Integrand size = 29, antiderivative size = 280 \[ \int (g+h x)^2 \sqrt {a+c x^2} \left (d+e x+f x^2\right ) \, dx=\frac {\left (8 c^2 d g^2+a^2 f h^2-2 a c \left (f g^2+h (2 e g+d h)\right )\right ) x \sqrt {a+c x^2}}{16 c^2}-\frac {(f g-2 e h) (g+h x)^2 \left (a+c x^2\right )^{3/2}}{10 c h}+\frac {f (g+h x)^3 \left (a+c x^2\right )^{3/2}}{6 c h}-\frac {\left (8 \left (2 a h^2 (2 f g+e h)+c g \left (f g^2-2 h (e g+5 d h)\right )\right )-3 h \left (5 (2 c d-a f) h^2-2 c g (f g-2 e h)\right ) x\right ) \left (a+c x^2\right )^{3/2}}{120 c^2 h}+\frac {a \left (8 c^2 d g^2+a^2 f h^2-2 a c \left (f g^2+h (2 e g+d h)\right )\right ) \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{16 c^{5/2}} \]
-1/10*(-2*e*h+f*g)*(h*x+g)^2*(c*x^2+a)^(3/2)/c/h+1/6*f*(h*x+g)^3*(c*x^2+a) ^(3/2)/c/h-1/120*(16*a*h^2*(e*h+2*f*g)+8*c*g*(f*g^2-2*h*(5*d*h+e*g))-3*h*( 5*(-a*f+2*c*d)*h^2-2*c*g*(-2*e*h+f*g))*x)*(c*x^2+a)^(3/2)/c^2/h+1/16*a*(8* c^2*d*g^2+a^2*f*h^2-2*a*c*(f*g^2+h*(d*h+2*e*g)))*arctanh(x*c^(1/2)/(c*x^2+ a)^(1/2))/c^(5/2)+1/16*(8*c^2*d*g^2+a^2*f*h^2-2*a*c*(f*g^2+h*(d*h+2*e*g))) *x*(c*x^2+a)^(1/2)/c^2
Time = 0.72 (sec) , antiderivative size = 245, normalized size of antiderivative = 0.88 \[ \int (g+h x)^2 \sqrt {a+c x^2} \left (d+e x+f x^2\right ) \, dx=\frac {\sqrt {a+c x^2} \left (-a^2 h (64 f g+32 e h+15 f h x)+2 a c \left (5 d h (16 g+3 h x)+f x \left (15 g^2+16 g h x+5 h^2 x^2\right )+e \left (40 g^2+30 g h x+8 h^2 x^2\right )\right )+4 c^2 x \left (5 d \left (6 g^2+8 g h x+3 h^2 x^2\right )+x \left (2 e \left (10 g^2+15 g h x+6 h^2 x^2\right )+f x \left (15 g^2+24 g h x+10 h^2 x^2\right )\right )\right )\right )}{240 c^2}-\frac {a \left (8 c^2 d g^2+a^2 f h^2-2 a c \left (f g^2+h (2 e g+d h)\right )\right ) \log \left (-\sqrt {c} x+\sqrt {a+c x^2}\right )}{16 c^{5/2}} \]
(Sqrt[a + c*x^2]*(-(a^2*h*(64*f*g + 32*e*h + 15*f*h*x)) + 2*a*c*(5*d*h*(16 *g + 3*h*x) + f*x*(15*g^2 + 16*g*h*x + 5*h^2*x^2) + e*(40*g^2 + 30*g*h*x + 8*h^2*x^2)) + 4*c^2*x*(5*d*(6*g^2 + 8*g*h*x + 3*h^2*x^2) + x*(2*e*(10*g^2 + 15*g*h*x + 6*h^2*x^2) + f*x*(15*g^2 + 24*g*h*x + 10*h^2*x^2)))))/(240*c ^2) - (a*(8*c^2*d*g^2 + a^2*f*h^2 - 2*a*c*(f*g^2 + h*(2*e*g + d*h)))*Log[- (Sqrt[c]*x) + Sqrt[a + c*x^2]])/(16*c^(5/2))
Time = 0.52 (sec) , antiderivative size = 267, normalized size of antiderivative = 0.95, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {2185, 27, 687, 27, 676, 211, 224, 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.
| \(\displaystyle \int \sqrt {a+c x^2} (g+h x)^2 \left (d+e x+f x^2\right ) \, dx\) |
| \(\Big \downarrow \) 2185 |
| \(\displaystyle \frac {\int 3 h (g+h x)^2 ((2 c d-a f) h-c (f g-2 e h) x) \sqrt {c x^2+a}dx}{6 c h^2}+\frac {f \left (a+c x^2\right )^{3/2} (g+h x)^3}{6 c h}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int (g+h x)^2 ((2 c d-a f) h-c (f g-2 e h) x) \sqrt {c x^2+a}dx}{2 c h}+\frac {f \left (a+c x^2\right )^{3/2} (g+h x)^3}{6 c h}\) |
| \(\Big \downarrow \) 687 |
| \(\displaystyle \frac {\frac {\int c (g+h x) \left (h (10 c d g-3 a f g-4 a e h)+\left (5 (2 c d-a f) h^2-2 c g (f g-2 e h)\right ) x\right ) \sqrt {c x^2+a}dx}{5 c}-\frac {1}{5} \left (a+c x^2\right )^{3/2} (g+h x)^2 (f g-2 e h)}{2 c h}+\frac {f \left (a+c x^2\right )^{3/2} (g+h x)^3}{6 c h}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{5} \int (g+h x) \left (h (10 c d g-3 a f g-4 a e h)+\left (5 (2 c d-a f) h^2-2 c g (f g-2 e h)\right ) x\right ) \sqrt {c x^2+a}dx-\frac {1}{5} \left (a+c x^2\right )^{3/2} (g+h x)^2 (f g-2 e h)}{2 c h}+\frac {f \left (a+c x^2\right )^{3/2} (g+h x)^3}{6 c h}\) |
| \(\Big \downarrow \) 676 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {5 h \left (a^2 f h^2-2 a c \left (h (d h+2 e g)+f g^2\right )+8 c^2 d g^2\right ) \int \sqrt {c x^2+a}dx}{4 c}-\frac {2 \left (a+c x^2\right )^{3/2} \left (2 a h^2 (e h+2 f g)-2 c g h (5 d h+e g)+c f g^3\right )}{3 c}+\frac {h x \left (a+c x^2\right )^{3/2} \left (5 h^2 (2 c d-a f)-2 c g (f g-2 e h)\right )}{4 c}\right )-\frac {1}{5} \left (a+c x^2\right )^{3/2} (g+h x)^2 (f g-2 e h)}{2 c h}+\frac {f \left (a+c x^2\right )^{3/2} (g+h x)^3}{6 c h}\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {5 h \left (a^2 f h^2-2 a c \left (h (d h+2 e g)+f g^2\right )+8 c^2 d g^2\right ) \left (\frac {1}{2} a \int \frac {1}{\sqrt {c x^2+a}}dx+\frac {1}{2} x \sqrt {a+c x^2}\right )}{4 c}-\frac {2 \left (a+c x^2\right )^{3/2} \left (2 a h^2 (e h+2 f g)-2 c g h (5 d h+e g)+c f g^3\right )}{3 c}+\frac {h x \left (a+c x^2\right )^{3/2} \left (5 h^2 (2 c d-a f)-2 c g (f g-2 e h)\right )}{4 c}\right )-\frac {1}{5} \left (a+c x^2\right )^{3/2} (g+h x)^2 (f g-2 e h)}{2 c h}+\frac {f \left (a+c x^2\right )^{3/2} (g+h x)^3}{6 c h}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {5 h \left (a^2 f h^2-2 a c \left (h (d h+2 e g)+f g^2\right )+8 c^2 d g^2\right ) \left (\frac {1}{2} a \int \frac {1}{1-\frac {c x^2}{c x^2+a}}d\frac {x}{\sqrt {c x^2+a}}+\frac {1}{2} x \sqrt {a+c x^2}\right )}{4 c}-\frac {2 \left (a+c x^2\right )^{3/2} \left (2 a h^2 (e h+2 f g)-2 c g h (5 d h+e g)+c f g^3\right )}{3 c}+\frac {h x \left (a+c x^2\right )^{3/2} \left (5 h^2 (2 c d-a f)-2 c g (f g-2 e h)\right )}{4 c}\right )-\frac {1}{5} \left (a+c x^2\right )^{3/2} (g+h x)^2 (f g-2 e h)}{2 c h}+\frac {f \left (a+c x^2\right )^{3/2} (g+h x)^3}{6 c h}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {5 h \left (\frac {a \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{2 \sqrt {c}}+\frac {1}{2} x \sqrt {a+c x^2}\right ) \left (a^2 f h^2-2 a c \left (h (d h+2 e g)+f g^2\right )+8 c^2 d g^2\right )}{4 c}-\frac {2 \left (a+c x^2\right )^{3/2} \left (2 a h^2 (e h+2 f g)-2 c g h (5 d h+e g)+c f g^3\right )}{3 c}+\frac {h x \left (a+c x^2\right )^{3/2} \left (5 h^2 (2 c d-a f)-2 c g (f g-2 e h)\right )}{4 c}\right )-\frac {1}{5} \left (a+c x^2\right )^{3/2} (g+h x)^2 (f g-2 e h)}{2 c h}+\frac {f \left (a+c x^2\right )^{3/2} (g+h x)^3}{6 c h}\) |
(f*(g + h*x)^3*(a + c*x^2)^(3/2))/(6*c*h) + (-1/5*((f*g - 2*e*h)*(g + h*x) ^2*(a + c*x^2)^(3/2)) + ((-2*(c*f*g^3 - 2*c*g*h*(e*g + 5*d*h) + 2*a*h^2*(2 *f*g + e*h))*(a + c*x^2)^(3/2))/(3*c) + (h*(5*(2*c*d - a*f)*h^2 - 2*c*g*(f *g - 2*e*h))*x*(a + c*x^2)^(3/2))/(4*c) + (5*h*(8*c^2*d*g^2 + a^2*f*h^2 - 2*a*c*(f*g^2 + h*(2*e*g + d*h)))*((x*Sqrt[a + c*x^2])/2 + (a*ArcTanh[(Sqrt [c]*x)/Sqrt[a + c*x^2]])/(2*Sqrt[c])))/(4*c))/5)/(2*c*h)
3.1.79.3.1 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)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 )), x] + Simp[2*a*(p/(2*p + 1)) Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ {a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
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[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x _Symbol] :> Simp[(e*f + d*g)*((a + c*x^2)^(p + 1)/(2*c*(p + 1))), x] + (Sim p[e*g*x*((a + c*x^2)^(p + 1)/(c*(2*p + 3))), x] - Simp[(a*e*g - c*d*f*(2*p + 3))/(c*(2*p + 3)) Int[(a + c*x^2)^p, x], x]) /; FreeQ[{a, c, d, e, f, g , p}, x] && !LeQ[p, -1]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[g*(d + e*x)^m*((a + c*x^2)^(p + 1)/(c*(m + 2*p + 2)) ), x] + Simp[1/(c*(m + 2*p + 2)) Int[(d + e*x)^(m - 1)*(a + c*x^2)^p*Simp [c*d*f*(m + 2*p + 2) - a*e*g*m + c*(e*f*(m + 2*p + 2) + d*g*m)*x, x], x], x ] /; FreeQ[{a, c, d, e, f, g, p}, x] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p]) && !(IGtQ[m, 0] && Eq Q[f, 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 = 0.69 (sec) , antiderivative size = 272, normalized size of antiderivative = 0.97
| method | result | size |
| default | \(d \,g^{2} \left (\frac {x \sqrt {c \,x^{2}+a}}{2}+\frac {a \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )}{2 \sqrt {c}}\right )+f \,h^{2} \left (\frac {x^{3} \left (c \,x^{2}+a \right )^{\frac {3}{2}}}{6 c}-\frac {a \left (\frac {x \left (c \,x^{2}+a \right )^{\frac {3}{2}}}{4 c}-\frac {a \left (\frac {x \sqrt {c \,x^{2}+a}}{2}+\frac {a \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )}{2 \sqrt {c}}\right )}{4 c}\right )}{2 c}\right )+\left (e \,h^{2}+2 f g h \right ) \left (\frac {x^{2} \left (c \,x^{2}+a \right )^{\frac {3}{2}}}{5 c}-\frac {2 a \left (c \,x^{2}+a \right )^{\frac {3}{2}}}{15 c^{2}}\right )+\frac {\left (2 d g h +g^{2} e \right ) \left (c \,x^{2}+a \right )^{\frac {3}{2}}}{3 c}+\left (d \,h^{2}+2 e g h +f \,g^{2}\right ) \left (\frac {x \left (c \,x^{2}+a \right )^{\frac {3}{2}}}{4 c}-\frac {a \left (\frac {x \sqrt {c \,x^{2}+a}}{2}+\frac {a \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )}{2 \sqrt {c}}\right )}{4 c}\right )\) | \(272\) |
| risch | \(-\frac {\left (-40 f \,h^{2} c^{2} x^{5}-48 c^{2} e \,h^{2} x^{4}-96 c^{2} f g h \,x^{4}-10 a f \,h^{2} c \,x^{3}-60 c^{2} d \,h^{2} x^{3}-120 c^{2} e g h \,x^{3}-60 c^{2} f \,g^{2} x^{3}-16 a e \,h^{2} c \,x^{2}-32 a f g h c \,x^{2}-160 c^{2} d g h \,x^{2}-80 c^{2} e \,g^{2} x^{2}+15 a^{2} f \,h^{2} x -30 a d \,h^{2} x c -60 a e g h x c -30 a f \,g^{2} x c -120 c^{2} d \,g^{2} x +32 a^{2} e \,h^{2}+64 a^{2} f g h -160 a c d g h -80 a c e \,g^{2}\right ) \sqrt {c \,x^{2}+a}}{240 c^{2}}+\frac {a \left (a^{2} f \,h^{2}-2 a c d \,h^{2}-4 a c e g h -2 a c f \,g^{2}+8 c^{2} d \,g^{2}\right ) \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )}{16 c^{\frac {5}{2}}}\) | \(283\) |
d*g^2*(1/2*x*(c*x^2+a)^(1/2)+1/2*a/c^(1/2)*ln(x*c^(1/2)+(c*x^2+a)^(1/2)))+ f*h^2*(1/6*x^3*(c*x^2+a)^(3/2)/c-1/2*a/c*(1/4*x*(c*x^2+a)^(3/2)/c-1/4*a/c* (1/2*x*(c*x^2+a)^(1/2)+1/2*a/c^(1/2)*ln(x*c^(1/2)+(c*x^2+a)^(1/2)))))+(e*h ^2+2*f*g*h)*(1/5*x^2*(c*x^2+a)^(3/2)/c-2/15*a/c^2*(c*x^2+a)^(3/2))+1/3*(2* d*g*h+e*g^2)*(c*x^2+a)^(3/2)/c+(d*h^2+2*e*g*h+f*g^2)*(1/4*x*(c*x^2+a)^(3/2 )/c-1/4*a/c*(1/2*x*(c*x^2+a)^(1/2)+1/2*a/c^(1/2)*ln(x*c^(1/2)+(c*x^2+a)^(1 /2))))
Time = 0.46 (sec) , antiderivative size = 595, normalized size of antiderivative = 2.12 \[ \int (g+h x)^2 \sqrt {a+c x^2} \left (d+e x+f x^2\right ) \, dx=\left [-\frac {15 \, {\left (4 \, a^{2} c e g h - 2 \, {\left (4 \, a c^{2} d - a^{2} c f\right )} g^{2} + {\left (2 \, a^{2} c d - a^{3} f\right )} h^{2}\right )} \sqrt {c} \log \left (-2 \, c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) - 2 \, {\left (40 \, c^{3} f h^{2} x^{5} + 80 \, a c^{2} e g^{2} - 32 \, a^{2} c e h^{2} + 48 \, {\left (2 \, c^{3} f g h + c^{3} e h^{2}\right )} x^{4} + 10 \, {\left (6 \, c^{3} f g^{2} + 12 \, c^{3} e g h + {\left (6 \, c^{3} d + a c^{2} f\right )} h^{2}\right )} x^{3} + 32 \, {\left (5 \, a c^{2} d - 2 \, a^{2} c f\right )} g h + 16 \, {\left (5 \, c^{3} e g^{2} + a c^{2} e h^{2} + 2 \, {\left (5 \, c^{3} d + a c^{2} f\right )} g h\right )} x^{2} + 15 \, {\left (4 \, a c^{2} e g h + 2 \, {\left (4 \, c^{3} d + a c^{2} f\right )} g^{2} + {\left (2 \, a c^{2} d - a^{2} c f\right )} h^{2}\right )} x\right )} \sqrt {c x^{2} + a}}{480 \, c^{3}}, \frac {15 \, {\left (4 \, a^{2} c e g h - 2 \, {\left (4 \, a c^{2} d - a^{2} c f\right )} g^{2} + {\left (2 \, a^{2} c d - a^{3} f\right )} h^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) + {\left (40 \, c^{3} f h^{2} x^{5} + 80 \, a c^{2} e g^{2} - 32 \, a^{2} c e h^{2} + 48 \, {\left (2 \, c^{3} f g h + c^{3} e h^{2}\right )} x^{4} + 10 \, {\left (6 \, c^{3} f g^{2} + 12 \, c^{3} e g h + {\left (6 \, c^{3} d + a c^{2} f\right )} h^{2}\right )} x^{3} + 32 \, {\left (5 \, a c^{2} d - 2 \, a^{2} c f\right )} g h + 16 \, {\left (5 \, c^{3} e g^{2} + a c^{2} e h^{2} + 2 \, {\left (5 \, c^{3} d + a c^{2} f\right )} g h\right )} x^{2} + 15 \, {\left (4 \, a c^{2} e g h + 2 \, {\left (4 \, c^{3} d + a c^{2} f\right )} g^{2} + {\left (2 \, a c^{2} d - a^{2} c f\right )} h^{2}\right )} x\right )} \sqrt {c x^{2} + a}}{240 \, c^{3}}\right ] \]
[-1/480*(15*(4*a^2*c*e*g*h - 2*(4*a*c^2*d - a^2*c*f)*g^2 + (2*a^2*c*d - a^ 3*f)*h^2)*sqrt(c)*log(-2*c*x^2 - 2*sqrt(c*x^2 + a)*sqrt(c)*x - a) - 2*(40* c^3*f*h^2*x^5 + 80*a*c^2*e*g^2 - 32*a^2*c*e*h^2 + 48*(2*c^3*f*g*h + c^3*e* h^2)*x^4 + 10*(6*c^3*f*g^2 + 12*c^3*e*g*h + (6*c^3*d + a*c^2*f)*h^2)*x^3 + 32*(5*a*c^2*d - 2*a^2*c*f)*g*h + 16*(5*c^3*e*g^2 + a*c^2*e*h^2 + 2*(5*c^3 *d + a*c^2*f)*g*h)*x^2 + 15*(4*a*c^2*e*g*h + 2*(4*c^3*d + a*c^2*f)*g^2 + ( 2*a*c^2*d - a^2*c*f)*h^2)*x)*sqrt(c*x^2 + a))/c^3, 1/240*(15*(4*a^2*c*e*g* h - 2*(4*a*c^2*d - a^2*c*f)*g^2 + (2*a^2*c*d - a^3*f)*h^2)*sqrt(-c)*arctan (sqrt(-c)*x/sqrt(c*x^2 + a)) + (40*c^3*f*h^2*x^5 + 80*a*c^2*e*g^2 - 32*a^2 *c*e*h^2 + 48*(2*c^3*f*g*h + c^3*e*h^2)*x^4 + 10*(6*c^3*f*g^2 + 12*c^3*e*g *h + (6*c^3*d + a*c^2*f)*h^2)*x^3 + 32*(5*a*c^2*d - 2*a^2*c*f)*g*h + 16*(5 *c^3*e*g^2 + a*c^2*e*h^2 + 2*(5*c^3*d + a*c^2*f)*g*h)*x^2 + 15*(4*a*c^2*e* g*h + 2*(4*c^3*d + a*c^2*f)*g^2 + (2*a*c^2*d - a^2*c*f)*h^2)*x)*sqrt(c*x^2 + a))/c^3]
Time = 0.57 (sec) , antiderivative size = 484, normalized size of antiderivative = 1.73 \[ \int (g+h x)^2 \sqrt {a+c x^2} \left (d+e x+f x^2\right ) \, dx=\begin {cases} \sqrt {a + c x^{2}} \left (\frac {f h^{2} x^{5}}{6} + \frac {x^{4} \left (c e h^{2} + 2 c f g h\right )}{5 c} + \frac {x^{3} \left (\frac {a f h^{2}}{6} + c d h^{2} + 2 c e g h + c f g^{2}\right )}{4 c} + \frac {x^{2} \left (a e h^{2} + 2 a f g h - \frac {4 a \left (c e h^{2} + 2 c f g h\right )}{5 c} + 2 c d g h + c e g^{2}\right )}{3 c} + \frac {x \left (a d h^{2} + 2 a e g h + a f g^{2} - \frac {3 a \left (\frac {a f h^{2}}{6} + c d h^{2} + 2 c e g h + c f g^{2}\right )}{4 c} + c d g^{2}\right )}{2 c} + \frac {2 a d g h + a e g^{2} - \frac {2 a \left (a e h^{2} + 2 a f g h - \frac {4 a \left (c e h^{2} + 2 c f g h\right )}{5 c} + 2 c d g h + c e g^{2}\right )}{3 c}}{c}\right ) + \left (a d g^{2} - \frac {a \left (a d h^{2} + 2 a e g h + a f g^{2} - \frac {3 a \left (\frac {a f h^{2}}{6} + c d h^{2} + 2 c e g h + c f g^{2}\right )}{4 c} + c d g^{2}\right )}{2 c}\right ) \left (\begin {cases} \frac {\log {\left (2 \sqrt {c} \sqrt {a + c x^{2}} + 2 c x \right )}}{\sqrt {c}} & \text {for}\: a \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {c x^{2}}} & \text {otherwise} \end {cases}\right ) & \text {for}\: c \neq 0 \\\sqrt {a} \left (d g^{2} x + \frac {f h^{2} x^{5}}{5} + \frac {x^{4} \left (e h^{2} + 2 f g h\right )}{4} + \frac {x^{3} \left (d h^{2} + 2 e g h + f g^{2}\right )}{3} + \frac {x^{2} \cdot \left (2 d g h + e g^{2}\right )}{2}\right ) & \text {otherwise} \end {cases} \]
Piecewise((sqrt(a + c*x**2)*(f*h**2*x**5/6 + x**4*(c*e*h**2 + 2*c*f*g*h)/( 5*c) + x**3*(a*f*h**2/6 + c*d*h**2 + 2*c*e*g*h + c*f*g**2)/(4*c) + x**2*(a *e*h**2 + 2*a*f*g*h - 4*a*(c*e*h**2 + 2*c*f*g*h)/(5*c) + 2*c*d*g*h + c*e*g **2)/(3*c) + x*(a*d*h**2 + 2*a*e*g*h + a*f*g**2 - 3*a*(a*f*h**2/6 + c*d*h* *2 + 2*c*e*g*h + c*f*g**2)/(4*c) + c*d*g**2)/(2*c) + (2*a*d*g*h + a*e*g**2 - 2*a*(a*e*h**2 + 2*a*f*g*h - 4*a*(c*e*h**2 + 2*c*f*g*h)/(5*c) + 2*c*d*g* h + c*e*g**2)/(3*c))/c) + (a*d*g**2 - a*(a*d*h**2 + 2*a*e*g*h + a*f*g**2 - 3*a*(a*f*h**2/6 + c*d*h**2 + 2*c*e*g*h + c*f*g**2)/(4*c) + c*d*g**2)/(2*c ))*Piecewise((log(2*sqrt(c)*sqrt(a + c*x**2) + 2*c*x)/sqrt(c), Ne(a, 0)), (x*log(x)/sqrt(c*x**2), True)), Ne(c, 0)), (sqrt(a)*(d*g**2*x + f*h**2*x** 5/5 + x**4*(e*h**2 + 2*f*g*h)/4 + x**3*(d*h**2 + 2*e*g*h + f*g**2)/3 + x** 2*(2*d*g*h + e*g**2)/2), True))
Time = 0.20 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.09 \[ \int (g+h x)^2 \sqrt {a+c x^2} \left (d+e x+f x^2\right ) \, dx=\frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} f h^{2} x^{3}}{6 \, c} + \frac {1}{2} \, \sqrt {c x^{2} + a} d g^{2} x - \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} a f h^{2} x}{8 \, c^{2}} + \frac {\sqrt {c x^{2} + a} a^{2} f h^{2} x}{16 \, c^{2}} + \frac {a d g^{2} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{2 \, \sqrt {c}} + \frac {a^{3} f h^{2} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{16 \, c^{\frac {5}{2}}} + \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} e g^{2}}{3 \, c} + \frac {2 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} d g h}{3 \, c} + \frac {{\left (2 \, f g h + e h^{2}\right )} {\left (c x^{2} + a\right )}^{\frac {3}{2}} x^{2}}{5 \, c} + \frac {{\left (f g^{2} + 2 \, e g h + d h^{2}\right )} {\left (c x^{2} + a\right )}^{\frac {3}{2}} x}{4 \, c} - \frac {{\left (f g^{2} + 2 \, e g h + d h^{2}\right )} \sqrt {c x^{2} + a} a x}{8 \, c} - \frac {{\left (f g^{2} + 2 \, e g h + d h^{2}\right )} a^{2} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{8 \, c^{\frac {3}{2}}} - \frac {2 \, {\left (2 \, f g h + e h^{2}\right )} {\left (c x^{2} + a\right )}^{\frac {3}{2}} a}{15 \, c^{2}} \]
1/6*(c*x^2 + a)^(3/2)*f*h^2*x^3/c + 1/2*sqrt(c*x^2 + a)*d*g^2*x - 1/8*(c*x ^2 + a)^(3/2)*a*f*h^2*x/c^2 + 1/16*sqrt(c*x^2 + a)*a^2*f*h^2*x/c^2 + 1/2*a *d*g^2*arcsinh(c*x/sqrt(a*c))/sqrt(c) + 1/16*a^3*f*h^2*arcsinh(c*x/sqrt(a* c))/c^(5/2) + 1/3*(c*x^2 + a)^(3/2)*e*g^2/c + 2/3*(c*x^2 + a)^(3/2)*d*g*h/ c + 1/5*(2*f*g*h + e*h^2)*(c*x^2 + a)^(3/2)*x^2/c + 1/4*(f*g^2 + 2*e*g*h + d*h^2)*(c*x^2 + a)^(3/2)*x/c - 1/8*(f*g^2 + 2*e*g*h + d*h^2)*sqrt(c*x^2 + a)*a*x/c - 1/8*(f*g^2 + 2*e*g*h + d*h^2)*a^2*arcsinh(c*x/sqrt(a*c))/c^(3/ 2) - 2/15*(2*f*g*h + e*h^2)*(c*x^2 + a)^(3/2)*a/c^2
Time = 0.28 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.12 \[ \int (g+h x)^2 \sqrt {a+c x^2} \left (d+e x+f x^2\right ) \, dx=\frac {1}{240} \, \sqrt {c x^{2} + a} {\left ({\left (2 \, {\left ({\left (4 \, {\left (5 \, f h^{2} x + \frac {6 \, {\left (2 \, c^{4} f g h + c^{4} e h^{2}\right )}}{c^{4}}\right )} x + \frac {5 \, {\left (6 \, c^{4} f g^{2} + 12 \, c^{4} e g h + 6 \, c^{4} d h^{2} + a c^{3} f h^{2}\right )}}{c^{4}}\right )} x + \frac {8 \, {\left (5 \, c^{4} e g^{2} + 10 \, c^{4} d g h + 2 \, a c^{3} f g h + a c^{3} e h^{2}\right )}}{c^{4}}\right )} x + \frac {15 \, {\left (8 \, c^{4} d g^{2} + 2 \, a c^{3} f g^{2} + 4 \, a c^{3} e g h + 2 \, a c^{3} d h^{2} - a^{2} c^{2} f h^{2}\right )}}{c^{4}}\right )} x + \frac {16 \, {\left (5 \, a c^{3} e g^{2} + 10 \, a c^{3} d g h - 4 \, a^{2} c^{2} f g h - 2 \, a^{2} c^{2} e h^{2}\right )}}{c^{4}}\right )} - \frac {{\left (8 \, a c^{2} d g^{2} - 2 \, a^{2} c f g^{2} - 4 \, a^{2} c e g h - 2 \, a^{2} c d h^{2} + a^{3} f h^{2}\right )} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + a} \right |}\right )}{16 \, c^{\frac {5}{2}}} \]
1/240*sqrt(c*x^2 + a)*((2*((4*(5*f*h^2*x + 6*(2*c^4*f*g*h + c^4*e*h^2)/c^4 )*x + 5*(6*c^4*f*g^2 + 12*c^4*e*g*h + 6*c^4*d*h^2 + a*c^3*f*h^2)/c^4)*x + 8*(5*c^4*e*g^2 + 10*c^4*d*g*h + 2*a*c^3*f*g*h + a*c^3*e*h^2)/c^4)*x + 15*( 8*c^4*d*g^2 + 2*a*c^3*f*g^2 + 4*a*c^3*e*g*h + 2*a*c^3*d*h^2 - a^2*c^2*f*h^ 2)/c^4)*x + 16*(5*a*c^3*e*g^2 + 10*a*c^3*d*g*h - 4*a^2*c^2*f*g*h - 2*a^2*c ^2*e*h^2)/c^4) - 1/16*(8*a*c^2*d*g^2 - 2*a^2*c*f*g^2 - 4*a^2*c*e*g*h - 2*a ^2*c*d*h^2 + a^3*f*h^2)*log(abs(-sqrt(c)*x + sqrt(c*x^2 + a)))/c^(5/2)
Timed out. \[ \int (g+h x)^2 \sqrt {a+c x^2} \left (d+e x+f x^2\right ) \, dx=\int {\left (g+h\,x\right )}^2\,\sqrt {c\,x^2+a}\,\left (f\,x^2+e\,x+d\right ) \,d x \]