Integrand size = 29, antiderivative size = 326 \[ \int \frac {\left (a+c x^2\right )^{3/2} \left (d+e x+f x^2\right )}{g+h x} \, dx=\frac {\left (8 \left (c g^2+a h^2\right ) \left (f g^2-e g h+d h^2\right )-h \left (4 c d g h^2+(f g-e h) \left (4 c g^2+3 a h^2\right )\right ) x\right ) \sqrt {a+c x^2}}{8 h^5}+\frac {\left (4 \left (f g^2-e g h+d h^2\right )-3 h (f g-e h) x\right ) \left (a+c x^2\right )^{3/2}}{12 h^3}+\frac {f \left (a+c x^2\right )^{5/2}}{5 c h}-\frac {\left (3 a^2 h^4 (f g-e h)+8 c^2 g^3 \left (f g^2-h (e g-d h)\right )+12 a c g h^2 \left (f g^2-h (e g-d h)\right )\right ) \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{8 \sqrt {c} h^6}-\frac {\left (c g^2+a h^2\right )^{3/2} \left (f g^2-e g h+d h^2\right ) \text {arctanh}\left (\frac {a h-c g x}{\sqrt {c g^2+a h^2} \sqrt {a+c x^2}}\right )}{h^6} \]
1/12*(4*d*h^2-4*e*g*h+4*f*g^2-3*h*(-e*h+f*g)*x)*(c*x^2+a)^(3/2)/h^3+1/5*f* (c*x^2+a)^(5/2)/c/h-(a*h^2+c*g^2)^(3/2)*(d*h^2-e*g*h+f*g^2)*arctanh((-c*g* x+a*h)/(a*h^2+c*g^2)^(1/2)/(c*x^2+a)^(1/2))/h^6-1/8*(3*a^2*h^4*(-e*h+f*g)+ 8*c^2*g^3*(f*g^2-h*(-d*h+e*g))+12*a*c*g*h^2*(f*g^2-h*(-d*h+e*g)))*arctanh( x*c^(1/2)/(c*x^2+a)^(1/2))/h^6/c^(1/2)+1/8*(8*(a*h^2+c*g^2)*(d*h^2-e*g*h+f *g^2)-h*(4*c*d*g*h^2+(-e*h+f*g)*(3*a*h^2+4*c*g^2))*x)*(c*x^2+a)^(1/2)/h^5
Time = 1.29 (sec) , antiderivative size = 360, normalized size of antiderivative = 1.10 \[ \int \frac {\left (a+c x^2\right )^{3/2} \left (d+e x+f x^2\right )}{g+h x} \, dx=\frac {\frac {h \sqrt {a+c x^2} \left (24 a^2 f h^4+a c h^2 \left (5 h (-32 e g+32 d h+15 e h x)+f \left (160 g^2-75 g h x+48 h^2 x^2\right )\right )+2 c^2 \left (f \left (60 g^4-30 g^3 h x+20 g^2 h^2 x^2-15 g h^3 x^3+12 h^4 x^4\right )+5 h \left (2 d h \left (6 g^2-3 g h x+2 h^2 x^2\right )+e \left (-12 g^3+6 g^2 h x-4 g h^2 x^2+3 h^3 x^3\right )\right )\right )\right )}{c}-240 \left (-c g^2-a h^2\right )^{3/2} \left (f g^2+h (-e g+d h)\right ) \arctan \left (\frac {\sqrt {c} (g+h x)-h \sqrt {a+c x^2}}{\sqrt {-c g^2-a h^2}}\right )+\frac {15 \left (3 a^2 h^4 (f g-e h)+12 a c g h^2 \left (f g^2+h (-e g+d h)\right )+8 c^2 \left (f g^5+g^3 h (-e g+d h)\right )\right ) \log \left (-\sqrt {c} x+\sqrt {a+c x^2}\right )}{\sqrt {c}}}{120 h^6} \]
((h*Sqrt[a + c*x^2]*(24*a^2*f*h^4 + a*c*h^2*(5*h*(-32*e*g + 32*d*h + 15*e* h*x) + f*(160*g^2 - 75*g*h*x + 48*h^2*x^2)) + 2*c^2*(f*(60*g^4 - 30*g^3*h* x + 20*g^2*h^2*x^2 - 15*g*h^3*x^3 + 12*h^4*x^4) + 5*h*(2*d*h*(6*g^2 - 3*g* h*x + 2*h^2*x^2) + e*(-12*g^3 + 6*g^2*h*x - 4*g*h^2*x^2 + 3*h^3*x^3)))))/c - 240*(-(c*g^2) - a*h^2)^(3/2)*(f*g^2 + h*(-(e*g) + d*h))*ArcTan[(Sqrt[c] *(g + h*x) - h*Sqrt[a + c*x^2])/Sqrt[-(c*g^2) - a*h^2]] + (15*(3*a^2*h^4*( f*g - e*h) + 12*a*c*g*h^2*(f*g^2 + h*(-(e*g) + d*h)) + 8*c^2*(f*g^5 + g^3* h*(-(e*g) + d*h)))*Log[-(Sqrt[c]*x) + Sqrt[a + c*x^2]])/Sqrt[c])/(120*h^6)
Time = 0.88 (sec) , antiderivative size = 345, normalized size of antiderivative = 1.06, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.379, Rules used = {2185, 27, 682, 27, 682, 27, 719, 224, 219, 488, 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.
| \(\displaystyle \int \frac {\left (a+c x^2\right )^{3/2} \left (d+e x+f x^2\right )}{g+h x} \, dx\) |
| \(\Big \downarrow \) 2185 |
| \(\displaystyle \frac {\int \frac {5 c h (d h-(f g-e h) x) \left (c x^2+a\right )^{3/2}}{g+h x}dx}{5 c h^2}+\frac {f \left (a+c x^2\right )^{5/2}}{5 c h}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(d h-(f g-e h) x) \left (c x^2+a\right )^{3/2}}{g+h x}dx}{h}+\frac {f \left (a+c x^2\right )^{5/2}}{5 c h}\) |
| \(\Big \downarrow \) 682 |
| \(\displaystyle \frac {\frac {\int \frac {c \left (a h \left (f g^2-h (e g-4 d h)\right )-\left (4 c d g h^2+(f g-e h) \left (4 c g^2+3 a h^2\right )\right ) x\right ) \sqrt {c x^2+a}}{g+h x}dx}{4 c h^2}+\frac {\left (a+c x^2\right )^{3/2} \left (4 \left (d h^2-e g h+f g^2\right )-3 h x (f g-e h)\right )}{12 h^2}}{h}+\frac {f \left (a+c x^2\right )^{5/2}}{5 c h}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {\left (a h \left (f g^2-h (e g-4 d h)\right )-\left (4 c d g h^2+(f g-e h) \left (4 c g^2+3 a h^2\right )\right ) x\right ) \sqrt {c x^2+a}}{g+h x}dx}{4 h^2}+\frac {\left (a+c x^2\right )^{3/2} \left (4 \left (d h^2-e g h+f g^2\right )-3 h x (f g-e h)\right )}{12 h^2}}{h}+\frac {f \left (a+c x^2\right )^{5/2}}{5 c h}\) |
| \(\Big \downarrow \) 682 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {c \left (a h \left (a \left (5 f g^2-h (5 e g-8 d h)\right ) h^2+4 c \left (f g^4-g^2 h (e g-d h)\right )\right )-\left (3 a^2 (f g-e h) h^4+12 a c g \left (f g^2-h (e g-d h)\right ) h^2+8 c^2 \left (f g^5-g^3 h (e g-d h)\right )\right ) x\right )}{(g+h x) \sqrt {c x^2+a}}dx}{2 c h^2}+\frac {\sqrt {a+c x^2} \left (8 \left (a h^2+c g^2\right ) \left (d h^2-e g h+f g^2\right )-h x \left (\left (3 a h^2+4 c g^2\right ) (f g-e h)+4 c d g h^2\right )\right )}{2 h^2}}{4 h^2}+\frac {\left (a+c x^2\right )^{3/2} \left (4 \left (d h^2-e g h+f g^2\right )-3 h x (f g-e h)\right )}{12 h^2}}{h}+\frac {f \left (a+c x^2\right )^{5/2}}{5 c h}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {a h \left (a \left (5 f g^2-h (5 e g-8 d h)\right ) h^2+4 c \left (f g^4-g^2 h (e g-d h)\right )\right )-\left (3 a^2 (f g-e h) h^4+12 a c g \left (f g^2-h (e g-d h)\right ) h^2+8 c^2 \left (f g^5-g^3 h (e g-d h)\right )\right ) x}{(g+h x) \sqrt {c x^2+a}}dx}{2 h^2}+\frac {\sqrt {a+c x^2} \left (8 \left (a h^2+c g^2\right ) \left (d h^2-e g h+f g^2\right )-h x \left (\left (3 a h^2+4 c g^2\right ) (f g-e h)+4 c d g h^2\right )\right )}{2 h^2}}{4 h^2}+\frac {\left (a+c x^2\right )^{3/2} \left (4 \left (d h^2-e g h+f g^2\right )-3 h x (f g-e h)\right )}{12 h^2}}{h}+\frac {f \left (a+c x^2\right )^{5/2}}{5 c h}\) |
| \(\Big \downarrow \) 719 |
| \(\displaystyle \frac {\frac {\frac {\frac {8 \left (a h^2+c g^2\right )^2 \left (d h^2-e g h+f g^2\right ) \int \frac {1}{(g+h x) \sqrt {c x^2+a}}dx}{h}-\frac {\left (3 a^2 h^4 (f g-e h)+12 a c g h^2 \left (f g^2-h (e g-d h)\right )+8 c^2 \left (f g^5-g^3 h (e g-d h)\right )\right ) \int \frac {1}{\sqrt {c x^2+a}}dx}{h}}{2 h^2}+\frac {\sqrt {a+c x^2} \left (8 \left (a h^2+c g^2\right ) \left (d h^2-e g h+f g^2\right )-h x \left (\left (3 a h^2+4 c g^2\right ) (f g-e h)+4 c d g h^2\right )\right )}{2 h^2}}{4 h^2}+\frac {\left (a+c x^2\right )^{3/2} \left (4 \left (d h^2-e g h+f g^2\right )-3 h x (f g-e h)\right )}{12 h^2}}{h}+\frac {f \left (a+c x^2\right )^{5/2}}{5 c h}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {\frac {\frac {\frac {8 \left (a h^2+c g^2\right )^2 \left (d h^2-e g h+f g^2\right ) \int \frac {1}{(g+h x) \sqrt {c x^2+a}}dx}{h}-\frac {\left (3 a^2 h^4 (f g-e h)+12 a c g h^2 \left (f g^2-h (e g-d h)\right )+8 c^2 \left (f g^5-g^3 h (e g-d h)\right )\right ) \int \frac {1}{1-\frac {c x^2}{c x^2+a}}d\frac {x}{\sqrt {c x^2+a}}}{h}}{2 h^2}+\frac {\sqrt {a+c x^2} \left (8 \left (a h^2+c g^2\right ) \left (d h^2-e g h+f g^2\right )-h x \left (\left (3 a h^2+4 c g^2\right ) (f g-e h)+4 c d g h^2\right )\right )}{2 h^2}}{4 h^2}+\frac {\left (a+c x^2\right )^{3/2} \left (4 \left (d h^2-e g h+f g^2\right )-3 h x (f g-e h)\right )}{12 h^2}}{h}+\frac {f \left (a+c x^2\right )^{5/2}}{5 c h}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {\frac {\frac {8 \left (a h^2+c g^2\right )^2 \left (d h^2-e g h+f g^2\right ) \int \frac {1}{(g+h x) \sqrt {c x^2+a}}dx}{h}-\frac {\text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right ) \left (3 a^2 h^4 (f g-e h)+12 a c g h^2 \left (f g^2-h (e g-d h)\right )+8 c^2 \left (f g^5-g^3 h (e g-d h)\right )\right )}{\sqrt {c} h}}{2 h^2}+\frac {\sqrt {a+c x^2} \left (8 \left (a h^2+c g^2\right ) \left (d h^2-e g h+f g^2\right )-h x \left (\left (3 a h^2+4 c g^2\right ) (f g-e h)+4 c d g h^2\right )\right )}{2 h^2}}{4 h^2}+\frac {\left (a+c x^2\right )^{3/2} \left (4 \left (d h^2-e g h+f g^2\right )-3 h x (f g-e h)\right )}{12 h^2}}{h}+\frac {f \left (a+c x^2\right )^{5/2}}{5 c h}\) |
| \(\Big \downarrow \) 488 |
| \(\displaystyle \frac {\frac {\frac {-\frac {8 \left (a h^2+c g^2\right )^2 \left (d h^2-e g h+f g^2\right ) \int \frac {1}{c g^2+a h^2-\frac {(a h-c g x)^2}{c x^2+a}}d\frac {a h-c g x}{\sqrt {c x^2+a}}}{h}-\frac {\text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right ) \left (3 a^2 h^4 (f g-e h)+12 a c g h^2 \left (f g^2-h (e g-d h)\right )+8 c^2 \left (f g^5-g^3 h (e g-d h)\right )\right )}{\sqrt {c} h}}{2 h^2}+\frac {\sqrt {a+c x^2} \left (8 \left (a h^2+c g^2\right ) \left (d h^2-e g h+f g^2\right )-h x \left (\left (3 a h^2+4 c g^2\right ) (f g-e h)+4 c d g h^2\right )\right )}{2 h^2}}{4 h^2}+\frac {\left (a+c x^2\right )^{3/2} \left (4 \left (d h^2-e g h+f g^2\right )-3 h x (f g-e h)\right )}{12 h^2}}{h}+\frac {f \left (a+c x^2\right )^{5/2}}{5 c h}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {\frac {-\frac {\text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right ) \left (3 a^2 h^4 (f g-e h)+12 a c g h^2 \left (f g^2-h (e g-d h)\right )+8 c^2 \left (f g^5-g^3 h (e g-d h)\right )\right )}{\sqrt {c} h}-\frac {8 \left (a h^2+c g^2\right )^{3/2} \left (d h^2-e g h+f g^2\right ) \text {arctanh}\left (\frac {a h-c g x}{\sqrt {a+c x^2} \sqrt {a h^2+c g^2}}\right )}{h}}{2 h^2}+\frac {\sqrt {a+c x^2} \left (8 \left (a h^2+c g^2\right ) \left (d h^2-e g h+f g^2\right )-h x \left (\left (3 a h^2+4 c g^2\right ) (f g-e h)+4 c d g h^2\right )\right )}{2 h^2}}{4 h^2}+\frac {\left (a+c x^2\right )^{3/2} \left (4 \left (d h^2-e g h+f g^2\right )-3 h x (f g-e h)\right )}{12 h^2}}{h}+\frac {f \left (a+c x^2\right )^{5/2}}{5 c h}\) |
(f*(a + c*x^2)^(5/2))/(5*c*h) + (((4*(f*g^2 - e*g*h + d*h^2) - 3*h*(f*g - e*h)*x)*(a + c*x^2)^(3/2))/(12*h^2) + (((8*(c*g^2 + a*h^2)*(f*g^2 - e*g*h + d*h^2) - h*(4*c*d*g*h^2 + (f*g - e*h)*(4*c*g^2 + 3*a*h^2))*x)*Sqrt[a + c *x^2])/(2*h^2) + (-(((3*a^2*h^4*(f*g - e*h) + 12*a*c*g*h^2*(f*g^2 - h*(e*g - d*h)) + 8*c^2*(f*g^5 - g^3*h*(e*g - d*h)))*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/(Sqrt[c]*h)) - (8*(c*g^2 + a*h^2)^(3/2)*(f*g^2 - e*g*h + d*h^2)* ArcTanh[(a*h - c*g*x)/(Sqrt[c*g^2 + a*h^2]*Sqrt[a + c*x^2])])/h)/(2*h^2))/ (4*h^2))/h
3.1.92.3.1 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_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
Int[1/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> -Subst[ Int[1/(b*c^2 + a*d^2 - x^2), x], x, (a*d - b*c*x)/Sqrt[a + b*x^2]] /; FreeQ [{a, b, c, d}, x]
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[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[g/e Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Simp[(e*f - d*g)/e Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && !IGtQ[m, 0]
Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] : > With[{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[f*(d + e*x) ^(m + q - 1)*((a + b*x^2)^(p + 1)/(b*e^(q - 1)*(m + q + 2*p + 1))), x] + Si mp[1/(b*e^q*(m + q + 2*p + 1)) Int[(d + e*x)^m*(a + b*x^2)^p*ExpandToSum[ b*e^q*(m + q + 2*p + 1)*Pq - b*f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x )^(q - 2)*(a*e^2*(m + q - 1) - b*d^2*(m + q + 2*p + 1) - 2*b*d*e*(m + q + p )*x), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, d , e, m, p}, x] && PolyQ[Pq, x] && NeQ[b*d^2 + a*e^2, 0] && !(EqQ[d, 0] && True) && !(IGtQ[m, 0] && RationalQ[a, b, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))
Time = 0.63 (sec) , antiderivative size = 563, normalized size of antiderivative = 1.73
| method | result | size |
| risch | \(\frac {\left (24 c^{2} f \,h^{4} x^{4}+30 c^{2} e \,h^{4} x^{3}-30 c^{2} f g \,h^{3} x^{3}+48 a c f \,h^{4} x^{2}+40 c^{2} d \,h^{4} x^{2}-40 c^{2} e g \,h^{3} x^{2}+40 c^{2} f \,g^{2} h^{2} x^{2}+75 a c e \,h^{4} x -75 a c f g \,h^{3} x -60 c^{2} d g \,h^{3} x +60 c^{2} e \,g^{2} h^{2} x -60 c^{2} f \,g^{3} h x +24 a^{2} f \,h^{4}+160 a c d \,h^{4}-160 a c e g \,h^{3}+160 a c f \,g^{2} h^{2}+120 c^{2} d \,g^{2} h^{2}-120 c^{2} e \,g^{3} h +120 c^{2} f \,g^{4}\right ) \sqrt {c \,x^{2}+a}}{120 c \,h^{5}}+\frac {\frac {\left (3 a^{2} e \,h^{5}-3 a^{2} f g \,h^{4}-12 a c d g \,h^{4}+12 a c e \,g^{2} h^{3}-12 a c f \,g^{3} h^{2}-8 c^{2} d \,g^{3} h^{2}+8 c^{2} e \,g^{4} h -8 c^{2} f \,g^{5}\right ) \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )}{h \sqrt {c}}-\frac {\left (8 a^{2} d \,h^{6}-8 a^{2} e g \,h^{5}+8 a^{2} f \,g^{2} h^{4}+16 a c d \,g^{2} h^{4}-16 g^{3} a c e \,h^{3}+16 a c f \,g^{4} h^{2}+8 c^{2} d \,g^{4} h^{2}-8 g^{5} c^{2} e h +8 g^{6} c^{2} f \right ) \ln \left (\frac {\frac {2 a \,h^{2}+2 c \,g^{2}}{h^{2}}-\frac {2 c g \left (x +\frac {g}{h}\right )}{h}+2 \sqrt {\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}\, \sqrt {\left (x +\frac {g}{h}\right )^{2} c -\frac {2 c g \left (x +\frac {g}{h}\right )}{h}+\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}}{x +\frac {g}{h}}\right )}{h^{2} \sqrt {\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}}}{8 h^{5}}\) | \(563\) |
| default | \(\frac {e h \left (\frac {x \left (c \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {c \,x^{2}+a}}{2}+\frac {a \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )}{2 \sqrt {c}}\right )}{4}\right )+\frac {f h \left (c \,x^{2}+a \right )^{\frac {5}{2}}}{5 c}-f g \left (\frac {x \left (c \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {c \,x^{2}+a}}{2}+\frac {a \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )}{2 \sqrt {c}}\right )}{4}\right )}{h^{2}}+\frac {\left (d \,h^{2}-e g h +f \,g^{2}\right ) \left (\frac {\left (\left (x +\frac {g}{h}\right )^{2} c -\frac {2 c g \left (x +\frac {g}{h}\right )}{h}+\frac {a \,h^{2}+c \,g^{2}}{h^{2}}\right )^{\frac {3}{2}}}{3}-\frac {c g \left (\frac {\left (2 c \left (x +\frac {g}{h}\right )-\frac {2 c g}{h}\right ) \sqrt {\left (x +\frac {g}{h}\right )^{2} c -\frac {2 c g \left (x +\frac {g}{h}\right )}{h}+\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}}{4 c}+\frac {\left (\frac {4 c \left (a \,h^{2}+c \,g^{2}\right )}{h^{2}}-\frac {4 c^{2} g^{2}}{h^{2}}\right ) \ln \left (\frac {-\frac {c g}{h}+c \left (x +\frac {g}{h}\right )}{\sqrt {c}}+\sqrt {\left (x +\frac {g}{h}\right )^{2} c -\frac {2 c g \left (x +\frac {g}{h}\right )}{h}+\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}\right )}{8 c^{\frac {3}{2}}}\right )}{h}+\frac {\left (a \,h^{2}+c \,g^{2}\right ) \left (\sqrt {\left (x +\frac {g}{h}\right )^{2} c -\frac {2 c g \left (x +\frac {g}{h}\right )}{h}+\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}-\frac {\sqrt {c}\, g \ln \left (\frac {-\frac {c g}{h}+c \left (x +\frac {g}{h}\right )}{\sqrt {c}}+\sqrt {\left (x +\frac {g}{h}\right )^{2} c -\frac {2 c g \left (x +\frac {g}{h}\right )}{h}+\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}\right )}{h}-\frac {\left (a \,h^{2}+c \,g^{2}\right ) \ln \left (\frac {\frac {2 a \,h^{2}+2 c \,g^{2}}{h^{2}}-\frac {2 c g \left (x +\frac {g}{h}\right )}{h}+2 \sqrt {\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}\, \sqrt {\left (x +\frac {g}{h}\right )^{2} c -\frac {2 c g \left (x +\frac {g}{h}\right )}{h}+\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}}{x +\frac {g}{h}}\right )}{h^{2} \sqrt {\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}}\right )}{h^{2}}\right )}{h^{3}}\) | \(643\) |
1/120/c*(24*c^2*f*h^4*x^4+30*c^2*e*h^4*x^3-30*c^2*f*g*h^3*x^3+48*a*c*f*h^4 *x^2+40*c^2*d*h^4*x^2-40*c^2*e*g*h^3*x^2+40*c^2*f*g^2*h^2*x^2+75*a*c*e*h^4 *x-75*a*c*f*g*h^3*x-60*c^2*d*g*h^3*x+60*c^2*e*g^2*h^2*x-60*c^2*f*g^3*h*x+2 4*a^2*f*h^4+160*a*c*d*h^4-160*a*c*e*g*h^3+160*a*c*f*g^2*h^2+120*c^2*d*g^2* h^2-120*c^2*e*g^3*h+120*c^2*f*g^4)*(c*x^2+a)^(1/2)/h^5+1/8/h^5*((3*a^2*e*h ^5-3*a^2*f*g*h^4-12*a*c*d*g*h^4+12*a*c*e*g^2*h^3-12*a*c*f*g^3*h^2-8*c^2*d* g^3*h^2+8*c^2*e*g^4*h-8*c^2*f*g^5)/h*ln(x*c^(1/2)+(c*x^2+a)^(1/2))/c^(1/2) -(8*a^2*d*h^6-8*a^2*e*g*h^5+8*a^2*f*g^2*h^4+16*a*c*d*g^2*h^4-16*a*c*e*g^3* h^3+16*a*c*f*g^4*h^2+8*c^2*d*g^4*h^2-8*c^2*e*g^5*h+8*c^2*f*g^6)/h^2/((a*h^ 2+c*g^2)/h^2)^(1/2)*ln((2*(a*h^2+c*g^2)/h^2-2*c*g/h*(x+1/h*g)+2*((a*h^2+c* g^2)/h^2)^(1/2)*((x+1/h*g)^2*c-2*c*g/h*(x+1/h*g)+(a*h^2+c*g^2)/h^2)^(1/2)) /(x+1/h*g)))
Timed out. \[ \int \frac {\left (a+c x^2\right )^{3/2} \left (d+e x+f x^2\right )}{g+h x} \, dx=\text {Timed out} \]
\[ \int \frac {\left (a+c x^2\right )^{3/2} \left (d+e x+f x^2\right )}{g+h x} \, dx=\int \frac {\left (a + c x^{2}\right )^{\frac {3}{2}} \left (d + e x + f x^{2}\right )}{g + h x}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 632 vs. \(2 (299) = 598\).
Time = 0.26 (sec) , antiderivative size = 632, normalized size of antiderivative = 1.94 \[ \int \frac {\left (a+c x^2\right )^{3/2} \left (d+e x+f x^2\right )}{g+h x} \, dx=-\frac {\sqrt {c x^{2} + a} c f g^{3} x}{2 \, h^{4}} + \frac {\sqrt {c x^{2} + a} c e g^{2} x}{2 \, h^{3}} - \frac {\sqrt {c x^{2} + a} c d g x}{2 \, h^{2}} - \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} f g x}{4 \, h^{2}} - \frac {3 \, \sqrt {c x^{2} + a} a f g x}{8 \, h^{2}} + \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} e x}{4 \, h} + \frac {3 \, \sqrt {c x^{2} + a} a e x}{8 \, h} - \frac {c^{\frac {3}{2}} f g^{5} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{h^{6}} + \frac {c^{\frac {3}{2}} e g^{4} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{h^{5}} - \frac {c^{\frac {3}{2}} d g^{3} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{h^{4}} - \frac {3 \, a \sqrt {c} f g^{3} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{2 \, h^{4}} + \frac {3 \, a \sqrt {c} e g^{2} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{2 \, h^{3}} - \frac {3 \, a \sqrt {c} d g \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{2 \, h^{2}} - \frac {3 \, a^{2} f g \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{8 \, \sqrt {c} h^{2}} + \frac {3 \, a^{2} e \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{8 \, \sqrt {c} h} + \frac {{\left (a + \frac {c g^{2}}{h^{2}}\right )}^{\frac {3}{2}} f g^{2} \operatorname {arsinh}\left (\frac {c g x}{\sqrt {a c} {\left | h x + g \right |}} - \frac {a h}{\sqrt {a c} {\left | h x + g \right |}}\right )}{h^{3}} - \frac {{\left (a + \frac {c g^{2}}{h^{2}}\right )}^{\frac {3}{2}} e g \operatorname {arsinh}\left (\frac {c g x}{\sqrt {a c} {\left | h x + g \right |}} - \frac {a h}{\sqrt {a c} {\left | h x + g \right |}}\right )}{h^{2}} + \frac {{\left (a + \frac {c g^{2}}{h^{2}}\right )}^{\frac {3}{2}} d \operatorname {arsinh}\left (\frac {c g x}{\sqrt {a c} {\left | h x + g \right |}} - \frac {a h}{\sqrt {a c} {\left | h x + g \right |}}\right )}{h} + \frac {\sqrt {c x^{2} + a} c f g^{4}}{h^{5}} - \frac {\sqrt {c x^{2} + a} c e g^{3}}{h^{4}} + \frac {\sqrt {c x^{2} + a} c d g^{2}}{h^{3}} + \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} f g^{2}}{3 \, h^{3}} + \frac {\sqrt {c x^{2} + a} a f g^{2}}{h^{3}} - \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} e g}{3 \, h^{2}} - \frac {\sqrt {c x^{2} + a} a e g}{h^{2}} + \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} d}{3 \, h} + \frac {\sqrt {c x^{2} + a} a d}{h} + \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} f}{5 \, c h} \]
-1/2*sqrt(c*x^2 + a)*c*f*g^3*x/h^4 + 1/2*sqrt(c*x^2 + a)*c*e*g^2*x/h^3 - 1 /2*sqrt(c*x^2 + a)*c*d*g*x/h^2 - 1/4*(c*x^2 + a)^(3/2)*f*g*x/h^2 - 3/8*sqr t(c*x^2 + a)*a*f*g*x/h^2 + 1/4*(c*x^2 + a)^(3/2)*e*x/h + 3/8*sqrt(c*x^2 + a)*a*e*x/h - c^(3/2)*f*g^5*arcsinh(c*x/sqrt(a*c))/h^6 + c^(3/2)*e*g^4*arcs inh(c*x/sqrt(a*c))/h^5 - c^(3/2)*d*g^3*arcsinh(c*x/sqrt(a*c))/h^4 - 3/2*a* sqrt(c)*f*g^3*arcsinh(c*x/sqrt(a*c))/h^4 + 3/2*a*sqrt(c)*e*g^2*arcsinh(c*x /sqrt(a*c))/h^3 - 3/2*a*sqrt(c)*d*g*arcsinh(c*x/sqrt(a*c))/h^2 - 3/8*a^2*f *g*arcsinh(c*x/sqrt(a*c))/(sqrt(c)*h^2) + 3/8*a^2*e*arcsinh(c*x/sqrt(a*c)) /(sqrt(c)*h) + (a + c*g^2/h^2)^(3/2)*f*g^2*arcsinh(c*g*x/(sqrt(a*c)*abs(h* x + g)) - a*h/(sqrt(a*c)*abs(h*x + g)))/h^3 - (a + c*g^2/h^2)^(3/2)*e*g*ar csinh(c*g*x/(sqrt(a*c)*abs(h*x + g)) - a*h/(sqrt(a*c)*abs(h*x + g)))/h^2 + (a + c*g^2/h^2)^(3/2)*d*arcsinh(c*g*x/(sqrt(a*c)*abs(h*x + g)) - a*h/(sqr t(a*c)*abs(h*x + g)))/h + sqrt(c*x^2 + a)*c*f*g^4/h^5 - sqrt(c*x^2 + a)*c* e*g^3/h^4 + sqrt(c*x^2 + a)*c*d*g^2/h^3 + 1/3*(c*x^2 + a)^(3/2)*f*g^2/h^3 + sqrt(c*x^2 + a)*a*f*g^2/h^3 - 1/3*(c*x^2 + a)^(3/2)*e*g/h^2 - sqrt(c*x^2 + a)*a*e*g/h^2 + 1/3*(c*x^2 + a)^(3/2)*d/h + sqrt(c*x^2 + a)*a*d/h + 1/5* (c*x^2 + a)^(5/2)*f/(c*h)
Exception generated. \[ \int \frac {\left (a+c x^2\right )^{3/2} \left (d+e x+f x^2\right )}{g+h x} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:index.cc index_m i_lex_is_greater E rror: Bad Argument Value
Timed out. \[ \int \frac {\left (a+c x^2\right )^{3/2} \left (d+e x+f x^2\right )}{g+h x} \, dx=\int \frac {{\left (c\,x^2+a\right )}^{3/2}\,\left (f\,x^2+e\,x+d\right )}{g+h\,x} \,d x \]