4 c e^2 (e f-3 d g) x^2-a e^2 (2 e f+d g)-e \left (3 a g e^2+c d (2 e f-15 d g)\right ) x}{\sqrt {f+g x} \sqrt {c x^2+a}}dx}{5 c g}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {5 c g d^3-4 c e^2 (e f-3 d g) x^2-a e^2 (2 e f+d g)-e \left (3 a g e^2+c d (2 e f-15 d g)\right ) x}{\sqrt {f+g x} \sqrt {c x^2+a}}dx}{5 c g}+\frac {2 e^2 \sqrt {a+c x^2} (d+e x) \sqrt {f+g x}}{5 c g}\) |
| \(\Big \downarrow \) 2185 |
| \(\displaystyle \frac {\frac {2 \int \frac {c g \left (g \left (15 c d^3 g-a e^2 (2 e f+15 d g)\right )-e \left (9 a e^2 g^2-c \left (8 e^2 f^2-30 d e g f+45 d^2 g^2\right )\right ) x\right )}{2 \sqrt {f+g x} \sqrt {c x^2+a}}dx}{3 c g^2}-\frac {8 e^2 \sqrt {a+c x^2} \sqrt {f+g x} (e f-3 d g)}{3 g}}{5 c g}+\frac {2 e^2 \sqrt {a+c x^2} (d+e x) \sqrt {f+g x}}{5 c g}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {g \left (15 c d^3 g-a e^2 (2 e f+15 d g)\right )-e \left (9 a e^2 g^2-c \left (8 e^2 f^2-30 d e g f+45 d^2 g^2\right )\right ) x}{\sqrt {f+g x} \sqrt {c x^2+a}}dx}{3 g}-\frac {8 e^2 \sqrt {a+c x^2} \sqrt {f+g x} (e f-3 d g)}{3 g}}{5 c g}+\frac {2 e^2 \sqrt {a+c x^2} (d+e x) \sqrt {f+g x}}{5 c g}\) |
| \(\Big \downarrow \) 599 |
| \(\displaystyle \frac {-\frac {2 \int -\frac {a e^2 (7 e f-15 d g) g^2-c \left (8 e^3 f^3-30 d e^2 g f^2+45 d^2 e g^2 f-15 d^3 g^3\right )-e \left (9 a e^2 g^2-c \left (8 e^2 f^2-30 d e g f+45 d^2 g^2\right )\right ) (f+g x)}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{3 g^3}-\frac {8 e^2 \sqrt {a+c x^2} \sqrt {f+g x} (e f-3 d g)}{3 g}}{5 c g}+\frac {2 e^2 \sqrt {a+c x^2} (d+e x) \sqrt {f+g x}}{5 c g}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {2 \int \frac {a e^2 (7 e f-15 d g) g^2-c \left (8 e^3 f^3-30 d e^2 g f^2+45 d^2 e g^2 f-15 d^3 g^3\right )-e \left (9 a e^2 g^2-c \left (8 e^2 f^2-30 d e g f+45 d^2 g^2\right )\right ) (f+g x)}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{3 g^3}-\frac {8 e^2 \sqrt {a+c x^2} \sqrt {f+g x} (e f-3 d g)}{3 g}}{5 c g}+\frac {2 e^2 \sqrt {a+c x^2} (d+e x) \sqrt {f+g x}}{5 c g}\) |
| \(\Big \downarrow \) 1511 |
| \(\displaystyle \frac {-\frac {2 \left (-\frac {\left (-e \sqrt {a g^2+c f^2} \left (9 a e^2 g^2-c \left (45 d^2 g^2-30 d e f g+8 e^2 f^2\right )\right )+a \sqrt {c} e^2 g^2 (7 e f-15 d g)-c^{3/2} \left (-15 d^3 g^3+45 d^2 e f g^2-30 d e^2 f^2 g+8 e^3 f^3\right )\right ) \int \frac {1}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{\sqrt {c}}-\frac {e \sqrt {a g^2+c f^2} \left (9 a e^2 g^2-c \left (45 d^2 g^2-30 d e f g+8 e^2 f^2\right )\right ) \int \frac {1-\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{\sqrt {c}}\right )}{3 g^3}-\frac {8 e^2 \sqrt {a+c x^2} \sqrt {f+g x} (e f-3 d g)}{3 g}}{5 c g}+\frac {2 e^2 \sqrt {a+c x^2} (d+e x) \sqrt {f+g x}}{5 c g}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle \frac {-\frac {2 \left (-\frac {e \sqrt {a g^2+c f^2} \left (9 a e^2 g^2-c \left (45 d^2 g^2-30 d e f g+8 e^2 f^2\right )\right ) \int \frac {1-\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{\sqrt {c}}-\frac {\sqrt [4]{a g^2+c f^2} \left (\frac {\sqrt {c} (f+g x)}{\sqrt {a g^2+c f^2}}+1\right ) \sqrt {\frac {a+\frac {c f^2}{g^2}-\frac {2 c f (f+g x)}{g^2}+\frac {c (f+g x)^2}{g^2}}{\left (a+\frac {c f^2}{g^2}\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {a g^2+c f^2}}+1\right )^2}} \left (-e \sqrt {a g^2+c f^2} \left (9 a e^2 g^2-c \left (45 d^2 g^2-30 d e f g+8 e^2 f^2\right )\right )+a \sqrt {c} e^2 g^2 (7 e f-15 d g)-c^{3/2} \left (-15 d^3 g^3+45 d^2 e f g^2-30 d e^2 f^2 g+8 e^3 f^3\right )\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt [4]{c f^2+a g^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} f}{\sqrt {c f^2+a g^2}}+1\right )\right )}{2 c^{3/4} \sqrt {a+\frac {c f^2}{g^2}-\frac {2 c f (f+g x)}{g^2}+\frac {c (f+g x)^2}{g^2}}}\right )}{3 g^3}-\frac {8 e^2 \sqrt {a+c x^2} \sqrt {f+g x} (e f-3 d g)}{3 g}}{5 c g}+\frac {2 e^2 \sqrt {a+c x^2} (d+e x) \sqrt {f+g x}}{5 c g}\) |
| \(\Big \downarrow \) 1509 |
| \(\displaystyle \frac {2 (d+e x) \sqrt {f+g x} \sqrt {c x^2+a} e^2}{5 c g}+\frac {-\frac {8 (e f-3 d g) \sqrt {f+g x} \sqrt {c x^2+a} e^2}{3 g}-\frac {2 \left (-\frac {e \sqrt {c f^2+a g^2} \left (9 a e^2 g^2-c \left (8 e^2 f^2-30 d e g f+45 d^2 g^2\right )\right ) \left (\frac {\sqrt [4]{c f^2+a g^2} \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right ) \sqrt {\frac {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}{\left (\frac {c f^2}{g^2}+a\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt [4]{c f^2+a g^2}}\right )|\frac {1}{2} \left (\frac {\sqrt {c} f}{\sqrt {c f^2+a g^2}}+1\right )\right )}{\sqrt [4]{c} \sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}-\frac {\sqrt {f+g x} \sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}{\left (\frac {c f^2}{g^2}+a\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right )}\right )}{\sqrt {c}}-\frac {\sqrt [4]{c f^2+a g^2} \left (a \sqrt {c} e^2 (7 e f-15 d g) g^2-c^{3/2} \left (8 e^3 f^3-30 d e^2 g f^2+45 d^2 e g^2 f-15 d^3 g^3\right )-e \sqrt {c f^2+a g^2} \left (9 a e^2 g^2-c \left (8 e^2 f^2-30 d e g f+45 d^2 g^2\right )\right )\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right ) \sqrt {\frac {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}{\left (\frac {c f^2}{g^2}+a\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt [4]{c f^2+a g^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} f}{\sqrt {c f^2+a g^2}}+1\right )\right )}{2 c^{3/4} \sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}\right )}{3 g^3}}{5 c g}\) |
(2*e^2*(d + e*x)*Sqrt[f + g*x]*Sqrt[a + c*x^2])/(5*c*g) + ((-8*e^2*(e*f - 3*d*g)*Sqrt[f + g*x]*Sqrt[a + c*x^2])/(3*g) - (2*(-((e*Sqrt[c*f^2 + a*g^2] *(9*a*e^2*g^2 - c*(8*e^2*f^2 - 30*d*e*f*g + 45*d^2*g^2))*(-((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])))/Sqrt[c]) - ((c*f^2 + a*g^2)^(1/4)*(a*Sqrt[c]*e^2*g^2*(7*e *f - 15*d*g) - c^(3/2)*(8*e^3*f^3 - 30*d*e^2*f^2*g + 45*d^2*e*f*g^2 - 15*d ^3*g^3) - e*Sqrt[c*f^2 + a*g^2]*(9*a*e^2*g^2 - c*(8*e^2*f^2 - 30*d*e*f*g + 45*d^2*g^2)))*(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)*Sqrt[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^(3/4)*Sqrt[a + (c*f^2)/g^2 - (2*c*f*(f + g*x))/g^2 + (c*( f + g*x)^2)/g^2])))/(3*g^3))/(5*c*g)
3.7.45.3.1 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^2*(d + e*x)^(m - 2)*Sqrt[f + g*x]*(Sqrt[a + c*x^2]/(c*g*(2*m - 1))), x] - Simp[1/(c*g*(2*m - 1)) Int[((d + e*x)^(m - 3)/(Sqrt[f + g*x]*Sqrt[a + c*x^2]))*Simp[a*e^2*(d*g + 2*e*f*(m - 2)) - c*d^ 3*g*(2*m - 1) + e*(e*(a*e*g*(2*m - 3)) + c*d*(2*e*f - 3*d*g*(2*m - 1)))*x + 2*e^2*(c*e*f - 3*c*d*g)*(m - 1)*x^2, x], x], x] /; FreeQ[{a, c, d, e, f, g }, x] && IntegerQ[2*m] && GeQ[m, 2]
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 = 2.98 (sec) , antiderivative size = 711, normalized size of antiderivative = 1.56
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (g x +f \right ) \left (c \,x^{2}+a \right )}\, \left (\frac {2 e^{3} x \sqrt {c g \,x^{3}+c f \,x^{2}+a g x +f a}}{5 c g}+\frac {2 \left (3 d \,e^{2}-\frac {4 f \,e^{3}}{5 g}\right ) \sqrt {c g \,x^{3}+c f \,x^{2}+a g x +f a}}{3 c g}+\frac {2 \left (d^{3}-\frac {2 f a \,e^{3}}{5 c g}-\frac {a \left (3 d \,e^{2}-\frac {4 f \,e^{3}}{5 g}\right )}{3 c}\right ) \left (\frac {f}{g}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}}\, F\left (\sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\right )}{\sqrt {c g \,x^{3}+c f \,x^{2}+a g x +f a}}+\frac {2 \left (3 d^{2} e -\frac {3 e^{3} a}{5 c}-\frac {2 f \left (3 d \,e^{2}-\frac {4 f \,e^{3}}{5 g}\right )}{3 g}\right ) \left (\frac {f}{g}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}}\, \left (\left (-\frac {f}{g}-\frac {\sqrt {-a c}}{c}\right ) E\left (\sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\right )+\frac {\sqrt {-a c}\, F\left (\sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\right )}{c}\right )}{\sqrt {c g \,x^{3}+c f \,x^{2}+a g x +f a}}\right )}{\sqrt {g x +f}\, \sqrt {c \,x^{2}+a}}\) | \(711\) |
| risch | \(\text {Expression too large to display}\) | \(1082\) |
| default | \(\text {Expression too large to display}\) | \(2950\) |
((g*x+f)*(c*x^2+a))^(1/2)/(g*x+f)^(1/2)/(c*x^2+a)^(1/2)*(2/5*e^3/c/g*x*(c* g*x^3+c*f*x^2+a*g*x+a*f)^(1/2)+2/3*(3*d*e^2-4/5*f/g*e^3)/c/g*(c*g*x^3+c*f* x^2+a*g*x+a*f)^(1/2)+2*(d^3-2/5*f*a/c/g*e^3-1/3*a/c*(3*d*e^2-4/5*f/g*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)*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* d^2*e-3/5*e^3/c*a-2/3*f/g*(3*d*e^2-4/5*f/g*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 = 0.12 (sec) , antiderivative size = 319, normalized size of antiderivative = 0.70 \[ \int \frac {(d+e x)^3}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx=-\frac {2 \, {\left ({\left (8 \, c e^{3} f^{3} - 30 \, c d e^{2} f^{2} g + 3 \, {\left (15 \, c d^{2} e - a e^{3}\right )} f g^{2} - 45 \, {\left (c d^{3} - a d e^{2}\right )} g^{3}\right )} \sqrt {c g} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c f^{2} - 3 \, a g^{2}\right )}}{3 \, c g^{2}}, -\frac {8 \, {\left (c f^{3} + 9 \, a f g^{2}\right )}}{27 \, c g^{3}}, \frac {3 \, g x + f}{3 \, g}\right ) + 3 \, {\left (8 \, c e^{3} f^{2} g - 30 \, c d e^{2} f g^{2} + 9 \, {\left (5 \, c d^{2} e - a e^{3}\right )} g^{3}\right )} \sqrt {c g} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c f^{2} - 3 \, a g^{2}\right )}}{3 \, c g^{2}}, -\frac {8 \, {\left (c f^{3} + 9 \, a f g^{2}\right )}}{27 \, c g^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c f^{2} - 3 \, a g^{2}\right )}}{3 \, c g^{2}}, -\frac {8 \, {\left (c f^{3} + 9 \, a f g^{2}\right )}}{27 \, c g^{3}}, \frac {3 \, g x + f}{3 \, g}\right )\right ) - 3 \, {\left (3 \, c e^{3} g^{3} x - 4 \, c e^{3} f g^{2} + 15 \, c d e^{2} g^{3}\right )} \sqrt {c x^{2} + a} \sqrt {g x + f}\right )}}{45 \, c^{2} g^{4}} \]
-2/45*((8*c*e^3*f^3 - 30*c*d*e^2*f^2*g + 3*(15*c*d^2*e - a*e^3)*f*g^2 - 45 *(c*d^3 - a*d*e^2)*g^3)*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) + 3*(8*c* e^3*f^2*g - 30*c*d*e^2*f*g^2 + 9*(5*c*d^2*e - a*e^3)*g^3)*sqrt(c*g)*weiers trassZeta(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*(3*c*e^3*g^3*x - 4*c*e^3*f*g^2 + 15* c*d*e^2*g^3)*sqrt(c*x^2 + a)*sqrt(g*x + f))/(c^2*g^4)
\[ \int \frac {(d+e x)^3}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx=\int \frac {\left (d + e x\right )^{3}}{\sqrt {a + c x^{2}} \sqrt {f + g x}}\, dx \]
\[ \int \frac {(d+e x)^3}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx=\int { \frac {{\left (e x + d\right )}^{3}}{\sqrt {c x^{2} + a} \sqrt {g x + f}} \,d x } \]
\[ \int \frac {(d+e x)^3}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx=\int { \frac {{\left (e x + d\right )}^{3}}{\sqrt {c x^{2} + a} \sqrt {g x + f}} \,d x } \]
Timed out. \[ \int \frac {(d+e x)^3}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx=\int \frac {{\left (d+e\,x\right )}^3}{\sqrt {f+g\,x}\,\sqrt {c\,x^2+a}} \,d x \]