Integrand size = 38, antiderivative size = 897 \[ \int \frac {A+B x}{(d+e x)^{3/2} \sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx=-\frac {2 (B d-A e) \left (2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g\right ) \left (b-\sqrt {b^2-4 a c}+2 c x\right ) \sqrt {\frac {(e f-d g) \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{\left (2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g\right ) (d+e x)}} E\left (\arcsin \left (\frac {\sqrt {2 c d-b e+\sqrt {b^2-4 a c} e} \sqrt {f+g x}}{\sqrt {2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g} \sqrt {d+e x}}\right )|\frac {\left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right ) \left (2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g\right )}{\left (2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e\right ) \left (2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g\right )}\right )}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} \left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right ) \sqrt {2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g} (e f-d g) \sqrt {\frac {(e f-d g) \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{\left (2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g\right ) (d+e x)}} \sqrt {a+b x+c x^2}}+\frac {2 \left (b B-2 A c+B \sqrt {b^2-4 a c}\right ) \left (b-\sqrt {b^2-4 a c}+2 c x\right ) \sqrt {\frac {(e f-d g) \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{\left (2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g\right ) (d+e x)}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2 c d-b e+\sqrt {b^2-4 a c} e} \sqrt {f+g x}}{\sqrt {2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g} \sqrt {d+e x}}\right ),\frac {\left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right ) \left (2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g\right )}{\left (2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e\right ) \left (2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g\right )}\right )}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} \left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right ) \sqrt {2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g} \sqrt {\frac {(e f-d g) \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{\left (2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g\right ) (d+e x)}} \sqrt {a+b x+c x^2}} \] Output:
-2*(-A*e+B*d)*(2*c*f-(b+(-4*a*c+b^2)^(1/2))*g)*(b-(-4*a*c+b^2)^(1/2)+2*c*x )*((-d*g+e*f)*(b+(-4*a*c+b^2)^(1/2)+2*c*x)/(2*c*f-(b+(-4*a*c+b^2)^(1/2))*g )/(e*x+d))^(1/2)*EllipticE(((-4*a*c+b^2)^(1/2)*e-b*e+2*c*d)^(1/2)*(g*x+f)^ (1/2)/(2*c*f-(b-(-4*a*c+b^2)^(1/2))*g)^(1/2)/(e*x+d)^(1/2),((2*c*d-(b+(-4* a*c+b^2)^(1/2))*e)*(2*c*f-(b-(-4*a*c+b^2)^(1/2))*g)/(2*c*d-(b-(-4*a*c+b^2) ^(1/2))*e)/(2*c*f-(b+(-4*a*c+b^2)^(1/2))*g))^(1/2))/(2*c*d-(b-(-4*a*c+b^2) ^(1/2))*e)^(1/2)/(2*c*d-(b+(-4*a*c+b^2)^(1/2))*e)/(2*c*f-(b-(-4*a*c+b^2)^( 1/2))*g)^(1/2)/(-d*g+e*f)/((-d*g+e*f)*(b-(-4*a*c+b^2)^(1/2)+2*c*x)/(2*c*f- (b-(-4*a*c+b^2)^(1/2))*g)/(e*x+d))^(1/2)/(c*x^2+b*x+a)^(1/2)+2*(b*B-2*A*c+ B*(-4*a*c+b^2)^(1/2))*(b-(-4*a*c+b^2)^(1/2)+2*c*x)*((-d*g+e*f)*(b+(-4*a*c+ b^2)^(1/2)+2*c*x)/(2*c*f-(b+(-4*a*c+b^2)^(1/2))*g)/(e*x+d))^(1/2)*Elliptic F(((-4*a*c+b^2)^(1/2)*e-b*e+2*c*d)^(1/2)*(g*x+f)^(1/2)/(2*c*f-(b-(-4*a*c+b ^2)^(1/2))*g)^(1/2)/(e*x+d)^(1/2),((2*c*d-(b+(-4*a*c+b^2)^(1/2))*e)*(2*c*f -(b-(-4*a*c+b^2)^(1/2))*g)/(2*c*d-(b-(-4*a*c+b^2)^(1/2))*e)/(2*c*f-(b+(-4* a*c+b^2)^(1/2))*g))^(1/2))/(2*c*d-(b-(-4*a*c+b^2)^(1/2))*e)^(1/2)/(2*c*d-( b+(-4*a*c+b^2)^(1/2))*e)/(2*c*f-(b-(-4*a*c+b^2)^(1/2))*g)^(1/2)/((-d*g+e*f )*(b-(-4*a*c+b^2)^(1/2)+2*c*x)/(2*c*f-(b-(-4*a*c+b^2)^(1/2))*g)/(e*x+d))^( 1/2)/(c*x^2+b*x+a)^(1/2)
Leaf count is larger than twice the leaf count of optimal. \(3628\) vs. \(2(897)=1794\).
Time = 36.09 (sec) , antiderivative size = 3628, normalized size of antiderivative = 4.04 \[ \int \frac {A+B x}{(d+e x)^{3/2} \sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx=\text {Result too large to show} \] Input:
Integrate[(A + B*x)/((d + e*x)^(3/2)*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2]), x]
Output:
(-2*e*(-(B*d) + A*e)*Sqrt[f + g*x]*(a + b*x + c*x^2))/((c*d^2 - b*d*e + a* e^2)*(e*f - d*g)*Sqrt[d + e*x]*Sqrt[a + x*(b + c*x)]) + (Sqrt[a + b*x + c* x^2]*((-2*(-(B*d) + A*e)*(d + e*x)^(5/2)*(c + (c*d^2)/(d + e*x)^2 - (b*d*e )/(d + e*x)^2 + (a*e^2)/(d + e*x)^2 - (2*c*d)/(d + e*x) + (b*e)/(d + e*x)) *(g + (e*f)/(d + e*x) - (d*g)/(d + e*x)))/(Sqrt[((d + e*x)^2*(c*(-1 + d/(d + e*x))^2 + (e*(b - (b*d)/(d + e*x) + (a*e)/(d + e*x)))/(d + e*x)))/e^2]* Sqrt[f + ((d + e*x)*(g - (d*g)/(d + e*x)))/e]) + (2*(c*d^2 - b*d*e + a*e^2 )*(-(e*f) + d*g)*(d + e*x)^(3/2)*Sqrt[(c + (c*d^2)/(d + e*x)^2 - (b*d*e)/( d + e*x)^2 + (a*e^2)/(d + e*x)^2 - (2*c*d)/(d + e*x) + (b*e)/(d + e*x))*(g + (e*f)/(d + e*x) - (d*g)/(d + e*x))]*(-((B*(-1/2*(2*c*d - b*e - Sqrt[b^2 *e^2 - 4*a*c*e^2])/(c*d^2 - b*d*e + a*e^2) + (d + e*x)^(-1))*Sqrt[(-1/2*(2 *c*d - b*e + Sqrt[b^2*e^2 - 4*a*c*e^2])/(c*d^2 - b*d*e + a*e^2) + (d + e*x )^(-1))/((2*c*d - b*e - Sqrt[b^2*e^2 - 4*a*c*e^2])/(2*(c*d^2 - b*d*e + a*e ^2)) - (2*c*d - b*e + Sqrt[b^2*e^2 - 4*a*c*e^2])/(2*(c*d^2 - b*d*e + a*e^2 )))]*Sqrt[(-(g/(-(e*f) + d*g)) + (d + e*x)^(-1))/((2*c*d - b*e - Sqrt[b^2* e^2 - 4*a*c*e^2])/(2*(c*d^2 - b*d*e + a*e^2)) - g/(-(e*f) + d*g))]*Ellipti cF[ArcSin[Sqrt[(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2] + (2*c*d^2)/(d + e* x) - (2*b*d*e)/(d + e*x) + (2*a*e^2)/(d + e*x))/Sqrt[(b^2 - 4*a*c)*e^2]]/S qrt[2]], (2*Sqrt[(b^2 - 4*a*c)*e^2]*(e*f - d*g))/(-2*c*d*e*f + b*e^2*f + e *Sqrt[(b^2 - 4*a*c)*e^2]*f + b*d*e*g - 2*a*e^2*g - d*Sqrt[(b^2 - 4*a*c)...
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.
\(\displaystyle \int \frac {A+B x}{(d+e x)^{3/2} \sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx\) |
\(\Big \downarrow \) 2154 |
\(\displaystyle \left (A-\frac {B d}{e}\right ) \int \frac {1}{(d+e x)^{3/2} \sqrt {f+g x} \sqrt {c x^2+b x+a}}dx+\int \frac {B}{e \sqrt {d+e x} \sqrt {f+g x} \sqrt {c x^2+b x+a}}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \left (A-\frac {B d}{e}\right ) \int \frac {1}{(d+e x)^{3/2} \sqrt {f+g x} \sqrt {c x^2+b x+a}}dx+\frac {B \int \frac {1}{\sqrt {d+e x} \sqrt {f+g x} \sqrt {c x^2+b x+a}}dx}{e}\) |
\(\Big \downarrow \) 1280 |
\(\displaystyle \left (A-\frac {B d}{e}\right ) \int \frac {1}{(d+e x)^{3/2} \sqrt {f+g x} \sqrt {c x^2+b x+a}}dx-\frac {2 B (d+e x) \sqrt {\frac {\left (a+b x+c x^2\right ) (e f-d g)^2}{(d+e x)^2 \left (a g^2-b f g+c f^2\right )}} \int \frac {1}{\sqrt {\frac {\left (c d^2-b e d+a e^2\right ) (f+g x)^2}{\left (c f^2-g (b f-a g)\right ) (d+e x)^2}-\frac {(2 c d f+2 a e g-b (e f+d g)) (f+g x)}{\left (c f^2-b g f+a g^2\right ) (d+e x)}+1}}d\frac {\sqrt {f+g x}}{\sqrt {d+e x}}}{e \sqrt {a+b x+c x^2} (e f-d g)}\) |
\(\Big \downarrow \) 1281 |
\(\displaystyle \left (A-\frac {B d}{e}\right ) \left (\frac {e \int \frac {\sqrt {f+g x}}{(d+e x)^{3/2} \sqrt {c x^2+b x+a}}dx}{e f-d g}-\frac {g \int \frac {1}{\sqrt {d+e x} \sqrt {f+g x} \sqrt {c x^2+b x+a}}dx}{e f-d g}\right )-\frac {2 B (d+e x) \sqrt {\frac {\left (a+b x+c x^2\right ) (e f-d g)^2}{(d+e x)^2 \left (a g^2-b f g+c f^2\right )}} \int \frac {1}{\sqrt {\frac {\left (c d^2-b e d+a e^2\right ) (f+g x)^2}{\left (c f^2-g (b f-a g)\right ) (d+e x)^2}-\frac {(2 c d f+2 a e g-b (e f+d g)) (f+g x)}{\left (c f^2-b g f+a g^2\right ) (d+e x)}+1}}d\frac {\sqrt {f+g x}}{\sqrt {d+e x}}}{e \sqrt {a+b x+c x^2} (e f-d g)}\) |
\(\Big \downarrow \) 1280 |
\(\displaystyle \left (A-\frac {B d}{e}\right ) \left (\frac {2 g (d+e x) \sqrt {\frac {\left (a+b x+c x^2\right ) (e f-d g)^2}{(d+e x)^2 \left (a g^2-b f g+c f^2\right )}} \int \frac {1}{\sqrt {\frac {\left (c d^2-b e d+a e^2\right ) (f+g x)^2}{\left (c f^2-g (b f-a g)\right ) (d+e x)^2}-\frac {(2 c d f+2 a e g-b (e f+d g)) (f+g x)}{\left (c f^2-b g f+a g^2\right ) (d+e x)}+1}}d\frac {\sqrt {f+g x}}{\sqrt {d+e x}}}{\sqrt {a+b x+c x^2} (e f-d g)^2}+\frac {e \int \frac {\sqrt {f+g x}}{(d+e x)^{3/2} \sqrt {c x^2+b x+a}}dx}{e f-d g}\right )-\frac {2 B (d+e x) \sqrt {\frac {\left (a+b x+c x^2\right ) (e f-d g)^2}{(d+e x)^2 \left (a g^2-b f g+c f^2\right )}} \int \frac {1}{\sqrt {\frac {\left (c d^2-b e d+a e^2\right ) (f+g x)^2}{\left (c f^2-g (b f-a g)\right ) (d+e x)^2}-\frac {(2 c d f+2 a e g-b (e f+d g)) (f+g x)}{\left (c f^2-b g f+a g^2\right ) (d+e x)}+1}}d\frac {\sqrt {f+g x}}{\sqrt {d+e x}}}{e \sqrt {a+b x+c x^2} (e f-d g)}\) |
\(\Big \downarrow \) 1292 |
\(\displaystyle \left (A-\frac {B d}{e}\right ) \left (\frac {2 g (d+e x) \sqrt {\frac {\left (a+b x+c x^2\right ) (e f-d g)^2}{(d+e x)^2 \left (a g^2-b f g+c f^2\right )}} \int \frac {1}{\sqrt {\frac {\left (c d^2-b e d+a e^2\right ) (f+g x)^2}{\left (c f^2-g (b f-a g)\right ) (d+e x)^2}-\frac {(2 c d f+2 a e g-b (e f+d g)) (f+g x)}{\left (c f^2-b g f+a g^2\right ) (d+e x)}+1}}d\frac {\sqrt {f+g x}}{\sqrt {d+e x}}}{\sqrt {a+b x+c x^2} (e f-d g)^2}+\frac {e \int \frac {\sqrt {f+g x}}{(d+e x)^{3/2} \sqrt {c x^2+b x+a}}dx}{e f-d g}\right )-\frac {2 B (d+e x) \sqrt {\frac {\left (a+b x+c x^2\right ) (e f-d g)^2}{(d+e x)^2 \left (a g^2-b f g+c f^2\right )}} \int \frac {1}{\sqrt {\frac {\left (c d^2-b e d+a e^2\right ) (f+g x)^2}{\left (c f^2-g (b f-a g)\right ) (d+e x)^2}-\frac {(2 c d f+2 a e g-b (e f+d g)) (f+g x)}{\left (c f^2-b g f+a g^2\right ) (d+e x)}+1}}d\frac {\sqrt {f+g x}}{\sqrt {d+e x}}}{e \sqrt {a+b x+c x^2} (e f-d g)}\) |
\(\Big \downarrow \) 1416 |
\(\displaystyle \left (A-\frac {B d}{e}\right ) \left (\frac {g \sqrt [4]{c f^2-g (b f-a g)} (d+e x) \sqrt {\frac {(e f-d g)^2 \left (c x^2+b x+a\right )}{\left (c f^2-b g f+a g^2\right ) (d+e x)^2}} \left (\frac {\sqrt {c d^2-b e d+a e^2} (f+g x)}{\sqrt {c f^2-g (b f-a g)} (d+e x)}+1\right ) \sqrt {\frac {\frac {\left (c d^2-b e d+a e^2\right ) (f+g x)^2}{\left (c f^2-g (b f-a g)\right ) (d+e x)^2}-\frac {(2 c d f+2 a e g-b (e f+d g)) (f+g x)}{\left (c f^2-b g f+a g^2\right ) (d+e x)}+1}{\left (\frac {\sqrt {c d^2-b e d+a e^2} (f+g x)}{\sqrt {c f^2-g (b f-a g)} (d+e x)}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c d^2-b e d+a e^2} \sqrt {f+g x}}{\sqrt [4]{c f^2-b g f+a g^2} \sqrt {d+e x}}\right ),\frac {1}{4} \left (\frac {2 c d f+2 a e g-b (e f+d g)}{\sqrt {c d^2-e (b d-a e)} \sqrt {c f^2-g (b f-a g)}}+2\right )\right )}{\sqrt [4]{c d^2-b e d+a e^2} (e f-d g)^2 \sqrt {c x^2+b x+a} \sqrt {\frac {\left (c d^2-b e d+a e^2\right ) (f+g x)^2}{\left (c f^2-g (b f-a g)\right ) (d+e x)^2}-\frac {(2 c d f+2 a e g-b (e f+d g)) (f+g x)}{\left (c f^2-b g f+a g^2\right ) (d+e x)}+1}}+\frac {e \int \frac {\sqrt {f+g x}}{(d+e x)^{3/2} \sqrt {c x^2+b x+a}}dx}{e f-d g}\right )-\frac {B \sqrt [4]{c f^2-g (b f-a g)} (d+e x) \sqrt {\frac {(e f-d g)^2 \left (c x^2+b x+a\right )}{\left (c f^2-b g f+a g^2\right ) (d+e x)^2}} \left (\frac {\sqrt {c d^2-b e d+a e^2} (f+g x)}{\sqrt {c f^2-g (b f-a g)} (d+e x)}+1\right ) \sqrt {\frac {\frac {\left (c d^2-b e d+a e^2\right ) (f+g x)^2}{\left (c f^2-g (b f-a g)\right ) (d+e x)^2}-\frac {(2 c d f+2 a e g-b (e f+d g)) (f+g x)}{\left (c f^2-b g f+a g^2\right ) (d+e x)}+1}{\left (\frac {\sqrt {c d^2-b e d+a e^2} (f+g x)}{\sqrt {c f^2-g (b f-a g)} (d+e x)}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c d^2-b e d+a e^2} \sqrt {f+g x}}{\sqrt [4]{c f^2-b g f+a g^2} \sqrt {d+e x}}\right ),\frac {1}{4} \left (\frac {2 c d f+2 a e g-b (e f+d g)}{\sqrt {c d^2-e (b d-a e)} \sqrt {c f^2-g (b f-a g)}}+2\right )\right )}{e \sqrt [4]{c d^2-b e d+a e^2} (e f-d g) \sqrt {c x^2+b x+a} \sqrt {\frac {\left (c d^2-b e d+a e^2\right ) (f+g x)^2}{\left (c f^2-g (b f-a g)\right ) (d+e x)^2}-\frac {(2 c d f+2 a e g-b (e f+d g)) (f+g x)}{\left (c f^2-b g f+a g^2\right ) (d+e x)}+1}}\) |
Input:
Int[(A + B*x)/((d + e*x)^(3/2)*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2]),x]
Output:
$Aborted
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/(Sqrt[(d_.) + (e_.)*(x_)]*Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_.) + (b_.) *(x_) + (c_.)*(x_)^2]), x_Symbol] :> Simp[-2*(d + e*x)*(Sqrt[(e*f - d*g)^2* ((a + b*x + c*x^2)/((c*f^2 - b*f*g + a*g^2)*(d + e*x)^2))]/((e*f - d*g)*Sqr t[a + b*x + c*x^2])) Subst[Int[1/Sqrt[1 - (2*c*d*f - b*e*f - b*d*g + 2*a* e*g)*(x^2/(c*f^2 - b*f*g + a*g^2)) + (c*d^2 - b*d*e + a*e^2)*(x^4/(c*f^2 - b*f*g + a*g^2))], x], x, Sqrt[f + g*x]/Sqrt[d + e*x]], x] /; FreeQ[{a, b, c , d, e, f, g}, x]
Int[1/(((d_.) + (e_.)*(x_))^(3/2)*Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_.) + (b_ .)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Simp[-g/(e*f - d*g) Int[1/(Sqrt[d + e*x]*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2]), x], x] + Simp[e/(e*f - d*g) Int[Sqrt[f + g*x]/((d + e*x)^(3/2)*Sqrt[a + b*x + c*x^2]), x], x] /; FreeQ[ {a, b, c, d, e, f, g}, x]
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x _) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Unintegrable[(d + e*x)^m*(f + g*x)^n* (a + b*x + c*x^2)^p, x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x]
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[(Px_)*((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b _.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[PolynomialQuotient[Px, d + e*x, x]*(d + e*x)^(m + 1)*(f + g*x)^n*(a + b*x + c*x^2)^p, x] + Simp[Polyn omialRemainder[Px, d + e*x, x] Int[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x ^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && PolynomialQ[Px, x ] && LtQ[m, 0] && !IntegerQ[n] && IntegersQ[2*m, 2*n, 2*p]
Leaf count of result is larger than twice the leaf count of optimal. \(3140\) vs. \(2(803)=1606\).
Time = 14.86 (sec) , antiderivative size = 3141, normalized size of antiderivative = 3.50
method | result | size |
elliptic | \(\text {Expression too large to display}\) | \(3141\) |
default | \(\text {Expression too large to display}\) | \(70227\) |
Input:
int((B*x+A)/(e*x+d)^(3/2)/(g*x+f)^(1/2)/(c*x^2+b*x+a)^(1/2),x,method=_RETU RNVERBOSE)
Output:
((g*x+f)*(c*x^2+b*x+a)*(e*x+d))^(1/2)/(g*x+f)^(1/2)/(c*x^2+b*x+a)^(1/2)/(e *x+d)^(1/2)*(2*(c*e*g*x^3+b*e*g*x^2+c*e*f*x^2+a*e*g*x+b*e*f*x+a*e*f)/(a*d* e^2*g-a*e^3*f-b*d^2*e*g+b*d*e^2*f+c*d^3*g-c*d^2*e*f)*(A*e-B*d)/((x+d/e)*(c *e*g*x^3+b*e*g*x^2+c*e*f*x^2+a*e*g*x+b*e*f*x+a*e*f))^(1/2)+2*(B/e+1/e*(a*e ^2*g-b*d*e*g+b*e^2*f+c*d^2*g-c*d*e*f)*(A*e-B*d)/(a*d*e^2*g-a*e^3*f-b*d^2*e *g+b*d*e^2*f+c*d^3*g-c*d^2*e*f)-(a*e*g+b*e*f)/(a*d*e^2*g-a*e^3*f-b*d^2*e*g +b*d*e^2*f+c*d^3*g-c*d^2*e*f)*(A*e-B*d))*(-f/g+d/e)*((-d/e-1/2/c*(-b+(-4*a *c+b^2)^(1/2)))*(x+f/g)/(f/g-d/e)/(x-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2) *(x-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))^2*((1/2/c*(-b+(-4*a*c+b^2)^(1/2))+f/g)* (x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(-1/2*(b+(-4*a*c+b^2)^(1/2))/c+f/g)/(x-1/ 2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2)*((1/2/c*(-b+(-4*a*c+b^2)^(1/2))+f/g)*( x+d/e)/(f/g-d/e)/(x-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2)/(-d/e-1/2/c*(-b+ (-4*a*c+b^2)^(1/2)))/(1/2/c*(-b+(-4*a*c+b^2)^(1/2))+f/g)/(c*e*g*(x+f/g)*(x -1/2/c*(-b+(-4*a*c+b^2)^(1/2)))*(x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)*(x+d/e))^ (1/2)*EllipticF(((-d/e-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))*(x+f/g)/(f/g-d/e)/(x -1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2),((1/2/c*(-b+(-4*a*c+b^2)^(1/2))+1/2 *(b+(-4*a*c+b^2)^(1/2))/c)*(-f/g+d/e)/(1/2*(b+(-4*a*c+b^2)^(1/2))/c-f/g)/( d/e+1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2))+2*((b*e*g-c*d*g+c*e*f)*(A*e-B*d )/(a*d*e^2*g-a*e^3*f-b*d^2*e*g+b*d*e^2*f+c*d^3*g-c*d^2*e*f)-(2*b*e*g+2*c*e *f)/(a*d*e^2*g-a*e^3*f-b*d^2*e*g+b*d*e^2*f+c*d^3*g-c*d^2*e*f)*(A*e-B*d)...
\[ \int \frac {A+B x}{(d+e x)^{3/2} \sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx=\int { \frac {B x + A}{\sqrt {c x^{2} + b x + a} {\left (e x + d\right )}^{\frac {3}{2}} \sqrt {g x + f}} \,d x } \] Input:
integrate((B*x+A)/(e*x+d)^(3/2)/(g*x+f)^(1/2)/(c*x^2+b*x+a)^(1/2),x, algor ithm="fricas")
Output:
integral(sqrt(c*x^2 + b*x + a)*(B*x + A)*sqrt(e*x + d)*sqrt(g*x + f)/(c*e^ 2*g*x^5 + (c*e^2*f + (2*c*d*e + b*e^2)*g)*x^4 + a*d^2*f + ((2*c*d*e + b*e^ 2)*f + (c*d^2 + 2*b*d*e + a*e^2)*g)*x^3 + ((c*d^2 + 2*b*d*e + a*e^2)*f + ( b*d^2 + 2*a*d*e)*g)*x^2 + (a*d^2*g + (b*d^2 + 2*a*d*e)*f)*x), x)
\[ \int \frac {A+B x}{(d+e x)^{3/2} \sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx=\int \frac {A + B x}{\left (d + e x\right )^{\frac {3}{2}} \sqrt {f + g x} \sqrt {a + b x + c x^{2}}}\, dx \] Input:
integrate((B*x+A)/(e*x+d)**(3/2)/(g*x+f)**(1/2)/(c*x**2+b*x+a)**(1/2),x)
Output:
Integral((A + B*x)/((d + e*x)**(3/2)*sqrt(f + g*x)*sqrt(a + b*x + c*x**2)) , x)
\[ \int \frac {A+B x}{(d+e x)^{3/2} \sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx=\int { \frac {B x + A}{\sqrt {c x^{2} + b x + a} {\left (e x + d\right )}^{\frac {3}{2}} \sqrt {g x + f}} \,d x } \] Input:
integrate((B*x+A)/(e*x+d)^(3/2)/(g*x+f)^(1/2)/(c*x^2+b*x+a)^(1/2),x, algor ithm="maxima")
Output:
integrate((B*x + A)/(sqrt(c*x^2 + b*x + a)*(e*x + d)^(3/2)*sqrt(g*x + f)), x)
\[ \int \frac {A+B x}{(d+e x)^{3/2} \sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx=\int { \frac {B x + A}{\sqrt {c x^{2} + b x + a} {\left (e x + d\right )}^{\frac {3}{2}} \sqrt {g x + f}} \,d x } \] Input:
integrate((B*x+A)/(e*x+d)^(3/2)/(g*x+f)^(1/2)/(c*x^2+b*x+a)^(1/2),x, algor ithm="giac")
Output:
integrate((B*x + A)/(sqrt(c*x^2 + b*x + a)*(e*x + d)^(3/2)*sqrt(g*x + f)), x)
Timed out. \[ \int \frac {A+B x}{(d+e x)^{3/2} \sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx=\int \frac {A+B\,x}{\sqrt {f+g\,x}\,{\left (d+e\,x\right )}^{3/2}\,\sqrt {c\,x^2+b\,x+a}} \,d x \] Input:
int((A + B*x)/((f + g*x)^(1/2)*(d + e*x)^(3/2)*(a + b*x + c*x^2)^(1/2)),x)
Output:
int((A + B*x)/((f + g*x)^(1/2)*(d + e*x)^(3/2)*(a + b*x + c*x^2)^(1/2)), x )
\[ \int \frac {A+B x}{(d+e x)^{3/2} \sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx=\int \frac {B x +A}{\left (e x +d \right )^{\frac {3}{2}} \sqrt {g x +f}\, \sqrt {c \,x^{2}+b x +a}}d x \] Input:
int((B*x+A)/(e*x+d)^(3/2)/(g*x+f)^(1/2)/(c*x^2+b*x+a)^(1/2),x)
Output:
int((B*x+A)/(e*x+d)^(3/2)/(g*x+f)^(1/2)/(c*x^2+b*x+a)^(1/2),x)