Integrand size = 25, antiderivative size = 315 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{d-f x^2} \, dx=-\frac {(5 b+2 c x) \sqrt {a+b x+c x^2}}{4 f}-\frac {\left (8 c^2 d+3 b^2 f+12 a c f\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 \sqrt {c} f^2}+\frac {\left (c d-b \sqrt {d} \sqrt {f}+a f\right )^{3/2} \text {arctanh}\left (\frac {b \sqrt {d}-2 a \sqrt {f}+\left (2 c \sqrt {d}-b \sqrt {f}\right ) x}{2 \sqrt {c d-b \sqrt {d} \sqrt {f}+a f} \sqrt {a+b x+c x^2}}\right )}{2 \sqrt {d} f^2}+\frac {\left (c d+b \sqrt {d} \sqrt {f}+a f\right )^{3/2} \text {arctanh}\left (\frac {b \sqrt {d}+2 a \sqrt {f}+\left (2 c \sqrt {d}+b \sqrt {f}\right ) x}{2 \sqrt {c d+b \sqrt {d} \sqrt {f}+a f} \sqrt {a+b x+c x^2}}\right )}{2 \sqrt {d} f^2} \] Output:
-1/4*(2*c*x+5*b)*(c*x^2+b*x+a)^(1/2)/f-1/8*(12*a*c*f+3*b^2*f+8*c^2*d)*arct anh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))/c^(1/2)/f^2+1/2*(c*d-b*d^(1 /2)*f^(1/2)+a*f)^(3/2)*arctanh(1/2*(b*d^(1/2)-2*a*f^(1/2)+(2*c*d^(1/2)-b*f ^(1/2))*x)/(c*d-b*d^(1/2)*f^(1/2)+a*f)^(1/2)/(c*x^2+b*x+a)^(1/2))/d^(1/2)/ f^2+1/2*(c*d+b*d^(1/2)*f^(1/2)+a*f)^(3/2)*arctanh(1/2*(b*d^(1/2)+2*a*f^(1/ 2)+(2*c*d^(1/2)+b*f^(1/2))*x)/(c*d+b*d^(1/2)*f^(1/2)+a*f)^(1/2)/(c*x^2+b*x +a)^(1/2))/d^(1/2)/f^2
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 1.58 (sec) , antiderivative size = 729, normalized size of antiderivative = 2.31 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{d-f x^2} \, dx=\frac {-\sqrt {c} f (5 b+2 c x) \sqrt {a+x (b+c x)}+\left (8 c^2 d+3 b^2 f+12 a c f\right ) \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a}-\sqrt {a+x (b+c x)}}\right )-2 \sqrt {c} \text {RootSum}\left [c^2 d-b^2 f+4 \sqrt {a} b f \text {$\#$1}-2 c d \text {$\#$1}^2-4 a f \text {$\#$1}^2+d \text {$\#$1}^4\&,\frac {-c^3 d^2 \log (x)+b^2 c d f \log (x)-2 a c^2 d f \log (x)+2 a b^2 f^2 \log (x)-a^2 c f^2 \log (x)+c^3 d^2 \log \left (-\sqrt {a}+\sqrt {a+b x+c x^2}-x \text {$\#$1}\right )-b^2 c d f \log \left (-\sqrt {a}+\sqrt {a+b x+c x^2}-x \text {$\#$1}\right )+2 a c^2 d f \log \left (-\sqrt {a}+\sqrt {a+b x+c x^2}-x \text {$\#$1}\right )-2 a b^2 f^2 \log \left (-\sqrt {a}+\sqrt {a+b x+c x^2}-x \text {$\#$1}\right )+a^2 c f^2 \log \left (-\sqrt {a}+\sqrt {a+b x+c x^2}-x \text {$\#$1}\right )-4 \sqrt {a} b c d f \log (x) \text {$\#$1}-4 a^{3/2} b f^2 \log (x) \text {$\#$1}+4 \sqrt {a} b c d f \log \left (-\sqrt {a}+\sqrt {a+b x+c x^2}-x \text {$\#$1}\right ) \text {$\#$1}+4 a^{3/2} b f^2 \log \left (-\sqrt {a}+\sqrt {a+b x+c x^2}-x \text {$\#$1}\right ) \text {$\#$1}+c^2 d^2 \log (x) \text {$\#$1}^2+b^2 d f \log (x) \text {$\#$1}^2+2 a c d f \log (x) \text {$\#$1}^2+a^2 f^2 \log (x) \text {$\#$1}^2-c^2 d^2 \log \left (-\sqrt {a}+\sqrt {a+b x+c x^2}-x \text {$\#$1}\right ) \text {$\#$1}^2-b^2 d f \log \left (-\sqrt {a}+\sqrt {a+b x+c x^2}-x \text {$\#$1}\right ) \text {$\#$1}^2-2 a c d f \log \left (-\sqrt {a}+\sqrt {a+b x+c x^2}-x \text {$\#$1}\right ) \text {$\#$1}^2-a^2 f^2 \log \left (-\sqrt {a}+\sqrt {a+b x+c x^2}-x \text {$\#$1}\right ) \text {$\#$1}^2}{-\sqrt {a} b f+c d \text {$\#$1}+2 a f \text {$\#$1}-d \text {$\#$1}^3}\&\right ]}{4 \sqrt {c} f^2} \] Input:
Integrate[(a + b*x + c*x^2)^(3/2)/(d - f*x^2),x]
Output:
(-(Sqrt[c]*f*(5*b + 2*c*x)*Sqrt[a + x*(b + c*x)]) + (8*c^2*d + 3*b^2*f + 1 2*a*c*f)*ArcTanh[(Sqrt[c]*x)/(Sqrt[a] - Sqrt[a + x*(b + c*x)])] - 2*Sqrt[c ]*RootSum[c^2*d - b^2*f + 4*Sqrt[a]*b*f*#1 - 2*c*d*#1^2 - 4*a*f*#1^2 + d*# 1^4 & , (-(c^3*d^2*Log[x]) + b^2*c*d*f*Log[x] - 2*a*c^2*d*f*Log[x] + 2*a*b ^2*f^2*Log[x] - a^2*c*f^2*Log[x] + c^3*d^2*Log[-Sqrt[a] + Sqrt[a + b*x + c *x^2] - x*#1] - b^2*c*d*f*Log[-Sqrt[a] + Sqrt[a + b*x + c*x^2] - x*#1] + 2 *a*c^2*d*f*Log[-Sqrt[a] + Sqrt[a + b*x + c*x^2] - x*#1] - 2*a*b^2*f^2*Log[ -Sqrt[a] + Sqrt[a + b*x + c*x^2] - x*#1] + a^2*c*f^2*Log[-Sqrt[a] + Sqrt[a + b*x + c*x^2] - x*#1] - 4*Sqrt[a]*b*c*d*f*Log[x]*#1 - 4*a^(3/2)*b*f^2*Lo g[x]*#1 + 4*Sqrt[a]*b*c*d*f*Log[-Sqrt[a] + Sqrt[a + b*x + c*x^2] - x*#1]*# 1 + 4*a^(3/2)*b*f^2*Log[-Sqrt[a] + Sqrt[a + b*x + c*x^2] - x*#1]*#1 + c^2* d^2*Log[x]*#1^2 + b^2*d*f*Log[x]*#1^2 + 2*a*c*d*f*Log[x]*#1^2 + a^2*f^2*Lo g[x]*#1^2 - c^2*d^2*Log[-Sqrt[a] + Sqrt[a + b*x + c*x^2] - x*#1]*#1^2 - b^ 2*d*f*Log[-Sqrt[a] + Sqrt[a + b*x + c*x^2] - x*#1]*#1^2 - 2*a*c*d*f*Log[-S qrt[a] + Sqrt[a + b*x + c*x^2] - x*#1]*#1^2 - a^2*f^2*Log[-Sqrt[a] + Sqrt[ a + b*x + c*x^2] - x*#1]*#1^2)/(-(Sqrt[a]*b*f) + c*d*#1 + 2*a*f*#1 - d*#1^ 3) & ])/(4*Sqrt[c]*f^2)
Time = 0.78 (sec) , antiderivative size = 321, normalized size of antiderivative = 1.02, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {1309, 27, 2144, 27, 1092, 219, 1366, 25, 27, 1154, 219}
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.
| \(\displaystyle \int \frac {\left (a+b x+c x^2\right )^{3/2}}{d-f x^2} \, dx\) |
| \(\Big \downarrow \) 1309 |
| \(\displaystyle \frac {\int \frac {5 d b^2+16 (c d+a f) x b+\left (3 f b^2+8 c^2 d+12 a c f\right ) x^2+4 a (c d+2 a f)}{4 \sqrt {c x^2+b x+a} \left (d-f x^2\right )}dx}{2 f}-\frac {(5 b+2 c x) \sqrt {a+b x+c x^2}}{4 f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {5 d b^2+16 (c d+a f) x b+\left (3 f b^2+8 c^2 d+12 a c f\right ) x^2+4 a (c d+2 a f)}{\sqrt {c x^2+b x+a} \left (d-f x^2\right )}dx}{8 f}-\frac {(5 b+2 c x) \sqrt {a+b x+c x^2}}{4 f}\) |
| \(\Big \downarrow \) 2144 |
| \(\displaystyle \frac {-\frac {\int -\frac {8 \left (c^2 d^2+2 a c f d+f \left (f a^2+b^2 d\right )+2 b f (c d+a f) x\right )}{\sqrt {c x^2+b x+a} \left (d-f x^2\right )}dx}{f}-\frac {\left (12 a c f+3 b^2 f+8 c^2 d\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{f}}{8 f}-\frac {(5 b+2 c x) \sqrt {a+b x+c x^2}}{4 f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {8 \int \frac {c^2 d^2+2 a c f d+f \left (f a^2+b^2 d\right )+2 b f (c d+a f) x}{\sqrt {c x^2+b x+a} \left (d-f x^2\right )}dx}{f}-\frac {\left (12 a c f+3 b^2 f+8 c^2 d\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{f}}{8 f}-\frac {(5 b+2 c x) \sqrt {a+b x+c x^2}}{4 f}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {\frac {8 \int \frac {c^2 d^2+2 a c f d+f \left (f a^2+b^2 d\right )+2 b f (c d+a f) x}{\sqrt {c x^2+b x+a} \left (d-f x^2\right )}dx}{f}-\frac {2 \left (12 a c f+3 b^2 f+8 c^2 d\right ) \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{f}}{8 f}-\frac {(5 b+2 c x) \sqrt {a+b x+c x^2}}{4 f}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {8 \int \frac {c^2 d^2+2 a c f d+f \left (f a^2+b^2 d\right )+2 b f (c d+a f) x}{\sqrt {c x^2+b x+a} \left (d-f x^2\right )}dx}{f}-\frac {\text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (12 a c f+3 b^2 f+8 c^2 d\right )}{\sqrt {c} f}}{8 f}-\frac {(5 b+2 c x) \sqrt {a+b x+c x^2}}{4 f}\) |
| \(\Big \downarrow \) 1366 |
| \(\displaystyle \frac {\frac {8 \left (\frac {\sqrt {f} \left (a f+b \sqrt {d} \sqrt {f}+c d\right )^2 \int \frac {1}{\sqrt {f} \left (\sqrt {d}-\sqrt {f} x\right ) \sqrt {c x^2+b x+a}}dx}{2 \sqrt {d}}-\frac {\sqrt {f} \left (a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d\right )^2 \int -\frac {1}{\sqrt {f} \left (\sqrt {f} x+\sqrt {d}\right ) \sqrt {c x^2+b x+a}}dx}{2 \sqrt {d}}\right )}{f}-\frac {\text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (12 a c f+3 b^2 f+8 c^2 d\right )}{\sqrt {c} f}}{8 f}-\frac {(5 b+2 c x) \sqrt {a+b x+c x^2}}{4 f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {8 \left (\frac {\sqrt {f} \left (a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d\right )^2 \int \frac {1}{\sqrt {f} \left (\sqrt {f} x+\sqrt {d}\right ) \sqrt {c x^2+b x+a}}dx}{2 \sqrt {d}}+\frac {\sqrt {f} \left (a f+b \sqrt {d} \sqrt {f}+c d\right )^2 \int \frac {1}{\sqrt {f} \left (\sqrt {d}-\sqrt {f} x\right ) \sqrt {c x^2+b x+a}}dx}{2 \sqrt {d}}\right )}{f}-\frac {\text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (12 a c f+3 b^2 f+8 c^2 d\right )}{\sqrt {c} f}}{8 f}-\frac {(5 b+2 c x) \sqrt {a+b x+c x^2}}{4 f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {8 \left (\frac {\left (a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d\right )^2 \int \frac {1}{\left (\sqrt {f} x+\sqrt {d}\right ) \sqrt {c x^2+b x+a}}dx}{2 \sqrt {d}}+\frac {\left (a f+b \sqrt {d} \sqrt {f}+c d\right )^2 \int \frac {1}{\left (\sqrt {d}-\sqrt {f} x\right ) \sqrt {c x^2+b x+a}}dx}{2 \sqrt {d}}\right )}{f}-\frac {\text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (12 a c f+3 b^2 f+8 c^2 d\right )}{\sqrt {c} f}}{8 f}-\frac {(5 b+2 c x) \sqrt {a+b x+c x^2}}{4 f}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {\frac {8 \left (-\frac {\left (a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d\right )^2 \int \frac {1}{4 \left (-\sqrt {d} \sqrt {f} b+c d+a f\right )-\frac {\left (-2 \sqrt {f} a+\left (2 c \sqrt {d}-b \sqrt {f}\right ) x+b \sqrt {d}\right )^2}{c x^2+b x+a}}d\left (-\frac {-2 \sqrt {f} a+\left (2 c \sqrt {d}-b \sqrt {f}\right ) x+b \sqrt {d}}{\sqrt {c x^2+b x+a}}\right )}{\sqrt {d}}-\frac {\left (a f+b \sqrt {d} \sqrt {f}+c d\right )^2 \int \frac {1}{4 \left (\sqrt {d} \sqrt {f} b+c d+a f\right )-\frac {\left (2 \sqrt {f} a+\left (\sqrt {f} b+2 c \sqrt {d}\right ) x+b \sqrt {d}\right )^2}{c x^2+b x+a}}d\left (-\frac {2 \sqrt {f} a+\left (\sqrt {f} b+2 c \sqrt {d}\right ) x+b \sqrt {d}}{\sqrt {c x^2+b x+a}}\right )}{\sqrt {d}}\right )}{f}-\frac {\text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (12 a c f+3 b^2 f+8 c^2 d\right )}{\sqrt {c} f}}{8 f}-\frac {(5 b+2 c x) \sqrt {a+b x+c x^2}}{4 f}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {8 \left (\frac {\left (a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d\right )^{3/2} \text {arctanh}\left (\frac {-2 a \sqrt {f}+x \left (2 c \sqrt {d}-b \sqrt {f}\right )+b \sqrt {d}}{2 \sqrt {a+b x+c x^2} \sqrt {a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d}}\right )}{2 \sqrt {d}}+\frac {\left (a f+b \sqrt {d} \sqrt {f}+c d\right )^{3/2} \text {arctanh}\left (\frac {2 a \sqrt {f}+x \left (b \sqrt {f}+2 c \sqrt {d}\right )+b \sqrt {d}}{2 \sqrt {a+b x+c x^2} \sqrt {a f+b \sqrt {d} \sqrt {f}+c d}}\right )}{2 \sqrt {d}}\right )}{f}-\frac {\text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (12 a c f+3 b^2 f+8 c^2 d\right )}{\sqrt {c} f}}{8 f}-\frac {(5 b+2 c x) \sqrt {a+b x+c x^2}}{4 f}\) |
Input:
Int[(a + b*x + c*x^2)^(3/2)/(d - f*x^2),x]
Output:
-1/4*((5*b + 2*c*x)*Sqrt[a + b*x + c*x^2])/f + (-(((8*c^2*d + 3*b^2*f + 12 *a*c*f)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(Sqrt[c]*f )) + (8*(((c*d - b*Sqrt[d]*Sqrt[f] + a*f)^(3/2)*ArcTanh[(b*Sqrt[d] - 2*a*S qrt[f] + (2*c*Sqrt[d] - b*Sqrt[f])*x)/(2*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a* f]*Sqrt[a + b*x + c*x^2])])/(2*Sqrt[d]) + ((c*d + b*Sqrt[d]*Sqrt[f] + a*f) ^(3/2)*ArcTanh[(b*Sqrt[d] + 2*a*Sqrt[f] + (2*c*Sqrt[d] + b*Sqrt[f])*x)/(2* Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + b*x + c*x^2])])/(2*Sqrt[d]))) /f)/(8*f)
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_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2 Subst[I nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a , b, c}, x]
Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym bol] :> Simp[-2 Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c , d, e}, x]
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_.) + (f_.)*(x_)^2)^(q_), x _Symbol] :> Simp[(b*(3*p + 2*q) + 2*c*(p + q)*x)*(a + b*x + c*x^2)^(p - 1)* ((d + f*x^2)^(q + 1)/(2*f*(p + q)*(2*p + 2*q + 1))), x] - Simp[1/(2*f*(p + q)*(2*p + 2*q + 1)) Int[(a + b*x + c*x^2)^(p - 2)*(d + f*x^2)^q*Simp[b^2* d*(p - 1)*(2*p + q) - (p + q)*(b^2*d*(1 - p) - 2*a*(c*d - a*f*(2*p + 2*q + 1))) - (2*b*(c*d - a*f)*(1 - p)*(2*p + q) - 2*(p + q)*b*(2*c*d*(2*p + q) - (c*d + a*f)*(2*p + 2*q + 1)))*x + (b^2*f*p*(1 - p) + 2*c*(p + q)*(c*d*(2*p - 1) - a*f*(4*p + 2*q - 1)))*x^2, x], x], x] /; FreeQ[{a, b, c, d, f, q}, x ] && NeQ[b^2 - 4*a*c, 0] && GtQ[p, 1] && NeQ[p + q, 0] && NeQ[2*p + 2*q + 1 , 0] && !IGtQ[p, 0] && !IGtQ[q, 0]
Int[((g_.) + (h_.)*(x_))/(((a_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + ( f_.)*(x_)^2]), x_Symbol] :> With[{q = Rt[(-a)*c, 2]}, Simp[(h/2 + c*(g/(2*q ))) Int[1/((-q + c*x)*Sqrt[d + e*x + f*x^2]), x], x] + Simp[(h/2 - c*(g/( 2*q))) Int[1/((q + c*x)*Sqrt[d + e*x + f*x^2]), x], x]] /; FreeQ[{a, c, d , e, f, g, h}, x] && NeQ[e^2 - 4*d*f, 0] && PosQ[(-a)*c]
Int[(Px_)/(((a_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symbol] :> With[{A = Coeff[Px, x, 0], B = Coeff[Px, x, 1], C = Coeff[Px, x, 2]}, Simp[C/c Int[1/Sqrt[d + e*x + f*x^2], x], x] + Simp[1/c Int[(A* c - a*C + B*c*x)/((a + c*x^2)*Sqrt[d + e*x + f*x^2]), x], x]] /; FreeQ[{a, c, d, e, f}, x] && PolyQ[Px, x, 2]
Leaf count of result is larger than twice the leaf count of optimal. \(541\) vs. \(2(245)=490\).
Time = 2.50 (sec) , antiderivative size = 542, normalized size of antiderivative = 1.72
| method | result | size |
| risch | \(-\frac {\left (2 c x +5 b \right ) \sqrt {c \,x^{2}+b x +a}}{4 f}-\frac {\frac {\left (12 a c f +3 b^{2} f +8 c^{2} d \right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{f \sqrt {c}}-\frac {4 \left (2 \sqrt {d f}\, a b f +2 \sqrt {d f}\, b c d -a^{2} f^{2}-2 a c d f -b^{2} d f -c^{2} d^{2}\right ) \ln \left (\frac {\frac {-2 b \sqrt {d f}+2 a f +2 c d}{f}+\frac {\left (-2 c \sqrt {d f}+f b \right ) \left (x +\frac {\sqrt {d f}}{f}\right )}{f}+2 \sqrt {\frac {-b \sqrt {d f}+a f +c d}{f}}\, \sqrt {c \left (x +\frac {\sqrt {d f}}{f}\right )^{2}+\frac {\left (-2 c \sqrt {d f}+f b \right ) \left (x +\frac {\sqrt {d f}}{f}\right )}{f}+\frac {-b \sqrt {d f}+a f +c d}{f}}}{x +\frac {\sqrt {d f}}{f}}\right )}{\sqrt {d f}\, f \sqrt {\frac {-b \sqrt {d f}+a f +c d}{f}}}-\frac {4 \left (2 \sqrt {d f}\, a b f +2 \sqrt {d f}\, b c d +a^{2} f^{2}+2 a c d f +b^{2} d f +c^{2} d^{2}\right ) \ln \left (\frac {\frac {2 b \sqrt {d f}+2 a f +2 c d}{f}+\frac {\left (2 c \sqrt {d f}+f b \right ) \left (x -\frac {\sqrt {d f}}{f}\right )}{f}+2 \sqrt {\frac {b \sqrt {d f}+a f +c d}{f}}\, \sqrt {c \left (x -\frac {\sqrt {d f}}{f}\right )^{2}+\frac {\left (2 c \sqrt {d f}+f b \right ) \left (x -\frac {\sqrt {d f}}{f}\right )}{f}+\frac {b \sqrt {d f}+a f +c d}{f}}}{x -\frac {\sqrt {d f}}{f}}\right )}{\sqrt {d f}\, f \sqrt {\frac {b \sqrt {d f}+a f +c d}{f}}}}{8 f}\) | \(542\) |
| default | \(\text {Expression too large to display}\) | \(1477\) |
Input:
int((c*x^2+b*x+a)^(3/2)/(-f*x^2+d),x,method=_RETURNVERBOSE)
Output:
-1/4*(2*c*x+5*b)*(c*x^2+b*x+a)^(1/2)/f-1/8/f*(1/f*(12*a*c*f+3*b^2*f+8*c^2* d)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))/c^(1/2)-4*(2*(d*f)^(1/2)*a* b*f+2*(d*f)^(1/2)*b*c*d-a^2*f^2-2*a*c*d*f-b^2*d*f-c^2*d^2)/(d*f)^(1/2)/f/( 1/f*(-b*(d*f)^(1/2)+a*f+c*d))^(1/2)*ln((2/f*(-b*(d*f)^(1/2)+a*f+c*d)+1/f*( -2*c*(d*f)^(1/2)+f*b)*(x+(d*f)^(1/2)/f)+2*(1/f*(-b*(d*f)^(1/2)+a*f+c*d))^( 1/2)*(c*(x+(d*f)^(1/2)/f)^2+1/f*(-2*c*(d*f)^(1/2)+f*b)*(x+(d*f)^(1/2)/f)+1 /f*(-b*(d*f)^(1/2)+a*f+c*d))^(1/2))/(x+(d*f)^(1/2)/f))-4*(2*(d*f)^(1/2)*a* b*f+2*(d*f)^(1/2)*b*c*d+a^2*f^2+2*a*c*d*f+b^2*d*f+c^2*d^2)/(d*f)^(1/2)/f/( (b*(d*f)^(1/2)+a*f+c*d)/f)^(1/2)*ln((2*(b*(d*f)^(1/2)+a*f+c*d)/f+(2*c*(d*f )^(1/2)+f*b)/f*(x-(d*f)^(1/2)/f)+2*((b*(d*f)^(1/2)+a*f+c*d)/f)^(1/2)*(c*(x -(d*f)^(1/2)/f)^2+(2*c*(d*f)^(1/2)+f*b)/f*(x-(d*f)^(1/2)/f)+(b*(d*f)^(1/2) +a*f+c*d)/f)^(1/2))/(x-(d*f)^(1/2)/f)))
Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{d-f x^2} \, dx=\text {Timed out} \] Input:
integrate((c*x^2+b*x+a)^(3/2)/(-f*x^2+d),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{d-f x^2} \, dx=- \int \frac {a \sqrt {a + b x + c x^{2}}}{- d + f x^{2}}\, dx - \int \frac {b x \sqrt {a + b x + c x^{2}}}{- d + f x^{2}}\, dx - \int \frac {c x^{2} \sqrt {a + b x + c x^{2}}}{- d + f x^{2}}\, dx \] Input:
integrate((c*x**2+b*x+a)**(3/2)/(-f*x**2+d),x)
Output:
-Integral(a*sqrt(a + b*x + c*x**2)/(-d + f*x**2), x) - Integral(b*x*sqrt(a + b*x + c*x**2)/(-d + f*x**2), x) - Integral(c*x**2*sqrt(a + b*x + c*x**2 )/(-d + f*x**2), x)
Exception generated. \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{d-f x^2} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((c*x^2+b*x+a)^(3/2)/(-f*x^2+d),x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(((c*sqrt(4*d*f))/(2*f^2)>0)', se e `assume?
Exception generated. \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{d-f x^2} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((c*x^2+b*x+a)^(3/2)/(-f*x^2+d),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{d-f x^2} \, dx=\int \frac {{\left (c\,x^2+b\,x+a\right )}^{3/2}}{d-f\,x^2} \,d x \] Input:
int((a + b*x + c*x^2)^(3/2)/(d - f*x^2),x)
Output:
int((a + b*x + c*x^2)^(3/2)/(d - f*x^2), x)
\[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{d-f x^2} \, dx=\int \frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{-f \,x^{2}+d}d x \] Input:
int((c*x^2+b*x+a)^(3/2)/(-f*x^2+d),x)
Output:
int((c*x^2+b*x+a)^(3/2)/(-f*x^2+d),x)