Integrand size = 26, antiderivative size = 349 \[ \int \frac {x \left (a+b x+c x^2\right )^{3/2}}{d-f x^2} \, dx=-\frac {\left (8 c^2 d+b^2 f+8 a c f+2 b c f x\right ) \sqrt {a+b x+c x^2}}{8 c f^2}-\frac {\left (a+b x+c x^2\right )^{3/2}}{3 f}-\frac {b \left (24 c^2 d-b^2 f+12 a c f\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{16 c^{3/2} f^2}-\frac {\left (c d-b \sqrt {d} \sqrt {f}+a f\right )^{3/2} \text {arctanh}\left (\frac {b \sqrt {d}-2 a \sqrt {f}+\left (2 c \sqrt {d}-b \sqrt {f}\right ) x}{2 \sqrt {c d-b \sqrt {d} \sqrt {f}+a f} \sqrt {a+b x+c x^2}}\right )}{2 f^{5/2}}+\frac {\left (c d+b \sqrt {d} \sqrt {f}+a f\right )^{3/2} \text {arctanh}\left (\frac {b \sqrt {d}+2 a \sqrt {f}+\left (2 c \sqrt {d}+b \sqrt {f}\right ) x}{2 \sqrt {c d+b \sqrt {d} \sqrt {f}+a f} \sqrt {a+b x+c x^2}}\right )}{2 f^{5/2}} \]
-1/3*(c*x^2+b*x+a)^(3/2)/f-1/16*b*(12*a*c*f-b^2*f+24*c^2*d)*arctanh(1/2*(2 *c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))/c^(3/2)/f^2-1/2*arctanh(1/2*(b*d^(1/2 )-2*a*f^(1/2)+x*(2*c*d^(1/2)-b*f^(1/2)))/(c*x^2+b*x+a)^(1/2)/(c*d+a*f-b*d^ (1/2)*f^(1/2))^(1/2))*(c*d+a*f-b*d^(1/2)*f^(1/2))^(3/2)/f^(5/2)+1/2*arctan h(1/2*(b*d^(1/2)+2*a*f^(1/2)+x*(2*c*d^(1/2)+b*f^(1/2)))/(c*x^2+b*x+a)^(1/2 )/(c*d+a*f+b*d^(1/2)*f^(1/2))^(1/2))*(c*d+a*f+b*d^(1/2)*f^(1/2))^(3/2)/f^( 5/2)-1/8*(2*b*c*f*x+8*a*c*f+b^2*f+8*c^2*d)*(c*x^2+b*x+a)^(1/2)/c/f^2
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.98 (sec) , antiderivative size = 619, normalized size of antiderivative = 1.77 \[ \int \frac {x \left (a+b x+c x^2\right )^{3/2}}{d-f x^2} \, dx=\frac {-2 \sqrt {c} \sqrt {a+x (b+c x)} \left (3 b^2 f+2 c f (16 a+7 b x)+8 c^2 \left (3 d+f x^2\right )\right )+3 b \left (-24 c^2 d+b^2 f-12 a c f\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )-24 c^{3/2} \text {RootSum}\left [b^2 d-a^2 f-4 b \sqrt {c} d \text {$\#$1}+4 c d \text {$\#$1}^2+2 a f \text {$\#$1}^2-f \text {$\#$1}^4\&,\frac {2 b^2 c d^2 \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )-a c^2 d^2 \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )+a b^2 d f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )-2 a^2 c d f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )-a^3 f^2 \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )-4 b c^{3/2} d^2 \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}-4 a b \sqrt {c} d f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}+c^2 d^2 \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2+b^2 d f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2+2 a c d f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2+a^2 f^2 \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{b \sqrt {c} d-2 c d \text {$\#$1}-a f \text {$\#$1}+f \text {$\#$1}^3}\&\right ]}{48 c^{3/2} f^2} \]
(-2*Sqrt[c]*Sqrt[a + x*(b + c*x)]*(3*b^2*f + 2*c*f*(16*a + 7*b*x) + 8*c^2* (3*d + f*x^2)) + 3*b*(-24*c^2*d + b^2*f - 12*a*c*f)*ArcTanh[(b + 2*c*x)/(2 *Sqrt[c]*Sqrt[a + x*(b + c*x)])] - 24*c^(3/2)*RootSum[b^2*d - a^2*f - 4*b* Sqrt[c]*d*#1 + 4*c*d*#1^2 + 2*a*f*#1^2 - f*#1^4 & , (2*b^2*c*d^2*Log[-(Sqr t[c]*x) + Sqrt[a + b*x + c*x^2] - #1] - a*c^2*d^2*Log[-(Sqrt[c]*x) + Sqrt[ a + b*x + c*x^2] - #1] + a*b^2*d*f*Log[-(Sqrt[c]*x) + Sqrt[a + b*x + c*x^2 ] - #1] - 2*a^2*c*d*f*Log[-(Sqrt[c]*x) + Sqrt[a + b*x + c*x^2] - #1] - a^3 *f^2*Log[-(Sqrt[c]*x) + Sqrt[a + b*x + c*x^2] - #1] - 4*b*c^(3/2)*d^2*Log[ -(Sqrt[c]*x) + Sqrt[a + b*x + c*x^2] - #1]*#1 - 4*a*b*Sqrt[c]*d*f*Log[-(Sq rt[c]*x) + Sqrt[a + b*x + c*x^2] - #1]*#1 + c^2*d^2*Log[-(Sqrt[c]*x) + Sqr t[a + b*x + c*x^2] - #1]*#1^2 + b^2*d*f*Log[-(Sqrt[c]*x) + Sqrt[a + b*x + c*x^2] - #1]*#1^2 + 2*a*c*d*f*Log[-(Sqrt[c]*x) + Sqrt[a + b*x + c*x^2] - # 1]*#1^2 + a^2*f^2*Log[-(Sqrt[c]*x) + Sqrt[a + b*x + c*x^2] - #1]*#1^2)/(b* Sqrt[c]*d - 2*c*d*#1 - a*f*#1 + f*#1^3) & ])/(48*c^(3/2)*f^2)
Time = 0.94 (sec) , antiderivative size = 369, normalized size of antiderivative = 1.06, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {1354, 27, 2140, 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.
| \(\displaystyle \int \frac {x \left (a+b x+c x^2\right )^{3/2}}{d-f x^2} \, dx\) |
| \(\Big \downarrow \) 1354 |
| \(\displaystyle \frac {\int \frac {3 \sqrt {c x^2+b x+a} \left (b f x^2+2 (c d+a f) x+b d\right )}{2 \left (d-f x^2\right )}dx}{3 f}-\frac {\left (a+b x+c x^2\right )^{3/2}}{3 f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\sqrt {c x^2+b x+a} \left (b f x^2+2 (c d+a f) x+b d\right )}{d-f x^2}dx}{2 f}-\frac {\left (a+b x+c x^2\right )^{3/2}}{3 f}\) |
| \(\Big \downarrow \) 2140 |
| \(\displaystyle \frac {-\frac {\int -\frac {b f^2 \left (-f b^2+24 c^2 d+12 a c f\right ) x^2+16 c f \left (d f b^2+(c d+a f)^2\right ) x+b d f \left (f b^2+8 c^2 d+20 a c f\right )}{4 \sqrt {c x^2+b x+a} \left (d-f x^2\right )}dx}{2 c f^2}-\frac {\sqrt {a+b x+c x^2} \left (8 a c f+b^2 f+2 b c f x+8 c^2 d\right )}{4 c f}}{2 f}-\frac {\left (a+b x+c x^2\right )^{3/2}}{3 f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {b f^2 \left (-f b^2+24 c^2 d+12 a c f\right ) x^2+16 c f \left (d f b^2+(c d+a f)^2\right ) x+b d f \left (f b^2+8 c^2 d+20 a c f\right )}{\sqrt {c x^2+b x+a} \left (d-f x^2\right )}dx}{8 c f^2}-\frac {\sqrt {a+b x+c x^2} \left (8 a c f+b^2 f+2 b c f x+8 c^2 d\right )}{4 c f}}{2 f}-\frac {\left (a+b x+c x^2\right )^{3/2}}{3 f}\) |
| \(\Big \downarrow \) 2144 |
| \(\displaystyle \frac {\frac {-b f \left (12 a c f+b^2 (-f)+24 c^2 d\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx-\frac {\int -\frac {16 c f^2 \left (2 b d (c d+a f)+\left (d f b^2+(c d+a f)^2\right ) x\right )}{\sqrt {c x^2+b x+a} \left (d-f x^2\right )}dx}{f}}{8 c f^2}-\frac {\sqrt {a+b x+c x^2} \left (8 a c f+b^2 f+2 b c f x+8 c^2 d\right )}{4 c f}}{2 f}-\frac {\left (a+b x+c x^2\right )^{3/2}}{3 f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {16 c f \int \frac {2 b d (c d+a f)+\left (d f b^2+(c d+a f)^2\right ) x}{\sqrt {c x^2+b x+a} \left (d-f x^2\right )}dx-b f \left (12 a c f+b^2 (-f)+24 c^2 d\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{8 c f^2}-\frac {\sqrt {a+b x+c x^2} \left (8 a c f+b^2 f+2 b c f x+8 c^2 d\right )}{4 c f}}{2 f}-\frac {\left (a+b x+c x^2\right )^{3/2}}{3 f}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {\frac {16 c f \int \frac {2 b d (c d+a f)+\left (d f b^2+(c d+a f)^2\right ) x}{\sqrt {c x^2+b x+a} \left (d-f x^2\right )}dx-2 b f \left (12 a c f+b^2 (-f)+24 c^2 d\right ) \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{8 c f^2}-\frac {\sqrt {a+b x+c x^2} \left (8 a c f+b^2 f+2 b c f x+8 c^2 d\right )}{4 c f}}{2 f}-\frac {\left (a+b x+c x^2\right )^{3/2}}{3 f}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {16 c f \int \frac {2 b d (c d+a f)+\left (d f b^2+(c d+a f)^2\right ) x}{\sqrt {c x^2+b x+a} \left (d-f x^2\right )}dx-\frac {b f \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (12 a c f+b^2 (-f)+24 c^2 d\right )}{\sqrt {c}}}{8 c f^2}-\frac {\sqrt {a+b x+c x^2} \left (8 a c f+b^2 f+2 b c f x+8 c^2 d\right )}{4 c f}}{2 f}-\frac {\left (a+b x+c x^2\right )^{3/2}}{3 f}\) |
| \(\Big \downarrow \) 1366 |
| \(\displaystyle \frac {\frac {16 c f \left (\frac {1}{2} \left (a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d\right )^2 \int -\frac {1}{\sqrt {f} \left (\sqrt {f} x+\sqrt {d}\right ) \sqrt {c x^2+b x+a}}dx+\frac {1}{2} \left (a f+b \sqrt {d} \sqrt {f}+c d\right )^2 \int \frac {1}{\sqrt {f} \left (\sqrt {d}-\sqrt {f} x\right ) \sqrt {c x^2+b x+a}}dx\right )-\frac {b f \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (12 a c f+b^2 (-f)+24 c^2 d\right )}{\sqrt {c}}}{8 c f^2}-\frac {\sqrt {a+b x+c x^2} \left (8 a c f+b^2 f+2 b c f x+8 c^2 d\right )}{4 c f}}{2 f}-\frac {\left (a+b x+c x^2\right )^{3/2}}{3 f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {16 c f \left (\frac {1}{2} \left (a f+b \sqrt {d} \sqrt {f}+c d\right )^2 \int \frac {1}{\sqrt {f} \left (\sqrt {d}-\sqrt {f} x\right ) \sqrt {c x^2+b x+a}}dx-\frac {1}{2} \left (a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d\right )^2 \int \frac {1}{\sqrt {f} \left (\sqrt {f} x+\sqrt {d}\right ) \sqrt {c x^2+b x+a}}dx\right )-\frac {b f \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (12 a c f+b^2 (-f)+24 c^2 d\right )}{\sqrt {c}}}{8 c f^2}-\frac {\sqrt {a+b x+c x^2} \left (8 a c f+b^2 f+2 b c f x+8 c^2 d\right )}{4 c f}}{2 f}-\frac {\left (a+b x+c x^2\right )^{3/2}}{3 f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {16 c f \left (\frac {\left (a f+b \sqrt {d} \sqrt {f}+c d\right )^2 \int \frac {1}{\left (\sqrt {d}-\sqrt {f} x\right ) \sqrt {c x^2+b x+a}}dx}{2 \sqrt {f}}-\frac {\left (a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d\right )^2 \int \frac {1}{\left (\sqrt {f} x+\sqrt {d}\right ) \sqrt {c x^2+b x+a}}dx}{2 \sqrt {f}}\right )-\frac {b f \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (12 a c f+b^2 (-f)+24 c^2 d\right )}{\sqrt {c}}}{8 c f^2}-\frac {\sqrt {a+b x+c x^2} \left (8 a c f+b^2 f+2 b c f x+8 c^2 d\right )}{4 c f}}{2 f}-\frac {\left (a+b x+c x^2\right )^{3/2}}{3 f}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {\frac {16 c f \left (\frac {\left (a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d\right )^2 \int \frac {1}{4 \left (-\sqrt {d} \sqrt {f} b+c d+a f\right )-\frac {\left (-2 \sqrt {f} a+\left (2 c \sqrt {d}-b \sqrt {f}\right ) x+b \sqrt {d}\right )^2}{c x^2+b x+a}}d\left (-\frac {-2 \sqrt {f} a+\left (2 c \sqrt {d}-b \sqrt {f}\right ) x+b \sqrt {d}}{\sqrt {c x^2+b x+a}}\right )}{\sqrt {f}}-\frac {\left (a f+b \sqrt {d} \sqrt {f}+c d\right )^2 \int \frac {1}{4 \left (\sqrt {d} \sqrt {f} b+c d+a f\right )-\frac {\left (2 \sqrt {f} a+\left (\sqrt {f} b+2 c \sqrt {d}\right ) x+b \sqrt {d}\right )^2}{c x^2+b x+a}}d\left (-\frac {2 \sqrt {f} a+\left (\sqrt {f} b+2 c \sqrt {d}\right ) x+b \sqrt {d}}{\sqrt {c x^2+b x+a}}\right )}{\sqrt {f}}\right )-\frac {b f \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (12 a c f+b^2 (-f)+24 c^2 d\right )}{\sqrt {c}}}{8 c f^2}-\frac {\sqrt {a+b x+c x^2} \left (8 a c f+b^2 f+2 b c f x+8 c^2 d\right )}{4 c f}}{2 f}-\frac {\left (a+b x+c x^2\right )^{3/2}}{3 f}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {16 c f \left (\frac {\left (a f+b \sqrt {d} \sqrt {f}+c d\right )^{3/2} \text {arctanh}\left (\frac {2 a \sqrt {f}+x \left (b \sqrt {f}+2 c \sqrt {d}\right )+b \sqrt {d}}{2 \sqrt {a+b x+c x^2} \sqrt {a f+b \sqrt {d} \sqrt {f}+c d}}\right )}{2 \sqrt {f}}-\frac {\left (a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d\right )^{3/2} \text {arctanh}\left (\frac {-2 a \sqrt {f}+x \left (2 c \sqrt {d}-b \sqrt {f}\right )+b \sqrt {d}}{2 \sqrt {a+b x+c x^2} \sqrt {a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d}}\right )}{2 \sqrt {f}}\right )-\frac {b f \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (12 a c f+b^2 (-f)+24 c^2 d\right )}{\sqrt {c}}}{8 c f^2}-\frac {\sqrt {a+b x+c x^2} \left (8 a c f+b^2 f+2 b c f x+8 c^2 d\right )}{4 c f}}{2 f}-\frac {\left (a+b x+c x^2\right )^{3/2}}{3 f}\) |
-1/3*(a + b*x + c*x^2)^(3/2)/f + (-1/4*((8*c^2*d + b^2*f + 8*a*c*f + 2*b*c *f*x)*Sqrt[a + b*x + c*x^2])/(c*f) + (-((b*f*(24*c^2*d - b^2*f + 12*a*c*f) *ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/Sqrt[c]) + 16*c*f *(-1/2*((c*d - b*Sqrt[d]*Sqrt[f] + a*f)^(3/2)*ArcTanh[(b*Sqrt[d] - 2*a*Sqr t[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])])/Sqrt[f] + ((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[f])))/(8*c* f^2))/(2*f)
3.1.86.3.1 Defintions of rubi rules used
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[((g_.) + (h_.)*(x_))*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_) + (f _.)*(x_)^2)^(q_), x_Symbol] :> Simp[h*(a + b*x + c*x^2)^p*((d + f*x^2)^(q + 1)/(2*f*(p + q + 1))), x] - Simp[1/(2*f*(p + q + 1)) Int[(a + b*x + c*x^ 2)^(p - 1)*(d + f*x^2)^q*Simp[h*p*(b*d) + a*(-2*g*f)*(p + q + 1) + (2*h*p*( c*d - a*f) + b*(-2*g*f)*(p + q + 1))*x + (h*p*((-b)*f) + c*(-2*g*f)*(p + q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, f, g, h, q}, x] && NeQ[b^2 - 4* a*c, 0] && GtQ[p, 0] && NeQ[p + q + 1, 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_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_) + (f_.)*(x_)^2)^(q_ ), x_Symbol] :> With[{A = Coeff[Px, x, 0], B = Coeff[Px, x, 1], C = Coeff[P x, x, 2]}, Simp[(B*c*f*(2*p + 2*q + 3) + C*(b*f*p) + 2*c*C*f*(p + q + 1)*x) *(a + b*x + c*x^2)^p*((d + f*x^2)^(q + 1)/(2*c*f^2*(p + q + 1)*(2*p + 2*q + 3))), x] - Simp[1/(2*c*f^2*(p + q + 1)*(2*p + 2*q + 3)) Int[(a + b*x + c *x^2)^(p - 1)*(d + f*x^2)^q*Simp[p*(b*d)*(C*((-b)*f)*(q + 1) - c*((-B)*f)*( 2*p + 2*q + 3)) + (p + q + 1)*(b^2*C*d*f*p + a*c*(C*(2*d*f) + f*(-2*A*f)*(2 *p + 2*q + 3))) + (2*p*(c*d - a*f)*(C*((-b)*f)*(q + 1) - c*((-B)*f)*(2*p + 2*q + 3)) + (p + q + 1)*((-b)*c*(C*(-4*d*f)*(2*p + q + 2) + f*(2*C*d + 2*A* f)*(2*p + 2*q + 3))))*x + (p*((-b)*f)*(C*((-b)*f)*(q + 1) - c*((-B)*f)*(2*p + 2*q + 3)) + (p + q + 1)*(C*f^2*p*(b^2 - 4*a*c) - c^2*(C*(-4*d*f)*(2*p + q + 2) + f*(2*C*d + 2*A*f)*(2*p + 2*q + 3))))*x^2, x], x], x]] /; FreeQ[{a, b, c, d, f, q}, x] && PolyQ[Px, x, 2] && GtQ[p, 0] && NeQ[p + q + 1, 0] && NeQ[2*p + 2*q + 3, 0] && !IGtQ[p, 0] && !IGtQ[q, 0]
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. \(601\) vs. \(2(275)=550\).
Time = 0.73 (sec) , antiderivative size = 602, normalized size of antiderivative = 1.72
| method | result | size |
| risch | \(-\frac {\left (8 f \,c^{2} x^{2}+14 b c f x +32 a c f +3 b^{2} f +24 c^{2} d \right ) \sqrt {c \,x^{2}+b x +a}}{24 c \,f^{2}}-\frac {\frac {b \left (12 a c f -b^{2} f +24 c^{2} d \right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{\sqrt {c}}-\frac {8 c \left (\sqrt {d f}\, a^{2} f^{2}+2 \sqrt {d f}\, a c d f +\sqrt {d f}\, b^{2} d f +\sqrt {d f}\, c^{2} d^{2}+2 a b d \,f^{2}+2 b c \,d^{2} f \right ) \ln \left (\frac {\frac {2 b \sqrt {d f}+2 f a +2 c d}{f}+\frac {\left (2 c \sqrt {d f}+b f \right ) \left (x -\frac {\sqrt {d f}}{f}\right )}{f}+2 \sqrt {\frac {b \sqrt {d f}+f a +c d}{f}}\, \sqrt {\left (x -\frac {\sqrt {d f}}{f}\right )^{2} c +\frac {\left (2 c \sqrt {d f}+b f \right ) \left (x -\frac {\sqrt {d f}}{f}\right )}{f}+\frac {b \sqrt {d f}+f a +c d}{f}}}{x -\frac {\sqrt {d f}}{f}}\right )}{\sqrt {d f}\, f \sqrt {\frac {b \sqrt {d f}+f a +c d}{f}}}-\frac {8 c \left (\sqrt {d f}\, a^{2} f^{2}+2 \sqrt {d f}\, a c d f +\sqrt {d f}\, b^{2} d f +\sqrt {d f}\, c^{2} d^{2}-2 a b d \,f^{2}-2 b c \,d^{2} f \right ) \ln \left (\frac {\frac {-2 b \sqrt {d f}+2 f a +2 c d}{f}+\frac {\left (-2 c \sqrt {d f}+b f \right ) \left (x +\frac {\sqrt {d f}}{f}\right )}{f}+2 \sqrt {\frac {-b \sqrt {d f}+f a +c d}{f}}\, \sqrt {\left (x +\frac {\sqrt {d f}}{f}\right )^{2} c +\frac {\left (-2 c \sqrt {d f}+b f \right ) \left (x +\frac {\sqrt {d f}}{f}\right )}{f}+\frac {-b \sqrt {d f}+f a +c d}{f}}}{x +\frac {\sqrt {d f}}{f}}\right )}{\sqrt {d f}\, f \sqrt {\frac {-b \sqrt {d f}+f a +c d}{f}}}}{16 f^{2} c}\) | \(602\) |
| default | \(\text {Expression too large to display}\) | \(1473\) |
-1/24*(8*c^2*f*x^2+14*b*c*f*x+32*a*c*f+3*b^2*f+24*c^2*d)*(c*x^2+b*x+a)^(1/ 2)/c/f^2-1/16/f^2/c*(b*(12*a*c*f-b^2*f+24*c^2*d)*ln((1/2*b+c*x)/c^(1/2)+(c *x^2+b*x+a)^(1/2))/c^(1/2)-8*c*((d*f)^(1/2)*a^2*f^2+2*(d*f)^(1/2)*a*c*d*f+ (d*f)^(1/2)*b^2*d*f+(d*f)^(1/2)*c^2*d^2+2*a*b*d*f^2+2*b*c*d^2*f)/(d*f)^(1/ 2)/f/((b*(d*f)^(1/2)+f*a+c*d)/f)^(1/2)*ln((2*(b*(d*f)^(1/2)+f*a+c*d)/f+(2* c*(d*f)^(1/2)+b*f)/f*(x-(d*f)^(1/2)/f)+2*((b*(d*f)^(1/2)+f*a+c*d)/f)^(1/2) *((x-(d*f)^(1/2)/f)^2*c+(2*c*(d*f)^(1/2)+b*f)/f*(x-(d*f)^(1/2)/f)+(b*(d*f) ^(1/2)+f*a+c*d)/f)^(1/2))/(x-(d*f)^(1/2)/f))-8*c*((d*f)^(1/2)*a^2*f^2+2*(d *f)^(1/2)*a*c*d*f+(d*f)^(1/2)*b^2*d*f+(d*f)^(1/2)*c^2*d^2-2*a*b*d*f^2-2*b* c*d^2*f)/(d*f)^(1/2)/f/(1/f*(-b*(d*f)^(1/2)+f*a+c*d))^(1/2)*ln((2/f*(-b*(d *f)^(1/2)+f*a+c*d)+1/f*(-2*c*(d*f)^(1/2)+b*f)*(x+(d*f)^(1/2)/f)+2*(1/f*(-b *(d*f)^(1/2)+f*a+c*d))^(1/2)*((x+(d*f)^(1/2)/f)^2*c+1/f*(-2*c*(d*f)^(1/2)+ b*f)*(x+(d*f)^(1/2)/f)+1/f*(-b*(d*f)^(1/2)+f*a+c*d))^(1/2))/(x+(d*f)^(1/2) /f)))
Timed out. \[ \int \frac {x \left (a+b x+c x^2\right )^{3/2}}{d-f x^2} \, dx=\text {Timed out} \]
\[ \int \frac {x \left (a+b x+c x^2\right )^{3/2}}{d-f x^2} \, dx=- \int \frac {a x \sqrt {a + b x + c x^{2}}}{- d + f x^{2}}\, dx - \int \frac {b x^{2} \sqrt {a + b x + c x^{2}}}{- d + f x^{2}}\, dx - \int \frac {c x^{3} \sqrt {a + b x + c x^{2}}}{- d + f x^{2}}\, dx \]
-Integral(a*x*sqrt(a + b*x + c*x**2)/(-d + f*x**2), x) - Integral(b*x**2*s qrt(a + b*x + c*x**2)/(-d + f*x**2), x) - Integral(c*x**3*sqrt(a + b*x + c *x**2)/(-d + f*x**2), x)
Exception generated. \[ \int \frac {x \left (a+b x+c x^2\right )^{3/2}}{d-f x^2} \, dx=\text {Exception raised: ValueError} \]
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. \[ \int \frac {x \left (a+b x+c x^2\right )^{3/2}}{d-f x^2} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
Timed out. \[ \int \frac {x \left (a+b x+c x^2\right )^{3/2}}{d-f x^2} \, dx=\int \frac {x\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{d-f\,x^2} \,d x \]