Integrand size = 26, antiderivative size = 520 \[ \int \frac {(d+e x) \sqrt {a+c x^2}}{\sqrt {f+g x}} \, dx=-\frac {2 \sqrt {f+g x} (4 e f-5 d g-3 e g x) \sqrt {a+c x^2}}{15 g^2}-\frac {4 \left (\sqrt {c} f-\sqrt {-a} g\right ) \sqrt {\sqrt {c} f+\sqrt {-a} g} \left (3 a e g^2+c f (4 e f-5 d g)\right ) \sqrt {1-\frac {\sqrt {c} (f+g x)}{\sqrt {c} f-\sqrt {-a} g}} \sqrt {1-\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} E\left (\arcsin \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt {\sqrt {c} f+\sqrt {-a} g}}\right )|\frac {\sqrt {c} f+\sqrt {-a} g}{\sqrt {c} f-\sqrt {-a} g}\right )}{15 c^{3/4} g^4 \sqrt {a+c x^2}}-\frac {4 \sqrt {\sqrt {c} f+\sqrt {-a} g} \left (3 \sqrt {-a} a e g^2+a \sqrt {c} g (e f-5 d g)+\sqrt {-a} c f (4 e f-5 d g)\right ) \sqrt {1-\frac {\sqrt {c} (f+g x)}{\sqrt {c} f-\sqrt {-a} g}} \sqrt {1-\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt {\sqrt {c} f+\sqrt {-a} g}}\right ),\frac {\sqrt {c} f+\sqrt {-a} g}{\sqrt {c} f-\sqrt {-a} g}\right )}{15 c^{3/4} g^3 \sqrt {a+c x^2}} \] Output:
-2/15*(g*x+f)^(1/2)*(-3*e*g*x-5*d*g+4*e*f)*(c*x^2+a)^(1/2)/g^2-4/15*(c^(1/ 2)*f-(-a)^(1/2)*g)*(c^(1/2)*f+(-a)^(1/2)*g)^(1/2)*(3*a*e*g^2+c*f*(-5*d*g+4 *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)*EllipticE(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^4/(c*x^2+a)^(1/2)-4/15*(c^(1/2)*f+(-a)^(1/2)*g)^(1/2)*( 3*(-a)^(1/2)*a*e*g^2+a*c^(1/2)*g*(-5*d*g+e*f)+(-a)^(1/2)*c*f*(-5*d*g+4*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^(3/4)/g^3/(c*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 26.25 (sec) , antiderivative size = 545, normalized size of antiderivative = 1.05 \[ \int \frac {(d+e x) \sqrt {a+c x^2}}{\sqrt {f+g x}} \, dx=\frac {\sqrt {f+g x} \left (\frac {2 (-4 e f+5 d g+3 e g x) \left (a+c x^2\right )}{g^2}+\frac {4 \left (g^2 \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}} \left (3 a e g^2+c f (4 e f-5 d g)\right ) \left (a+c x^2\right )-\sqrt {c} \left (i \sqrt {c} f-\sqrt {a} g\right ) \left (3 a e g^2+c f (4 e f-5 d g)\right ) \sqrt {\frac {g \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{f+g x}} \sqrt {-\frac {\frac {i \sqrt {a} g}{\sqrt {c}}-g x}{f+g x}} (f+g x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}{\sqrt {f+g x}}\right )|\frac {\sqrt {c} f-i \sqrt {a} g}{\sqrt {c} f+i \sqrt {a} g}\right )+\sqrt {a} \sqrt {c} g \left (\sqrt {c} f+i \sqrt {a} g\right ) \left (3 i \sqrt {a} e g+\sqrt {c} (-4 e f+5 d g)\right ) \sqrt {\frac {g \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{f+g x}} \sqrt {-\frac {\frac {i \sqrt {a} g}{\sqrt {c}}-g x}{f+g x}} (f+g x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}{\sqrt {f+g x}}\right ),\frac {\sqrt {c} f-i \sqrt {a} g}{\sqrt {c} f+i \sqrt {a} g}\right )\right )}{c g^4 \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}} (f+g x)}\right )}{15 \sqrt {a+c x^2}} \] Input:
Integrate[((d + e*x)*Sqrt[a + c*x^2])/Sqrt[f + g*x],x]
Output:
(Sqrt[f + g*x]*((2*(-4*e*f + 5*d*g + 3*e*g*x)*(a + c*x^2))/g^2 + (4*(g^2*S qrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(3*a*e*g^2 + c*f*(4*e*f - 5*d*g))*(a + c*x ^2) - Sqrt[c]*(I*Sqrt[c]*f - Sqrt[a]*g)*(3*a*e*g^2 + c*f*(4*e*f - 5*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))]*(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)/(Sqrt[c]*f + I*S qrt[a]*g)] + Sqrt[a]*Sqrt[c]*g*(Sqrt[c]*f + I*Sqrt[a]*g)*((3*I)*Sqrt[a]*e* g + Sqrt[c]*(-4*e*f + 5*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))]*(f + g*x)^(3/2)*Ellipti cF[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*g^4*Sqrt[-f - (I*Sqrt[a]*g)/ Sqrt[c]]*(f + g*x))))/(15*Sqrt[a + c*x^2])
Time = 0.92 (sec) , antiderivative size = 698, normalized size of antiderivative = 1.34, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {682, 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.
| \(\displaystyle \int \frac {\sqrt {a+c x^2} (d+e x)}{\sqrt {f+g x}} \, dx\) |
| \(\Big \downarrow \) 682 |
| \(\displaystyle \frac {4 \int -\frac {c \left (a g (e f-5 d g)-\left (3 a e g^2+c f (4 e f-5 d g)\right ) x\right )}{2 \sqrt {f+g x} \sqrt {c x^2+a}}dx}{15 c g^2}-\frac {2 \sqrt {a+c x^2} \sqrt {f+g x} (-5 d g+4 e f-3 e g x)}{15 g^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 \int \frac {a g (e f-5 d g)-\left (3 a e g^2+c f (4 e f-5 d g)\right ) x}{\sqrt {f+g x} \sqrt {c x^2+a}}dx}{15 g^2}-\frac {2 \sqrt {a+c x^2} \sqrt {f+g x} (-5 d g+4 e f-3 e g x)}{15 g^2}\) |
| \(\Big \downarrow \) 599 |
| \(\displaystyle \frac {4 \int -\frac {(4 e f-5 d g) \left (c f^2+a g^2\right )-\left (3 a e g^2+c f (4 e f-5 d g)\right ) (f+g x)}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{15 g^4}-\frac {2 \sqrt {a+c x^2} \sqrt {f+g x} (-5 d g+4 e f-3 e g x)}{15 g^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {4 \int \frac {(4 e f-5 d g) \left (c f^2+a g^2\right )-\left (3 a e g^2+c f (4 e f-5 d g)\right ) (f+g x)}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{15 g^4}-\frac {2 \sqrt {a+c x^2} \sqrt {f+g x} (-5 d g+4 e f-3 e g x)}{15 g^2}\) |
| \(\Big \downarrow \) 1511 |
| \(\displaystyle \frac {4 \left (\frac {\sqrt {a g^2+c f^2} \left (-\sqrt {c} \sqrt {a g^2+c f^2} (4 e f-5 d g)+3 a e g^2+c f (4 e f-5 d g)\right ) \int \frac {1}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{\sqrt {c}}-\frac {\sqrt {a g^2+c f^2} \left (3 a e g^2+c f (4 e f-5 d g)\right ) \int \frac {1-\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{\sqrt {c}}\right )}{15 g^4}-\frac {2 \sqrt {a+c x^2} \sqrt {f+g x} (-5 d g+4 e f-3 e g x)}{15 g^2}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle \frac {4 \left (\frac {\left (a g^2+c f^2\right )^{3/4} \left (\frac {\sqrt {c} (f+g x)}{\sqrt {a g^2+c f^2}}+1\right ) \sqrt {\frac {a+\frac {c f^2}{g^2}-\frac {2 c f (f+g x)}{g^2}+\frac {c (f+g x)^2}{g^2}}{\left (a+\frac {c f^2}{g^2}\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {a g^2+c f^2}}+1\right )^2}} \left (-\sqrt {c} \sqrt {a g^2+c f^2} (4 e f-5 d g)+3 a e g^2+c f (4 e f-5 d g)\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt [4]{c f^2+a g^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} f}{\sqrt {c f^2+a g^2}}+1\right )\right )}{2 c^{3/4} \sqrt {a+\frac {c f^2}{g^2}-\frac {2 c f (f+g x)}{g^2}+\frac {c (f+g x)^2}{g^2}}}-\frac {\sqrt {a g^2+c f^2} \left (3 a e g^2+c f (4 e f-5 d g)\right ) \int \frac {1-\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{\sqrt {c}}\right )}{15 g^4}-\frac {2 \sqrt {a+c x^2} \sqrt {f+g x} (-5 d g+4 e f-3 e g x)}{15 g^2}\) |
| \(\Big \downarrow \) 1509 |
| \(\displaystyle \frac {4 \left (\frac {\left (a g^2+c f^2\right )^{3/4} \left (\frac {\sqrt {c} (f+g x)}{\sqrt {a g^2+c f^2}}+1\right ) \sqrt {\frac {a+\frac {c f^2}{g^2}-\frac {2 c f (f+g x)}{g^2}+\frac {c (f+g x)^2}{g^2}}{\left (a+\frac {c f^2}{g^2}\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {a g^2+c f^2}}+1\right )^2}} \left (-\sqrt {c} \sqrt {a g^2+c f^2} (4 e f-5 d g)+3 a e g^2+c f (4 e f-5 d g)\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt [4]{c f^2+a g^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} f}{\sqrt {c f^2+a g^2}}+1\right )\right )}{2 c^{3/4} \sqrt {a+\frac {c f^2}{g^2}-\frac {2 c f (f+g x)}{g^2}+\frac {c (f+g x)^2}{g^2}}}-\frac {\sqrt {a g^2+c f^2} \left (3 a e g^2+c f (4 e f-5 d g)\right ) \left (\frac {\sqrt [4]{a g^2+c f^2} \left (\frac {\sqrt {c} (f+g x)}{\sqrt {a g^2+c f^2}}+1\right ) \sqrt {\frac {a+\frac {c f^2}{g^2}-\frac {2 c f (f+g x)}{g^2}+\frac {c (f+g x)^2}{g^2}}{\left (a+\frac {c f^2}{g^2}\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {a g^2+c f^2}}+1\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt [4]{c f^2+a g^2}}\right )|\frac {1}{2} \left (\frac {\sqrt {c} f}{\sqrt {c f^2+a g^2}}+1\right )\right )}{\sqrt [4]{c} \sqrt {a+\frac {c f^2}{g^2}-\frac {2 c f (f+g x)}{g^2}+\frac {c (f+g x)^2}{g^2}}}-\frac {\sqrt {f+g x} \sqrt {a+\frac {c f^2}{g^2}-\frac {2 c f (f+g x)}{g^2}+\frac {c (f+g x)^2}{g^2}}}{\left (a+\frac {c f^2}{g^2}\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {a g^2+c f^2}}+1\right )}\right )}{\sqrt {c}}\right )}{15 g^4}-\frac {2 \sqrt {a+c x^2} \sqrt {f+g x} (-5 d g+4 e f-3 e g x)}{15 g^2}\) |
Input:
Int[((d + e*x)*Sqrt[a + c*x^2])/Sqrt[f + g*x],x]
Output:
(-2*Sqrt[f + g*x]*(4*e*f - 5*d*g - 3*e*g*x)*Sqrt[a + c*x^2])/(15*g^2) + (4 *(-((Sqrt[c*f^2 + a*g^2]*(3*a*e*g^2 + c*f*(4*e*f - 5*d*g))*(-((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)^(3/4)*(3*a*e*g^2 + c*f*(4*e *f - 5*d*g) - Sqrt[c]*(4*e*f - 5*d*g)*Sqrt[c*f^2 + a*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])))/(15*g^4)
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_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*c*d*(2*p + 1) + g*c*e*(m + 2*p + 1)*x)*((a + c*x^2)^p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), x] + Simp[2*(p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))) Int[(d + e*x) ^m*(a + c*x^2)^(p - 1)*Simp[f*a*c*e^2*(m + 2*p + 2) + a*c*d*e*g*m - (c^2*f* d*e*(m + 2*p + 2) - g*(c^2*d^2*(2*p + 1) + a*c*e^2*(m + 2*p + 1)))*x, x], x ], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && GtQ[p, 0] && (IntegerQ[p] || ! RationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) && !ILtQ[m + 2*p, 0] && (Intege rQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
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]
Time = 2.25 (sec) , antiderivative size = 678, normalized size of antiderivative = 1.30
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (g x +f \right ) \left (c \,x^{2}+a \right )}\, \left (\frac {2 e x \sqrt {c g \,x^{3}+c f \,x^{2}+a g x +a f}}{5 g}+\frac {2 \left (c d -\frac {4 c f e}{5 g}\right ) \sqrt {c g \,x^{3}+c f \,x^{2}+a g x +a f}}{3 c g}+\frac {2 \left (a d -\frac {2 a f e}{5 g}-\frac {a \left (c d -\frac {4 c f e}{5 g}\right )}{3 c}\right ) \left (\frac {f}{g}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\right )}{\sqrt {c g \,x^{3}+c f \,x^{2}+a g x +a f}}+\frac {2 \left (\frac {2 a e}{5}-\frac {2 f \left (c d -\frac {4 c f e}{5 g}\right )}{3 g}\right ) \left (\frac {f}{g}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}}\, \left (\left (-\frac {f}{g}-\frac {\sqrt {-a c}}{c}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\right )+\frac {\sqrt {-a c}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\right )}{c}\right )}{\sqrt {c g \,x^{3}+c f \,x^{2}+a g x +a f}}\right )}{\sqrt {g x +f}\, \sqrt {c \,x^{2}+a}}\) | \(678\) |
| risch | \(\frac {2 \left (3 e g x +5 d g -4 e f \right ) \sqrt {g x +f}\, \sqrt {c \,x^{2}+a}}{15 g^{2}}+\frac {2 \left (\frac {2 \left (3 a e \,g^{2}-5 c d f g +4 c e \,f^{2}\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 {10 a d \,g^{2} \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 a e f 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 )}}{15 g^{2} \sqrt {g x +f}\, \sqrt {c \,x^{2}+a}}\) | \(834\) |
| default | \(\text {Expression too large to display}\) | \(1828\) |
Input:
int((e*x+d)*(c*x^2+a)^(1/2)/(g*x+f)^(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)*(2/5*e/g*x*(c*g*x^ 3+c*f*x^2+a*g*x+a*f)^(1/2)+2/3*(c*d-4/5*c*f/g*e)/c/g*(c*g*x^3+c*f*x^2+a*g* x+a*f)^(1/2)+2*(a*d-2/5*a*f/g*e-1/3*a/c*(c*d-4/5*c*f/g*e))*(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*(2/5*a*e-2/3*f/g*(c *d-4/5*c*f/g*e))*(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 = 0.10 (sec) , antiderivative size = 268, normalized size of antiderivative = 0.52 \[ \int \frac {(d+e x) \sqrt {a+c x^2}}{\sqrt {f+g x}} \, dx=-\frac {2 \, {\left (2 \, {\left (4 \, c e f^{3} - 5 \, c d f^{2} g + 6 \, a e f g^{2} - 15 \, a d g^{3}\right )} \sqrt {c g} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c f^{2} - 3 \, a g^{2}\right )}}{3 \, c g^{2}}, -\frac {8 \, {\left (c f^{3} + 9 \, a f g^{2}\right )}}{27 \, c g^{3}}, \frac {3 \, g x + f}{3 \, g}\right ) + 6 \, {\left (4 \, c e f^{2} g - 5 \, c d f g^{2} + 3 \, a e g^{3}\right )} \sqrt {c g} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c f^{2} - 3 \, a g^{2}\right )}}{3 \, c g^{2}}, -\frac {8 \, {\left (c f^{3} + 9 \, a f g^{2}\right )}}{27 \, c g^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c f^{2} - 3 \, a g^{2}\right )}}{3 \, c g^{2}}, -\frac {8 \, {\left (c f^{3} + 9 \, a f g^{2}\right )}}{27 \, c g^{3}}, \frac {3 \, g x + f}{3 \, g}\right )\right ) - 3 \, {\left (3 \, c e g^{3} x - 4 \, c e f g^{2} + 5 \, c d g^{3}\right )} \sqrt {c x^{2} + a} \sqrt {g x + f}\right )}}{45 \, c g^{4}} \] Input:
integrate((e*x+d)*(c*x^2+a)^(1/2)/(g*x+f)^(1/2),x, algorithm="fricas")
Output:
-2/45*(2*(4*c*e*f^3 - 5*c*d*f^2*g + 6*a*e*f*g^2 - 15*a*d*g^3)*sqrt(c*g)*we ierstrassPInverse(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*(4*c*e*f^2*g - 5*c*d*f*g^2 + 3*a*e*g^3)*s qrt(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*(3*c*e*g^3*x - 4*c*e* f*g^2 + 5*c*d*g^3)*sqrt(c*x^2 + a)*sqrt(g*x + f))/(c*g^4)
\[ \int \frac {(d+e x) \sqrt {a+c x^2}}{\sqrt {f+g x}} \, dx=\int \frac {\sqrt {a + c x^{2}} \left (d + e x\right )}{\sqrt {f + g x}}\, dx \] Input:
integrate((e*x+d)*(c*x**2+a)**(1/2)/(g*x+f)**(1/2),x)
Output:
Integral(sqrt(a + c*x**2)*(d + e*x)/sqrt(f + g*x), x)
\[ \int \frac {(d+e x) \sqrt {a+c x^2}}{\sqrt {f+g x}} \, dx=\int { \frac {\sqrt {c x^{2} + a} {\left (e x + d\right )}}{\sqrt {g x + f}} \,d x } \] Input:
integrate((e*x+d)*(c*x^2+a)^(1/2)/(g*x+f)^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(c*x^2 + a)*(e*x + d)/sqrt(g*x + f), x)
\[ \int \frac {(d+e x) \sqrt {a+c x^2}}{\sqrt {f+g x}} \, dx=\int { \frac {\sqrt {c x^{2} + a} {\left (e x + d\right )}}{\sqrt {g x + f}} \,d x } \] Input:
integrate((e*x+d)*(c*x^2+a)^(1/2)/(g*x+f)^(1/2),x, algorithm="giac")
Output:
integrate(sqrt(c*x^2 + a)*(e*x + d)/sqrt(g*x + f), x)
Timed out. \[ \int \frac {(d+e x) \sqrt {a+c x^2}}{\sqrt {f+g x}} \, dx=\int \frac {\sqrt {c\,x^2+a}\,\left (d+e\,x\right )}{\sqrt {f+g\,x}} \,d x \] Input:
int(((a + c*x^2)^(1/2)*(d + e*x))/(f + g*x)^(1/2),x)
Output:
int(((a + c*x^2)^(1/2)*(d + e*x))/(f + g*x)^(1/2), x)
\[ \int \frac {(d+e x) \sqrt {a+c x^2}}{\sqrt {f+g x}} \, dx=\frac {2 \sqrt {g x +f}\, \sqrt {c \,x^{2}+a}\, a e g +2 \sqrt {g x +f}\, \sqrt {c \,x^{2}+a}\, c e f x -3 \left (\int \frac {\sqrt {g x +f}\, \sqrt {c \,x^{2}+a}\, x^{2}}{c g \,x^{3}+c f \,x^{2}+a g x +a f}d x \right ) a c e \,g^{2}+5 \left (\int \frac {\sqrt {g x +f}\, \sqrt {c \,x^{2}+a}\, x^{2}}{c g \,x^{3}+c f \,x^{2}+a g x +a f}d x \right ) c^{2} d f g -4 \left (\int \frac {\sqrt {g x +f}\, \sqrt {c \,x^{2}+a}\, x^{2}}{c g \,x^{3}+c f \,x^{2}+a g x +a f}d x \right ) c^{2} e \,f^{2}-\left (\int \frac {\sqrt {g x +f}\, \sqrt {c \,x^{2}+a}}{c g \,x^{3}+c f \,x^{2}+a g x +a f}d x \right ) a^{2} e \,g^{2}+5 \left (\int \frac {\sqrt {g x +f}\, \sqrt {c \,x^{2}+a}}{c g \,x^{3}+c f \,x^{2}+a g x +a f}d x \right ) a c d f g -2 \left (\int \frac {\sqrt {g x +f}\, \sqrt {c \,x^{2}+a}}{c g \,x^{3}+c f \,x^{2}+a g x +a f}d x \right ) a c e \,f^{2}}{5 c f g} \] Input:
int((e*x+d)*(c*x^2+a)^(1/2)/(g*x+f)^(1/2),x)
Output:
(2*sqrt(f + g*x)*sqrt(a + c*x**2)*a*e*g + 2*sqrt(f + g*x)*sqrt(a + c*x**2) *c*e*f*x - 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)*a*c*e*g**2 + 5*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**2*d*f*g - 4*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**2*e*f** 2 - 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**2*e*g**2 + 5*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*c*d*f*g - 2*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*c*e*f**2)/(5*c*f*g)