Integrand size = 28, antiderivative size = 489 \[ \int \frac {(d+e x)^2}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx=\frac {2 e^2 \sqrt {f+g x} \sqrt {a+c x^2}}{3 c g}+\frac {4 e \left (\sqrt {c} f-\sqrt {-a} g\right ) \sqrt {\sqrt {c} f+\sqrt {-a} g} (e f-3 d g) \sqrt {1-\frac {\sqrt {c} (f+g x)}{\sqrt {c} f-\sqrt {-a} g}} \sqrt {1-\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} E\left (\arcsin \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt {\sqrt {c} f+\sqrt {-a} g}}\right )|\frac {\sqrt {c} f+\sqrt {-a} g}{\sqrt {c} f-\sqrt {-a} g}\right )}{3 c^{3/4} g^3 \sqrt {a+c x^2}}+\frac {2 \sqrt {\sqrt {c} f+\sqrt {-a} g} \left (3 c d^2 g-a e^2 g+2 \sqrt {-a} \sqrt {c} e (e f-3 d g)\right ) \sqrt {1-\frac {\sqrt {c} (f+g x)}{\sqrt {c} f-\sqrt {-a} g}} \sqrt {1-\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt {\sqrt {c} f+\sqrt {-a} g}}\right ),\frac {\sqrt {c} f+\sqrt {-a} g}{\sqrt {c} f-\sqrt {-a} g}\right )}{3 c^{5/4} g^2 \sqrt {a+c x^2}} \] Output:
2/3*e^2*(g*x+f)^(1/2)*(c*x^2+a)^(1/2)/c/g+4/3*e*(c^(1/2)*f-(-a)^(1/2)*g)*( c^(1/2)*f+(-a)^(1/2)*g)^(1/2)*(-3*d*g+e*f)*(1-c^(1/2)*(g*x+f)/(c^(1/2)*f-( -a)^(1/2)*g))^(1/2)*(1-c^(1/2)*(g*x+f)/(c^(1/2)*f+(-a)^(1/2)*g))^(1/2)*Ell ipticE(c^(1/4)*(g*x+f)^(1/2)/(c^(1/2)*f+(-a)^(1/2)*g)^(1/2),((c^(1/2)*f+(- a)^(1/2)*g)/(c^(1/2)*f-(-a)^(1/2)*g))^(1/2))/c^(3/4)/g^3/(c*x^2+a)^(1/2)+2 /3*(c^(1/2)*f+(-a)^(1/2)*g)^(1/2)*(3*c*d^2*g-a*e^2*g+2*(-a)^(1/2)*c^(1/2)* e*(-3*d*g+e*f))*(1-c^(1/2)*(g*x+f)/(c^(1/2)*f-(-a)^(1/2)*g))^(1/2)*(1-c^(1 /2)*(g*x+f)/(c^(1/2)*f+(-a)^(1/2)*g))^(1/2)*EllipticF(c^(1/4)*(g*x+f)^(1/2 )/(c^(1/2)*f+(-a)^(1/2)*g)^(1/2),((c^(1/2)*f+(-a)^(1/2)*g)/(c^(1/2)*f-(-a) ^(1/2)*g))^(1/2))/c^(5/4)/g^2/(c*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 25.10 (sec) , antiderivative size = 473, normalized size of antiderivative = 0.97 \[ \int \frac {(d+e x)^2}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx=\frac {2 \sqrt {f+g x} \left (e^2 g^2 \left (a+c x^2\right )-\frac {2 e g^2 (e f-3 d g) \left (a+c x^2\right )}{f+g x}-2 i c e \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}} (e f-3 d g) \sqrt {\frac {g \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{f+g x}} \sqrt {-\frac {\frac {i \sqrt {a} g}{\sqrt {c}}-g x}{f+g x}} \sqrt {f+g x} E\left (i \text {arcsinh}\left (\frac {\sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}{\sqrt {f+g x}}\right )|\frac {\sqrt {c} f-i \sqrt {a} g}{\sqrt {c} f+i \sqrt {a} g}\right )+\frac {g \left (3 i c d^2 g-i a e^2 g+2 \sqrt {a} \sqrt {c} e (e f-3 d g)\right ) \sqrt {\frac {g \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{f+g x}} \sqrt {-\frac {\frac {i \sqrt {a} g}{\sqrt {c}}-g x}{f+g x}} \sqrt {f+g x} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}{\sqrt {f+g x}}\right ),\frac {\sqrt {c} f-i \sqrt {a} g}{\sqrt {c} f+i \sqrt {a} g}\right )}{\sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}\right )}{3 c g^3 \sqrt {a+c x^2}} \] Input:
Integrate[(d + e*x)^2/(Sqrt[f + g*x]*Sqrt[a + c*x^2]),x]
Output:
(2*Sqrt[f + g*x]*(e^2*g^2*(a + c*x^2) - (2*e*g^2*(e*f - 3*d*g)*(a + c*x^2) )/(f + g*x) - (2*I)*c*e*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(e*f - 3*d*g)*Sqr t[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*Sqrt[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)] + (g*((3*I)*c*d^2*g - I*a*e^2*g + 2*Sqrt[a]*Sqrt[c]*e*(e*f - 3*d*g)) *Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[ c] - g*x)/(f + g*x))]*Sqrt[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*Sq rt[a]*g)])/Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]))/(3*c*g^3*Sqrt[a + c*x^2])
Time = 1.58 (sec) , antiderivative size = 697, normalized size of antiderivative = 1.43, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {728, 25, 2004, 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.
| \(\displaystyle \int \frac {(d+e x)^2}{\sqrt {a+c x^2} \sqrt {f+g x}} \, dx\) |
| \(\Big \downarrow \) 728 |
| \(\displaystyle \frac {2 e^2 \sqrt {a+c x^2} \sqrt {f+g x}}{3 c g}-\frac {\int -\frac {-2 c e^2 (e f-3 d g) x^2-e \left (a g e^2+c d (2 e f-9 d g)\right ) x+d \left (3 c d^2-a e^2\right ) g}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+a}}dx}{3 c g}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {-2 c e^2 (e f-3 d g) x^2-e \left (a g e^2+c d (2 e f-9 d g)\right ) x+d \left (3 c d^2-a e^2\right ) g}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+a}}dx}{3 c g}+\frac {2 e^2 \sqrt {a+c x^2} \sqrt {f+g x}}{3 c g}\) |
| \(\Big \downarrow \) 2004 |
| \(\displaystyle \frac {\int \frac {\left (3 c d^2-a e^2\right ) g-2 c e (e f-3 d g) x}{\sqrt {f+g x} \sqrt {c x^2+a}}dx}{3 c g}+\frac {2 e^2 \sqrt {a+c x^2} \sqrt {f+g x}}{3 c g}\) |
| \(\Big \downarrow \) 599 |
| \(\displaystyle \frac {2 e^2 \sqrt {a+c x^2} \sqrt {f+g x}}{3 c g}-\frac {2 \int \frac {a e^2 g^2-c \left (2 e^2 f^2-6 d e g f+3 d^2 g^2\right )+2 c e (e f-3 d g) (f+g x)}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{3 c g^3}\) |
| \(\Big \downarrow \) 1511 |
| \(\displaystyle \frac {2 e^2 \sqrt {a+c x^2} \sqrt {f+g x}}{3 c g}-\frac {2 \left (\left (2 \sqrt {c} e \sqrt {a g^2+c f^2} (e f-3 d g)+a e^2 g^2-c \left (3 d^2 g^2-6 d e f g+2 e^2 f^2\right )\right ) \int \frac {1}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}-2 \sqrt {c} e \sqrt {a g^2+c f^2} (e f-3 d g) \int \frac {1-\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}\right )}{3 c g^3}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle \frac {2 e^2 \sqrt {a+c x^2} \sqrt {f+g x}}{3 c g}-\frac {2 \left (\frac {\sqrt [4]{a g^2+c f^2} \left (\frac {\sqrt {c} (f+g x)}{\sqrt {a g^2+c f^2}}+1\right ) \sqrt {\frac {a+\frac {c f^2}{g^2}-\frac {2 c f (f+g x)}{g^2}+\frac {c (f+g x)^2}{g^2}}{\left (a+\frac {c f^2}{g^2}\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {a g^2+c f^2}}+1\right )^2}} \left (2 \sqrt {c} e \sqrt {a g^2+c f^2} (e f-3 d g)+a e^2 g^2-c \left (3 d^2 g^2-6 d e f g+2 e^2 f^2\right )\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt [4]{c f^2+a g^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} f}{\sqrt {c f^2+a g^2}}+1\right )\right )}{2 \sqrt [4]{c} \sqrt {a+\frac {c f^2}{g^2}-\frac {2 c f (f+g x)}{g^2}+\frac {c (f+g x)^2}{g^2}}}-2 \sqrt {c} e \sqrt {a g^2+c f^2} (e f-3 d g) \int \frac {1-\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}\right )}{3 c g^3}\) |
| \(\Big \downarrow \) 1509 |
| \(\displaystyle \frac {2 e^2 \sqrt {a+c x^2} \sqrt {f+g x}}{3 c g}-\frac {2 \left (\frac {\sqrt [4]{a g^2+c f^2} \left (\frac {\sqrt {c} (f+g x)}{\sqrt {a g^2+c f^2}}+1\right ) \sqrt {\frac {a+\frac {c f^2}{g^2}-\frac {2 c f (f+g x)}{g^2}+\frac {c (f+g x)^2}{g^2}}{\left (a+\frac {c f^2}{g^2}\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {a g^2+c f^2}}+1\right )^2}} \left (2 \sqrt {c} e \sqrt {a g^2+c f^2} (e f-3 d g)+a e^2 g^2-c \left (3 d^2 g^2-6 d e f g+2 e^2 f^2\right )\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt [4]{c f^2+a g^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} f}{\sqrt {c f^2+a g^2}}+1\right )\right )}{2 \sqrt [4]{c} \sqrt {a+\frac {c f^2}{g^2}-\frac {2 c f (f+g x)}{g^2}+\frac {c (f+g x)^2}{g^2}}}-2 \sqrt {c} e \sqrt {a g^2+c f^2} (e f-3 d g) \left (\frac {\sqrt [4]{a g^2+c f^2} \left (\frac {\sqrt {c} (f+g x)}{\sqrt {a g^2+c f^2}}+1\right ) \sqrt {\frac {a+\frac {c f^2}{g^2}-\frac {2 c f (f+g x)}{g^2}+\frac {c (f+g x)^2}{g^2}}{\left (a+\frac {c f^2}{g^2}\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {a g^2+c f^2}}+1\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt [4]{c f^2+a g^2}}\right )|\frac {1}{2} \left (\frac {\sqrt {c} f}{\sqrt {c f^2+a g^2}}+1\right )\right )}{\sqrt [4]{c} \sqrt {a+\frac {c f^2}{g^2}-\frac {2 c f (f+g x)}{g^2}+\frac {c (f+g x)^2}{g^2}}}-\frac {\sqrt {f+g x} \sqrt {a+\frac {c f^2}{g^2}-\frac {2 c f (f+g x)}{g^2}+\frac {c (f+g x)^2}{g^2}}}{\left (a+\frac {c f^2}{g^2}\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {a g^2+c f^2}}+1\right )}\right )\right )}{3 c g^3}\) |
Input:
Int[(d + e*x)^2/(Sqrt[f + g*x]*Sqrt[a + c*x^2]),x]
Output:
(2*e^2*Sqrt[f + g*x]*Sqrt[a + c*x^2])/(3*c*g) - (2*(-2*Sqrt[c]*e*(e*f - 3* d*g)*Sqrt[c*f^2 + a*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])) + ((c*f^2 + a*g^ 2)^(1/4)*(a*e^2*g^2 + 2*Sqrt[c]*e*(e*f - 3*d*g)*Sqrt[c*f^2 + a*g^2] - c*(2 *e^2*f^2 - 6*d*e*f*g + 3*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)]*Ell ipticF[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^(1/4)*Sqrt[a + (c*f^2)/g^2 - (2*c*f*( f + g*x))/g^2 + (c*(f + g*x)^2)/g^2])))/(3*c*g^3)
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[(u_)*((d_) + (e_.)*(x_))^(q_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.) , x_Symbol] :> Int[u*(d + e*x)^(p + q)*(a/d + (c/e)*x)^p, x] /; FreeQ[{a, b , c, d, e, q}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p]
Time = 4.73 (sec) , antiderivative size = 614, normalized size of antiderivative = 1.26
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (g x +f \right ) \left (c \,x^{2}+a \right )}\, \left (\frac {2 e^{2} \sqrt {c g \,x^{3}+c f \,x^{2}+a g x +a f}}{3 c g}+\frac {2 \left (d^{2}-\frac {a \,e^{2}}{3 c}\right ) \left (\frac {f}{g}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\right )}{\sqrt {c g \,x^{3}+c f \,x^{2}+a g x +a f}}+\frac {2 \left (2 d e -\frac {2 e^{2} f}{3 g}\right ) \left (\frac {f}{g}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}}\, \left (\left (-\frac {f}{g}-\frac {\sqrt {-a c}}{c}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\right )+\frac {\sqrt {-a c}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\right )}{c}\right )}{\sqrt {c g \,x^{3}+c f \,x^{2}+a g x +a f}}\right )}{\sqrt {g x +f}\, \sqrt {c \,x^{2}+a}}\) | \(614\) |
| risch | \(\frac {2 e^{2} \sqrt {g x +f}\, \sqrt {c \,x^{2}+a}}{3 c g}-\frac {\left (-\frac {4 c e \left (3 d g -e f \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 ) \operatorname {EllipticE}\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}\, \operatorname {EllipticF}\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 +a f}}+\frac {2 a \,e^{2} g \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}}}\, \operatorname {EllipticF}\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 +a f}}-\frac {6 c \,d^{2} g \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}}}\, \operatorname {EllipticF}\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 +a f}}\right ) \sqrt {\left (g x +f \right ) \left (c \,x^{2}+a \right )}}{3 c g \sqrt {g x +f}\, \sqrt {c \,x^{2}+a}}\) | \(820\) |
| default | \(\text {Expression too large to display}\) | \(1769\) |
Input:
int((e*x+d)^2/(g*x+f)^(1/2)/(c*x^2+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
((g*x+f)*(c*x^2+a))^(1/2)/(g*x+f)^(1/2)/(c*x^2+a)^(1/2)*(2/3*e^2/c/g*(c*g* x^3+c*f*x^2+a*g*x+a*f)^(1/2)+2*(d^2-1/3*a*e^2/c)*(f/g-(-a*c)^(1/2)/c)*((x+ f/g)/(f/g-(-a*c)^(1/2)/c))^(1/2)*((x-(-a*c)^(1/2)/c)/(-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*(2*d*e-2/3*e^2*f/g)*(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))))
Time = 0.12 (sec) , antiderivative size = 246, normalized size of antiderivative = 0.50 \[ \int \frac {(d+e x)^2}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx=\frac {2 \, {\left (3 \, \sqrt {c x^{2} + a} \sqrt {g x + f} c e^{2} g^{2} + {\left (2 \, c e^{2} f^{2} - 6 \, c d e f g + 3 \, {\left (3 \, c d^{2} - a e^{2}\right )} g^{2}\right )} \sqrt {c g} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c f^{2} - 3 \, a g^{2}\right )}}{3 \, c g^{2}}, -\frac {8 \, {\left (c f^{3} + 9 \, a f g^{2}\right )}}{27 \, c g^{3}}, \frac {3 \, g x + f}{3 \, g}\right ) + 6 \, {\left (c e^{2} f g - 3 \, c d e g^{2}\right )} \sqrt {c g} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c f^{2} - 3 \, a g^{2}\right )}}{3 \, c g^{2}}, -\frac {8 \, {\left (c f^{3} + 9 \, a f g^{2}\right )}}{27 \, c g^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c f^{2} - 3 \, a g^{2}\right )}}{3 \, c g^{2}}, -\frac {8 \, {\left (c f^{3} + 9 \, a f g^{2}\right )}}{27 \, c g^{3}}, \frac {3 \, g x + f}{3 \, g}\right )\right )\right )}}{9 \, c^{2} g^{3}} \] Input:
integrate((e*x+d)^2/(g*x+f)^(1/2)/(c*x^2+a)^(1/2),x, algorithm="fricas")
Output:
2/9*(3*sqrt(c*x^2 + a)*sqrt(g*x + f)*c*e^2*g^2 + (2*c*e^2*f^2 - 6*c*d*e*f* g + 3*(3*c*d^2 - a*e^2)*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) + 6* (c*e^2*f*g - 3*c*d*e*g^2)*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) ))/(c^2*g^3)
\[ \int \frac {(d+e x)^2}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx=\int \frac {\left (d + e x\right )^{2}}{\sqrt {a + c x^{2}} \sqrt {f + g x}}\, dx \] Input:
integrate((e*x+d)**2/(g*x+f)**(1/2)/(c*x**2+a)**(1/2),x)
Output:
Integral((d + e*x)**2/(sqrt(a + c*x**2)*sqrt(f + g*x)), x)
\[ \int \frac {(d+e x)^2}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx=\int { \frac {{\left (e x + d\right )}^{2}}{\sqrt {c x^{2} + a} \sqrt {g x + f}} \,d x } \] Input:
integrate((e*x+d)^2/(g*x+f)^(1/2)/(c*x^2+a)^(1/2),x, algorithm="maxima")
Output:
integrate((e*x + d)^2/(sqrt(c*x^2 + a)*sqrt(g*x + f)), x)
\[ \int \frac {(d+e x)^2}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx=\int { \frac {{\left (e x + d\right )}^{2}}{\sqrt {c x^{2} + a} \sqrt {g x + f}} \,d x } \] Input:
integrate((e*x+d)^2/(g*x+f)^(1/2)/(c*x^2+a)^(1/2),x, algorithm="giac")
Output:
integrate((e*x + d)^2/(sqrt(c*x^2 + a)*sqrt(g*x + f)), x)
Timed out. \[ \int \frac {(d+e x)^2}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx=\int \frac {{\left (d+e\,x\right )}^2}{\sqrt {f+g\,x}\,\sqrt {c\,x^2+a}} \,d x \] Input:
int((d + e*x)^2/((f + g*x)^(1/2)*(a + c*x^2)^(1/2)),x)
Output:
int((d + e*x)^2/((f + g*x)^(1/2)*(a + c*x^2)^(1/2)), x)
\[ \int \frac {(d+e x)^2}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx=\frac {2 \sqrt {g x +f}\, \sqrt {c \,x^{2}+a}\, d e -3 \left (\int \frac {\sqrt {g x +f}\, \sqrt {c \,x^{2}+a}\, x^{2}}{c g \,x^{3}+c f \,x^{2}+a g x +a f}d x \right ) c d e g +\left (\int \frac {\sqrt {g x +f}\, \sqrt {c \,x^{2}+a}\, x^{2}}{c g \,x^{3}+c f \,x^{2}+a g x +a f}d x \right ) c \,e^{2} f -\left (\int \frac {\sqrt {g x +f}\, \sqrt {c \,x^{2}+a}}{c g \,x^{3}+c f \,x^{2}+a g x +a f}d x \right ) a d e g +\left (\int \frac {\sqrt {g x +f}\, \sqrt {c \,x^{2}+a}}{c g \,x^{3}+c f \,x^{2}+a g x +a f}d x \right ) c \,d^{2} f}{c f} \] Input:
int((e*x+d)^2/(g*x+f)^(1/2)/(c*x^2+a)^(1/2),x)
Output:
(2*sqrt(f + g*x)*sqrt(a + c*x**2)*d*e - 3*int((sqrt(f + g*x)*sqrt(a + c*x* *2)*x**2)/(a*f + a*g*x + c*f*x**2 + c*g*x**3),x)*c*d*e*g + int((sqrt(f + g *x)*sqrt(a + c*x**2)*x**2)/(a*f + a*g*x + c*f*x**2 + c*g*x**3),x)*c*e**2*f - int((sqrt(f + g*x)*sqrt(a + c*x**2))/(a*f + a*g*x + c*f*x**2 + c*g*x**3 ),x)*a*d*e*g + int((sqrt(f + g*x)*sqrt(a + c*x**2))/(a*f + a*g*x + c*f*x** 2 + c*g*x**3),x)*c*d**2*f)/(c*f)