Integrand size = 28, antiderivative size = 410 \[ \int \frac {(d+e x)^2 \sqrt {f+g x}}{\sqrt {a+c x^2}} \, dx=\frac {2 e (e f+7 d g) \sqrt {f+g x} \sqrt {a+c x^2}}{15 c g}+\frac {2 e (d+e x) \sqrt {f+g x} \sqrt {a+c x^2}}{5 c}+\frac {2 \sqrt {-a} \left (9 a e^2 g^2+c \left (2 e^2 f^2-10 d e f g-15 d^2 g^2\right )\right ) \sqrt {f+g x} \sqrt {1+\frac {c x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{15 c^{3/2} g^2 \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {a+c x^2}}-\frac {4 \sqrt {-a} e (e f-5 d g) \left (c f^2+a g^2\right ) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {1+\frac {c x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{15 c^{3/2} g^2 \sqrt {f+g x} \sqrt {a+c x^2}} \]
2/15*e*(7*d*g+e*f)*(g*x+f)^(1/2)*(c*x^2+a)^(1/2)/c/g+2/5*e*(e*x+d)*(g*x+f) ^(1/2)*(c*x^2+a)^(1/2)/c+2/15*(9*a*e^2*g^2+c*(-15*d^2*g^2-10*d*e*f*g+2*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^2/(c*x^2+a)^(1/2)/((g*x+f)*c^(1/2)/(g*(-a)^(1/2)+f*c^(1/2)))^(1/2 )-4/15*e*(-5*d*g+e*f)*(a*g^2+c*f^2)*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^2 /(g*x+f)^(1/2)/(c*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 25.09 (sec) , antiderivative size = 591, normalized size of antiderivative = 1.44 \[ \int \frac {(d+e x)^2 \sqrt {f+g x}}{\sqrt {a+c x^2}} \, dx=\frac {\sqrt {f+g x} \left (\frac {2 e \left (a+c x^2\right ) (10 d g+e (f+3 g x))}{c g}+\frac {(f+g x) \left (\frac {2 g^2 \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}} \left (-9 a e^2 g^2+c \left (-2 e^2 f^2+10 d e f g+15 d^2 g^2\right )\right ) \left (a+c x^2\right )}{(f+g x)^2}+\frac {2 \sqrt {c} \left (-i \sqrt {c} f+\sqrt {a} g\right ) \left (-9 a e^2 g^2+c \left (-2 e^2 f^2+10 d e f g+15 d^2 g^2\right )\right ) \sqrt {\frac {g \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{f+g x}} \sqrt {-\frac {\frac {i \sqrt {a} g}{\sqrt {c}}-g x}{f+g x}} E\left (i \text {arcsinh}\left (\frac {\sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}{\sqrt {f+g x}}\right )|\frac {\sqrt {c} f-i \sqrt {a} g}{\sqrt {c} f+i \sqrt {a} g}\right )}{\sqrt {f+g x}}+\frac {2 \sqrt {c} g \left (\sqrt {c} f+i \sqrt {a} g\right ) \left (15 i c d^2 g-9 i a e^2 g+2 \sqrt {a} \sqrt {c} e (e f-5 d g)\right ) \sqrt {\frac {g \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{f+g x}} \sqrt {-\frac {\frac {i \sqrt {a} g}{\sqrt {c}}-g x}{f+g x}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}{\sqrt {f+g x}}\right ),\frac {\sqrt {c} f-i \sqrt {a} g}{\sqrt {c} f+i \sqrt {a} g}\right )}{\sqrt {f+g x}}\right )}{c^2 g^3 \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}\right )}{15 \sqrt {a+c x^2}} \]
(Sqrt[f + g*x]*((2*e*(a + c*x^2)*(10*d*g + e*(f + 3*g*x)))/(c*g) + ((f + g *x)*((2*g^2*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(-9*a*e^2*g^2 + c*(-2*e^2*f^2 + 10*d*e*f*g + 15*d^2*g^2))*(a + c*x^2))/(f + g*x)^2 + (2*Sqrt[c]*((-I)*S qrt[c]*f + Sqrt[a]*g)*(-9*a*e^2*g^2 + c*(-2*e^2*f^2 + 10*d*e*f*g + 15*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)/Sqr t[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)] )/Sqrt[f + g*x] + (2*Sqrt[c]*g*(Sqrt[c]*f + I*Sqrt[a]*g)*((15*I)*c*d^2*g - (9*I)*a*e^2*g + 2*Sqrt[a]*Sqrt[c]*e*(e*f - 5*d*g))*Sqrt[(g*((I*Sqrt[a])/S qrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*E llipticF[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^3*S qrt[-f - (I*Sqrt[a]*g)/Sqrt[c]])))/(15*Sqrt[a + c*x^2])
Time = 1.16 (sec) , antiderivative size = 764, normalized size of antiderivative = 1.86, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.321, Rules used = {735, 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.
\(\displaystyle \int \frac {(d+e x)^2 \sqrt {f+g x}}{\sqrt {a+c x^2}} \, dx\) |
\(\Big \downarrow \) 735 |
\(\displaystyle \frac {2 e \sqrt {a+c x^2} (d+e x) \sqrt {f+g x}}{5 c}-\frac {\int -\frac {5 c f d^2+c e (e f+7 d g) x^2-a e (2 e f+d g)-\left (3 a e^2 g-c d (8 e f+5 d g)\right ) x}{\sqrt {f+g x} \sqrt {c x^2+a}}dx}{5 c}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\int \frac {5 c f d^2+c e (e f+7 d g) x^2-a e (2 e f+d g)-\left (3 a e^2 g-c d (8 e f+5 d g)\right ) x}{\sqrt {f+g x} \sqrt {c x^2+a}}dx}{5 c}+\frac {2 e \sqrt {a+c x^2} (d+e x) \sqrt {f+g x}}{5 c}\) |
\(\Big \downarrow \) 2185 |
\(\displaystyle \frac {\frac {2 \int \frac {c g \left (g \left (15 c d^2 f-a e (7 e f+10 d g)\right )-\left (9 a e^2 g^2+c \left (2 e^2 f^2-10 d e g f-15 d^2 g^2\right )\right ) x\right )}{2 \sqrt {f+g x} \sqrt {c x^2+a}}dx}{3 c g^2}+\frac {2 e \sqrt {a+c x^2} \sqrt {f+g x} (7 d g+e f)}{3 g}}{5 c}+\frac {2 e \sqrt {a+c x^2} (d+e x) \sqrt {f+g x}}{5 c}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {\int \frac {g \left (15 c d^2 f-a e (7 e f+10 d g)\right )-\left (9 a e^2 g^2+c \left (2 e^2 f^2-10 d e g f-15 d^2 g^2\right )\right ) x}{\sqrt {f+g x} \sqrt {c x^2+a}}dx}{3 g}+\frac {2 e \sqrt {a+c x^2} \sqrt {f+g x} (7 d g+e f)}{3 g}}{5 c}+\frac {2 e \sqrt {a+c x^2} (d+e x) \sqrt {f+g x}}{5 c}\) |
\(\Big \downarrow \) 599 |
\(\displaystyle \frac {\frac {2 e \sqrt {a+c x^2} \sqrt {f+g x} (7 d g+e f)}{3 g}-\frac {2 \int -\frac {2 e (e f-5 d g) \left (c f^2+a g^2\right )-\left (9 a e^2 g^2+c \left (2 e^2 f^2-10 d e g f-15 d^2 g^2\right )\right ) (f+g x)}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{3 g^3}}{5 c}+\frac {2 e \sqrt {a+c x^2} (d+e x) \sqrt {f+g x}}{5 c}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\frac {2 \int \frac {2 e (e f-5 d g) \left (c f^2+a g^2\right )-\left (9 a e^2 g^2+c \left (2 e^2 f^2-10 d e g f-15 d^2 g^2\right )\right ) (f+g x)}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{3 g^3}+\frac {2 e \sqrt {a+c x^2} \sqrt {f+g x} (7 d g+e f)}{3 g}}{5 c}+\frac {2 e \sqrt {a+c x^2} (d+e x) \sqrt {f+g x}}{5 c}\) |
\(\Big \downarrow \) 1511 |
\(\displaystyle \frac {\frac {2 e \sqrt {a+c x^2} \sqrt {f+g x} (7 d g+e f)}{3 g}-\frac {2 \left (\frac {\sqrt {a g^2+c f^2} \left (-2 \sqrt {c} e \sqrt {a g^2+c f^2} (e f-5 d g)+9 a e^2 g^2+c \left (-15 d^2 g^2-10 d e f g+2 e^2 f^2\right )\right ) \int \frac {1}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{\sqrt {c}}-\frac {\sqrt {a g^2+c f^2} \left (9 a e^2 g^2+c \left (-15 d^2 g^2-10 d e f g+2 e^2 f^2\right )\right ) \int \frac {1-\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{\sqrt {c}}\right )}{3 g^3}}{5 c}+\frac {2 e \sqrt {a+c x^2} (d+e x) \sqrt {f+g x}}{5 c}\) |
\(\Big \downarrow \) 1416 |
\(\displaystyle \frac {\frac {2 e \sqrt {a+c x^2} \sqrt {f+g x} (7 d g+e f)}{3 g}-\frac {2 \left (\frac {\left (a g^2+c f^2\right )^{3/4} \left (\frac {\sqrt {c} (f+g x)}{\sqrt {a g^2+c f^2}}+1\right ) \sqrt {\frac {a+\frac {c f^2}{g^2}-\frac {2 c f (f+g x)}{g^2}+\frac {c (f+g x)^2}{g^2}}{\left (a+\frac {c f^2}{g^2}\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {a g^2+c f^2}}+1\right )^2}} \left (-2 \sqrt {c} e \sqrt {a g^2+c f^2} (e f-5 d g)+9 a e^2 g^2+c \left (-15 d^2 g^2-10 d e f g+2 e^2 f^2\right )\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt [4]{c f^2+a g^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} f}{\sqrt {c f^2+a g^2}}+1\right )\right )}{2 c^{3/4} \sqrt {a+\frac {c f^2}{g^2}-\frac {2 c f (f+g x)}{g^2}+\frac {c (f+g x)^2}{g^2}}}-\frac {\sqrt {a g^2+c f^2} \left (9 a e^2 g^2+c \left (-15 d^2 g^2-10 d e f g+2 e^2 f^2\right )\right ) \int \frac {1-\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{\sqrt {c}}\right )}{3 g^3}}{5 c}+\frac {2 e \sqrt {a+c x^2} (d+e x) \sqrt {f+g x}}{5 c}\) |
\(\Big \downarrow \) 1509 |
\(\displaystyle \frac {\frac {2 e \sqrt {a+c x^2} \sqrt {f+g x} (7 d g+e f)}{3 g}-\frac {2 \left (\frac {\left (a g^2+c f^2\right )^{3/4} \left (\frac {\sqrt {c} (f+g x)}{\sqrt {a g^2+c f^2}}+1\right ) \sqrt {\frac {a+\frac {c f^2}{g^2}-\frac {2 c f (f+g x)}{g^2}+\frac {c (f+g x)^2}{g^2}}{\left (a+\frac {c f^2}{g^2}\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {a g^2+c f^2}}+1\right )^2}} \left (-2 \sqrt {c} e \sqrt {a g^2+c f^2} (e f-5 d g)+9 a e^2 g^2+c \left (-15 d^2 g^2-10 d e f g+2 e^2 f^2\right )\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt [4]{c f^2+a g^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} f}{\sqrt {c f^2+a g^2}}+1\right )\right )}{2 c^{3/4} \sqrt {a+\frac {c f^2}{g^2}-\frac {2 c f (f+g x)}{g^2}+\frac {c (f+g x)^2}{g^2}}}-\frac {\sqrt {a g^2+c f^2} \left (9 a e^2 g^2+c \left (-15 d^2 g^2-10 d e f g+2 e^2 f^2\right )\right ) \left (\frac {\sqrt [4]{a g^2+c f^2} \left (\frac {\sqrt {c} (f+g x)}{\sqrt {a g^2+c f^2}}+1\right ) \sqrt {\frac {a+\frac {c f^2}{g^2}-\frac {2 c f (f+g x)}{g^2}+\frac {c (f+g x)^2}{g^2}}{\left (a+\frac {c f^2}{g^2}\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {a g^2+c f^2}}+1\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt [4]{c f^2+a g^2}}\right )|\frac {1}{2} \left (\frac {\sqrt {c} f}{\sqrt {c f^2+a g^2}}+1\right )\right )}{\sqrt [4]{c} \sqrt {a+\frac {c f^2}{g^2}-\frac {2 c f (f+g x)}{g^2}+\frac {c (f+g x)^2}{g^2}}}-\frac {\sqrt {f+g x} \sqrt {a+\frac {c f^2}{g^2}-\frac {2 c f (f+g x)}{g^2}+\frac {c (f+g x)^2}{g^2}}}{\left (a+\frac {c f^2}{g^2}\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {a g^2+c f^2}}+1\right )}\right )}{\sqrt {c}}\right )}{3 g^3}}{5 c}+\frac {2 e \sqrt {a+c x^2} (d+e x) \sqrt {f+g x}}{5 c}\) |
(2*e*(d + e*x)*Sqrt[f + g*x]*Sqrt[a + c*x^2])/(5*c) + ((2*e*(e*f + 7*d*g)* Sqrt[f + g*x]*Sqrt[a + c*x^2])/(3*g) - (2*(-((Sqrt[c*f^2 + a*g^2]*(9*a*e^2 *g^2 + c*(2*e^2*f^2 - 10*d*e*f*g - 15*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 + (Sqr t[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)*(9*a*e^2*g^2 - 2*Sqrt[c]*e*(e*f - 5*d*g)*Sqrt[c*f^2 + a*g^2] + c*(2*e^2*f^2 - 10*d*e*f*g - 15*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)
3.7.37.3.1 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]))
Leaf count of result is larger than twice the leaf count of optimal. \(699\) vs. \(2(338)=676\).
Time = 2.09 (sec) , antiderivative size = 700, normalized size of antiderivative = 1.71
method | result | size |
elliptic | \(\frac {\sqrt {\left (g x +f \right ) \left (c \,x^{2}+a \right )}\, \left (\frac {2 e^{2} x \sqrt {c g \,x^{3}+c f \,x^{2}+a g x +f a}}{5 c}+\frac {2 \left (2 d e g +\frac {1}{5} e^{2} f \right ) \sqrt {c g \,x^{3}+c f \,x^{2}+a g x +f a}}{3 c g}+\frac {2 \left (d^{2} f -\frac {2 e^{2} f a}{5 c}-\frac {\left (2 d e g +\frac {1}{5} e^{2} f \right ) a}{3 c}\right ) \left (\frac {f}{g}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}}\, F\left (\sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\right )}{\sqrt {c g \,x^{3}+c f \,x^{2}+a g x +f a}}+\frac {2 \left (d^{2} g +2 d e f -\frac {3 e^{2} a g}{5 c}-\frac {2 \left (2 d e g +\frac {1}{5} e^{2} f \right ) f}{3 g}\right ) \left (\frac {f}{g}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}}\, \left (\left (-\frac {f}{g}-\frac {\sqrt {-a c}}{c}\right ) E\left (\sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\right )+\frac {\sqrt {-a c}\, F\left (\sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\right )}{c}\right )}{\sqrt {c g \,x^{3}+c f \,x^{2}+a g x +f a}}\right )}{\sqrt {g x +f}\, \sqrt {c \,x^{2}+a}}\) | \(700\) |
risch | \(\text {Expression too large to display}\) | \(1073\) |
default | \(\text {Expression too large to display}\) | \(2470\) |
((g*x+f)*(c*x^2+a))^(1/2)/(g*x+f)^(1/2)/(c*x^2+a)^(1/2)*(2/5*e^2/c*x*(c*g* x^3+c*f*x^2+a*g*x+a*f)^(1/2)+2/3*(2*d*e*g+1/5*e^2*f)/c/g*(c*g*x^3+c*f*x^2+ a*g*x+a*f)^(1/2)+2*(d^2*f-2/5*e^2/c*f*a-1/3*(2*d*e*g+1/5*e^2*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*(d^2*g+2* d*e*f-3/5*e^2/c*a*g-2/3*(2*d*e*g+1/5*e^2*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+(-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 = 0.10 (sec) , antiderivative size = 300, normalized size of antiderivative = 0.73 \[ \int \frac {(d+e x)^2 \sqrt {f+g x}}{\sqrt {a+c x^2}} \, dx=\frac {2 \, {\left (2 \, {\left (c e^{2} f^{3} - 5 \, c d e f^{2} g - 15 \, a d e g^{3} + 3 \, {\left (5 \, c d^{2} - 2 \, a e^{2}\right )} f g^{2}\right )} \sqrt {c g} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c f^{2} - 3 \, a g^{2}\right )}}{3 \, c g^{2}}, -\frac {8 \, {\left (c f^{3} + 9 \, a f g^{2}\right )}}{27 \, c g^{3}}, \frac {3 \, g x + f}{3 \, g}\right ) + 3 \, {\left (2 \, c e^{2} f^{2} g - 10 \, c d e f g^{2} - 3 \, {\left (5 \, c d^{2} - 3 \, a e^{2}\right )} g^{3}\right )} \sqrt {c g} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c f^{2} - 3 \, a g^{2}\right )}}{3 \, c g^{2}}, -\frac {8 \, {\left (c f^{3} + 9 \, a f g^{2}\right )}}{27 \, c g^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c f^{2} - 3 \, a g^{2}\right )}}{3 \, c g^{2}}, -\frac {8 \, {\left (c f^{3} + 9 \, a f g^{2}\right )}}{27 \, c g^{3}}, \frac {3 \, g x + f}{3 \, g}\right )\right ) + 3 \, {\left (3 \, c e^{2} g^{3} x + c e^{2} f g^{2} + 10 \, c d e g^{3}\right )} \sqrt {c x^{2} + a} \sqrt {g x + f}\right )}}{45 \, c^{2} g^{3}} \]
2/45*(2*(c*e^2*f^3 - 5*c*d*e*f^2*g - 15*a*d*e*g^3 + 3*(5*c*d^2 - 2*a*e^2)* f*g^2)*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*(2*c*e^2*f^2*g - 10*c* d*e*f*g^2 - 3*(5*c*d^2 - 3*a*e^2)*g^3)*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), weierstrassPInver se(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^2*g^3*x + c*e^2*f*g^2 + 10*c*d*e*g^3)*sqrt(c*x^2 + a)*sqrt(g*x + f))/(c^2*g^3)
\[ \int \frac {(d+e x)^2 \sqrt {f+g x}}{\sqrt {a+c x^2}} \, dx=\int \frac {\left (d + e x\right )^{2} \sqrt {f + g x}}{\sqrt {a + c x^{2}}}\, dx \]
\[ \int \frac {(d+e x)^2 \sqrt {f+g x}}{\sqrt {a+c x^2}} \, dx=\int { \frac {{\left (e x + d\right )}^{2} \sqrt {g x + f}}{\sqrt {c x^{2} + a}} \,d x } \]
\[ \int \frac {(d+e x)^2 \sqrt {f+g x}}{\sqrt {a+c x^2}} \, dx=\int { \frac {{\left (e x + d\right )}^{2} \sqrt {g x + f}}{\sqrt {c x^{2} + a}} \,d x } \]
Timed out. \[ \int \frac {(d+e x)^2 \sqrt {f+g x}}{\sqrt {a+c x^2}} \, dx=\int \frac {\sqrt {f+g\,x}\,{\left (d+e\,x\right )}^2}{\sqrt {c\,x^2+a}} \,d x \]