Integrand size = 29, antiderivative size = 419 \[ \int \frac {1}{(g x)^{3/2} (d+e x) \sqrt {a+b x+c x^2}} \, dx=-\frac {2 \sqrt {a+b x+c x^2}}{a d g \sqrt {g x}}+\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {g x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {1+\frac {b+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {b^2-4 a c}}{b+\sqrt {b^2-4 a c}}\right )}{a d g^2 \sqrt {-\frac {c x}{b+\sqrt {b^2-4 a c}}} \sqrt {a+b x+c x^2}}-\frac {4 \sqrt {2} \sqrt {b^2-4 a c} e \sqrt {-\frac {c x}{b+\sqrt {b^2-4 a c}}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticPi}\left (-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e},\arcsin \left (\frac {\sqrt {1+\frac {b+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {b^2-4 a c}}{b+\sqrt {b^2-4 a c}}\right )}{d \left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right ) g \sqrt {g x} \sqrt {a+b x+c x^2}} \] Output:
-2*(c*x^2+b*x+a)^(1/2)/a/d/g/(g*x)^(1/2)+2^(1/2)*(-4*a*c+b^2)^(1/2)*(g*x)^ (1/2)*(-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(1/2)*EllipticE(1/2*(1+(2*c*x+b)/(-4 *a*c+b^2)^(1/2))^(1/2)*2^(1/2),2^(1/2)*((-4*a*c+b^2)^(1/2)/(b+(-4*a*c+b^2) ^(1/2)))^(1/2))/a/d/g^2/(-c*x/(b+(-4*a*c+b^2)^(1/2)))^(1/2)/(c*x^2+b*x+a)^ (1/2)-4*2^(1/2)*(-4*a*c+b^2)^(1/2)*e*(-c*x/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*( -c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(1/2)*EllipticPi(1/2*(1+(2*c*x+b)/(-4*a*c+b ^2)^(1/2))^(1/2)*2^(1/2),-2*(-4*a*c+b^2)^(1/2)*e/(2*c*d-(b+(-4*a*c+b^2)^(1 /2))*e),2^(1/2)*((-4*a*c+b^2)^(1/2)/(b+(-4*a*c+b^2)^(1/2)))^(1/2))/d/(2*c* d-(b+(-4*a*c+b^2)^(1/2))*e)/g/(g*x)^(1/2)/(c*x^2+b*x+a)^(1/2)
Result contains complex when optimal does not.
Time = 25.24 (sec) , antiderivative size = 424, normalized size of antiderivative = 1.01 \[ \int \frac {1}{(g x)^{3/2} (d+e x) \sqrt {a+b x+c x^2}} \, dx=-\frac {i \sqrt {1+\frac {2 a}{\left (b+\sqrt {b^2-4 a c}\right ) x}} x^{5/2} \sqrt {\frac {4 a+2 b x-2 \sqrt {b^2-4 a c} x}{b x-\sqrt {b^2-4 a c} x}} \left (\left (-b+\sqrt {b^2-4 a c}\right ) d E\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {a}{b+\sqrt {b^2-4 a c}}}}{\sqrt {x}}\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )+\left (b d-\sqrt {b^2-4 a c} d+2 a e\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {a}{b+\sqrt {b^2-4 a c}}}}{\sqrt {x}}\right ),\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )-2 a e \operatorname {EllipticPi}\left (\frac {\left (b+\sqrt {b^2-4 a c}\right ) d}{2 a e},i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {a}{b+\sqrt {b^2-4 a c}}}}{\sqrt {x}}\right ),\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )\right )}{2 a \sqrt {\frac {a}{b+\sqrt {b^2-4 a c}}} d^2 (g x)^{3/2} \sqrt {a+x (b+c x)}} \] Input:
Integrate[1/((g*x)^(3/2)*(d + e*x)*Sqrt[a + b*x + c*x^2]),x]
Output:
((-1/2*I)*Sqrt[1 + (2*a)/((b + Sqrt[b^2 - 4*a*c])*x)]*x^(5/2)*Sqrt[(4*a + 2*b*x - 2*Sqrt[b^2 - 4*a*c]*x)/(b*x - Sqrt[b^2 - 4*a*c]*x)]*((-b + Sqrt[b^ 2 - 4*a*c])*d*EllipticE[I*ArcSinh[(Sqrt[2]*Sqrt[a/(b + Sqrt[b^2 - 4*a*c])] )/Sqrt[x]], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])] + (b*d - Sqrt [b^2 - 4*a*c]*d + 2*a*e)*EllipticF[I*ArcSinh[(Sqrt[2]*Sqrt[a/(b + Sqrt[b^2 - 4*a*c])])/Sqrt[x]], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])] - 2*a*e*EllipticPi[((b + Sqrt[b^2 - 4*a*c])*d)/(2*a*e), I*ArcSinh[(Sqrt[2]*S qrt[a/(b + Sqrt[b^2 - 4*a*c])])/Sqrt[x]], (b + Sqrt[b^2 - 4*a*c])/(b - Sqr t[b^2 - 4*a*c])]))/(a*Sqrt[a/(b + Sqrt[b^2 - 4*a*c])]*d^2*(g*x)^(3/2)*Sqrt [a + x*(b + c*x)])
Time = 1.22 (sec) , antiderivative size = 550, normalized size of antiderivative = 1.31, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {1288, 2009}
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.
| \(\displaystyle \int \frac {1}{(g x)^{3/2} (d+e x) \sqrt {a+b x+c x^2}} \, dx\) |
| \(\Big \downarrow \) 1288 |
| \(\displaystyle \int \left (\frac {1}{d (g x)^{3/2} \sqrt {a+b x+c x^2}}-\frac {e}{d g \sqrt {g x} (d+e x) \sqrt {a+b x+c x^2}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt [4]{c} \sqrt {g x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+b x+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{a^{3/4} d g^2 \sqrt {x} \sqrt {a+b x+c x^2}}-\frac {2 \sqrt [4]{c} \sqrt {g x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+b x+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{a^{3/4} d g^2 \sqrt {x} \sqrt {a+b x+c x^2}}-\frac {\sqrt {2} e \sqrt {\sqrt {b^2-4 a c}-b} \sqrt {\frac {2 c x}{b-\sqrt {b^2-4 a c}}+1} \sqrt {\frac {2 c x}{\sqrt {b^2-4 a c}+b}+1} \operatorname {EllipticPi}\left (\frac {\left (b-\sqrt {b^2-4 a c}\right ) e}{2 c d},\arcsin \left (\frac {\sqrt {2} \sqrt {c} \sqrt {g x}}{\sqrt {\sqrt {b^2-4 a c}-b} \sqrt {g}}\right ),\frac {b-\sqrt {b^2-4 a c}}{b+\sqrt {b^2-4 a c}}\right )}{\sqrt {c} d^2 g^{3/2} \sqrt {a+b x+c x^2}}+\frac {2 \sqrt {c} \sqrt {g x} \sqrt {a+b x+c x^2}}{a d g^2 \left (\sqrt {a}+\sqrt {c} x\right )}-\frac {2 \sqrt {a+b x+c x^2}}{a d g \sqrt {g x}}\) |
Input:
Int[1/((g*x)^(3/2)*(d + e*x)*Sqrt[a + b*x + c*x^2]),x]
Output:
(-2*Sqrt[a + b*x + c*x^2])/(a*d*g*Sqrt[g*x]) + (2*Sqrt[c]*Sqrt[g*x]*Sqrt[a + b*x + c*x^2])/(a*d*g^2*(Sqrt[a] + Sqrt[c]*x)) - (2*c^(1/4)*Sqrt[g*x]*(S qrt[a] + Sqrt[c]*x)*Sqrt[(a + b*x + c*x^2)/(Sqrt[a] + Sqrt[c]*x)^2]*Ellipt icE[2*ArcTan[(c^(1/4)*Sqrt[x])/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(a^ (3/4)*d*g^2*Sqrt[x]*Sqrt[a + b*x + c*x^2]) + (c^(1/4)*Sqrt[g*x]*(Sqrt[a] + Sqrt[c]*x)*Sqrt[(a + b*x + c*x^2)/(Sqrt[a] + Sqrt[c]*x)^2]*EllipticF[2*Ar cTan[(c^(1/4)*Sqrt[x])/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(a^(3/4)*d* g^2*Sqrt[x]*Sqrt[a + b*x + c*x^2]) - (Sqrt[2]*Sqrt[-b + Sqrt[b^2 - 4*a*c]] *e*Sqrt[1 + (2*c*x)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*x)/(b + Sqrt[b^ 2 - 4*a*c])]*EllipticPi[((b - Sqrt[b^2 - 4*a*c])*e)/(2*c*d), ArcSin[(Sqrt[ 2]*Sqrt[c]*Sqrt[g*x])/(Sqrt[-b + Sqrt[b^2 - 4*a*c]]*Sqrt[g])], (b - Sqrt[b ^2 - 4*a*c])/(b + Sqrt[b^2 - 4*a*c])])/(Sqrt[c]*d^2*g^(3/2)*Sqrt[a + b*x + c*x^2])
Int[((f_.) + (g_.)*(x_))^(n_)/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Int[ExpandIntegrand[1/(Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2]), (f + g*x)^(n + 1/2)/(d + e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && IntegerQ[n + 1/2]
Leaf count of result is larger than twice the leaf count of optimal. \(818\) vs. \(2(371)=742\).
Time = 3.96 (sec) , antiderivative size = 819, normalized size of antiderivative = 1.95
| method | result | size |
| risch | \(-\frac {2 \sqrt {c \,x^{2}+b x +a}}{a d g \sqrt {g x}}+\frac {\left (\frac {\left (b +\sqrt {-4 a c +b^{2}}\right ) \sqrt {2}\, \sqrt {\frac {\left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) c}{b +\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {-\frac {2 c x}{b +\sqrt {-4 a c +b^{2}}}}\, \left (\left (-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \operatorname {EllipticE}\left (\sqrt {2}\, \sqrt {\frac {\left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) c}{b +\sqrt {-4 a c +b^{2}}}}, \frac {\sqrt {-\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right )}{c \left (-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}}{2}\right )+\frac {\left (-b +\sqrt {-4 a c +b^{2}}\right ) \operatorname {EllipticF}\left (\sqrt {2}\, \sqrt {\frac {\left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) c}{b +\sqrt {-4 a c +b^{2}}}}, \frac {\sqrt {-\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right )}{c \left (-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}}{2}\right )}{2 c}\right )}{\sqrt {c g \,x^{3}+b g \,x^{2}+a g x}}-\frac {a \left (b +\sqrt {-4 a c +b^{2}}\right ) \sqrt {2}\, \sqrt {\frac {\left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) c}{b +\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {-\frac {2 c x}{b +\sqrt {-4 a c +b^{2}}}}\, \operatorname {EllipticPi}\left (\sqrt {2}\, \sqrt {\frac {\left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) c}{b +\sqrt {-4 a c +b^{2}}}}, -\frac {b +\sqrt {-4 a c +b^{2}}}{2 c \left (\frac {d}{e}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}, \frac {\sqrt {-\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right )}{c \left (-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}}{2}\right )}{c \sqrt {c g \,x^{3}+b g \,x^{2}+a g x}\, \left (\frac {d}{e}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}\right ) \sqrt {x g \left (c \,x^{2}+b x +a \right )}}{d a g \sqrt {g x}\, \sqrt {c \,x^{2}+b x +a}}\) | \(819\) |
| elliptic | \(\frac {\sqrt {x g \left (c \,x^{2}+b x +a \right )}\, \left (-\frac {2 \left (c g \,x^{2}+b g x +a g \right )}{g^{2} a d \sqrt {x \left (c g \,x^{2}+b g x +a g \right )}}+\frac {\left (b +\sqrt {-4 a c +b^{2}}\right ) \sqrt {2}\, \sqrt {\frac {\left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) c}{b +\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {-\frac {2 c x}{b +\sqrt {-4 a c +b^{2}}}}\, \left (\left (-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \operatorname {EllipticE}\left (\sqrt {2}\, \sqrt {\frac {\left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) c}{b +\sqrt {-4 a c +b^{2}}}}, \frac {\sqrt {-\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right )}{c \left (-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}}{2}\right )+\frac {\left (-b +\sqrt {-4 a c +b^{2}}\right ) \operatorname {EllipticF}\left (\sqrt {2}\, \sqrt {\frac {\left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) c}{b +\sqrt {-4 a c +b^{2}}}}, \frac {\sqrt {-\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right )}{c \left (-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}}{2}\right )}{2 c}\right )}{a d g \sqrt {c g \,x^{3}+b g \,x^{2}+a g x}}-\frac {\left (b +\sqrt {-4 a c +b^{2}}\right ) \sqrt {2}\, \sqrt {\frac {\left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) c}{b +\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {-\frac {2 c x}{b +\sqrt {-4 a c +b^{2}}}}\, \operatorname {EllipticPi}\left (\sqrt {2}\, \sqrt {\frac {\left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) c}{b +\sqrt {-4 a c +b^{2}}}}, -\frac {b +\sqrt {-4 a c +b^{2}}}{2 c \left (\frac {d}{e}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}, \frac {\sqrt {-\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right )}{c \left (-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}}{2}\right )}{g d c \sqrt {c g \,x^{3}+b g \,x^{2}+a g x}\, \left (\frac {d}{e}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}\right )}{\sqrt {g x}\, \sqrt {c \,x^{2}+b x +a}}\) | \(838\) |
| default | \(\text {Expression too large to display}\) | \(2112\) |
Input:
int(1/(g*x)^(3/2)/(e*x+d)/(c*x^2+b*x+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
-2*(c*x^2+b*x+a)^(1/2)/a/d/g/(g*x)^(1/2)+1/d/a*((b+(-4*a*c+b^2)^(1/2))*2^( 1/2)*((x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*((x -1/2/c*(-b+(-4*a*c+b^2)^(1/2)))/(-1/2*(b+(-4*a*c+b^2)^(1/2))/c-1/2/c*(-b+( -4*a*c+b^2)^(1/2))))^(1/2)*(-2*c*x/(b+(-4*a*c+b^2)^(1/2)))^(1/2)/(c*g*x^3+ b*g*x^2+a*g*x)^(1/2)*((-1/2*(b+(-4*a*c+b^2)^(1/2))/c-1/2/c*(-b+(-4*a*c+b^2 )^(1/2)))*EllipticE(2^(1/2)*((x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(b+(-4*a*c+b ^2)^(1/2))*c)^(1/2),1/2*(-2*(b+(-4*a*c+b^2)^(1/2))/c/(-1/2*(b+(-4*a*c+b^2) ^(1/2))/c-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2))+1/2/c*(-b+(-4*a*c+b^2)^(1 /2))*EllipticF(2^(1/2)*((x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(b+(-4*a*c+b^2)^( 1/2))*c)^(1/2),1/2*(-2*(b+(-4*a*c+b^2)^(1/2))/c/(-1/2*(b+(-4*a*c+b^2)^(1/2 ))/c-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2)))-a*(b+(-4*a*c+b^2)^(1/2))/c*2^ (1/2)*((x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*(( x-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))/(-1/2*(b+(-4*a*c+b^2)^(1/2))/c-1/2/c*(-b+ (-4*a*c+b^2)^(1/2))))^(1/2)*(-2*c*x/(b+(-4*a*c+b^2)^(1/2)))^(1/2)/(c*g*x^3 +b*g*x^2+a*g*x)^(1/2)/(d/e-1/2*(b+(-4*a*c+b^2)^(1/2))/c)*EllipticPi(2^(1/2 )*((x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(b+(-4*a*c+b^2)^(1/2))*c)^(1/2),-1/2*( b+(-4*a*c+b^2)^(1/2))/c/(d/e-1/2*(b+(-4*a*c+b^2)^(1/2))/c),1/2*(-2*(b+(-4* a*c+b^2)^(1/2))/c/(-1/2*(b+(-4*a*c+b^2)^(1/2))/c-1/2/c*(-b+(-4*a*c+b^2)^(1 /2))))^(1/2)))/g*(x*g*(c*x^2+b*x+a))^(1/2)/(g*x)^(1/2)/(c*x^2+b*x+a)^(1/2)
Timed out. \[ \int \frac {1}{(g x)^{3/2} (d+e x) \sqrt {a+b x+c x^2}} \, dx=\text {Timed out} \] Input:
integrate(1/(g*x)^(3/2)/(e*x+d)/(c*x^2+b*x+a)^(1/2),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {1}{(g x)^{3/2} (d+e x) \sqrt {a+b x+c x^2}} \, dx=\int \frac {1}{\left (g x\right )^{\frac {3}{2}} \left (d + e x\right ) \sqrt {a + b x + c x^{2}}}\, dx \] Input:
integrate(1/(g*x)**(3/2)/(e*x+d)/(c*x**2+b*x+a)**(1/2),x)
Output:
Integral(1/((g*x)**(3/2)*(d + e*x)*sqrt(a + b*x + c*x**2)), x)
\[ \int \frac {1}{(g x)^{3/2} (d+e x) \sqrt {a+b x+c x^2}} \, dx=\int { \frac {1}{\sqrt {c x^{2} + b x + a} {\left (e x + d\right )} \left (g x\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/(g*x)^(3/2)/(e*x+d)/(c*x^2+b*x+a)^(1/2),x, algorithm="maxima")
Output:
integrate(1/(sqrt(c*x^2 + b*x + a)*(e*x + d)*(g*x)^(3/2)), x)
\[ \int \frac {1}{(g x)^{3/2} (d+e x) \sqrt {a+b x+c x^2}} \, dx=\int { \frac {1}{\sqrt {c x^{2} + b x + a} {\left (e x + d\right )} \left (g x\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/(g*x)^(3/2)/(e*x+d)/(c*x^2+b*x+a)^(1/2),x, algorithm="giac")
Output:
integrate(1/(sqrt(c*x^2 + b*x + a)*(e*x + d)*(g*x)^(3/2)), x)
Timed out. \[ \int \frac {1}{(g x)^{3/2} (d+e x) \sqrt {a+b x+c x^2}} \, dx=\int \frac {1}{{\left (g\,x\right )}^{3/2}\,\left (d+e\,x\right )\,\sqrt {c\,x^2+b\,x+a}} \,d x \] Input:
int(1/((g*x)^(3/2)*(d + e*x)*(a + b*x + c*x^2)^(1/2)),x)
Output:
int(1/((g*x)^(3/2)*(d + e*x)*(a + b*x + c*x^2)^(1/2)), x)
\[ \int \frac {1}{(g x)^{3/2} (d+e x) \sqrt {a+b x+c x^2}} \, dx=\int \frac {1}{\left (g x \right )^{\frac {3}{2}} \left (e x +d \right ) \sqrt {c \,x^{2}+b x +a}}d x \] Input:
int(1/(g*x)^(3/2)/(e*x+d)/(c*x^2+b*x+a)^(1/2),x)
Output:
int(1/(g*x)^(3/2)/(e*x+d)/(c*x^2+b*x+a)^(1/2),x)